id stringlengths 1 6 | url stringlengths 16 1.82k | content stringlengths 37 9.64M |
|---|---|---|
8600 | https://www.youtube.com/watch?v=bNxtxiN-FtI | Intermediate Algebra Lecture 11.2: Solving Quadratic Equations with the Quadratic Formula
Professor Leonard
1040000 subscribers
512 likes
Description
33488 views
Posted: 1 May 2012
Intermediate Algebra Lecture 11.2: Solving Quadratic Equations with the Quadratic Formula
42 comments
Transcript:
we're gonna learn something and 11.2 call the quadratic formula however I'm gonna keep one of these on the board because what we're going to do in this example we're going to solve a quadratic equation once and for all which means we're going to do universally and there might create a formula you can use all the time that works just of the coefficients it's a pretty awesome idea I'm so glad I had it ve quadratic formula do you agree that any quadratic equation can be written as ax squared plus BX plus C equals zero do you agree that any quadratic can be written that way it says an x squared term an X term and the constant yeah everything can be written as once on off to one side and zero on the other side you with me still okay what we're gonna do is we're gonna use completing the square to create a formula I'm going to show you the process what it wants you to do right now if you'd like to if you like to take notes on this but really just wants you following you'll never have to show me this I just need you to understand where it comes from you got it so take notes if you want that I'm going to move pretty fast I just want you following I want you to follow each step and see oh this is where it comes from because I always hated it when my teachers would give me formulas but never exactly explain how you got them like well fine I can use that but does it make any sense what we're doing here is we're gonna solve this in general by completing the square notice I haven't told you a is what B is 1 C is on the solve anyway some good like that you know this is by completing the square we're gonna follow the steps to completely sweater right now but steps to completing the square said first off you need to get your constant term off to the other side you follow what's my constant term here CV no subtract see so I get ax squared plus BX equals negative C true the next step was that number two said you need to somehow get rid of that coefficient what's the coefficient I need to get rid of here so I'm going to divide everything by a what happens is I'm gonna get an x squared plus B over a X notice how that's a fraction equals negative C over a are you still okay so far be sure so constant term gone divided by that coefficient no problem I have that next up was the moneymaker step it said take that middle term this one take half of it now let me show you something if you in case you never saw this before if you want to take half of a fraction for instance that you want to take half of one-third half that one-third be the same thing as multiplying by 1/2 right that's going to be 1/6 in other words to take half of this fraction you just put a 2 there and multiply by 2 on the denominator that takes half of it you're basically divided by 2 do you okay with that one so if I want to take 1/2 of B over a what I'm gonna do they say o be over a half of that is B over 2a that's half of B over a or you can that one so I take half of this term and I square it that's exactly what we did here we took half of this divided by another 2 we divided by another 2 here then we square it we got this thing and we added that to both sides so I'm gonna add so notice this is B squared Oh what's this piece of B squared over over how much raisins for a for a for a squared you agree that this is this remember this is the key step here I've taken half of this check I've squared it check I've added it to both sides well not yet I'm going to add this to both sides so I'm adding this B squared over 4a squared to both sides well this is going to give me watch carefully on the left hand side I have x squared plus B over ax plus B squared over 4a squared equals on the right hand side I have negative C over a plus B squared over 4a squared that was the hardest part how many feel okay what's hardest part no role are you guys okay with it where it's coming from so got blue the constant divided by this coefficient taking this middle term just like this 1 divided by 2 that means you just have another two of the denominators squared it that means you're gonna get a square root of 4a squared added it to both sides that's the idea we've added that to both sides here we have x squared plus B over ax plus B squared over 4a squared Fang we get negative C over a plus B squared over 4a squared got it the next step is I'm only this alone for a second I'm just gonna work on this left-hand side for one step I'm going to factor it now according to this how we factor this thing you try to do a diamond problem on this one you're crazy all right you can't you can't do that you have to know what a perfect square trinomial is this is a perfect square trinomial what it is is X plus that middle term divided by two so in our case we're gonna have X plus what's gonna go there what did you see it it's on the board what's gonna go what's gonna go right there it's on the board right now B over 2a do you see why the over - it is it's whatever you get before you square it on the right hand side guess it looks a little nasty I'm going to work on that in just a bit on the right hand side I have negative C over a plus B squared over 4a squared raise your hand feel okayed getting down that far good all right let's work on the right hand side cuz right now the thing is dis ugly how do you combine two fractions now fortunately in Chapter seven you learned how to find a common denominator this our LCD here would be 4a squared you see that so what I'm gonna do to find a common founder to get our common denominator I'm going to multiply this by 4a over 4a do you see why you left hand side I'm gonna keep that the same for just awhile right hand side I get notice so I still have a negative I've got a negative or AC DC with negative voices coming from all over 4a squared plus B squared over 4a squared am i successful do I know I'm successful doing do I have common denominators what I'm asking ya do I can make that into one fraction now when I do it I know that my denominator is gonna be 4a squared the truth on my numerator I've got negative 4ac I've got plus B squared you with me but I'm gonna write it differently watching the work carefully I'm gonna write the B squared first I'm gonna write the minus 4ac st. is that okay with you be sure so it me squared first - remember you don't have to write this down you never know how to do this in my class if you want all great right now if you'd like to look at it later that'd be fantastic but it really do need to following here and be with me okay that's more important are you okay getting on the far right now we're good enough to take the square root of both sides we have this the way we wanted this right here by the way that little piece is called the discriminant can you say discriminant it's not discriminate okay we're not discriminating but this is discriminant what's gonna tell you is how many solutions you're gonna get and what type the article so we're good enough right now to take the square root on both sides of course when you take the square root of both sides what must you have hey you're gonna have two solutions here just like before they're gonna have a plus or minus now I know this doesn't look that great what you're gonna see in a minute we're gonna be able to work with it on the left hand side what happens to the square and square root gone notice that we're using like everything we've learned in this class up to this wall up pretty much everything we've learned design qualities to get here all the algebra X plus B over 2a those are gone on the right-hand side I had plus or minus the square root b squared minus 4ac all divided by 4a squared so far so good now you're gonna see something that we've done before just just looking really good right here on this side I'm going to split this up as a square root of the numerator over the square root of the denominator we've actually done that so already and then actually I'm going to show that step to you too so I'm just splits up what's the square root of 4a squared how much is the square root of 4a squared how much maybe we've done that in this fight tonight cool plus or minus the square root b squared minus 4ac divided by 2a where the square root go you did it right there - ASA quick for a square gives you when you take the square root to 8 you know what there's only two more steps first step sell for X how am i solved that price of course you mean by both sides I know this we haven't had this on the board but does this look familiar compared to one of the problems we just completed kind of cool right we actually have a common denominator already we're gonna put this together as one fraction when you put it together as one fraction what you're going to get is notice that negative I can associated that with the numerator I'm gonna do negative B plus or minus is gonna stay on the top of my fraction just like you did before the square root of b squared minus 4ac all divided by notice how this is a common denominator now all divided by what's going to be on the bottom negative B plus or minus B squared square root of b squared minus 4ac all divided by 2a that's it don't trust me trust me this way is easier I'll prove it to you in five seconds yes you cannot simplify that because it's connected with the subtraction you can't slip that up that only works with multiplication if it's not connected by subtraction or addition okay no why was the song smiley no no this isn't some recipe this right here completed the square right and complete the square once and for all which means you never have to do this again which means that if you look at this this is gonna be very interesting to you it will break when you right now they will be interesting does the 70 X isn't it this just has coefficients notice a b and c are numbers right numbers this means if you look here and you find your numbers and you plug them in you automatically get your answer x equals that's your answer is that awesome yes you no work no work just plug in numbers now I can tell you this is right here a lot of people screw this thing up you know why what you do memorize it I'm gonna teach you how to do that just say but also also the signs positive and negative your positive and negative signs kill you guys they kill you they murder your problems I see it all the time maybe further your problems you got to know that when you square something you're not going to have a negative anymore right even if the negatives in there you swear you're not gonna have a negative anymore this is B squared in reasons -4 is that you're gonna get is over here you got nobody have a negative and you're subtracting it becomes a plus right when you have a negative in there's no negative over there are two negatives you're actually subtracting that you know that the a a here is a number you have to multiply that thing negative negative that can change it into a positive right there so you have to watch your signs very careful that's the only thing that can really happen let's do one example I'll give you a way to memorize this formula so I'm going to leave the quadratic formula on the board over there we're gonna identify that any of these problems can be solved with that for example 2x squared plus 9x plus 10 equals 0 all you need to be able to do is use the quadratic formula find your a B and C now first thing could you do this with factoring I don't know maybe maybe you could but it has an extra step on it right that's not going to be that funny could you do this with completing the square absolutely every problem can be done by completing the square that that works all the time however I'm going to tell you some of them if you can't factor it in less than 10 seconds and it has the number up front you're definitely going to be completing the square if you can't factor it in ten seconds you have two options here the factory or quadratic formula if you can't factor it in ten seconds in fact at ten seconds ready to go one two ten have you factored it okay let's skip it then go the quadratic formula I'm serious I'm serious if you cannot factor prom in ten seconds you're wasting too much time line factor should be quick or you skip something else that works faster you with me that's the idea don't use completing the square then what takes way too long all I need to do is find your a your B you're seeing everybody what's your a yes too deep just the coefficient this has no variables in it folks are there the x's are their interests no ABC are the coefficients of your problem a is the number B is the number and C's the number what is your beef what is your seat make sure you're done right yep we're gonna take these numbers plug them in there do you see your abs and C's we know our ABC we're gonna take these numbers plug them in for these letters let's see if we can go x equals negative B what's negative B in this case then I have to have what folks assign it as a formula it does not change and then a square root and the square root you're going to put and I want you all to do this just to make it habit of it parenthesis squared that's going to be in parentheses means whatever you have you're gonna square positive or negative what goes inside the parenthesis 9b squared B squared minus 4 4 doesn't change 4hd means 4 times a times C how much is your a here lately time to see what's C and all over 2 times what to 2 times 2 relation have you see where this problems come from keep your hand up if you understood where this formula is coming from that's kind of important too right you know where it's government's between the square so we know our a B and C when you plug those things in or we'll work this thing out lift that thing out tomorrow will do maybe three or four examples when you done now would you like to learn a song on how to complete this thing I'll do it but you're gonna do it too and if you don't how much for the video and wants you here it is now don't say you know the lyrics because the lyrics are on the board that's it here's how it good and you're gonna hate me you're gonna hate me around Christmastime I promise you will because here's what you never forget this it's gonna be in your head forever here's your lyrics this is your your pass out of class if you're doing good what you to leave if not moving your all day it goes so don't laugh x equals minus b plus or minus radical b squared minus 4ac all divided by 2a a little more time that you guys have coming Christmas music x equals minus B plus or minus radical B squared minus 2a you like it download them together ready wanna see my you ready the words are here 1 2 3 x equals minus b plus or minus radical b squared minus 4ac divided by 2 hey one more time on your own so job - it's not too late - hey okay alright so if you remember from last class working with the quadratic formula last time we kind of invented this formula together hopefully you remember how to do that you don't need to do it yourself but remember where this formula came from this is basically we completed the square once and for all on a general equation and that gave us this nice formula it has no accident just a B and C which are the numbers they correspond to coefficients in an any quadratic equation so when we do our problem of course the only thing you need to be able to do to get this right is to be able to identify a B and C when you are at a standard form so notice do you have to have zero on one side for the Stork yes yes you do otherwise the science for your apse will change do you see that as having negative over here we don't want that we want the correct size for each of these has to be the correct order x squared X number the 0 is got to be on one side that's how you find your a B and C budget of your paper bet you sure so just like most of the quadratics we are going to get a zero on one side and all everything else on the other side the only time we did not do that was really completing the square that was a special case you need to be focused up here folks all right that's it so we found our a is 2 or B is 9 or C is tennis we take those two numbers we substitute it into that formula and we get down the Vista far I think that's where we stop less time is that where we stopped last time let's continue because I told you it as long as you everyone seemed able to find a B and C everybody seemed able to do that almost everybody's gonna be able to plug them in here correctly unless you just completely forget the song and talk different completely forget out of substitute numbers this is going to be right or 95% of you but for 95% you'll be just fine on this the problem happens when you start doing the ax mathematics science happened simple math error stats and that's how this thing gets messed up so we're gonna go nice and slowly you're not gonna rush through this stuff all right you're gonna have plenty of time on your test don't rush it don't rush it go slowly you should calculate if you want to I don't care what you use I just make sure your your your simple mathematics here is accurate so we're gonna go through we know the negative nine you're gonna leave that so we go x equals negative x plus or minus radical nine squared 18 or 81 which one notice that you're spraying this number it will always be positive for this portion if you get a negative number here well chances are you probably got something actually but you've definitely done something wrong if you get a negative after you square something right now this yeah that can't be negative we're not squaring and using over here this is minus how much does that mean all over the square root of 2 times 2 I'm sorry all over 2 times 2 which is 4 still ok so far to go and step by step we've just done this was called a what was the inside part of our radical called the inside it's usually called the radicand but in our case for the special name for the B squared months for our discriminant that's right this is called the discriminant we'll talk about that in just a bit but get that name down in your hand okay what next yeah but you probably do that math right those are gonna be two numbers 81-82 you 1 negative 5 plus or minus radical don't skip out on these steps it's important to get all these steps down over 4 we're going slowly that way we don't make any math there's it's a novel art product it's it's simple math I mean you know how to square numbers and I multiplied subtracting 2 by it you know how to do that just to make sure that you accurately how much is the square root of 1 so don't leave it as a square root of 1 oh my gosh no no no we're actually gonna get two real nice numbers out of this so the square root of 1 gives us x equals negative 9 plus or minus 1 over 4 I'm not just doing this for your benefit showing you all the steps this is how we want to see it I want to see each and every one of these steps that's going to make you more accurate hey do I get two solutions that plus or minus gives us two solutions so we're going to get all the street upstream downstream you're gonna write out both of these before you do your mouth so check it out you get negative 9 plus 1 over 4 just like we did it before right plus 1 and we're going to get an 8 and 9 minus 1 the negative nine that's that one doesn't change but we get the plus and we get the minus upstream downstream do you guys have the support hey kid you subtract and add this one from the negative nine can you do that problem Jesus your boat so what's an agent nine plus one community so what don't hold back on that you can do that the only time you can't do that is if you have an eye over here or square root which is the same thing so don't forget you know how to do this type of problem you guys need to be so negative 9 plus 1 times negative n plus 1 how much is negative 9 minus 1 in tandem for so don't let don't make this harder than it is this is not a hard thing you're just doing simple mathematics here just because this is a quadratic formula doesn't mean you can't add a number because I mean you can't subtract a number okay so don't stop don't stop here don't stop there don't stop here if you can put those together then do it if you can't we'll okay if this happened to still be a square root a little okay you can't do that but it's a whole number like we have well shoot we can put those things together it's a real nice antigen this what's negative 8 over 4 can you reduce that what you're in have to know chemicals so what's the hardest part of the toilet this is not far as part that's probably the easiest part yeah clicking this in is probably an easy part I quarter doing the mathematics in getting all the way down to here ironically that's the hardest part and to be honest with you doing this step ad-libs in math lady Athens are allowed to request you do this stuff you do the addition subtraction multiplication division so basically once we've done this man we made this stuff a lot easier this is easier to factory it's easier to clean the square it's looking at numbers and plug it in the formula you gets right with this now what we're gonna do today is try a couple more examples I'll give you some different forms the different ways this can look but we can still use our quadratic formula okay so the next example still kinda basic basic here okay let's jump look at the board the first thing I need to realize is well first question I have for you is this in the appropriate form is that how I want it why or why not do I have zero on one side are they in the correct order then great that's fine what I want you to do right now is find a B and C write it out explicitly a equals blank B equals write that up remember that when you find a B and C it goes with the sign in front of that number so with that in mind how much is a everybody how much is BT's at five and negative five and C do I have any X's when I do a B and C you see 18 see our numbers there coefficients so you should never ever have an X in this formula what this is doing is taking the values of a B and C the coefficients using them in such a way that you meet your solutions without completing the square that did that for us okay what are you supposed to do now okay I want to show you how to do this because the first time we have some negative numbers so when I'm taking this and plugging it into my or substitute into my quadratic formula I'm going to have x equals x equals negative b we are going to write everything out explicitly without doing any math in our head when we do math in our head that's typically where the sign errors come in so when we when we do this watch what I want you to write up here we're gonna write negative B that native is going to be there regardless of what B is truth so when I write the negative then I write how much B is how much this be socially negative 5 and negative negative 5 wait okay good from this he says negative whatever that number is negative negative 5 then what do I write okay that's gonna be there it's fully then the square root that's a big square root covering everything there inside of that I have B squared how much is begin notice this is why I wanted you to run one of them why do I have prints and run that I told you that last time I want you to practice that because when you get to something like this look at the board legs this right here that right there is different than that right there those are different things you've recovered that a couple times this right here will be negative 25 do you see it that would be negative 25 you square the 5 and then you'd apply the multiplication of negative 1 you can get negative 25 I've covered that they just a couple times in here this right here is different that says negative 5 squared that's what you want to show you want to show that number squared whatever it is you're squaring it so in this case I really need negative 25 what are you going to get put it in parenthesis practice that make sure you have in parenthesis right because that negatives that also use squared then what do I have listen gentlemen let's keep playing along anymore mine is just like that much more x times 1 times 3 because a is 3 and then x see how it's with see this is the only place you put your seat you put B 2 places if you put a 2 places you put your C only one place all over all the way how does what you put you a to place so we're gonna double-check before you anything else this is a good thing to do trust me don't just assume that you did this right the first time make sure you did it right we have a negative B check plus or minus okay square root got it we have B squared that's negative 5 squared I've had that in parentheses making sure all of these being squared minus 4 times a times C got it all over 2 times a that's appropriate that's exactly 1 now we work slowly going through this thing doing step by step realizing that we need to go slowly normativity signs credit so let's start how much is negative negative 5 votes so I'm gonna write positive 5 the plus and minus I don't deal with that too like the last step so if you guys are so now I need your attention up here I anticipate how much is negative 5 squared plus related then watch what happens I do have a - I'm gonna write this out so you can see it so that when I doing this in our head identify - but how much is 4 times 3 times negative 2 is that like that is that like that won't like this is that like that yeah don't lose those negatives are those minuses this - is this minus that negative is that negative what's that gonna do to my problem am I gonna subtract or am I going to bad okay you know what happens I'm gonna give you a problem like this on the test what happens here is people give me one of two things they give me 49 on the unique one is that gonna make your problem different can you see how people would get one it happens all the time people do that they go 25 minus 24 and they give you one or they do it correctly and realize oh I'm subtracting a negative number I'm subtracting negative I've written it out I see the minuses right next to each other that's gonna make it plus of course we have all over six so this gives you 5 plus or minus the square root of 25 minus negative 24 that is 49 over 6 hey am I done am I gonna leave it like that what can you do next and we can simplify that right don't leave the square root of you simplify square root of 49 up17 okay am I going to leave it like that no definitely not how many systems are we getting right as your solutions five plus 7 over 6 5 minus 7 over 6 the vibe doesn't change will we have a + n we have a - how are we gonna leave them like this is this as far as we can go no no should do it I mean actually do the mathematics you can do 5 plus 7 right you can do 5 minus 7 do those two things you're going to need your two solutions in much better terms than this and wave it in terms of this this is something you could actually work with so we'd have x equals 12 over 6 and x equals negative 2 over 6 you see where those numbers are coming from folks 12 over 6 negative 2 over 6 and we're gonna do what now good so we're gonna simplify these fractions to turn negative 1/3 would you be happy we'll give it to support oh I missed it I erased lyrics to a song oh that's good did you like the song again on the Internet what's that thingy poem okay it's fun I'm discriminating you against you by and maybe Human Rights the I taught you a song yeah I won you know way back but did your daughter not believe him her daughter didn't believe that I sang in class you said that's something only kindergarteners do when I teach college there you go who is not here for the song shame on you shame on you know I'm gonna see him you know one time you can have a video you can watch it forever you're going to sing it again actually I should make the people who weren't here come up here and sing a song I don't think I'm going to because they'd have a mutiny but anyway okay we're gonna do two more problems together I'll give you a couple to learn own but honestly it's the same thing over and over you just need to know how to manipulate these equations to get them in the right form so first question is is this the same as the first two exams we've done what's wrong with this okay so if you try to look at the board here ropes like I don't papers go around but you still need to pay attention here if you try to do this right now you get a speed of us are 8 is 3 B is negative 8 and you can hear C is 2 do you have a right that's rough you know people get confused on that ledge was real quickly go all it's almost right and eat no problem but there's no zero or there's no zero you cannot use the quadratic formula so do the first step right now get that in the correct form what feels right here plus 2 or minus 2 as you subtract the two from both sides okay what's the next thing good you can use a paper good news quadratic formula a B and C looking at the morni real good with me please would you do a completing the square on this problem okay this student which could you you could would it be easy what would make this hard to do the completing the square that makes it hard and when you divide it that makes it really hard okay are you gonna want to do this completing the school you're not gonna want to do that could you factor it you could try I don't know if you can or not but you could try you have negative eight you can have negative six right that's not factorable so you'd spend a ten seconds trying to do that don't flick it the stuff that you know already you do know how to factor things done problem answer step right you do know how to do that in the next section if you try to do quadratic formula all the time it's gonna work every time listen listen if the quadratic formula works every time no matter what without fails it will work for you but sometimes veterans easier if I give you something like this could you use quadratic formula absolutely it's factoring easier definitely yeah positive 6 negative 1 so you've X plus 6 X minus 1 that's way quicker than the quadratic formula so don't forget that you can factor stuff you should try that first if that works spend 10 seconds doing it if you if you can't do it in 10 seconds and then the quadratic formula is quicker but if you can factor it in 10 seconds shoot do that does that make sense to you so do not lose the stuff you've already learned in this class okay now you said a B and C sure because I can't factor that in 10 seconds it's not a factorable so we're gonna go ahead and we're gonna find our a B and C to do that now write it down explicitly for me if you've already done that try plugging that in the quadratic okay how much is your ad ladies and gentlemen one person goes my how much is my be a my seat now after that once we find our a B and C we should be pretty good I mean that's it right it's on the right form we're gonna take those numbers and plug them into the quadratic formula without doing any math that's what I want you to practice right now you should have several parentheses up here I want you to plug that in here without doing anything except plugging in those numbers so go ahead take about ten seconds and do that now notice that once you get the hang of this this actually just go pretty quick I mean this is just a formula you just plug your substitute that's nice okay x equals well we're gonna have negative B in our case negative B is negative negative 8 you should have a parenthesis and then we have a plus and minus and we have a radical because that radical is going to be an employed reticle that you ever do then we have B squared if B is negative 8 like this in this case we've got negative 8 squared then we have a minus and then we have 4 times a times C that's 4 times 3 times C which is negative 2 and then don't forget to the body you know what some people are going so fast sometimes they just give me this and I forget that there's anything down here I want to just put a 2 it's not just a two all the time it's 2 times 8 3 raise you can if you have exactly that on your paper that's what I want to see that's worth partial credit that's worth at least a few points because you understand how to use a quadratic formula now if we did this right this is worth like maybe 3 or 4 points out of 10 maybe three people at least that's something but I do need to get correct answers right because this is useless unless you can manipulate to get right but from here there's actually don't matter see going on there's only prions we're going on there's just changing size there's just squaring numbers multiplying subtracting maybe figuring out a square root but you do that in three L's for 2/3 so let's go ahead and go real slowly through this we're gonna do negative negative 8 how much is that so possibly plus or minus square what's the first thing in our square root folks the first thing in their square roots gonna be no not from this part now can you get a negative within your square root yes sure if you've done all this math you might get a negative but this first part will always be positive because you're squaring a number okay then we have a minus sign what are we subtracting negative okay good so you should have a minus and then the negative G of minus than a negative and they all divided up above that let's continue going how much is 64 minus negative 24 we're not gonna get 40 are we subtracting or actually adding I am ready plus or minus you said AJ over six done can you simplify the radical like we've done in any other stuff now I can't say what's screwing of 88 you go on seven well that's not gonna work right but we can't find out what that is what number goes into 88 or four-and-twenty to give it up you sure so this is 8 plus or minus notice that this is 4 times 22 that's going to give you to route 22 do you see where the two route 22 is coming from lays engine all over 6 are you done now we can you could write this as two solutions now but it's not good enough okay this answer will never be in the back of the book this answer is not simplified what you need to do on these problems you actually need to factor if you can I want you to notice that by north of 60 per second notice how to could be divided out of that do you see that you with me this is really this is like 8 plus 2 root 22 and 8 minus root 2 or 22 right 2 is a common factor here not the 22 of you boy you finished that part but this can be factored as to watch carefully 4 plus or minus square root of 22 why hey wait a minute how did I get the four I thought you did do but I'd like to factory me sir what about you where did this to go i factored it I divided it away why didn't I do that there well that's got a square root I'm already taken away to two members two times so if I divide this whole thing by two effect on the to the two skull on the point two stays the same so i factor this by 2 and divide that by 2 I get 4 all over my lunch hey hey look it can you simplify this now yes absolutely absolutely you can so when I simplify this this is 2 times this whole expression I can simplify 2 times something that becomes a 1 this becomes a how much change so x equals the 1 doesn't do much I have 4 plus or minus the square root of 22 over 30 I am almost done almost done do I have 2 solutions like serve that put some eyes due to two solutions I rather it's all simplified as much as I can I just want you to write it out so x equals will follow the stream for upstream plus root 22 over 3 and for downstream 4 minus square root of 22 oh those are the two solutions that you should get that's as simple as possible that is what would be in the back of your book okay so you're doing this problem listen this happens a lot when you're doing completing the square in quadratic form if you own the problem yellow be done you just on the back you checking your answer the back hook by the way right hope you are and you're dinging what am I doing wrong maybe nothing maybe just haven't gone for now if you write this this doesn't look anything like that all every number is different except the 22 so if you get this ago I was doing well either you messed up a sign you plugged in the ABC wrong or given simplified so check it whether your name is your correct check whether you did your math correctly but then if you did we'll try some fun fact about the number and simplify it yes I like that okay so make sure you are factoring I did put up your answers for the test review last night like 8 o'clock you're welcome and you'll notice sometimes I give you two answers the first one is what you should have done initially and then there's a little arrow and the arrow says that's what I want maybe there is go further give me that how do people get one we talked about so far so you're putting things in the correct form first and then we're just working it out are there any question on this cause I have two races to go on it's pretty straightforward though isn't it pretty straightforward now you knew I was gonna do something to you right here that it's not fun you knew that right you knew because I'm building these up that of course I'm making it more difficult it's Friday it's Friday after all so ruin your good Friday with rain and with friendships yeah you get a triple whammy today you get rain you give fractions and you get a homework up Friday Friday Friday Friday oh my gosh hey personally let's let's check a couple things there's two things you check first thing is is this all to one side answer on the other side like I wanted okay yeah this is true is it in the correct order which means x squared X number yeah that's how you need to have it to find a B and C how much is your name how much is your B how much you see so could I take these numbers less afford could I take these numbers and plug them into this one six would go there and there one negative one half would go here and here I'm going to get the one-fourth of that and then make me want to be going there and truth do you want to do that do you have on one work with fractions no is this an equation booyah what can you do with the equations chapter seven what can you do with the equations can you always get rid of a fraction with equations oh my gosh don't do that listen if you plug fractions into this thing two things are going to happen you're going to get it wrong or you're gonna send it to a menace amount of time doing this to get the right answer a tremendous amount of time because two infractions folks you have one half here on the top of it you'd have a complex fraction you need to work with that you want to work with that no you don't do that you can find a comment others all day long and manipulating this to get the correct thing when we're gonna be able to solve this first well not solve it but we're gonna be able to get rid of those fractions first and then have something very simple a deal you with me on that so the first thing I need you to do when you get when you in a problem like this in your Testament you will have it can be either very easy or very hard if you don't clear the fractions it's very hard if you clear the fractions first it's very easy so first they eliminate fractions okay how do I illuminate fractions LCD so I need that can you find the LCD it includes all manometers how much is it is that a denominator so even though we're this a 4 would it would it pay would it be part of this if this was like a a 4 or if this how would this how this if this was a 5 would it change my LCD no because this is like a 5 only 1 right so don't let that confuse you if this is a whole number no problem it's like a whole number over one that plays no part in your LCD you're just looking at the denominators here just that denominators now this was if this is 1 over 5 would that be a part of your LCD absolutely if you have a number over here 1 you won't because you're gonna put in the proper form first aren't you so with that in mind how much does your LCD please it says yeah we don't need 12 do we just need 6 but make it worse than it is so take I said what do you do with that 6 multiply it by how many things one thing to think 3 thing for things just do if I multiply eight looking for if i multiply here and here does that work notice how it multiplied two terms I've left out the rest of my opinion what else would even will fly by on that little one okay I don't like 666 I'm gonna have another one well yeah technically technically you had to do that you know technically you got to do that because this is an equation and when you multiply one side you have to multiply the other side now fortunately for us how much is this going to be yeah don't make the mistake of it 6 please don't do that 0 times anything is 0 right so yes you are multiplying this but if you do this correctly everything's on one side anyway this is still going to be 0 you follow it's not everybody so that's gonna be 0 how much is this going to be perfect next where is this kind of simplify yeah it has to you using an LCD that's gonna be - what and then this is gonna give you how much hey does that look a lot better oh my gosh it's so much better that's way way better you could even try it that you can't factor this you'd spend ten seconds student you have negative three negative six you try to factor wondered if I didn't intend you can't do it then use quadratic formulas so that's the idea here you don't want to do completing the square that's an odd number so do the quadratic formula to find your a your B and your C for me real quick this lesson will actually do all the way together how much is your a how much is your B I'm glad you got the a but a it's not 0 that is 1 and C is so we're going to plug this in we'll get x equals negative b plus or minus the square root of b squared minus 4 times a times D all over 2 times 8 so negative negative 3 plus or minus negative 3 squared minus 4 times 1 times maybe 6 all over 2 times 1 if you show me that at least I know you know what you're doing right if you get that far you do know what you're doing after that would be a math error basic back there so we're going to do this what's the first thing I'm gonna write ladies and gentlemen well it's completely we should have no I do this at this point how much and then sure inside my radical the first thing gives me 9 minus 1 minus negative 24 30 all over 2 times 1 gives you 2 you still okay so far so we're going to have x equals 3 plus or minus the square root what's inside my square root at this point am i subtracting should I have a negative or should have a positive and how much over 2 now at this point personally how we can feel Kate you'd know that for good for you at this point you try to simplify that can you simplify that now it's 3 times 11 that's two primes you can do that can't simplify this can't Factor anything else you can't Factor anything here you cannot factor of 3 this 3 is okay this one's inside the radical you can't do that it's gonna be outside the radical 2 factor so in this point well well this is kind of nice you're actually pretty much done and you just write your two solutions do you see your two solutions right about you're done you've got x equals 3 plus square root 33 over 2 you get 3 minus square root 33 over 2 that's it I'd like you to do one of your own before you go any further again you're gonna have one very much like this on your test so yes Salem's tomorrow so making sure we have the correct form that means that it goes x squared X number you're making sure that everything's on one side you need to have a zero somewhere on one seven equation if you use quadratic form you have to have a zero if not is not going to work next thing should you ever plug in fractions to your quadratic formula never thought never never let that be ridiculous you have an equation here get rid of your fractions first otherwise it makes it way too hard so in our case we are trying to get rid of fractions as chapter seven stuff what are you going to multiply by to read your fractions good so here x 8 and here might be how about the - yeah yeah technically yes now fortunately for us that's a zero seven times anything is zero not eight so we have a zero on one side of our equation here you get nicely x squared minus two X minus 16 gives you zero how many we were able to make it that far good for you now in normal situations well this is of course a quadratic formula section so you're probably using that but most of the time you're going to check to see if you can factor that you would check that you'd have negative two on the top maybe 16 on the bottom I'm not seeing anything that I can factor out of that to use for adding the negative to multiply the negative 16 I don't see it but I'd try for 10 seconds if you can't do it then you move on to quadratic form a is B is negative 2 and C is negative 16 did you make it that far books good deal now we got to plug it in properly we're gonna have x equals negative negative 2 plus or minus the square root of negative 2 in parenthesis squared minus 4 times a times C all over 2 times 8 check your work against my words you have exactly that of your paper hopefully we both did it right so negative negative 2 well that's 2 plus or minus the square root of negative 2 squared is positive 4 - you're 4 times 16 that gives you 64 all over - we should have been made at that for exhume Bette's you know 4 minus negative 6 for the truth well that's going to give us 2 plus or minus the square - how much you since that might print my eggs not ready can I break up can I simplify the square root of 68 in 74 and Sisodia if you try to abide by the perfect square the only one that works here is 4 so you're gonna do the 4 times 17 or you're going to get to root 17 did you get to route 17 if not well we're going simplification if you made it this far you should be able to make it down the simplification so this is 2 plus or minus 2 root 17 over 2 right side of the room can you go any further what can you do affected how much you know this please watch what you can't do this is one of those comments say to see in the calculus what you can't do you're not watching you're gonna make this mistake you can't do that cantina that's gonna by addition and subtraction it's a double whammy you're double messing this up and two negatives don't make it right here you have to factor in order to simplify you have to factor have to can't cross these twos out either by the stretch so you have to do one plus or minus the square root of 17 notice the difference now that's being multiplied by the entire expression that's what you use in the flesh those are not going what's on the denominator of a fraction do you need to write the one i really you've got one plus the square root of 17 you've got one minus the square root of 17 and that's as good as you can do which may have feel okay with this problem now I'm going to take a moment right now to talk about what we're doing because a lot of times people can do the math they just don't understand exactly what we are doing here what you're doing when you're finding these two solutions you're identifying part of the graph do you understand what type of graph and next where it makes you remember the problem the u-shaped graph what you're doing when you're finding this when you find these two solutions you find out where that graph crosses the x-axis where they grab us of the x-axis so the options you have are you can how many spots are you normally good across the x-axis up with the prevalent well if you go down you have to come up somewhere right and it's not gonna go back down it's gonna go up or if you're upside down you're gonna cross the two spots that's what you're doing when you find two solutions you're finding where according to the x-axis you're crossing here and here or here and here are you with them now there are a couple more options could you not cross at all that doesn't cross at all yet we're still getting two solutions however you know what's gonna happen hope you listen here you're gonna get complex numbers complex numbers means you don't cross it all that means you have two solutions there's not real solutions this would be two real solutions this would be two complex solutions or that is there every case where you can get one one solution would be you cross it but you don't come back up is that possible but is there in other cases they're gonna auction for you not a site you can't do that can you possibly touch this at one point what if this is resting right on does that hit it at one point or upside down therefore your three options you have the option where you cross it twice those are two reals if the option we don't cross it at all too complex you also have the option where you touch it at one spot the birthday is on that ground this is called a double root it's technically two real roots they just happen at the same point so we're gonna say this is one real now we're going to talk about the discriminant and what gives you these things so look up at the discriminate hey if this is look at the board with three please if this is positive that means you're adding and subtracting some number some real number you with me some real number if you add as a peck a real number you're going to get plus and minus a real number that gives you two real solutions these things are real solutions follow them if this is negative what's the square root of a negative number what's it going to give you I square root of negative is I that's going to give you two complex solutions you'd be adding and subtracting the complex that would be this case what's the only time you can get one solution how much remember if you add and subtract a real you get two reals if you hadn't spent accomplish you get to Publix what does this have to be to add and subtract it to still get only one thing it's it again I heard someone say zero if you add and subtract nothing you're just going to get one number you're gonna get just be negative B over 2a that would be your one solution if this happens to be a zero it does happen from time to time now we're going to talk about one more example this one I'll show you what to do with it we'll talk about how to apply it to one instance and that'll be our section so first thing what would you have to do here if I did everything in one side that's either this we can either subtract an X but you're gonna have to deal with all negatives that's gonna suck it under otherwise these over here add them so add 4x squared and add 4 to both sides you're going to get 4x squared plus X plus 4 equals 0 do you see why I put it in this order you got to have that in that correct order so 4x squared plus X and then plus 4 how much is your 8b c well I could do that x equals negative well so a minus radical B squared minus 4 a c e all divided by 2 do you see where those numbers are coming from and you get the song epithet are you okay I'm getting that this would be kind of a routine at this point plug them in correctly I just need you to see what happens on this one example you're going to get negative 1 plus or minus the square root of 1 squared gives you 1 minus 4 times 4 times 4 is 64 all over nature hey you tell me something ladies and gentlemen on the inside of my radical can you tell me what I have wait 63 do you say 63 you said negative 63 it's gonna make a difference forget about it is it gonna make a difference absolutely one of them would be real one of them is going to be now the square root of negative six you fit something we should have focused on with our eyes you know the negative gives you an eye 63 is nine times seven that's gonna be 307 be with me if not work on that again this is 307 don't forget about your eye your eyes you can't really see the problem hahaha too much really actually the train this morning skidding puppy into puppies so this is the negative one that's RI 9 times 7 that's 3 root 7 307 you can't simplify any more than this you're just going to write out your two solutions negative 1 plus 3i root 7 over 8 negative 1 minus 3 I 3 to 7 over 8 which is rich in half people kids what we talked about so forth so just a lot of manipulation there's only one thing I got to do it's going to take you ten seconds but I promise student help you on your tests and your homework do you remember Pythagorean theorem if I give you something like this not so I get a to speak easier two and twenty could you set it up Pythagorean theorem says a leg squared plus a leg squared notice the parentheses equals the hypotenuse squared okay with that what would you do now for the last shirt x squared plus x squared plus 4x no simple terms you're going to foil plus four equals 400 what now combine like terms and get everything to one side get everything once that you need is zero to use a quadratic formula we'd have 2x squared plus 4x minus 396 equals zero yes no how could you use the quadratic formula yeah maybe divide everything by two divide everything by two it lowers it by a factor of two and then use the quadratic formula you're gonna have some big numbers but it will work also you're going to get two solutions one positive and one negative is the negative gonna work in this situation we're gonna omit the negative because you can't have a negative distance so you're gonna take the positive one that will be your answer for X did you did you follow this by going through and then the same same stuff so that's it how we will feel given what we talked about good deal |
8601 | https://ecampusontario.pressbooks.pub/app/uploads/sites/2599/2023/01/EnhancedChem-F2023-Chapter15.pdf | CHAPTER 15: REACTIONS IN AQUEOUS SOLUTIONS | 910 CHAPTER 15: REACTIONS IN AQUEOUS SOLUTIONS Enhanced Introductory College Chemistry by Gregory Anderson; Caryn Fahey; Jackie MacDonald; Adrienne Richards; Samantha Sullivan Sauer; J.R. van Haarlem; and David Wegman; Chapter Contents • 15.1 Salts • 15.2 Electrolytes • 15.3 Precipitation Reactions • 15.4 Describing Reactions in Solutions by Writing Molecular, Complete Ionic, and Net Ionic Equations • Summary • Review Except where otherwise noted, this OER is licensed under CC BY 4.0 ( licenses/by/4.0/) Please visit the web version of Enhanced Introductory College Chemistry ( to access the complete book, interactive activities and ancillary resources. In this chapter, you will learn about • The properties of salts and give examples 911 | CHAPTER 15: REACTIONS IN AQUEOUS SOLUTIONS • The properties of electrolytes and classifying them as strong or weak. • Precipitation reactions in aqueous solutions • Writing molecular equations, complete ionic equations, net ionic equations (for neutralization, precipitation, and gas formation reactions) • Concepts relating to the ionization of water To better support your learning, you should be familiar with the following concepts before starting this chapter: • Nomenclature for ionic compounds: naming ionic compounds and writing its chemical formula • How to use your scientifc calculator including the following functions (buttons): scientifc notation. • Polyatomic ions • Review double displacement reactions and balancing chemical equations • Solubility concepts • Properties of water We have learned in the previous section(s) that solutions are of extreme importance in everyday life. The food we eat, the liquids we drink, the fuids in our body, the air we breathe, and the household products we use, such as household cleaners, hand sanitizers, and medications, are all solutions. We often focus on solutions only being table salt dissolving in water when, actually, particles like pollutants suspended in the air are also solutions. This particular example is a gaseous solution. Metal alloys, such as brass (which is mix of copper and zinc), is an example of a solid solution. This chapter will focus on reactions occurring specifcally in aqueous solutions. Water is by far the most important liquid solvent, partly because it is plentiful and partly because of its unique properties. In your body, in other living systems, and in the outside environment a tremendous number of reactions take place in aqueous solutions. Consequently this section, as well as signifcant portions of many other sections of this text, are devoted to developing an understanding of reactions which occur in water. Since ionic compounds and polar covalent compounds constitute the main classes which are appreciably soluble in water, reactions in aqueous solutions usually involve these types of substances. There are three important classes of reactions CHAPTER 15: REACTIONS IN AQUEOUS SOLUTIONS | 912 which occur in aqueous solution: precipitation reactions, acid-base reactions, and oxidation-reduction (redox) reactions. • Precipitation reactions are useful for detecting the presence of various ions and for determining the concentrations of solutions. • Acid-base reactions and redox reactions are similar in that something is being transferred from one species to another. ◦ Acid-base reactions involve proton transfers, whereas redox reactions involve electron transfers. ◦ Redox reactions are somewhat more complicated, though, because proton transfers and other bond-making and bond-breaking processes occur at the same time as electron transfer. Below are demonstrations of each of the types of reactions. Some of these reactions will be explained in detail in this chapter; others will be covered in more depth in later chapters. The frst video demonstrates precipitation reaction between aqueous solutions of silver nitrate and sodium chloride. Watch Double Displacement Reaction of AgNO3 and NaCl (41 sec) ( eGG3EI4mwok) The next video demonstrates an acid and base reacting with aluminum (in the form of a soda can) at room temperature. When aluminum reacts with hydrochloric acid, it yields aqueous aluminum chloride and col ourless hydrogen gas. When aluminum reacts with sodium hydroxide it forms water soluble sodium aluminate and hydrogen gas as products. Watch Coke Cans in Acid and Base – Periodic Table of Videos (2min 49sec) ( WnPrtYUKke8) The last video shows a displacement reaction of zinc metal in aqueous copper (II) sulfate solution. Watch Displacement Reaction of Metals – Zinc in Copper (II) Sulfate – with explanation at micro level (5min 45 sec) ( Attribution & References Except where otherwise noted, this section is adapted by Jackie MacDonald from “11.1: Prelude to Aqueous Phase Reactions ( Book%3A_ChemPRIME_(Moore_et_al.)/11%3A_Reactions_in_Aqueous_Solutions/ 11.01%3A_Prelude_to_Aqueous_Phase_Reactions)” In ChemPRIME (LibreTexts CHEMISTRY) by Ed Vitz, John W. Moore, Justin Shorb, Xavier Prat-Resina, Tim Wendorf, & Adam Hahn, licensed under CC BY-NC-SA 4.0. / Adaptations and additions to content was updated for student comprehension. 913 | 15.1 SALTS 15.1 SALTS Learning Objectives By the end of this section, you will be able to: • Defne the term salt and identify common salts • Explain how a salt is formed Salts are found all around us and within our human body. In nature, these ionic compounds are abundant in rocks and minerals of the Earth’s mantel but also found in seawater and other bodies of water. We have already learned about salts when discussing ionic compounds. Salts are a chemical compound formed when ions form ionic bonds. In these reactions, one atom gives up one or more electrons, and thus becomes positively charged, whereas the other accepts one or more electrons and becomes negatively charged; overall the ionic compound has no net charge. Salts typically are crystalline, odourless, colourless/transparent or white, and have high melting and boiling points. As we learned in the previous chapter on solutions, many salts are soluble in water; however, some are not. If a given salt is soluble in water, it completely dissociates into ions other than a hydrogen ion (H+) or hydroxide ion (OH –) and forms an aqueous solution. This fact is elemental in distinguishing salts from acids and bases. Salts are derived from the neutralization reaction of an acid and base. Since acids and bases always contain either a metal cation or a cation derived from ammonium (NH4 +) and a nonmetal anion, the two can combine to form a salt. For instance, when you mix aqueous solutions of hydrochloric acid and sodium hydroxide, aqueous sodium chloride and water are formed. HCl(aq)+NaOH(aq) → NaCl(aq)+H2O(l) If the aqueous solution of sodium chloride undergoes a distillation process, a solid, crystalline NaCl salt would remain. Concepts related to acids and bases will be discussed in more detail in the next chapter. A common salt, NaCl, also known as table salt, dissociates completely in water (Figure 15.1a). The positive and negative regions on the water molecule (the hydrogen and oxygen ends respectively) attract the negative chloride and positive sodium ions, pulling them away from each other. Non-polar and polar covalently bonded compounds break apart into molecules in solution; however, salts dissociate into ions. These ions are 15.1 SALTS | 914 electrolytes, which will be discussed in the next section. This ionic property is critical to the function of ions in the human body when transmitting nerve impulses and prompting muscle contraction. Figure 15.1a Dissociation of Sodium Chloride in Water: Notice that the solid crystals of sodium chloride dissociate not into molecules of NaCl, but into Na+ cations and Cl– anions, each completely surrounded by water molecules (credit: Anatomy and Physiology 2e (Open Stax), CC BY 4.0 ( Many other salts are important in the body. For example, bile salts produced by the liver help break apart dietary fats, and calcium phosphate salts, Ca3(PO4)2 (s), form the mineral portion of teeth and bones. Some other commons salts include: • Na2CO3 (sodium bicarbonate): Baking soda • MgSO4 – (magnesium sulfate): Epsom salt • Fe2O3 (iron (III) oxide): Rust • KNO3 (potassium nitrate): component of crop fertilizer, used in rocket propellants, freworks, gunpowder Indigenous Perspective: Bone Bone has two components: organic matter (which is mostly Type I collagen) and about 70% inorganic matter. The inorganic material of bone consists of calcium phosphate mineral crystals 915 | 15.1 SALTS known as hydroxyapatite, Ca5(PO4)3(OH). This mineral is embedded throughout the organic bone matrix to give bone its structural stability and strength. One of the composite materials traditionally important to Inuit has been bone. This comes from a variety of natural sources, including caribou antler and walrus tusk. An important use of this strong carvable material is in snow goggles, called ilgaak or iggaak. Wearing these human-made goggles are invaluable in preventing snow blindness (Rayner-Canham et al., 2016, Composite Materials section). Figure 15.1b Ilgaak/Iggaak – Inuit snow goggles: Snow goggles such as the ones illustrated are commonly made from caribou antler with caribou sinew for a strap (credit: work by CambridgeBayWeather, PD) Not only are salts crucial compounds necessary for sustaining life, but they are also invaluable constituents of chemical reactions that occur around us on a daily basis. Watch What are Salts? (5min 9s) ( Attribution & References Except where otherwise noted, this section is adapted by Jackie MacDonald from “2.4 Inorganic Compounds Essential to Human Functioning” In Anatomy and Physiology 2e (Open Stax) by J. Gordon Betts, Kelly A. Young, James A. Wise, Eddie Johnson, Brandon Poe, Dean H. Kruse, Oksana Korol, Jody E. Johnson, Mark Womble, Peter DeSaix is licensed under CC BY 4.0 ( Access for free at Anatomy and Physiology 2e (OpenStax). / Adaptations and additions to content was updated for student comprehension. References Rayner-Canham, G., Taylor, R., & Lee, Y.-R. (2016, February). Making chemistry relevant to Indigenous 15.1 SALTS | 916 Peoples. Chem 13 News Magazine. Retrieved December 15, 2022, from chem13-news-magazine/february-2016/feature/making-chemistry-relevant-indigenous-peoples 917 | 15.2 ELECTROLYTES 15.2 ELECTROLYTES Learning Objectives By the end of this section, you will be able to: • Defne electrolyte and give examples of electrolytes • Relate electrolyte strength to solute-solvent attractive forces Pure water (purifed water) is a pure substance made up of just three atoms: two hydrogen and one oxygen atom, H2O. Since pure water has no constituents other than these atoms, it does not have any taste or smell, and it doesn’t conduct electricity on its own. However, water can become a medium for conducting electricity. When ionic substances (salts) are dissolved in water, they undergo either a physical or a chemical change that yields free ions moving independently in the aqueous solution. This feature permits them to carry positive or negative electrical charges from one place to another and the solution can conduct an electrical current. These substances constitute an important class of compounds called electrolytes. Salts that ionize in aqueous solution are great conductors of electricity and are known as electrolytes. Substances that do not yield ions when dissolved are called nonelectrolytes. If the physical or chemical process that generates the ions is essentially 100% efcient (all of the dissolved compound yields ions), then the substance is known as a strong electrolyte. If only a relatively small fraction of the dissolved substance undergoes the ion-producing process, it is called a weak electrolyte. Substances may be identifed as strong, weak, or nonelectrolytes by measuring the electrical conductance of an aqueous solution containing the substance. To conduct electricity, a substance must contain freely mobile, charged species. Most familiar is the conduction of electricity through metallic wires, in which case the mobile, charged entities are electrons. Solutions may also conduct electricity if they contain dissolved ions, with conductivity increasing as ion concentration increases. Applying a voltage to electrodes immersed in a solution permits assessment of the relative concentration of dissolved ions, either quantitatively, by measuring the electrical current fow, or qualitatively, by observing the brightness of a light bulb included in the circuit (Figure 15.2a). 15.2 ELECTROLYTES | 918 Figure 15.2a Solutions Containing Nonelectrolytes, Strong Electrolytes, or Weak Electrolytes: Solutions of nonelectrolytes such as ethanol do not contain dissolved ions and cannot conduct electricity. Solutions of electrolytes contain ions that permit the passage of electricity. The conductivity of an electrolyte solution is related to the strength of the electrolyte (credit: Chemistry (OpenStax), CC BY 4.0). Ionic Electrolytes Water and other polar molecules are attracted to ions, as shown in Figure 15.2b. The electrostatic attraction between an ion and a molecule with a dipole is called an ion-dipole attraction. These attractions play an important role in the dissolution (ionization) of ionic compounds in water. Figure 15.2b Representation of what Happens when Potassium Chloride is Dissolved in Water: As potassium chloride (KCl) dissolves in water, the ions are hydrated. The polar water molecules are attracted by the charges on the K+ and Cl− ions. Water molecules in front of and behind the ions are not shown (credit: Chemistry (OpenStax), CC BY 4.0). When ionic compounds dissolve in water, the ions in the solid separate and disperse uniformly throughout the solution because water molecules surround and solvate the ions, reducing the strong electrostatic forces between them. This process represents a physical change known as dissociation, which was discussed in the 919 | 15.2 ELECTROLYTES previous chapter – Solutions. Under most conditions, ionic compounds will dissociate nearly completely when dissolved, and so they are classifed as strong electrolytes. Let us consider what happens at the microscopic level when we add solid KCl to water. Ion-dipole forces attract the positive (hydrogen) end of the polar water molecules to the negative chloride ions at the surface of the solid, and they attract the negative (oxygen) ends to the positive potassium ions. The water molecules penetrate between individual K+ and Cl−ions and surround them, reducing the strong interionic forces that bind the ions together and letting them move of into solution as solvated ions, as Figure 15.2b shows. The reduction of the electrostatic attraction permits the independent motion of each hydrated ion in a dilute solution, resulting in an increase in the disorder of the system as the ions change from their fxed and ordered positions in the crystal to mobile and much more disordered states in solution. This increased disorder is responsible for the dissolution of many ionic compounds, including KCl, which dissolve with the absorption of heat. In other cases, the electrostatic attractions between the ions in a crystal are so large, or the ion-dipole attractive forces between the ions and water molecules are so weak, that the increase in disorder cannot compensate for the energy required to separate the ions, and the crystal is insoluble. Such is the case for salt compounds such as calcium carbonate (limestone), calcium phosphate (the inorganic component of bone), and iron oxide (rust). Covalent Electrolytes As mentioned previously, pure water is an extremely poor conductor of electricity because it is only very slightly ionized—only about two out of every 1 billion molecules ionize at 25 °C. The ionization of water will be discussed in more depth later in this chapter and these concepts will be important in understanding the behaviours of acids and bases. In brief, water ionizes when one molecule of water gives up a proton to another molecule of water, yielding hydronium and hydroxide ions. In some cases, we fnd that solutions prepared from covalent compounds conduct electricity because the solute molecules react chemically with the solvent to produce ions. For example, pure hydrogen chloride is a gas consisting of covalently bonded HCl molecules. This gas contains no ions. Hydrogen chloride gas is very soluble in water and is dissolved in water to prepare hydrochloric acid. When we dissolve hydrogen chloride gas in water, we fnd that the solution is a very good conductor. The water molecules play an essential part in forming ions. However, it is important to note that solutions of hydrogen chloride in many other solvents, such as benzene, do not conduct electricity and do not contain ions. 15.2 ELECTROLYTES | 920 Figure 15.2c Reaction between Water and Hydrogen Chloride Gas to form Hydrochloric Acid: Hydrogen chloride gas dissolves in water and the ions react with water, transferring H+ ions to form hydronium ions (H3O+) and chloride ions (Cl−), which illustrates the ionization of this strong acid (credit: Chemistry (OpenStax), CC BY 4.0). Hydrogen chloride gas dissolves in water and the ions react with water, transferring H+ ions to form hydronium ions (H3O+) and chloride ions (Cl−). This illustrates the ionization of a strong acid. This reaction is essentially 100% complete for HCl (i.e., it is a strong acid and, consequently, a strong electrolyte). Likewise, weak acids and bases that only react partially generate relatively low concentrations of ions when dissolved in water and are classifed as weak electrolytes. You will learn more about classifying the strength of acids and bases in later chapters. For a summary Watch Aqueous Solutions, Dissolving, and Solvation (14min 6sec). ( Attribution & References Except where otherwise noted, this page is adapted by Jackie MacDonald from “11.2 Electrolytes ( In General Chemistry 1 & 2 by Rice University, a derivative of Chemistry (Open Stax) by Paul Flowers, Klaus Theopold, Richard Langley & William R. Robinson and is licensed under CC BY 4.0. Access for free at Chemistry (OpenStax) ( / Adaptations and additions were made to content in this section were made or student comprehension. 921 | 15.3 PRECIPITATION REACTIONS 15.3 PRECIPITATION REACTIONS Learning Objectives By the end of this section, you will be able to: • Predict the solubility of common inorganic compounds by using solubility rules • Defne precipitation reactions • Recognize and identify examples of precipitation reactions • Apply the solubility rules of common inorganic compounds to predict the products formed when two aqueous solutions are mixed Scientists have found it convenient (or even necessary) to classify chemical interactions by identifying common patterns of reactivity. This section of this chapter will focus on a specifc type of double displacement reaction called a precipitation reaction. Precipitation Reactions and Solubility Rules Solubility of Inorganic Compounds The idea of solubility was introduced in the solutions chapter. The extent to which a substance may be dissolved in water, or any solvent, is quantitatively expressed as its solubility, defned as the maximum concentration of a substance that can be achieved under specifed conditions. Substances with relatively large solubilities are said to be soluble and are found as dissolved ions in aqueous solution. A substance will precipitate when solution conditions are such that its concentration exceeds its solubility. Substances with relatively low solubilities are said to be insoluble, and these are the substances that readily precipitate from solution to form a solid (s). For purposes of predicting the identities of solids formed by precipitation reactions, one may simply refer to the solubility guidelines for many ionic compounds in Table 15.3a to predict whether a precipitation reaction will occur when solutions of soluble ionic compounds are mixed together. First, it is important to become familiar with using the solubility table to determine if a given salt 15.3 PRECIPITATION REACTIONS | 922 will dissolve (aq), or not (s) in aqueous solution. If a salt is said to be insoluble, or has low solubility, or is slightly soluble, it will form a precipitate – a solid – and the symbol (s) will be used to represent that observation. If a salt is soluble, the salt will dissociate (ionize) in aqueous solution, and the symbol (aq) will be used to show that chemistry. It is important to mention that Table 15.3a does not include every possible soluble, insoluble salt combination. Since there are other possibilities, Table 14.2a and Table 14.2b can also be referenced to determine solubility of inorganic compounds. Table 15.3a Solubility Guidelines for Inorganic Compounds in Water at 25°C Negative Ion (Anion) Positive Ion (Cation) Solubility Phase, Phase Symbol All + Li+, Na+, K+, Rb+, Cs+ NH4 Soluble aqueous, (aq) Chloride (Cl – ), Bromide (Br – ), Iodide (I – ) Ag+, Pb2+, Hg2 2+, Cu+ Low solubility solid, (s) Chloride (Cl – ), Bromide (Br – ), Iodide (I – ) All others Soluble aqueous, (aq) F – compounds with group 2 metal cations, Li+ , Al3+, Pb2+, Fe2+and Fe3+ Low solubility solid, (s) F – All others Soluble aqueous, (aq) Hydroxide (OH – ) Li+, Na+, K+, Rb+, NH4 +, Sr2+, Ba2+ Soluble aqueous, (aq) Hydroxide (OH – ) All others Low solubility solid, (s) – NO3 – , NO2 All (exception: AgNO2 is insoluble) Soluble aqueous, (aq) 3-), Phosphate (PO4 2-) Carbonate (CO3 + Na+, K+, Rb+, NH4 Soluble aqueous, (aq) 3-), Phosphate (PO4 2-) Carbonate (CO3 All others Low solubility solid, (s) 2-) Sulphate (SO4 Ca2+, Sr2+, Ba2+, Ag+, Pb2+ Low solubility solid, (s) 2-) Sulphate (SO4 All others Soluble aqueous, (aq) Source: “Table 15.3a Solubility Guidelines for Inorganic Compounds in Water at 25°C” was created by Jackie MacDonald, CC BY-NC-SA 4.0. Using the Solubility Table Guidelines to Determine Solubility To qualitatively determine the solubility of a salt in aqueous solution, reference Table 15.3a and follow the steps outline below: 923 | 15.3 PRECIPITATION REACTIONS 1. Identify the negative ion in the salt; locate the ion in the frst column of Table 15.3a. 2. Determine the positive ion in the salt by moving into the second column, same row as your anion. 3. Finally, move horizontally in that row into the solubility column (column 3) to determine whether that compound is soluble or insoluble. 4. Identify the symbol used to represent its solubility Example 15.3a Use the solubility rules to determine the solubility of the following substances. Categorize each as soluble or insoluble and also write the symbol used to show its solubility. 1. Na2S 2. AgBr 3. Mg(Cl)2 Solution 1. Soluble, (aq) – Since S is not listed as an anion, it falls into the “all” option. It’s cation is a Na+ cation, so it is soluble, aq. 2. Insoluble, (s) – When Br– forms a salt with Ag+, it forms an insoluble salt, s. 3. Soluble, (aq) – When Cl– forms a salt with Mg2+, it fall into the “all others” category for cations, and it forms a soluble salt, aq. Source: Example 15.3a created by Jackie MacDonald and David McCuaig, CC BY-NC-SA 4.0. Exercise 15.3a Check Your Learning Exercise (Text Version) From the options provided, identify the salts that will form a precipitate in aqueous solution at 25 degrees Celsius. 1. Fe(NO3)2 2. BaSO4 3. AgF 4. Mg(OH)2 15.3 PRECIPITATION REACTIONS | 924 5. AgNO2 6. ZnCl2 7. PbI2 8. KBr 9. LiClO3 10. AgBr 11. KOH 12. (NH4)PO4 13. CaCO3 Check Your Answers1 Source: “Exercise 15.3a” created by Jackie MacDonald and David McCuaig, licensed under CC BY-NC-SA 4.0. Watch the video Precipitation Reactions ( starting at 6min 31sec until 9min05sec. It demonstrates how to use a similar solubility table to determine solubility. For a more in depth solubility table that is online and printable, link to “Solubility Rules Chart” by MilliporeSigma. Source: Section titled “How to Use the Solubility Table Guidelines to Determine Solubility” was created by Jackie MacDonald and David McCuaig and is licensed under CC BY-NC-SA 4.0. Precipitation Reactions A precipitation reaction is one in which dissolved substances react to form one (or more) solid products. Many reactions of this type involve the exchange of ions between ionic compounds in aqueous solution and are sometimes referred to as double displacement, double replacement, or metathesis reactions. These reactions are common in nature and are responsible for the formation of coral reefs in ocean waters and kidney stones in animals. They are used widely in industry for production of a number of commodity and specialty chemicals. Precipitation reactions also play a central role in many chemical analysis techniques, including spot tests used to identify metal ions and gravimetric methods for determining the composition of matter. A vivid example of precipitation is observed when aqueous solutions of potassium iodide and lead nitrate are mixed, resulting in the formation of solid lead iodide: 925 | 15.3 PRECIPITATION REACTIONS 2KI(aq) + Pb(NO3)2(aq) → PbI2(s) + 2KNO3(aq) This observation is consistent with the solubility guidelines: The only insoluble compound among all those involved is lead iodide, one of the exceptions to the general solubility of iodide salts. Lead iodide is a bright yellow solid that was formerly used as an artist’s pigment known as iodine yellow (Figure 15.3a). The properties of pure PbI2 crystals make them useful for fabrication of X-ray and gamma ray detectors. Figure 15.3a Formation of Precipitate Lead(II) Iodide during a Precipitation Reaction: The precipitation reaction producing Lead(II) iodide is shown. It is known as “golden rain” because of the yellow hexagonal crystals forming throughout in the aqueous solution and the solid crystals settle at the bottom of the beaker. This picture was taken after cooling a heated lead(II) nitrate and potassium iodide solution on a Bunsen burner. (credit: work by Der Kreole, CC BY-SA 3.0) The solubility table may be used to predict whether a precipitation reaction will occur when solutions of soluble ionic compounds are mixed together. One merely needs to identify all the ions present in the solution and then consider if possible cation/anion pairing could result in an insoluble compound. For example, mixing aqueous solutions of silver nitrate and sodium chloride will yield a solution containing Ag+, NO3 −, Na+, and Cl−ions. Aside from the two ionic compounds originally present in the solutions, AgNO3 and NaCl, two additional ionic compounds may be derived from this collection of ions: NaNO3 and AgCl. The solubility table can be used to determine if either of these salt combinations are insoluble in aqueous solution. Insoluble salts will precipitate out of the solution to form a solid (s). The solubility table indicates all nitrate salts are soluble, so sodium nitrate (NaNO3) will remain ions in solution. However, silver chloride, AgCl, is one of the exceptions to the general solubility rules of chloride salts, and this combination of ions will form a solid in aqueous solution. Therefore, a precipitation reaction is predicted to occur, as described by the following molecular equation: NaCl(aq) + AgNO3(aq) → AgCl(s) + NaNO3(aq) Watch Precipitation Reactions (10mins 13sec) ( 15.3 PRECIPITATION REACTIONS | 926 Example 15.3b Predict the result of mixing reasonably concentrated solutions of the following ionic compounds. If a reaction occurs and a precipitation is expected, write a balanced molecular equation for the following two reactions. a. potassium sulfate and barium nitrate b. lithium chloride and silver acetate) Solution a. The two possible products for this combination are KNO3 and BaSO4. The solubility guidelines indicate KNO3 is soluble and BaSO4 is insoluble, so a precipitation reaction is expected. The balanced molecular equation for this reaction is K2SO4(aq) + Ba(NO3)2 (aq) → BaSO4(s) + 2KNO3(aq) b. The two possible products for this combination are LiC2H3O2 and AgCl. The solubility guidelines indicate LiC2H3O2 is soluble and AgCl is insoluble, and so a precipitation reaction is expected. The balanced molecular equation for this reaction is LiCl (aq) + AgC2H3O2(aq) → AgCl(s) + LiC2H3O2(aq) 927 | 15.3 PRECIPITATION REACTIONS Exercise 15.3b Predict the result of mixing reasonably concentrated aqueous solutions of the following ionic compounds. If a reaction occurs and a precipitation is expected, write a balanced molecular equation for the following reaction between lead(II) nitrate and ammonium carbonate. Check Your Answer2 Exercise 15.3c Which solution(s) could be used to precipitate the barium ion, Ba2+, in a aqueous solution barium nitrate: sodium chloride, sodium hydroxide, or sodium sulfate? If a reaction occurs and a precipitation is expected, write a balanced molecular equation for that reaction Check Your Answer3 Links to Interactive Learning Tools Explore Precipitation Reactions ( Precipitation-Reactions) from the Physics Classroom ( Attribution & References Except where otherwise noted, this page is adapted by Jackie MacDonald from “6.2 Precipitation Reactions ( In 15.3 PRECIPITATION REACTIONS | 928 CHEM 1114 – Introduction to Chemistry (BCcampus, Pressbooks) by Shirley Wacowich-Sgarbi and Langara Chemistry Department is licensed under CC BY-NC-SA 4.0. / Adaptations and additions to content in this section were made for student comprehension. Notes 1. The salts that form precipitates are PbI2, AgBr, CaCO3, Mg(OH)2, BaSO4, AgNO2 2. The two possible products for this combination are PbCO3 and NH4NO3. The solubility guidelines indicate NH4NO3 is soluble and PbCO3 is insoluble, and so a precipitation reaction is expected. The balanced molecular equation for this reaction is Pb(NO3)2(aq) + (NH4)2CO3(aq) → PbCO3(s) + 2NH4NO3(aq) 3. The solubility table can be used to determine which ionic compounds containing barium will be insoluble in aqueous solution. There are three possibilities to consider: BaCl2, Ba(OH)2, and BaSO4. The solubility guidelines indicate BaCl2 and Ba(OH)2 are soluble in water and will not form a precipitate; however, BaSO4 is insoluble, so this combination will result in a precipitation reaction. The balanced molecular equation for this reaction is Ba(NO3)2(aq) + Na2SO4(aq) → BaSO4(s) + 2NaNO3(aq) 929 | 15.4 DESCRIBING REACTIONS IN SOLUTIONS BY WRITING MOLECULAR, COMPLETE IONIC, AND NET IONIC EQUATIONS 15.4 DESCRIBING REACTIONS IN SOLUTIONS BY WRITING MOLECULAR, COMPLETE IONIC, AND NET IONIC EQUATIONS Learning Objectives By the end of this section, you will be able to: • Write and balance chemical equations in molecular, complete ionic, and net ionic formats. Ionic Compounds in Solution We have learned that one important aspect about ionic compounds that difers from molecular compounds has to do with dissolving in a liquid, such as water. When molecular compounds, such as sugar, dissolve in water, the individual molecules drift apart from each other. When ionic compounds dissolve, the ions physically separate from each other. We can use a chemical equation to represent this process—for example, with NaCl: Figure 15.4a Chemical Equation Illustrating Sodium Chloride Salt Ionizing in Water: NaCl will dissolve in water; each atom separates into their individual ions in the water solution. Na+ and Cl – ions disperse throughout the water and the salt has dissolved or ionized in the aqueous solution. (credit: work by Shirley Wacowich-Sgarbi, CC BY-NC-SA 4.0; / Adapted by Jackie MacDonald). 15.4 DESCRIBING REACTIONS IN SOLUTIONS BY WRITING MOLECULAR, COMPLETE IONIC, AND NET IONIC EQUATIONS | 930 Figure 15.4b Water Hydration of a Crystal of Salt (NaCl): When immersed in water, the attractive forces of the water molecules to the NaCl separates the NaCl molecule to disperse into its separate ions. The negatively-charged oxygens of the water molecules are attracted to the positively-charged sodium ions. The positively-charged side of the water molecules (the hydrogen atoms) are attracted to the negatively-charged chloride ions. Water molecules pull the sodium and chloride ions apart, breaking the ionic bond that held them together. After the salt compounds are pulled apart, the sodium and chloride atoms are surrounded by water molecules. Once this happens, the salt is dissolved, resulting in a homogeneous solution. (credit: work by Andrea Hazard, CC-BY-SA 4.0) When NaCl dissolves in water, the ions separate and go their own way in solution; the ions are now written with their respective charges, and the (aq) phase label emphasizes that they are dissolved (Figure 15.4a and Figure 15.4b). When an ionic compound dissociates in water, water molecules surround each ion and separate it from the rest of the solid. Each ion goes its own way in solution. All ionic compounds that dissolve behave this way. (This behaviour was frst suggested by the Swedish chemist Svante August Arrhenius [1859–1927] as part of his PhD dissertation in 1884. Interestingly, his PhD examination team had a hard time believing that ionic compounds would behave like this, so they gave Arrhenius a barely passing grade. Later, this work was cited when Arrhenius was awarded the Nobel Prize in Chemistry.) Keep in mind that when the ions separate, all of the ions separate. Thus, when CaCl2 dissolves, the one Ca2+ ion and the two Cl−ions separate from each other: CaCl2(s) →Ca2+(aq) + Cl–(aq) + Cl–(aq) which is simplifed to CaCl2(s) → Ca2+(aq) + 2Cl–(aq) That is, the two chloride ions go of on their own. They do not remain as Cl2 (that would be elemental chlorine; these are chloride ions); they do not stick together to make Cl2 −or Cl2 2−. They become dissociated ions in their own right. It is important to note that polyatomic ions retain their overall identity when they are dissolved. It is important to understand and write the chemical equation that represents the dissociation of ions, as it is a crucial concept when writing complete ionic chemical equations, and net ionic equations. 931 | 15.4 DESCRIBING REACTIONS IN SOLUTIONS BY WRITING MOLECULAR, COMPLETE IONIC, AND NET IONIC EQUATIONS Example 15.4a Write the chemical equation that represents the dissociation of each ionic compound. a. KBr b. Na2SO4 c. (NH4)3PO4 Solution a. KBr(s) → K+(aq) + Br−(aq) b. Not only do the two sodium ions go their own way, but the polyatomic sulfate ion stays together as the sulfate ion. The dissolving equation is Na2SO4(s) → 2Na+(aq) + SO42−(aq) c. Not only do the three ammonium ions stay together, but the one phosphate ion stays together as well since they are both polyatomic ions. The dissolving equation is (NH4)3PO4(s) → 3NH4+(aq) + PO43−(aq) 15.4 DESCRIBING REACTIONS IN SOLUTIONS BY WRITING MOLECULAR, COMPLETE IONIC, AND NET IONIC EQUATIONS | 932 Exercise 15.4a Exercise 15.4a.1 1) Write the chemical equation that represents the dissociation of (NH4)2S. Check Your Answer1 Exercise 15.4a.2 2) Write the chemical equation that represents the dissociation of iron(III) sulfate, Fe2(SO4)3. Check Your Answer2 Equations for Ionic Reactions Given the abundance of water on earth, it stands to reason that a great many chemical reactions take place in aqueous media. When ions are involved in these reactions, the chemical equations may be written with various levels of detail appropriate to their intended use. To illustrate this, consider a reaction between ionic compounds taking place in an aqueous solution. When aqueous solutions of CaCl2 and AgNO3 are mixed, a precipitation reaction takes place producing aqueous Ca(NO3)2 and solid AgCl as shown in the following molecular equation: CaCl2(aq) + 2AgNO3(aq) → Ca(NO3)2(aq) + 2AgCl(s) This balanced equation, derived by means describe in previous sections, is called a molecular equation because it doesn’t explicitly represent the ionic species that are present in solution. When ionic compounds dissolve in water, they may dissociate into their constituent ions, which are subsequently dispersed homogenously throughout the resulting solution. Ionic compounds dissolved in water are, therefore, more realistically represented as dissociated ions, in this case: CaCl2(aq) → Ca2+(aq) + 2Cl−(aq) 2AgNO3(aq) → 2Ag+(aq) + 2NO3 −(aq) Ca(NO3)2(aq) → Ca2+(aq) + 2NO3 −(aq) Unlike these three ionic compounds, AgCl does not dissolve in water to a signifcant extent, as signifed by its physical state notation, s. 933 | 15.4 DESCRIBING REACTIONS IN SOLUTIONS BY WRITING MOLECULAR, COMPLETE IONIC, AND NET IONIC EQUATIONS Explicitly representing all dissolved ions results in a complete ionic equation. In this particular case, the formulas for the dissolved ionic compounds are replaced by formulas for their dissociated ions: Ca2+(aq) + 2Cl−(aq) + 2Ag+(aq) + 2NO3 −(aq) → Ca2+(aq) + 2NO3 −(aq) + 2AgCl(s) NOTE: the order in which the products are written, does not matter. The solid may be written frst or second. Examining this complete ionic equation shows that two chemical species are present in identical forms on both sides of the arrow, Ca2+(aq) and NO3 −(aq). These spectator ions—ions whose presence is required to maintain charge neutrality—are neither chemically nor physically changed by the process, and so they may be eliminated from the equation to yield a more concise representation called a net ionic equation. The net ionic equation summarizes the chemical change that occurred in the chemical reaction. Below, the complete ionic equation is shown frst. The spectator ions (Ca2+(aq) and 2NO3 –(aq)) are bolded in blue. Then, the other reactants and products are included in the net ionic equation, which is shown second. Complete Ionic Equation: Ca2+(aq) + 2Cl−(aq) + 2Ag+(aq) + 2NO3 −(aq) →Ca2+(aq) + 2NO3 −(aq)+ 2AgCl(s) Net Ionic Equation: 2Cl−(aq) + 2Ag+(aq) → 2AgCl(s) which can be simplifed to the smallest possible integers as coefcients and written as Cl−(aq) + Ag+(aq) → AgCl(s) This net ionic equation indicates that solid silver chloride may be produced from dissolved chloride and silver ions, regardless of the source of these ions. These molecular and complete ionic equations provide additional information, namely, the ionic compounds used as sources of Cl−and Ag+. The above reaction is a precipitation reaction. Watch How to Write and Balance Net Ionic Equations (6min 18sec) ( PXRH_IrN11Y). Example 15.4b The molecular equations for two chemical reactions are given below: Write the complete ionic equation for each chemical reaction. a. KBr(aq) + AgC2H3O2(aq) → KC2H3O2(aq) + AgBr(s) b. MgSO4(aq) + Ba(NO3)2(aq) → Mg(NO3)2(aq) + BaSO4(s) Solution For any ionic compound that is aqueous, we will write the compound as separated ions. a. The complete ionic equation is K+(aq) + Br−(aq) + Ag+(aq) + C2H3O2−(aq) → K+(aq) + C2H3O2−(aq) + AgBr(s) 15.4 DESCRIBING REACTIONS IN SOLUTIONS BY WRITING MOLECULAR, COMPLETE IONIC, AND NET IONIC EQUATIONS | 934 b. The complete ionic equation is Mg2+(aq) + SO42−(aq) + Ba2+(aq) + 2NO3−(aq) → Mg2+(aq) + 2NO3−(aq) + BaSO4(s) Exercise 15.4b Write the complete ionic equation for the following chemical chemical reaction: CaCl2(aq) + Pb(NO3)2(aq) → Ca(NO3)2(aq) + PbCl2(s) Check Your Answer3 Example 15.4c Write the net ionic equation for each chemical reaction below. Identify the spectator ions in each of these reactions. a. K+(aq) + Br−(aq) + Ag+(aq) + C2H3O2−(aq) → K+(aq) + C2H3O2−(aq) + AgBr(s) b. Mg2+(aq) + SO42−(aq) + Ba2+(aq) + 2 NO3−(aq) → Mg2+(aq) + 2 NO3−(aq) + BaSO4(s) Solution a. In the frst equation, the K+(aq) and C2H3O2−(aq) ions are spectator ions (bolded in blue), so they are cancelled. K+(aq) + Br−(aq) + Ag+(aq) +C2H3O2−(aq) → K+(aq) + C2H3O2−(aq) + AgBr(s); The net ionic equation is Br−(aq) + Ag+(aq) → AgBr(s) b. In the second equation, the Mg2+(aq) and NO3−(aq) ions are spectator ions (bolded in blue), so they are cancelled. Mg2+(aq) + SO42−(aq) + Ba2+(aq) +2NO3−(aq) → Mg2+(aq) + 2NO3−(aq) + BaSO4(s); The net ionic equation is SO42−(aq) + Ba2+(aq) → BaSO4(s) 935 | 15.4 DESCRIBING REACTIONS IN SOLUTIONS BY WRITING MOLECULAR, COMPLETE IONIC, AND NET IONIC EQUATIONS Exercise 15.4c Write the net ionic equation for the chemical reaction below. Identify the spectator ions in this reaction. Ca2+(aq) + 2Cl−(aq) + Pb2+(aq) + 2NO3−(aq) → Ca2+(aq) + 2NO3−(aq) + PbCl2(s) Check Your Answer4 Example 15.4d When aqueous barium chloride is mixed with an aqueous solution of sodium sulfate, the mixture reacts to yield solid barium sulfate and aqueous sodium chloride. Write balanced molecular, complete ionic, and net ionic equations for this process. Name the spectator ions. Solution Begin by identifying formulas for the reactants and products and arranging them properly in chemical equation form: BaCl2(aq) + Na2SO4(aq) → BaSO4(s) + NaCl(aq) (unbalanced equation) Balance is achieved easily in this case by changing the coeffcient for NaCl to 2, resulting in the molecular equation for this reaction: BaCl2(aq) + Na2SO4(aq) → BaSO4(s) + 2NaCl(aq) (balanced equation) The dissolved ionic compounds, BaCl2, Na2SO4, and NaCl, can be represented as dissociated ions to yield the complete ionic equation: Ba2+(aq) + 2Cl−(aq) + 2Na+(aq) + SO42-(aq) → BaSO4(s) + 2Na+(aq) + 2Cl−(aq) Finally, identify the spectator ion(s), in this case Na+(aq) and Cl– (aq), and remove them from each side of the equation to generate the net ionic equation: Ba2+(aq) + SO42-(aq) → BaSO4(s) 15.4 DESCRIBING REACTIONS IN SOLUTIONS BY WRITING MOLECULAR, COMPLETE IONIC, AND NET IONIC EQUATIONS | 936 Exercise 15.4d When an aqueous solution of AgNO3 is added to an aqueous solution of CaCl2, insoluble AgCl precipitates. Write three equations (complete molecular equation, complete ionic equation, and net ionic equation) that describe this process. Check Your Answer5 Net Ionic Equations of Gas Forming Reactions Sometimes a gas will be involved as one of the reactants or products in a solution reaction. Net ionic equations can be written for these chemical reactions, as well. For example, 2HCl(aq) + Na2S(aq) →H2S(g) + 2NaCl(aq) In this example, since hydrochloric acid fully dissociates in aqueous solution, the complete ionic equation would be: 2H+(aq) + 2Cl–(aq) + 2Na+(aq) + S2-(aq) → H2S(g) + 2Na+(aq) + 2Cl–(aq) Removing the spectator ions we obtain the net ionic equation: 2H+(aq) + S2-(aq) → H2S(g) Exercise 15.4e When carbon dioxide gas combines with an aqueous solution of sodium hydroxide, the mixture reacts to yield aqueous sodium carbonate and liquid water. Write balanced molecular, complete ionic, and net ionic equations for this process. Check Your Answer6 Net Ionic Equations of Single Replacement Reactions 937 | 15.4 DESCRIBING REACTIONS IN SOLUTIONS BY WRITING MOLECULAR, COMPLETE IONIC, AND NET IONIC EQUATIONS Involving Aqueous Solutions Recall, that during a single replacement reaction (SR) there is an exchange of elements; typically cations; an element becomes a compound and a compound becomes an element. SR reactions are always oxidation reduction (redox) reactions. These types of reactions will be discussed more in the Oxidation Reduction chapter, but we will cover the basics of writing net ionic equations for redox reactions undergoing single replacement. Consider the following SR reaction between iron metal and copper(II) nitrate: Balanced Molecular Equation: Fe(s) + Cu(NO3)2(aq) → Fe(NO3)2(aq) + Cu(s) Complete Ionic Equation: Fe(s) + Cu2+(aq) + 2NO3 –(aq) → Fe2+(aq) + 2NO3 –(aq) + Cu(s) Since nitrate ions are dissociated in the reactant and product side of the complete ionic equation, these are removed to produce the net ionic equation. Net Ionic Equation: Fe(s) + Cu2+(aq) → Fe2+(aq) + Cu(s) Exercise 15.4f Write the complete ionic and net ionic equations for the following chemical reaction. Ni(s) + 2HCl(aq) → NiCl2(aq) + H2(g) Check Your Answer7 Net Ionic Equations of Neutralization Reactions A neutralization reaction is a type of double displacement/replacement reaction. There is a double exchange of two cations, or two anions to form two new compounds. Neutralization reactions will be discussed more in the Acids and Bases chapter, but we will cover the basics of writing net ionic equations for neutralization reactions, now. In a neutralization reaction, the reactants are an acid and a base, and the products are often a salt (soluble or insoluble) and water, and neither reactant is the water itself: acid + base → salt + water Consider the following reaction between a strong acid, hydrochloric acid, and strong base, sodium hydroxide. The balanced molecular equation can be used to generate a complete ionic equation, and its net ionic equation: Balanced Molecular Equation: HCl(aq) + NaOH(aq) → NaCl(aq) + H2O(l) 15.4 DESCRIBING REACTIONS IN SOLUTIONS BY WRITING MOLECULAR, COMPLETE IONIC, AND NET IONIC EQUATIONS | 938 Complete Ionic Equation: H+(aq) + Cl–(aq) + Na+(aq) + OH–(aq) → Na+(aq) + Cl –(aq) + H2O(l) Since nitrate ions are dissociated in the reactant and product side of the complete ionic equation, these are removed to produce the net ionic equation. Net Ionic Equation: H+(aq) + OH–(aq) → H2O(l) Links to Interactive Learning Tools Explore Net Ionic Equations ( Precipitation-Reactions) from the Physics Classroom ( Attribution & References Except where otherwise noted, this page is adapted from “4.1 Writing and Balancing Chemical Equations ( equations/)” In CHEM 1114 – Introduction to Chemistry by Shirley Wacowich-Sgarbi and Langara Chemistry Department is licensed under CC BY-NC-SA 4.0. Adaptations and additions to content in this section were made by Jackie MacDonald for student comprehension. Notes 1. ( NH4)2S (s) → 2NH4 +(aq) + S2−(aq) 2. Fe2(SO4)3 (s) → 2Fe3+(aq) + 3SO4 2−(aq) 3. Complete ionic equation is Ca2+(aq) + 2Cl−(aq) + Pb2+(aq) + 2NO3 −(aq) →Ca2+(aq) + 2NO3 −(aq) + PbCl2(s) 4. Ca2+(aq) and NO3 −(aq) ions are spectator ions (bolded in blue), so they are cancelled. Ca2+(aq) + 2Cl−(aq) + Pb2+(aq) + 2NO3 −(aq) →Ca2+(aq) + 2NO3 −(aq) + PbCl2(s); The net ionic equation is Pb2+(aq) + 2Cl−(aq) → PbCl2(s) 5. Begin by identifying formulas for the reactants and products and arranging them properly in chemical equation form: AgNO3(aq) + CaCl2(aq) → AgCl(s) + Ca(NO3)2(aq) (unbalanced equation) Balancing this reaction results in the molecular equation for this reaction: 2AgNO3(aq) + CaCl2(aq) → 2AgCl(s) + Ca(NO3)2(aq) (balanced equation) The dissolved ionic compounds, AgNO3, CaCl2, and Ca(NO3)2, can be represented as dissociated ions to yield the complete ionic equation: 2Ag+(aq) + 2NO3 −(aq) + Ca2+(aq) + 2Cl-(aq) → 2AgCl(s) + Ca2+(aq) + 2NO3 −(aq) Finally, identify the spectator ion(s), in this case Ca2+(aq) and NO3 −(aq), and remove them from each side of the equation to generate the net ionic equation: 2Ag+(aq) + 2Cl-(aq) → 2AgCl(s) which can be simplifed to the smallest possible integers as coefcients and written as Ag+(aq) + Cl-(aq) → AgCl(s) 6. Begin by identifying formulas for the reactants and products and arranging them properly in chemical equation form: 939 | 15.4 DESCRIBING REACTIONS IN SOLUTIONS BY WRITING MOLECULAR, COMPLETE IONIC, AND NET IONIC EQUATIONS CO2(g) + NaOH(aq) → Na2CO3(aq) + H2O(l) (unbalanced) Balance is achieved easily in this case by changing the coefcient for NaOH to 2, resulting in the molecular equation for this reaction: CO2(g) + 2NaOH(aq) → Na2CO3(aq) + H2O(l) (balanced) The two dissolved ionic compounds, NaOH and Na2CO3, can be represented as dissociated ions to yield the complete ionic equation: CO2(g) + 2Na+(aq) +2OH-(aq) → 2Na+(aq) + CO3 2-(aq) + H2O(l) Finally, identify the spectator ion(s), in this case Na+(aq), and remove it from each side of the equation to generate the net ionic equation: CO2(g) + 2OH-(aq) → CO3 2-(aq) + H2O(l) To review a video of this solution watch How to Write the Net Ionic Equation for NaOH + CO2 = Na2CO3 + H2O (2mins 38sec). ( QKXeIdCVqYQ) 7. Complete Ionic Equation: Ni(s) + 2H+(aq) + 2Cl-(aq) → Ni2+(aq) + 2Cl-(aq) + H2(g) Since chloride ions are dissociated in both the reactant and product side of the complete ionic equation, these are removed to produce the net ionic equation. Net Ionic Equation: Ni(s) + 2H+(aq) → Ni2+(aq) + H2(g) CHAPTER 15 - SUMMARY | 940 CHAPTER 15 - SUMMARY 15.1 Salts Salts are a chemical compound formed when ions form ionic bonds. In these reactions, one atom gives up one or more electrons, and thus becomes positively charged, whereas the other accepts one or more electrons and becomes negatively charged; overall the ionic compound has no net charge. Salts typically are crystalline, odourless, colourless/transparent or white, and have high melting and boiling points. Many salts are soluble in water; however, some are not. If a given salt is soluble in water, it completely dissociates into ions other than a hydrogen ion (H+) or hydroxide ion (OH –) and forms an aqueous solution. This fact is elemental in distinguishing salts from acids and bases. Salts are derived from the neutralization reaction of an acid and base. Since acids and bases always contain either a metal cation or a cation derived from ammonium (NH4 +) and a nonmetal anion, respectively, the two can combine to form a salt. 15.2 Electrolytes Substances that dissolve in water to yield ions are called electrolytes. Electrolytes may be covalent compounds that chemically react with water to produce ions (for example, acids and bases), or they may be ionic compounds that dissociate to yield their constituent cations and anions, when dissolved. Dissolution of an ionic compound is facilitated by ion-dipole attractions between the ions of the compound and the polar water molecules. Soluble ionic substances and strong acids ionize completely and are strong electrolytes, while weak acids and bases ionize to only a small extent and are weak electrolytes. Nonelectrolytes are substances that do not produce ions when dissolved in water. 15.3 Precipitation Reactions Chemical reactions are classifed according to similar patterns of behaviour. Precipitation is one type of chemical reaction which involves the formation of one or more insoluble products. Precipitation reactions, also called double displacement reactions can be summarized with the following reaction equation: AB(aq) + CD (aq) → AD(s)+CB(aq) or (s) The formation of the solid from combining two aqueous solutions is the DRIVING FORCE of the reaction (the factor that makes the reaction go). A precipitation reaction can be predicted to occur with the help of a solubility table. 941 | CHAPTER 15 - SUMMARY 15.4 Describing Reactions in Solutions by Writing Molecular, Complete Ionic, and Net Ionic Equations Chemical equations are symbolic representations of chemical and physical changes. Chemical reactions in aqueous solution that involve ionic reactants or products may be represented more realistically by complete ionic equations and, more succinctly, by net ionic equations. Complete ionic and net ionic equations can be used to illustrate what is happening during precipitation reactions, neutralization reactions, gas evolving reactions, and single replacement reactions when these reactions occur in aqueous solutions. Attributions and References This page is adapted by Jackie MacDonald from: • Salts (15.1) is adapted from “2.4 Inorganic Compounds Essential to Human Functioning ( human-functioning)” In Anatomy and Physiology 2e (Open Stax) by J. Gordon Betts, Kelly A. Young, James A. Wise, Eddie Johnson, Brandon Poe, Dean H. Kruse, Oksana Korol, Jody E. Johnson, Mark Womble, Peter DeSaix is licensed under CC BY 4.0 ( Access for free at Anatomy and Physiology 2e (OpenStax) ( physiology-2e/pages/1-introduction). • Electrolytes (15.2) is adapted from “Ch. 11 Summary ( 11-summary)” In Chemistry 2e (Open Stax) by Paul Flowers, Klaus Theopold, Richard Langley & William R. Robinson is licensed under CC BY 4.0. Access for free at Chemistry 2e (OpenStax). ( • Precipitation Reactions (15.3) is adapted from “6.2 Precipitation Reactions ( metathesis%20reactions.)” In CHEM 1114 – Introduction to Chemistry (BCcampus, Pressbooks) by Shirley Wacowich-Sgarbi and Langara Chemistry Department is licensed under CC BY-NC-SA 4.0. • Describing Reactions in Solutions by Writing Molecular, Complete Ionic, and Net Ionic Equations (15.4) is adapted from “4. 1 Writing and Balancing Chemical Equations ( chemical-equations/)” In CHEM 1114 – Introduction to Chemistry by Shirley Wacowich-Sgarbi and Langara Chemistry Department is licensed under CC BY-NC-SA 4.0. Adaptations to aid in student comprehension by Jackie MacDonald. CHAPTER 15 - REVIEW | 942 CHAPTER 15 - REVIEW 15.1 Salts 1. Explain how a salt is formed. Check answers: 1 2. What salt will form when you combine hydrobromic acid and lithium hydroxide? Check answers: 2 3. What salt will form when you combine sulfuric acid, H2SO4, and sodium hydroxide? Check answers: 3 15.2 Electrolytes 1. Diferentiate between strong and weak electrolytes. Check Answers: 4 2. Explain why the ions Na+ and Cl−are strongly solvated in water but not in hexane, a solvent composed of nonpolar molecules. Check Answer: 5 3. Explain why solutions of HBr in benzene (a nonpolar solvent) are nonconductive, while solutions in water (a polar solvent) are conductive. 4. Consider the solutions presented: a. Which of the following sketches, shown in the fgure below, best represents the ions in a solution of Fe(NO3)3(aq)? b. Write a balanced chemical equation showing the products of the dissolution of Fe(NO3)3. Check Answer: 6 5. Compare the processes that occur when methanol (CH3OH), hydrogen chloride (HCl), and sodium hydroxide (NaOH) dissolve in water. Write equations and prepare sketches showing the form in which each of these compounds is present in its respective solution. 943 | CHAPTER 15 - REVIEW 6. What is the expected electrical conductivity of the following solutions? a. NaOH(aq) b. HCl(aq) c. C6H12O6(aq) (glucose) d. NH3(l) Check Answer: 7 7. Why are most solid ionic compounds electrically nonconductive, whereas aqueous solutions of ionic compounds are good conductors? Would you expect a liquid (molten) ionic compound to be electrically conductive or nonconductive? Explain. 8. Indicate the most important type of intermolecular attraction responsible for solvation in each of the following solutions: a. the solutions in Figure 14.2c b. methanol, CH3OH, dissolved in ethanol, C2H5OH c. methane, CH4, dissolved in benzene, C6H6 d. the polar halocarbon CF2Cl2 dissolved in the polar halocarbon CF2ClCFCl2 e. O2(l) in N2(l) Check Answer: 8 15.3 Precipitation Reactions 1. What are the general characteristics that help you recognize double replacement reactions? Check Answers: 9 2. What are the general characteristics that help you recognize a precipitation reaction? Check Answers: 10 3. Assuming that the following is a precipitation reaction, determine the products (and identify their phases, aq or s) and write the balanced chemical equation. Zn(NO3)2 + NaOH →? Check Answers: 11 4. Assuming that the following is a precipitation reaction, determine the products (and identify their phases, aq or s) and write the balanced chemical equation. MgCl2 + NaOH → Check Answers: 12 5. Use the solubility table to predict if the following double replacement reaction will occur and, if so, write a balanced chemical equation. Pb(NO3)2 + KBr →? Check Answers: 13 6. Use the solubility table to predict if the following double replacement reaction will occur and, if so, write a balanced chemical equation. CHAPTER 15 - REVIEW | 944 KCl + Na2CO3 →? Check Answers:14 7. Which solution could be used to precipitate the barium ion, Ba2+, in a water sample: sodium chloride, sodium hydroxide, or sodium sulfate? What is the formula for the expected precipitate? Check Answers: 15 15.4 Net Ionic Equations 1. From the balanced molecular equations, write the complete ionic and net ionic equations for the following: (a) K 2C2O4(aq) + Ba(OH)2 → 2KOH(aq) + BaC2O2(s) (b) Pb(NO3)2(aq) + H2SO4(aq) → PbSO4(s) +2HNO3(aq) (c) CaCO 3(s) + H2SO4(aq) → CaSO4(s) + CO2(g) + H2O(l) Check Answers: 16 2. Predict the result of mixing reasonably concentrated solutions of the following ionic compounds. If precipitation is expected, write a balanced net ionic equation for the reaction. (a) potassium sulfate and barium nitrate (b) lithium chloride and silver acetate (c) lead nitrate and ammonium carbonate Check Answers: 17 3. From the balanced molecular equation, write the net ionic equations for the following reaction: 2HCl(aq) + Ba(OH)2(aq) → BaCl2(aq) + H2O(l) Check Answers: 18 4. From the balanced molecular equation, write the net ionic equations for the following reaction: 2AgNO3(aq) + Cu(s) → Cu(NO3)2(aq) + 2Ag(s) Check Answers: 19 Attribution & References Except where otherwise noted, this page is adapted by Jackie MacDonald from: • 15.1 Salts & 15.2 Electrolytes, 15.3 Precipitation Reactions (#2, 4, 6), 15.4 Net Ionic Equations (#3 & #4) – Questions created by Jackie MacDonald, licensed under CC BY 4.0 • 15.3 Precipitation Reactions – Questions 1, 3, 5, 7 – adapted from “4.2 Classifying Chemical Reactions“In CHEM 1114 – Introduction to Chemistry by Shirley Wacowich-Sgarbi and Langara Chemistry Department is licensed under CC BY-NC-SA 4.0. • 15.4 Net Ionic Equations, questions 1 & 2 are adapted from “4.1 Writing and Balancing Chemical 945 | CHAPTER 15 - REVIEW Equations ( balancing-chemical-equations/)” and “4.2 Classifying Chemical Reactions“In CHEM 1114 – Introduction to Chemistry by Shirley Wacowich-Sgarbi and Langara Chemistry Department is licensed under CC BY-NC-SA 4.0.Adaptations and additions were made to content from these sections for student comprehension. Notes 1. A salt is formed in a neutralization reaction of an acid and a base. Acids and bases always contain either a metal cation or a cation derived from ammonium (NH4 +) and a nonmetal anion, the two can combine to form a salt. 2. the anion from the HBr is Br-. The cation from LiOH is Li+. The salt formed is LiBr, lithium bromide. 3. the anion from the H2SO4 is SO4 2-. The cation from NaOH is Na+. The salt formed is Na2SO4, sodium sulfate. 4. A strong electrolyte is a salt that fully ionizes (100%) in the aqueous solution and are great conductors of electricity. Whereas, with a weak electrolyte only a relatively small fraction of the dissolved substance undergoes the ion-producing process and therefore is a poor conductor of electricity. 5. Crystals of NaCl dissolve in water, a polar liquid with a very large dipole moment, and the individual ions become strongly solvated. Hexane is a nonpolar liquid with a dipole moment of zero and, therefore, does not signifcantly interact with the ions of the NaCl crystals. 6. (a) Fe(NO3)3 is a strong electrolyte, thus it should completely dissociate into Fe3+ and ( ) ions. Therefore, (z) best represents the solution. (b) 7. (a) high conductivity (solute is an ionic compound that will dissociate when dissolved); (b) high conductivity (solute is a strong acid and will ionize completely when dissolved); (c) nonconductive (solute is a covalent compound, neither acid nor base, unreactive towards water); (d) low conductivity (solute is a weak base and will partially ionize when dissolved) 8. (a) ion-dipole; (b) hydrogen bonds; (c) dispersion forces; (d) dipole-dipole attractions; (e) dispersion forces 9. A double replacement reaction occurs when parts of two ionic compounds are exchanged, making two new compounds. A characteristic of a double-replacement equation is that there are two compounds as reactants and two diferent compounds as products. 10. In a precipitation reactions, the reactants will be dissolved substances in aqueous solutions and will react to form one (or more) solid products. Many reactions of this type involve the exchange of ions between ionic compounds in aqueous solution and a type of double displacement/replacement reaction. 11. Zn(NO3)2(aq) + 2NaOH(aq) → 2NaNO3(aq) + Zn(OH)2(s) 12. MgCl2(aq) + 2NaOH(aq) → 2NaCl(aq) + Mg(OH)2(s) 13. The reaction will occur since a precipitate will form. Pb(NO3)2(aq) + 2KBr(aq) → 2KNO3(aq) + PbBr2(s) 14. No reaction occurs; both products are soluble in water. 2KCl(aq) + Na2CO3(aq) → 2NaCl(aq) + K2CO3(aq) 15. sodium sulfate, BaSO4 16. (a) complete ionic: 2K+(aq) + C2O4 2−(aq) + Ba2+(aq) + 2OH−(aq) → 2K+(aq) + 2OH−(aq) + BaC2O4(s) net ionic: Ba2+(aq) + C2O4 2−(aq) → BaC2O4(s) (b) complete ionic: Pb2+(aq) + 2NO3 −(aq) + 2H+(aq) + SO4 2−(aq) → PbSO4(s) + 2H+(aq) + 2NO3 −(aq) net ionic: Pb2+(aq) + SO4 2−(aq) → PbSO4(s) (c) complete ionic: CaCO3(s) + 2H+(aq) + SO4 2−(aq) → CaSO4(s) + CO2(g) + H2O(l) net ionic: CaCO3(s) + 2H+(aq) + SO4 2−(aq) → CaSO4(s) + CO2(g) + H2O(l) CHAPTER 15 - REVIEW | 946 17. (a) The two possible products for this combination are KNO3 and BaSO4. The solubility guidelines indicate BaSO4 is insoluble, and so a precipitation reaction is expected. The net ionic equation for this reaction isBa2+(aq) + SO4 2−(aq) →BaSO 4(s) (b) The two possible products for this combination are LiC2H3O2 and AgCl. The solubility guidelines indicate AgCl is insoluble, and so a precipitation reaction is expected. The net ionic equation for this reaction isAg+(aq) + Cl−(aq) → AgCl(s) (c) The two possible products for this combination are PbCO3 and NH4NO3. The solubility guidelines indicate PbCO3 is insoluble, and so a precipitation reaction is expected. The net ionic equation for this reaction isPb2+(aq) + CO3 2−(aq) → PbCO3(s) 18. H+(aq) + OH-(aq) →H2O(l) 19. 2Ag+(aq) + Cu(s) → Cu2+(aq) + 2Ag(s) |
8602 | https://lup.lub.lu.se/record/a81ab5a2-d22f-49a0-b7db-ce92b22d70ca | Trypsin activation of porcine procolipase. Kinetics of activation and effects on lipid binding | Lund University Publications
Skip to main content
LUND UNIVERSITY|LUND UNIVERSITY LIBRARIES
Lund University Publications
LUND UNIVERSITY LIBRARIES
Register publications|Statistics|Marked list 0|Saved searches 0Advanced
HomePublicationsDepartments
Trypsin activation of porcine procolipase. Kinetics of activation and effects on lipid binding
Mark
Wieloch, TadeuszLU (1985) In FEBS Letters 185(1). p.63-66 Abstract
The kinetics of trypsin activation of pancreatic procolipase was investigated and the pH dependence of the binding of procolipase and colipase to a tributyrine-bile salt interface studied. The K m was 0.06 mM and K cat 8 s-1, and was of the same order of magnitude as for the activation of pancreatic zymogens. At basic pH values colipase had a higher affinity for the tributyrine-bile salt interface as compared to procolipase. The trypsin activation of procolipase ensures a rapid degradation of dietary lipids in the intestine.
Links
Research Portal page
DOI
Google Scholar find title
Please use this url to cite or link to this publication:
DetailsBibTeX
authorWieloch, TadeuszLU organization
Laboratory for Experimental Brain Research (research group)
publishing date 1985-06-03 type Contribution to journal publication status published subject
Medicinal Chemistry
keywordsColipase, Kinetics, Lipid binding, Procolipase, Trypsin activation in FEBS Letters volume 185 issue 1 pages 4 pages publisher Wiley-Blackwell external identifiers
scopus:0021873193
pmid:3838943
ISSN0014-5793DOI10.1016/0014-5793(85)80741-9languageEnglishLU publication?yes id a81ab5a2-d22f-49a0-b7db-ce92b22d70ca date added to LUP 2019-06-13 17:23:52 date last changed 2025-04-04 14:08:01
@article{a81ab5a2-d22f-49a0-b7db-ce92b22d70ca,
abstract = {{The kinetics of trypsin activation of pancreatic procolipase was investigated and the pH dependence of the binding of procolipase and colipase to a tributyrine-bile salt interface studied. The Km was 0.06 mM and Kcat 8 s-1, and was of the same order of magnitude as for the activation of pancreatic zymogens. At basic pH values colipase had a higher affinity for the tributyrine-bile salt interface as compared to procolipase. The trypsin activation of procolipase ensures a rapid degradation of dietary lipids in the intestine.}},
author = {{Wieloch, Tadeusz}},
issn = {{0014-5793}},
keywords = {{Colipase; Kinetics; Lipid binding; Procolipase; Trypsin activation}},
language = {{eng}},
month = {{06}},
number = {{1}},
pages = {{63--66}},
publisher = {{Wiley-Blackwell}},
series = {{FEBS Letters}},
title = {{Trypsin activation of porcine procolipase. Kinetics of activation and effects on lipid binding}},
url = {{
doi = {{10.1016/0014-5793(85)80741-9}},
volume = {{185}},
year = {{1985}},
}
Built with LibreCatUser guide|About accessibility|Contact |
8603 | https://en.wikipedia.org/wiki/Circular_reasoning | Contents
Circular reasoning
Part ofa serieson
Pyrrhonism
PyrrhonistsPyrrhoTimon of PhliusAenesidemusAgrippa the SkepticSextus Empiricus
ConceptsAtaraxiaAcatalepsyAdiaphoraAporiaDogmaEpoché
Similar philosophiesEmpiric schoolEpilogismAcademic skepticism
Modern influenceMünchhausen trilemmaBenson MatesRobert Fogelin
Philosophy portal
vte
Circular reasoning (Latin: circulus in probando, "circle in proving"; also known as circular logic) is a logical fallacy in which the reasoner begins with what they are trying to end with. Circular reasoning is not a formal logical fallacy, but a pragmatic defect in an argument whereby the premises are just as much in need of proof or evidence as the conclusion. As a consequence, the argument becomes a matter of faith and fails to persuade those who do not already accept it. Other ways to express this are that there is no reason to accept the premises unless one already believes the conclusion, or that the premises provide no independent ground or evidence for the conclusion. Circular reasoning is closely related to begging the question, and in modern usage the two generally refer to the same thing.
Circular reasoning is often of the form: "A is true because B is true; B is true because A is true." Circularity can be difficult to detect if it involves a longer chain of propositions.
An example of circular reasoning is: "Alkaline water is healthy because it results in health benefits, and it has health benefits because it is healthy".
History
The problem of circular reasoning has been noted in Western philosophy at least as far back as the Pyrrhonist philosopher Agrippa who includes the problem of circular reasoning among his Five Tropes of Agrippa. The Pyrrhonist philosopher Sextus Empiricus described the problem of circular reasoning as "the reciprocal trope":
The reciprocal trope occurs when what ought to be confirmatory of the object under investigation needs to be made convincing by the object under investigation; then, being unable to take either in order to establish the other, we suspend judgement about both.
The problem of induction
Joel Feinberg and Russ Shafer-Landau note that "using the scientific method to judge the scientific method is circular reasoning". Scientists attempt to discover the laws of nature and to predict what will happen in the future, based on those laws. The laws of nature are arrived at through inductive reasoning. David Hume's problem of induction demonstrates that one must appeal to the principle of the uniformity of nature if they seek to justify their implicit assumption that laws which held true in the past will also hold true in the future. Since the principle of the uniformity of nature is itself an inductive principle, any justification for induction must be circular. But as Bertrand Russell observed, "The method of 'postulating' what we want has many advantages; they are the same as the advantages of theft over honest toil".
See also
References
External links
vteCommonfallacies(list)
Formal | Inpropositional logicAffirming a disjunctAffirming the consequentDenying the antecedentArgument from fallacyMasked manMathematical fallacyInquantificational logicExistentialIllicit conversionProof by exampleQuantifier shiftSyllogistic fallacyAffirmative conclusion from a negative premiseNegative conclusion from affirmative premisesExclusive premisesExistentialNecessityFour termsIllicit majorIllicit minorUndistributed middle | Inpropositional logic | Affirming a disjunctAffirming the consequentDenying the antecedentArgument from fallacyMasked manMathematical fallacy | Inquantificational logic | ExistentialIllicit conversionProof by exampleQuantifier shift | Syllogistic fallacy | Affirmative conclusion from a negative premiseNegative conclusion from affirmative premisesExclusive premisesExistentialNecessityFour termsIllicit majorIllicit minorUndistributed middle
Inpropositional logic | Affirming a disjunctAffirming the consequentDenying the antecedentArgument from fallacyMasked manMathematical fallacy
Inquantificational logic | ExistentialIllicit conversionProof by exampleQuantifier shift
Syllogistic fallacy | Affirmative conclusion from a negative premiseNegative conclusion from affirmative premisesExclusive premisesExistentialNecessityFour termsIllicit majorIllicit minorUndistributed middle
Informal | EquivocationEquivocationFalse equivalenceFalse attributionQuoting out of contextLoki's WagerNo true ScotsmanReificationMap–territory relationQuestion-beggingCircular reasoning/Begging the questionLoaded languageLeading questionCompound question/Loaded question/Complex questionNo true ScotsmanCorrelative-basedFalse dilemmaPerfect solutionDenying the correlativeSuppressed correlativeIllicit transferenceCompositionDivisionEcologicalSecundum quidAccidentConverse accidentFaulty generalizationAnecdotal evidenceSampling biasCherry pickingMcNamaraBase rate/ConjunctionDouble countingFalse analogySlothful inductionOverwhelming exceptionAmbiguityAccentFalse precisionMoving the goalpostsQuoting out of contextSlippery slopeSorites paradoxSyntactic ambiguityQuestionable causeAnimisticFurtiveCorrelation implies causationCum hocPost hocGambler'sInverseRegressionSingle causeSlippery slopeTexas sharpshooterAppealsLaw/LegalityStone/Proof by assertionConsequencesArgumentum ad baculumWishful thinkingEmotionChildrenFearFlatteryNoveltyPityRidiculeIn-group favoritismInvented here/Not invented hereIsland mentalityLoyaltyParade of horriblesSpiteStirring symbolsWisdom of repugnanceGenetic fallacyAd hominemAppeal to motiveAssociationReductio ad HitlerumGodwin's lawReductio ad StalinumBulverismPoisoning the wellToneTu quoqueWhataboutismAuthorityAccomplishmentIpse dixitPoverty/WealthEtymologyNatureTradition/NoveltyChronological snobberyOtherfallaciesof relevanceArgumentsAd nauseamSealioningArgument from anecdoteArgument from silenceArgument to moderationArgumentum ad populumClichéThe Four Great ErrorsI'm entitled to my opinionIgnoratio elenchiInvincible ignoranceMoralistic/NaturalisticMotte-and-bailey fallacyPsychologist's fallacyRationalizationRed herringTwo wrongs make a rightSpecial pleadingStraw man | Equivocation | EquivocationFalse equivalenceFalse attributionQuoting out of contextLoki's WagerNo true ScotsmanReificationMap–territory relation | Question-begging | Circular reasoning/Begging the questionLoaded languageLeading questionCompound question/Loaded question/Complex questionNo true Scotsman | Correlative-based | False dilemmaPerfect solutionDenying the correlativeSuppressed correlative | Illicit transference | CompositionDivisionEcological | Secundum quid | AccidentConverse accident | Faulty generalization | Anecdotal evidenceSampling biasCherry pickingMcNamaraBase rate/ConjunctionDouble countingFalse analogySlothful inductionOverwhelming exception | Ambiguity | AccentFalse precisionMoving the goalpostsQuoting out of contextSlippery slopeSorites paradoxSyntactic ambiguity | Questionable cause | AnimisticFurtiveCorrelation implies causationCum hocPost hocGambler'sInverseRegressionSingle causeSlippery slopeTexas sharpshooter | Appeals | Law/LegalityStone/Proof by assertionConsequencesArgumentum ad baculumWishful thinkingEmotionChildrenFearFlatteryNoveltyPityRidiculeIn-group favoritismInvented here/Not invented hereIsland mentalityLoyaltyParade of horriblesSpiteStirring symbolsWisdom of repugnance | Consequences | Argumentum ad baculumWishful thinking | Emotion | ChildrenFearFlatteryNoveltyPityRidiculeIn-group favoritismInvented here/Not invented hereIsland mentalityLoyaltyParade of horriblesSpiteStirring symbolsWisdom of repugnance | Genetic fallacy | Ad hominemAppeal to motiveAssociationReductio ad HitlerumGodwin's lawReductio ad StalinumBulverismPoisoning the wellToneTu quoqueWhataboutismAuthorityAccomplishmentIpse dixitPoverty/WealthEtymologyNatureTradition/NoveltyChronological snobbery | Ad hominem | Appeal to motiveAssociationReductio ad HitlerumGodwin's lawReductio ad StalinumBulverismPoisoning the wellToneTu quoqueWhataboutism | AuthorityAccomplishmentIpse dixitPoverty/WealthEtymologyNatureTradition/NoveltyChronological snobbery | Otherfallaciesof relevance | ArgumentsAd nauseamSealioningArgument from anecdoteArgument from silenceArgument to moderationArgumentum ad populumClichéThe Four Great ErrorsI'm entitled to my opinionIgnoratio elenchiInvincible ignoranceMoralistic/NaturalisticMotte-and-bailey fallacyPsychologist's fallacyRationalizationRed herringTwo wrongs make a rightSpecial pleadingStraw man | Arguments | Ad nauseamSealioningArgument from anecdoteArgument from silenceArgument to moderationArgumentum ad populum | ClichéThe Four Great ErrorsI'm entitled to my opinionIgnoratio elenchiInvincible ignoranceMoralistic/NaturalisticMotte-and-bailey fallacyPsychologist's fallacyRationalizationRed herringTwo wrongs make a rightSpecial pleadingStraw man
Equivocation | EquivocationFalse equivalenceFalse attributionQuoting out of contextLoki's WagerNo true ScotsmanReificationMap–territory relation
Question-begging | Circular reasoning/Begging the questionLoaded languageLeading questionCompound question/Loaded question/Complex questionNo true Scotsman
Correlative-based | False dilemmaPerfect solutionDenying the correlativeSuppressed correlative
Illicit transference | CompositionDivisionEcological
Secundum quid | AccidentConverse accident
Faulty generalization | Anecdotal evidenceSampling biasCherry pickingMcNamaraBase rate/ConjunctionDouble countingFalse analogySlothful inductionOverwhelming exception
Ambiguity | AccentFalse precisionMoving the goalpostsQuoting out of contextSlippery slopeSorites paradoxSyntactic ambiguity
Questionable cause | AnimisticFurtiveCorrelation implies causationCum hocPost hocGambler'sInverseRegressionSingle causeSlippery slopeTexas sharpshooter
Appeals | Law/LegalityStone/Proof by assertionConsequencesArgumentum ad baculumWishful thinkingEmotionChildrenFearFlatteryNoveltyPityRidiculeIn-group favoritismInvented here/Not invented hereIsland mentalityLoyaltyParade of horriblesSpiteStirring symbolsWisdom of repugnance | Consequences | Argumentum ad baculumWishful thinking | Emotion | ChildrenFearFlatteryNoveltyPityRidiculeIn-group favoritismInvented here/Not invented hereIsland mentalityLoyaltyParade of horriblesSpiteStirring symbolsWisdom of repugnance
Consequences | Argumentum ad baculumWishful thinking
Emotion | ChildrenFearFlatteryNoveltyPityRidiculeIn-group favoritismInvented here/Not invented hereIsland mentalityLoyaltyParade of horriblesSpiteStirring symbolsWisdom of repugnance
Genetic fallacy | Ad hominemAppeal to motiveAssociationReductio ad HitlerumGodwin's lawReductio ad StalinumBulverismPoisoning the wellToneTu quoqueWhataboutismAuthorityAccomplishmentIpse dixitPoverty/WealthEtymologyNatureTradition/NoveltyChronological snobbery | Ad hominem | Appeal to motiveAssociationReductio ad HitlerumGodwin's lawReductio ad StalinumBulverismPoisoning the wellToneTu quoqueWhataboutism | AuthorityAccomplishmentIpse dixitPoverty/WealthEtymologyNatureTradition/NoveltyChronological snobbery
Ad hominem | Appeal to motiveAssociationReductio ad HitlerumGodwin's lawReductio ad StalinumBulverismPoisoning the wellToneTu quoqueWhataboutism
AuthorityAccomplishmentIpse dixitPoverty/WealthEtymologyNatureTradition/NoveltyChronological snobbery
Otherfallaciesof relevance | ArgumentsAd nauseamSealioningArgument from anecdoteArgument from silenceArgument to moderationArgumentum ad populumClichéThe Four Great ErrorsI'm entitled to my opinionIgnoratio elenchiInvincible ignoranceMoralistic/NaturalisticMotte-and-bailey fallacyPsychologist's fallacyRationalizationRed herringTwo wrongs make a rightSpecial pleadingStraw man | Arguments | Ad nauseamSealioningArgument from anecdoteArgument from silenceArgument to moderationArgumentum ad populum | ClichéThe Four Great ErrorsI'm entitled to my opinionIgnoratio elenchiInvincible ignoranceMoralistic/NaturalisticMotte-and-bailey fallacyPsychologist's fallacyRationalizationRed herringTwo wrongs make a rightSpecial pleadingStraw man
Arguments | Ad nauseamSealioningArgument from anecdoteArgument from silenceArgument to moderationArgumentum ad populum
ClichéThe Four Great ErrorsI'm entitled to my opinionIgnoratio elenchiInvincible ignoranceMoralistic/NaturalisticMotte-and-bailey fallacyPsychologist's fallacyRationalizationRed herringTwo wrongs make a rightSpecial pleadingStraw man
Category
Inpropositional logic | Affirming a disjunctAffirming the consequentDenying the antecedentArgument from fallacyMasked manMathematical fallacy
Inquantificational logic | ExistentialIllicit conversionProof by exampleQuantifier shift
Syllogistic fallacy | Affirmative conclusion from a negative premiseNegative conclusion from affirmative premisesExclusive premisesExistentialNecessityFour termsIllicit majorIllicit minorUndistributed middle
Equivocation | EquivocationFalse equivalenceFalse attributionQuoting out of contextLoki's WagerNo true ScotsmanReificationMap–territory relation
Question-begging | Circular reasoning/Begging the questionLoaded languageLeading questionCompound question/Loaded question/Complex questionNo true Scotsman
Correlative-based | False dilemmaPerfect solutionDenying the correlativeSuppressed correlative
Illicit transference | CompositionDivisionEcological
Secundum quid | AccidentConverse accident
Faulty generalization | Anecdotal evidenceSampling biasCherry pickingMcNamaraBase rate/ConjunctionDouble countingFalse analogySlothful inductionOverwhelming exception
Ambiguity | AccentFalse precisionMoving the goalpostsQuoting out of contextSlippery slopeSorites paradoxSyntactic ambiguity
Questionable cause | AnimisticFurtiveCorrelation implies causationCum hocPost hocGambler'sInverseRegressionSingle causeSlippery slopeTexas sharpshooter
Appeals | Law/LegalityStone/Proof by assertionConsequencesArgumentum ad baculumWishful thinkingEmotionChildrenFearFlatteryNoveltyPityRidiculeIn-group favoritismInvented here/Not invented hereIsland mentalityLoyaltyParade of horriblesSpiteStirring symbolsWisdom of repugnance | Consequences | Argumentum ad baculumWishful thinking | Emotion | ChildrenFearFlatteryNoveltyPityRidiculeIn-group favoritismInvented here/Not invented hereIsland mentalityLoyaltyParade of horriblesSpiteStirring symbolsWisdom of repugnance
Consequences | Argumentum ad baculumWishful thinking
Emotion | ChildrenFearFlatteryNoveltyPityRidiculeIn-group favoritismInvented here/Not invented hereIsland mentalityLoyaltyParade of horriblesSpiteStirring symbolsWisdom of repugnance
Genetic fallacy | Ad hominemAppeal to motiveAssociationReductio ad HitlerumGodwin's lawReductio ad StalinumBulverismPoisoning the wellToneTu quoqueWhataboutismAuthorityAccomplishmentIpse dixitPoverty/WealthEtymologyNatureTradition/NoveltyChronological snobbery | Ad hominem | Appeal to motiveAssociationReductio ad HitlerumGodwin's lawReductio ad StalinumBulverismPoisoning the wellToneTu quoqueWhataboutism | AuthorityAccomplishmentIpse dixitPoverty/WealthEtymologyNatureTradition/NoveltyChronological snobbery
Ad hominem | Appeal to motiveAssociationReductio ad HitlerumGodwin's lawReductio ad StalinumBulverismPoisoning the wellToneTu quoqueWhataboutism
AuthorityAccomplishmentIpse dixitPoverty/WealthEtymologyNatureTradition/NoveltyChronological snobbery
Otherfallaciesof relevance | ArgumentsAd nauseamSealioningArgument from anecdoteArgument from silenceArgument to moderationArgumentum ad populumClichéThe Four Great ErrorsI'm entitled to my opinionIgnoratio elenchiInvincible ignoranceMoralistic/NaturalisticMotte-and-bailey fallacyPsychologist's fallacyRationalizationRed herringTwo wrongs make a rightSpecial pleadingStraw man | Arguments | Ad nauseamSealioningArgument from anecdoteArgument from silenceArgument to moderationArgumentum ad populum | ClichéThe Four Great ErrorsI'm entitled to my opinionIgnoratio elenchiInvincible ignoranceMoralistic/NaturalisticMotte-and-bailey fallacyPsychologist's fallacyRationalizationRed herringTwo wrongs make a rightSpecial pleadingStraw man
Arguments | Ad nauseamSealioningArgument from anecdoteArgument from silenceArgument to moderationArgumentum ad populum
ClichéThe Four Great ErrorsI'm entitled to my opinionIgnoratio elenchiInvincible ignoranceMoralistic/NaturalisticMotte-and-bailey fallacyPsychologist's fallacyRationalizationRed herringTwo wrongs make a rightSpecial pleadingStraw man
Consequences | Argumentum ad baculumWishful thinking
Emotion | ChildrenFearFlatteryNoveltyPityRidiculeIn-group favoritismInvented here/Not invented hereIsland mentalityLoyaltyParade of horriblesSpiteStirring symbolsWisdom of repugnance
Ad hominem | Appeal to motiveAssociationReductio ad HitlerumGodwin's lawReductio ad StalinumBulverismPoisoning the wellToneTu quoqueWhataboutism
AuthorityAccomplishmentIpse dixitPoverty/WealthEtymologyNatureTradition/NoveltyChronological snobbery
Arguments | Ad nauseamSealioningArgument from anecdoteArgument from silenceArgument to moderationArgumentum ad populum
ClichéThe Four Great ErrorsI'm entitled to my opinionIgnoratio elenchiInvincible ignoranceMoralistic/NaturalisticMotte-and-bailey fallacyPsychologist's fallacyRationalizationRed herringTwo wrongs make a rightSpecial pleadingStraw man |
8604 | https://coconote.app/notes/c0e31c53-501c-4a2a-b278-7a93cf393fd5 | Understanding the Malate-Aspartate Shuttle | Coconote
Coconote AI notes AI voice & video notes
Try for free
⚡
Understanding the Malate-Aspartate Shuttle
Nov 10, 2024
📄 View transcript
🤓 Take quiz
🃏 Review flashcards
Lecture Notes: Malate-Aspartate Shuttle
Background: Cellular Respiration
Stages of Cellular Respiration:
Glycolysis:
Occurs in the cytosol.
1 molecule of glucose → 2 molecules of pyruvate.
Generates 2 NADH.
Oxidation of Pyruvate:
Oxidative decarboxylation of pyruvate → Acetyl Coenzyme A.
Occurs in the mitochondrial matrix.
Generates 2 NADH (from 2 pyruvate molecules).
TCA Cycle (Krebs/Citric Acid Cycle):
Oxidation of Acetyl Coenzyme A to CO2.
Electrons passed to coenzymes NAD+ and FAD → NADH and FADH2.
Occurs in the mitochondrial matrix.
Electron Transport Chain & Oxidative Phosphorylation:
NADH and FADH2 give up electrons to synthesize ATP.
NADH from mitochondrial matrix is used, but cytosolic NADH cannot cross into the matrix.
NADH Oxidation and ATP Synthesis Problem
Cytosolic NADH:
Generated during glycolysis.
Cannot cross mitochondrial membrane, needs oxidation to regenerate NAD+.
Two pathways for NADH oxidation:
Anaerobic Pathway: Pyruvate → Lactate (via lactate dehydrogenase), NADH is oxidized to NAD+.
Aerobic Pathway: NADH should be oxidized via electron transport chain.
Malate-Aspartate Shuttle
Function: Transfers electrons from cytosolic NADH to mitochondrial NADH without direct transport of NADH.
Process Overview:
Oxaloacetate in the cytosol acquires electrons from NADH → Reduced to Malate (via malate dehydrogenase enzyme).
Malate transported into the mitochondrial matrix.
Malate converted back to Oxaloacetate in the matrix, electrons transferred to mitochondrial NAD+ → NADH.
NADH then enters electron transport chain, yielding 2.5 ATP per NADH.
Oxaloacetate Recycling:
Cannot cross the mitochondrial membrane.
Transaminated to Aspartate (via aspartate aminotransferase enzyme) with amino group donated by glutamate.
Aspartate can cross back to the cytosol.
Glutamate and Alpha-Ketoglutarate Cycle:
Glutamate donates an amino group to form aspartate → Converted to alpha-ketoglutarate.
Alpha-ketoglutarate and malate are exchanged between cytosol and mitochondria via an antiporter.
Alpha-ketoglutarate is transaminated back to glutamate in the cytosol.
Electron Transport and ATP Synthesis
NADH Regeneration in Mitochondria:
Electrons passed through complexes I, III, IV, and cytochrome c.
Final electron acceptor is oxygen.
Protons pumped into intermembrane space drive ATP synthesis via ATP synthase (complex V).
1 cytosolic NADH yields 2.5 ATP via the shuttle.
Conclusion
The Malate-Aspartate Shuttle efficiently transfers reducing equivalents from cytosolic NADH to mitochondria, enabling continued ATP production and glycolysis function.
Operates prominently in the heart, liver, and kidneys.
Important for maintaining cellular energy balance.
📄 Full transcript
Lecture Notes: Malate-Aspartate Shuttle
Background: Cellular Respiration
Stages of Cellular Respiration:
Glycolysis:
Occurs in the cytosol.
1 molecule of glucose → 2 molecules of pyruvate.
Generates 2 NADH.
Oxidation of Pyruvate:
Oxidative decarboxylation of pyruvate → Acetyl Coenzyme A.
Occurs in the mitochondrial matrix.
Generates 2 NADH (from 2 pyruvate molecules).
TCA Cycle (Krebs/Citric Acid Cycle):
Oxidation of Acetyl Coenzyme A to CO2.
Electrons passed to coenzymes NAD+ and FAD → NADH and FADH2.
Occurs in the mitochondrial matrix.
Electron Transport Chain & Oxidative Phosphorylation:
NADH and FADH2 give up electrons to synthesize ATP.
NADH from mitochondrial matrix is used, but cytosolic NADH cannot cross into the matrix.
NADH Oxidation and ATP Synthesis Problem
Cytosolic NADH:
Generated during glycolysis.
Cannot cross mitochondrial membrane, needs oxidation to regenerate NAD+.
Two pathways for NADH oxidation:
Anaerobic Pathway: Pyruvate → Lactate (via lactate dehydrogenase), NADH is oxidized to NAD+.
Aerobic Pathway: NADH should be oxidized via electron transport chain.
Malate-Aspartate Shuttle
Function: Transfers electrons from cytosolic NADH to mitochondrial NADH without direct transport of NADH.
Process Overview:
Oxaloacetate in the cytosol acquires electrons from NADH → Reduced to Malate (via malate dehydrogenase enzyme).
Malate transported into the mitochondrial matrix.
Malate converted back to Oxaloacetate in the matrix, electrons transferred to mitochondrial NAD+ → NADH.
NADH then enters electron transport chain, yielding 2.5 ATP per NADH.
Oxaloacetate Recycling:
Cannot cross the mitochondrial membrane.
Transaminated to Aspartate (via aspartate aminotransferase enzyme) with amino group donated by glutamate.
Aspartate can cross back to the cytosol.
Glutamate and Alpha-Ketoglutarate Cycle:
Glutamate donates an amino group to form aspartate → Converted to alpha-ketoglutarate.
Alpha-ketoglutarate and malate are exchanged between cytosol and mitochondria via an antiporter.
Alpha-ketoglutarate is transaminated back to glutamate in the cytosol.
Electron Transport and ATP Synthesis
NADH Regeneration in Mitochondria:
Electrons passed through complexes I, III, IV, and cytochrome c.
Final electron acceptor is oxygen.
Protons pumped into intermembrane space drive ATP synthesis via ATP synthase (complex V).
1 cytosolic NADH yields 2.5 ATP via the shuttle.
Conclusion
The Malate-Aspartate Shuttle efficiently transfers reducing equivalents from cytosolic NADH to mitochondria, enabling continued ATP production and glycolysis function.
Operates prominently in the heart, liver, and kidneys.
Important for maintaining cellular energy balance. |
8605 | https://math.libretexts.org/Bookshelves/Algebra/Intermediate_Algebra_1e_(OpenStax)/07%3A_Rational_Expressions_and_Functions/7.07%3A_Solve_Rational_Inequalities | 7.7: Solve Rational Inequalities - Mathematics LibreTexts
Skip to main content
Table of Contents menu
search Search build_circle Toolbar fact_check Homework cancel Exit Reader Mode
school Campus Bookshelves
menu_book Bookshelves
perm_media Learning Objects
login Login
how_to_reg Request Instructor Account
hub Instructor Commons
Search
Search this book
Submit Search
x
Text Color
Reset
Bright Blues Gray Inverted
Text Size
Reset
+-
Margin Size
Reset
+-
Font Type
Enable Dyslexic Font - [x]
Downloads expand_more
Download Page (PDF)
Download Full Book (PDF)
Resources expand_more
Periodic Table
Physics Constants
Scientific Calculator
Reference expand_more
Reference & Cite
Tools expand_more
Help expand_more
Get Help
Feedback
Readability
x
selected template will load here
Error
This action is not available.
chrome_reader_mode Enter Reader Mode
7: Rational Expressions and Functions
Intermediate Algebra 1e (OpenStax)
{ "7.7E:_Exercises" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1" }
{ "7.01:_Rational_Expressions_and_Functions" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "7.02:_Multiply_and_Divide_Rational_Expressions" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "7.03:_Add_and_Subtract_Rational_Expressions" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "7.04:_Simplify_Complex_Rational_Expressions" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "7.05:_Solve_Rational_Equations" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "7.06:Solve_Applications_with_Rational_Equations" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "7.07:_Solve_Rational_Inequalities" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "7.0E:_7.E:_Chapter_Exercise" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1" }
{ "00:_Front_Matter" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "01:_Foundations" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "02:_Solving_Linear_Equations" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "03:_Graphs_and_Functions" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "04:_Systems_of_Linear_Equations" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "05:_Polynomial_and_Polynomial_Functions" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "06:_Factoring" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "07:_Rational_Expressions_and_Functions" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "08:_Roots_and_Radicals" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "09:_Quadratic_Equations_and_Functions" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "10:_Exponential_and_Logarithmic_Functions" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "11:_Conics" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "12:_Sequences_Series_and_Binomial_Theorem" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "zz:_Back_Matter" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1" }
Mon, 14 Feb 2022 01:05:34 GMT
7.7: Solve Rational Inequalities
5164
5164
admin
{ }
Anonymous
Anonymous
2
false
false
[ "article:topic", "authorname:openstax", "license:ccby", "showtoc:no", "program:openstax", "licenseversion:40", "source@ ]
[ "article:topic", "authorname:openstax", "license:ccby", "showtoc:no", "program:openstax", "licenseversion:40", "source@ ]
Search site Search Search Go back to previous article
Sign in
Username Password Sign in
Sign in
Sign in
Forgot password
Expand/collapse global hierarchy
1. Home
2. Bookshelves
3. Algebra
4. Intermediate Algebra 1e (OpenStax)
5. 7: Rational Expressions and Functions
6. 7.7: Solve Rational Inequalities
Expand/collapse global location
7.7: Solve Rational Inequalities
Last updated Feb 14, 2022
Save as PDF
7.6E: Exercises
7.7E: Exercises
Page ID 5164
OpenStax
OpenStax
( \newcommand{\kernel}{\mathrm{null}\,})
Table of contents
1. Learning Objectives
2. Be Prepared
3. Solve Rational Inequalities
1. Rational Inequality
2. Example 7.7.1
3. Exercise 7.7.1
4. Exercise 7.7.2
5. How to Solve a rational inequality
6. Example 7.7.2
7. Exercise 7.7.3
8. Exercise 7.7.4
9. Example 7.7.3
10. Exercise 7.7.5
11. Exercise 7.7.6
12. Example 7.7.4
13. Exercise 7.7.7
14. Exercise 7.7.8
Solve an Inequality with Rational Functions
Example 7.7.5
Exercise 7.7.9
Exercise 7.7.10
Example 7.7.6
Exercise 7.7.11
Exercise 7.7.12
Learning Objectives
Solve rational inequalities
Solve an inequality with rational functions
Be Prepared
Before you get started, take this readiness quiz.
Find the value of x−5 when ⓐ x=6 ⓑ x=−3 ⓒ x=5
If you missed this problem, review Example 1.2.16.
2. Solve: 8−2x<12
If you missed this problem, review Example 2.6.13.
3. Write in interval notation: −3≤x<5
If you missed this problem, review Example 2.6.4.
Solve Rational Inequalities
We learned to solve linear inequalities after learning to solve linear equations. The techniques were very much the same with one major exception. When we multiplied or divided by a negative number, the inequality sign reversed.
Having just learned to solve rational equations we are now ready to solve rational inequalities. A rational inequality is an inequality that contains a rational expression.
Rational Inequality
A rational inequality is an inequality that contains a rational expression.
Inequalities such as 3 2x>1,2x x−3<4,2x−3 x−6≥x, and 1 4−2 x 2≤3 x are rational inequalities as they each contain a rational expression.
When we solve a rational inequality, we will use many of the techniques we used solving linear inequalities. We especially must remember that when we multiply or divide by a negative number, the inequality sign must reverse.
Another difference is that we must carefully consider what value might make the rational expression undefined and so must be excluded.
When we solve an equation and the result is x=3, we know there is one solution, which is 3.
When we solve an inequality and the result is x>3, we know there are many solutions. We graph the result to better help show all the solutions, and we start with 3. Three becomes a critical point and then we decide whether to shade to the left or right of it. The numbers to the right of 3 are larger than 3, so we shade to the right.
To solve a rational inequality, we first must write the inequality with only one quotient on the left and 0 on the right.
Next we determine the critical points to use to divide the number line into intervals. A critical point is a number which make the rational expression zero or undefined.
We then will evaluate the factors of the numerator and denominator, and find the quotient in each interval. This will identify the interval, or intervals, that contains all the solutions of the rational inequality.
We write the solution in interval notation being careful to determine whether the endpoints are included.
Example 7.7.1
Solve and write the solution in interval notation: x−1 x+3≥0
Solution
Step 1. Write the inequality as one quotient on the left and zero on the right.
Our inequality is in this form.x−1 x+3≥0
Step 2. Determine the critical points—the points where the rational expression will be zero or undefined.
The rational expression will be zero when the numerator is zero. Since x−1=0 when x=1, then 1 is a critical point.
The rational expression will be undefined when the denominator is zero. Since x+3=0 when x=−3, then -3 is a critical point.
The critical points are 1 and -3.
Step 3. Use the critical points to divide the number line into intervals.
The number line is divided into three intervals:
(−∞,−3)(−3,1)(1,∞)
Step 4. Test a value in each interval. Above the number line show the sign of each factor of the rational expression in each interval. Below the number line show the sign of the quotient.
To find the sign of each factor in an interval, we choose any point in that interval and use it as a test point. Any point in the interval will give the expression the same sign, so we can choose any point in the interval.
Interval(−∞,−3)
The number -4 is in the interval (−∞,−3). Test x=−4 in the expression in the numerator and the denominator.
The numerator:
x−1−4−1−5 Negative
The denominator:
x+3−4+3−1 Negative
Above the number line, mark the factor x−1 negative and mark the factor x+3 negative.
Since a negative divided by a negative is positive, mark the quotient positive in the interval (−∞,−3).
Interval(−3,1)
The number 0 is in the interval (−3,1). Test x=0.
The numerator:
x−1 0−1−1 Negative
The denominator:
x+3 0+3 3 Positive
Above the number line, mark the factor x−1 negative and mark x+3 positive.
Since a negative divided by a positive is negative, the quotient is marked negative in the interval (−3,1).
Interval(1,∞)
The number 2 is in the interval (1,∞). Test x=2.
The numerator:
x−1 2−1 1 Positive
The denominator:
x+3 2+3 5 Positive
Above the number line, mark the factor x−1 positive and mark x+3 positive.
Since a positive divided by a positive is positive, mark the quotient positive in the interval (1,∞).
Step 5. Determine the intervals where the inequality is correct. Write the solution in interval notation.
We want the quotient to be greater than or equal to zero, so the numbers in the intervals (−∞,−3) and (1,∞)are solutions.
But what about the critical points?
The critical point x=−3 makes the denominator 0, so it must be excluded from the solution and we mark it with a parenthesis.
The critical point x=1 makes the whole rational expression 0. The inequality requires that the rational expression be greater than or equal to. So, 1 is part of the solution and we will mark it with a bracket.
Recall that when we have a solution made up of more than one interval we use the union symbol, ∪, to connect the two intervals. The solution in interval notation is (−∞,−3)∪[1,∞).
Exercise 7.7.1
Solve and write the solution in interval notation: x−2 x+4≥0
Answer
(−∞,−4)∪[2,∞)
Exercise 7.7.2
Solve and write the solution in interval notation: x+2 x−4≥0
Answer
(−∞,−2]∪(4,∞)
We summarize the steps for easy reference.
How to Solve a rational inequality
Step 1. Write the inequality as one quotient on the left and zero on the right.
Step 2. Determine the critical points–the points where the rational expression will be zero or undefined.
Step 3. Use the critical points to divide the number line into intervals.
Step 4. Test a value in each interval. Above the number line show the sign of each factor of the numerator and denominator in each interval. Below the number line show the sign of the quotient.
Step 5. Determine the intervals where the inequality is correct. Write the solution in interval notation.
The next example requires that we first get the rational inequality into the correct form.
Example 7.7.2
Solve and write the solution in interval notation: 4x x−6<1
Solution
4x x−6<1
Subtract 1 to get zero on the right.
4x x−6−1<0
Rewrite 1 as a fraction using the LCD.
4x x−6−x−6 x−6<0
Subtract the numerators and place the difference over the common denominator.
4x−(x−6)x−6<0
Simplify.
3x+6 x−6<0
Factor the numerator to show all factors.
3(x+2)x−6<0
Find the critical points.
The quotient will be zero when the numerator is zero. The quotient is undefined when the denominator is zero.
x+2=0 x−6=0 x=−2 x=6
Use the critical points to divide the number line into intervals.
Test a value in each interval.
| | (−∞,−2) | (−2,6) | (6,∞) |
--- --- |
| x+2) | x+2 -3+2 -1 - | x+2 0+2 2 + | x+2 7+2 9 + |
| x−6 | x-6 -3-6 -9 - | x-6 0-6 -6 - | x-6 7-6 1 + |
Above the number line show the sign of each factor of the rational expression in each interval. Below the number line show the sign of the quotient.
Determine the intervals where the inequality is correct. We want the quotient to be negative, so the solution includes the points between −2 and 6. Since the inequality is strictly less than, the endpoints are not included.
We write the solution in interval notation as (−2, 6).
Exercise 7.7.3
Solve and write the solution in interval notation: 3x x−3<1.
Answer
(−3 2,3)
Exercise 7.7.4
Solve and write the solution in interval notation: 3x x−4<2.
Answer
(−8,4)
In the next example, the numerator is always positive, so the sign of the rational expression depends on the sign of the denominator.
Example 7.7.3
Solve and write the solution in interval notation: 5 x 2−2x−15>0.
Solution
The inequality is in the correct form.
5 x 2−2x−15>0
Factor the denominator.
5(x+3)(x−5)>0
Find the critical points. The quotient is 0 when the numerator is 0. Since the numerator is always 5, the quotient cannot be 0.
The quotient will be undefined when the denominator is zero.
(x+3)(x−5)=0 x=−3,x=5
Use the critical points to divide the number line into intervals.
Test values in each interval. Above the number line show the sign of each factor of the denominator in each interval. Below the number line, show the sign of the quotient.
Write the solution in interval notation.
(−∞,−3)∪(5,∞)
Exercise 7.7.5
Solve and write the solution in interval notation: 1 x 2+2x−8>0.
Answer
(−∞,−4)∪(2,∞)
Exercise 7.7.6
Solve and write the solution in interval notation: 3 x 2+x−12>0.
Answer
(−∞,−4)∪(3,∞)
The next example requires some work to get it into the needed form.
Example 7.7.4
Solve and write the solution in interval notation: 1 3−2 x 2<5 3x.
Solution
1 3−2 x 2<5 3x
Subtract 5 3x to get zero on the right.
1 3−2 x 2−5 3x<0
Rewrite to get each fraction with the LCD 3x 2.
1⋅x 2 3⋅x 2−2⋅3 x 2⋅3−5⋅x 3x−x<0
Simplify.
x 2 3x 2−6 3x 2−5x 3x 2<0
Subtract the numerators and place the difference over the common denominator.
x 2−5x−6 3x 2<0
Factor the numerator.
(x−6)(x+1)3x 2<0
Find the critical points.
3x 2=0 x−6=0 x+1=0 x=0 x=6 x=−1
Use the critical points to divide the number line into intervals.
Above the number line show the sign of each factor in each interval. Below the number line, show the sign of the quotient.
Since, 0 is excluded, the solution is the two (−1,0)∪(0,6) intervals, (−1,0) and (0,6).
Exercise 7.7.7
Solve and write the solution in interval notation: 1 2+4 x 2<3 x.
Answer
(2,4)
Exercise 7.7.8
Solve and write the solution in interval notation: 1 3+6 x 2<3 x.
Answer
(3,6)
Solve an Inequality with Rational Functions
When working with rational functions, it is sometimes useful to know when the function is greater than or less than a particular value. This leads to a rational inequality.
Example 7.7.5
Given the function R(x)=x+3 x−5, find the values of x that make the function less than or equal to 0.
Solution
We want the function to be less than or equal to 0.
R(x)≤0
Substitute the rational expression for R(x).
x+3 x−5≤0 x≠5
Find the critical points.
x+3=0 x−5=0 x=−3 x=5
Use the critical points to divide the number line into intervals.
Test values in each interval. Above the number line, show the sign of each factor in each interval. Below the number line, show the sign of the quotient. Write the solution in interval notation. Since 5 is excluded we, do not include it in the interval.
[−3,5)
Exercise 7.7.9
Given the function R(x)=x−2 x+4, find the values of x that make the function less than or equal to 0.
Answer
(−4,2]
Exercise 7.7.10
Given the function R(x)=x+1 x−4, find the values of x that make the function less than or equal to 0.
Answer
[−1,4)
In economics, the function C(x) is used to represent the cost of producing x units of a commodity. The average cost per unit can be found by dividing C(x) by the number of items x. Then, the average cost per unit is (c(x)=\dfrac{C(x)}{x}).
Example 7.7.6
The function C(x)=10x+3000 represents the cost to produce x, number of items. Find:
The average cost function, c(x)
How many items should be produced so that the average cost is less than $40.
Solution
C(x)=10x+3000
The average cost function is c(x)=C(x)x). To find the average cost function, divide the cost function by x.
c(x)=C(x)x c(x)=10x+3000 x
The average cost function is c(x)=10x+3000 x
We want the function c(x) to be less than 40.
c(x)<40
Substitute the rational expression forc(x).
10x+3000 x<40,x≠0
Subtract 40 to get 0 on the right.
10x+3000 x−40<0
Rewrite the left side as one quotient by finding the LCD and performing the subtraction.
10x+3000 x−40(x x)<0 10x+3000 x−40x x<0 10x+3000−40x x<0−30x+3000 x<0
Factor the numerator to show all factors.
−30(x−100)x<0−30(x−100)=0 x=0
Find the critical points.
−30≠0 x−100=0 x=100
More than 100 items must be produced to keep the average cost below $40 per item.
Exercise 7.7.11
The function C(x)=20x+6000 represents the cost to produce x, number of items. Find:
How many items should be produced so that the average cost is less than $60.
Answer
1. c(x)=20x+6000 x
2. More than 150 items must be produced to keep the average cost below $60 per item.
Exercise 7.7.12
The function C(x)=5x+900 represents the cost to produce x, number of items. Find:
How many items should be produced so that the average cost is less than $20.
Answer
1. c(x)=5x+900 x
2. More than 60 items must be produced to keep the average cost below $20 per item.
This page titled 7.7: Solve Rational Inequalities is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform.
Back to top
7.6E: Exercises
7.7E: Exercises
Was this article helpful?
Yes
No
Recommended articles
7.7E: Exercises
7.6: Solve Applications with Rational Equations
7.E: Chapter 7 Review Exercises
7.1: Rational Expressions and Functions
7.2: Multiply and Divide Rational Expressions
Article typeSection or PageAuthorOpenStaxLicenseCC BYLicense Version4.0OER program or PublisherOpenStaxShow Page TOCno
Tags
source@
© Copyright 2025 Mathematics LibreTexts
Powered by CXone Expert ®
?
The LibreTexts libraries arePowered by NICE CXone Expertand are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Privacy Policy. Terms & Conditions. Accessibility Statement.For more information contact us atinfo@libretexts.org.
Support Center
How can we help?
Contact Support Search the Insight Knowledge Base Check System Status×
contents readability resources tools
☰ |
8606 | https://www.saskoer.ca/app/uploads/sites/313/2020/05/F-Distribution-Table.pdf | F Distribution Table 1 d.f.N. = degrees of freedom in numerator d.f.D. = degrees of freedom in denominator α = 0.005 d.f.N. d.f.D. 1 2 3 4 5 6 7 8 9 10 12 15 20 24 30 40 60 120 ∞ 1 16211 20000 21615 22500 23056 23437 23715 23925 24091 24224 24426 24630 24836 24940 25044 25148 25253 25359 25465 2 198.5 199.0 199.2 199.2 199.3 199.3 199.4 199.4 199.4 199.4 199.4 199.4 199.4 199.5 199.5 199.5 199.5 199.5 199.5 3 55.55 49.80 47.47 46.19 45.39 44.84 44.43 44.13 43.88 43.69 43.39 43.08 42.78 42.62 42.47 42.31 42.15 41.99 41.83 4 31.33 26.28 24.26 23.15 22.46 21.97 21.62 21.35 21.14 20.97 20.70 20.44 20.17 20.03 19.89 19.75 19.61 19.47 19.32 5 22.78 18.31 16.53 15.56 14.94 14.51 14.20 13.96 13.77 13.62 13.38 13.15 12.90 12.78 12.66 12.53 12.40 12.27 12.14 6 18.63 14.54 12.92 12.03 11.46 11.07 10.79 10.57 10.39 10.25 10.03 9.81 9.59 9.47 9.36 9.24 9.12 9.00 8.88 7 16.24 12.40 10.88 10.05 9.52 9.16 8.89 8.68 8.51 8.38 8.18 7.97 7.75 7.65 7.53 7.42 7.31 7.19 7.08 8 14.69 11.04 9.60 8.81 8.30 7.95 7.69 7.50 7.34 7.21 7.01 6.81 6.61 6.50 6.40 6.29 6.18 6.06 5.95 9 13.61 10.11 8.72 7.96 7.47 7.13 6.88 6.69 6.54 6.42 6.23 6.03 5.83 5.73 5.62 5.52 5.41 5.30 5.19 10 12.83 9.43 8.08 7.34 6.87 6.54 6.30 6.12 5.97 5.85 5.66 5.47 5.27 5.17 5.07 4.97 4.86 4.75 4.64 11 12.23 8.91 7.60 6.88 6.42 6.10 5.86 5.68 5.54 5.42 5.24 5.05 4.86 4.76 4.65 4.55 4.44 4.34 4.23 12 11.75 8.51 7.23 6.52 6.07 5.76 5.52 5.35 5.20 5.09 4.91 4.72 4.53 4.43 4.33 4.23 4.12 4.01 3.90 13 11.37 8.19 6.93 6.23 5.79 5.48 5.25 5.08 4.94 4.82 4.64 4.46 4.27 4.17 4.07 3.97 3.87 3.76 3.65 14 11.06 7.92 6.68 6.00 5.56 5.26 5.03 4.86 4.72 4.60 4.43 4.25 4.06 3.96 3.86 3.76 3.66 3.55 3.44 15 10.80 7.70 6.48 5.80 5.37 5.07 4.85 4.67 4.54 4.42 4.25 4.07 3.88 3.79 3.69 3.58 3.48 3.37 3.26 16 10.58 7.51 6.30 5.64 5.21 4.91 4.69 4.52 4.38 4.27 4.10 3.92 3.73 3.64 3.54 3.44 3.33 3.22 3.11 17 10.38 7.35 6.16 5.50 5.07 4.78 4.56 4.39 4.25 4.14 3.97 3.79 3.61 3.51 3.41 3.31 3.21 3.10 2.98 18 10.22 7.21 6.03 5.37 4.96 4.66 4.44 4.28 4.14 4.03 3.86 3.68 3.50 3.40 3.30 3.20 3.10 2.99 2.87 19 10.07 7.09 5.92 5.27 4.85 4.56 4.34 4.18 4.04 3.93 3.76 3.59 3.40 3.31 3.21 3.11 3.00 2.89 2.78 20 9.94 6.99 5.82 5.17 4.76 4.47 4.26 4.09 3.96 3.85 3.68 3.50 3.32 3.22 3.12 3.02 2.92 2.81 2.69 21 9.83 6.89 5.73 5.09 4.68 4.39 4.18 4.01 3.88 3.77 3.60 3.43 3.24 3.15 3.05 2.95 2.84 2.73 2.61 22 9.73 6.81 5.65 5.02 4.61 4.32 4.11 3.94 3.81 3.70 3.54 3.36 3.18 3.08 2.98 2.88 2.77 2.66 2.55 23 9.63 6.73 5.58 4.95 4.54 4.26 4.05 3.88 3.75 3.64 3.47 3.30 3.12 3.02 2.92 2.82 2.71 2.60 2.48 24 9.55 6.66 5.52 4.89 4.49 4.20 3.99 3.83 3.69 3.59 3.42 3.25 3.06 2.97 2.87 2.77 2.66 2.55 2.43 25 9.48 6.60 5.46 4.84 4.43 4.15 3.94 3.78 3.63 3.54 3.37 3.20 3.01 2.92 2.82 2.72 2.61 2.50 2.38 26 9.41 6.54 5.41 4.79 4.38 4.10 3.89 3.73 3.60 3.49 3.33 3.15 2.97 2.87 2.77 2.67 2.56 2.45 2.33 27 9.34 6.49 5.36 4.74 4.34 4.06 3.85 3.69 3.56 3.45 3.28 3.11 2.93 2.83 2.73 2.63 2.52 2.41 2.25 28 9.28 6.44 5.32 4.70 4.30 4.02 3.81 3.65 3.52 3.41 3.25 3.07 2.89 2.79 2.69 2.59 2.48 2.37 2.29 29 9.23 6.40 5.28 4.66 4.26 3.98 3.77 3.61 3.48 3.38 3.21 3.04 2.86 2.76 2.66 2.56 2.45 2.33 2.24 30 9.18 6.35 5.24 4.62 4.23 3.95 3.74 3.58 3.45 3.34 3.18 3.01 2.82 2.73 2.63 2.52 2.42 2.30 2.18 40 8.83 6.07 4.98 4.37 3.99 3.71 3.51 3.35 3.22 3.12 2.95 2.78 2.60 2.50 2.40 2.30 2.18 2.06 1.93 60 8.49 5.79 4.73 4.14 3.76 3.49 3.29 3.13 3.01 2.90 2.74 2.57 2.39 2.29 2.19 2.08 1.96 1.83 1.69 120 8.18 5.54 4.50 3.92 3.55 3.28 3.09 2.93 2.81 2.71 2.54 2.37 2.19 2.09 1.98 1.87 1.75 1.61 1.43 ∞ 7.88 5.30 4.28 3.72 3.35 3.09 2.90 2.74 2.62 2.52 2.36 2.19 2.00 1.90 1.79 1.67 1.53 1.36 1.00 F Distribution Table 2 α = 0.01 d.f.N. d.f.D. 1 2 3 4 5 6 7 8 9 10 12 15 20 24 30 40 60 120 ∞ 1 4052 4999.5 5403 5625 5764 5859 5928 5982 6022 6056 6106 6157 6209 6235 6261 6287 6313 6339 6366 2 98.50 99.00 99.17 99.25 99.30 99.33 99.36 99.37 99.39 99.40 99.42 99.43 99.45 99.46 99.47 99.47 99,48 99.49 99.50 3 34.12 30.82 29.46 28.71 28.24 27.91 27.67 27.49 27.35 27.23 27.05 26.87 26.69 26.60 26.50 26.41 26.32 26.22 26.13 4 21.20 18.00 16.69 15.98 15.52 15.21 14.98 14.80 14.66 14.55 14.37 14.20 14.02 13.93 13.84 13.75 13.65 13.56 13.46 5 16.26 13.27 12.06 11.39 10.97 10.67 10.46 10.29 10.16 10.05 9.89 9.72 9.55 9.47 9.38 9.29 9.20 9.11 9.02 6 13.75 10.92 9.78 9.15 8.75 8.47 8.26 8.10 7.98 7.87 7.72 7.56 7.40 7.31 7.23 7.14 7.06 6.97 6.88 7 12.25 9.55 8.45 7.85 7.46 7.19 6.99 6.84 6.72 6.62 6.47 6.31 6.16 6.07 5.99 5.91 5.82 5.74 5.65 8 11.26 8.65 7.59 7.01 6.63 6.37 6.18 6.03 5.91 5.81 5.67 5.52 5.36 5.28 5.20 5.12 5.03 4.95 4.86 9 10.56 8.02 6.99 6.42 6.06 5.80 5.61 5.47 5.35 5.26 5.11 4.96 4.81 4.73 4.65 4.57 4.48 4.40 4.31 10 10.04 7.56 6.55 5.99 5.64 5.39 5.20 5.06 4.94 4.85 4.71 4.56 4.41 4.33 4.25 4.17 4.08 4.00 3.91 11 9.65 7.21 6.22 5.67 5.32 5.07 4.89 4.74 4.63 4.54 4.40 4.25 4.10 4.02 3.94 3.86 3.78 3.69 3.60 12 9.33 6.93 5.95 5.41 5.06 4.82 4.64 4.50 4.39 4.30 4.16 4.01 3.86 3.78 3.70 3.62 3.54 3.45 3.36 13 9.07 6.70 5.74 5.21 4.86 4.62 4.44 4.30 4.19 4.10 3.96 3.82 3.66 3.59 3.51 3.43 3.34 3.25 3.17 14 8.86 6.51 5.56 5.04 4.69 4.46 4.28 4.14 4.03 3.94 3.80 3.66 3.51 3.43 3.35 3.27 3.18 3.09 3.00 15 8.68 6.36 5.42 4.89 4.56 4.32 4.14 4.00 3.89 3.80 3.67 3.52 3.37 3.29 3.21 3.13 3.05 2.96 2.87 16 8.53 6.23 5.29 4.77 4.44 4.20 4.03 3.89 3.78 3.69 3.55 3.41 3.26 3.18 3.10 3.02 2.93 2.84 2.75 17 8.40 6.11 5.18 4.67 4.34 4.10 3.93 3.79 3.68 3.59 3.46 3.31 3.16 3.08 3.00 2.92 2.83 2.75 2.65 18 8.29 6.01 5.09 4.58 4.25 4.01 3.84 3.71 3.60 3.51 3.37 3.23 3.08 3.00 2.92 2.84 2.75 2.66 2.57 19 8.18 5.93 5.01 4.50 4.17 3.94 3.77 3.63 3.52 3.43 3.30 3.15 3.00 2.92 2.84 2.76 2.67 2.58 2.49 20 8.10 5.85 4.94 4.43 4.10 3.87 3.70 3.56 3.46 3.37 3.23 3.09 2.94 2.86 2.78 2.69 2.61 2.52 2.42 21 8.02 5.78 4.87 4.37 4.04 3.81 3.64 3.51 3.40 3.31 3.17 3.03 2.88 2.80 2.72 2.64 2.55 2.46 2.36 22 7.95 5.72 4.82 4.31 3.99 3.76 3.59 3.45 3.35 3.26 3.12 2.98 2.83 2.75 2.67 2.58 2.50 2.40 2.31 23 7.88 5.66 4.76 4.26 3.94 3.71 3.54 3.41 3.30 3.21 3.07 2.93 2.78 2.70 2.62 2.54 2.45 2.35 2.26 24 7.82 5.61 4.72 4.22 3.90 3.67 3.50 3.36 3.26 3.17 3.03 2.89 2.74 2.66 2.58 2.49 2.40 2.31 2.21 25 7.77 5.57 4.68 4.18 3.85 3.63 3.46 3.32 3.22 3.13 2.99 2.85 2.70 2.62 2.54 2.45 2.36 2.27 2.17 26 7.72 5.53 4.64 4.14 3.82 3.59 3.42 3.29 3.18 3.09 2.96 2.81 2.66 2.58 2.50 2.42 2.33 2.23 2.13 27 7.68 5.49 4.60 4.11 3.78 3.56 3.39 3.26 3.15 3.06 2.93 2.78 2.63 2.55 2.47 2.38 2.29 2.20 2.10 28 7.64 5.45 4.57 4.07 3.75 3.53 3.36 3.23 3.12 3.03 2.90 2.75 2.60 2.52 2.44 2.35 2.26 2.17 2.06 29 7.60 5.42 4.54 4.04 3.73 3.50 3.33 3.20 3.09 3.00 2.87 2.73 2.57 2.49 2.41 2.33 2.23 2.14 2.03 30 7.56 5.39 4.51 4.02 3.70 3.47 3.30 3.17 3.07 2.98 2.84 2.70 2.55 2.47 2.39 2.30 2.21 2.11 2.01 40 7.31 5.18 4.31 3.83 3.51 3.29 3.12 2.99 2.89 2.80 2.66 2.52 2.37 2.29 2.20 2.11 2.02 1.92 1.80 60 7.08 4.98 4.13 3.65 3.34 3.12 2.95 2.82 2.72 2.63 2.50 2.35 2.20 2.12 2.03 1.94 1.84 1.73 1.60 120 6.85 4.79 3.95 3.48 3.17 2.96 2.79 2.66 2.56 2.47 2.34 2.19 2.03 1.95 1.86 1.76 1.66 1.53 1.38 ∞ 6.63 4.61 3.78 3.32 3.02 2.80 2.64 2.51 2.41 2.32 2.18 2.04 1.88 1.79 1.70 1.59 1.47 1.32 1.00 F Distribution Table 3 α = 0.025 d.f.N. d.f.D. 1 2 3 4 5 6 7 8 9 10 12 15 20 24 30 40 60 120 ∞ 1 647.8 799.5 864.2 899.6 921.8 937.1 948.2 956.7 963.3 968.6 976.7 984.9 993.1 997.2 1001 1006 1010 1014 1018 2 38.51 39.00 39.17 39.25 39.30 39.33 39.36 39.37 39.39 39.40 39.41 39.43 39.45 39.46 39.46 39.47 39.48 39.49 39.50 3 17.44 16.04 15.44 15.10 14.88 14.73 14.62 14.54 14.47 14.42 14.34 14.25 14.17 14.12 14.08 14.04 13.99 13.95 13.90 4 12.22 10.65 9.98 9.60 9.36 9.20 9.07 8.98 8.90 8.84 8.75 8.66 8.56 8.51 8.46 8.41 8.36 8.31 8.26 5 10.01 8.43 7.76 7.39 7.15 6.98 6.85 6.76 6.68 6.62 6.52 6.43 6.33 6.28 6.23 6.18 6.12 6.07 6.02 6 8.81 7.26 6.60 6.23 5.99 5.82 5.70 5.60 5.52 5.46 5.37 5.27 5.17 5.12 5.07 5.01 4.96 4.90 4.85 7 8.07 6.54 5.89 5.52 5.29 5.12 4.99 4.90 4.82 4.76 4.67 4.57 4.47 4.42 4.36 4.31 4.25 4.20 4.14 8 7.57 6.06 5.42 5.05 4.82 4.65 4.53 4.43 4.36 4.30 4.20 4.10 4.00 3.95 3.89 3.84 3.78 3.73 3.67 9 7.21 5.71 5.08 4.72 4.48 4.32 4.20 4.10 4.03 3.96 3.87 3.77 3.67 3.61 3.56 3.51 3.45 3.39 3.33 10 6.94 5.46 4.83 4.47 4.24 4.07 3.95 3.85 3.78 3.72 3.62 3.52 3.42 3.37 3.31 3.26 3.20 3.14 3.08 11 6.72 5.26 4.63 4.28 4.04 3.88 3.76 3.66 3.59 3.53 3.43 3.33 3.23 3.17 3.12 3.06 3.00 2.94 2.88 12 6.55 5.10 4.47 4.12 3.89 3.73 3.61 3.51 3.44 3.37 3.28 3.18 3.07 3.02 2.96 2.91 2.85 2.79 2.72 13 6.41 4.97 4.35 4.00 3.77 3.60 3.48 3.39 3.31 3.25 3.15 3.05 2.95 2.89 2.84 2.78 2.72 2.66 2.60 14 6.30 4.86 4.24 3.89 3.66 3.50 3.38 3.29 3.21 3.15 3.05 2.95 2.84 2.79 2.73 2.67 2.61 2.55 2.49 15 6.20 4.77 4.15 3.80 3.58 3.41 3.29 3.20 3.12 3.06 2.96 2.86 2.76 2.70 2.64 2.59 2.52 2.46 2.40 16 6.12 4.69 4.08 3.73 3.50 3.34 3.22 3.12 3.05 2.99 2.89 2.79 2.68 2.63 2.57 2.51 2.45 2.38 2.32 17 6.04 4.62 4.01 3.66 3.44 3.28 3.16 3.06 2.98 2.92 2.82 2.72 2.62 2.56 2.50 2.44 2.38 2.32 2.25 18 5.98 4.56 3.95 3.61 3.38 3.22 3.10 3.01 2.93 2.87 2.77 2.67 2.56 2.50 2.44 2.38 2.32 2.26 2.19 19 5.92 4.51 3.90 3.56 3.33 3.17 3.05 2.96 2.88 2.82 2.72 2.62 2.51 2.45 2.39 2.33 2.27 2.20 2.13 20 5.87 4.46 3.86 3.51 3.29 3.13 3.01 2.91 2.84 2.77 2.68 2.57 2.46 2.41 2.35 2.29 2.22 2.16 2.09 21 5.83 4.42 3.82 3.48 3.25 3.09 2.97 2.87 2.80 2.73 2.64 2.53 2.42 2.37 2.31 2.25 2.18 2.11 2.04 22 5.79 4.38 3.78 3.44 3.22 3.05 2.93 2.84 2.76 2.70 2.60 2.50 2.39 2.33 2.27 2.21 2.14 2.08 2.00 23 5.75 4.35 3.75 3.41 3.18 3.02 2.90 2.81 2.73 2.67 2.57 2.47 2.36 2.30 2.24 2.18 2.11 2.04 1.97 24 5.72 4.32 3.72 3.38 3.15 2.99 2.87 2.78 2.70 2.64 2.54 2.44 2.33 2.27 2.21 2.15 2.08 2.01 1.94 25 5.69 4.29 3.69 3.35 3.13 2.97 2.85 2.75 2.68 2.61 2.51 2.41 2.30 2.24 2.18 2.12 2.05 1.98 1.91 26 5.66 4.27 3.67 3.33 3.10 2.94 2.82 2.73 2.65 2.59 2.49 2.39 2.28 2.22 2.16 2.09 2.03 1.95 1.88 27 5.63 4.24 3.65 3.31 3.08 2.92 2.80 2.71 2.63 2.57 2.47 2.36 2.25 2.19 2.13 2.07 2.00 1.93 1.85 28 5.61 4.22 3.63 3.29 3.06 2.90 2.78 2.69 2.61 2.55 2.45 2.34 2.23 2.17 2.11 2.05 1.98 1.91 1.83 29 5.59 4.20 3.61 3.27 3.04 2.88 2.76 2.67 2.59 2.53 2.43 2.32 2.21 2.15 2.09 2.03 1.96 1.89 1.81 30 5.57 4.18 3.69 3.25 3.03 2.87 2.75 2.65 2.57 2.51 2.41 2.31 2.20 2.14 2.07 2.01 1.94 1.87 1.79 40 5.42 4.05 3.46 3.13 2.90 2.74 2.62 2.53 2.45 2.39 2.29 2.18 2.07 2.01 1.94 1.88 1.80 1.72 1.64 60 5.29 3.93 3.34 3.01 2.79 2.63 2.51 2.41 2.33 2.27 2.17 2.06 1.94 1.88 1.82 1.74 1.67 1.58 1.48 120 5.15 3.80 3.23 2.89 2.67 2.52 2.39 2.30 2.22 2.16 2.05 1.94 1.82 1.76 1.69 1.61 1.53 1.43 1.31 ∞ 5.02 3.69 3.12 2.79 2.57 2.41 2.29 2.19 2.11 2.05 1.94 1.83 1.71 1.64 1.57 1.48 1.39 1.27 1.00 F Distribution Table 4 α = 0.05 d.f.N. d.f.D. 1 2 3 4 5 6 7 8 9 10 12 15 20 24 30 40 60 120 ∞ 1 161.4 199.5 215.7 224.6 230.2 234.0 236.8 238.9 240.5 241.9 243.9 245.9 248.0 249.1 250.1 251.1 252.2 253.3 254.3 2 18.51 19.00 19.16 19.25 19.30 19.33 19.35 19.37 19.38 19.40 19.41 19.43 19.45 19.45 19.46 19.47 19.48 19.49 19.50 3 10.13 9.55 9.28 9.12 9.01 8.94 8.89 8.85 8.81 8.79 8.74 8.70 8.66 8.64 8.62 8.59 8.57 8.55 8.53 4 7.71 6.94 6.59 6.39 6.26 6.16 6.09 6.04 6.00 5.96 5.91 5.86 5.80 5.77 5.75 5.72 5.69 5.66 5.63 5 6.61 5.79 5.41 5.19 5.05 4.95 4.88 4.82 4.77 4.74 4.68 4.62 4.56 4.53 4.50 4.46 4.43 4.40 4.36 6 5.99 5.14 4.76 4.53 4.39 4.28 4.21 4.15 4.10 4.06 4.00 3.94 3.87 3.84 3.81 3.77 3.74 3.70 3.67 7 5.59 4.74 4.35 4.12 3.97 3.87 3.79 3.73 3.68 3.64 3.57 3.51 3.44 3.41 3.38 3.34 3.30 3.27 3.23 8 5.32 4.46 4.07 3.84 3.69 3.58 3.50 3.44 3.39 3.35 3.28 3.22 3.15 3.12 3.08 3.04 3.01 2.97 2.93 9 5.12 4.26 3.86 3.63 3.48 3.37 3.29 3.23 3.18 3.14 3.07 3.01 2.94 2.90 2.86 2.83 2.79 2.75 2.71 10 4.96 4.10 3.71 3.48 3.33 3.22 3.14 3.07 3.02 2.98 2.91 2.85 2.77 2.74 2.70 2.66 2.62 2.58 2.54 11 4.84 3.98 3.59 3.36 3.20 3.09 3.01 2.95 2.90 2.85 2.79 2.72 2.65 2.61 2.57 2.53 2.49 2.45 2.40 12 4.75 3.89 3.49 3.26 3.11 3.00 2.91 2.85 2.80 2.75 2.69 2.62 2.54 2.51 2.47 2.43 2.38 2.34 2.30 13 4.67 3.81 3.41 3.18 3.03 2.92 2.83 2.77 2.71 2.67 2.60 2.53 2.46 2.42 2.38 2.34 2.30 2.25 2.21 14 4.60 3.74 3.34 3.11 2.96 2.85 2.76 2.70 2.65 2.60 2.53 2.46 2.39 2.35 2.31 2.27 2.22 2.18 2.13 15 4.54 3.68 3.29 3.06 2.90 2.79 2.71 2.64 2.59 2.54 2.48 2.40 2.33 2.29 2.25 2.20 2.16 2.11 2.07 16 4.49 3.63 3.24 3.01 2.85 2.74 2.66 2.59 2.54 2.49 2.42 2.35 2.28 2.24 2.19 2.15 2.11 2.06 2.01 17 4.45 3.59 3.20 2.96 2.81 2.70 2.61 2.55 2.49 2.45 2.38 2.31 2.23 2.19 2.15 2.10 2.06 2.01 1.96 18 4.41 3.55 3.16 2.93 2.77 2.66 2.58 2.51 2.46 2.41 2.34 2.27 2.19 2.15 2.11 2.06 2.02 1.97 1.92 19 4.38 3.52 3.13 2.90 2.74 2.63 2.54 2.48 2.42 2.38 2.31 2.23 2.16 2.11 2.07 2.03 1.98 1.93 1.88 20 4.35 3.49 3.10 2.87 2.71 2.60 2.51 2.45 2.39 2.35 2.28 2.20 2.12 2.08 2.04 1.99 1.95 1.90 1.84 21 4.32 3.47 3.07 2.84 2.68 2.57 2.49 2.42 2.37 2.32 2.25 2.18 2.10 2.05 2.01 1.96 1.92 1.87 1.81 22 4.30 3.44 3.05 2.82 2.66 2.55 2.46 2.40 2.34 2.30 2.23 2.15 2.07 2.03 1.98 1.94 1.89 1.84 1.78 23 4.28 3.42 3.03 2.80 2.64 2.53 2.44 2.37 2.32 2.27 2.20 2.13 2.05 2.01 1.96 1.91 1.86 1.81 1.76 24 4.26 3.40 3.01 2.78 2.62 2.51 2.42 2.36 2.30 2.25 2.18 2.11 2.03 1.98 1.94 1.89 1.84 1.79 1.73 25 4.24 3.39 2.99 2.76 2.60 2.49 2.40 2.34 2.28 2.24 2.16 2.09 2.01 1.96 1.92 1.87 1.82 1.77 1.71 26 4.23 3.37 2.98 2.74 2.59 2.47 2.39 2.32 2.27 2.22 2.15 2.07 1.99 1.95 1.90 1.85 1.80 1.75 1.69 27 4.21 3.35 2.96 2.73 2.57 2.46 2.37 2.31 2.25 2.20 2.13 2.06 1.97 1.93 1.88 1.84 1.79 1.73 1.67 28 4.20 3.34 2.95 2.71 2.56 2.45 2.36 2.29 2.24 2.19 2.12 2.04 1.96 1.91 1.87 1.82 1.77 1.71 1.65 29 4.18 3.33 2.93 2.70 2.55 2.43 2.35 2.28 2.22 2.18 2.10 2.03 1.94 1.90 1.85 1.81 1.75 1.70 1.64 30 4.17 3.32 2.92 2.69 2.53 2.42 2.33 2.27 2.21 2.16 2.09 2.01 1.93 1.89 1.84 1.79 1.74 1.68 1.62 40 4.08 3.23 2.84 2.61 2.45 2.34 2.25 2.18 2.12 2.08 2.00 1.92 1.84 1.79 1.74 1.69 1.64 1.58 1.51 60 4.00 3.15 2.76 2.52 2.37 2.25 2.17 2.10 2.04 1.99 1.92 1.84 1.75 1.70 1.65 1.59 1.53 1.47 1.39 120 3.92 3.07 2.68 2.45 2.29 2.17 2.09 2.02 1.96 1.91 1.83 1.75 1.66 1.61 1.55 1.50 1.43 1.35 1.25 ∞ 3.84 3.00 2.60 2.37 2.21 2.10 2.01 1.94 1.88 1.83 1.75 1.67 1.57 1.52 1.46 1.39 1.32 1.22 1.00 F Distribution Table 5 α = 0.10 d.f.N. d.f.D. 1 2 3 4 5 6 7 8 9 10 12 15 20 24 30 40 60 120 ∞ 1 39.86 49.50 53.59 55.83 57.24 58.20 58.91 59.44 59.86 60.19 60.71 61.22 61.74 62.00 62.26 62.53 62.79 63.06 63.33 2 8.53 9.00 9.16 9.24 9.29 9.33 9.35 9.37 9.38 9.39 9.41 9.42 9.44 9.45 9.46 9.47 9.47 9.48 9.49 3 5.54 5.46 5.39 5.34 5.31 5.28 5.27 5.25 5.24 5.23 5.22 5.20 5.18 5.18 5.17 5.16 5.15 5.14 5.13 4 4.54 4.32 4.19 4.11 4.05 4.01 3.98 3.95 3.94 3.92 3.90 3.87 3.84 3.83 3.82 3.80 3.79 3.78 3.76 5 4.06 3.78 3.62 3.52 3.45 3.40 3.37 3.34 3.32 3.30 3.27 3.24 3.21 3.19 3.17 3.16 3.14 3.12 3.10 6 3.78 3.46 3.29 3.18 3.11 3.05 3.01 2.98 2.96 2.94 2.90 2.87 2.84 2.82 2.80 2.78 2.76 2.74 2.72 7 3.59 3.26 3.07 2.96 2.88 2.83 2.78 2.75 2.72 2.70 2.67 2.63 2.59 2.58 2.56 2.54 2.51 2.49 2.47 8 3.46 3.11 2.92 2.81 2.73 2.67 2.62 2.59 2.56 2.54 2.50 2.46 2.42 2.40 2.38 2.36 2.34 2.32 2.29 9 3.36 3.01 2.81 2.69 2.61 2.55 2.51 2.47 2.44 2.42 2.38 2.34 2.30 2.28 2.25 2.23 2.21 2.18 2.16 10 3.29 2.92 2.73 2.61 2.52 2.46 2.41 2.38 2.35 2.32 2.28 2.24 2.20 2.18 2.16 2.13 2.11 2.08 2.06 11 3.23 2.86 2.66 2.54 2.45 2.39 2.34 2.30 2.27 2.25 2.21 2.17 2.12 2.10 2.08 2.05 2.03 2.00 1.97 12 3.18 2.81 2.61 2.48 2.39 2.33 2.28 2.24 2.21 2.19 2.15 2.10 2.06 2.04 2.01 1.99 1.96 1.93 1.90 13 3.14 2.76 2.56 2.43 2.35 2.28 2.23 2.20 2.16 2.14 2.10 2.05 2.01 1.98 1.96 1.93 1.90 1.88 1.85 14 3.10 2.73 2.52 2.39 2.31 2.24 2.19 2.15 2.12 2.10 2.05 2.01 1.96 1.94 1.91 1.89 1.86 1.83 1.80 15 3.07 2.70 2.49 2.36 2.27 2.21 2.16 2.12 2.09 2.06 2.02 1.97 1.92 1.90 1.87 1.85 1.82 1.79 1.76 16 3.05 2.67 2.46 2.33 2.24 2.18 2.13 2.09 2.06 2.03 1.99 1.94 1.89 1.87 1.84 1.81 1.78 1.75 1.72 17 3.03 2.64 2.44 2.31 2.22 2.15 2.10 2.06 2.03 2.00 1.96 1.91 1.86 1.84 1.81 1.78 1.75 1.72 1.69 18 3.01 2.62 2.42 2.29 2.20 2.13 2.08 2.04 2.00 1.98 1.93 1.89 1.84 1.81 1.78 1.75 1.72 1.69 1.66 19 2.99 2.61 2.40 2.27 2.18 2.11 2.06 2.02 1.98 1.96 1.91 1.86 1.81 1.79 1.76 1.73 1.70 1.67 1.63 20 2.97 2.59 2.38 2.25 2.16 2.09 2.04 2.00 1.96 1.94 1.89 1.84 1.79 1.77 1.74 1.71 1.68 1.64 1.61 21 2.96 2.57 2.36 2.23 2.14 2.08 2.02 1.98 1.95 1.92 1.87 1.83 1.78 1.75 1.72 1.69 1.66 1.62 1.59 22 2.95 2.56 2.35 2.22 2.13 2.06 2.01 1.97 1.93 1.90 1.86 1.81 1.76 1.73 1.70 1.67 1.64 1.60 1.57 23 2.94 2.55 2.34 2.21 2.11 2.05 1.99 1.95 1.92 1.89 1.84 1.80 1.74 1.72 1.69 1.66 1.62 1.59 1.55 24 2.93 2.54 2.33 2.19 2.10 2.04 1.98 1.94 1.91 1.88 1.83 1.78 1.73 1.70 1.67 1.64 1.61 1.57 1.53 25 2.92 2.53 2.32 2.18 2.09 2.02 1.97 1.93 1.89 1.87 1.82 1.77 1.72 1.69 1.66 1.63 1.59 1.56 1.52 26 2.91 2.52 2.31 2.17 2.08 2.01 1.96 1.92 1.88 1.86 1.81 1.76 1.71 1.68 1.65 1.61 1.58 1.54 1.50 27 2.90 2.51 2.30 2.17 2.07 2.00 1.95 1.91 1.87 1.85 1.80 1.75 1.70 1.67 1.64 1.60 1.57 1.53 1.49 28 2.89 2.50 2.29 2.16 2.06 2.00 1.94 1.90 1.87 1.84 1.79 1.74 1.69 1.66 1.63 1.59 1.56 1.52 1.48 29 2.89 2.50 2.28 2.15 2.06 1.99 1.93 1.89 1.86 1.83 1.78 1.73 1.68 1.65 1.62 1.58 1.55 1.51 1.47 30 2.88 2.49 2.28 2.14 2.05 1.98 1.93 1.88 1.85 1.82 1.77 1.72 1.67 1.64 1.61 1.57 1.54 1.50 1.46 40 2.84 2.44 2.23 2.09 2.00 1.93 1.87 1.83 1.79 1.76 1.71 1.66 1.61 1.57 1.54 1.51 1.47 1.42 1.38 60 2.79 2.39 2.18 2.04 1.95 1.87 1.82 1.77 1.74 1.71 1.66 1.60 1.54 1.51 1.48 1.44 1.40 1.35 1.29 120 2.75 2.35 2.13 1.99 1.90 1.82 1.77 1.72 1.68 1.65 1.60 1.55 1.48 1.45 1.41 1.37 1.32 1.26 1.19 ∞ 2.71 2.30 2.08 1.94 1.85 1.77 1.72 1.67 1.63 1.60 1.55 1.49 1.42 1.38 1.34 1.30 1.24 1.17 1.00 G. Sarty & University of Saskatchewan Distance Education Unit |
8607 | https://arxiv.org/pdf/1111.6423 | arXiv:1111.6423v1 [cond-mat.soft] 28 Nov 2011
Structure and dynamics of ring polymers: entanglement effects because of solution density and ring topology
Angelo Rosa ∗
SISSA - Scuola Internazionale Superiore di Studi Avanzati and IIT - Italian Institute of Technology (SISSA unit) Via Bonomea 265, 34136 Trieste (Italy)
Enzo Orlandini †
Dipartimento di Fisica and Sezione INFN, Universit` a di Padova, Via Marzolo 8, 35131 Padova (Italy)
Luca Tubiana ‡ and Cristian Micheletti §
SISSA - Scuola Internazionale Superiore di Studi Avanzati, Via Bonomea 265, 34136 Trieste (Italy)
(Dated: August 15, 2018) The effects of entanglement in solutions and melts of unknotted ring polymers have been addressed by several theoretical and numerical studies. The system properties have been typically profiled as a function of ring contour length at fixed solution density. Here, we use a different approach to investigate numerically the equilibrium and kinetic properties of solutions of model ring polymers. Specifically, the ring contour length is maintained fixed, while the interplay of inter- and intra-chain entanglement is modulated by varying both solution density (from infinite dilution up to ≈ 40% volume occupancy) and ring topology (by considering unknotted and trefoil-knotted chains). The equilibrium metric properties of rings with either topology are found to be only weakly affected by the increase of solution density. Even at the highest density, the average ring size, shape anisotropy and length of the knotted region differ at most by 40% from those of isolated rings. Conversely, kinetics are strongly affected by the degree of inter-chain entanglement: for both unknots and trefoils the characteristic times of ring size relaxation, reorientation and diffusion change by one order of magnitude across the considered range of concentrations. Yet, significant topology-dependent differences in kinetics are observed only for very dilute solutions (much below the ring overlap threshold). For knotted rings, the slowest kinetic process is found to correspond to the diffusion of the knotted region along the ring backbone.
I. INTRODUCTION
Characterizing the equilibrium and kinetic properties of semi-dilute solutions of cyclic polymers (rings) is one of the major remaining challenges in theoretical and experimental polymer physics [1–7]. One aspect of these systems that is very actively investigated regards how intra-chain and inter-chain entanglement reverberate on physical properties of rings solutions. Entanglement effects have been extensively investigated, and are hence well understood, for linear, i.e. open, polymer chains. However, unlike linear chains, physical properties of dense solutions of rings strongly depend on their preparation history during which the ring’s geometrical entanglement is locked permanently in topological constraints such as links and knots. Because of these constraints, the same concepts that proved so useful in the characterization of dense solution of linear chains are usually not easily, nor profitably, transferred to the case of circular polymers. A chief example, that paradigmatically highlights the qualitative differences of systems of closed versus open chains, is offered by the so-called Edwards-DeGennes reptation model [8, 9] for the kinetics of dense linear polymers. According to this model, each chain is confined inside a tube-like region resulting from the excluded-volume interactions with the neighboring chains. The motion of each chain therefore consists of a one-dimensional diffusion along the tube centerline resulting from the “inchworm movements” of small sub-chain loops. This type of local motion is found also in polymer rings moving through a fixed array of obstacles . However, the asymptotic standard diffusive behavior of polymer chains in melt depends on the ongoing process of “tube renewal” which results from the fact that the two polymer ends are practically free and hence capable of exploring and realizing new configurations [8–13]. This mechanism, which has received striking experimental confirmations as being the dominant kinetic process in dense polymer melts , explicitly builds on the linear character of the chains in solution. Hence, it is not applicable, at least in its
∗anrosa76@gmail.com
†orlandini@pd.infn.it
‡tubiana@sissa.it
§michelet@sissa.it
2conventional formulation, to ring polymers. The latter must consequently relax and move in dense solution according to different kinetic mechanisms , which are only recently being characterized by means of computationally-intensive molecular dynamics simulations [6, 7, 15–19]. Ring topology affects strongly also equilibrium properties of the chains, and in particular metric ones [20–24]. In fact, the closure condition not only introduces correlations in the backbone orientation that are of longer range than for linear chains [25, 26], but it preserves the knotted topology of the ring. If chain self-crossing is disallowed (no bond is cut and rejoined) then the ring knotted or unknotted state, trapped at the time of circularization, is maintained at all times. The ring self-entanglement associated to a given topological state is so important that it affects the scaling of metric properties of isolated rings, as first argued by Des Cloizeaux , and later by Grosberg and Deguchi [28, 29] and other studies (see cited references in ). In particular, Cates and Deutsch and M¨ uller et al. [3, 4] have suggested that strong inter-chain topological interactions lead to pronounced chain compaction of unlinked and unknotted rings in semi-dilute solutions, and that the typical chain configuration should resemble a branched polymer (akin to a lattice animal) [3, 4]. These seminal studies spurred several theoretical and computational efforts where equilibrium and kinetic properties of ring solutions were studied at fixed density – almost invariably corresponding to melt conditions – and for increasing ring length. These approaches, have provided considerable insight into the expected asymptotic scaling of the ring’s metric properties [6, 7, 18, 19]. In particular, recent evidence [6, 7] suggests that the branched polymer regime is likely at a crossover between the Gaussian regime for short rings and the “crumpled globule” regime for long ones . In the latter regime, each ring portion is highly compact and minimally intermingling with the rest of the chain , similarly to dense systems of long biopolymers in vivo, such as eukaryotic chromosomes . Compared to these studies, here we use stochastic molecular dynamics simulations to investigate complementary aspects of the effects that intra- and inter-chain topological constraints have on kinetic and equilibrium properties of solutions of model ring polymers. Specifically, we consider systems in which the contour length of the rings is fixed while the solution density is varied so to cover an appreciable range of inter-chain entanglement. The study is carried out for monodisperse solutions of unknotted rings as well as trefoil-knotted ones. To have a better insight into topological effects on the statics and dynamics of knotted rings we look also at the knotted portion of the ring. This is in general a formidable task and we succeeded in tackling it by relying on a recently-introduced effective and transparent procedure for locating knots [33, 34]. To the best of our knowledge this is the first time that a systematic off-lattice study of kinetic and equilibrium properties of rings with non-trivial topology and of their knotted portions has been carried out. By first exploring the metric and shape properties of equilibrated rings in solution we find that their behavior depends appreciably on ring topology and that not only it differs from the one seen for linear polymers in solutions but even from other physical realizations of dense ring polymers. Specifically, unlike what has been observed in confined [33–35] and collapsed knotted rings [36, 37], the average size of the knotted portion of a ring in solution is only weakly affected by the increase (for increasing monomer concentration) of the system’s geometrical entanglement. In particular, by increasing the solution density, no crossover to delocalized knots and no multiscale behavior of the entanglement is observed, unlike the case of spherical confinement . The presence of topological constraints makes the dynamics of rings in solution an even richer phenomenon whose understanding requires a full knowledge of the relationship between the spatial motion of a ring in a melt and the motion of its knotted portion along the ring backbone. By using the knot location tool previously mentioned, and considering dynamic observables commonly used in linear polymer contexts we show that several time scales are at play: the autocorrelation time of ring configurations, the time required by the rings to diffuse over regions comparable to their average size and – for knotted rings – the time required by the knot to diffuse over the ring contour. Various relaxation properties of the entire ring are found to change by one order of magnitude across the considered density range. The slowest kinetic process is associated with the diffusion along the backbone of the knotted region. This property, that to the best of our knowledge has not been pointed out before, seamlessly integrates with the other kinetic aspects thus offering a consistent picture for polymer relaxation in solutions of topologically constrained rings.
II. MODEL AND METHODS A. The model
To model the rings in solution, we use the bead-spring polymer model introduced by Kremer and Grest . The model accounts for the connectivity, bending rigidity, excluded volume and topology conservation of polymer chains. 3Specifically, the intra-chain energy consists of the following terms:
Hintra =
n
∑
i=1
[UF EN E (i, i + 1) + (1)
Ubr (i, i + 1 , i + 2) + (2)
n
∑
j=i+1
ULJ (i, j )] (3) where n is the total number of beads per ring, and i and j run over the indices of the beads. The latter are assumed to be numbered consecutively along the ring from one chosen reference monomer. The modulo-n indexing is implicitly assumed because of the ring periodicity. From now on we shall take the nominal bead diameter, σ, as the unit length and adopt the following notation: the position of the center of the ith beads is indicated by ~ri while the pairwise vector distance of beads i and j is denoted as ~di,j = ~rj − ~ri and its norm simply as di,j .With this notation the chain connectivity term, UF EN E (i, i + 1) is expressed as:
UF EN E (i, i + 1) =
− k
2
R20 ln
[
1 −
( di,i +1
R0
)2]
, d i,i +1 ≤ R0
0, d i,i +1 > R 0
(4) where R0 = 1 .5σ, k = 30 .0ǫ/σ 2 and the thermal energy kB T equals 1 .0ǫ . The bending energy has instead the standard Kratky-Porod form (discretized worm-like chain):
Ubr (i, i + 1 , i + 2) = KB T ξ p
σ
(
1 − ~di,i +1 · ~di+1 ,i +2
di,i +1 di+1 ,i +2
)
(5) where ξp = 4 .5σ is the nominal persistence length of the chain . Polymer chains are significantly bent by thermal fluctuations at contour lengths larger than ℓK , where ℓK = 2 ξp = 9 .0σ is the Kuhn length of the chain [8, 39]. The excluded volume interaction between distinct beads (including consecutive ones) corresponds to a purely repulsive Lennard-Jones potential:
ULJ (i, j ) =
{ 4ǫ[( σ/d i,j )12 − (σ/d i,j )6 + 1 /4] , d i,j ≤ σ21/6
0, d i,j > σ 21/6 . (6) This repulsive interaction controls the inter-chain excluded volume too:
Hinter =
N−1
∑
I=1
N
∑
J=I+1
ULJ (i, j ) (7) where N is the number of rings in solution and the index i [j] runs over the beads in chain I [J].
B. Simulation details
We consider solutions of N = 64 rings, each consisting of n = 216 beads (i.e., the ring contour length, Lc = 216 σ
corresponds to 24 ℓK ) at six different monomer densities, ρ: ρσ 3 = 0 .010 , 0.025 , 0.050 < ρ ∗σ3, ρσ 3 = 0 .100 and ρσ 3 =0.200 , 0.400 > ρ ∗σ3. Notice, that the chosen densities cover the cross-over to the overlap monomer concentration,
ρ∗σ3 ≈ 0.1 (see Eq. 10 below). For reference, we also consider isolated rings, and solutions of N = 21 rings in very dilute conditions ( ρσ 3 = 0 .003). The equilibrium and kinetic properties of these systems are studied using fixed-volume and constant-temperature Molecular Dynamics (MD) simulations. The system dynamics is integrated with the LAMMPS engine with Langevin thermostat (target temperature = 1 .0 LJ-units). The elementary integration time step is chosen equal to ∆t = 0 .012 τMD , where τMD = σ(m/ǫ )1/2 is the Lennard-Jones time, m is the bead mass and the friction coefficient,
γ, corresponds to γ/m = 0 .5τ −1
MD
. We stress that, because we are dealing with model polymers, the solvent is not explicitly included in our simulation. The polymer-solvent interaction is effectively accounted for only through the Langevin thermostat and related friction term. 4
(a) (c) (b) (d)
FIG. 1. Typical configurations of ring polymer solutions, at ρσ 3= 0 .025 (a) and ρσ 3= 0 .4 (b), and corresponding selected rings with trefoil-knotted (c and d, right) and unknotted topologies (c and d, left).
Preparation of initial configurations – At each density, we consider monodisperse solutions, consisting either of unknotted or trefoil-knotted rings. In the following we shall often use the standard shorthand notation UN and 3 1 to denote unknotted and trefoil-knotted topologies, respectively. The system initialization consists of placing a template unknotted or trefoil-knotted ring at the center of a cubic cell with periodic boundary conditions. The template configuration is replicated 4 times along each spatial direction. The linear dimension of the cubic cell is large enough to avoid overlap and linking between the template copies and yields an initial monomer density of about ρσ 3 ≈ 5 · 10 −3. Prior to the production runs, the cell is first evolved at constant pressure until the desired target density is reached and is subsequently equilibrated for a time span of about 10 5τMD . At all considered densities, this time span exceeds the time required by one ring to diffuse over distances larger than its typical size. Production runs have a typical duration of 6 · 10 6τMD . Typical equilibrated configurations at two different densities are shown in Fig. 1.
C. Ring shape and size
As customary, the salient metric properties (shape and size) of a given ring are characterized through the gyration tensor, Q. The entries of this 3 × 3 matrix are given by:
Qα,β = 1
n
n
∑
i=1
(~ri,α − ~rCM,α )( ~ri,β − ~rCM,β ), (8) where α and β run over the three Cartesian components and ~rCM = 1
n
∑ni=1 ~ri is the spatial location of the ring center of mass. The non-negative eigenvalues of Q, ranked with decreasing magnitude, Λ 1, Λ 2 and Λ 3 correspond to the square length of the principal axes of the ring gyration ellipsoid. Accordingly, their relative magnitude conveniently captures the overall spatial anisotropy of the rings, while their sum yields the ring square radius of gyration:
R2
g
= T r Q =
3
∑
α=1
Λα, (9) where T r is the trace operator. To measure the typical size and anisotropy of the ensemble of rings in solution, we compute the averages 〈R2
g
〉 and 〈Λi=1 ,2,3〉 where the brackets 〈... 〉 denote averaging over all rings of all system snapshots at a given solution density. 5
(a) (b) (c)
FIG. 2. Knotted portions in a non-trivially embedded trefoil knot (a), as identified by two different search approaches. (b) Top-down approach: the red arc is the shortest continuously knotted portion. It can not be shortened without loosing the 31topology. (c) Bottom-up approach: the red is the shortest arc with 3 1topology. Complementary arcs (in blue) in both approaches must be unknotted (see Ref. , for details).
D. Identifying the knotted portion of the rings
Because of the chain connectivity constraint, the global topological state of the rings is preserved in the course of the MD evolution. One way to characterize the interplay of geometrical and topological entanglement is by establishing the magnitude of the typical length of the ring portion that accommodates the knot [33, 36, 41]. From a mathematical point of view, the problem of locating the knot in a closed chain is ambiguous [30, 42, 43]. The difficulties are essentially two . First, to assess if the knot is contained in a given ring portion, which is an open arc, it is necessary to define a procedure for closing it by suitably bridging the arc ends. In fact, only after the arc is closed it is possible to characterize its topological state by using topological invariants. Viable closure procedure should introduce the least possible additional entanglement. This criterion is well satisfied by the minimally-interfering closure that some of us introduced recently and that will accordingly be used throughout our study. The second difficulty lies in the fact that the geometrical entanglement of a knotted ring can be so high that the identification of the region that accommodates the knot depends on the method used to search for it . Arguably, upper and lower bounds to the length of the region accommodating the knot are offered by the top-down and bottom-up searches illustrated in Fig. 2. Both methods were applied here and systematically yielded consistent results, see Supporting Fig. 1. Based on the robustness of the knot localization procedure for rings in solution, we shall limit considerations to the bottom-up search. This method returns the shortest arc having the same topology of the whole ring (a trefoil knot, in our case) while its complementary arc is unknotted, see Fig. 2c.
E. Surface accessible area
The rings surface accessible area was computed using the SAS routine of the GROMACS package [44, 45] using a probe sphere of diameter equal to the bead diameter, σ. The computed surface area includes contributions from inner cavities of the ring conformations that are large enough to accommodate the probe sphere.
III. RESULTS AND DISCUSSION A. Geometric properties of the rings
We first report on how various equilibrium metric properties of the rings depend on their topology and solution density. In a progression from global to finer aspects, we shall consider the rings size, shape, exposed surface area, the mean-square distance of points at increasing arclength separation and the degree of localization of the knot that they accommodate (in case of non-trivial topology). The rings consist of n = 216 beads of diameter σ with associated Kuhn length, ℓK = 9 σ. The considered monomer density of the solution, ρσ 3, spans the interval [0 .01 , 0.4]. This range spans from a dilute situation to one where the inter-chain entanglement is significant and yet the solution is isotropic . The lowest monomer density is much smaller than the one at which one expects significant ring overlap. The overlap density is estimated as ,
ρ∗ ≈ Lc/σ
〈R2
g
〉3/2 . (10) where 〈R2
g
〉 is the mean square radius of gyration of an isolated ring. Using the Zimm-Stockmayer estimate for the isolated ring size , 〈R2
g
〉 = Lc ℓK
12
≈ 162 σ yields ρ∗σ3 ≈ 0.1, which is an order of magnitude larger than the minimal monomer density considered here. 6Above this density, significant mutual entanglement of the chains is expected. For dense solutions of linear polymers, the intricacy of the melt [8, 9] and the resulting physical properties [11, 46] are captured by the entanglement length, Le. Roughly speaking, this quantity corresponds to the typical chain arclength separation between two consecutive topological constraints (known as entanglements ) arising from inter-chain un crossability. It has been recently proposed that the relationship tying Le, ℓK and the solution density, ρ is adequately captured by the following phenomenological expression:
Le
ℓK
≈ (0 .06 ( ρ σ ℓ 2
K
)) −2/5 + (0 .06 ( ρ σ ℓ 2
K
)) −2. (11) Using the previous expression, one has that the interval of ρ considered here corresponds to a wide range of Le for the equivalent system of linear chains. Indeed, for ρσ 3 going from 10 −2 to 0 .4 we have that Le/ℓ K spans from ≈ 400 to ≈ 1. In the latter situation, the entanglement length is approximately equal to the Kuhn length, and hence the system is at the crossover from the loosely- to the tightly-entangled regimes (i.e. the chain is nearly straight between two consecutive entanglements ). Higher densities are not considered, because for ρσ 3 >
∼ 0.5 the onset of the Onsager isotropic/nematic transition is expected to break the spatial isotropy of the solution.
〈R2
g〉[σ2]〈Λ1〉[σ2]〈Λ2〉[σ2]〈Λ3〉[σ2]
UN 175 .4±0.6116 .3±0.643 .9±0.115 .2±0.1
31120 .8±0.777 .9±0.730 .6±0.112 .3±0.1
FJR 168 .7±0.4108 .5±0.442 .1±0.218 .1±0.1
GR 161 .6±0.7106 .1±0.639 .7±0.116 .2±0.1
TABLE I. Average value of the mean square radius of gyration and eigenvalues of the gyration tensor for unknotted and trefoil-knotted isolated rings. Analogous quantities are also shown for an equivalent freely-jointed ring (FJR) and Gaussian ring (GR) of 24 bonds. The FJR bond length and the GR root-mean square bond length are equal to ℓK= 9 σ.
Following the outline set at the beginning of this section we first discuss how overall ring metric properties depend on the monomer density, ρ.For reference, we provide in Table I the average values of the square radius of gyration, 〈R2
g
〉, and the eigenvalues of the gyration tensor for isolated unknotted and trefoil-knotted rings. Analogous quantities are reported for an equivalent freely-jointed ring (FJR) and Gaussian ring (GR) of Lc/ℓ K = 24 bonds . The FJR data are obtained by a Markovian exploration of FJRs, while the GR data are obtained by stochastic molecular dynamics simulations. Table I conveys the effect that intra-chain constraints (excluded volume and fixed topology) have on the overall size and shape of rings at infinite dilution. Both types of constraints are absent in the FJR and the GR. Notice, that the FJR and GR quantities are typically within 10% of those of unknotted rings. The difference with analogous quantities for trefoil-knotted rings is substantially larger. These results reflect the fact that, for the considered values of Lc and
ℓK , unknotted rings dominate the equilibrium ensemble of infinitely-thin (FJR) or Gaussian rings with unrestricted topology. Trefoil-knotted rings therefore possess a degree of entanglement that would be atypical in rings circularized in equilibrium. The presence of the non-trivial topological constraint causes such rings to be tighter and slightly more isotropic than unknotted ones. This result is consistent with the intuitive notion that a certain arclength of trefoil-knotted rings is “used up” in the non-trivial entanglement (in fact the minimal ropelength required to tie trefoil-knotted rings is larger than for unknotted ones).
UN UN UN 313131
Monomer density, ρσ 3〈R2
g〉[σ2]〈Λ1〉/〈Λ3〉〈Λ2〉/〈Λ3〉〈R2
g〉[σ2]〈Λ1〉/〈Λ3〉〈Λ2〉/〈Λ3〉
0.010 165 .59 ±0.37 7.45 ±0.05 2.77 ±0.01 116 .07 ±0.01 6.118 ±0.001 2.4151 ±0.0004
0.025 150 .55 ±0.39 7.21 ±0.03 2.63 ±0.02 108 .23 ±0.08 5.91 ±0.01 2.335 ±0.003
0.050 134 .64 ±0.06 7.07 ±0.02 2.530 ±0.003 97 .26 ±0.05 5.80 ±0.01 2.266 ±0.006
0.100 115 .63 ±0.09 7.00 ±0.03 2.44 ±0.07 85 .62 ±0.19 5.95 ±0.02 2.244 ±0.004
0.200 98 .64 ±0.10 7.02 ±0.02 2.385 ±0.006 77 .94 ±0.22 6.29 ±0.03 2.260 ±0.008
0.400 86 .32 ±0.85 7.46 ±0.13 2.41 ±0.01 70 .11 ±0.06 6.84 ±0.03 2.318 ±0.008
TABLE II. Mean square radius of gyration, 〈R2
g〉and average shape of the simulated rings expressed as the ratios 〈Λ1〉/〈Λ3〉
and 〈Λ2〉/〈Λ3〉, where 〈Λα=1 ,2,3〉are the three average eigenvalues of the gyration tensor, Eq. 8. For very dilute conditions (ρσ 3= 0 .010), the calculated quantities are close to the theoretical values for ideal semi-flexible rings (see Table I).
Ring size. Table II reports the same metric properties of Table I, but calculated for rings in solution. The data suggest that, for both topologies, the ring size decreases with the monomer density, ρ. Specifically, going from 7
ρσ 3 = 10 −2 to ρσ 3 = 0 .4 it is seen that R2
g
decreases from the typical value of isolated rings down to about half of it. The decreasing trend of the average extension results from the non-concatenation constraint of the rings, consistent with previous numerical findings [3, 4]. It should also be noted that the monomer density attained by each ring is much smaller than the density of the entire solution. In fact, even in the densest case ρσ 3 = 0 .4 the individual ring density – defined as the number of ring monomers divided by the volume of the average gyration ellipsoid – is only
≈ 0.05 σ−3 for unknots and ≈ 0.06 σ−3 for trefoil-knotted rings. These findings suggest that an increase of solution density promotes extensive chain intermingling as opposed to tight compaction of the individual rings.
Ring shape. A further indication that an increase of the solution density affects only weakly the metric properties of each ring is given by looking at the typical ring shape. For this purpose, following previous studies of either closed or open chains [49–51], we calculate the average eigenvalues 〈Λα=1 ,2,3〉 of the gyration tensor, Eq. 8 which are shown in Table II. We note that the ratios 〈Λ1〉/〈Λ3〉 and 〈Λ2〉/〈Λ3〉 fluctuate by a few percent throughout the explored range of solution densities and remain close to the overall anisotropy expected for equivalent, isolated FJRs (see Table I).
Exposed surface. To complete the statistical characterization of ring conformations we next focus on the average exposed surface area of each ring. This is an important geometric indicator that aptly complements the radius of gyration [3, 4, 6, 7, 31] in describing the overall degree of ring compactness attained in denser and denser solutions. We stress that, because the purpose is to characterize the properties of individual rings, the exposed surface per bead is calculated separately for each ring, i.e. without taking into account the burial effect due to the surrounding chains. Furthermore, the exposed surface is calculated by taking into account voids (cavities) possibly present in a ring (see Methods, Sec. II E). The varying degree of exposure of the beads in a ring is illustrated in Fig. 3b: the profile pertains to the unknotted ring conformation in panel (a) (picked at ρσ 3 = 0 .4), where the beads are color coded according to the degree of exposure. For reference, in panel (b) it is also shown the value of the exposed surface per bead in a perfectly straight chain configuration. For the case shown in panels (a-b), highly exposed beads nearly approach this reference value, while the most buried ones have an exposed surface equal to about 1 /3 of the reference value. The distributions of the total exposed surface per ring are shown in panel (c): in particular, we notice that the ring exposed surface covers a moderately narrow interval. Nevertheless, the exposed area for trefoil-knotted rings is generally smaller than for the unknots (at the same monomer concentration, ρ), and the mean value of the exposed surface per bead (averaged over all beads of all rings in various configurations at fixed density) is a decreasing function of solution density. Finally, it is interesting to correlate the decrease of the surface area with the decrease of the ring radius of gyration observed for increasing density. This relationship is shown in panel (d) of Fig. 3. By taking into account the different scales and offsets of the two axes in the graph it is realized that unknotted and trefoil rings at the same monomer concentration differ appreciably by average size (see also Table II) though not by exposed surface. Secondly, the density-dependent decrease of the radius of gyration is not paralleled by an analogous decrease of the exposed surface. In fact, with respect to the diluted case, the latter diminishes by less than 5% up to ρσ 3 = 0 .2 and by less than 10% up to ρσ 3 = 0 .4. We note that simple dimensional considerations would have suggested, instead, a proportional dependence of the surface accessible area on 〈R2
g
〉. The failure of the simple dimensional analysis suggests that, despite ring sizes that are, on average, smaller at higher densities, the corresponding configurations are still sufficiently exposed to the solvent and maintain around each bead enough “free room” to accommodate the probe sphere used to measure the accessible surface.
Geometry of ring portions. The results presented so far address properties of entire rings. We shall next examine how various ring portions, or arcs, are affected by intra-chain and inter-chain entanglement. We first report on how the mean square end-to-end distance, R2
ee
(ℓ) of arcs of contour length ℓ, depends on the solution density and ring topology. Furthermore, for rings with non-trivial topology we shall identify the smallest arc accommodating the knot and examine how its contour length depends on the monomer density. The density-dependent behavior of R2
ee
(ℓ) is shown in Fig. 4. The data for knotted and unknotted rings are presented in two separate panels. For reference, in each panel we include the graph for the mean square end-to-end distance for a worm-like chain (WLC):
R2
ee,W LC
(ℓ) = ℓ2
K
2
[ 2ℓ
ℓK
exp
(
− 2ℓ
ℓK
)
− 1
]
. (12) The analogous expression for a worm-like ring (WLR) is closely approximated by:
R2
ee,W LR
(ℓ) =
(
1
R2
ee,W LC
(ℓ) + 1
R2
ee,W LC
(Lc − ℓ)
)−1
. (13) The latter expression is obtained by matching analytically the exact, small-ℓ (stiff) and large-ℓ (flexible) limiting behaviors for a ring polymer. 8
a) b) 0 50 100 150 200 index
01234567bead exposed area (σ 2 )
c)
1000 1050 1100 1150 1200 1250 1300
ρ = 0.01
ρ = 0.4
unknots trefoils
ρ = 0.1
00.01 0.02 0.03 0.04 0.05 0.06 ring exposed area ( σ2)Probability density ( 1/ σ )2
d)
60 80 100 120 140 160 180 < R g
2
average ring exposed area (σ ) 2
1000 1050 1100 1150 1200 1250 1300
unknots trefoils
ρ = 0.01 0.025 0.05 0.1 0.2 0.4 ρ = 0.01 0.025 0.05 0.1 0.2 0.4
FIG. 3. (a) The shaded region outlines the accessible surface of the shown ring configuration. This is calculated using the SAS routine of the GROMACS package [44, 45] with a probe sphere of diameter equal to the bead diameter, σ. Monomer beads are colored from red to cyan for increasing degree of their surface exposure, shown in panel (b) with the same coloring scheme. (c) Probability distribution functions of the total accessible surface per ring, for different solution densities and chain topologies. (d) Average accessible surface per ring as a function of the ring average size, for different solution densities and chain topologies. (a)
10 0
10 1
10 2
10 3
10 -2 10 -1
Unknot
R2ee, WLC (solid) R2ee, WLR (dashed)
ρ = 0.010
ρ = 0.025
ρ = 0.050
ρ = 0.100
ρ = 0.200
ρ = 0.400
R2ee [σ2]
(b)
10 0
10 1
10 2
10 3
10 -2 10 -1
31
R2ee, WLC (solid) R2ee, WLR (dashed)
R2ee [σ2]
l / L c
FIG. 4. Average square spatial distance R2
ee
between ring monomers as a function of their (normalized) contour length separation, ℓ/L c, along the chain: results for (a) unknotted and (b) trefoil-knotted ring polymers at increasing monomer densities. 9Fig. 4 shows that for both unknotted and trefoil-knotted rings at all densities, the WLC (and WLR) behavior is followed closely only for arclengths smaller than about ℓK (corresponding to ℓ/L c ∼ 0.04). Beyond this arclength, the WLR behavior is followed well by unknotted rings in dilute solutions. Noticeable differences from the WLR trend occur for both unknotted and knotted rings at densities larger than ρ∗σ3 = 0 .1. The deviations grow with: (i) increasing solution density at fixed topology and arclength separation; (ii) increasing arclength separation at fixed topology and density; (iii) changing ring topology from unknotted to trefoil-knotted at fixed density and arclength separation. In summary, departures from the WLR trend are seen upon increasing either the inter-chain (solution density) or the intra-chain (topological) entanglements.
Properties of the knotted portion of a ring. We conclude the analysis of the equilibrium properties by examining how density influences the length of the ring portion accommodating the knot, Lknot . The analysis is motivated by recent studies which showed that the typical value of Lknot is substantially larger for collapsed rings than for unconstrained or swollen ones [34, 36, 37]. In addition, for knotted rings subject to spherical confinement, it was seen that the increasing geometrical entanglement acquires a complex multiscale character so that very different values of
Lknot are obtained upon locating the knotted portion using bottom-up or top-down approaches (see Methods, Sec. II D). These properties are expected to significantly impact the physical behavior of the chains (see e.g. Ref. for a discussion of knot localization effects in a biological context, namely the ejection of tightly packed - and highly entangled - DNA from a viral capsid).
Monomer density, ρσ 3〈Lknot 〉〈Lknot 〉/L c
0.010 88 .8±0.20.41 ±0.0010
0.025 90 .2±0.30.42 ±0.0010
0.050 89 .7±0.10.42 ±0.0005
0.100 85 .9±0.10.40 ±0.0005
0.200 79 .9±0.10.37 ±0.0005
0.400 72 .9±0.10.34 ±0.0005
TABLE III. Average size of the knotted portion of 3 1ring polymers, 〈Lknot 〉as a function of monomer density, ρ. On average, the knot occupies a large fraction of the polymer chain.
In the present context, the average value of Lknot remains fairly constant at all densities, see Table III. However, it should be noted that above the overlap density, ρ∗, the probability distribution function for Lknot acquires a rounder and broader shape, see Fig. ?? . The results are robust across different methods used for locating the knot (see Supporting Fig. 1 for a comparison of top-down and bottom-up approaches). In conclusion, the properties of knots in dense solutions differ qualitatively from the two most typical conditions (ring collapse or ring spatial confinement) leading to high monomer density.
0
0.5
1
1.5
2
2.5
3
3.5
4
00.2 0.4 0.6 0.8 1
ρ= 0.010
ρ= 0.025
ρ= 0.050
ρ= 0.100
ρ= 0.200
ρ= 0.400
Lknot / L c
FIG. 5. Probability distribution function of the normalized knot length, Lknot /L c, at different monomer concentrations.
B. Dynamical properties
The growing level of entanglement found in solutions of increasing density is expected to strongly impact the ring dynamics. By analogy with the case of linear polymers melts [8, 11, 13, 38, 52], multiple dynamical regimes are 10
(a)
10 -3
10 -2
10 -1
10 0
02000 4000 6000 8000 10000
Unknot
31
(■)ρσ 3= 0.003 (❏) Isolated
t [ τMD ]
CRg(t)
(b)
10 -3
10 -2
10 -1
10 0
02000 4000 6000 8000 10000
(■)ρσ 3= 0.4 (❏)ρσ 3= 0.01
t [ τMD ]
CRg(t)
FIG. 6. Radius of gyration time correlation function CRg (t) (Eq. 14) for unknots (red symbols) and trefoils (green symbols). (a): Result for isolated rings (open symbols) and rings in very dilute conditions with ρσ 3 = 0 .003 (filled symbols). The long time behavior of CRg (t) for 3 1 rings deviates from the CRg (t) for the unknots, and shows a longer exponential tail, in agreement with previous studies of isolated ring polymers on lattice [20, 24, 54]. (b): Results for ρσ 3 = 0 .01 (open symbols) and ρσ 3 = 0 .4(filled symbols).
expected. For the latter system, it is known that at times longer than the slowest chain relaxation time, the motion of the chain center of mass follows the standard diffusion while at smaller times the kinetics is dominated by the slow chain reptation inside the tube created by inter-chain constraints . We accordingly monitor several observables: 1. The typical time scale for ring diffusion in space; 2. The time scales dominating the fluctuations in rings size and orientation; 3. Finally, for rings with non-trivial topology, we examine how the knotted portion moves both in space and along the ring backbone. Some of the above mentioned quantities have been considered before in studies on melts of unknotted rings [3, 4, 7, 53], in lattice models of isolated knotted rings [20, 24, 54], and in polymer models of knot diffusion along DNA [55, 56]. The present context therefore offers an opportunity to examine the impact of both solution density and ring topology on dynamics and compare the time-scales associated to the various phenomena.
Time autocorrelation function for the radius of gyration. We start by considering the time autocorrelation function of an internal (i.e. independent of the ring absolute space position and orientation) quantity, namely the radius of gyration:
CRg (t) = 〈Rg (t) Rg (0) 〉 − 〈 Rg 〉2
〈R2
g
〉 − 〈 Rg 〉2 . (14) where the brackets, 〈... 〉, denote the average over simulation time and over rings. The behavior of CRg (t) was considered in recent lattice studies of isolated , unconstrained knotted and unknotted rings [24, 54]. These investigations showed that CRg decays with a characteristic time, τRg , that is larger for knotted rings than for unknotted ones [24, 54]. The effect reflects the enhanced self-hindrance of rings with non-trivial topology; consider, for instance, that the minimal number of crossings observed in any two-dimensional projection of an unknotted ring is zero, while it is three for trefoil-knotted rings (which hence must necessarily wind on themselves more than unknots). Investigating the decay properties of CRg (t) in the present context serves a twofold purpose. On the one hand, it can clarify whether the above-mentioned lattice results apply to off-lattice contexts, too. On the other hand, by analyzing the dependence of CRg (t) on solution density, it is possible to assess if, and to what extent, the increased intra-chain and inter-chain entanglement affect differently the size relaxation times of unknotted and knotted rings. According to this plan we first compute CRg from simulations of isolated knotted and unknotted rings. The results are shown in Fig. 6a (open symbols) and fully support the lattice results regarding the slower relaxation of trefoils compared to unknots. For rings in solutions, at all considered densities ( ρσ 3 = 0 .01 − 0.4), one is surprised to find that the CRg curves for trefoils and unknots display much smaller differences compared to the isolated case, see Fig. 11 6b. Indeed, a behavior quantitatively similar to the isolated case is found only at densities lower than ρσ 3 = 0 .003, as shown in Fig. 6a. The effect is quantitatively captured by calculating the effective relaxation time, τRg , given by the integral of CRg (t), see Table IV. The data in the table illustrate vividly that the density-dependent increase of the intra- and inter-chain entanglement obliterates differences in the average kinetic behavior of rings with different topological state. For both unknots and trefoils, τRg has an order of magnitude increase over the considered range of solution density, ρ. Notice that upon increasing ρ, the value of τRg for unknots overtakes the one of trefoils (see also the order of the red and green curves in panels a and b in Fig. 6).
UN 31 UN 31 UN 31
Monomer density, ρσ 3 τRg τRg τdiam τdiam τCM τCM
Single ring (6 .6 ± 0.6) · 10 2 (7 .9 ± 0.6) · 10 2 (2 .8 ± 0.1) · 10 3 (1 .6 ± 0.1) · 10 3 3.1 · 10 3 2.2 · 10 3
0.003 (7 .0 ± 0.6) · 10 2 (7 .0 ± 0.6) · 10 2 (2 .7 ± 0.1) · 10 3 (1 .6 ± 0.1) · 10 3 3.4 · 10 3 2.4 · 10 3
0.010 (8 .1 ± 0.6) · 10 2 (7 .3 ± 0.6) · 10 2 (2 .7 ± 0.4) · 10 3 (1 .6 ± 0.3) · 10 3 4.8 · 10 3 3.6 · 10 3
0.025 (1 .1 ± 0.1) · 10 3 (7 .8 ± 0.6) · 10 2 (2 .8 ± 0.4) · 10 3 (1 .6 ± 0.3) · 10 3 4.8 · 10 3 3.6 · 10 3
0.050 (1 .6 ± 0.1) · 10 3 (1 .0 ± 0.1) · 10 3 (3 .1 ± 0.4) · 10 3 (1 .7 ± 0.3) · 10 3 6.0 · 10 3 3.6 · 10 3
0.100 (2 .7 ± 0.1) · 10 3 (1 .5 ± 0.1) · 10 3 (3 .9 ± 0.4) · 10 3 (2 .2 ± 0.3) · 10 3 7.2 · 10 3 4.8 · 10 3
0.200 (4 .7 ± 0.1) · 10 3 (2 .6 ± 0.1) · 10 3 (6 .1 ± 0.4) · 10 3 (3 .3 ± 0.3) · 10 3 1.2 · 10 4 7.2 · 10 3
0.400 (1 .25 ± 0.01) · 10 4 (7 .3 ± 0.1) · 10 3 (1 .59 ± 0.04) · 10 4 (9 .2 ± 0.4) · 10 3 3.9 · 10 4 2.2 · 10 4
TABLE IV. Correlation times τRg and τdiam , calculated by numerical integration of the respective time correlation functions
CRg (t) (Eq. 14) and Cdiam (t) (Eq. 15). Numerical integration is limited to the time interval where correlation functions are
10 −2. τCM is defined as the typical time required by rings to transverse a region of linear size comparable to the ring average gyration radius, i.e. δr 2
CM
(τCM ) = 〈R2
g
〉 (see Eq. 16). All times are expressed in units of τM D .
Reorientation time. We now turn to consider kinetic properties that do depend on the absolute orientation of rings in space. For linear polymers, it is customary to consider the autocorrelation function of the end-to-end vector which, for closed chains, admits several generalizations . The one considered here is the time correlation of the ring diameter vector:
Cdiam (t) = 〈~d(t) · ~d(0) 〉
〈| ~d|2〉 , (15) where ~d is the vector joining two monomers that are “diametrically opposite” on the ring backbone, i.e. monomers with the largest possible chemical distance, Lc/2. Here the brackets 〈· · ·〉 denote a multiple average: over the dynamical trajectory and over each pair of diametrically opposite beads for every ring in solution. Notice that Cdiam is sensitive to both changes in modulus of the diameter vector and to its absolute orientation. For this reason, τdiam , the characteristic decay time of Cdiam (t) is customarily referred to as the reorientation time. Fig. 7 portrays Cdiam (t) for unknots (red symbols) and 3 1 knots (green symbols) at different monomer concentra-tions. Analogously to τRg , we define the corresponding τdiam as the numerical integral of Cdiam (t). Final results are reported in Table IV. As for CRg (t), the asymptotic decay time of Cdiam (t) depends on ring topology and is faster for trefoils than unknots. To the best of our knowledge this effect, that holds at all densities, was neither pointed out nor addressed before. Extending this analysis and considerations to rings with more complicated knot types could represent an interesting avenue for further investigations.
Diffusion of the ring center of mass. Because trefoils are more compact than unknots they should diffuse faster in the solution, analogously to what they do in other types of media [57, 58]. This intuitive expectation is indeed confirmed by inspecting Fig. 8, which portrays the mean square displacement of the ring center of mass, δr 2
CM
(t), for time lags of increasing duration, t:
δr 2
CM
(t) ≡ 〈 (~rCM (t) − ~rCM (0)) 2〉. (16) The displacement data given in Fig. 8 pertain to ρσ 3 = 0 .2; similar plots are obtained for different values of ρ. It is seen that over the time-span covered by the simulations, both unknotted and trefoil rings diffuse over distances that exceed by several orders of magnitude their average size. For traveled distances larger than the average ring size, the motion of the center of mass follows standard diffusion. The corresponding diffusion time τCM , defined as
δr 2
CM
(τCM ) = 〈R2
g
〉, is given in Table IV. Notice, that the values of τCM are comparable to the values of τdiam at the same solution density, consistently with the case of linear polymers . 12
(a)
10 -2
10 -1
10 0
05000 10000 15000 20000
Unknot
31
t [ τMD ]
Cdiam (t)
(b)
10 -2
10 -1
10 0
05000 10000 15000 20000
t [ τMD ]
Cdiam (t)
FIG. 7. Ring diameter time correlation function Cdiam (t) (Eq. 15) for the unknots (red symbols) and trefoils (green symbols). (a): ρσ 3 = 0 .01 (open symbols) and ρσ 3 = 0 .025 (filled symbols). (b): ρσ 3 = 0 .1 (open symbols) and ρσ 3 = 0 .4 (filled symbols). (a)
10 0
10 1
10 2
10 3
10 4
10 5
10 110 210 310 410 510 6
ρσ 3 = 0.2
t [ τMD ]
δ rCM 2(t) (+) [ σ2 ]
Unknot
31
∼t
(b)
FIG. 8. (a): Comparison between the time mean-square displacement of the rings center of mass ( δr 2
rCM
(t), Eq. 16) for unknots (red symbols) and trefoils (green symbols) and the corresponding average square gyration radii (horizontal lines), for
ρσ 3 = 0 .2. Within the considered time-span, rings diffuse over distances much larger than their typical sizes. (b): Diffusive motion of the center of mass of a randomly selected trefoil, for ρσ 3 = 0 .2. The red bar denotes the average ring size, 〈R2
g
〉1/2.Corresponding plots at different monomer densities look qualitatively similar.
Backbone motion of the knot. Finally, we examine the dynamics in space and along the ring backbone of the knotted portion of the trefoils. This is a computationally-demanding task, because the numerically-costly identification of the knotted portion must be carried out for each ring of all sampled system snapshots. For simplicity, the instantaneous position of the knot on the ring is taken to coincide with the chemical coordinate of the midpoint of the knotted portion (defined with respect to the absolute monomer indexing of the ring set at the beginning of the simulation). The instantaneous knot midpoint position, sk(t) is recorded at fixed time intervals, ∆t = 120 τMD . The incremental knot displacement between time snapshots n (= t
∆t
) and n + 1, dk(n), is computed using the expression:
dk(n) =
{[
sk(t + ∆ t) − sk(t) + 3
2 Lc
]
mod Lc
}
− Lc
2 (17) where “mod” denotes the modulus operation. The previous expression ensures that the displacement is mapped in the [−Lc/2, L c/2] interval. The value of ∆ t is sufficiently small that, at all considered densities, the occurrence of large “jumps” (say equal or larger than Lc/4 in modulus) is rare. This is illustrated in the inset of Fig. 9 (for ρσ 3 = 0 .1) which further highlights the Gaussian character of the distribution. 13
(a) (b)
0
50
100
150
200
250
300
350
400
0200 400 600 800 1000
t [ τMD ]
δsk2(t) [ σ2] 0
0.02
0.04
0.06
0.08
-100 -50 050 100
dk
FIG. 9. (a) Time behavior of the knot midpoint position along the ring contour. Midpoint coordinates are defined with respect to a reference monomer chosen at the beginning of the simulation. On the right, four ring configurations sampled at different times along the trajectory, with the corresponding knotted portions marked in red. (b) Mean square displacement of the knot along the ring contour, δs 2
k
(t) (Eq. 18), at density ρσ 3 = 0 .1. The proportional dependence on time (red line passing through the origin) can be expressed as 2 Dk t, where Dk is the one-dimensional diffusion coefficient: for the case shown, Dk = 0 .21 σ2 /τ M D
(see Table V, for a list of diffusion coefficients at all monomer concentrations). Inset : The Gaussian distribution (red curve) associated with the model one-dimensional diffusion process matches well the numerical distribution (black line) of incremental knot displacements, dk (Eq. 17).
The knot mean square displacement on the ring contour, δs 2
k
(t), is defined by
δs 2
k
(t = i · ∆t) =
〈
i−1
∑
j=0
dk(j)
2
〉
, (18) where the summation index runs over consecutive displacements, and the average 〈... 〉 is taken (1) over the trajectory and (2) over the rings ensemble. Fig. 9 illustrates that a linear relationship holds between δs 2
k
(t) and time, indicating that the knot sliding on the ring backbone can be treated as a one-dimensional diffusion process. The corresponding diffusion coefficients, Dk, calculated at different solution densities are given in Table V which also reports the typical time, τk, required by the knot to diffuse by its average length on the ring backbone. Interestingly, τk is almost unaffected by increasing monomer concentration. In addition, for a given solution density, τk exceeds all other characteristic times given in Table IV.
Monomer density, ρ Dk [σ2/τ M D ] τk [τM D ]
0.010 0.18 ± 0.01 2.1 · 10 4
0.025 0.19 ± 0.01 2.0 · 10 4
0.050 0.19 ± 0.01 2.0 · 10 4
0.100 0.21 ± 0.01 1.8 · 10 4
0.200 0.23 ± 0.01 1.6 · 10 4
0.400 0.16 ± 0.02 2.3 · 10 4
TABLE V. Diffusion coefficients, Dk for the motion of the knotted portion along the ring contour length. The value of Dk
corresponds to half the slope of the line that, passing through the origin, provides the best interpolation to δs 2
k
(t) vs t. The typical diffusion (sliding) time, τk , is defined as the time required to diffuse by a backbone distance equal to the typical knot contour length ( ≈ 40% of the ring contour length, see Table III).
Motion of the knotted portion in space. The above observation indicates that the knot location on the ring backbone is practically quenched as a ring diffuses in space over a distance equal to its size. This suggests that the three-dimensional motion of the knotted portion should be akin to the one of a fixed, tagged, portion of the ring of comparable size. This observation was verified by considering the time dependence of the mean-square displacement of the knot center of mass, defined (analogously to Eq. 16) as:
δr 2
kCM
(t) = 〈(~rkCM (t) − ~rkCM (0)) 2〉. (19) 14
(a)
10 0
10 1
10 2
10 3
10 4
10 5
10 110 210 310 410 510 6
ρσ 3 = 0.1
t [ τMD ]
δrCM 2(t) [ σ2]
δrkCM 2(t) [ σ2]
∼t
∼t0.7
31
(b)
10 0
10 1
10 2
10 3
10 4
10 5
10 110 210 310 410 510 6
t [ τMD ]
δrCM 2(t) [ σ2]
Tagged ring subportion (=0.4 L c) [ σ2]
∼t
∼t0.7
Unknot
FIG. 10. (a) Comparison between the mean-square displacement of the center of mass of trefoils ( δr 2
CM (t), Eq. 16, green symbols) and the center of mass of their knotted portion ( δr 2
kCM (t), Eq. 19, brown symbols). The motion of the knotted portion is sub-diffusive at short times, and follows the chain global displacement at larger times. (b) Comparison between the mean-square displacement of the center of mass of unknots ( δr 2
CM (t), Eq. 16, green symbols) and the center of mass of a tagged portion (brown symbols) of linear size equal to the average knot length on 3 1rings ( ≈0.4Lc, Table III).
Fig. 10 (left panel) shows that the diffusive behavior of the observable ~rkCM sets in at times larger than the relaxation time of the whole chain (compare to Fig. 8), while a subdiffusive behavior ∼ tα with α ≈ 0.7 is seen at smaller times. Notice that the same exponent, α ≈ 0.7 was previously observed by looking at the stochastic dynamics of the center of mass of the knotted portion in isolated self-avoiding polygons on a cubic lattice . Analogous plots at different monomer densities show the same subdiffusive behavior with an exponent α that decreases slightly with ρ (not shown). We have then calculated the mean square displacement of a randomly-picked ring portion spanning 40% of the contour length (the typical size of the knotted portion, Table III) of the unknotted ring. As shown in Fig. 10b it was found that the tagged portion moves very compatibly with the motion of the knotted part, showing an analogous crossover from a diffusive to a subdiffusive behavior. This suggests that the underlying mechanism of the subdiffusive regime of the “quenched” knotted part is similar to the one that governs the subdiffusive behavior of any other (topologically trivial) tagged subregion of the ring. Note, however, that the effective length of the rings considered here is very small compared to the one of the lattice rings studied in , and no simple argument is presently available to predict if the
α ≈ 0.7 exponent is expected to be maintained for increasing contour lengths where the size of the knotted portion is supposed to be negligible with respect to the one of the whole ring. In conclusion, the above results provide a vivid picture of the key features of the dynamics of the knotted region. Specifically, they illustrate that the displacement of the knot center of mass follows “passively” the one of the whole ring because of the very long times required by the knot to slide along the ring.
IV. SUMMARY AND CONCLUSIONS
We reported on a systematic computational study of the equilibrium and dynamics of solutions of unconcatenated ring polymers with different knot topology. Specifically, molecular dynamics simulations (with no explicit treatment of the solvent) at fixed-volume and constant temperature were carried out on bead-spring models of semiflexible unknotted and trefoil-knotted rings for several solution densities. This framework was used to explore the extent to which the interplay between topological constraints (knotting) and the geometrical self (intra-chain) and mutual (inter-chain) entanglements affect both the equilibrium and the dynamical properties of the rings. The study complements previous investigations of dense solutions of unlinked and unknotted ring polymers of various contour lengths. Regarding the equilibrium metric properties it is found that changes of the inter-chain or intra-chain entanglement operated by varying the solution density and/or ring topology, affect modestly the average ring size and shape compared to the infinitely-diluted case. Specifically, the root mean radius of gyration of both unknotted and trefoil rings at the highest solution densities (occupied volume fraction equal to 0.4) are at most about 40% smaller than in the unconstrained case. At all densities, trefoils are smaller (between 10% and 20% in linear size) and slightly more globular than unknots. Yet the average exposed surface of rings with different topology is practically the same and about constant at all densities. These results offer an interesting insight regarding the compactness of the rings 15 and, in particular, they indicate that the moderate decrease in ring size following the increasing intra- and inter-chain entanglement does not preclude the persistence of voids and cavities within the rings convex hull so that the exposed surface area is about the same as for unconstrained isolated rings. The weak dependence of the overall ring geometric features on solution density prompts the question of whether, for knotted rings, the metric properties of the knotted ring portion are weakly affected too. By knotted ring portion we refer to the shortest arc that accommodates the knot. This question is addressed by using the effective and transparent knot-location algorithm that was used recently to show that in isolated rings the length of the knotted region varies dramatically with increasing spherical confinement. At variance with the latter case, it is found that the length of the average size of the knotted ring portion is practically insensitive to density variations. While ring equilibrium metric properties are only weakly affected by variations of solution density and ring topology, the opposite holds for kinetic properties. The characteristic times of ring size relaxation, reorientation, and center of mass diffusion change by one order of magnitude across the considered density range. Further topology-dependent aspects of ring kinetics were highlighted by monitoring various kinetic observables of the ring knotted portion. Specifically we focused on how such region displaces in space and along the ring contour. The backbone motion is found to follow standard one-dimensional diffusion and the corresponding diffusion coefficient is smaller (much smaller at low concentrations) than the one of the ring center of mass. Consequently, for time scales over which a ring moves appreciably in space, its knotted portion remains practically quenched and, as we verified, diffuses in space as any other equally-long tagged portion of the ring. In summary, the results presented here offer novel insights into the impact of intra-chain and inter-chain entangle-ment in solutions of ring polymers. The mild dependence of equilibrium metric properties on both the above-mentioned effects has no parallel with the behavior of dense systems of rings obtained by three-dimensional spatial (spherical) confinement. This suggests that an interesting novel avenue to address in future work would be to examine analogous effects for dense systems of rings obtained by two-and one-dimensional confinement (slabs and channels), which are increasingly adopted for advanced polymer micromanipulations. By converse, the sensitive dependence of various ring kinetic properties on solution density and ring topology suggests that both effects should be relevant to the studies of rheology and transport properties in semi-dilute solutions of circular polymers.
V. ACKNOWLEDGMENTS
Numerical calculations were partly carried out in CINECA, and partly under the HPC-Europa2 project (project number: 228398) with the support of the European Commission Capacities Area-Research Infrastructures Initiative. This work made use of the facilities of HECToR, the UK’s national high performance computing service, which is provided by UoE HPCx Ltd at the University of Edinburgh, Cray Inc and NAG Ltd, and funded by the Office of Science and Technology through EPSRC’s High End Computing Programme. Reprinted with permission from A. Rosa et al., Macromolecules , 2011, 44 (21), pp. 8668-8680. Copyright (2011) American Chemical Society. 16
M. Rubinstein, Phys. Rev. Lett. 57 , 3023 (1986). M. E. Cates and J. M. Deutsch, J. Phys. (Paris) 47 , 2121 (1986). M. M¨ uller, J. P. Wittmer, and M. E. Cates, Phys. Rev. E 53 , 5063 (1996). M. M¨ uller, J. P. Wittmer, and M. E. Cates, Phys. Rev. E 61 , 4078 (2000). M. Kapnistos, M. Lang, D. Vlassopoulos, W. Pyckhout-Hintzen, D. Richter, D. Cho, T. Chang, and M. Rubinstein, Nature Materials 7, 997 (2008). J. D. Halverson, W. B. Lee, G. S. Grest, A. Y. Grosberg, and K. Kremer, J. Chem. Phys. 134 , 204904 (2011). J. D. Halverson, W. B. Lee, G. S. Grest, A. Y. Grosberg, and K. Kremer, J. Chem. Phys. 134 , 204905 (2011). M. Doi and S. F. Edwards, The Theory of Polymer Dynamics (Oxford University Press, New York, 1986). P.-G. de Gennes, J. Chem. Phys. 55 , 572 (1971). R. Auhl, R. Everaers, G. S. Grest, K. Kremer, and S. J. Plimpton, J. Chem. Phys. 119 , 12718 (2003). R. Everaers, S. K. Sukumaran, G. S. Grest, C. Svaneborg, A. Sivasubramanian, and K. Kremer, Science 303 , 823 (2004). C. Tzoumanekas and D. N. Theodorou, Macromolecules 39 , 4592 (2006). F. Lahmar, C. Tzoumanekas, D. N. Theodorou, and B. Rousseau, Macromolecules 42 , 7485 (2009). S. A. Wischnewski, M. Monkenbusch, L. Willner, D. Richter, and G. Kali, Phys. Rev. Lett. 90 , 058302 (2003). K. Hur, R. G. Winkler, and D. Y. Yoon, Macromolecules 39 , 3975 (2006). G. Tsolou, N. Stratikis, C. Baig, P. S. Stephanou, and V. G. Mavrantzas, Macromolecules 43 , 10692 (2010). K. Hur, C. Jeong, R. G. Winkler, N. Lacevic, R. H. Gee, and D. Y. Yoon, Macromolecules 44 , 2311 (2011). D. Reith, L. Mirny, and P. Virnau, Prog. Theor. Phys. in press (2011). D. Reith, A. Milchev, P. Virnau, and K. Binder, Europhys. Lett. 95 , 28003 (2011). S. R. Quake, Phys. Rev. Lett. 73 , 3317 (1994). N. T. Moore, R. C. Lua, and A. Y. Grosberg, Proc. Natl. Acad. Sci. USA 101 , 13431 (2004). E.-G. Kim and M. L. Klein, Macromolecules 116 , 5333 (2004). E. Orlandini and S. G. Whittington, Rev. Mod. Phys. 79 , 611 (2007). E. Orlandini, A. L. Stella, C. Vanderzande, and F. Zonta, J. Phys. A: Math. Theor. 41 , 122002 (2008). G. Witz, K. Rechendorff, J. Adamcik, and G. Dietler, Phys. Rev. Lett. 101 , 4 (2008). F. Drube, K. Alim, G. Witz, G. Dietler, and E. Frey, Nano Lett. 10 , 1445 (2010). J. des Cloizeaux, J. Phys. Lett. 42 , L433 (1981). A. Y. Grosberg, Polym. Sci., Ser. A 51 , 70 (2009). H. Matsuda, A. Yao, H. Tsukahara, T. Deguchi, K. Furuta, and T. Inami, Phys. Rev. E 68 , 011102 (2003). C. Micheletti, D. Marenduzzo, and E. Orlandini, Physics Reports 504 , 1 (2011). A. Grosberg, Y. Rabin, S. Havlin, and A. Neer, Europhys. Lett. 23 , 373 (1993). A. Rosa and R. Everaers, PLoS Comput. Biol. 4, e1000153 (2008). L. Tubiana, E. Orlandini, and C. Micheletti, Progress Theor. Phys. in press, cond-mat arXiv/1103.0475 (2011). L. Tubiana, E. Orlandini, and C. Micheletti, Phys. Rev. Lett. (to appear), cond-mat arXiv/1106.1872 (2011). D. Marenduzzo, E. Orlandini, A. Stasiak, D. W. Sumners, L. Tubiana, and C. Micheletti, Proc. Natl. Acad. Sci. USA
106 , 22269 (2009). B. Marcone, E. Orlandini, A. L. Stella, and F. Zonta, Phys. Rev. E 75 , 041105 (2007). M. Baiesi, E. Orlandini, A. L. Stella, and F. Zonta, Phys. Rev. Lett. 106 , 258301 (2011). K. Kremer and G. S. Grest, J. Chem. Phys. 92 , 5057 (1990). M. Rubinstein and R. H. Colby, Polymer Physics (Oxford University Press, New York, 2003). S. Plimpton, J. Comp. Phys. 117 , 119 (1995). V. Katritch, W. K. Olson, A. Vologodskii, J. Dubochet, and A. Stasiak, Phys. Rev. E 61 , 5545 (2000). K. C. Millett, A. Dobay, and A. Stasiak, Macromolecules 38 , 601 (2005). P. Virnau, Y. Kantor, and M. Kardar, J. Am. Chem. Soc. 127 , 15102 (2005). F. Eisenhaber, P. Lijnzaad, P. Argos, C. Sander, and M. Scharf, J. Comp. Chem. 16 , 273 (1995). B. Hess, C. Kutzner, D. van der Spoel, and E. Lindahl, J. Chem. Theory Comput. 4, 435 (2008). N. Uchida, G. S. Grest, and R. Everaers, J. Chem. Phys. 128 , 044902 (2008). B. H. Zimm and W. H. Stockmayer, J. Chem. Phys. 17 , 1301 (1949). D. C. Morse, Macromolecules 31 , 7030 (1998). M. Bishop and J. P. J. Michels, J. Chem. Phys. 84 , 444 (1986). J. Rudnick and G. Gaspari, Science 237 , 384 (1987). K. Alim and E. Frey, Phys. Rev. Lett. 99 , 198102 (2007). C. Tzoumanekas, F. Lahmar, B. Rousseau, and D. N. Theodorou, Macromolecules 42 , 7474 (2009). T. Vettorel, A. Y. Grosberg, and K. Kremer, Phys. Biol. 6, 025013 (2009). P.-Y. Lai, Phys. Rev. E 66 , 021805 (2002). X. R. Bao, H. J. Lee, and S. R. Quake, Phys. Rev. Lett. 91 , 265506 (2003). A. Vologodskii, Biophys. J. 90 , 1594 (2006). O. Gonzalez, A. B. A. Graf, and J. H. Maddocks, J. Fluid Mech. 519 , 133 (2004). C. Weber, A. Stasiak, P. D. L. Rios, and G. Dietler, Biophys. J. 90 , 3100 (2006). 17
SUPPORTING FIG. 1. Three-dimensional histogram comparing top-down and bottom-up search schemes of the length, Lknot
of the knotted portion of the chain (see Methods, Sec. IID): the histogram reports the number of times the pair of values given by (Lbottom −up knot , L top −down knot
) occur in the course of the simulation. It is evident that the two algorithms measure the same knotted portion in the majority of cases. The case shown corresponds to ring polymers at monomer concentration ρ = 0 .4. Analogous results are found at the other concentrations. 10 0
10 1
10 2
10 3
10 4
10 5
10 1 10 2 10 3 10 4 10 5 10 6
ρσ 3 = 0.2
t [ τMD ]
δ rCM 2(t) (+) [ σ2 ]
Unknot
31
∼t |
8608 | https://engineering.purdue.edu/wcchew/ece255f17/ECE%20255%20f17%20latex%20pdf%20files/ece255LectureOct12.pdf | ECE 255 12 October 2017 In this lecture, the basic configurations of BJT amplifiers will be studied.
Previously, it has been shown that with the transistor DC biased at the ap-propriate point, linear relations can be derived between the small voltage and current signals. With these linear relations, the principles of linear systems can be applied to solve for the node voltages and branch currents. Moreover, they can be easily handled by commercial software such as SPICE for highly com-plex circuits as long as they are linear. These large complex circuit problems can be cast into solving a set of linear algebraic equations which can be solved efficiently by computers.
The take home message here is that linear problems are a lot simpler to solve compared to nonlinear problems. Hence, nonlinear problems are linearized with the small signal approximations before they are solved.
1 The Three Basic Configurations The three basic configurations of a BJT are (a) common emitter (CE), (b) common base (CB), (c) common collector (CC) or emitter follower. These basic configurations are shown in Figure 1.
The replacement of the basic configurations with equivalent circuit models and the small signal approximations convert the original nonlinear problems into linear ones, greatly simplifying their analysis.
2 Characterizing Amplifiers An amplifier can be denoted by a functional block as expressed in Figure 2(a), where a triangle block encapsulates the details of the small-signal equivalent circuit model as shown in Figure 2(b). The equivalent circuit model indicates that the amplifier has a finite internal impedance, Rin. Moreover, it has a finite output resistance Ro, which can be found by the test-current method, as shown in Figure Figure 2(c).
The internal input resistance can be found from Rin = vi ii (2.1) Printed on November 2, 2017 at 15 : 49: W.C. Chew and Z.H. Chen.
1 (a) (b) (c) Figure 1: Basic configurations of a transistor amplifier: (a) common emitter (CE), (b) common base (CB), and (c) common collector (CC) (Courtesy of Sedra and Smith).
The amplifier is also defined with an open-circuit voltage gain Avo defined as Avo = vo vi RL=∞ (2.2) The output resistance can be measured by setting vi = 0 using the test-current method, and it is given by Ro = vx ix (2.3) With the load resistor RL connected, the actual output voltage in Figure 2(b) is vo = RL RL + Ro Avovi (2.4) Hence, the actual voltage gain of the amplifier proper also called the ter-minal voltage gain, when a finite load RL is added, is Av given by Av = vo vi = Avo RL RL + Ro (2.5) and the overall voltage gain of the entire circuit is given by Gv = vo vsig = vo vi vi vsig (2.6) Using (2.5) and the fact that vi vsig = Rin Rin + Rsig give Gv = Rin Rin + Rsig RL RL + Ro Avo (2.7) 2 Figure 2: (a) The characterization of an amplifier, with a functional block rep-resenting the small-signal model, including Rsig for the source and RL for the load. (b) The equivalent small signal model, and (c) the definition of the output resistance Ro using the test-current method (Courtesy of Sedra and Smith).
3 Figure 3: A common emitter (CE) amplifier (a) and its equivalent circuit hybrid-π model (b) (Courtesy of Sedra and Smith).
3 Common Emitter Amplifier This is the most popular amplifier design, and by cascading a number of them, the aggregate gain of the amplifier circuit can be greatly increased.
3.1 Characteristic Parameters of the CE Amplifier Figure 3 shows the BJT CE amplifier and its small-signal equivalent circuit model. It is seen, after using the voltage-divider formula, that vi = rπ rπ + Rsig vsig, and vo = −gmviRC (3.1) where gm, the transconductance, is given by IC/VT , and ic = gmvi have been used. Then Avo = vo vi = −gmRC (3.2) With the load resistance RL, then the voltage gain proper (terminal voltage gain) Av = −gm(RC||RL) (3.3) and the overall voltage gain Gv = vo vsig = vi vsig vo vi = − rπ rπ + Rsig gm(RC||RL) (3.4) 4 When the transconductance gm of a transistor is large, this can be a large number. In the above vi = vbe, the base-emitter voltage. This voltage is reduced when Rsig is large, decreasing Gv.
3.2 Final Remarks on CE Amplifier 1. The CE amplifiers gives high input impedance (draws little current), and a moderately high output resistance (easier to match), and high voltage gain (a desirable feature of an amplifier).
2. The input resistance of the CE amplifier is Rin = rπ = β/gm = βVT /IC is inversely proportional to IC. Hence, Rin can be increased by decreasing IC, but that will lower gm and reduce the gain of the amplifier, a trade-off.
3. Reducing RC reduces the output resistance of a CE amplifier, but unfortu-nately, the voltage gain is also reduced. Alternate design can be employed to reduce the output resistance (to be discussed later).
4. A CE amplifier suffers from poor high frequency performance, as most transistor amplifiers do.
4 Common Emitter Amplifier with an Emitter Resistance It is seen that the input resistance of the circuit is Rin = vi ib , with ib = ie β + 1 (4.1) With ie = vi/(re + Re), then Rin = (β + 1)(re + Re) (4.2) This is known as the resistance-reflection rule, because the input resistance is amplified by the factor β + 1, which can be large. Therefore, including an emitter resistance Re can greatly increase Rin.
More specifically, Rin(with Re) Rin(without Re) = re + Re re = 1 + Re re ≈1 + gmRe (4.3) after using the fact that re = α/gm ≈1/gm. Hence the input resistance Rin can be greatly increased with Re.
The open-circuit voltage gain is given by Avo = vo/vi. Since vo = −icRC = −αieRC (4.4) 5 Figure 4: A CE amplifier with an emitter resistance: (top) detail circuit, (bot-tom) equivalent circuit hybrid-π model (Courtesy of Sedra and Smith).
6 and that ie = vi/(re + Re), then Avo = −α RC re + Re (4.5) Using gm = α/re, the above can be rewritten as Avo = −α re RC 1 + RC re !
= − gmRC 1 + Re/re ≈− gmRC 1 + gmRe (4.6) where gm ≈1/re has been used in the last approximation. Hence, including Re reduces the voltage gain, which also reduces Rin, a tradeofffor designers.
The output resistance Ro = RC (4.7) With a load resistor RL connected to the amplifier output, then the voltage gain proper Av = Avo RL RL + Ro = −α RC re + Re RL RL + Ro = −αRC||RL re + Re (4.8) which is similar in spirit to (4.5) where RC is now replaced by RC||RL.
The overall voltage gain Gv is Gv = vi vsig Av = Rin Rin + Rsig −αRC||RL re + Re = −β RC||RL Rsig + (β + 1)(re + Re) (4.9) where (4.2) for Rin and that α = β/(β + 1) have been used. Notice that the voltage gain is lowered with the presence of Re.
Moreover, the presence of Re reduces nonlinearity, because vπ, which is the base-emitter voltage vbe, is reduced, namely, vπ vi = re re + Re ≈ 1 1 + gmRe (4.10) where gm ≈1/re has been used.
Hence, it seen that vπ compared to vi is reduced by a substantial factor, improving the linearity of the amplifier. This is because iC = ICevbe/VT , a nonlinear relationship that has been linearized with a small signal approximation. The smaller vbe/Vt is, the better the small-signal or linearization approximation.
4.1 Summary of the CE Amplifier with Emitter Resis-tance 1. The input resistance Rin is increased by a factor of 1 + gmRe as seen in (4.3).
2. The base to collector voltage gain, Avo, is reduced by a factor of 1+gmRe as seen in (4.6).
7 3. For the same nonlinear distortion, the input signal can be increased by a factor of 1 + gmRe compared to without Re.
4. The voltage gain is less dependent on β.
5. As shall be shown later, the high-frequency response of this design is improved.
In general, the addition of the emitter resistance Re gives rise to a “negative” feedback factor 1 + gmRe that reduces voltage gain, but improves linearity, and high-frequency response. Because of the negative-feedback action of Re, it is also called the emitter degenerate resistance.
5 Common-Base (CB) Amplifier The common-base amplifier is shown in Figure 5. The input resistance Rin is given by Rin = re = α gm ≈1/gm (5.1) The voltage gain is Avo = vo vi = αieRC reie = α re RC = gmRC (5.2) The output resistance here is Ro = RC. When a load resistance RL is connected, then the voltage gain proper (terminal voltage gain) Av = vo vi = gmRC||RL (5.3) and the overall voltage gain is Gv = vo vsig = vi vsig vo vi = re Rsig + re α re RC||RL = α RC||RL Rsig + re ≈RC||RL Rsig + re (5.4) 5.1 Summary of the CB Amplifier 1. The CB amplifier has a low input resistance ≈1/gm. This is undesirable as it will draw large current when driven by a voltage input.
2. The voltage gain of the CB amplifier can be made similar in magnitude to that of the CE amplifier if RC||RL can be made large compared to Rsig + Re.
3. The output resistance can be made large since Ro = RC.
4. The CB amplifier has good high frequency performance as shall be shown later.
8 Figure 5: A common-base (CB) amplifier (a) with biasing detail omitted, (b) equivalent circuit using the T model for the BJT (Courtesy of Sedra and Smith).
6 The Emitter Follower The emitter follower can be used as a voltage buffer. It is also the common-collector amplifier: one of the three basic configurations of transistor amplifiers.
Figure 6 shows the use of a voltage buffer concept in a circuit with an amplifier of unit gain.
Figure 7 shows the circuit diagram for the the common-collector amplifier.
The input impedance is given by Rin = vi ib (6.1) Using that ib = ie/(β + 1) and that ie = vi/(re + RL), one gets Rin = (β + 1)(re + RL) (6.2) which also agrees with the resistance-reflection rule. It is this reflection rule that makes the amplifier or the emitter follower to have a large input impedance, and hence, its usefulness as a buffer.
The voltage gain proper (terminal voltage gain) Av is Av = vo vi = RL RL + re ∼1, RL →∞ (6.3) (When an amplifier is loaded with an output resistor RL, the above voltage gain is usually called the voltage gain proper or the terminal voltage gain.) Hence, it works well as an emitter follower since vo = vi.
To find Ro, one finds the Th´ evenin equivalence of the amplifier when it is driven by an input voltage source vi. Then Ro is just the Th´ evenin resistor, 9 Figure 6: The schematics of a unit gain voltage buffer amplifier (Courtesy of Sedra and Smith).
10 which in this case is re. Hence, Ro = re (6.4) Since re is usually small, this amplifier has a low output impedance.
To find the overall voltage gain Gv, first one finds that vi vsig = Rin Rin + Rsig = (β + 1)(re + RL) (β + 1)(re + RL) + Rsig (6.5) Then the overall voltage gain is Gv = vo vsig = vi vsig vo vi = vi vsig Av = (β + 1)RL (β + 1)(re + RL) + Rsig ∼1, RL →∞ (6.6) This clearly indicates that Gv, the overall voltage gain is less than one but can approach one for large RL.
One can think of the emitter follower just as a voltage divider. Two simpler versions of the equivalent voltage divider are presented in Figure 8.
6.1 Th´ evenin Equivalence of the Emitter Follower The Th´ evenin equivalence of the emitter follower can be found. Let us first assume that we do not know what the voltage source is. Since this is a linear problem in small signals in Figure 9(a), the voltage source is indicated by Gvovsig with Rout yet to be determined as indicated in the figure. By setting RL to be infinitely large, making it an open circuit, and using the equivalent circuit as shown in Figure 8(b), it is seen that Gvo = 1 and that Rout = re + Rsig/(β + 1) (6.7) as shown in Figures 9(b) and Figures 9(c). The more detail circuit is shown in Figure 9(d).
11 Figure 7: A common-collector (CC) amplifier or emitter follower: (a) with biasing detail omitted, (b) equivalent circuit using the T model for the BJT (Courtesy of Sedra and Smith).
12 Figure 8: Two ways of representing the emitter follower as a voltage divider (Courtesy of Sedra and Smith).
Figure 9: Way to find the Th´ evenin equivalent of the small-signal model of the emitter follower as shown in (a) and (b) to find the open circuit voltage, (c) to find the equivalent Th´ evenin resistance, and (d) the full circuit model for small signals. In (d), notice that the impedances are different looking to the right or to the left of the transistor (Courtesy of Sedra and Smith).
13 6.2 Summary Table The summary table, Table 7.5 is from Sedra and Smith. Notice that Av, the voltage gain proper (also called terminal voltage gain), becomes Avo, the open-circuit voltage gain when RL →∞. The CB amplifier has a low input impedance, while the emitter follower has a low output impedance.
14 15 16 Appendix A Maximum Power Transfer Theorem Let us assume that a signal source can be modeled with a Th´ evenin equivalence.
So when the signal source is connected to a load, it can be modeled by the circuit shown in Figure 10.
The current flowing in the circuit is I = VS/(RS + RL) (A.1) The power transferred to the load is given by PL = I2RL = V 2 S RL (RS + RL)2 (A.2) RL in the above can be varied so that PL is maximized. Without loss of gener-ality, one can assume that VS = 1, and find that dPL dRL = 1 (RS + RL)2 −2 RL (RS + RL)3 = 0 (A.3) at the maximum point. The above evaluates to RL = RS. In other words, the load impedance should be matched to the source impedance.
The theorem can be generalized to time-harmonic sources and complex impedances. In this case, the matching condition is that ZL = Z∗ S (A.4) where ∗implies the complex conjugation, and ZL is the complex load impedance, and ZS is the complex source impedance.
17 Figure 10: (Top) Simple circuit where a load RL is connected to the Th´ evenin equivalence of the source. (Bottom) The plot of power transferred to the load PL versus the change of RL (Courtesy of Sedra and Smith).
18 |
8609 | https://fiveable.me/key-terms/principles-physics-i/centripetal-force | Centripetal force - (Principles of Physics I) - Vocab, Definition, Explanations | Fiveable | Fiveable
new!Printable guides for educators Printable guides for educators. Bring Fiveable to your classroom
ap study content toolsprintablespricing
my subjectsupgrade
All Key Terms
Principles of Physics I
Centripetal force
🍏principles of physics i review
key term - Centripetal force
Citation:
MLA
Definition
Centripetal force is the net force that acts on an object moving in a circular path, directed toward the center of the circle. This force is essential for maintaining uniform circular motion and plays a key role in the analysis of rotating systems and the dynamics of objects in orbit. Understanding centripetal force helps to explain various phenomena, such as the forces experienced by satellites and planets, and how they maintain their paths around larger celestial bodies.
5 Must Know Facts For Your Next Test
Centripetal force can arise from various sources, including tension, gravity, and friction, depending on the situation.
The formula for centripetal force is given by ( F_c = \frac{mv^2}{r} ), where ( m ) is mass, ( v ) is the tangential velocity, and ( r ) is the radius of the circular path.
In uniform circular motion, the speed of the object remains constant while its direction changes continuously due to centripetal force.
Centripetal acceleration, which is directed toward the center of the circular path, is calculated using the formula ( a_c = \frac{v^2}{r} ), showing how acceleration relates to both speed and radius.
When analyzing orbital motion, centripetal force is provided by gravitational attraction between two bodies, ensuring that planets and satellites maintain their orbits.
Review Questions
How does centripetal force contribute to uniform circular motion, and what happens if this force is removed?
Centripetal force is crucial for keeping an object in uniform circular motion by continuously pulling it toward the center of its circular path. If this force is removed, such as when a car goes around a turn too fast and loses traction, the object will no longer follow a curved path and will instead move off in a straight line according to Newton's first law of motion. This demonstrates that without centripetal force, an object's inertia takes over and it travels in a straight line.
Discuss how gravitational force acts as centripetal force in the context of orbital motion.
In orbital motion, gravitational force between two bodies acts as the centripetal force required to keep one body in a stable orbit around another. For example, Earth's gravity pulls satellites towards its center, providing the necessary centripetal force that keeps them in orbit. The balance between gravitational pull and the satellite's inertia creates a stable orbit, showing how these forces interact to maintain celestial mechanics.
Evaluate how changes in mass or radius affect centripetal force required for an object to maintain circular motion.
To maintain circular motion, both mass and radius influence the required centripetal force significantly. If the mass of an object increases while maintaining constant speed and radius, the centripetal force must also increase proportionally to keep it on its circular path. Conversely, if the radius increases while mass and speed remain constant, the required centripetal force decreases because it's easier for the object to travel along a larger circle. This interplay highlights how forces change dynamically based on an object's characteristics and path.
Related terms
centrifugal force:
An apparent force that acts outward on an object moving in a circular path, perceived in a rotating reference frame.
angular velocity:
A measure of how quickly an object rotates or revolves around a center point, typically expressed in radians per second.
gravitational force:
The attractive force between two masses, which governs the motion of planets, moons, and other celestial bodies within gravitational fields.
"Centripetal force" also found in:
Subjects (3)
AP Human Geography
AP Physics 2
Intro to Astronomy
Study Content & Tools
Study GuidesPractice QuestionsGlossaryScore Calculators
Company
Get $$ for referralsPricingTestimonialsFAQsEmail us
Resources
AP ClassesAP Classroom
every AP exam is fiveable
history
🌎 ap world history🇺🇸 ap us history🇪🇺 ap european history
social science
✊🏿 ap african american studies🗳️ ap comparative government🚜 ap human geography💶 ap macroeconomics🤑 ap microeconomics🧠 ap psychology👩🏾⚖️ ap us government
english & capstone
✍🏽 ap english language📚 ap english literature🔍 ap research💬 ap seminar
arts
🎨 ap art & design🖼️ ap art history🎵 ap music theory
science
🧬 ap biology🧪 ap chemistry♻️ ap environmental science🎡 ap physics 1🧲 ap physics 2💡 ap physics c: e&m⚙️ ap physics c: mechanics
math & computer science
🧮 ap calculus ab♾️ ap calculus bc📊 ap statistics💻 ap computer science a⌨️ ap computer science p
world languages
🇨🇳 ap chinese🇫🇷 ap french🇩🇪 ap german🇮🇹 ap italian🇯🇵 ap japanese🏛️ ap latin🇪🇸 ap spanish language💃🏽 ap spanish literature
go beyond AP
high school exams
✏️ PSAT🎓 Digital SAT🎒 ACT
honors classes
🍬 honors algebra II🐇 honors biology👩🏽🔬 honors chemistry💲 honors economics⚾️ honors physics📏 honors pre-calculus📊 honors statistics🗳️ honors us government🇺🇸 honors us history🌎 honors world history
college classes
👩🏽🎤 arts👔 business🎤 communications🏗️ engineering📓 humanities➗ math🧑🏽🔬 science💶 social science
RefundsTermsPrivacyCCPA
© 2025 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
every AP exam is fiveable
Study Content & Tools
Study GuidesPractice QuestionsGlossaryScore Calculators
Company
Get $$ for referralsPricingTestimonialsFAQsEmail us
Resources
AP ClassesAP Classroom
history
🌎 ap world history🇺🇸 ap us history🇪🇺 ap european history
social science
✊🏿 ap african american studies🗳️ ap comparative government🚜 ap human geography💶 ap macroeconomics🤑 ap microeconomics🧠 ap psychology👩🏾⚖️ ap us government
english & capstone
✍🏽 ap english language📚 ap english literature🔍 ap research💬 ap seminar
arts
🎨 ap art & design🖼️ ap art history🎵 ap music theory
science
🧬 ap biology🧪 ap chemistry♻️ ap environmental science🎡 ap physics 1🧲 ap physics 2💡 ap physics c: e&m⚙️ ap physics c: mechanics
math & computer science
🧮 ap calculus ab♾️ ap calculus bc📊 ap statistics💻 ap computer science a⌨️ ap computer science p
world languages
🇨🇳 ap chinese🇫🇷 ap french🇩🇪 ap german🇮🇹 ap italian🇯🇵 ap japanese🏛️ ap latin🇪🇸 ap spanish language💃🏽 ap spanish literature
go beyond AP
high school exams
✏️ PSAT🎓 Digital SAT🎒 ACT
honors classes
🍬 honors algebra II🐇 honors biology👩🏽🔬 honors chemistry💲 honors economics⚾️ honors physics📏 honors pre-calculus📊 honors statistics🗳️ honors us government🇺🇸 honors us history🌎 honors world history
college classes
👩🏽🎤 arts👔 business🎤 communications🏗️ engineering📓 humanities➗ math🧑🏽🔬 science💶 social science
RefundsTermsPrivacyCCPA
© 2025 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
Study Content & Tools
Study GuidesPractice QuestionsGlossaryScore Calculators
Company
Get $$ for referralsPricingTestimonialsFAQsEmail us
Resources
AP ClassesAP Classroom
every AP exam is fiveable
history
🌎 ap world history🇺🇸 ap us history🇪🇺 ap european history
social science
✊🏿 ap african american studies🗳️ ap comparative government🚜 ap human geography💶 ap macroeconomics🤑 ap microeconomics🧠 ap psychology👩🏾⚖️ ap us government
english & capstone
✍🏽 ap english language📚 ap english literature🔍 ap research💬 ap seminar
arts
🎨 ap art & design🖼️ ap art history🎵 ap music theory
science
🧬 ap biology🧪 ap chemistry♻️ ap environmental science🎡 ap physics 1🧲 ap physics 2💡 ap physics c: e&m⚙️ ap physics c: mechanics
math & computer science
🧮 ap calculus ab♾️ ap calculus bc📊 ap statistics💻 ap computer science a⌨️ ap computer science p
world languages
🇨🇳 ap chinese🇫🇷 ap french🇩🇪 ap german🇮🇹 ap italian🇯🇵 ap japanese🏛️ ap latin🇪🇸 ap spanish language💃🏽 ap spanish literature
go beyond AP
high school exams
✏️ PSAT🎓 Digital SAT🎒 ACT
honors classes
🍬 honors algebra II🐇 honors biology👩🏽🔬 honors chemistry💲 honors economics⚾️ honors physics📏 honors pre-calculus📊 honors statistics🗳️ honors us government🇺🇸 honors us history🌎 honors world history
college classes
👩🏽🎤 arts👔 business🎤 communications🏗️ engineering📓 humanities➗ math🧑🏽🔬 science💶 social science
RefundsTermsPrivacyCCPA
© 2025 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
every AP exam is fiveable
Study Content & Tools
Study GuidesPractice QuestionsGlossaryScore Calculators
Company
Get $$ for referralsPricingTestimonialsFAQsEmail us
Resources
AP ClassesAP Classroom
history
🌎 ap world history🇺🇸 ap us history🇪🇺 ap european history
social science
✊🏿 ap african american studies🗳️ ap comparative government🚜 ap human geography💶 ap macroeconomics🤑 ap microeconomics🧠 ap psychology👩🏾⚖️ ap us government
english & capstone
✍🏽 ap english language📚 ap english literature🔍 ap research💬 ap seminar
arts
🎨 ap art & design🖼️ ap art history🎵 ap music theory
science
🧬 ap biology🧪 ap chemistry♻️ ap environmental science🎡 ap physics 1🧲 ap physics 2💡 ap physics c: e&m⚙️ ap physics c: mechanics
math & computer science
🧮 ap calculus ab♾️ ap calculus bc📊 ap statistics💻 ap computer science a⌨️ ap computer science p
world languages
🇨🇳 ap chinese🇫🇷 ap french🇩🇪 ap german🇮🇹 ap italian🇯🇵 ap japanese🏛️ ap latin🇪🇸 ap spanish language💃🏽 ap spanish literature
go beyond AP
high school exams
✏️ PSAT🎓 Digital SAT🎒 ACT
honors classes
🍬 honors algebra II🐇 honors biology👩🏽🔬 honors chemistry💲 honors economics⚾️ honors physics📏 honors pre-calculus📊 honors statistics🗳️ honors us government🇺🇸 honors us history🌎 honors world history
college classes
👩🏽🎤 arts👔 business🎤 communications🏗️ engineering📓 humanities➗ math🧑🏽🔬 science💶 social science
RefundsTermsPrivacyCCPA
© 2025 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
0 |
8610 | https://www.youtube.com/watch?v=aJghRnDi-VY | Why can't we group the terms in 1-1+1-1+... ?
Mu Prime Math
48200 subscribers
170 likes
Description
3239 views
Posted: 5 Feb 2022
We can group the terms in the divergent infinite sum 1-1+1-1+... to get a value of zero or a value of one, even though the series doesn't converge. This video gives an explanation of why grouping terms in Grandi's series doesn't give the correct answer.
Calculus Problems playlist:
0:00 Why the sum diverges
7:33 Why grouping fails
12:57 When grouping is valid
Subscribe to see more new math videos!
Music: C418 - Pr Department
28 comments
Transcript:
Why the sum diverges so we are going to talk about the infinite sum 1 minus 1 plus 1 minus 1 plus 1 and so on to infinity now one way that we can approach finding the value of this sum is grouping the terms so i can look at this 1 minus 1 right here and put that in parentheses and then after that we'll have another plus 1 minus 1. i can put that in parentheses as well and after that we would have a plus one minus one again so i can put that group in parentheses and we can keep doing that to infinity now one minus one is just equal to zero so this is a zero this is also one minus one is zero this is a zero and so on so we're just adding up infinitely many zeros and that means that the answer right here is just zero now there's another way that we could group the terms what i'm going to do is put the first one here by itself so we'll have a one out here and after this one there's a minus one plus one so we could group those two together so here's the minus one plus one and after that after that plus one there's another minus one plus one and after that there's another minus one plus one so we can keep doing that to infinity so this one stays as just a one but then after that minus one plus one is just a zero and then plus zero and then plus zero and so on to infinity so now we have one plus a bunch of zeros and that's going to leave us with just the value of one but wait a minute the first time we got zero the second time we get one zero is not equal to one so we just tried to evaluate the sum using grouping and we got two different answers so it looks like something is going a little bit wrong here and in this video i want to talk about why this grouping method for finding the sum doesn't actually give us the right answer for the series 1 minus 1 plus 1 minus 1 and so on let's start from the beginning our goal is to find the value of 1 minus 1 plus 1 minus 1 and so on to infinity but we don't really know how to add up infinitely many things we do know how to add up finitely many things so maybe we should start by taking finite sums and see if that can tell us anything about the infinite sum so there are infinitely many terms in this series that's what this dot dot dot is saying is that there are more and more terms out to infinity but maybe we can start by adding up the first few terms at a time if we look at just the first term that's going to be a one if we look at the first two terms we have one minus one which is just zero if we look at the first three terms we have one minus one plus one which is going to be a one and after that one minus one plus one minus one is zero and we could keep doing this on to infinity now for each of these sums we've only added up the first few terms of the series we've only looked at a part of the infinite sum so these values right here are called partial sums they're what we get when we add up the first few terms of a series without adding up all the infinitely many terms and we denote the partial sum by sn where the number down here in the subscript is the number of terms that we've added up so in this first example we only looked at the first term so that's s1 here we added up the first two terms so that's s2 here's the first three terms so that's s3 and the next one is s4 and so on now that we have these partial sums we can start to talk about the infinite sum that we're really trying to find if this infinite sum has a value we would kind of expect that as we add up more and more terms in the sequence the partial sums will get closer and closer to the actual value of the series so we should be able to get really close to the value of this infinite sum as long as we take a big enough partial sum so maybe it takes us until s1 billion to get really really close to that infinite sum but eventually we should get there now we use that idea that the partial sums should get really close to the infinite sum as our definition of the infinite sum so the way we define the value of this infinite sum up here is we say this infinite sum equals whatever the partial sums approach as n approaches infinity so if we want to use fancy math language for that we can say that the value of this sum is the limit as n goes to infinity of s n so this is just saying as we make this subscript really really big what value do the partial sums approach and that's our definition of the infinite sum so our original problem was about finding a sum of infinitely many numbers but now that we have this definition of the infinite sum we can just look at the sequence of partial sums s1 s2 s3 s4 those are all just numbers and we have to figure out what those numbers approach as n approaches infinity so let's take a look at those partial sums if we look at s1 we get the value of one s2 equals zero s3 equals one s4 equals zero and if we keep calculating the partial sums we continue the pattern where we have 1 0 1 0 1 0 and so on to infinity so the partial sums are always going to alternate back and forth between a value of 1 and a value of 0. so if we have this alternating sequence that's just one zero one zero and so on does this sequence approach any value maybe the sequence approaches one there are infinitely many ones in the sequence well the problem with that is there are also infinitely many zeros and zero is always a fixed distance away from one it's not getting closer at all so this sequence isn't going to approach one and it doesn't approach zero either because there are infinitely many ones and those are also a fixed distance away from zero so you might think that this sequence of partial sums doesn't approach any value at all and you would be correct this sequence doesn't approach any number and as a result the limit as n approaches infinity of that sequence of partial sums does not exist and since that's our definition of the infinite sum this infinite sum up here does not exist either Why grouping fails so now we know that the infinite sum up here doesn't have a value because the limit of the sequence of partial sums does not exist now let's go back and look at the groupings that we talked about at the beginning of the video if we group these pairs of terms 1 minus 1 or negative 1 plus 1 this first sum gives us the answer of 0 and the second one gives us the answer of one so what goes wrong when we do the grouping like this well both of these are also infinite sums so we can use the same method that we did for finding the infinite sum up here we can look at the sequence of partial sums and think about what that sequence approaches as n approaches infinity so let's look at the partial sums of this first series the first term in the series is this 1 minus 1 in parentheses so the first partial sum is going to be 1 minus 1 which we know is 0. but notice that this 1 minus 1 gives us the first two terms of the original sum up here so if we look at our original sequence of partial sums from this series this partial sum is s2 it's that one minus one now the second partial sum if we add up two terms in this grouping that's going to give us two groups of one minus one so we'll have one minus one plus one minus one and that's zero again but notice that now from the original series we have four terms so this is going to be s4 down here if we look at the third partial sum of this series we're going to have three groups of two and that's going to give us six terms so we'll have s6 and that's another zero so as we do this look at the sequence of partial sums over here the index down here this subscript is going to go 2 4 6 8 10 it's only going to be the even index partial sums we're never going to get s1 because s1 is when we just look at the first one in the sum but that one is inside of a group so we always consider it with the minus one similarly we're never going to get s3 because s3 happens when we cut off at this plus one but when we group the terms that plus one always has to be with the minus one so we're going to skip every single odd index partial sum and that means we're going to skip every time the partial sum equals 1. what happens here is instead of looking at the sequence of partial sums for this original series we're looking at a subsequence which means we just pick out some of the terms in the sequence and we ignore the rest of them but of course we can have a subsequence that converges to zero without the entire sequence equaling zero because if we look over here at just the even indexes of course those are all going to be zero and so if we look at this subsequence that's just the even index partial sums that subsequence does approach zero which means that this series right here the one minus one in parentheses plus one minus one in parentheses and so on this series does equal zero no mistakes there the problem is that this subsequence doesn't have the same value as the original sequence because the original sequence doesn't converge the same thing is happening if we look at this second grouping let's look at the partial sums of this grouping down here the first term here is just the one by itself so that's going to be s1 because that's the first term in this infinite series up here now if we look at the second partial sum in this series we have the one and then we also have this second grouping which is minus one plus one so that's going to give us a value of one but notice that we've added up the first three terms of the original series so this is s3 and if we add up the first three terms of this series down here we're gonna get minus one plus one and then another minus one plus one so in total we're adding up five of the terms from the original series so this is s5 and if we look at the second grouping the partial sums we're going to get are s1 s3 s5 s7 and so on so we're only going to get the odd indexes we're not going to get any of the even index partial sums so just like this sum equals zero the second sum equals 1 that's completely correct but this sum is not the same as the original sum because we can have a sequence that doesn't approach anything and still have a subsequence that approaches a specific value When grouping is valid now after this explanation of why the grouping doesn't work in the case of this sum you might start to lose faith in grouping you might start to think well this must not be a valid method at all but it turns out there is one case where we can use grouping and it will give us the right answer and that situation is if the sum exists so we can use some fancy math to prove that if the sum exists if the original sum does converge to some value then every grouping will give the correct answer and the reason for this is that if we have a sequence that converges then every subsequence has to converge to that same value so if you know that a sum converges if you already know that it exists and your goal is just to find what it equals you're totally fine to use grouping and you will get the right answer but before you use grouping you have to make sure that this sum converges because if you have a sequence that doesn't converge it can have subsequences that do converge but that doesn't give you the value of the original infinite sum so that's what's going on with the infinite sum 1 minus 1 plus 1 minus 1 and so on if we look at the sequence of partial sums we can see that it doesn't approach any specific value but if we pick a subsequence of just the odd terms then we get a subsequence that approaches one and if we look at just the even terms we get a subsequence that approaches zero so this is a sequence that doesn't exist but it has multiple convergent subsequences and that's why when we want to group terms for an infinite series first we have to know that that sum actually exists [Music] |
8611 | https://www.journals.uchicago.edu/doi/full/10.1086/726012 | Revisiting the Role of Genetic Variation in Adaptation | The American Naturalist: Vol 202, No 4
Skip to main content
Search
Search
This journal
This journal
Anywhere
Quick Search in Journals
Search Search
Quick Search anywhere
Search Search
Advanced Search
Log In | RegisterLog in | Register
0
Skip main navigation
menuDrawerCloseText menuDrawerOpenTextHome
Subscriber/member
Institutions
Member subscriptions
Member renewals
Recommend to your library
Purchase back issues
Browse issues
All issues
Member access
Forthcoming
Ahead of Print
Just Accepted
Contributors
Submit manuscript or review
Instructions for authors
Why submit to AN?
Editorial policy
Authors' rights
Open access at Chicago
Obtaining permissions
Resources for authors
About
About AN
Editorial Board
Contact the editorial office
Abstracting and indexing
Advertise in AN
ASN
About the ASN
Join ASN
Manage ASN membership
Member directory
Visit the ASN website
Donate to ASN
HomeThe American NaturalistVolume 202, Number 4
Previous articleNext article
Open Access Symposium
Revisiting the Role of Genetic Variation in Adaptation
Deepa Agashe,
Mrudula Sane,and
Shivansh Singhal
Deepa Agashe
National Centre for Biological Sciences, Gandhi Krishi Vignana Kendra (GKVK) Campus, Bellary Road, Bangalore 560065, India
†Corresponding author; email: dagashe@ncbs.res.in.
Search for more articles by this author
,
Mrudula Sane
National Centre for Biological Sciences, Gandhi Krishi Vignana Kendra (GKVK) Campus, Bellary Road, Bangalore 560065, India
Search for more articles by this author
,and
Shivansh Singhal
National Centre for Biological Sciences, Gandhi Krishi Vignana Kendra (GKVK) Campus, Bellary Road, Bangalore 560065, India
Search for more articles by this author
PDF
PDF PLUS
Abstract
Full Text
Add to favorites
Download Citation
Track Citations
Permissions
Reprints
Share on
Facebook
X
LinkedIn
Reddit
Bluesky
Email
Print
Sections
Abstract
Introduction
Part I: Introduction of Genetic Variation—Mutation Rate and Bias
Part II: The Phenotypic and Fitness Effects of Genetic Variation
Part III: The Fate of Genetic Variation in Natural Populations
Summary
Data and Code Availability
Notes
Literature Cited
More
Details
Figures
References
Cited by
Abstract
Evolutionary biologists have thought about the role of genetic variation during adaptation for a very long time—before we understood the organization of the genetic code, the provenance of genetic variation, and how such variation influenced the phenotypes on which natural selection acts. Half a century after the discovery of the structure of DNA and the unraveling of the genetic code, we have a rich understanding of these problems and the means to both delve deeper and widen our perspective across organisms and natural populations. The 2022 Vice Presidential Symposium of the American Society of Naturalists highlighted examples of recent insights into the role of genetic variation in adaptive processes, which are compiled in this special section. The work was conducted in different parts of the world, included theoretical and empirical studies with diverse organisms, and addressed distinct aspects of how genetic variation influences adaptation. In our introductory article to the special section, we discuss some important recent insights about the generation and maintenance of genetic variation, its impacts on phenotype and fitness, its fate in natural populations, and its role in driving adaptation. By placing the special section articles in the broader context of recent developments, we hope that this overview will also serve as a useful introduction to the field.
Introduction
Naturalists have been fascinated with phenotypic variation for centuries. Naturally, how this variation arises has been a central focus of biological investigation (Tilman 2000). The source of variation became apparent only in the twentieth century with the discovery of DNA as the genetic material and its structure, and with subsequent work linking variation in the information encoded in DNA to variation in phenotypes and traits (Olby 2003). What followed was an era of rapid progress in understanding the molecular processes involved in replicating, repairing, and decoding genetic information, and we now understand in detail the mechanisms that introduce genetic variation. Advances in diverse fields such as molecular biology, cell biology, ecology, and genetics continue to drive new discoveries about the nature, extent, and impact of genetic variation in the context of adaptation (Sella and Barton 2015; Bomblies and Peichel 2022). This has allowed our thinking about genetic variation to become more mechanistically informed and, at the same time, more easily connected to larger-scale repercussions. In the past few decades, we have thus made remarkable progress in determining the phenotypic as well as fitness consequences of genetic variation, allowing us to predict how such variation may drive evolutionary change (Wortel et al. 2023). Hence, a deeper understanding of genetic variation is important not only for evolutionary biology but also, more broadly, for biodiversity and ecosystem services that benefit human society at large (Stange et al. 2021).
The idea that genetic variation is critical for adaptation is deceptively simple and, for current students of evolutionary biology, quite intuitive. But exactly where does this variation come from, and what is the rate at which it is generated? Are all types of genetic variants similar, or do some matter more than others for adaptation? How and when does genetic variation influence phenotype and fitness, and what determines its relevance for adaptation? Finally, what is the role of old and new variation during adaptation, and what is the fate of genetic variants in natural populations? These questions arose in the very early days of evolutionary thinking, and spurred by technological and other advances in the twenty-first century, they continue to yield rich and novel insights with higher resolution, causal mechanisms, and greater generality (table 1).
Table 1.
Major advances in understanding the role of genetic variation in adaptation and underlying technological and other causes that facilitated the advance
| Advance | Cause |
:---: |
| Cheap sequencing | Genetic resourcesa | High-throughput analysisb | Macromolecule structurec | Computational power | New theory | Large datasets | New study organisms |
| Higher resolution | X | X | | | | | X | |
| Higher sample size | X | X | X | | | | X | X |
| Greater generality | X | X | X | | | X | X | X |
| Molecular mechanism | X | X | X | X | X | X | | |
| New methods | | X | X | X | X | | | X |
a For example, large-scale directed mutagenesis, genome databases, inbred lines, tools for genetic manipulation.
b For example, microfluidics, liquid handling, flow cytometry.
c For example, folding and structure of RNA and proteins.
View Table Image
The 2022 Vice Presidential Symposium of the American Society of Naturalists focused on updating our understanding of the role of genetic variation in light of these advances. This special section includes invited symposium contributions that span a range of methods, study organisms, and core questions (table 2). In this introductory article, we provide a broad overview of important new insights from the past few decades (fig. 1). We focus on variation introduced by mutations and discuss three aspects of the role of genetic variation in the process and outcome of adaptation. We first discuss how mutational variation is introduced and briefly summarize new understanding of how mutation rate and mutation type influence adaptation (part I). Next, we summarize advances in our understanding of the phenotypic and fitness consequences of genetic variation and how they affect adaptive outcomes (part II). Finally, we discuss some key factors that determine the persistence and fate of genetic variation in natural populations (part III). In each part, we focus on a small (and subjective) selection of questions, point readers to relevant reviews, and summarize key insights from recent work when such reviews are not available. We also highlight questions in need of empirical data, which recent technical advances now place within our reach. We hope that this article provides readers with a broad road map of current understanding and future trajectories of the role of genetic variation in adaptation.
Table 2.
Summary of the articles included in this special section
| Article | Summary | Study system |
:---:
| Tuffaha et al. 2023 | Changes in mutation bias enhance invasion of mutator genotypes by altering the distribution of fitness effects | Theory, bacteria |
| Gitschlag et al. 2023 | Associations between mutation rate and selection coefficients can be generated by mutation and selection | Theory |
| Miller et al. 2023 | Mutations have similar phenotypic effects across an environmental gradient | Zebrafish |
| Lin et al. 2023 | Adaptation to novel environments is driven by new alleles of large effect, but adaptation to previously experienced environments is driven by ancestral alleles of small effect | Wild banana |
| Petak et al. 2023 | Local adaptation to environmental fluctuations is driven by regulatory changes with low pleiotropy | Purple sea urchin |
View Table Image
Figure 1.
Summary of the broad context for this article. Shown is a schematic outlining key concepts in evolutionary biology that are the focus of this article. Starting from top left: factors that influence the amount and type of genetic variation available within a population; factors that mediate the phenotypic effect of genetic variants (mutations); other factors that determine individual phenotype; factors that mediate the phenotype-fitness relationship; and finally, factors that determine how the fitness consequences of mutations feed back into the amount and type of genetic variation in a population. Parts I and II of this article discuss the factors highlighted in boldface type, and part III gives an overview of how some of these factors influence the adaptive role and fate of genetic variation in natural populations.
View Large ImageDownload PowerPoint
Part I: Introduction of Genetic Variation—Mutation Rate and Bias
Mutations in genomes reflect spontaneous chemical change or damage to DNA or errors in DNA replication and/or subsequent repair. Most organisms have several mechanisms to correct such errors (Maki 2002; Friedberg 2003, 2005). Those that escape correction are inherited by the progeny, and these mutations are the most common source of genetic variation in most species. For a long time, the preeminent view was that evolutionary change emerges from shifts in frequencies of alleles available as “standing variation” rather than from new mutations, so that the influence of the mutation process itself was limited (Stoltzfus 2021; Beatty 2022). However, a large body of work now shows adaptation from new (as opposed to standing) variation in laboratory experiments and natural populations, and substantial sequencing data suggest that adaptive variation observed in nature reflects signatures of both mutational and selective processes. In recent years, the importance of both the rate of introduction and the type of new variation has come to be better appreciated, and their interaction with other evolutionary processes is being explored. In part I of this article, we first discuss new insights into the roles of mutation rate and bias in driving evolutionary outcomes. We then consider synonymous variation as a special category, to highlight the ongoing radical shift in our understanding of its role in evolution.
Mutation Rate
Quantifying mutation rates is important for understanding the role of genetic variation in adaptation because it determines the supply of beneficial mutations and contributes to shaping their evolutionary fate. In natural populations, new mutations may immediately face strong selection, increasing in frequency (under positive selection) or being eliminated (under negative selection). Alternatively, if mutations are neutral or nearly neutral or are linked with beneficial mutations, they may persist in populations as standing genetic variation (SGV). Therefore, the mutational supply rate also governs the amount of SGV that is available to populations experiencing new environments (the expected amount of polymorphism is ∼2 N e μ for haploids and ∼4 N e μ for diploids, where N e is effective population size and μ is the per-generation mutation rate). The mutation rate thus determines the amount of genetic variation in a population, whether (and how much) de novo versus standing variation is available for adaptation (see part III), and ultimately the capacity to respond to selection.
Compared with earlier methods, extensive genome sequencing in the past few years has led to more accurate and direct estimates of mutation rates across many taxa (Katju and Bergthorsson 2018). Mutation rates vary nearly a thousandfold across cellular organisms, from as low as 10−11 per site per generation to as high as 10−8 per site per generation (Katju and Bergthorsson 2018). Much of this work—especially in combination with mutation accumulation experiments that minimize selection—has focused on microbes and other small, tractable laboratory-based model organisms. However, recent large-scale whole-genome sequencing of parent-offspring trios also demonstrates variation in germline mutation rates across vertebrates (Bergeron et al. 2023). A broad evolutionary explanation for the evolution of mutation rates, known as the drift-barrier hypothesis, suggests that such variation in mutation rates across species is largely explained by variation in effective population size, which determines the efficiency of selection relative to the impact of genetic drift. Increasing replication fidelity provides diminishing returns, so in small populations selection cannot improve fidelity beyond the barrier imposed by genetic drift (Lynch et al. 2016). Results from diverse clades are consistent with this hypothesis (Wei et al. 2022; Bergeron et al. 2023).
Within species, mutation rates also vary across different parts of genomes, particularly between coding and noncoding regions and within coding regions depending on levels of gene expression and sequence context (Hodgkinson and Eyre-Walker 2011; Martincorena and Luscombe 2013). Analyses of some bacteria, the yeast Saccharomyces cerevisiae, and mammalian cells have elucidated molecular details of how various molecular pathways function individually and in concert to maintain genomic integrity and drive observed mutation rates (Friedberg et al. 2004; Sancar et al. 2004; Smeenk and Attikum 2013; Yi and He 2013). We now know that transient changes in expression of genes in DNA replication or repair pathways under stressful conditions (Foster 2008; Maharjan and Ferenci 2017) and loss of function of genes involved in DNA replication and repair (Katju and Bergthorsson 2018) leads to “mutator” phenotypes with substantially elevated mutation rates. Thus, DNA repair genes effectively serve as modifiers of mutational supply rates.
Mutators have been studied intensively by both theoreticians (James and Jain 2016; Jain and James 2017) and experimentalists (Raynes and Sniegowski 2014), to determine population genetic and demographic factors that promote or hamper their spread and to test the impacts of changing mutation rate on adaptation. Mutator genotypes that are linked with beneficial variants rise in frequency by genetic hitchhiking, increasing the overall mutation rate and allowing the accumulation of substantial genetic variation (Raynes and Sniegowski 2014). Factors such as population bottlenecks, clonal interference (competition between independent beneficial mutations), and increased levels of recombination may hinder the spread of mutators, while population substructure, strong selection (e.g., under stressful environments), and increased ploidy can promote mutator success (Raynes and Sniegowski 2014). Importantly, mutators are expected to eventually decline in populations due to rapid accumulation of deleterious mutations that imposes a genetic load and decreases mutator fitness (fig. 2A). However, much of this work has focused on asexual organisms, and our understanding of the impacts of mutators in sexually reproducing organisms is sparse. Exchange and reorganization of genetic material via recombination or horizontal gene transfer is generally expected to reduce mutator success (Tenaillon et al. 2000; Koonin et al. 2001; Gogarten and Townsend 2005; Denamur and Matic 2006), but experimental evidence for these ideas is missing. Hence, to better understand the mutation rate and its modifiers in sexual organisms, we require studies where the rates of mutation and genetic exchange can be independently modulated. Overall, work in the past few decades shows that the rate at which genetic variation is introduced into populations is itself a dynamic quantity.
Figure 2.
Illustration of some key concepts and terms used in this article. Shaded boxes indicate mutations that are expected to be most relevant for adaptation. Env = environment.
View Large ImageDownload PowerPoint
Mutation Type (Bias)
Thus far, we have discussed the evolutionary impact of mutation rate without distinguishing between different types of mutations. Mutations come in many forms, ranging from point mutations (single base pair changes) to small insertions-deletions (involving up to a few base pairs) to large-scale structural rearrangements involving thousands of base pairs. The latter include gene duplication (leading to copy number variation) as well as whole-genome duplication (WGD; causing polyploidy). These are common across eukaryotes and can drive evolutionary innovation and specialization because redundant gene copies face relaxed selection (Taylor and Raes 2004; Sémon and Wolfe 2007), also allowing the accumulation of genetic diversity for future evolution (e.g., Dorant et al. 2020). While we know of many examples of adaptive gene duplications, they can also be neutral or deleterious (e.g., because of dosage imbalance), and paralogs can eventually become nonfunctional pseudogenes (Kondrashov 2012; Kuzmin et al. 2022). The effects of WGD have been best studied in plants, where polyploidy can facilitate diversification and trait evolution (Baduel et al. 2018; Soltis and Soltis 2021). Several other types of structural variants also have important adaptive roles but remain relatively poorly understood and understudied (Mérot et al. 2020). Overall, despite a large body of general theory and specific empirical work on the evolutionary impacts of gene and genome duplication, we have relatively low predictive power about the immediate fitness consequences of such events, especially compared with point mutations.
Within point mutations, changes can be further categorized as transitions (i.e., purine-to-purine or pyrimidine-to-pyrimidine changes) or transversions (i.e., purine-to-pyrimidine or pyrimidine-to-purine changes), synonymous (mutations that do not alter the amino acid) or nonsynonymous (mutations that change the amino acid), and so on. The frequency distribution of different types of mutations is called the mutation spectrum, with most organisms showing spectra that are skewed toward specific types of mutations, for example, a pervasive transition bias (Katju and Bergthorsson 2018; Goldberg and Harris 2021). Such biases are thought to arise from the action of specific DNA replication and repair pathways (Foster et al. 2015; Katju and Bergthorsson 2018). Importantly, the prevalence of biased mutation spectra implies that during evolution, some mutational steps are more likely to be sampled than others and may therefore be overrepresented among the observed pool of fixed adaptive mutations (Stoltzfus and McCandlish 2017; Payne et al. 2019; Cano et al. 2022; fig. 2B). Furthermore, if different types of mutations have distinct fitness effects, mutational biases may also determine the outcomes of adaptation. For instance, regulatory mutations were proposed to have larger adaptive effect sizes than coding mutations, although this was not broadly supported (Hoekstra and Coyne 2007; Hendry 2013). In general, the rate and trajectory of adaptation can thus be determined not just by how much new genetic variation is sampled (mutation rate) but also by which kind of genetic variation can be sampled (mutation bias).
The importance of sampling of new genetic variation in determining evolutionary outcomes has become clear in the past few decades, with large amounts of sequence data (Stoltzfus 2021) suggesting that the course of evolution is driven by both the arrival of new variation and the selection that determines its fate (Yampolsky and Stoltzfus 2001; Stoltzfus and Yampolsky 2009; McCandlish and Stoltzfus 2014). Simulations of adaptive walks on fitness landscapes demonstrate the impacts of mutation biases on adaptation, specifically on protein evolution (Stoltzfus 2006), choice of evolutionary pathway (Schaper and Louis 2014), navigability of the fitness landscape (Cano and Payne 2020), interactions between mutation supply rate and mutation bias (Yampolsky and Stoltzfus 2001; Gomez et al. 2020; Soares et al. 2021; Cano et al. 2022), the predictability of evolutionary trajectories (Cano and Payne 2020), and the supply of beneficial mutations by altering the sampling of mutational space (Sane et al. 2023). Empirical evidence for the importance of mutation bias in adaptation has also accumulated, particularly in microbes. For instance, the repeatability of mutations in the bacteriophage phiX174 is best explained by a model incorporating nonuniform mutation rates as well as mutational effect sizes (Rokyta et al. 2005). Similarly, “mutational hotspots” in the genome of Pseudomonas fluorescens influence both the identity of loci where adaptive mutations fix and the type of adaptive mutations (Horton et al. 2021). Strain-specific mutation bias (in different mutator strains) allows Escherichia coli populations to sample distinct antibiotic resistance mutations with different fitness effect sizes (Couce et al. 2013, 2015), generally altering the distribution of mutational effect sizes and enhancing access to new beneficial mutations (Sane et al. 2023). A recent meta-analysis of adaptive substitutions in different microbes shows that the mutation spectrum of adaptive substitutions broadly mirrors the underlying mutation spectrum (Cano et al. 2022). The broad generality of such effects is further supported by studies showing that mutation bias is an important determinant of the genetic basis of adaptation in several organisms (Streisfeld and Rausher 2010; Storz et al. 2019), and genomic regions prone to mutation drive parallel evolution across populations (e.g., in stickleback fish; Xie et al. 2019). Thus, mutation biases can determine both the prevalence and the evolutionary fate of different types of mutations.
Large changes in mutation bias are often accompanied by changes in mutation rates because altered DNA repair activity affects both quantities (Lee et al. 2012; Foster et al. 2015; Katju and Bergthorsson 2018; Sane et al. 2023). Previous work has suggested that mutator bacteria that oppose an ancestral mutation bias can sample more beneficial mutations (Sane et al. 2023) and that mutators are more likely to emerge if they sample large-effect beneficial mutations (Couce et al. 2013). In this special section, Tuffaha et al. (2023) connect these two findings and ask: how does concomitant change in mutation rate and bias affect the emergence and success of mutators? The authors use both theory and simulations of whole populations undergoing adaptation to understand the impacts of bias shifts on the distribution of fitness effects (DFE; also called the distribution of mutation effects) and the fate of mutations that change mutation rate, mutation bias, or both. During 100,000 generations of adaptive evolution, the fraction of beneficial mutations declined faster for the class of mutations favored by the ancestor, compared with mutations favored by the bias-shifted strain. Beneficial mutations in the latter mutational class also had larger effect sizes. Thus, a bias shift favoring previously undersampled mutations should allow populations to sample more beneficial mutations of larger effect size. Indeed, simulations show that bias-shifted strains had a higher probability of invading an ancestral population across a wide range of conditions, and the difference in invasion probability was greater when the bias shift was more extreme and when adaptive potential was lower. These findings suggest that even a modest change in the ancestral mutation bias can alter invasion success, especially later during adaptation. Importantly, a bias shift could extend the range of mutation rates at which mutators can successfully invade a wild-type population, suggesting that such mutators may face lower deleterious genetic load than expected. This study therefore highlights the role of the type of genetic variation in determining the evolutionary fates of genotypes under selection.
In summary, theoretical and empirical studies in recent years have demonstrated the role of mutational processes—particularly mutation bias—in influencing adaptive outcomes. The empirical work has been possible to a large extent because new technologies such as sequencing have become cheaper, more reliable, and more accessible. The collection of more, richer, and more diverse genetic data from many different model and nonmodel organisms has in turn improved datasets for comparative genomics, allowing scientists to uncover mutational patterns in genomes. Further analyses of mutation rates and biases in more taxa, especially in natural populations, will help us to better understand how the processes that give rise to genetic variation shape evolution.
Synonymous Variation
When the nature of the genetic code was discovered in the 1960s, it became clear that many amino acids were encoded by multiple codons. Mutations that switched between codons for the same amino acid (synonymous mutations) would maintain the protein sequence and were thus assumed to be neutral and invisible to natural selection. Indeed, this assumption was involved in developing our understanding of evolutionary forces that act on genetic variation (King and Jukes 1969). Additionally, the assumption of neutrality of synonymous variation served as the basis for methods to test for signatures of selection on gene sequences. For instance, the McDonald and Kreitman test identifies regions under selection by comparing the rates of substitution and polymorphism at nonsynonymous versus synonymous sites (McDonald and Kreitman 1991). However, the discovery that synonymous codon use varied within and across organisms had already suggested that all synonyms are not equivalent and therefore not necessarily neutral (Fiers et al. 1976; Grantham 1978; Grantham et al. 1980). From these early observations to recent work, a large body of comparative and experimental evidence, both indirect and direct, has led to an ongoing shift in our understanding of the evolutionary impacts of synonymous variation (Agashe 2022). Substantial evidence indicates that synonymous mutations may often evolve under selection. Below, we focus on experimental evidence that illustrates the evolutionary impacts of synonymous mutations.
Direct evidence for the evolutionary impacts of synonymous mutations comes from studies showing that (i) artificially generated synonymous mutations in a focal gene or gene region have measurable phenotypic and fitness effects and (ii) synonymous mutations arise repeatedly in laboratory evolution experiments and evolve under selection. As early as 1987, Hoekema and colleagues reported that replacing codons at the 5′ end of a gene in Saccharomyces cerevisiae with synonymous rarely used codons reduced gene expression and protein levels (Hoekema et al. 1987). Decades later, experiments with Drosophila melanogaster showed that increasing the number of rare codons in the adh (alcohol dehydrogenase) gene reduced transcription and alcohol tolerance of larvae (Carlini and Stephan 2003; Carlini 2004). Subsequently, several studies in microbes have shown that synonymous mutations alter fitness (Bailey et al. 2021; Agashe 2022). These effects of synonymous changes arise via diverse mechanisms, including messenger RNA folding, splicing, and expression; translation speed, accuracy, and efficiency; and protein folding, solubility, and localization (Sauna et al. 2007; Hershberg and Petrov 2008; Plotkin and Kudla 2010; Sauna and Kimchi-Sarfaty 2011; Hunt et al. 2014; Liu 2020; Callens et al. 2021). Many studies have also reported significant fitness impacts of mutations obtained from genetic screens, site-directed mutagenesis, and mutation accumulation experiments, showing that the DFE of synonymous and nonsynonymous changes is much more similar than previously assumed (Bailey et al. 2021). More recently, several laboratory evolution experiments have demonstrated strong beneficial effects of rapidly and parallelly evolved synonymous mutations (Agashe 2022). Finally, accumulated evidence from comparative genomics has identified widespread genomic signatures of selection on synonymous mutations (Hunt et al. 2014), especially in humans, where synonymous mutations can play a significant role in disease phenotypes (Sauna and Kimchi-Sarfaty 2011; Hunt et al. 2014). Together, this body of work shows that the fitness effects of individual synonymous mutations can have important evolutionary impacts, although several important questions remain about their generality and magnitude across evolutionary timescales. Overall, synonymous genetic variants deserve more careful consideration while thinking about the role of genetic variation in adaptation.
Part II: The Phenotypic and Fitness Effects of Genetic Variation
Genetic variants are often broadly classified in terms of their combined or interactive phenotypic effects (e.g., additive, epistatic, or dominance). This classification is useful both while modeling and empirically testing the evolutionary effects of genetic variation. Another approach to genotype-phenotype associations is quantitative genetics, where one identifies genes that govern complex polygenic traits (quantitative trait loci [QTL]) or mutations (or genetic markers) associated with specific phenotypic differences (genome-wide association studies [GWAS]). All of these methods take a phenotype-centric view, with the goal of understanding factors underlying a particular phenotype. A different approach is to estimate the statistical DFE of mutations, which describes how mutations will generally influence fitness and, by implication, the phenotype (the “genotype-phenotype map”; fig. 2C). Although all of these approaches offer complementary and useful perspectives on the links between genotype and phenotype, we focus here on DFEs, which directly address the consequences of genetic variation.
The DFE of new mutations is key for predicting how fast populations can adapt in a given environment (based on the proportion and effect size of beneficial mutations) as well as how much neutral or nearly neutral genetic variation can persist in the population and may later become available for selection in new environments (Eyre-Walker and Keightley 2007; Lanfear et al. 2014). To estimate the shape and other properties of the DFE, biologists have made increasingly large collections of mutants whose phenotypic effects can be investigated simultaneously (Eyre-Walker and Keightley 2007; Chen et al. 2022). This has been best studied in fast-growing acellular or unicellular organisms, such as viruses, bacteria, and some yeasts. Increasingly, such studies involve larger multicellular organisms with more complex life histories, such as fruit flies, worms, and fish. In part II of this article, we discuss important new insights derived from these efforts, focusing on two sets of questions. First, we discuss some general properties of the DFE, factors that influence its shape, and the implications for adaptation. Next, we discuss recent insights into pleiotropic effects of mutations that can alter the DFE across environments and influence the process and outcome of adaptation.
The DFE of Mutations
The DFE is often inferred indirectly in populations, via observed distributions of allele frequencies (site frequency spectra [SFS]) and polymorphisms that are expected to reflect fitness effect sizes of the alleles (Eyre-Walker and Keightley 2007; fig. 2C). However, this requires several assumptions—including the independence of mutational effects and specifying the broad form of the DFE (e.g., a gamma distribution)—such that the specific parameters of the distribution can be estimated using the SFS. Recent methods have attempted to use low-frequency variants to infer the DFE for haplotypes, incorporating linked alleles (Vecchyo et al. 2022), or to infer the DFE using GWAS data (Simons et al. 2022). More direct estimation of the DFE relies on generating libraries of mutants (via either directed or random mutagenesis), mixing the mutants, and measuring allele frequency change over time (Kemble et al. 2019). A variation of this strategy involves using mutation accumulation experiments to generate random mutations and then using observed allele frequency changes under selection to infer the DFE (Böndel et al. 2022). Some studies focus on the beneficial mutations observed during experimental evolution—especially in microbes—and directly measure their impact on fitness to infer the DFE of beneficial mutations (Sniegowski and Gerrish 2010). More rarely, mutation accumulation experiments followed by next-generation sequencing and fitness assays are used to estimate the DFE of randomly sampled mutations, before selection can substantially alter it (Sane et al. 2018, 2023). All of these methods have different limitations imposed by logistical challenges and statistical power: the inability to detect lethal mutations, imprecision in estimating the shape of the beneficial part of the DFE, or limited relevance under natural conditions. Nonetheless, together these studies demonstrate that multiple factors influence the precise shape of the inferred or estimated DFE, converging on two key parameters: the genetic background (due to epistasis; i.e., effects of interactions between mutations) and the environment (which influences the selective effect of mutations; Kemble et al. 2019).
DFEs tend to be bimodal or multimodal, with a large peak representing neutral or nearly neutral mutations, and another peak with highly deleterious or lethal mutations. Importantly, mutations that are most likely to drive adaptation—strongly beneficial mutations—are generally rare, although the exact proportion varies across studies. A growing body of empirical work with mutation accumulation lines (where the effect of selection is minimized) suggests that beneficial mutations may be more common than previously anticipated (Shaw et al. 2000, 2003; Zeyl and DeVisser 2001; Joseph and Hall 2004; Dickinson 2008; Sniegowski and Gerrish 2010; Böndel et al. 2022; Sane et al. 2023). However, this is an area that awaits more empirical research. In parallel, it is also important to determine the fraction of neutral or nearly neutral mutations, which may persist in natural populations as SGV and drive subsequent adaptation to new environments (see part III). In this special section, Miller et al. (2023) tested mutational impacts on swimming speed in zebra fish, which is an important determinant of fitness in the wild. As expected, some mutations increased swimming speed while others decreased it. However, the average effect of mutations was neutral, indicating that the DFE is not substantially skewed toward deleterious mutations. Such studies of mutational effects in environments relevant for natural populations remain rare, and hence it is currently difficult to broadly generalize these findings.
Apart from bimodality, another general feature of DFEs is that large-effect mutations (whether beneficial or deleterious) are typically—though not always—rare. In other words, the selection coefficient is negatively correlated with mutation rate. Such a relationship is expected, for at least two reasons. First, most mutations are thought to have weak phenotypic effects. This is because many mutations may have minimal effects on protein structure and function (e.g., if they are synonymous; but see part I) or because organisms have a high degree of mutational robustness, such that single mutations have generally small effect sizes. Second, if populations are reasonably well adapted in an environment, we expect fewer large-effect beneficial mutations. However, in this special section, Gitschlag et al. (2023) show that the actions of mutation bias and selection can themselves generate negative (and sometimes positive) relationships between the mutation rate and selection coefficient, even if no such relationship existed to begin with. Associations between mutation rate and fitness effects may also arise when only a few beneficial mutations are available in a given environment. Importantly, the authors devise a general mathematical model to analyze these effects under a strong-selection/weak-mutation regime, where the effects of mutation rate and selection are more easily differentiated. Finally, using the distribution of mutation rates and selection coefficients in a cancer dataset, they find that among clinically prevalent TP53 (tumor suppressor) variants, multinucleotide variants—which have lower mutation rates—tend to have a greater growth advantage. These results imply that inferring DFEs using an observed negative relationship between allelic effect size and frequency may be problematic. Thus, we need to better understand how adaptation shapes such relationships and what information the observed associations may carry.
Overall, despite significant recent progress, direct empirical estimates of the DFE are still limited to a few species, genes, or environments, with only some cases with very large mutational datasets. The impact of genetic background and epistasis in shaping the DFE is also poorly understood. Finally, logistical challenges in capturing lethal mutations continue to pose an important limitation. Accurately estimating DFEs thus remains a problem in need of new methods and analyses as well as broader taxonomic sampling.
Mutational Pleiotropy
We now discuss how pleiotropy influences the fate and impact of genetic variation. Pleiotropy has been discussed in various forms for more than 100 years and has acquired many definitions and meanings (Stearns 2010). It is also challenging to define and quantify pleiotropy in ways that allow for robust and easy comparison across studies (Wagner and Zhang 2011). Of interest here are instances when a single gene or mutation influences multiple phenotypes or the same phenotype in multiple environmental contexts. Such pleiotropy is important to understand and predict adaptation: it can determine the success and mechanism of adaptation under diverse selection pressures (Fisher 1930; Orr 2000), and it directly influences the fate of variants in new environments (see part III). As a result, considerations of pleiotropy have had a powerful impact on many aspects of evolutionary and ecological thinking. For instance, pleiotropy has often been invoked as an important constraint on trait evolution (Orr 2000). However, direct evidence of such impacts is surprisingly scarce. For example, pleiotropy (specifically antagonistic pleiotropy; see below) was thought to explain widespread dietary specialization of insect herbivores, but there is little evidence supporting a role for such genetic constraints (Hardy et al. 2020).
Before thinking about the evolutionary impact of pleiotropy, we must first ask whether pleiotropy is observed frequently. Do most genes show high levels of pleiotropy? That is, do most genes affect multiple traits? While early evolutionary models often assumed such “universal” pleiotropy, functional genomics and QTL mapping studies in the past few decades suggested more modularity, with a given gene affecting only a small set of traits or pathways (Wagner and Zhang 2011; fig. 2D). On the other hand, many complex traits, such as human disease phenotypes, are significantly associated with a large number of loci across the genome—not only genes in known disease pathways or modules (Boyle et al. 2017). GWAS analyses also show that highly polygenic traits are prevalent (e.g., Sinnott-Armstrong et al. 2021), suggesting extensive pleiotropy rather than functions organized into discrete modules. The distinction is important because at one extreme, phenotype may be influenced by a small number of genes (mutations) of very large effect; on the other hand, many genes (mutations) of small effect may determine quantitative polygenic traits. A clear conciliation of the wide range of observed patterns, as well as their interpretation and underlying causes, is not yet at hand.
To determine the impact of pleiotropy on adaptation, we must also consider the nature of pleiotropy (fig. 2E). Synergistic pleiotropy indicates cases where a mutation has similar effects across traits or environments (e.g., a mutation is beneficial in two different environments). Thus, synergistic pleiotropy has the potential to facilitate adaptation, since variation that arises or fixes in one environment may also be beneficial in another context. In contrast, antagonistic pleiotropy reflects cases where a mutation has opposite effects in different environments (i.e., it is beneficial in one context but deleterious in another). Such pleiotropy can hinder adaptation because previously beneficial mutations can impose substantial fitness costs in the new environment. Unfortunately, discussions of pleiotropy often do not distinguish explicitly between synergistic and antagonistic pleiotropy, even though this distinction is critical to predict adaptive outcomes in new environments. Several studies show that antagonistic pleiotropy is relatively infrequent, whereas synergistic pleiotropy is more common. For instance, analysis of mutation accumulation lines of Caenorhabditis elegans as well as Drosophila melanogaster showed that mutational effects are strongly positively correlated across environments (Fry and Heinsohn 2002; Baer et al. 2006). Mutational variation among natural strains of the predatory ameba Dictyostelium discoideum also had prey-specific fitness effects, with low antagonistic pleiotropy (Stewart et al. 2022). Similarly, an analysis of effects of single Escherichia coli mutations in different carbon sources found very few antagonistically pleiotropic mutations, whereas beneficial synergistic pleiotropy was more common (Sane et al. 2018); this result mirrors the outcomes of a prior study focusing only on beneficial mutations (Ostrowski et al. 2005).
Most of the studies discussed above tested pleiotropic effects in discrete pairs of environments (e.g., hot vs. cold). What happens when organisms experience spatial environmental gradients (e.g., a gradual shift in temperature)? In the zebrafish study discussed earlier, Miller et al. (2023) induced germline mutations in males and analyzed the effect on swimming speed of their offspring across temperatures. They found little evidence for antagonistic pleiotropy in swimming speed at different temperatures. Thus, mutations that increased swimming speed at a particular temperature were also more likely to increase speed at other temperatures. Their results suggest that synergistic beneficial pleiotropy is common and may facilitate rapid adaptation even under complex (and probably commonly encountered) spatially explicit scenarios, such as environmental temperature gradients. Thus, the lack of support for a strong constraining role of pleiotropy may stem from the fact that mutations rarely show antagonistic pleiotropy. In the future, studies combining mutagenesis experiments with measurement of phenotypic traits relevant for fitness in natural environments will be important to connect what we understand about mutational effects in the laboratory to the evolutionary fate of genetic variation in natural populations.
Part III: The Fate of Genetic Variation in Natural Populations
In parts I and II, we drew primarily on theoretical and laboratory studies to discuss the types and fitness effects of mutations and their impact on evolution. We now summarize progress in and current understanding of the role and fate of genetic (allelic) variation in natural populations. To begin thinking about this, we first ask: how much genetic diversity do natural populations harbor? Much of the variation observed in the wild was initially thought to be a consequence of selection (Mayr 1972; Clarke 1979) because variants that led populations away from optimum trait values should be rapidly purged, eroding genetic diversity. However, populations still harbor substantial genetic variation, especially when large population size and/or high mutation rate allows for significant mutational input (as mentioned in part I, polymorphism levels are proportional to N e μ). Beginning in the 1970s with allozyme markers, it was clear that natural populations harbor much higher genetic diversity than expected under strong purifying selection (Ellegren and Galtier 2016; Allendorf 2017). With further advances in technology and lower sequencing costs, many recent studies have shown that natural populations harbor large amounts of genomic variation despite evolving under selection (Leffler et al. 2012; Ellegren and Galtier 2016). Importantly, this includes substantial amounts of additive genetic variance that is critical for evolutionary change (Hendry et al. 2018). Once genetic variation is generated by processes such as mutation and recombination (fig. 1), what factors determine its persistence in natural populations, and what are its fate and role during adaptation?
Persistence of Genetic Variation
The most obvious route for a specific genetic variant to persist in a population is direct selection favoring the variant. However, strong selection on a single allele will erode diversity at and near the selected site, so this cannot generally explain the generation and maintenance of extensive polymorphism. A large body of theoretical work has suggested several mechanisms that can maintain polymorphism, such as balancing selection (heterozygote advantage or frequency-dependent selection) or spatial variation in selection pressure (Slatkin 1978; Charlesworth et al. 1997). Recent analyses of natural populations find support for all of these (Carius et al. 2001; Kazancıoğlu and Arnqvist 2014; Luo et al. 2019). Temporal fluctuation in selection pressures is also predicted to maintain genetic variability if different alleles are adaptive in distinct environments experienced by a population (Wittmann et al. 2017). While there is ample evidence of phenotypic, metabolic, or genomic heterogeneity driven by temporally fluctuating environments in clonal microbial laboratory populations (Schreiber and Ackermann 2020), direct evidence from natural populations is rare because it is very difficult to distinguish the effect of confounding factors. For instance, while studies of stickleback fish, monkeyflowers, and sessile oak populations reported temporal fluctuations in allele frequencies, none of them could demonstrate a causal role for temporally variable selection (Garcia-Elfring et al. 2021; Kelly 2022; Saleh et al. 2022).
Genetic variants may also persist in populations in the absence of direct selection. First, variants that are neutral or only slightly deleterious in the ancestral environment may increase in frequency due to random genetic drift (Barrett and Schluter 2008). For example, ancestral marine stickleback populations harbored genetic variants that were neutral under high salinity but later drove adaptation to acidic or basic conditions as the fish colonized new habitats (Haenel et al. 2022). Similarly, as weedy rice (Oryza sativa f. spontanea) transitioned from wild to domesticated and then back to the wild as feral rice, variation that persisted through domestication was reused for secondary adaptation in the wild (Zhou et al. 2021). Second, an influx of alleles from other populations may maintain genetic diversity in focal populations (Hedrick 2013). A good example comes from altitudinal adaptation in the vinous parrotbill in Taiwan, driven by alleles from migrant birds from the mainland (Lai et al. 2019). Similar patterns of adaptation driven by gene flow are observed in many marine fish and invertebrates (Hellberg 2009; Nielsen et al. 2009; Sanford and Kelly 2011). Third, some genetic variants (neutral or mildly deleterious) may rise to high frequencies by hitchhiking with other alleles that were beneficial in a previously experienced environment (Kelly 2022). Such hitchhiking can be reflected as hard sweeps, with a single causal allele on a specific genomic background, or soft sweeps, when multiple adaptive alleles at the same locus increase in frequency simultaneously. In either case, linked variation is swept to higher frequencies (fig. 2F). While hard sweeps will decrease overall genetic diversity, soft sweeps can contribute to the maintenance of high levels of SGV (Hermisson and Pennings 2005; Stephan 2019; Charlesworth and Jensen 2021). In European butterflies, substantial diversity is maintained by such selection at linked sites (Mackintosh et al. 2019). A special case of such linkage occurs with large structural mutations like inversions, translocations, and transversions, which can reduce or eliminate recombination locally and allow genetic variation to accumulate (Frankham 2012). However, there is substantial variability in the amount and nature of genetic variation associated with structural variants (Kapun and Flatt 2019), and the precise role of balancing or divergent selection and reduced recombination in maintaining diversity within structural variants is poorly understood (Wellenreuther and Bernatchez 2018).
Overall, current evidence supports several processes and mechanisms that may drive the persistence of genetic variation in natural populations. As work on nonmodel wild populations accumulates, we will be better able to determine the relative importance of different mechanisms in determining the levels of persistent genetic diversity.
Role and Fate of Genetic Variants during Adaptation to New Environments
We now consider the fate of genetic variants when natural populations adapt to new conditions under strong selection (Nadeau and Jiggins 2010; Bomblies and Peichel 2022). Identifying causal adaptive variants in natural populations is challenging because a large amount of genetic data is necessary to identify rare or small-effect variants, and linking genotype to phenotype to fitness in the wild is not trivial. Adaptive variants are often categorized as preexisting variation (SGV) or variation that arises after encountering a new selection pressure (de novo mutation [DNM]; Barrett and Schluter 2008; fig. 2F). Variation can also be introduced into populations by migration; depending on the rate of migration and selection, such variants can contribute to SGV (Blanquart et al. 2012). Strikingly, in natural populations of sexually outcrossing species (representing the bulk of relevant recent and ongoing research in natural populations), adaptation often seems to occur from SGV rather than DNM (Alves et al. 2019; Gaines et al. 2019; Lai et al. 2019; Marques et al. 2019; Kersten et al. 2021; Moan et al. 2021; Nakamura et al. 2021; Mittan-Moreau et al. 2022; Small et al. 2022). However, given the paucity of studies of wild microbial and other haploid asexual systems, the role of SGV versus DNM during evolution is still debated (Beatty 2022). Nonetheless, because alleles representing DNM and SGV have distinct evolutionary histories and initial frequencies at the onset of new selection pressure, they are predicted to have distinct evolutionary fates and impacts (Orr and Betancourt 2001; Hermisson and Pennings 2005; Jensen 2014). Hence, to understand the broad underlying principles that determine the role and fate of variants during adaptation in natural populations, it is useful to consider the ancestral context of genetic variants, which distinguishes SGV and DNM.
Initial allele frequency is an important determinant of the fate of variation in natural populations. Depending on population size and mutation rate, SGV may be present in populations at higher frequencies and may therefore drive adaptation more often and faster than DNM, which originates as a single copy in the population (Barrett and Schluter 2008; Olson-Manning et al. 2012). In addition to the speed of adaptation, high initial allele frequency of adaptive SGV can also facilitate genetic parallelism, because independently evolving populations with shared ancestry are more likely to use these alleles for adaptation (MacPherson and Nuismer 2017). Such parallelism is important because it makes evolution (and allelic fate) more predictable. For instance, marine stickleback fish adapting to freshwater habitats with different pH values repeatedly used the same genetic variants, which were present in moderate frequencies in the ancestral population (Haenel et al. 2019). Similarly, adaptation to the myxoma virus in rabbit populations separated by thousands of kilometers occurred via similar genomic variants, likely present as SGV in the ancestral population (Alves et al. 2019). Thus, the presence of preexisting adaptive SGV may allow high levels of parallelism in natural populations. In contrast, because DNM arises after the change in selection pressure, parallelism via DNM would require high rates of mutation supply and strong selection (e.g., as observed in laboratory evolution experiments with microbes; Schenk et al. 2022) or strong mutation bias that increases the sampling of specific adaptive mutations (Storz et al. 2019; note that mutation bias will also promote parallelism during adaptation via SGV).
A second factor determining the fate of a genetic variant is its fitness effect size. Theory predicts that when a population is far from the fitness optimum in a new environment, large-effect DNMs are more likely to drive adaptation than small-effect mutations (Orr 1998). In contrast, adaptation via SGV is more likely to lead to the fixation of many variants of small-effect size (Orr and Betancourt 2001; Matuszewski et al. 2015). Analyses of natural populations include examples of various combinations of effect sizes and adaptive loci derived from either SGV or DNM (Connallon and Hodgins 2021; Bomblies and Peichel 2022), without a clear preponderance of small-effect SGV or large-effect DNM. In this special section, Lin et al. (2023) present a study with wild banana plants found in continental China and Taiwan under diverse climatic conditions. Climatic variables related to temperature are similar across the two regions, while those related to precipitation are more distinct. The authors analyzed the genomes of extant banana populations across different regions of Taiwan and compared genetic variation in these populations to the presumed ancestral populations from mainland China. They found that the Taiwan populations harbor single-nucleotide polymorphisms (SNPs) that are significantly associated with temperature-related factors, and many of these persist in the ancestral populations from China (thus, likely representing SGV). In contrast, SNPs associated with precipitation differences are found only in the Taiwanese populations and probably represent new mutations that arose during colonization. The effect sizes of SGV were much lower than those of DNM, supporting the idea that effect size can play an important role in the fate and impact of variation on adaptation. While the relationship between effect size and ancestry of the variant may vary across organisms, it is clear that many traits show signatures consistent with polygenic adaptation (Barghi et al. 2020). However, demonstrating the genetic architecture of a trait does not necessarily tell us whether and how it evolved under selection (Barrett and Hoekstra 2011), and hence the impact of effect size on allelic fate in natural populations remains unclear.
As discussed in part II, allelic fate should also be determined by pleiotropic effects of mutations. In general, variants that survived through ancestral environmental fluctuations are more likely to be neutral or mildly deleterious in new environments (i.e., they are less likely to show antagonistic pleiotropy). Indeed, analyses of loci underlying adaptive traits in plants has suggested that antagonistic pleiotropy is not very common, and instead adaptive variants are often neutral in other environments (i.e., they show conditional neutrality; Anderson et al. 2011; Weng et al. 2022). Given that SGV is by definition “pretested” by populations, it may be more likely to show weak pleiotropy and therefore more likely to drive adaptation in new environments compared with DNM (Barrett and Schluter 2008). In this special section, a study of purple sea urchins provides rare support for this hypothesis, suggesting that weak pleiotropy of SGV may assist in adaptation to spatially variable environments (Petak et al. 2023). Urchin populations that experienced distinct pH extremes have SNPs that are tightly correlated with environmental pH, despite a lack of overall population structure. The SNPs associated with pH variability also occur in genes that are mostly expressed during late developmental stages and are connected to only a few other genes within the predicted gene interaction network. As a result, these genes should have low antagonistic pleiotropy. Finally, the SNPs associated with pH adaptation are also enriched in putative regulatory regions, which are less likely to have pleiotropic effects (Carroll 2008). Thus, this study highlights a case where widespread adaptation to fluctuating environments has occurred via low-pleiotropy regulatory mutations. To uncover broad patterns of the relationship among allelic ancestry, pleiotropy, and role in adaptation, we need data from more such studies in natural populations.
Summary
We have tried to provide here an overview of some recent advances in and updates to our understanding of the amount, provenance, effect, and fate of genetic variation relevant for adaptation (table 3). Although there is increasing integration across diverse areas of biology, it is clear that there are still large opaque blocks in our view of adaptation, which reduce our ability to synthesize our understanding across organisms, timescales, and levels of resolution. For instance, the nature and rate of introduction of new genetic variation via mutations remains poorly explored in natural populations, especially for multicellular eukaryotes. In contrast, analyses of the role and fate of genetic variation in wild populations are largely focused on plants, animals, and specific microbes of importance to human health (e.g., pathogens). While many specific open questions remain, efforts to gather empirical data across a breadth of taxa and including both molecular and fitness analyses may be most fruitful in allowing a broader synthesis. In that spirit, the articles in this special section represent examples of theoretical and empirical work that offer some clarity (table 2), and we hope that more analyses in the coming decades will lead to a richer understanding of genetic variation.
Table 3.
Important recent insights discussed in this article
Part I: Introduction of Genetic Variation—Mutation Rate and Bias
•Mutation rate is dynamic and variable and can evolve under selection
•Mutation biases are pervasive and may determine adaptive outcomes
•Synonymous mutations can be adaptive and may evolve under selection due to several mechanisms
Part II: The Phenotypic and Fitness Effects of Genetic Variation
•The statistical distribution of fitness effects of mutations is typically bimodal, with many highly deleterious and neutral or nearly neutral mutations; beneficial mutations are generally rare, but the proportion varies across studies
•The nature and incidence of mutational pleiotropy varies across studies
•Many traits are quantitative and are governed by several genes of small effect size, somewhat at odds with observations of largely modular gene networks
Part III: The Fate of Genetic Variation in Natural Populations
•Once generated via mutation, genetic variation can persist via several routes
•Adaptive substitutions comprise standing genetic variation more often than de novo mutations, but it is unclear whether this pattern reflects biological principles or logistical limitations
View Table Image
Acknowledgments
We thank Shyamsunder Buddh, Katrina McGuigan, Krushnamegh Kunte, Cheng-Ruei Lee, Pratibha Sanjenbam, Arlin Stoltzfus, Marwa Tuffaha, Lindi Wahl, and two reviewers for discussion and comments on the manuscript. We acknowledge funding and support from the National Centre for Biological Sciences (NCBS-TIFR) and the Department of Atomic Energy, government of India (project identification RTI 4006).
Statement of Authorship
All authors contributed to conceptualization, referencing, writing, review, and editing. D.A. prepared figures and tables with input from M.S. and S.S.
Data and Code Availability
No new data or code has been generated for this study.
Notes
ORCIDs: Agashe, Sane, Singhal,
This article originated as part of the 2022 Vice Presidential Symposium at the annual meetings of the American Society of Naturalists in Cleveland, Ohio.
Literature Cited
Agashe, D. 2022. Evolutionary forces that generate SNPs: the evolutionary impacts of synonymous mutations. Pages 15–36 in Z. E. Sauna and C. Kimchi-Sarfaty, eds. Single nucleotide polymorphisms: human variation and a coming revolution in biology and medicine. Springer, Cham.
First citation in article Google Scholar
Allendorf, F. W. 2017. Genetics and the conservation of natural populations: allozymes to genomes. Molecular Ecology 26:420–430.
First citation in article CrossrefMedlineGoogle Scholar
Alves, J. M., M. Carneiro, J. Y. Cheng, A. L. de Matos, M. M. Rahman, L. Loog, P. F. Campos, et al. 2019. Parallel adaptation of rabbit populations to myxoma virus. Science 363:1319–1326.
First citation in article CrossrefMedlineGoogle Scholar
Anderson, J. T., J. H. Willis, and T. Mitchell-Olds. 2011. Evolutionary genetics of plant adaptation. Trends in Genetics 27:258–266.
First citation in article CrossrefMedlineGoogle Scholar
Baduel, P., S. Bray, M. Vallejo-Marin, F. Kolář, and L. Yant. 2018. The “polyploid hop”: shifting challenges and opportunities over the evolutionary lifespan of genome duplications. Frontiers in Ecology and Evolution 6:117.
First citation in article CrossrefGoogle Scholar
Baer, C. F., N. Phillips, D. Ostrow, A. Avalos, D. Blanton, A. Boggs, T. Keller, et al. 2006. Cumulative effects of spontaneous mutations for fitness in caenorhabditis: role of genotype, environment and stress. Genetics 174:1387–1395.
First citation in article CrossrefMedlineGoogle Scholar
Bailey, S. F., L. A. A. Morales, and R. Kassen. 2021. Effects of synonymous mutations beyond codon bias: the evidence for adaptive synonymous substitutions from microbial evolution experiments. Genome Biology and Evolution 13:evab141.
First citation in article Google Scholar
Barghi, N., J. Hermisson, and C. Schlötterer. 2020. Polygenic adaptation: a unifying framework to understand positive selection. Nature Reviews Genetics 21:769–781.
First citation in article CrossrefMedlineGoogle Scholar
Barrett, R. D. H., and H. E. Hoekstra. 2011. Molecular spandrels: tests of adaptation at the genetic level. Nature Reviews Genetics 12:767–780.
First citation in article CrossrefMedlineGoogle Scholar
Barrett, R. D. H., and D. Schluter. 2008. Adaptation from standing genetic variation. Trends in Ecology and Evolution 23:38–44.
First citation in article CrossrefMedlineGoogle Scholar
Beatty, J. 2022. The synthesis and the two scenarios. Evolution 76:6–14.
First citation in article CrossrefMedlineGoogle Scholar
Bergeron, L. A., S. Besenbacher, J. Zheng, P. Li, M. F. Bertelsen, B. Quintard, J. I. Hoffman, et al. 2023. Evolution of the germline mutation rate across vertebrates. Nature 615:285–291.
First citation in article CrossrefMedlineGoogle Scholar
Blanquart, F., S. Gandon, and S. L. Nuismer. 2012. The effects of migration and drift on local adaptation to a heterogeneous environment. Journal of Evolutionary Biology 25:1351–1363.
First citation in article CrossrefMedlineGoogle Scholar
Bomblies, K., and C. L. Peichel. 2022. Genetics of adaptation. Proceedings of the National Academy of Sciences of the USA 119:e2122152119.
First citation in article CrossrefMedlineGoogle Scholar
Böndel, K. B., T. Samuels, R. J. Craig, R. W. Ness, N. Colegrave, and P. D. Keightley. 2022. The distribution of fitness effects of spontaneous mutations in Chlamydomonas reinhardtii inferred using frequency changes under experimental evolution. PLoS Genetics 18:e1009840.
First citation in article CrossrefMedlineGoogle Scholar
Boyle, E. A., Y. I. Li, and J. K. Pritchard. 2017. An expanded view of complex traits: from polygenic to omnigenic. Cell 169:1177–1186.
First citation in article CrossrefMedlineGoogle Scholar
Callens, M., L. Pradier, M. Finnegan, C. Rose, and S. Bedhomme. 2021. Read between the lines: diversity of non-translational selection pressures on local codon usage. Genome Biology and Evolution 13:evab097.
First citation in article Google Scholar
Cano, A. V., and J. L. Payne. 2020. Mutation bias interacts with composition bias to influence adaptive evolution. PLoS Computational Biology 16:e1008296.
First citation in article CrossrefMedlineGoogle Scholar
Cano, A. V., H. Rozhoňová, A. Stoltzfus, D. M. McCandlish, and J. L. Payne. 2022. Mutation bias shapes the spectrum of adaptive substitutions. Proceedings of the National Academy of Sciences of the USA 119:e2119720119.
First citation in article CrossrefMedlineGoogle Scholar
Carius, H. J., T. J. Little, and D. Ebert. 2001. Genetic variation in a host-parasite association: potential for coevolution and frequency-dependent selection. Evolution 55:1136–1145.
First citation in article CrossrefMedlineGoogle Scholar
Carlini, D. B. 2004. Experimental reduction of codon bias in the Drosophila alcohol dehydrogenase gene results in decreased ethanol tolerance of adult flies. Journal of Evolutionary Biology 17:779–785.
First citation in article CrossrefMedlineGoogle Scholar
Carlini, D. B., and W. Stephan. 2003. In vivo introduction of unpreferred synonymous codons into the Drosophila Adh gene results in reduced levels of ADH protein. Genetics 163:239–243.
First citation in article CrossrefMedlineGoogle Scholar
Carroll, S. B. 2008. Evo-devo and an expanding evolutionary synthesis: a genetic theory of morphological evolution. Cell 134:25–36.
First citation in article CrossrefMedlineGoogle Scholar
Charlesworth, B., and J. D. Jensen. 2021. Effects of selection at linked sites on patterns of genetic variability. Annual Review of Ecology, Evolution, and Systematics 52:177–197.
First citation in article CrossrefMedlineGoogle Scholar
Charlesworth, B., M. Nordborg, and D. Charlesworth. 1997. The effects of local selection, balanced polymorphism and background selection on equilibrium patterns of genetic diversity in subdivided populations. Genetics Research 70:155–174.
First citation in article CrossrefMedlineGoogle Scholar
Chen, J., T. Bataillon, S. Glémin, and M. Lascoux. 2022. What does the distribution of fitness effects of new mutations reflect? insights from plants. New Phytologist 233:1613–1619.
First citation in article CrossrefMedlineGoogle Scholar
Clarke, B. C. 1979. The evolution of genetic diversity. Proceedings of the Royal Society B 205:453–474.
First citation in article CrossrefMedlineGoogle Scholar
Connallon, T., and K. A. Hodgins. 2021. Allen Orr and the genetics of adaptation. Evolution 75:2624–2640.
First citation in article CrossrefMedlineGoogle Scholar
Couce, A., J. R. Guelfo, and J. Blázquez. 2013. Mutational spectrum drives the rise of mutator bacteria. PLoS Genetics 9:e1003167.
First citation in article CrossrefMedlineGoogle Scholar
Couce, A., A. Rodríguez-Rojas, and J. Blázquez. 2015. Bypass of genetic constraints during mutator evolution to antibiotic resistance. Proceedings of the Royal Society B 282:20142698.
First citation in article CrossrefMedlineGoogle Scholar
Denamur, E., and I. Matic. 2006. Evolution of mutation rates in bacteria. Molecular Microbiology 60:820–827.
First citation in article CrossrefMedlineGoogle Scholar
Dickinson, W. J. 2008. Synergistic fitness interactions and a high frequency of beneficial changes among mutations accumulated under relaxed selection in Saccharomyces cerevisiae. Genetics 178:1571–1578.
First citation in article CrossrefMedlineGoogle Scholar
Dorant, Y., H. Cayuela, K. Wellband, M. Laporte, Q. Rougemont, C. Mérot, E. Normandeau, et al. 2020. Copy number variants outperform SNPs to reveal genotype–temperature association in a marine species. Molecular Ecology 29:4765–4782.
First citation in article CrossrefMedlineGoogle Scholar
Ellegren, H., and N. Galtier. 2016. Determinants of genetic diversity. Nature Reviews Genetics 17:422–433.
First citation in article CrossrefMedlineGoogle Scholar
Eyre-Walker, A., and P. D. Keightley. 2007. The distribution of fitness effects of new mutations. Nature Reviews Genetics 8:610–618.
First citation in article CrossrefMedlineGoogle Scholar
Fiers, W., R. Contreras, F. Duerinck, G. Haegeman, D. Iserentant, J. Merregaert, W. M. Jou, et al. 1976. Complete nucleotide sequence of bacteriophage MS2 RNA: primary and secondary structure of the replicase gene. Nature 260:500–507.
First citation in article CrossrefMedlineGoogle Scholar
Fisher, R. A. 1930. The genetical theory of natural selection. Clarendon, London.
First citation in article Google Scholar
Foster, P. L. 2008. Stress-induced mutagenesis in bacteria. Critical Reviews in Biochemistry and Molecular Biology 42:373–397.
First citation in article CrossrefGoogle Scholar
Foster, P. L., H. Lee, E. Popodi, J. P. Townes, and H. Tang. 2015. Determinants of spontaneous mutation in the bacterium Escherichia coli as revealed by whole-genome sequencing. Proceedings of the National Academy of Sciences of the USA 112:E5990–E5999.
First citation in article CrossrefMedlineGoogle Scholar
Frankham, R. 2012. How closely does genetic diversity in finite populations conform to predictions of neutral theory? large deficits in regions of low recombination. Heredity 108:167–178.
First citation in article CrossrefMedlineGoogle Scholar
Friedberg, E. C. 2003. DNA damage and repair. Nature 421:436–440.
First citation in article CrossrefMedlineGoogle Scholar
———. 2005. DNA repair and mutagenesis. E. C. Friedberg, G. C. Walker, W. Siede, and R. D. Wood, eds. ASM Press, Washington, DC.
First citation in article Google Scholar
Friedberg, E. C., L. D. McDaniel, and R. A. Schultz. 2004. The role of endogenous and exogenous DNA damage and mutagenesis. Current Opinion in Genetics and Development 14:5–10.
First citation in article CrossrefMedlineGoogle Scholar
Fry, J. D., and S. L. Heinsohn. 2002. Environment dependence of mutational parameters for viability in Drosophila melanogaster. Genetics 161:1155–1167.
First citation in article CrossrefMedlineGoogle Scholar
Gaines, T. A., E. L. Patterson, and P. Neve. 2019. Molecular mechanisms of adaptive evolution revealed by global selection for glyphosate resistance. New Phytologist 223:1770–1775.
First citation in article CrossrefMedlineGoogle Scholar
Garcia-Elfring, A., A. Paccard, T. J. Thurman, B. A. Wasserman, E. P. Palkovacs, A. P. Hendry, and R. D. H. Barrett. 2021. Using seasonal genomic changes to understand historical adaptation to new environments: parallel selection on stickleback in highly-variable estuaries. Molecular Ecology 30:2054–2064.
First citation in article CrossrefMedlineGoogle Scholar
Gitschlag, B. L., A. V. Cano, J. L. Payne, D. M. McCandlish, and A. Stoltzfus. Mutation and selection induce correlations between selection coefficients and mutation rates. American Naturalist 202:534–557.
First citation in article Google Scholar
Gogarten, J. P., and J. P. Townsend. 2005. Horizontal gene transfer, genome innovation and evolution. Nature Reviews Microbiology 3:679–687.
First citation in article CrossrefMedlineGoogle Scholar
Goldberg, M. E., and K. Harris. 2021. Mutational signatures of replication timing and epigenetic modification persist through the global divergence of mutation spectra across the great ape phylogeny. Genome Biology and Evolution 14:evab104.
First citation in article Google Scholar
Gomez, K., J. Bertram, and J. Masel. 2020. Mutation bias can shape adaptation in large asexual populations experiencing clonal interference. Proceedings of the Royal Society B 287:20201503.
First citation in article CrossrefMedlineGoogle Scholar
Grantham, R. 1978. Viral, prokaryote and eukaryote genes contrasted by mRNA sequence indexes. FEBS Letters 95:1–11.
First citation in article CrossrefMedlineGoogle Scholar
Grantham, R., C. Gautier, M. Gouy, R. Mercier, and A. Pavé. 1980. Codon catalog usage and the genome hypothesis. Nucleic Acids Research 8:197–197.
First citation in article CrossrefMedlineGoogle Scholar
Haenel, Q., L. Guerard, A. D. C. MacColl, and D. Berner. 2022. The maintenance of standing genetic variation: gene flow vs. selective neutrality in Atlantic stickleback fish. Molecular Ecology 31:811–821.
First citation in article CrossrefMedlineGoogle Scholar
Haenel, Q., M. Roesti, D. Moser, A. D. C. MacColl, and D. Berner. 2019. Predictable genome-wide sorting of standing genetic variation during parallel adaptation to basic versus acidic environments in stickleback fish. Evolution Letters 3:28–42.
First citation in article CrossrefMedlineGoogle Scholar
Hardy, N. B., C. Kaczvinsky, G. Bird, and B. B. Normark. 2020. What we don’t know about diet-breadth evolution in herbivorous insects. Annual Review of Ecology, Evolution, and Systematics 51:103–122.
First citation in article CrossrefGoogle Scholar
Hedrick, P. W. 2013. Adaptive introgression in animals: examples and comparison to new mutation and standing variation as sources of adaptive variation. Molecular Ecology 22:4606–4618.
First citation in article CrossrefMedlineGoogle Scholar
Hellberg, M. E. 2009. Gene flow and isolation among populations of marine animals. Annual Review of Ecology, Evolution, and Systematics 40:291–310.
First citation in article CrossrefGoogle Scholar
Hendry, A. P. 2013. Key questions in the genetics and genomics of eco-evolutionary dynamics. Heredity 111:456–466.
First citation in article CrossrefMedlineGoogle Scholar
Hendry, A. P., D. J. Schoen, M. E. Wolak, and J. M. Reid. 2018. The contemporary evolution of fitness. Annual Review of Ecology, Evolution, and Systematics 49:457–476.
First citation in article CrossrefGoogle Scholar
Hermisson, J., and P. S. Pennings. 2005. Soft sweeps. Genetics 169:2335–2352.
First citation in article CrossrefMedlineGoogle Scholar
Hershberg, R., and D. A. Petrov. 2008. Selection on codon bias. Annual Review of Genetics 42:287–299.
First citation in article CrossrefMedlineGoogle Scholar
Hodgkinson, A., and A. Eyre-Walker. 2011. Variation in the mutation rate across mammalian genomes. Nature Reviews Genetics 12:756–766.
First citation in article CrossrefMedlineGoogle Scholar
Hoekema, A., R. A. Kastelein, M. Vasser, and H. A. de Boer. 1987. Codon replacement in the PGK1 gene of Saccharomyces cerevisiae: experimental approach to study the role of biased codon usage in gene expression. Molecular and Cellular Biology 7:2914–2924.
First citation in article MedlineGoogle Scholar
Hoekstra, H. E., and J. A. Coyne. 2007. The locus of evolution: evo devo and the genetics of adaptation. Evolution 61:995–1016.
First citation in article CrossrefMedlineGoogle Scholar
Horton, J. S., L. M. Flanagan, R. W. Jackson, N. K. Priest, and T. B. Taylor. 2021. A mutational hotspot that determines highly repeatable evolution can be built and broken by silent genetic changes. Nature Communications 12:6092.
First citation in article CrossrefMedlineGoogle Scholar
Hunt, R. C., V. L. Simhadri, M. Iandoli, Z. E. Sauna, and C. Kimchi-Sarfaty. 2014. Exposing synonymous mutations. Trends in Genetics 30:308–321.
First citation in article CrossrefMedlineGoogle Scholar
Jain, K., and A. James. 2017. Fixation probability of a nonmutator in a large population of asexual mutators. Journal of Theoretical Biology 433:85–93.
First citation in article CrossrefMedlineGoogle Scholar
James, A., and K. Jain. 2016. Fixation probability of rare nonmutator and evolution of mutation rates. Ecology and Evolution 6:755–764.
First citation in article CrossrefMedlineGoogle Scholar
Jensen, J. D. 2014. On the unfounded enthusiasm for soft selective sweeps. Nature Communications 5:5281.
First citation in article CrossrefMedlineGoogle Scholar
Joseph, S. B., and D. W. Hall. 2004. Spontaneous mutations in diploid Saccharomyces cerevisiae more beneficial than expected. Genetics 168:1817–1825.
First citation in article CrossrefMedlineGoogle Scholar
Kapun, M., and T. Flatt. 2019. The adaptive significance of chromosomal inversion polymorphisms in Drosophila melanogaster. Molecular Ecology 28:1263–1282.
First citation in article CrossrefMedlineGoogle Scholar
Katju, V., and U. Bergthorsson. 2018. Old trade, new tricks: insights into the spontaneous mutation process from the partnering of classical mutation accumulation experiments with high-throughput genomic approaches. Genome Biology and Evolution 11:136–165.
First citation in article CrossrefGoogle Scholar
Kazancıoğlu, E., and G. Arnqvist. 2014. The maintenance of mitochondrial genetic variation by negative frequency-dependent selection. Ecology Letters 17:22–27.
First citation in article CrossrefMedlineGoogle Scholar
Kelly, J. K. 2022. The genomic scale of fluctuating selection in a natural plant population. Evolution Letters 6:506–521.
First citation in article CrossrefMedlineGoogle Scholar
Kemble, H., P. Nghe, and O. Tenaillon. 2019. Recent insights into the genotype–phenotype relationship from massively parallel genetic assays. Evolutionary Applications 12:1721–1742.
First citation in article CrossrefMedlineGoogle Scholar
Kersten, S., J. Chang, C. D. Huber, Y. Voichek, C. Lanz, T. Hagmaier, P. Lang, et al. 2021. Standing genetic variation fuels rapid evolution of herbicide resistance in blackgrass. bioRxiv,
First citation in article Google Scholar
King, J. L., and T. H. Jukes. 1969. Non-Darwinian evolution. Science 164:788–798.
First citation in article CrossrefMedlineGoogle Scholar
Kondrashov, F. A. 2012. Gene duplication as a mechanism of genomic adaptation to a changing environment. Proceedings of the Royal Society B 279:5048–5057.
First citation in article CrossrefMedlineGoogle Scholar
Koonin, E. V., K. S. Makarova, and L. Aravind. 2001. Horizontal gene transfer in prokaryotes: quantification and classification. Annual Review of Microbiology 55:709–742.
First citation in article CrossrefMedlineGoogle Scholar
Kuzmin, E., J. S. Taylor, and C. Boone. 2022. Retention of duplicated genes in evolution. Trends in Genetics 38:59–72.
First citation in article CrossrefMedlineGoogle Scholar
Lai, Y.-T., C. K. L. Yeung, K. E. Omland, E.-L. Pang, Y. Hao, B.-Y. Liao, H.-F. Cao, et al. 2019. Standing genetic variation as the predominant source for adaptation of a songbird. Proceedings of the National Academy of Sciences of the USA 116:2152–2157.
First citation in article CrossrefMedlineGoogle Scholar
Lanfear, R., H. Kokko, and A. Eyre-Walker. 2014. Population size and the rate of evolution. Trends in Ecology and Evolution 29:33–41.
First citation in article CrossrefMedlineGoogle Scholar
Lee, H., E. Popodi, H. Tang, and P. L. Foster. 2012. Rate and molecular spectrum of spontaneous mutations in the bacterium Escherichia coli as determined by whole-genome sequencing. Proceedings of the National Academy of Sciences of the USA 109:E2774–E2783.
First citation in article CrossrefMedlineGoogle Scholar
Leffler, E. M., K. Bullaughey, D. R. Matute, W. K. Meyer, L. Ségurel, A. Venkat, P. Andolfatto, et al. 2012. Revisiting an old riddle: what determines genetic diversity levels within species? PLoS Biology 10:e1001388.
First citation in article CrossrefMedlineGoogle Scholar
Lin, Y.-P., C.-Y. Lu, and C.-R. Lee. The past contribution and future fate of genetic variants under climate change in an island population of Musa itinerans. American Naturalist 202:558–570.
First citation in article Google Scholar
Liu, Y. 2020. A code within the genetic code: codon usage regulates co-translational protein folding. Cell Communication and Signaling 18:145.
First citation in article CrossrefMedlineGoogle Scholar
Luo, Z., J. Brock, J. M. Dyer, T. Kutchan, D. Schachtman, M. Augustin, Y. Ge, et al. 2019. Genetic diversity and population structure of a Camelina sativa spring panel. Frontiers in Plant Science 10:184.
First citation in article CrossrefMedlineGoogle Scholar
Lynch, M., M. S. Ackerman, J.-F. Gout, H. Long, W. Sung, W. K. Thomas, and P. L. Foster. 2016. Genetic drift, selection and the evolution of the mutation rate. Nature 17:704–714.
First citation in article Google Scholar
Mackintosh, A., D. R. Laetsch, A. Hayward, B. Charlesworth, M. Waterfall, R. Vila, and K. Lohse. 2019. The determinants of genetic diversity in butterflies. Nature Communications 10:3466.
First citation in article CrossrefMedlineGoogle Scholar
MacPherson, A., and S. L. Nuismer. 2017. The probability of parallel genetic evolution from standing genetic variation. Journal of Evolutionary Biology 30:326–337.
First citation in article CrossrefMedlineGoogle Scholar
Maharjan, R. P., and T. Ferenci. 2017. A shifting mutational landscape in 6 nutritional states: stress-induced mutagenesis as a series of distinct stress input–mutation output relationships. PLoS Biology 15:e2001477-22.
First citation in article CrossrefMedlineGoogle Scholar
Maki, H. 2002. Origins of spontaneous mutations: specificity and directionality of base-substitution, frameshift, and sequence-substitution mutageneses. Annual Review of Genetics 36:279–303.
First citation in article CrossrefMedlineGoogle Scholar
Marques, D. A., J. I. Meier, and O. Seehausen. 2019. A combinatorial view on speciation and adaptive radiation. Trends in Ecology and Evolution 34:531–544.
First citation in article CrossrefMedlineGoogle Scholar
Martincorena, I., and N. M. Luscombe. 2013. Non-random mutation: the evolution of targeted hypermutation and hypomutation. BioEssays 35:123–130.
First citation in article CrossrefMedlineGoogle Scholar
Matuszewski, S., J. Hermisson, and M. Kopp. 2015. Catch me if you can: adaptation from standing genetic variation to a moving phenotypic optimum. Genetics 200:1255–1274.
First citation in article CrossrefMedlineGoogle Scholar
Mayr, E. 1972. The nature of the Darwinian revolution. Science 176:981–989.
First citation in article CrossrefMedlineGoogle Scholar
McCandlish, D. M., and A. Stoltzfus. 2014. Modeling evolution using the probability of fixation: history and implications. Quarterly Review of Biology 89:225–252.
First citation in article LinkGoogle Scholar
McDonald, J. H., and M. Kreitman. 1991. Adaptive protein evolution at the Adh locus in Drosophila. Nature 351:652–654.
First citation in article CrossrefMedlineGoogle Scholar
Mérot, C., R. A. Oomen, A. Tigano, and M. Wellenreuther. 2020. A roadmap for understanding the evolutionary significance of structural genomic variation. Trends in Ecology and Evolution 35:561–572.
First citation in article CrossrefMedlineGoogle Scholar
Miller, C. L., D. Sun, L. H. Thornton, and K. McGuigan. The contribution of mutation to variation in temperature-dependent sprint speed in zebrafish, Danio rerio. American Naturalist 202:519–533.
First citation in article Google Scholar
Mittan-Moreau, C. S., C. Kelehear, L. F. Toledo, J. Bacon, J. M. Guayasamin, A. Snyder, and K. R. Zamudio. 2022. Cryptic lineages and standing genetic variation across independent cane toad introductions. Molecular Ecology 31:6440–6456.
First citation in article CrossrefMedlineGoogle Scholar
Moan, A. L., D. Bekkevold, and J. Hemmer-Hansen. 2021. Evolution at two time frames: ancient structural variants involved in post-glacial divergence of the European plaice (Pleuronectes platessa). Heredity 126:668–683.
First citation in article CrossrefMedlineGoogle Scholar
Nadeau, N. J., and C. D. Jiggins. 2010. A golden age for evolutionary genetics? genomic studies of adaptation in natural populations. Trends in Genetics 26:484–492.
First citation in article CrossrefMedlineGoogle Scholar
Nakamura, H., M. Aibara, R. Kajitani, H. D. J. Mrosso, S. I. Mzighani, A. Toyoda, T. Itoh, et al. 2021. Genomic signatures for species-specific adaptation in Lake Victoria cichlids derived from large-scale standing genetic variation. Molecular Biology and Evolution 38:msab084.
First citation in article Google Scholar
Nielsen, E. E., J. Hemmer-Hansen, P. F. Larsen, and D. Bekkevold. 2009. Population genomics of marine fishes: identifying adaptive variation in space and time. Molecular Ecology 18:3128–3150.
First citation in article CrossrefMedlineGoogle Scholar
Olby, R. 2003. Quiet debut for the double helix. Nature 421:402–405.
First citation in article CrossrefMedlineGoogle Scholar
Olson-Manning, C. F., M. R. Wagner, and T. Mitchell-Olds. 2012. Adaptive evolution: evaluating empirical support for theoretical predictions. Nature Reviews Genetics 13:867–877.
First citation in article CrossrefMedlineGoogle Scholar
Orr, H. A. 1998. The population genetics of adaptation: the distribution of factors fixed during adaptive evolution. Evolution 52:935–949.
First citation in article CrossrefMedlineGoogle Scholar
———. 2000. Adaptation and the cost of complexity. Evolution 54:13–20.
First citation in article CrossrefMedlineGoogle Scholar
Orr, H. A., and A. J. Betancourt. 2001. Haldane’s sieve and adaptation from the standing genetic variation. Genetics 157:875–884.
First citation in article CrossrefMedlineGoogle Scholar
Ostrowski, E. A., D. E. Rozen, and R. E. Lenski. 2005. Pleiotropic effects of beneficial mutations in Escherichia coli. Evolution 59:2343–2352.
First citation in article CrossrefMedlineGoogle Scholar
Payne, J. L., F. Menardo, A. Trauner, S. Borrell, S. M. Gygli, C. Loiseau, S. Gagneux, et al. 2019. Transition bias influences the evolution of antibiotic resistance in Mycobacterium tuberculosis. PLoS Biology 17:e3000265-23.
First citation in article CrossrefMedlineGoogle Scholar
Petak, C., L. Frati, R. S. Brennan, and M. H. Pespeni. Whole-genome sequencing reveals that regulatory and low pleiotropy variants underlie local adaptation to environmental variability in purple sea urchins. American Naturalist 202:571–586.
First citation in article Google Scholar
Plotkin, J. B., and G. Kudla. 2010. Synonymous but not the same: the causes and consequences of codon bias. Nature Reviews Genetics 12:32–42.
First citation in article CrossrefMedlineGoogle Scholar
Raynes, Y., and P. D. Sniegowski. 2014. Experimental evolution and the dynamics of genomic mutation rate modifiers. Heredity 113:375–380.
First citation in article CrossrefMedlineGoogle Scholar
Rokyta, D. R., P. Joyce, S. B. Caudle, and H. A. Wichman. 2005. An empirical test of the mutational landscape model of adaptation using a single-stranded DNA virus. Nature Genetics 37:441–444.
First citation in article CrossrefMedlineGoogle Scholar
Saleh, D., J. Chen, J. Leplé, T. Leroy, L. Truffaut, B. Dencausse, C. Lalanne, et al. 2022. Genome-wide evolutionary response of European oaks during the Anthropocene. Evolution Letters 6:4–20.
First citation in article CrossrefMedlineGoogle Scholar
Sancar, A., L. A. Lindsey-Boltz, K. Ünsal-Kaçmaz, and S. Linn. 2004. Molecular mechanisms of mammalian DNA repair and the DNA damage checkpoints. Annual Review of Biochemistry 73:39–85.
First citation in article CrossrefMedlineGoogle Scholar
Sane, M., G. D. Diwan, B. A. Bhat, L. M. Wahl, and D. Agashe. 2023. Shifts in mutation spectra enhance access to beneficial mutations. Proceedings of the National Academy of Sciences of the USA 120:e2207355120.
First citation in article CrossrefMedlineGoogle Scholar
Sane, M., J. J. Miranda, and D. Agashe. 2018. Antagonistic pleiotropy for carbon use is rare in new mutations. Evolution 72:2202–2213.
First citation in article CrossrefMedlineGoogle Scholar
Sanford, E., and M. W. Kelly. 2011. Local adaptation in marine invertebrates. Marine Science 3:509–535.
First citation in article CrossrefGoogle Scholar
Sauna, Z. E., and C. Kimchi-Sarfaty. 2011. Understanding the contribution of synonymous mutations to human disease. Nature 12:683–691.
First citation in article Google Scholar
Sauna, Z. E., C. Kimchi-Sarfaty, S. V. Ambudkar, and M. M. Gottesman. 2007. Silent polymorphisms speak: how they affect pharmacogenomics and the treatment of cancer. Cancer Research 67:9609–9612.
First citation in article CrossrefMedlineGoogle Scholar
Schaper, S., and A. A. Louis. 2014. The arrival of the frequent: how bias in genotype-phenotype maps can steer populations to local optima. PLoS ONE 9:e86635.
First citation in article CrossrefMedlineGoogle Scholar
Schenk, M. F., M. P. Zwart, S. Hwang, P. Ruelens, E. Severing, J. Krug, and J. A. G. M. de Visser. 2022. Population size mediates the contribution of high-rate and large-benefit mutations to parallel evolution. Nature Ecology and Evolution 6:439–447.
First citation in article CrossrefMedlineGoogle Scholar
Schreiber, F., and M. Ackermann. 2020. Environmental drivers of metabolic heterogeneity in clonal microbial populations. Current Opinion in Biotechnology 62:202–211.
First citation in article CrossrefMedlineGoogle Scholar
Sella, G., and N. H. Barton. 2015. Thinking about the evolution of complex traits in the era of genome-wide association studies. Annual Review of Genomics and Human Genetics 20:1–33.
First citation in article Google Scholar
Sémon, M., and K. H. Wolfe. 2007. Consequences of genome duplication. Current Opinion in Genetics and Development 17:505–512.
First citation in article CrossrefMedlineGoogle Scholar
Shaw, R. G., D. L. Byers, and E. Darmo. 2000. Spontaneous mutational effects on reproductive traits of Arabidopsis thaliana. Genetics 155:369–378.
First citation in article CrossrefMedlineGoogle Scholar
Shaw, R. G., F. H. Shaw, and C. Geyer. 2003. What fraction of mutations reduces fitness? a reply to Keightley and Lynch. Evolution 57:686–689.
First citation in article CrossrefGoogle Scholar
Simons, Y. B., H. Mostafavi, C. J. Smith, J. K. Pritchard, and G. Sella. 2022. Simple scaling laws control the genetic architectures of human complex traits. bioRxiv,
First citation in article Google Scholar
Sinnott-Armstrong, N., S. Naqvi, M. Rivas, and J. K. Pritchard. 2021. GWAS of three molecular traits highlights core genes and pathways alongside a highly polygenic background. eLife 10:e58615.
First citation in article Google Scholar
Slatkin, M. 1978. Spatial patterns in the distributions of polygenic characters. Journal of Theoretical Biology 70:213–228.
First citation in article CrossrefMedlineGoogle Scholar
Small, S. T., C. Costantini, N. Sagnon, M. W. Guelbeogo, S. J. Emrich, A. D. Kern, M. C. Fontaine, et al. 2022. Standing genetic variation and chromosome differences drove rapid ecotype formation in a major malaria vector. bioRxiv,
First citation in article Google Scholar
Smeenk, G., and H. van Attikum. 2013. The chromatin response to DNA breaks: leaving a mark on genome integrity. Annual Review of Biochemistry 82:55–80.
First citation in article CrossrefMedlineGoogle Scholar
Sniegowski, P. D., and P. J. Gerrish. 2010. Beneficial mutations and the dynamics of adaptation in asexual populations. Philosophical Transactions of the Royal Society B 365:1255–1263.
First citation in article CrossrefMedlineGoogle Scholar
Soares, A. de, L. Wardil, L. B. Klaczko, and R. Dickman. 2021. Hidden role of mutations in the evolutionary process. Physical Review E 104:044413.
First citation in article CrossrefMedlineGoogle Scholar
Soltis, P. S., and D. E. Soltis. 2021. Plant genomes: markers of evolutionary history and drivers of evolutionary change. Plants, People, Planet 3:74–82.
First citation in article Google Scholar
Stange, M., R. D. H. Barrett, and A. P. Hendry. 2021. The importance of genomic variation for biodiversity, ecosystems and people. Nature Reviews Genetics 22:89–105.
First citation in article CrossrefMedlineGoogle Scholar
Stearns, F. W. 2010. One hundred years of pleiotropy: a retrospective. Genetics 186:767–773.
First citation in article CrossrefMedlineGoogle Scholar
Stephan, W. 2019. Selective sweeps. Genetics 211:5–13.
First citation in article CrossrefMedlineGoogle Scholar
Stewart, B., N. Gruenheit, A. Baldwin, R. Chisholm, D. Rozen, A. Harwood, J. B. Wolf, et al. 2022. The genetic architecture underlying prey-dependent performance in a microbial predator. Nature Communications 13:319.
First citation in article CrossrefMedlineGoogle Scholar
Stoltzfus, A. 2006. Mutation-biased adaptation in a protein NK model. Molecular Biology and Evolution 23:1852–1862.
First citation in article CrossrefMedlineGoogle Scholar
———. 2021. Mutation, randomness and evolution. Oxford University Press, Oxford.
First citation in article Google Scholar
Stoltzfus, A., and D. M. McCandlish. 2017. Mutational biases influence parallel adaptation. Molecular Biology and Evolution 34:2163–2172.
First citation in article CrossrefMedlineGoogle Scholar
Stoltzfus, A., and L. Y. Yampolsky. 2009. Climbing mount probable: mutation as a cause of nonrandomness in evolution. Journal of Heredity 100:637–647.
First citation in article CrossrefMedlineGoogle Scholar
Storz, J. F., C. Natarajan, A. V. Signore, C. C. Witt, D. M. McCandlish, and A. Stoltzfus. 2019. The role of mutation bias in adaptive molecular evolution: insights from convergent changes in protein function. Philosophical Transactions of the Royal Society B 374:20180238-10.
First citation in article CrossrefMedlineGoogle Scholar
Streisfeld, M. A., and M. D. Rausher. 2010. Population genetics, pleiotropy, and the preferential fixation of mutations during adaptive evolution. Evolution 65:629–642.
First citation in article CrossrefMedlineGoogle Scholar
Taylor, J. S., and J. Raes. 2004. Duplication and divergence: the evolution of new genes and old ideas. Annual Review of Genetics 38:615–643.
First citation in article CrossrefMedlineGoogle Scholar
Tenaillon, O., H. L. Nagard, B. Godelle, and F. Taddei. 2000. Mutators and sex in bacteria: conflict between adaptive strategies. Proceedings of the National Academy of Sciences of the USA 97:10465–10470.
First citation in article CrossrefMedlineGoogle Scholar
Tilman, D. 2000. Causes, consequences and ethics of biodiversity. Nature 405:208–211.
First citation in article CrossrefMedlineGoogle Scholar
Tuffaha, M. Z, S. Varakunan, D. Castellano, R. N. Gutenkunst, and L. M. Wahl. Shifts in mutation bias promote mutators by altering the distribution of fitness effects. American Naturalist 202:503–518.
First citation in article Google Scholar
Vecchyo, D. O.-D., K. E. Lohmueller, and J. Novembre. 2022. Haplotype-based inference of the distribution of fitness effects. Genetics 220:iyac002.
First citation in article Google Scholar
Wagner, G. P., and J. Zhang. 2011. The pleiotropic structure of the genotype–phenotype map: the evolvability of complex organisms. Nature 12:204–213.
First citation in article Google Scholar
Wei, W., W.-C. Ho, M. G. Behringer, S. F. Miller, G. Bcharah, and M. Lynch. 2022. Rapid evolution of mutation rate and spectrum in response to environmental and population-genetic challenges. Nature Communications 13:4752.
First citation in article CrossrefMedlineGoogle Scholar
Wellenreuther, M., and L. Bernatchez. 2018. Eco-evolutionary genomics of chromosomal inversions. Trends in Ecology and Evolution 33:427–440.
First citation in article CrossrefMedlineGoogle Scholar
Weng, X., T. Haque, L. Zhang, S. Razzaque, J. T. Lovell, J. D. Palacio-Mejía, P. Duberney, et al. 2022. A pleiotropic flowering time QTL exhibits gene-by-environment interaction for fitness in a perennial grass. Molecular Biology and Evolution 39:msac203.
First citation in article Google Scholar
Wittmann, M. J., A. O. Bergland, M. W. Feldman, P. S. Schmidt, and D. A. Petrov. 2017. Seasonally fluctuating selection can maintain polymorphism at many loci via segregation lift. Proceedings of the National Academy of Sciences of the USA 114:E9932–E9941.
First citation in article CrossrefMedlineGoogle Scholar
Wortel, M. T., D. Agashe, S. F. Bailey, C. Bank, K. Bisschop, T. Blankers, J. Cairns, et al. 2023. Towards evolutionary predictions: current promises and challenges. Evolutionary Applications 16:3–21.
First citation in article CrossrefMedlineGoogle Scholar
Xie, K. T., G. Wang, A. C. Thompson, J. I. Wucherpfennig, T. E. Reimchen, A. D. C. MacColl, D. Schluter, et al. 2019. DNA fragility in the parallel evolution of pelvic reduction in stickleback fish. Science 363:81–84.
First citation in article CrossrefMedlineGoogle Scholar
Yampolsky, L. Y., and A. Stoltzfus. 2001. Bias in the introduction of variation as an orienting factor in evolution. Evolution and Development 3:73–83.
First citation in article CrossrefMedlineGoogle Scholar
Yi, C., and C. He. 2013. DNA repair by reversal of DNA damage. Cold Spring Harbor Perspectives in Biology 5:a012575.
First citation in article CrossrefMedlineGoogle Scholar
Zeyl, C., and J. A. DeVisser. 2001. Estimates of the rate and distribution of fitness effects of spontaneous mutation in Saccharomyces cerevisiae. Genetics 157:53–61.
First citation in article CrossrefMedlineGoogle Scholar
Zhou, C., Y. Feng, G. Li, M. Wang, J. Jian, Y. Wang, W. Zhang, et al. 2021. The new is old: novel germination strategy evolved from standing genetic variation in weedy rice. Frontiers in Plant Science 12:699464.
First citation in article CrossrefMedlineGoogle Scholar
Editor: Volker H. W. Rudolf
Previous articleNext article
Details
Figures
References
Cited by
The American Naturalist
Volume 202, Number 4October 2023
Published for The American Society of Naturalists
Article DOI
Permissions
Request permissions
Views:5445
Total views on this site
Citations:6
Citations are reported from Crossref
See more details
X (20)
Mendeley (37)
History
Submitted February 21, 2023
Accepted April 17, 2023
Published online September 07, 2023
Keywords
mutation rate
mutation bias
distribution of fitness effects
pleiotropy
standing genetic variation
de novo mutations
© 2023 The University of Chicago. This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License (CC BY-NC 4.0), which permits non-commercial reuse of the work with attribution. For commercial use, contact journalpermissions@press.uchicago.edu.
PDF download
Literature Cited
Agashe, D. 2022. Evolutionary forces that generate SNPs: the evolutionary impacts of synonymous mutations. Pages 15–36 in Z. E. Sauna and C. Kimchi-Sarfaty, eds. Single nucleotide polymorphisms: human variation and a coming revolution in biology and medicine. Springer, Cham.
First citation in article Google Scholar
Allendorf, F. W. 2017. Genetics and the conservation of natural populations: allozymes to genomes. Molecular Ecology 26:420–430.
First citation in article CrossrefMedlineGoogle Scholar
Alves, J. M., M. Carneiro, J. Y. Cheng, A. L. de Matos, M. M. Rahman, L. Loog, P. F. Campos, et al. 2019. Parallel adaptation of rabbit populations to myxoma virus. Science 363:1319–1326.
First citation in article CrossrefMedlineGoogle Scholar
Anderson, J. T., J. H. Willis, and T. Mitchell-Olds. 2011. Evolutionary genetics of plant adaptation. Trends in Genetics 27:258–266.
First citation in article CrossrefMedlineGoogle Scholar
Baduel, P., S. Bray, M. Vallejo-Marin, F. Kolář, and L. Yant. 2018. The “polyploid hop”: shifting challenges and opportunities over the evolutionary lifespan of genome duplications. Frontiers in Ecology and Evolution 6:117.
First citation in article CrossrefGoogle Scholar
Baer, C. F., N. Phillips, D. Ostrow, A. Avalos, D. Blanton, A. Boggs, T. Keller, et al. 2006. Cumulative effects of spontaneous mutations for fitness in caenorhabditis: role of genotype, environment and stress. Genetics 174:1387–1395.
First citation in article CrossrefMedlineGoogle Scholar
Bailey, S. F., L. A. A. Morales, and R. Kassen. 2021. Effects of synonymous mutations beyond codon bias: the evidence for adaptive synonymous substitutions from microbial evolution experiments. Genome Biology and Evolution 13:evab141.
First citation in article Google Scholar
Barghi, N., J. Hermisson, and C. Schlötterer. 2020. Polygenic adaptation: a unifying framework to understand positive selection. Nature Reviews Genetics 21:769–781.
First citation in article CrossrefMedlineGoogle Scholar
Barrett, R. D. H., and H. E. Hoekstra. 2011. Molecular spandrels: tests of adaptation at the genetic level. Nature Reviews Genetics 12:767–780.
First citation in article CrossrefMedlineGoogle Scholar
Barrett, R. D. H., and D. Schluter. 2008. Adaptation from standing genetic variation. Trends in Ecology and Evolution 23:38–44.
First citation in article CrossrefMedlineGoogle Scholar
Beatty, J. 2022. The synthesis and the two scenarios. Evolution 76:6–14.
First citation in article CrossrefMedlineGoogle Scholar
Bergeron, L. A., S. Besenbacher, J. Zheng, P. Li, M. F. Bertelsen, B. Quintard, J. I. Hoffman, et al. 2023. Evolution of the germline mutation rate across vertebrates. Nature 615:285–291.
First citation in article CrossrefMedlineGoogle Scholar
Blanquart, F., S. Gandon, and S. L. Nuismer. 2012. The effects of migration and drift on local adaptation to a heterogeneous environment. Journal of Evolutionary Biology 25:1351–1363.
First citation in article CrossrefMedlineGoogle Scholar
Bomblies, K., and C. L. Peichel. 2022. Genetics of adaptation. Proceedings of the National Academy of Sciences of the USA 119:e2122152119.
First citation in article CrossrefMedlineGoogle Scholar
Böndel, K. B., T. Samuels, R. J. Craig, R. W. Ness, N. Colegrave, and P. D. Keightley. 2022. The distribution of fitness effects of spontaneous mutations in Chlamydomonas reinhardtii inferred using frequency changes under experimental evolution. PLoS Genetics 18:e1009840.
First citation in article CrossrefMedlineGoogle Scholar
Boyle, E. A., Y. I. Li, and J. K. Pritchard. 2017. An expanded view of complex traits: from polygenic to omnigenic. Cell 169:1177–1186.
First citation in article CrossrefMedlineGoogle Scholar
Callens, M., L. Pradier, M. Finnegan, C. Rose, and S. Bedhomme. 2021. Read between the lines: diversity of non-translational selection pressures on local codon usage. Genome Biology and Evolution 13:evab097.
First citation in article Google Scholar
Cano, A. V., and J. L. Payne. 2020. Mutation bias interacts with composition bias to influence adaptive evolution. PLoS Computational Biology 16:e1008296.
First citation in article CrossrefMedlineGoogle Scholar
Cano, A. V., H. Rozhoňová, A. Stoltzfus, D. M. McCandlish, and J. L. Payne. 2022. Mutation bias shapes the spectrum of adaptive substitutions. Proceedings of the National Academy of Sciences of the USA 119:e2119720119.
First citation in article CrossrefMedlineGoogle Scholar
Carius, H. J., T. J. Little, and D. Ebert. 2001. Genetic variation in a host-parasite association: potential for coevolution and frequency-dependent selection. Evolution 55:1136–1145.
First citation in article CrossrefMedlineGoogle Scholar
Carlini, D. B. 2004. Experimental reduction of codon bias in the Drosophila alcohol dehydrogenase gene results in decreased ethanol tolerance of adult flies. Journal of Evolutionary Biology 17:779–785.
First citation in article CrossrefMedlineGoogle Scholar
Carlini, D. B., and W. Stephan. 2003. In vivo introduction of unpreferred synonymous codons into the Drosophila Adh gene results in reduced levels of ADH protein. Genetics 163:239–243.
First citation in article CrossrefMedlineGoogle Scholar
Carroll, S. B. 2008. Evo-devo and an expanding evolutionary synthesis: a genetic theory of morphological evolution. Cell 134:25–36.
First citation in article CrossrefMedlineGoogle Scholar
Charlesworth, B., and J. D. Jensen. 2021. Effects of selection at linked sites on patterns of genetic variability. Annual Review of Ecology, Evolution, and Systematics 52:177–197.
First citation in article CrossrefMedlineGoogle Scholar
Charlesworth, B., M. Nordborg, and D. Charlesworth. 1997. The effects of local selection, balanced polymorphism and background selection on equilibrium patterns of genetic diversity in subdivided populations. Genetics Research 70:155–174.
First citation in article CrossrefMedlineGoogle Scholar
Chen, J., T. Bataillon, S. Glémin, and M. Lascoux. 2022. What does the distribution of fitness effects of new mutations reflect? insights from plants. New Phytologist 233:1613–1619.
First citation in article CrossrefMedlineGoogle Scholar
Clarke, B. C. 1979. The evolution of genetic diversity. Proceedings of the Royal Society B 205:453–474.
First citation in article CrossrefMedlineGoogle Scholar
Connallon, T., and K. A. Hodgins. 2021. Allen Orr and the genetics of adaptation. Evolution 75:2624–2640.
First citation in article CrossrefMedlineGoogle Scholar
Couce, A., J. R. Guelfo, and J. Blázquez. 2013. Mutational spectrum drives the rise of mutator bacteria. PLoS Genetics 9:e1003167.
First citation in article CrossrefMedlineGoogle Scholar
Couce, A., A. Rodríguez-Rojas, and J. Blázquez. 2015. Bypass of genetic constraints during mutator evolution to antibiotic resistance. Proceedings of the Royal Society B 282:20142698.
First citation in article CrossrefMedlineGoogle Scholar
Denamur, E., and I. Matic. 2006. Evolution of mutation rates in bacteria. Molecular Microbiology 60:820–827.
First citation in article CrossrefMedlineGoogle Scholar
Dickinson, W. J. 2008. Synergistic fitness interactions and a high frequency of beneficial changes among mutations accumulated under relaxed selection in Saccharomyces cerevisiae. Genetics 178:1571–1578.
First citation in article CrossrefMedlineGoogle Scholar
Dorant, Y., H. Cayuela, K. Wellband, M. Laporte, Q. Rougemont, C. Mérot, E. Normandeau, et al. 2020. Copy number variants outperform SNPs to reveal genotype–temperature association in a marine species. Molecular Ecology 29:4765–4782.
First citation in article CrossrefMedlineGoogle Scholar
Ellegren, H., and N. Galtier. 2016. Determinants of genetic diversity. Nature Reviews Genetics 17:422–433.
First citation in article CrossrefMedlineGoogle Scholar
Eyre-Walker, A., and P. D. Keightley. 2007. The distribution of fitness effects of new mutations. Nature Reviews Genetics 8:610–618.
First citation in article CrossrefMedlineGoogle Scholar
Fiers, W., R. Contreras, F. Duerinck, G. Haegeman, D. Iserentant, J. Merregaert, W. M. Jou, et al. 1976. Complete nucleotide sequence of bacteriophage MS2 RNA: primary and secondary structure of the replicase gene. Nature 260:500–507.
First citation in article CrossrefMedlineGoogle Scholar
Fisher, R. A. 1930. The genetical theory of natural selection. Clarendon, London.
First citation in article Google Scholar
Foster, P. L. 2008. Stress-induced mutagenesis in bacteria. Critical Reviews in Biochemistry and Molecular Biology 42:373–397.
First citation in article CrossrefGoogle Scholar
Foster, P. L., H. Lee, E. Popodi, J. P. Townes, and H. Tang. 2015. Determinants of spontaneous mutation in the bacterium Escherichia coli as revealed by whole-genome sequencing. Proceedings of the National Academy of Sciences of the USA 112:E5990–E5999.
First citation in article CrossrefMedlineGoogle Scholar
Frankham, R. 2012. How closely does genetic diversity in finite populations conform to predictions of neutral theory? large deficits in regions of low recombination. Heredity 108:167–178.
First citation in article CrossrefMedlineGoogle Scholar
Friedberg, E. C. 2003. DNA damage and repair. Nature 421:436–440.
First citation in article CrossrefMedlineGoogle Scholar
———. 2005. DNA repair and mutagenesis. E. C. Friedberg, G. C. Walker, W. Siede, and R. D. Wood, eds. ASM Press, Washington, DC.
First citation in article Google Scholar
Friedberg, E. C., L. D. McDaniel, and R. A. Schultz. 2004. The role of endogenous and exogenous DNA damage and mutagenesis. Current Opinion in Genetics and Development 14:5–10.
First citation in article CrossrefMedlineGoogle Scholar
Fry, J. D., and S. L. Heinsohn. 2002. Environment dependence of mutational parameters for viability in Drosophila melanogaster. Genetics 161:1155–1167.
First citation in article CrossrefMedlineGoogle Scholar
Gaines, T. A., E. L. Patterson, and P. Neve. 2019. Molecular mechanisms of adaptive evolution revealed by global selection for glyphosate resistance. New Phytologist 223:1770–1775.
First citation in article CrossrefMedlineGoogle Scholar
Garcia-Elfring, A., A. Paccard, T. J. Thurman, B. A. Wasserman, E. P. Palkovacs, A. P. Hendry, and R. D. H. Barrett. 2021. Using seasonal genomic changes to understand historical adaptation to new environments: parallel selection on stickleback in highly-variable estuaries. Molecular Ecology 30:2054–2064.
First citation in article CrossrefMedlineGoogle Scholar
Gitschlag, B. L., A. V. Cano, J. L. Payne, D. M. McCandlish, and A. Stoltzfus. Mutation and selection induce correlations between selection coefficients and mutation rates. American Naturalist 202:534–557.
First citation in article Google Scholar
Gogarten, J. P., and J. P. Townsend. 2005. Horizontal gene transfer, genome innovation and evolution. Nature Reviews Microbiology 3:679–687.
First citation in article CrossrefMedlineGoogle Scholar
Goldberg, M. E., and K. Harris. 2021. Mutational signatures of replication timing and epigenetic modification persist through the global divergence of mutation spectra across the great ape phylogeny. Genome Biology and Evolution 14:evab104.
First citation in article Google Scholar
Gomez, K., J. Bertram, and J. Masel. 2020. Mutation bias can shape adaptation in large asexual populations experiencing clonal interference. Proceedings of the Royal Society B 287:20201503.
First citation in article CrossrefMedlineGoogle Scholar
Grantham, R. 1978. Viral, prokaryote and eukaryote genes contrasted by mRNA sequence indexes. FEBS Letters 95:1–11.
First citation in article CrossrefMedlineGoogle Scholar
Grantham, R., C. Gautier, M. Gouy, R. Mercier, and A. Pavé. 1980. Codon catalog usage and the genome hypothesis. Nucleic Acids Research 8:197–197.
First citation in article CrossrefMedlineGoogle Scholar
Haenel, Q., L. Guerard, A. D. C. MacColl, and D. Berner. 2022. The maintenance of standing genetic variation: gene flow vs. selective neutrality in Atlantic stickleback fish. Molecular Ecology 31:811–821.
First citation in article CrossrefMedlineGoogle Scholar
Haenel, Q., M. Roesti, D. Moser, A. D. C. MacColl, and D. Berner. 2019. Predictable genome-wide sorting of standing genetic variation during parallel adaptation to basic versus acidic environments in stickleback fish. Evolution Letters 3:28–42.
First citation in article CrossrefMedlineGoogle Scholar
Hardy, N. B., C. Kaczvinsky, G. Bird, and B. B. Normark. 2020. What we don’t know about diet-breadth evolution in herbivorous insects. Annual Review of Ecology, Evolution, and Systematics 51:103–122.
First citation in article CrossrefGoogle Scholar
Hedrick, P. W. 2013. Adaptive introgression in animals: examples and comparison to new mutation and standing variation as sources of adaptive variation. Molecular Ecology 22:4606–4618.
First citation in article CrossrefMedlineGoogle Scholar
Hellberg, M. E. 2009. Gene flow and isolation among populations of marine animals. Annual Review of Ecology, Evolution, and Systematics 40:291–310.
First citation in article CrossrefGoogle Scholar
Hendry, A. P. 2013. Key questions in the genetics and genomics of eco-evolutionary dynamics. Heredity 111:456–466.
First citation in article CrossrefMedlineGoogle Scholar
Hendry, A. P., D. J. Schoen, M. E. Wolak, and J. M. Reid. 2018. The contemporary evolution of fitness. Annual Review of Ecology, Evolution, and Systematics 49:457–476.
First citation in article CrossrefGoogle Scholar
Hermisson, J., and P. S. Pennings. 2005. Soft sweeps. Genetics 169:2335–2352.
First citation in article CrossrefMedlineGoogle Scholar
Hershberg, R., and D. A. Petrov. 2008. Selection on codon bias. Annual Review of Genetics 42:287–299.
First citation in article CrossrefMedlineGoogle Scholar
Hodgkinson, A., and A. Eyre-Walker. 2011. Variation in the mutation rate across mammalian genomes. Nature Reviews Genetics 12:756–766.
First citation in article CrossrefMedlineGoogle Scholar
Hoekema, A., R. A. Kastelein, M. Vasser, and H. A. de Boer. 1987. Codon replacement in the PGK1 gene of Saccharomyces cerevisiae: experimental approach to study the role of biased codon usage in gene expression. Molecular and Cellular Biology 7:2914–2924.
First citation in article MedlineGoogle Scholar
Hoekstra, H. E., and J. A. Coyne. 2007. The locus of evolution: evo devo and the genetics of adaptation. Evolution 61:995–1016.
First citation in article CrossrefMedlineGoogle Scholar
Horton, J. S., L. M. Flanagan, R. W. Jackson, N. K. Priest, and T. B. Taylor. 2021. A mutational hotspot that determines highly repeatable evolution can be built and broken by silent genetic changes. Nature Communications 12:6092.
First citation in article CrossrefMedlineGoogle Scholar
Hunt, R. C., V. L. Simhadri, M. Iandoli, Z. E. Sauna, and C. Kimchi-Sarfaty. 2014. Exposing synonymous mutations. Trends in Genetics 30:308–321.
First citation in article CrossrefMedlineGoogle Scholar
Jain, K., and A. James. 2017. Fixation probability of a nonmutator in a large population of asexual mutators. Journal of Theoretical Biology 433:85–93.
First citation in article CrossrefMedlineGoogle Scholar
James, A., and K. Jain. 2016. Fixation probability of rare nonmutator and evolution of mutation rates. Ecology and Evolution 6:755–764.
First citation in article CrossrefMedlineGoogle Scholar
Jensen, J. D. 2014. On the unfounded enthusiasm for soft selective sweeps. Nature Communications 5:5281.
First citation in article CrossrefMedlineGoogle Scholar
Joseph, S. B., and D. W. Hall. 2004. Spontaneous mutations in diploid Saccharomyces cerevisiae more beneficial than expected. Genetics 168:1817–1825.
First citation in article CrossrefMedlineGoogle Scholar
Kapun, M., and T. Flatt. 2019. The adaptive significance of chromosomal inversion polymorphisms in Drosophila melanogaster. Molecular Ecology 28:1263–1282.
First citation in article CrossrefMedlineGoogle Scholar
Katju, V., and U. Bergthorsson. 2018. Old trade, new tricks: insights into the spontaneous mutation process from the partnering of classical mutation accumulation experiments with high-throughput genomic approaches. Genome Biology and Evolution 11:136–165.
First citation in article CrossrefGoogle Scholar
Kazancıoğlu, E., and G. Arnqvist. 2014. The maintenance of mitochondrial genetic variation by negative frequency-dependent selection. Ecology Letters 17:22–27.
First citation in article CrossrefMedlineGoogle Scholar
Kelly, J. K. 2022. The genomic scale of fluctuating selection in a natural plant population. Evolution Letters 6:506–521.
First citation in article CrossrefMedlineGoogle Scholar
Kemble, H., P. Nghe, and O. Tenaillon. 2019. Recent insights into the genotype–phenotype relationship from massively parallel genetic assays. Evolutionary Applications 12:1721–1742.
First citation in article CrossrefMedlineGoogle Scholar
Kersten, S., J. Chang, C. D. Huber, Y. Voichek, C. Lanz, T. Hagmaier, P. Lang, et al. 2021. Standing genetic variation fuels rapid evolution of herbicide resistance in blackgrass. bioRxiv,
First citation in article Google Scholar
King, J. L., and T. H. Jukes. 1969. Non-Darwinian evolution. Science 164:788–798.
First citation in article CrossrefMedlineGoogle Scholar
Kondrashov, F. A. 2012. Gene duplication as a mechanism of genomic adaptation to a changing environment. Proceedings of the Royal Society B 279:5048–5057.
First citation in article CrossrefMedlineGoogle Scholar
Koonin, E. V., K. S. Makarova, and L. Aravind. 2001. Horizontal gene transfer in prokaryotes: quantification and classification. Annual Review of Microbiology 55:709–742.
First citation in article CrossrefMedlineGoogle Scholar
Kuzmin, E., J. S. Taylor, and C. Boone. 2022. Retention of duplicated genes in evolution. Trends in Genetics 38:59–72.
First citation in article CrossrefMedlineGoogle Scholar
Lai, Y.-T., C. K. L. Yeung, K. E. Omland, E.-L. Pang, Y. Hao, B.-Y. Liao, H.-F. Cao, et al. 2019. Standing genetic variation as the predominant source for adaptation of a songbird. Proceedings of the National Academy of Sciences of the USA 116:2152–2157.
First citation in article CrossrefMedlineGoogle Scholar
Lanfear, R., H. Kokko, and A. Eyre-Walker. 2014. Population size and the rate of evolution. Trends in Ecology and Evolution 29:33–41.
First citation in article CrossrefMedlineGoogle Scholar
Lee, H., E. Popodi, H. Tang, and P. L. Foster. 2012. Rate and molecular spectrum of spontaneous mutations in the bacterium Escherichia coli as determined by whole-genome sequencing. Proceedings of the National Academy of Sciences of the USA 109:E2774–E2783.
First citation in article CrossrefMedlineGoogle Scholar
Leffler, E. M., K. Bullaughey, D. R. Matute, W. K. Meyer, L. Ségurel, A. Venkat, P. Andolfatto, et al. 2012. Revisiting an old riddle: what determines genetic diversity levels within species? PLoS Biology 10:e1001388.
First citation in article CrossrefMedlineGoogle Scholar
Lin, Y.-P., C.-Y. Lu, and C.-R. Lee. The past contribution and future fate of genetic variants under climate change in an island population of Musa itinerans. American Naturalist 202:558–570.
First citation in article Google Scholar
Liu, Y. 2020. A code within the genetic code: codon usage regulates co-translational protein folding. Cell Communication and Signaling 18:145.
First citation in article CrossrefMedlineGoogle Scholar
Luo, Z., J. Brock, J. M. Dyer, T. Kutchan, D. Schachtman, M. Augustin, Y. Ge, et al. 2019. Genetic diversity and population structure of a Camelina sativa spring panel. Frontiers in Plant Science 10:184.
First citation in article CrossrefMedlineGoogle Scholar
Lynch, M., M. S. Ackerman, J.-F. Gout, H. Long, W. Sung, W. K. Thomas, and P. L. Foster. 2016. Genetic drift, selection and the evolution of the mutation rate. Nature 17:704–714.
First citation in article Google Scholar
Mackintosh, A., D. R. Laetsch, A. Hayward, B. Charlesworth, M. Waterfall, R. Vila, and K. Lohse. 2019. The determinants of genetic diversity in butterflies. Nature Communications 10:3466.
First citation in article CrossrefMedlineGoogle Scholar
MacPherson, A., and S. L. Nuismer. 2017. The probability of parallel genetic evolution from standing genetic variation. Journal of Evolutionary Biology 30:326–337.
First citation in article CrossrefMedlineGoogle Scholar
Maharjan, R. P., and T. Ferenci. 2017. A shifting mutational landscape in 6 nutritional states: stress-induced mutagenesis as a series of distinct stress input–mutation output relationships. PLoS Biology 15:e2001477-22.
First citation in article CrossrefMedlineGoogle Scholar
Maki, H. 2002. Origins of spontaneous mutations: specificity and directionality of base-substitution, frameshift, and sequence-substitution mutageneses. Annual Review of Genetics 36:279–303.
First citation in article CrossrefMedlineGoogle Scholar
Marques, D. A., J. I. Meier, and O. Seehausen. 2019. A combinatorial view on speciation and adaptive radiation. Trends in Ecology and Evolution 34:531–544.
First citation in article CrossrefMedlineGoogle Scholar
Martincorena, I., and N. M. Luscombe. 2013. Non-random mutation: the evolution of targeted hypermutation and hypomutation. BioEssays 35:123–130.
First citation in article CrossrefMedlineGoogle Scholar
Matuszewski, S., J. Hermisson, and M. Kopp. 2015. Catch me if you can: adaptation from standing genetic variation to a moving phenotypic optimum. Genetics 200:1255–1274.
First citation in article CrossrefMedlineGoogle Scholar
Mayr, E. 1972. The nature of the Darwinian revolution. Science 176:981–989.
First citation in article CrossrefMedlineGoogle Scholar
McCandlish, D. M., and A. Stoltzfus. 2014. Modeling evolution using the probability of fixation: history and implications. Quarterly Review of Biology 89:225–252.
First citation in article LinkGoogle Scholar
McDonald, J. H., and M. Kreitman. 1991. Adaptive protein evolution at the Adh locus in Drosophila. Nature 351:652–654.
First citation in article CrossrefMedlineGoogle Scholar
Mérot, C., R. A. Oomen, A. Tigano, and M. Wellenreuther. 2020. A roadmap for understanding the evolutionary significance of structural genomic variation. Trends in Ecology and Evolution 35:561–572.
First citation in article CrossrefMedlineGoogle Scholar
Miller, C. L., D. Sun, L. H. Thornton, and K. McGuigan. The contribution of mutation to variation in temperature-dependent sprint speed in zebrafish, Danio rerio. American Naturalist 202:519–533.
First citation in article Google Scholar
Mittan-Moreau, C. S., C. Kelehear, L. F. Toledo, J. Bacon, J. M. Guayasamin, A. Snyder, and K. R. Zamudio. 2022. Cryptic lineages and standing genetic variation across independent cane toad introductions. Molecular Ecology 31:6440–6456.
First citation in article CrossrefMedlineGoogle Scholar
Moan, A. L., D. Bekkevold, and J. Hemmer-Hansen. 2021. Evolution at two time frames: ancient structural variants involved in post-glacial divergence of the European plaice (Pleuronectes platessa). Heredity 126:668–683.
First citation in article CrossrefMedlineGoogle Scholar
Nadeau, N. J., and C. D. Jiggins. 2010. A golden age for evolutionary genetics? genomic studies of adaptation in natural populations. Trends in Genetics 26:484–492.
First citation in article CrossrefMedlineGoogle Scholar
Nakamura, H., M. Aibara, R. Kajitani, H. D. J. Mrosso, S. I. Mzighani, A. Toyoda, T. Itoh, et al. 2021. Genomic signatures for species-specific adaptation in Lake Victoria cichlids derived from large-scale standing genetic variation. Molecular Biology and Evolution 38:msab084.
First citation in article Google Scholar
Nielsen, E. E., J. Hemmer-Hansen, P. F. Larsen, and D. Bekkevold. 2009. Population genomics of marine fishes: identifying adaptive variation in space and time. Molecular Ecology 18:3128–3150.
First citation in article CrossrefMedlineGoogle Scholar
Olby, R. 2003. Quiet debut for the double helix. Nature 421:402–405.
First citation in article CrossrefMedlineGoogle Scholar
Olson-Manning, C. F., M. R. Wagner, and T. Mitchell-Olds. 2012. Adaptive evolution: evaluating empirical support for theoretical predictions. Nature Reviews Genetics 13:867–877.
First citation in article CrossrefMedlineGoogle Scholar
Orr, H. A. 1998. The population genetics of adaptation: the distribution of factors fixed during adaptive evolution. Evolution 52:935–949.
First citation in article CrossrefMedlineGoogle Scholar
———. 2000. Adaptation and the cost of complexity. Evolution 54:13–20.
First citation in article CrossrefMedlineGoogle Scholar
Orr, H. A., and A. J. Betancourt. 2001. Haldane’s sieve and adaptation from the standing genetic variation. Genetics 157:875–884.
First citation in article CrossrefMedlineGoogle Scholar
Ostrowski, E. A., D. E. Rozen, and R. E. Lenski. 2005. Pleiotropic effects of beneficial mutations in Escherichia coli. Evolution 59:2343–2352.
First citation in article CrossrefMedlineGoogle Scholar
Payne, J. L., F. Menardo, A. Trauner, S. Borrell, S. M. Gygli, C. Loiseau, S. Gagneux, et al. 2019. Transition bias influences the evolution of antibiotic resistance in Mycobacterium tuberculosis. PLoS Biology 17:e3000265-23.
First citation in article CrossrefMedlineGoogle Scholar
Petak, C., L. Frati, R. S. Brennan, and M. H. Pespeni. Whole-genome sequencing reveals that regulatory and low pleiotropy variants underlie local adaptation to environmental variability in purple sea urchins. American Naturalist 202:571–586.
First citation in article Google Scholar
Plotkin, J. B., and G. Kudla. 2010. Synonymous but not the same: the causes and consequences of codon bias. Nature Reviews Genetics 12:32–42.
First citation in article CrossrefMedlineGoogle Scholar
Raynes, Y., and P. D. Sniegowski. 2014. Experimental evolution and the dynamics of genomic mutation rate modifiers. Heredity 113:375–380.
First citation in article CrossrefMedlineGoogle Scholar
Rokyta, D. R., P. Joyce, S. B. Caudle, and H. A. Wichman. 2005. An empirical test of the mutational landscape model of adaptation using a single-stranded DNA virus. Nature Genetics 37:441–444.
First citation in article CrossrefMedlineGoogle Scholar
Saleh, D., J. Chen, J. Leplé, T. Leroy, L. Truffaut, B. Dencausse, C. Lalanne, et al. 2022. Genome-wide evolutionary response of European oaks during the Anthropocene. Evolution Letters 6:4–20.
First citation in article CrossrefMedlineGoogle Scholar
Sancar, A., L. A. Lindsey-Boltz, K. Ünsal-Kaçmaz, and S. Linn. 2004. Molecular mechanisms of mammalian DNA repair and the DNA damage checkpoints. Annual Review of Biochemistry 73:39–85.
First citation in article CrossrefMedlineGoogle Scholar
Sane, M., G. D. Diwan, B. A. Bhat, L. M. Wahl, and D. Agashe. 2023. Shifts in mutation spectra enhance access to beneficial mutations. Proceedings of the National Academy of Sciences of the USA 120:e2207355120.
First citation in article CrossrefMedlineGoogle Scholar
Sane, M., J. J. Miranda, and D. Agashe. 2018. Antagonistic pleiotropy for carbon use is rare in new mutations. Evolution 72:2202–2213.
First citation in article CrossrefMedlineGoogle Scholar
Sanford, E., and M. W. Kelly. 2011. Local adaptation in marine invertebrates. Marine Science 3:509–535.
First citation in article CrossrefGoogle Scholar
Sauna, Z. E., and C. Kimchi-Sarfaty. 2011. Understanding the contribution of synonymous mutations to human disease. Nature 12:683–691.
First citation in article Google Scholar
Sauna, Z. E., C. Kimchi-Sarfaty, S. V. Ambudkar, and M. M. Gottesman. 2007. Silent polymorphisms speak: how they affect pharmacogenomics and the treatment of cancer. Cancer Research 67:9609–9612.
First citation in article CrossrefMedlineGoogle Scholar
Schaper, S., and A. A. Louis. 2014. The arrival of the frequent: how bias in genotype-phenotype maps can steer populations to local optima. PLoS ONE 9:e86635.
First citation in article CrossrefMedlineGoogle Scholar
Schenk, M. F., M. P. Zwart, S. Hwang, P. Ruelens, E. Severing, J. Krug, and J. A. G. M. de Visser. 2022. Population size mediates the contribution of high-rate and large-benefit mutations to parallel evolution. Nature Ecology and Evolution 6:439–447.
First citation in article CrossrefMedlineGoogle Scholar
Schreiber, F., and M. Ackermann. 2020. Environmental drivers of metabolic heterogeneity in clonal microbial populations. Current Opinion in Biotechnology 62:202–211.
First citation in article CrossrefMedlineGoogle Scholar
Sella, G., and N. H. Barton. 2015. Thinking about the evolution of complex traits in the era of genome-wide association studies. Annual Review of Genomics and Human Genetics 20:1–33.
First citation in article Google Scholar
Sémon, M., and K. H. Wolfe. 2007. Consequences of genome duplication. Current Opinion in Genetics and Development 17:505–512.
First citation in article CrossrefMedlineGoogle Scholar
Shaw, R. G., D. L. Byers, and E. Darmo. 2000. Spontaneous mutational effects on reproductive traits of Arabidopsis thaliana. Genetics 155:369–378.
First citation in article CrossrefMedlineGoogle Scholar
Shaw, R. G., F. H. Shaw, and C. Geyer. 2003. What fraction of mutations reduces fitness? a reply to Keightley and Lynch. Evolution 57:686–689.
First citation in article CrossrefGoogle Scholar
Simons, Y. B., H. Mostafavi, C. J. Smith, J. K. Pritchard, and G. Sella. 2022. Simple scaling laws control the genetic architectures of human complex traits. bioRxiv,
First citation in article Google Scholar
Sinnott-Armstrong, N., S. Naqvi, M. Rivas, and J. K. Pritchard. 2021. GWAS of three molecular traits highlights core genes and pathways alongside a highly polygenic background. eLife 10:e58615.
First citation in article Google Scholar
Slatkin, M. 1978. Spatial patterns in the distributions of polygenic characters. Journal of Theoretical Biology 70:213–228.
First citation in article CrossrefMedlineGoogle Scholar
Small, S. T., C. Costantini, N. Sagnon, M. W. Guelbeogo, S. J. Emrich, A. D. Kern, M. C. Fontaine, et al. 2022. Standing genetic variation and chromosome differences drove rapid ecotype formation in a major malaria vector. bioRxiv,
First citation in article Google Scholar
Smeenk, G., and H. van Attikum. 2013. The chromatin response to DNA breaks: leaving a mark on genome integrity. Annual Review of Biochemistry 82:55–80.
First citation in article CrossrefMedlineGoogle Scholar
Sniegowski, P. D., and P. J. Gerrish. 2010. Beneficial mutations and the dynamics of adaptation in asexual populations. Philosophical Transactions of the Royal Society B 365:1255–1263.
First citation in article CrossrefMedlineGoogle Scholar
Soares, A. de, L. Wardil, L. B. Klaczko, and R. Dickman. 2021. Hidden role of mutations in the evolutionary process. Physical Review E 104:044413.
First citation in article CrossrefMedlineGoogle Scholar
Soltis, P. S., and D. E. Soltis. 2021. Plant genomes: markers of evolutionary history and drivers of evolutionary change. Plants, People, Planet 3:74–82.
First citation in article Google Scholar
Stange, M., R. D. H. Barrett, and A. P. Hendry. 2021. The importance of genomic variation for biodiversity, ecosystems and people. Nature Reviews Genetics 22:89–105.
First citation in article CrossrefMedlineGoogle Scholar
Stearns, F. W. 2010. One hundred years of pleiotropy: a retrospective. Genetics 186:767–773.
First citation in article CrossrefMedlineGoogle Scholar
Stephan, W. 2019. Selective sweeps. Genetics 211:5–13.
First citation in article CrossrefMedlineGoogle Scholar
Stewart, B., N. Gruenheit, A. Baldwin, R. Chisholm, D. Rozen, A. Harwood, J. B. Wolf, et al. 2022. The genetic architecture underlying prey-dependent performance in a microbial predator. Nature Communications 13:319.
First citation in article CrossrefMedlineGoogle Scholar
Stoltzfus, A. 2006. Mutation-biased adaptation in a protein NK model. Molecular Biology and Evolution 23:1852–1862.
First citation in article CrossrefMedlineGoogle Scholar
———. 2021. Mutation, randomness and evolution. Oxford University Press, Oxford.
First citation in article Google Scholar
Stoltzfus, A., and D. M. McCandlish. 2017. Mutational biases influence parallel adaptation. Molecular Biology and Evolution 34:2163–2172.
First citation in article CrossrefMedlineGoogle Scholar
Stoltzfus, A., and L. Y. Yampolsky. 2009. Climbing mount probable: mutation as a cause of nonrandomness in evolution. Journal of Heredity 100:637–647.
First citation in article CrossrefMedlineGoogle Scholar
Storz, J. F., C. Natarajan, A. V. Signore, C. C. Witt, D. M. McCandlish, and A. Stoltzfus. 2019. The role of mutation bias in adaptive molecular evolution: insights from convergent changes in protein function. Philosophical Transactions of the Royal Society B 374:20180238-10.
First citation in article CrossrefMedlineGoogle Scholar
Streisfeld, M. A., and M. D. Rausher. 2010. Population genetics, pleiotropy, and the preferential fixation of mutations during adaptive evolution. Evolution 65:629–642.
First citation in article CrossrefMedlineGoogle Scholar
Taylor, J. S., and J. Raes. 2004. Duplication and divergence: the evolution of new genes and old ideas. Annual Review of Genetics 38:615–643.
First citation in article CrossrefMedlineGoogle Scholar
Tenaillon, O., H. L. Nagard, B. Godelle, and F. Taddei. 2000. Mutators and sex in bacteria: conflict between adaptive strategies. Proceedings of the National Academy of Sciences of the USA 97:10465–10470.
First citation in article CrossrefMedlineGoogle Scholar
Tilman, D. 2000. Causes, consequences and ethics of biodiversity. Nature 405:208–211.
First citation in article CrossrefMedlineGoogle Scholar
Tuffaha, M. Z, S. Varakunan, D. Castellano, R. N. Gutenkunst, and L. M. Wahl. Shifts in mutation bias promote mutators by altering the distribution of fitness effects. American Naturalist 202:503–518.
First citation in article Google Scholar
Vecchyo, D. O.-D., K. E. Lohmueller, and J. Novembre. 2022. Haplotype-based inference of the distribution of fitness effects. Genetics 220:iyac002.
First citation in article Google Scholar
Wagner, G. P., and J. Zhang. 2011. The pleiotropic structure of the genotype–phenotype map: the evolvability of complex organisms. Nature 12:204–213.
First citation in article Google Scholar
Wei, W., W.-C. Ho, M. G. Behringer, S. F. Miller, G. Bcharah, and M. Lynch. 2022. Rapid evolution of mutation rate and spectrum in response to environmental and population-genetic challenges. Nature Communications 13:4752.
First citation in article CrossrefMedlineGoogle Scholar
Wellenreuther, M., and L. Bernatchez. 2018. Eco-evolutionary genomics of chromosomal inversions. Trends in Ecology and Evolution 33:427–440.
First citation in article CrossrefMedlineGoogle Scholar
Weng, X., T. Haque, L. Zhang, S. Razzaque, J. T. Lovell, J. D. Palacio-Mejía, P. Duberney, et al. 2022. A pleiotropic flowering time QTL exhibits gene-by-environment interaction for fitness in a perennial grass. Molecular Biology and Evolution 39:msac203.
First citation in article Google Scholar
Wittmann, M. J., A. O. Bergland, M. W. Feldman, P. S. Schmidt, and D. A. Petrov. 2017. Seasonally fluctuating selection can maintain polymorphism at many loci via segregation lift. Proceedings of the National Academy of Sciences of the USA 114:E9932–E9941.
First citation in article CrossrefMedlineGoogle Scholar
Wortel, M. T., D. Agashe, S. F. Bailey, C. Bank, K. Bisschop, T. Blankers, J. Cairns, et al. 2023. Towards evolutionary predictions: current promises and challenges. Evolutionary Applications 16:3–21.
First citation in article CrossrefMedlineGoogle Scholar
Xie, K. T., G. Wang, A. C. Thompson, J. I. Wucherpfennig, T. E. Reimchen, A. D. C. MacColl, D. Schluter, et al. 2019. DNA fragility in the parallel evolution of pelvic reduction in stickleback fish. Science 363:81–84.
First citation in article CrossrefMedlineGoogle Scholar
Yampolsky, L. Y., and A. Stoltzfus. 2001. Bias in the introduction of variation as an orienting factor in evolution. Evolution and Development 3:73–83.
First citation in article CrossrefMedlineGoogle Scholar
Yi, C., and C. He. 2013. DNA repair by reversal of DNA damage. Cold Spring Harbor Perspectives in Biology 5:a012575.
First citation in article CrossrefMedlineGoogle Scholar
Zeyl, C., and J. A. DeVisser. 2001. Estimates of the rate and distribution of fitness effects of spontaneous mutation in Saccharomyces cerevisiae. Genetics 157:53–61.
First citation in article CrossrefMedlineGoogle Scholar
Zhou, C., Y. Feng, G. Li, M. Wang, J. Jian, Y. Wang, W. Zhang, et al. 2021. The new is old: novel germination strategy evolved from standing genetic variation in weedy rice. Frontiers in Plant Science 12:699464.
First citation in article CrossrefMedlineGoogle Scholar
Crossref reports the following articles citing this article:
Athina Karapli‐Petritsopoulou, Jasmin Josephine Heckelmann, Dörthe Becker, N. John Anderson, Dagmar FrischAltered Phenotypic Responses of Asexual Arctic Daphnia After 10 Years of Rapid Climate Change, Global Change Biology 31,no.3 3(Mar 2025). G. Jithu Priya, S. Sibi Sidharth, R. Rahul, Parthiban Brindha DeviGenetic Adaptations Inradioactive Environments: Lessons from Organisms Near Nuclear Waste Sites, (Dec 2024):307–324. Haoran Cai, Diogo Melo, David L. Des MaraisDisentangling variational bias: the roles of development, mutation, and selection, Trends in Genetics 41,no.1 1(Jan 2025):23–32. McKenze J. Moss, Laura M. Chamness, Patricia L. ClarkThe Effects of Codon Usage on Protein Structure and Folding, Annual Review of Biophysics 53,no.1 1(Jul 2024):87–108. Xudong Wu, Decai Xiang, Wei Zhang, Yu Ma, Guiying Zhao, Zongjun YinIdentification of Breed-Specific SNPs of Danish Large White Pig in Comparison with Four Chinese Local Pig Breed Genomes, Genes 15,no.5 5(May 2024):623. Erik D. Enbody, Ashley T. Sendell-Price, C. Grace Sprehn, Carl-Johan Rubin, Peter M. Visscher, B. Rosemary Grant, Peter R. Grant, Leif AnderssonCommunity-wide genome sequencing reveals 30 years of Darwin’s finch evolution, Science 381,no.6665 6665(Sep 2023).
Close Figure Viewer
Browse All Figures Return to Figure
Previous Figure Next Figure
Caption
back
The University of Chicago Press Books
Chicago Distribution Center
The University of Chicago
Accessibility
Open access at Chicago
Permissions
Statement of Publication Ethics
Diversity and Inclusion at the University of Chicago
Follow us on facebook
Follow us on Twitter
Contact us
Terms and Conditions
Privacy Notice
Media and advertising requests
Follow us on facebook
Follow us on Twitter
Follow us on BlueSky
✓
Thanks for sharing!
AddToAny
More…
Close crossmark popup |
8612 | https://www.quora.com/What-is-a-proof-that-every-positive-integer-can-be-written-as-the-sum-of-one-or-more-distinct-non-consecutive-Fibonacci-numbers | Something went wrong. Wait a moment and try again.
Fibonacci Series
Proofs of Mathematics For...
Number Sequences
Arithmetic Of Integers
Mathematical Proof
Number Theory
Series and Sequences
Fibonacci Numbers
5
What is a proof that every positive integer can be written as the sum of one or more distinct non-consecutive Fibonacci numbers?
B.A. with MMath in Mathematics, University of Cambridge (Graduated 2023)
·
Upvoted by
Horst H. von Brand
, PhD Computer Science & Mathematics, Louisiana State University (1987) · Author has 1.3K answers and 5.5M answer views
·
7y
We use something called a Brown’s criterion
, but before that let's define what's called a complete sequence .
Definition of Complete Sequence
A sequence of positive integers is called complete if and only if every positive integer n can be written as the sum of one or more distinct numbers in the sequence.
Now we can enter Brown’s criterion:
Lemma 1 Brown’s Criterion
A sequence {a1,a2,⋯} is complete if and only if
a1=1(1)
a1+a2+⋯+an−1≥an(2)
This fact is, I would say, trivial, but it serves as a framework for us to prove the statement i
Brown's Criterion
Complete Sequence
We use something called a Brown’s criterion
, but before that let's define what's called a complete sequence
.
Definition of Complete Sequence
A sequence of positive integers is called complete if and only if every positive integer n can be written as the sum of one or more distinct numbers in the sequence.
Now we can enter Brown’s criterion:
Lemma 1 Brown’s Criterion
A sequence {a1,a2,⋯} is complete if and only if
a1=1(1)
a1+a2+⋯+an−1≥an(2)
This fact is, I would say, trivial, but it serves as a framework for us to prove the statement in the question.
If you still want to prove it, you can use a strong mathematical induction: by supposing every integer before an and itself can be expressed as the sum of integers in the sequence, then every integer between an and 2an can be expressed as the sum of integers in the sequence. For any integers beyond 2an, because of (2), it can expressed as an+1 plus any integer less than it. Since it is trivial that this happens when n=1, by the principle of mathematical induction, this is true.
Now the question is equivalent to asking whether Fibonacci numbers form a complete sequence, and the way to check is to see if it fulfils the Brown’s Criterion.
For (1), it is trivial because in the question, we slightly from down the Fibonacci sequence to start with 1, 2, 3, 5, 8, …
For (2), here is one special property:
F1+F2+F3+⋯+Fn=Fn+2−1(3)
which can also be proved by mathematical induction or by observing that Fn=Fn+2−Fn+1 and that the sum is telescoping.
Based on (3),
F2+F3+⋯+Fn=Fn+2−2=Fn+1+(Fn−2)≥Fn+1−1
So the Fibonacci sequence does fulfil the Brown’s criterion and thus the proof is complete.
Footnotes
Brown's Criterion
Complete Sequence
Sponsored by Grammarly
Is your writing working as hard as your ideas?
Grammarly’s AI brings research, clarity, and structure—so your writing gets sharper with every step.
Gustavo Sánchez
M.S. in Physics & Mathematics, University of Buenos Aires (Graduated 1986)
·
Author has 3K answers and 2.2M answer views
·
7y
Let’s try by induction over n, the index of the sequence of the Fibonacci sequence. Let the Pn proposition be ,
“for every positive integer m≤Fn,
k=Fi1+Fi2+⋯+Fik
where i1,i2,…ik are different integers less or equal to n, and k≥1.
For n=1, P1 is obviously true, because F1=1, and the only m≤F1 is m=1. Obviously 1=F1, and therefore P1 is true.
Let’s assume that Pn is true for every n≤N. We should prove that it is true for N+1. In order to that, we should write every integer m between 1+FN and FN+1 as a sum of different Fibonacci’s numbers. These int
Let’s try by induction over n, the index of the sequence of the Fibonacci sequence. Let the Pn proposition be ,
“for every positive integer m≤Fn,
k=Fi1+Fi2+⋯+Fik
where i1,i2,…ik are different integers less or equal to n, and k≥1.
For n=1, P1 is obviously true, because F1=1, and the only m≤F1 is m=1. Obviously 1=F1, and therefore P1 is true.
Let’s assume that Pn is true for every n≤N. We should prove that it is true for N+1. In order to that, we should write every integer m between 1+FN and FN+1 as a sum of different Fibonacci’s numbers. These integers are written as
m=FN+c,
where c is a positive integer
c≤FN+1−FN=FN−1<FN.
As c satisfies the inductive hypothesis,
c=Fi1+Fi2+⋯+Fik,
therefore
m=Fi1+Fi2+⋯+Fik+FN.
Therefore, PN+1 is true.
Q.E.D.
Edit: Graeme McRae make me notice that I’ve misunderstood the question. What I proved is that every positive integer can be written as the sum of distict Fibonacci numbers, but I’ve dropped “non-consecutive”.
Krunoslav Popović
BS in Mathematics, University of Zagreb, Faculty of Science (Graduated 1990)
·
Author has 900 answers and 398.4K answer views
·
3y
Let's check one example:
F(n-2)=5, F(n-1)=8, F(n)=13. Suppose all numbers <8 have that property → numbers <5 have it too. So between 8 and 13 there are the numbers: 8,8+1,8+2,8+3, and 8+(5–1), which all having 'sum of distinct non-consecutive Fibonacci numbers’ property, as sum of consecutive Fibonacci's 8 +5 is not in that sequence.
This example can easily be generalized imho.
Related questions
What is a proof to show that every positive integer can be written as the sum of one or more distinct non-consecutive Fibonacci numbers, exclude F1, so that if Fn appears in the sum, then Fn+1 does not?
Is it true that every natural number can be written as a sum of Fibonacci numbers?
How do you prove that for every positive integer m there is a Fibonacci number divisible by m?
How can I rigorously prove that that for each Fibonacci number
F
4
n
is a multiple of 3 for every
n
≥
4
?
What a proof that the sum of any set of distinct non-consecutive Fibonacci numbers whose largest member is Fk is strictly less than Fk+1?
Amitabha Tripathi
have been teaching Discrete Mathematics for almost 40 years
·
Upvoted by
Michael
, Ph.D. Mathematics, University of Manchester (1982) and
Vance Faber
, Ph. D. Mathematics · Author has 4.7K answers and 13.9M answer views
·
4y
Originally Answered: The Fibonacci numbers are defined by the recurrence f_1 = f_2 = 1 and f_n = f_n - 1 + f_n - 2 for n >= 3. How can you prove that every positive integer can be written as the sum of Fibonacci numbers, no two of which are consecutive?
·
We prove that every positive integer n is a sum of one or more terms of the Fibonacci sequence, no terms being consecutive. We use induction on n.
The base case is n=1, and this equals f1. Assume the result for all positive integers less than n, and define k by fk≤n<fk+1. If n=fk, there is nothing more to prove. Otherwise, 0<n−fk<fk+1−fk=fk−1, and n−fk can be written as a sum of one or more distinct terms of the Fibonacci sequence, no terms being consecutive. Since n−fk<fk−1, none of these terms in the summand can equal fk−1. It follows that n=fk+(n−fk)=fk+f
We prove that every positive integer n is a sum of one or more terms of the Fibonacci sequence, no terms being consecutive. We use induction on n.
The base case is n=1, and this equals f1. Assume the result for all positive integers less than n, and define k by fk≤nk1+1>k2+2>…>kr+r. This completes the proof by mathematical induction.
Sponsored by CDW Corporation
Looking for tools to transform the way you connect?
Add energy to presentations with SMART devices from CDW, offering interactivity and engagement for all.
Assistant
Bot
·
1y
The statement that every positive integer can be expressed as the sum of one or more distinct non-consecutive Fibonacci numbers is a consequence of Zeckendorf's theorem. This theorem states that every positive integer can be uniquely represented as a sum of one or more distinct Fibonacci numbers such that no two of the Fibonacci numbers in the sum are consecutive.
Proof Outline
Fibonacci Sequence Definition:
The Fibonacci sequence is defined as:
[math]F_1 = 1[/math]
[math]F_2 = 2[/math]
[math]F_n = F_{n-1} + F_{n-2}[/math] for [math]n \geq 3[/math]
The first few Fibonacci numbers are: 1, 2, 3, 5, 8, 13, 21, 34, ...
Induction Base
The statement that every positive integer can be expressed as the sum of one or more distinct non-consecutive Fibonacci numbers is a consequence of Zeckendorf's theorem. This theorem states that every positive integer can be uniquely represented as a sum of one or more distinct Fibonacci numbers such that no two of the Fibonacci numbers in the sum are consecutive.
Proof Outline
Fibonacci Sequence Definition:
The Fibonacci sequence is defined as:
[math]F_1 = 1[/math]
[math]F_2 = 2[/math]
[math]F_n = F_{n-1} + F_{n-2}[/math] for [math]n \geq 3[/math]
The first few Fibonacci numbers are: 1, 2, 3, 5, 8, 13, 21, 34, ...
Induction Base Case:
For small integers, we can verify the representation:
[math]1 = F_1[/math]
[math]2 = F_2[/math]
[math]3 = F_3[/math]
[math]4 = F_1 + F_3[/math]
[math]5 = F_5[/math]
[math]6 = F_2 + F_5[/math]
[math]7 = F_3 + F_5[/math]
[math]8 = F_6[/math]
[math]9 = F_1 + F_6[/math]
[math]10 = F_2 + F_6[/math]
[math]11 = F_3 + F_6[/math]
[math]12 = F_1 + F_2 + F_6[/math]
[math]13 = F_7[/math]
[math]14 = F_1 + F_7[/math]
[math]15 = F_2 + F_7[/math]
[math]16 = F_3 + F_7[/math]
[math]17 = F_1 + F_4 + F_7[/math]
[math]18 = F_2 + F_4 + F_7[/math]
This shows that small integers can be represented as sums of distinct non-consecutive Fibonacci numbers.
Inductive Step:
Assume that for all integers up to [math]n[/math], we can express them as sums of distinct non-consecutive Fibonacci numbers. We need to show it holds for [math]n + 1[/math].
If [math]n + 1[/math] is not a Fibonacci number, we can find the largest Fibonacci number [math]F_k[/math] less than or equal to [math]n + 1[/math]. We can consider two cases:
If [math]n + 1 - F_k[/math] is not a Fibonacci number and does not use [math]F_{k-1}[/math], we can express [math]n + 1[/math] as [math]F_k + (n + 1 - F_k)[/math].
If [math]n + 1 - F_k[/math] is a Fibonacci number, we can consider [math]F_{k-2}[/math] instead, ensuring the distinctness and non-consecutiveness.
If [math]n + 1[/math] is a Fibonacci number, we can use [math]n + 1[/math] itself.
Conclusion:
By the principle of mathematical induction, every positive integer can be represented as a sum of distinct non-consecutive Fibonacci numbers.
Example
For instance, to express [math]19[/math]:
- The largest Fibonacci number less than 19 is 13.
- Subtracting, [math]19 - 13 = 6[/math], which can be expressed as [math]2 + 5[/math] (Fibonacci numbers).
- Therefore, [math]19 = 13 + 5 + 2[/math] (using [math]F_7, F_5, F_3[/math]), which are not consecutive.
Thus, we conclude that every positive integer can indeed be expressed as a sum of distinct non-consecutive Fibonacci numbers, confirming Zeckendorf's theorem.
Yannick Chevalier
M.S in Computer Science & Mathematics, Ecole Normale Superieure De Lyon (Graduated 1999)
·
Author has 480 answers and 400.5K answer views
·
4y
Originally Answered: The Fibonacci numbers are defined by the recurrence f_1 = f_2 = 1 and f_n = f_n - 1 + f_n - 2 for n >= 3. How can you prove that every positive integer can be written as the sum of Fibonacci numbers, no two of which are consecutive?
·
Assume this is not the case, and among the numbers for which it isn't possible, let n be the smallest one. It cannot itself be a Fibonacci number, so n is strictly greater than 1. So there are Fibonacci numbers strictly smaller than n. Let m be the greatest such number. Since n is minimal, the number n-m can be written as a sum of distinct non-consecutive Fibonacci numbers. Since the m is th greatest Fibonacci number smaller than n, n-m <m so m cannot occur in that sequence, and the predecessor of m also cannot occur, for otherwise you could replace m and its predecessor with its successor, ag
Assume this is not the case, and among the numbers for which it isn't possible, let n be the smallest one. It cannot itself be a Fibonacci number, so n is strictly greater than 1. So there are Fibonacci numbers strictly smaller than n. Let m be the greatest such number. Since n is minimal, the number n-m can be written as a sum of distinct non-consecutive Fibonacci numbers. Since the m is th greatest Fibonacci number smaller than n, n-m <m so m cannot occur in that sequence, and the predecessor of m also cannot occur, for otherwise you could replace m and its predecessor with its successor, again contradicting m is maximal. Add m to that sequence, and you contradict that n cannot be written as a sum of non-consecutive Fibonacci numbers.
Related questions
Is it possible to prove that every positive integer can be expressed as a sum of distinct terms from the Fibonacci numbers, with no consecutive terms?
What is a proof that no positive integer can be written in two different ways as the sum of distinct non-consecutive Fibonacci numbers?
What is the largest number that can be written as a sum of positive Fibonacci numbers?
Can you provide a proof that every integer can be expressed as a sum of Fibonacci numbers?
How do you prove that the sum of any 16 consecutive Fibonacci numbers is divisible by 21?
Fun Mathis
Curriculum Developer in Math Enrichment
·
Author has 86 answers and 76.5K answer views
·
4y
Originally Answered: The Fibonacci numbers are defined by the recurrence f_1 = f_2 = 1 and f_n = f_n - 1 + f_n - 2 for n >= 3. How can you prove that every positive integer can be written as the sum of Fibonacci numbers, no two of which are consecutive?
·
This is what is called the Zeckendorf representation of a positive integer [math]n[/math]. An inductive argument or a greedy algorithm shows that it exists for every [math]n[/math]. In other words, for every such [math]n[/math], there exists a finite set [math]A \subset \mathbb{Z}_{\geq 2} = {2, 3, 4, \ldots}[/math] such that no two elements of [math]A[/math] are consecutive, and
[math]\displaystyle n = \sum\limits_{k \in A} {F_k}[/math].
So if [math]1[/math] is a summand, it must be regarded as [math]F_2[/math] rather than as [math]F_1[/math], implying the absence of [math]F_3 = 2[/math] as a summand. The uniqueness of the Zeckendorf representation then follows from the fact that for all [math]k \geq 2[/math],
[math]\displaystyle F_k + F_{[/math]
This is what is called the Zeckendorf representation of a positive integer [math]n[/math]. An inductive argument or a greedy algorithm shows that it exists for every [math]n[/math]. In other words, for every such [math]n[/math], there exists a finite set [math]A \subset \mathbb{Z}_{\geq 2} = {2, 3, 4, \ldots}[/math] such that no two elements of [math]A[/math] are consecutive, and
[math]\displaystyle n = \sum\limits_{k \in A} {F_k}[/math].
So if [math]1[/math] is a summand, it must be regarded as [math]F_2[/math] rather than as [math]F_1[/math], implying the absence of [math]F_3 = 2[/math] as a summand. The uniqueness of the Zeckendorf representation then follows from the fact that for all [math]k \geq 2[/math],
[math]\displaystyle F_k + F_{k - 2} + F_{k - 4} + \cdots + F_{r(k)} < F_{k + 1}[/math],
where [math]r(k) = (k \bmod 2) + 2[/math] is either [math]2[/math] or [math]3[/math].
Suggested links: Wikipedia
, EoM
.
Footnotes
Zeckendorf's theorem - Wikipedia
Zeckendorf representation
Your response is private
Was this worth your time?
This helps us sort answers on the page.
Absolutely not
Definitely yes
Promoted by The Penny Hoarder
Lisa Dawson
Finance Writer at The Penny Hoarder
·
Updated Tue
What's some brutally honest advice that everyone should know?
Here’s the thing: I wish I had known these money secrets sooner. They’ve helped so many people save hundreds, secure their family’s future, and grow their bank accounts—myself included.
And honestly? Putting them to use was way easier than I expected. I bet you can knock out at least three or four of these right now—yes, even from your phone. Don’t wait like I did.
Cancel Your Car Insurance
You might not even realize it, but your car insurance company is probably overcharging you. In fact, they’re kind of counting on you not noticing. Luckily, this problem is easy to fix.
Don’t waste your time
Here’s the thing: I wish I had known these money secrets sooner. They’ve helped so many people save hundreds, secure their family’s future, and grow their bank accounts—myself included.
And honestly? Putting them to use was way easier than I expected. I bet you can knock out at least three or four of these right now—yes, even from your phone. Don’t wait like I did.
Cancel Your Car Insurance
You might not even realize it, but your car insurance company is probably overcharging you. In fact, they’re kind of counting on you not noticing. Luckily, this problem is easy to fix.
Don’t waste your time browsing insurance sites for a better deal. A company calledInsurify shows you all your options at once — people who do this save up to $996 per year.
If you tell them a bit about yourself and your vehicle, they’ll send you personalized quotes so you can compare them and find the best one for you.
Tired of overpaying for car insurance? It takes just five minutes to compare your options with Insurify andsee how much you could save on car insurance.
Ask This Company to Get a Big Chunk of Your Debt Forgiven
A company calledNational Debt Relief could convince your lenders to simply get rid of a big chunk of what you owe. No bankruptcy, no loans — you don’t even need to have good credit.
If you owe at least $10,000 in unsecured debt (credit card debt, personal loans, medical bills, etc.), National Debt Relief’s experts will build you a monthly payment plan. As your payments add up, they negotiate with your creditors to reduce the amount you owe. You then pay off the rest in a lump sum.
On average, you could become debt-free within 24 to 48 months. It takes less than a minute to sign up and see how much debt you could get rid of.
Set Up Direct Deposit — Pocket $300
When you set up direct deposit withSoFi Checking and Savings (Member FDIC), they’ll put up to $300 straight into your account. No… really. Just a nice little bonus for making a smart switch.
Why switch? With SoFi, you can earn up to 3.80% APY on savings and 0.50% on checking, plus a 0.20% APY boost for your first 6 months when you set up direct deposit or keep $5K in your account. That’s up to 4.00% APY total. Way better than letting your balance chill at 0.40% APY.
There’s no fees. No gotchas.Make the move to SoFi and get paid to upgrade your finances.
You Can Become a Real Estate Investor for as Little as $10
Take a look at some of the world’s wealthiest people. What do they have in common? Many invest in large private real estate deals. And here’s the thing: There’s no reason you can’t, too — for as little as $10.
An investment called the Fundrise Flagship Fund lets you get started in the world of real estate by giving you access to a low-cost, diversified portfolio of private real estate. The best part? You don’t have to be the landlord. The Flagship Fund does all the heavy lifting.
With an initial investment as low as $10, your money will be invested in the Fund, which already owns more than $1 billion worth of real estate around the country, from apartment complexes to the thriving housing rental market to larger last-mile e-commerce logistics centers.
Want to invest more? Many investors choose to invest $1,000 or more. This is a Fund that can fit any type of investor’s needs. Once invested, you can track your performance from your phone and watch as properties are acquired, improved, and operated. As properties generate cash flow, you could earn money through quarterly dividend payments. And over time, you could earn money off the potential appreciation of the properties.
So if you want to get started in the world of real-estate investing, it takes just a few minutes tosign up and create an account with the Fundrise Flagship Fund.
This is a paid advertisement. Carefully consider the investment objectives, risks, charges and expenses of the Fundrise Real Estate Fund before investing. This and other information can be found in the Fund’s prospectus. Read them carefully before investing.
Cut Your Phone Bill to $15/Month
Want a full year of doomscrolling, streaming, and “you still there?” texts, without the bloated price tag? Right now, Mint Mobile is offering unlimited talk, text, and data for just $15/month when you sign up for a 12-month plan.
Not ready for a whole year-long thing? Mint’s 3-month plans (including unlimited) are also just $15/month, so you can test the waters commitment-free.
It’s BYOE (bring your own everything), which means you keep your phone, your number, and your dignity. Plus, you’ll get perks like free mobile hotspot, scam call screening, and coverage on the nation’s largest 5G network.
Snag Mint Mobile’s $15 unlimited deal before it’s gone.
Get Up to $50,000 From This Company
Need a little extra cash to pay off credit card debt, remodel your house or to buy a big purchase?
We found a company willing to help.
Here’s how it works: If your credit score is at least 620, AmONE can help you borrow up to $50,000 (no collateral needed) with fixed rates starting at 6.40% and terms from 6 to 144 months.
AmONE won’t make you stand in line or call a bank. And if you’re worried you won’t qualify, it’s free tocheck online. It takes just two minutes, and it could save you thousands of dollars.
Totally worth it.
Get Paid $225/Month While Watching Movie Previews
If we told you that you could get paid while watching videos on your computer, you’d probably laugh.
It’s too good to be true, right?
But we’re serious. By signing up for a free account with InboxDollars, you could add up to $225 a month to your pocket. They’ll send you short surveys every day, which you can fill out while you watch someone bake brownies or catch up on the latest Kardashian drama.
No, InboxDollars won’t replace your full-time job, but it’s something easy you can do while you’re already on the couch tonight, wasting time on your phone.
Unlike other sites, InboxDollars pays you in cash — no points or gift cards. It’s already paid its users more than $56 million.
Signing up takes about one minute, and you’ll immediately receive a $5 bonus to get you started.
Earn $1000/Month by Reviewing Games and Products You Love
Okay, real talk—everything is crazy expensive right now, and let’s be honest, we could all use a little extra cash. But who has time for a second job?
Here’s the good news. You’re already playing games on your phone to kill time, relax, or just zone out. So why not make some extra cash while you’re at it?
WithKashKick, you can actually get paid to play. No weird surveys, no endless ads, just real money for playing games you’d probably be playing anyway. Some people are even making over $1,000 a month just doing this!
Oh, and here’s a little pro tip: If you wanna cash out even faster, spending $2 on an in-app purchase to skip levels can help you hit your first $50+ payout way quicker.
Once you’ve got $10, you can cash out instantly through PayPal—no waiting around, just straight-up money in your account.
Seriously, you’re already playing—might as well make some money while you’re at it.Sign up for KashKick and start earning now!
Doug Dillon
Ph.D. Mathematics
·
Author has 12.4K answers and 11.4M answer views
·
7y
I believe this is this is known as Zeckendorf’s Theorem. Check it out.
Zeckendorf's theorem - Wikipedia
Alon Amit
PhD in Mathematics; Mathcircler.
·
Upvoted by
James McElhatton Ph.D. (Glasgow, 1976)
, B.Sc. Mathematics & Chemistry, University of Malta (1967) and
Samuel Gomes da Silva
, Ph.D. Mathematics & Set Theory, University of São Paulo (2004) · Author has 8.8K answers and 173.5M answer views
·
8y
Related
Is it true that every natural number can be written as a sum of Fibonacci numbers?
Every natural number is a sum of Fibonacci numbers because 1 is a Fibonacci number, and adding enough 1’s will get you anywhere you want.
You probably meant to ask if every natural number is the sum of distinct Fibonacci numbers. The answer is Yes, and it’s even better to require that the representation avoids using two adjacent Fibonacci numbers. With that requirement it's still possible, and in fact in one and only one way. This is known as Zeckendorf’s theorem.
Finally, you're asking if the representation can be found using a greedy algorithm, which picks the largest Fibonacci number smaller
Every natural number is a sum of Fibonacci numbers because 1 is a Fibonacci number, and adding enough 1’s will get you anywhere you want.
You probably meant to ask if every natural number is the sum of distinct Fibonacci numbers. The answer is Yes, and it’s even better to require that the representation avoids using two adjacent Fibonacci numbers. With that requirement it's still possible, and in fact in one and only one way. This is known as Zeckendorf’s theorem.
Finally, you're asking if the representation can be found using a greedy algorithm, which picks the largest Fibonacci number smaller than the target number, subtracts it, and proceeds in the same way. The answer is again Yes, and the simplest proof of Zeckendorf’s theorem is to show that this does indeed work. With this as your hint, the proof of the existence part of the theorem becomes a routine exercise. Uniqueness is only slightly harder.
Promoted by Coverage.com
Johnny M
Master's Degree from Harvard University (Graduated 2011)
·
Updated Sep 9
Does switching car insurance really save you money, or is that just marketing hype?
This is one of those things that I didn’t expect to be worthwhile, but it was. You actually can save a solid chunk of money—if you use the right tool like this one. I ended up saving over $1,500/year, but I also insure four cars.
I tested several comparison tools and while some of them ended up spamming me with junk, there were a couple like Coverage.com and these alternatives that I now recommend to my friend.
Most insurance companies quietly raise your rate year after year. Nothing major, just enough that you don’t notice. They’re banking on you not shopping around—and to be honest, I didn’t.
This is one of those things that I didn’t expect to be worthwhile, but it was. You actually can save a solid chunk of money—if you use the right tool like this one. I ended up saving over $1,500/year, but I also insure four cars.
I tested several comparison tools and while some of them ended up spamming me with junk, there were a couple like Coverage.com and these alternatives that I now recommend to my friend.
Most insurance companies quietly raise your rate year after year. Nothing major, just enough that you don’t notice. They’re banking on you not shopping around—and to be honest, I didn’t. It always sounded like a hassle. Dozens of tabs, endless forms, phone calls I didn’t want to take.
But recently I decided to check so I used this quote tool, which compares everything in one place. It took maybe 2 minutes, tops. I just answered a few questions and it pulled up offers from multiple big-name providers, side by side. Prices, coverage details, even customer reviews—all laid out in a way that made the choice pretty obvious.
They claimed I could save over $1,000 per year. I ended up exceeding that number and I cut my monthly premium by over $100. That’s over $1200 a year. For the exact same coverage.
No phone tag. No junk emails. Just a better deal in less time than it takes to make coffee. Here’s the link to two comparison sites - the one I used and an alternative that I also tested.
If it’s been a while since you’ve checked your rate, do it. You might be surprised at how much you’re overpaying.
Greg Simay
Co-author, peer-reviewed papers in combinatorics, Erdos 2 (Prof. Chinn)
·
Author has 404 answers and 558.7K answer views
·
7y
Including the zeroth member, Fibonacci sequence begins as 0,1,1,2,…
Now the first and third members are non-consecutive and distinct; they are 1 and 2. With enough 1’s and 2’s you can sum to any positive integer you want.
Silvio Capobianco
Senior Researcher at Tallinn University of Technology (2009–present)
·
Author has 3.4K answers and 630K answer views
·
4y
Originally Answered: The Fibonacci numbers are defined by the recurrence f_1 = f_2 = 1 and f_n = f_n - 1 + f_n - 2 for n >= 3. How can you prove that every positive integer can be written as the sum of Fibonacci numbers, no two of which are consecutive?
·
Ah, Zeckendorf’s theorem! It is actually easy if you use a greedy algorithm: given
[math]n[/math]
, you take the largest
[math]k[/math]
such that
[math]f_k\leqslant{n}[/math]
, then you iterate on
[math]m=n-f_k[/math]
. If you exploit the fact that the algorithm is greedy, then you can use the Fibonacci recurrence to show that, if you choose
[math]f_k[/math]
, then you do
not
choose
[math]f_{k-1}[/math]
; you only need to be careful for the first values of
[math]n[/math]
, but after that, it’s simple. You need strong induction rather than induction, but it’s the same.
Junayed Ahmed Sunny
Studied Mathematics & Calculus at Coursera (Graduated 2019)
·
Author has 302 answers and 1.6M answer views
·
11mo
The Riemann Hypothesis involves the behavior of the Riemann zeta function, a complex mathematical function. Its proof requires a deep understanding of complex analysis and number theory. The hypothesis has connections to various mathematical fields, making it challenging to approach. Additionally, its complexity leads to many intricate details that need to be carefully addressed in any proof attempt, increasing the likelihood of errors. As a result, numerous attempts have been made, but the subtle nature of the problem combined with its interdisciplinarity have rendered successful proofs elusi
The Riemann Hypothesis involves the behavior of the Riemann zeta function, a complex mathematical function. Its proof requires a deep understanding of complex analysis and number theory. The hypothesis has connections to various mathematical fields, making it challenging to approach. Additionally, its complexity leads to many intricate details that need to be carefully addressed in any proof attempt, increasing the likelihood of errors. As a result, numerous attempts have been made, but the subtle nature of the problem combined with its interdisciplinarity have rendered successful proofs elusive.
Shardul Sajnekar
Author has 56 answers and 37.5K answer views
·
4y
Originally Answered: The Fibonacci numbers are defined by the recurrence f_1 = f_2 = 1 and f_n = f_n - 1 + f_n - 2 for n >= 3. How can you prove that every positive integer can be written as the sum of Fibonacci numbers, no two of which are consecutive?
·
I don't think it is possible. Integers like 1, 2, and 3 would disprove most of the proofs I can think of.
If consecutive means consecutive in the Fibonacci series then 3 can't be written as their sum, because we need 1 and 2 are the only ones which can add to 3.
If consecutive means consecutive in the number line, then too we can't write 3 as a sum.
And of course we cannot write 1 as a sum at all because 0 does not come in the Fibonacci series.
Related questions
What is a proof to show that every positive integer can be written as the sum of one or more distinct non-consecutive Fibonacci numbers, exclude F1, so that if Fn appears in the sum, then Fn+1 does not?
Is it true that every natural number can be written as a sum of Fibonacci numbers?
How do you prove that for every positive integer m there is a Fibonacci number divisible by m?
How can I rigorously prove that that for each Fibonacci number
is a multiple of 3 for every
?
What a proof that the sum of any set of distinct non-consecutive Fibonacci numbers whose largest member is Fk is strictly less than Fk+1?
Is it possible to prove that every positive integer can be expressed as a sum of distinct terms from the Fibonacci numbers, with no consecutive terms?
What is a proof that no positive integer can be written in two different ways as the sum of distinct non-consecutive Fibonacci numbers?
What is the largest number that can be written as a sum of positive Fibonacci numbers?
Can you provide a proof that every integer can be expressed as a sum of Fibonacci numbers?
How do you prove that the sum of any 16 consecutive Fibonacci numbers is divisible by 21?
What are Fibonacci numbers?
Can every positive integer be expressed as the sum of two numbers from the Fibonacci sequence? If so, how?
Are there any positive integers which have no multiple among the Fibonacci numbers? If yes, what is the sequence and is it infinite? If no, why not?
The Fibonacci numbers are defined by the recurrence f_1 = f_2 = 1 and f_n = f_n - 1 + f_n - 2 for n >= 3. How can you prove that every positive integer can be written as the sum of Fibonacci numbers, no two of which are consecutive?
Is it possible to represent any positive integer as a sum of two or more consecutive Fibonacci numbers? If yes, what is the smallest example?
About
·
Careers
·
Privacy
·
Terms
·
Contact
·
Languages
·
Your Ad Choices
·
Press
·
© Quora, Inc. 2025 |
8613 | https://www.wyzant.com/resources/answers/810162/find-the-vector-equation-for-the-line-of-intersection-of-the-planes-5-2-1-a | WYZANT TUTORING
Mark T.
Find the vector equation for the line of intersection of the planes 𝑥−5𝑦+2𝑧=−1 and 𝑥+2𝑧=0
Find the vector equation for the line of intersection of the planes 𝑥 −5𝑦 + 2𝑧 = −1 and 𝑥 + 2𝑧 = 0
| _ | | -10 |
x= | _ | + t | _ |
| 0 | | _ |
Matrices fill in the blanks
1 Expert Answer
Josh W. answered • 01/16/21
Math PhD Tutoring
One normal vector of the plane x-5y+2z=-1 is n1=<1, -5, 2> and one normal vector of the plane x+2z=0 is n2=<1,0,2>. Note that the intersection line of the planes x-5y+2z=-1 and x+2z=0 is perpendicular to both n1 and n2. So we can choose n=n1Xn2, the cross product of n1 and n2, as a direction vector of the intersection line.
| i j k |
Now n=<1,-5,2>X<1,0,2>= | 1 -5 2 | = (-10-0)i - (2-2)j +(0+5)k = <-10, 0 ,5> is a direction vector
| 1 0 2 |
of the line. Furthermore, we need to find one point on the line. To do so, we can set z=0 and get a system of linear equations in x and y: x-5y=-1 and x=0. Therefore, x=0, y=1/5. As a result, the point (0, 1/5, 0) is a point on the intersection line. Hence the vector form of the line is
r(t) = (0, 1/5, 0 ) + t(-10, 0, 5).
Still looking for help? Get the right answer, fast.
Get a free answer to a quick problem.
Most questions answered within 4 hours.
OR
Choose an expert and meet online.
No packages or subscriptions, pay only for the time you need.
RELATED TOPICS
RELATED QUESTIONS
CAN I SUBMIT A MATH EQUATION I'M HAVING PROBLEMS WITH?
Answers · 3
If i have rational function and it has a numerator that can be factored and the denominator is already factored out would I simplify by factoring the numerator?
Answers · 7
how do i find where a function is discontinuous if the bottom part of the function has been factored out?
Answers · 3
find the limit as it approaches -3 in the equation (6x+9)/x^4+6x^3+9x^2
Answers · 8
prove addition form for coshx
Answers · 4
RECOMMENDED TUTORS
Priti S.
Zachary C.
Geoffrey B.
find an online tutor
Download our free app
A link to the app was sent to your phone.
Get to know us
Learn with us
Work with us
Download our free app
Let’s keep in touch
Need more help?
Learn more about how it works
Tutors by Subject
Tutors by Location
IXL
Comprehensive K-12 personalized learning
Rosetta Stone
Immersive learning for 25 languages
Education.com
35,000 worksheets, games, and lesson plans
TPT
Marketplace for millions of educator-created resources
Vocabulary.com
Adaptive learning for English vocabulary
ABCya
Fun educational games for kids
SpanishDictionary.com
Spanish-English dictionary, translator, and learning
Inglés.com
Diccionario inglés-español, traductor y sitio de aprendizaje
Emmersion
Fast and accurate language certification |
8614 | https://schools.utah.gov/curr/mathematics/_mathematics_/_core/_math_core_guides_tab_/CoreGuidesGrade6ExpressionsEquations.pdf | Expressions and Equations Core Guide Grade 6 Use properties of operations to generate equivalent expressions (6.EE.1-4) Standard 6.EE.1: Write and evaluate numerical expressions involving whole-number exponents Concepts and Skills to Master • Understand the meaning of exponents and exponential notation. • Write numerical expressions involving whole-number exponents. • Evaluate numerical expressions involving whole-number exponents. Related Standards: Current Course Related Standards: Future Courses 6.EE.2, 6.EE.3, 6.EE.4 8.EE.1, 8.EE.2, 8.EE.3, 8.EE.4, 8.G.7, Foundational for all numerical exponent work Support for Teachers Critical Background Knowledge (Access Background Knowledge) • Interpret the products of whole numbers (3.OA.2) • Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols (5.OA.1) • Write and interpret simple numerical expression (5.OA.2) • Explain patterns in the number of zeros of the product when multiplying a number by powers of 10. Use whole-number exponents to denote powers of 10 (5.NBT.2) Academic Vocabulary Exponent, base, expression Resources Curriculum Resources: 6.EE.1 Page | 1 ADA Compliant 10/30/2019 Expressions and Equations Core Guide Grade 6 Use properties of operations to generate equivalent expressions (6.EE.1-4) Standard 6.EE.2: Write, read, and evaluate expressions in which letters represent numbers. a. Write expressions that record operations with numbers and with letters representing numbers. For example, express the calculation "Subtract y from 5" as 5 – y and express “Jane had $105.00 in her bank account. One year later, she had x dollars more. Write an expression that shows her new balance” as $105.00 + x. b. Identify parts of an expression using mathematical terms (for example, sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity and a sum of two terms. For example, describe the expression 2(8 + 7) as a product of two factors; view (8 + 7) as both a single entity and a sum of two terms. c. Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, applying the Order of Operations when there are no parentheses to specify a particular order. For example, use the formulas V = 𝑠𝑠3 and A = 6𝑠𝑠2 to find the volume and surface area of a cube with sides of length s = 1/2. Concepts and Skills to Master • Translate verbal expressions into numerical expressions and numerical expressions into verbal expressions. • Identify parts of an expression (sum, term, product, factor, quotient, coefficient). • View one or more parts of an expression to flexibly recognize the structure of the expression (see examples from b). • Evaluate expressions at specific values of their variables. Related Standards: Current Course Related Standards: Future Courses 6.EE.1, 6.EE.3, 6.EE.4, 6.EE.5, 6.EE.6 7.MP.7, 7.EE.1, 7.EE.2, 8.EE.2, 8.EE.7, I.F.IF.2, I.A.SSE.1, II.A.SSE.1, III.A.SSE.1 Support for Teachers Critical Background Knowledge (Access Background Knowledge) • Use addition and subtraction to solve word problems with unknowns in all positions. Use equations with a symbol for the unknown number to represent the problem (1.OA.1, 2.OA.1) • Determine the unknown whole number in an addition or subtraction equation (1.OA.8) or multiplication or division equation (3.OA.4) relating three whole numbers. • Represent multi-step word problems using equations with a letter standing for the unknown quantity (3.OA.8, 4.OA.3) • Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols (5.OA.1) • Write and interpret simple numerical expressions (5.OA.2) Academic Vocabulary expression, evaluate, variable, term, coefficient, sum, difference, product, factor, quotient, constant, term Resources Curriculum Resources: 6.EE.2 Page | 2 ADA Compliant 10/30/2019 Expressions and Equations Core Guide Grade 6 Use properties of operations to generate equivalent expressions (6.EE.1-4) Standard 6.EE.3: Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3(2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6(4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y. Concepts and Skills to Master • Understand that the properties used with numbers also apply to expressions with variables. • Apply the properties of operations with expressions involving variables to generate equivalent expressions. Related Standards: Current Course Related Standards: Future Courses 6.EE.2, 6.EE.4, 6.EE.5, 6.EE.6, 6.NS.4 7.MP.7, 7.EE.1, 7.EE.2, 8.EE.1, 8.EE.7, 8.NS.3, I.F.IF.2, I.A.SSE.1, II.A.SSE.1, II.A.SSE.2, II.A.SSE.3, III.A.SSE.1, III.A.SSE.2. III.A.APR.4, III.A.APR.6 Support for Teachers Critical Background Knowledge (Access Background Knowledge) • Apply properties of operations as strategies to add and subtract, commutative and associative (1.OA.3) • Apply properties of operations as strategies to multiply and divide, commutative, associative, and distributive (3.OA.5) • Interpret numerical expressions without evaluating them (5.OA.2) Academic Vocabulary Distributive property, expression, variable Resources Curriculum Resources: 6.EE.3 Page | 3 ADA Compliant 10/30/2019 Expressions and Equations Core Guide Grade 6 Use properties of operations to generate equivalent expressions (6.EE.1-4) Standard 6.EE.4: Identify when two expressions are equivalent. For example, the expressions y + y + y and 3y are equivalent because they name the same number, regardless of which number y represents. Concepts and Skills to Master • Identify when two expressions are equivalent. • Reason that two expressions are equivalent by combining like terms. • Understand how to manipulate an expression to identify a different, yet equivalent form. Related Standards: Current Course Related Standards: Future Courses 6.EE.2, 6.EE.3, 6.EE.5, 6.EE.6 7.MP.7, 7.EE.1, 7.EE.2, 8.NS.3, 8.EE.1, 8.EE.7, I.F.IF.2, I.A.SSE.1, II.A.SSE.1, II.A.SSE.2, II.A.SSE.3, III.A.SSE.1, III.A.SSE.2, III.A.APR.4, III.A.APR.6 Support for Teachers Critical Background Knowledge (Access Background Knowledge) • Apply properties of operations as strategies to add and subtract, commutative and associative (1.OA.3) • Apply properties of operations as strategies to multiply and divide, commutative, associative, and distributive (3.OA.5) • Interpret numerical expressions without evaluating them (5.OA.2) Academic Vocabulary expression, variable Resources Curriculum Resources: 6.EE.4 Page | 4 ADA Compliant 10/30/2019 Expressions and Equations Core Guide Grade 6 Use properties of operations to generate equivalent expressions (6.EE.5-8) Standard 6.EE.5: Understand solving an equation or inequality as a process of answering the question: Which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true. Concepts and Skills to Master • Understand that a solution is any value or values that make an equation or inequality true. • Use substitution to determine whether the given value makes the equation or inequality true. • Recognize that solutions to inequalities represent a range of possible values rather than a single solution. Related Standards: Current Course Related Standards: Future Courses 6.RP.3b, 6.EE.6, 6.EE.7, 6.EE.8 7.MP.7, 7.RP.3, 7.EE.3, 7.EE.4, 8.EE.2, 8.EE.7, 8.EE.8, 8.G.6, I.A.CED.1, I.A.REI.1, I.A.REI.3, I.A.REI.10, I.A.REI.12, II.A.CED.1, II.A.CED.4, II.A.REI.4, III.A.CED.1, III.A.CED.4, III.A.REI.2 Support for Teachers Critical Background Knowledge (Access Background Knowledge) • Understand the meaning of the equal sign (1.OA.7) • Compare numbers recording the results of comparisons with the symbols >, =, and < (1.NBT.3, 2.NBT.4, 4.NBT.2) Academic Vocabulary Equation, inequality, substitution Resources Curriculum Resources: 6.EE.5 Page | 5 ADA Compliant 10/30/2019 Expressions and Equations Core Guide Grade 6 Use properties of operations to generate equivalent expressions (6.EE.5-8) Standard 6.EE.6: Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set. Concepts and Skills to Master • Represent real-world scenarios with variable expressions, identifying what the variable represents. • Understand that a variable can represent a number or any number in a specified set of numbers Related Standards: Current Course Related Standards: Future Courses 6.RP.3b, 6.EE.2, 6.EE.5, 6.EE.7, 6.EE.8 7.MP.7, 7.RP.3, 7.EE.3, 7.EE.4, 8.EE.2, 8.EE.7, 8.EE.8, 8.G.6, I.A.CED.1, I.A.REI.1, I.A.REI.3, I.A.REI.10, I.A.REI.12, II.A.CED.1, II.A.CED.4, II.A.REI.4, III.A.CED.1, III.A.CED.4, III.A.REI.2 Support for Teachers Critical Background Knowledge (Access Background Knowledge) • Use addition and subtraction to solve word problems with unknowns in all positions. Use equations with a symbol for the unknown number to represent the problem (1.OA.1, 2.OA.1) • Represent multi-step word problems using equations with a letter standing for the unknown quantity (3.OA.8, 4.OA.3) • Write and interpret simple numerical expressions (5.OA.2) Academic Vocabulary Variable, expression Resources Curriculum Resources: 6.EE.6 Page | 6 ADA Compliant 10/30/2019 Expressions and Equations Core Guide Grade 6 Use properties of operations to generate equivalent expressions (6.EE.5-8) Standard 6.EE.7: Solve real-world and mathematical problems by writing and solving equations of the form x + a = b and ax = b for cases in which a, b and x are all non-negative rational numbers. Concepts and Skills to Master • Solve equations that represent real-world and mathematical problems. • Write and solve equations of the form x + a = b. • Write and solve equations of the form ax = b. Related Standards: Current Course Related Standards: Future Courses 6.RP.3b, 6.EE.5, 6.EE.7, 6.EE.8, 6.NS.1 7.MP.7, 7.RP.3, 7.EE.3, 7.EE.4, 8.EE.2, 8.EE.7, 8.EE.8, 8.G.6, I.A.CED.1, I.A.REI.1, I.A.REI.3, I.A.REI.10, II.A.CED.1, II.A.CED.4, II.A.REI.4, III.A.CED.1, III.A.CED.4, III.A.REI.2 Support for Teachers Critical Background Knowledge (Access Background Knowledge) • Add and subtract fractions and mixed numbers (4.NF.1, 4.NF.3, 5.NF.1, 5.NF.2) • Multiply fractions and mixed numbers (4.NF.4, 5.NF.4, 5.NF.6); Divide fractions (5.NF.7, 6.NS.1) • Add, subtract, multiply, and divide decimals to hundredths (5.NBT.7) • Write and interpret simple numerical expressions (5.OA.2) Academic Vocabulary Equation, rational Resources Curriculum Resources: 6.EE.7 Page | 7 ADA Compliant 10/30/2019 Expressions and Equations Core Guide Grade 6 Use properties of operations to generate equivalent expressions (6.EE.5-8) Standard 6.EE.8: Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams. Concepts and Skills to Master • Understand inequalities as a constraint or condition. • Write inequalities based on real-world and mathematical problems. • Recognize that inequalities of the form x > c or x < c have infinitely many solutions. • Represent solutions of inequalities on number line diagrams. Related Standards: Current Course Related Standards: Future Courses 6.RP.3b, 6.EE.5, 6.EE.7, 6.EE.8, 6.NS.6, 6.NS.7 7.MP.7, 7.RP.3, 7.EE.3, 7.EE.4, 8.EE.2, 8.EE.7, 8.EE.8, 8.G.6, I.A.CED.1, I.A.REI.3, I.A.REI.12, II.A.CED.1, II.A.REI.4, III.A.CED.1 Support for Teachers Critical Background Knowledge (Access Background Knowledge) • Compare numbers recording the results of comparisons with the symbols >, =, and < (1.NBT.3, 2.NBT.4, 4.NBT.2) • Represent whole numbers (2.MD.6), fractions (3.NF.2), and decimals (4.NF.6) on a number line. Academic Vocabulary inequality Resources Curriculum Resources: 6.EE.8 Page | 8 ADA Compliant 10/30/2019 Expressions and Equations Core Guide Grade 6 Use properties of operations to generate equivalent expressions (6.EE.9) Standard 6.EE.9: Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time. Concepts and Skills to Master • Utilize graphs and tables to recognize that a change in the independent variable creates a change in the dependent variable.
• Write an equation to express the relationship of the quantities in terms of the dependent and independent variables.
Related Standards: Current Course Related Standards: Future Courses 6.RP.1, 6.RP.2, 6.RP.3, 6.EE.2, 6.EE.5, 6.EE.6, 6.EE.7 Foundational for every situation involving two quantities. Support for Teachers Critical Background Knowledge (Access Background Knowledge) • Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself (4.OA.5) • Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane (5.OA.3) Academic Vocabulary Independent variable, dependent variable Resources Curriculum Resources: 6.EE.9 Page | 9 ADA Compliant 10/30/2019 |
8615 | https://w1.mtsu.edu/faculty/wding/math_behind_NN2022.pdf | 人工神经网络背后的数学原理！ The following article is from Datawhale Author 贾博文 Datawhale 一个专注于AI领域的开源组织，汇聚了众多优秀学习者，使命-for the learner，和学习… 本文约8000字，建议阅读22分钟 审稿人：阿泽，Datawhale成员，复旦大学计算机硕士，目前在携程担任高级算法工程师。 前言 提到人工智能算法，人工神经网络（ANN）是一个绕不过去的话题。但是对于新手，往往容易被 ANN中一堆复杂的概念公式搞得头大，最后只能做到感性的认识，而无法深入的理解。正好最近 笔者本人也在经历这个痛苦的过程，本着真理越辩越明的态度，索性坐下来认真的把这些头大的 问题梳理一番，试试看能不能搞清楚ANN背后的数学原理。 其实ANN 的流程概括来说倒不是很复杂，以最简单的前馈神经网络为例，无非就是 1. 搭建网络架构（含有待定参数） 2. 通过比较输出与标签的差值定义损失函数（自变量为待定参数的函数） 3. 随机给出一组初始参数 4. 给出一个（sgd）/ 一批（Minibatch）训练样例（包括ANN的输入值和输出值） 5. 前向传播得到预测标签和损失 6. 利用梯度下降算法从后往前调整网络参数（误差反向传播，BP） 7. 得到所有参数值 2022年02月21日 21:40 机器学习算法那些事 7/22/24, 8:40 AM 人工神经网络背后的数学原理！ 1/28 8. 得到ANN并使用 其中4、5、6里的内容是需要反复迭代的（注意6是双重循环）。 在ANN的一堆操作里，梯度下降算法是一个相对独立的过程，不妨就让我们从梯度下降算法开始 吧。 一、梯度下降到底在干什么 其实这个问题非常简单，只是大家被梯度下降复杂的过程搞蒙了，忘记了它的本质。梯度下降算 法自始至终都在干一件事——就是找到函数的极值点，当然确切的说是极小值点。但是，这种方 法不同于以往我们在高等数学里学到的找极值点的方法。那么我们首先就要问，求极值的经典方 法不香吗？ 1.1 求极值：传统的方法不香吗？ 要回答这个问题，让我们先快速回顾一下在中学和大学里学到的传统的求极值点的方法。 对于一元函数来说，极值可能出现在一阶导函数为0的点（驻点）或是导数不存在的点。 例如要找到 f(x) = x^2 + 3x 的极值点 求导得 dy/dx = 2x + 3 令 dy/dx = 0 就得到 x = -1.5 时，导函数为0。 注意，上面找到的只是可能的极值点，也就是极值存在的必要条件。还需要验证一下充分条件， 才能确定极值。这时，可以判断二阶导的正负性、或是判定一阶导在可能的极值点两边的正负情 况。回到我们的例子 7/22/24, 8:40 AM 人工神经网络背后的数学原理！ 2/28 当 x < -1.5 时，2x + 3 < 0 当 x > -1.5 时，2x + 3 > 0 说明函数在x = -1.5 附近先下降、后上升 该点是一个极小值点 对于二元函数 ，情况更复杂一些。首先要找出该函数的驻点和偏导数不存在的点，这些点 仍然只是可能的极值点。而二元函数的驻点需要同时满足两个偏导数为0的条件，即 显然，这里的驻点是需要解这样一个二元方程才能求得的。对于驻点分别求出其3个二阶偏导数的 值，再根据一些规则才能判断是不是极值点。 还需注意这个判断规则是不同于一元函数的，因为一元函数极值的充分条件只需要考察一个二阶 偏导数，而这里则需要综合考察二元函数的3个二阶偏导数，计算量明显增大了。对于偏导数不存 在的情形还需要特判。 综合以上，我们可以看出使用经典方法虽然能准确的解出极值点，但当函数自变量的个数很多 时，用这种方法求解极值点还真的不香。比如： 该方法方法不具有普适性。所谓普适性，就是不能简单的向多元推广。从一元到二元的例子可 以看出，函数的自变量个数每增加一元，就要研究新的求解方案。可以想象如果是三元函数， 其二阶偏导数的个数更多，则判断极值的充分条件还要来的更加复杂。而ANN中可能会求解 上亿元函数的极值点。 其次，这种方法需要解多元方程组，而且这些方程还不一定都是线性的。对于这种多元的非线 性方程组，我们的直观感受就是很难解出。事实上，虽然存在一些可供编程的数值计算解法， 但计算量大，且求出的是近似解，具有一定的局限性。 7/22/24, 8:40 AM 人工神经网络背后的数学原理！ 3/28 基于此，为了找出多元函数的极值点，我们还需另寻他法。这种方法要简单易行，特别是要能简 单的向任意元函数推广，而且这种方法要能够适应计算机数值计算的特点，毕竟我们这套程序肯 定是要放在电脑上跑的。而这就是传说中的梯度下降算法。 1.2 什么是梯度？ 梯度的概念其实也不难，但为了让尽可能多的人明白这一概念，我们还是从一元函数开始吧。不 过现在我们的目标是——用纯粹数值计算的方法，从函数上的某一点出发，找到函数的极值。这 里我们只考察极小值。 1.2.1 一元函数找极值：从枚举试探法到梯度下降法 以函数 为例，让我们看看如何找到极值。 7/22/24, 8:40 AM 人工神经网络背后的数学原理！ 4/28 既然是从函数上的某一点出发，那么不妨设想我们在x = 1 的地方，这个地方是不是极小值点 呢，我们可以试探一下。 向右走0.5，发现f(1.5) > f(1)，说明这个方向是上升的方向，不应该选择这个方向； 向左走0.5，发现f(0.5) < f(1)，说明这个方向是下降的方向，选择这个方向； 再向左走0.5，发现f(0) < f(0.5)，说明这个方向是下降的方向，选择这个方向； 再向左走0.5，发现f(-0.5) > f(0)，说明这个方向是上升的方向，不应该选择这个方向。 至此，我们可以将x = 0作为极小值点。 7/22/24, 8:40 AM 人工神经网络背后的数学原理！ 5/28 回顾这个过程，我们将寻找极小值点的过程抽象如下： 首先，选择一个方向 试着沿该方向走一小步，并据此判断该方向是否合理。如果合理，则走这一步；如果不合理， 换一个方向 反复重复第二步，直到找到极小值点 当然这里还有几点值得注意 第一，对于一元函数来说我们只有向左走或向右走两个选项。换句话说，每一步我们的选择是 有限的，是可以枚举的。因此，这个方法我把它称之为枚举试探法。 第二，判断方向其实不必这样试错，直接求导就好。如果某点的导数值 > 0,说明在该点处函 数是递增的，为了找到极小值，应该向左走；而如果导数值 < 0，则反之向右走即可。 第三，这种方法是不一定能找到极小值的，能不能找到极值点受选择的起始点以及每次前进的 步长这两个因素影响。 对于第二点，我们可以引出梯度的定义了。 梯度是一个向量，它总指向当前函数值增长最快的方向。而它的模长则是这个最快的增长率（导 数）的值。想要得到梯度向量，也很简单，它在x, y, z……等方向上的分量（坐标）就是相应的导 数值。于是我们求导就可以了。 对于一元函数，函数变化的方向只有两个，我们定义一种一维的向量来表示梯度，比如5i，-5i。i 前的数为正时，代表向量指向x轴正向；i前的数为负时，代表向量指向x轴负向。由下图可以看 出，按照上述定义规定的梯度向量自然的指向了函数增长的方向，是不是很神奇。 7/22/24, 8:40 AM 人工神经网络背后的数学原理！ 6/28 由于梯度的方向正是函数增长最快的方向，所以梯度的逆方向就成了函数下降最快的方向。当然 对于一元函数来说，没有最快的方向的概念，因为毕竟就两个方向而已，根本没得比。不过有了 梯度，我们就可以进一步简化上述寻找极小值点的过程： 首先，求出某点的梯度 沿梯度的反方向移动一小步 反复进行第一、二步，直到找到极小值点 仍以函数 ，起始点x = 1为例，让我们看看如何用梯度找到极值。 7/22/24, 8:40 AM 人工神经网络背后的数学原理！ 7/28 初始x = 1， 步长step = 0.5 #在我们的例子里，梯度的计算式为2xi。i是指向x轴正向的单位向量 求x = 1处的梯度为2i，梯度反方向为-i #注意这里我们只关注梯度的方向，至于梯度的模长则不必在意 沿此方向走一步，x新 = x旧 + step 负梯度方向上单位向量的坐标 = 1 + 0.5 (-1) = 0.5 求x = 0.5处的梯度为1i，梯度反方向为-i 沿此方向再走一步，x新 = 0.5 - 0.5 1 = 0 求x = 0处的梯度为0，说明到达极值点 以上就是用梯度找极小值点的过程，也就是梯度下降算法所做的事情，其实不难理解对吧。 可以看出，相比于枚举试探法，梯度下降法明显智能了许多，它直接给出了正确的方向，不需要 我们一步步试探了。此外，使用梯度下降法不必再关注具体的函数值，只需要把注意力放在导数 上，而且只关注一阶导数即可。 在后面，我们还将给上面提到的步长step换一个高大上的名字——学习率，这样就完全是机器学 习里的叫法了。 这里用到梯度的时候，我进行了单位化操作，其实也可以不进行这一步，这样当函数变化比较剧 烈的时候，移动的距离就比较多；函数变化比较平缓的时候，移动的距离就比较短。比如，在我 们这个例子里，只需一轮迭代就能得到结果了。 初始x = 1， 学习率step = 0.5 #在我们的例子里，梯度的计算式为2xi。i是指向x轴正向的单位向量 求x = 1处的梯度为2i，梯度反方向为-2i #注意这里我们既关注梯度的方向，也关注梯度的模长 沿此方向走一步，x新 = x旧 + step 负梯度的坐标 = 1 + 0.5 (-2) = 0 求x = 0处的梯度为0，说明到达极值点 好了说完了梯度，对于前面第三点提到的找不到极值的情形，我们举两个具体的例子 7/22/24, 8:40 AM 人工神经网络背后的数学原理！ 8/28 还是函数 ，如果起始点选为0.4，而学习率仍为0.5，在采用单位化梯度向量的情形下，则 无法找到事实上的极小值点 对于这种情况，我们可以通过减小学习率使结果尽可能精确，例如我们将学习率设置为0.1，就仍 然能得到精确的结果。事实上，在实际操作中，一般也会把学习率设置为0.1。 而对于这种有多个极值点的函数，这种方法是没法找到全部极值点的，更遑论找到全局的极值点 7/22/24, 8:40 AM 人工神经网络背后的数学原理！ 9/28 了。这时，我们可以在算法里加入一些随机性，使其有一定概率跳出可能陷入的局部极值点。 1.2.2 多元函数的梯度 前面说过梯度下降算法的好处之一在于可以很方便的向多维推广，现在我们以二元函数为例，看 看梯度是如何帮助我们找到极值点的。 这次我们的函数变成了 ，起始点选择为（-5, -5），学习率仍设置为0.5。现在 我们的目标是从这个点出发，找到该函数的极值点，我们知道这个极值点应该是（0, 0）。 7/22/24, 8:40 AM 人工神经网络背后的数学原理！ 10/28 这里与一元函数有几点不同： 7/22/24, 8:40 AM 人工神经网络背后的数学原理！ 11/28 首先，二元函数描述的是一个自变量和两个因变量之间的关系，也就是说函数的定义域是一个 二维平面，我们要找的极值点就在这个二维平面上。 其次，由于是在二维平面上寻找极值点，我们每一步可以选择的方向不再局限于一维时的向左 或向右，而是瞬间变成了无穷多个方向。因此，枚举试探法彻底宣告失效。还好我们有更智能 的梯度下降法。 一元梯度定义式里的导数现在已经换成了多元函数的偏导数。 好了，现在算法开始： 起始点坐标(-5,-5)， 学习率step = 0.5 #在我们的例子里，梯度的计算式为2xi + 2yj。i和j分别是指向x轴正向和y轴正向的单位向量 求点(-5,-5)处的梯度为-10i-10j，负梯度为10i+10j，写成坐标形式就是（10,10） 在点(-5,-5)处沿此梯度走一步 根据公式 向量坐标 = 终点坐标 - 起点坐标，得终点坐标 = 起点坐标 + 向量坐标 这里，终点坐标是（x新, y新），起点坐标是（x旧, y旧） = （-5， -5） 向量坐标是负梯度坐标 = （10,10），再考虑学习率step，就可以得到 （x新, y新） = （-5，-5） + 0.5 （10, 10） = （0, 0） 求（0, 0）处的梯度为零向量，说明到达极值点 将上述过程抽象，我们就得到了梯度下降算法的全部逻辑： 我们要找函数的极小值点（使函数取值尽可能小的那一组自变量），因为，梯度的方向是函数值 增长速度最快的方向，所以，沿着梯度的反方向函数值下降最快。 因此，只要沿着梯度的反方向一步步逼近就有可能找到那一组使函数取值尽可能小的自变量。 如何沿着梯度的反方向一步步逼近呢？ 7/22/24, 8:40 AM 人工神经网络背后的数学原理！ 12/28 我们随机指定一个起点坐标（一组自变量取值），然后沿着梯度的方向求出未知的终点坐标，梯 度是一个向量，本身也具有坐标 通过上面的迭代公式，无论是多少元的函数，它的一个个自变量们都会比较快的接近极值点（或 者其近似）。这样我们就可以找到一组自变量值，使得函数值尽可能的小。 1.2.3 小结 7/22/24, 8:40 AM 人工神经网络背后的数学原理！ 13/28 梯度的计算公式为 梯度是一个向量，它总指向当前函数值增长最快的方向，而它的模长则是这个最快的增长率 （导数）的值。 梯度下降法是一种通过数值计算求解函数极值点的方法 其过程概括来说就是顺着梯度的反方向一步步逼近可能的极值点 使用梯度下降法的理由在于求极值点的其他方法（如传统法、枚举试探法）不具有可计算性， 无法编程实现 梯度下降法可以很方便的向多元函数推广，利于编写程序 记住在这个过程中，我们要找的是极值点（使函数取极值的那一组自变量），而不是具体的极 值 梯度下降法的劣势在于不一定能找到全局最优解 二、人工神经网络（ANN） 如果是第一次听到人工神经网络这个名词，不免会觉得比较高大上，好像我们已经可以模仿神秘 的神经系统了。其实它只是一个数学模型而已。当然ANN的效果是令人眼前一亮的，好像计算机 一下子真的有了人的能力，可以识人、识物。 但其实稍加抽象便能发现，这个东西无非就是个分类器，它的输入是一张图片，或者确切的说就 是一堆代表像素点的数值，而输出则是一个类别。 所以说白了，所谓的人工神经网络其实就是一个超大规模的函数。 这就好比飞机和鸟儿的关系。让飞机飞起来靠的不是依葫芦画瓢造一个人工鸟，而是靠流体力学 中的原理建立数学模型，然后计算得出飞机的尺寸、造型，并设计相应的发动机。 7/22/24, 8:40 AM 人工神经网络背后的数学原理！ 14/28 2.1 神经元的数学模型 盗一张老师ppt里的图说明问题，可以看出ANN中的每一个节点（也就是所谓的神经元）就是这 样一个简单的线性函数模型。 当然通过激活函数我们可以制造一点非线性的因素，以提高模型的表达能力。这样的话下面的神 经元就代表这样一个函数 其中， ，这里w1, w2, w3, b都是参数，x1, x2, x3 是函数的输入，也就是因变量。 7/22/24, 8:40 AM 人工神经网络背后的数学原理！ 15/28 常用的激活函数在这里（仍然盗用老师的ppt，捂脸逃~） 7/22/24, 8:40 AM 人工神经网络背后的数学原理！ 16/28 以上就是所谓的人工神经元或者叫人造神经元，很多很多这样的神经元按一定规则相连就构成了 ANN，所以我才说ANN就是一个超大规模的函数而已。 是不是和你想象中的高大上的神经元大相径庭，但是我们现在所谓的人工智能其实就是这样的数 学模型而已。无论是简单的图像分类器还是战胜人类的AlphaGo，都是靠这样的数学计算算出来 结果的，而不是靠什么化腐朽为神奇的力量。 2.2 ANN是如何炼成的？ 7/22/24, 8:40 AM 人工神经网络背后的数学原理！ 17/28 知道了ANN的本质，现在就让我们看看得到一个ANN需要怎么做？这里，请留意我们会遇到不同 功能的函数，千万不要搞混了。 既然ANN是一个超大规模的函数，那么首先我们做的就是搭建起这个函数的架构，也就是设计人 工神经网络的架构。这时这个函数就有一堆参数待定了。接下来我们准备一堆训练数据训练 ANN，也就是把上面提到的待定参数都给他确定了。模型完成，可以使用。 显然，最关键的是第三步——确定未知参数。 这里首先解释训练数据，我们知道ANN是一个分类器也是一个函数，这个函数读取一些输入值， 经过复杂的计算后得到输出值，这些输出值可以被解释为类别。而训练数据就是输入值和最后的 输出值都已知的一组数据，换句话说就是已知一组函数的自变量和因变量的对应关系。 再说的明白点，我们的任务就是，已知函数的架构、函数的一组输入值和输出值，但不知道函数 的一些参数，现在要推出这些未知参数。我把这里我们要求出的这个函数称之为目标函数。于 是，一言以蔽之，我们的任务就是求出目标函数的未知参数。 为了完成这个任务，我们引出另一个重要的概念——损失函数。 2.2.1 损失函数 在这里，我们玩一点小心机。注意了，这里很关键！！！ 既然我们已知目标函数的一组输入和输出，而未知其参数，那么我们不妨将计就计将这些未知参 数直接视为因变量，而将目标函数的输入直接代入进去，这样我们不就得到了一个自变量是目标 函数的所有未知参数且函数整体完全已知的函数了吗？ 这时，如果能找到一组合适的未知参数，这个函数应该能输出和已知输入对应的输出完全一致的 值。 7/22/24, 8:40 AM 人工神经网络背后的数学原理！ 18/28 于是我们可以通过作差比较定义损失函数了 上图给出了损失函数的两种形式（除此之外还有交叉熵损失函数等其他类型），一般需要根据不 同的任务类型选取适当的损失函数。为了便于初学者理解，后文将以第二种均方误差的形式做讲 解。这里之所以出现了求和符号，是因为ANN的输出端可能对应了不止一个函数，这些函数可以 分别表示把一张图片分成不同类别的概率。后面我们引入一个直观的例子，一看便知。 这里一定要注意，损失函数看起来虽然还有目标函数的影子，但实际已经完全不同了。我们列表 比较一下 目标函数 损失函数 表现形式 f loss 生成方式 事先搭好框架，再通过训练得出待定 参数 将目标函数的输出与实际值作差 得到框架，然后代入一个具体的 训练样例（包括输入值与标签 值） 7/22/24, 8:40 AM 人工神经网络背后的数学原理！ 19/28 目标函数 损失函数 自变量 ——神经网络的输入值 （实际场合中可以是一张图片的所有 像素值） ——目标函数的 待定参数 函数值 （因变 量）含义 属于不同分类的概率 预测值与实际值的差值（越小越 好） 特点 我们最终想要得到的函数，可以用来 作图像分类。是线性函数与非线性函 数的组合，规模很大，自变量与参数 都很多 用来求出目标函数的过渡函数。 非负，最小值为0，一般要使用 梯度下降法找到极值点 举个例子看看函数变异的过程吧。设原函数为 ，这是一个关于 的二元函数，其中a, b, c均是常数，也可以叫待定参数。现在我们给出一组具体的函数输入值比 如，令 把它们代入函数，并且将a, b, c视为变量，则函数变成了关于a, b, c的 三元函数，记作 。 综上，求目标函数的过程，就变成了寻找损失函数极小值点的过程，而寻找极小值点不正可以用 上面介绍的梯度下降法实现吗？ 2.2.2 一个实例：关于链式求导和误差反向传播（BP） 行文至此，有关ANN的重要概念，我们还剩下链式求导和误差反向传播（BP）没有提及，让我们 用一个实例融会贯通一下。 考虑下面这个简单的ANN： 7/22/24, 8:40 AM 人工神经网络背后的数学原理！ 20/28 这个ANN只有4个神经元，分别是 。它输出两个目标函数，均是输入变量 的 函数，分别由神经元 输出。可以记为 这里给 加上帽子，表示这两个函数（即目标函数）的函数值是预测值，区别于训练数据给出 的实际标签值。而 均是目标函数的待定参数，这里我们假定神经元 7/22/24, 8:40 AM 人工神经网络背后的数学原理！ 21/28 均采用sigmoid激活函数，即 ，而神经元 不采用激活函数。 现在定义损失函数为 注意接下来我们会将具体的一组输入变量 带进去，这样损失函数就被视作以 为自变量的多元函数（具体的自变量变化过程参见上文描述）。其中， 是中间变量，它们均是以 为自变量的多元函数。 现在只要给出一个包含输入输出数据的训练样例，损失函数就成为不含未知参数的完全确定的函 数。而我们要做的就是找到这个损失函数的极小值。 按照梯度下降算法的推导，此时我们只要按照下面的步骤就可以找出这个极小值： 1. 随机指定一组初始参数 2. 计算Loss函数关于各个参数的偏导数，注意这一步要代入参数的具体数值，也就是说这一步得 到的是一个数 3. 按照梯度下降的公式更新各个参数值直到满足一定条件为止 其中2、3里的内容是需要反复迭代的。 现在，我们以其中的几个参数为例，看看在调整过程中会遇到什么新问题。 先试试调整 吧，这时我们需要求出损失函数对自变量 的偏导数值（注意是数值，不是表达 式），为此写出它的依赖关系： 7/22/24, 8:40 AM 人工神经网络背后的数学原理！ 22/28 7/22/24, 8:40 AM 人工神经网络背后的数学原理！ 23/28 这里，Loss函数依赖于变量 ，但 与 无关。回想多元函数求偏导数的规则，我们对求导 时 ，应将视为常数 。 而 依赖于变量 ，因此这里应按照复合函数求导法则，即传说中的链式求导法则，先让 Loss函数对变量 求导，再令 对 求导，即： 其实从神经网络的图中可以很清楚的看出求导链。 这里有几个要点： 首先，式子的每一项均加下标w，表示要将具体的一组 代入式子，得 到一个数值 其次，当g(u)表示sigmoid函数时，对其求导的结果就是 函数 对变量 求偏导时，虽然 也是函数，但它是关 于自变量 的函数，与 无关，因此视为常数。这样，对求 偏导的结果就是 等式右端最后得出的三项，在给出一个训练样例，并指定初始参数 后，是可以独立计算出结果的 求出损失函数对 的偏导数值，我们就可以按照梯度下降算法推导的公式，调整这个参数了！ 7/22/24, 8:40 AM 人工神经网络背后的数学原理！ 24/28 现在，再来看看靠前的参数是怎么调整的，我们以 和 为例 还是老规矩，对照神经网络图，先写出它的依赖关系： 7/22/24, 8:40 AM 人工神经网络背后的数学原理！ 25/28 可以看出 和 ，分别是函数 和 的变量，而函数 均与 和 有关，所以Loss函数需 要对 均求偏导。 依然按照链式求导法则对 求偏导，有： 对 求偏导，有： 注意红框圈出来的部分是不是有些眼熟？ 事实上，这一部分已经在调整后层参数的时候计算过了（请回看计算 时的计算公式）。因此在 编程时，可以让程序保存中间结果，这里直接拿来用。 现在纵观整个过程，我们惊奇的发现，对于ANN，当我们需要使用它时，是从最前面给出输入， 然后一步步往后计算得出这个庞大复杂函数的输出的；而当我们需要训练它时，则是从最后面的 7/22/24, 8:40 AM 人工神经网络背后的数学原理！ 26/28 参数开始，一步步向前求导，调整各个参数的。并且计算前面的参数时一般都会用到之前计算过 的中间结果。 这样，ANN调整参数的过程就可以看作是一个误差反向传播（BP）的过程。 之所以会这样反向传播，是因为神经网络中靠后的参数依赖的中间变量少、复合层数少，而靠前 的参数则经过层层复合，求导链会拉的很长。 2.2.3 最后的一点小问题 以上我们将求偏导的过程整个过了一遍，而求偏导只是梯度下降算法的一环。 程序跑起来之后，我们会对每一个训练样例，一遍遍的求偏导，直到基本上找到极小值点。 也就是说，按照梯度下降算法，每一个训练样例都会最终给出一组参数值。 一个训练集中显然会有多个训练样例，因此最终会得到好多组各不相同的参数值。 而ANN的训练目标是确定一组参数值，得到一个有一定效果的很复杂的函数， 怎么解决？ 对每一个参数，我们可以将不同训练样例得出的不同值求一个平均。也可以构建一个更大的损失 函数，即将每一个训练样例生成的损失函数求和，然后用梯度下降算法找到这个累计损失函数的 极小值点。 这样，我们就能通过训练，最终确定ANN这个大函数的所有待定参数，然后用它来做一些神奇的 事情。 7/22/24, 8:40 AM 人工神经网络背后的数学原理！ 27/28 7/22/24, 8:40 AM 人工神经网络背后的数学原理！ 28/28 |
8616 | https://gladwin.glk12.org/mod/book/tool/print/index.php?id=5875 | Tables
Skip to main content
Print book
Tables
Site:Gladwin
Course:Michigan Algebra I Sept. 2012
Book:Tables
Printed by:Guest user
Date:Sunday, September 28, 2025, 8:05 PM
Description
Tables
Table of contents
Introduction
Finding Differences
Example
Example Continued
Video Lesson
Guided Practice
Practice
Answer Keys
Sources
Introduction
In certain situations, the table of values for a polynomial function is known, but the equation is not. To model a polynomial function using a table, work the steps in finding the x-intercepts of a polynomial backwards. First, determine the zeros of the function by looking at the table. Second, turn each zero into a factor of the function. Finally, determine the factored form of the polynomial.
Example 1Determine the function that models the table below:
Step 1. Determine the zeros of the function.
(-5, 0), (1, 0), (3, 0)
Step 2. Turn each zero into a factor of the function.
Since x = -5, then (x + 5) is a factor of the polynomial.
Since x = 1, then (x - 1) is a factor of the polynomial.
Since x = 3, then (x- 3) is a factor of the polynomial.
Step 3. Determine the factored form of the polynomial. Use a point from the table to find value of a.
y = a (x + 5)(x - 1)(x - 3); point (0, 30)
30 = a (0 + 5)(0 - 1)(0 - 3)
30 = a (15)
2 = a
y = 2 ( x + 5)( x - 1)( x - 3)
Listen
Finding Differences
There is a method that can be used to determine the degree of a polynomial just by using a data table. This method is called finite differences. To find the exact equation for the polynomial function, you need to find the coefficients by solving a system of equations or using some other method. This will be covered in later mathematics classes.
If you have a set of data with equally spaced x-values, you can find the lowest possible degree of a polynomial function that fits the data, if there is one, by analyzing the differences in the y-values. This technique, the finite differences method,determines only the degree of the polynomial. The steps are as follows:
Method of Finite Differences Algorithm
List the terms as a sequence of numbers.
Find the differences between successive terms until a constant is reached for every difference.
Count the number of times it was necessary to subtract to get a common difference.
The number of subtractions is the degree of the polynomial describing the sequence.
Listen
Video Player is loading.
Play Video
Play
Mute
Current Time 0:00
/
Duration-:-
Loaded: 0%
Stream Type LIVE
Seek to live, currently playing live LIVE
Remaining Time--:-
1x
Playback Rate
Chapters
Chapters
Descriptions
descriptions off, selected
Captions
captions settings, opens captions settings dialog
captions off, selected
Audio Track
Picture-in-Picture
This is a modal window.
The media could not be loaded, either because the server or network failed or because the format is not supported.
Beginning of dialog window. Escape will cancel and close the window.
Text Color Transparency Background Color Transparency Window Color Transparency
Font Size Text Edge Style Font Family
Reset restore all settings to the default values Done
Close Modal Dialog
End of dialog window.
Example
The table below contains sets or ordered pairs from an unknown polynomial function. Use finite differences to determine the degree of the polynomial.
Step 1. Make sure the x-values are in equal increments.
The x-values of the table are increasing by one.
Step 2. Find the first difference of the table.
Since the first difference is not constant, the polynomial is not linear.
Video Player is loading. Play Video Play Mute Current Time 0:00 / Duration 0:00 Loaded: 0% Stream Type LIVE Seek to live, currently playing live LIVE Remaining Time-0:00 1x Playback Rate Chapters Chapters Descriptions descriptions off, selected Captions captions settings, opens captions settings dialog captions off, selected Audio Track Picture-in-Picture This is a modal window. The media could not be loaded, either because the server or network failed or because the format is not supported. Beginning of dialog window. Escape will cancel and close the window. Text Color Transparency Background Color Transparency Window Color Transparency Font Size Text Edge Style Font Family Reset restore all settings to the default values Done Close Modal Dialog End of dialog window.Listen
Example Continued
Step 3. Find the second difference.
Since the second difference is not constant, the polynomial is not a quadratic.
Step 4. Find the third difference.
Since the third difference is constant, this table can be modeled by a third degree polynomial.
Listen
Video Lesson
For a video lesson on finding the degree of a polynomial, select the following link:
Degree of Polynomials
Guided Practice
To solidify your understanding of writing a function from a graph & table, visit the following link to Holt, Rinehart and Winston Homework Help Online. It provides examples, video tutorials and interactive practice with answers available. The Practice and Problem Solving section has two parts. The first part offers practice with a complete video explanation for the type of problem with just a click of the video icon. The second part offers practice with the solution for each problem only a click of the light bulb away.
Guided Practice
Practice
Finite Differences Worksheet
Stacking Bricks Worksheet
Note: If Google Docs displays “Sorry, we were unable to retrieve the document for viewing,†refresh your browser.
Answer Keys
Finite Differences Answer Key
Stacking Bricks Answer Key
Note: If Google Docs displays “Sorry, we were unable to retrieve the document for viewing,†refresh your browser.
Sources
Boast, Steve. "Finding Differences Lab." 08/02/2000. webs.wichita.edu/facsme/cbl/algebra2/finitedif.pdf (accessed 08/12/2010).
Embracing Mathematics, Assessment & Technology in High Schools; A Michigan Mathematics & Science Partnership Grant Project
Holt, Rinehart & Winston, "Polynomial Functions." (accessed 9/8/2010).
Holt, Rinehart & Winston, "Using Finite Differences to Determine Degree." (accessed 9/8/2010).
Texas Instruments, "Count the Differences." (accessed 08/12/2010). |
8617 | https://www.hollywoodreporter.com/movies/movie-reviews/intersections-film-review-417090/ | Intersections: Film Review
Skip to main content
Plus Icon Click to expand the Mega Menu
Click to expand search form
Got a tip?
The Hollywood Reporter homepage
Newsletters
Subscribe
The Definitive Voice of Entertainment News
Subscribe for full access to The Hollywood Reporter
See My Options Sign Up
site categories
News
Film
TV
Music
Awards
Lifestyle
Business
International
Video
Charts
Lists
Plus Icon Click to expand the Mega Menu Menu
Click to expand search form
The Hollywood Reporter homepage
Read Next:
Paul McCartney Stands by Pride Flag, Debuts “Help!” at Santa Barbara Bowl ‘Got Back’ Tour Kickoff
Subscribe
The Definitive Voice of Entertainment News
Subscribe for full access to The Hollywood Reporter
See My Options Sign Up
Home
Movies
Movie Reviews
Intersections: Film Review
Roschdy Zem is the main attraction as David Marconi applies Besson formula to Sahara action thriller.
By Bernard Besserglik
Plus Icon
Bernard Besserglik
More Stories by Bernard Besserglik
‘Latifa: A Fighting Heart’ (‘Latifa, le coeur au combat’): Film Review
‘Money’: Film Review
‘Stalin’s Couch’ (‘Le Divan de Staline’): Film Review
View All
January 31, 2013 10:00am
Share on Facebook
Share on X
Share to Flipboard
Send an Email
Show additional share options
Share on LinkedIn
Share on Pinterest
Share on Reddit
Share on Tumblr
Share on Whats App
Print the Article
Post a Comment
Intersections Film Still - H 2013
Share on Facebook
Share on X
Share to Flipboard
Send an Email
Show additional share options
Share on LinkedIn
Share on Pinterest
Share on Reddit
Share on Tumblr
Share on Whats App
Print the Article
Post a Comment
After a 20-year absence since his directorial debut, writer David Marconiis back behind the camera with a LucBesson-backed action thriller set in the Sahara and starring French-Moroccan actor Roschdy Zem(Outside the Law). The usual Besson rules of rapid action and broad-brush characterization apply, but without an empathetic hero of the sort incarnated by Liam Neeson in the Taken franchise, Intersections may struggle to find an audience. On the plus side, the scenery is great, the characters are a spectacularly unlovely bunch and the implausibilities of the plot mount up at such a rate that it’s almost fun ticking them off as they come.
Related Stories
Movies
Warner Bros. Begins Production on Korean Remake of 'The Intern' Starring Choi Min-sik and Han So-hee
Movies
Park Chan-wook's Critically Acclaimed Comedy Thriller 'No Other Choice' Opens in First Place at Korea's Box Office
When New York hedge fund manager Scott Dolan (Frank Grillo) and his new bride Taylor (Jaimie Alexander) set out on a desert excursion for their honeymoon, it’s clear that villainy is afoot, because in no time at all Taylor is cheating on her husband with one of his colleagues, Travis (Charlie Bewley). After a car chase through the vast sandy wastes, the two men contrive to cause a multiple pile-up involving vehicles carrying diamond smuggler Omar (Moussa Maaskri) — being transferred to a prison under police escort — and a young French woman, Audrey (Marie-Josee Croze), who is carrying a small baby. The policeman dies in the crash but the numbers are made up by the arrival of the mysterious stranger Saleh (Zem) whose motorbike has broken down in the vicinity.
The Bottom Line Besson broad-brush formula applied to Sahara action thriller lacks warmth or plausibility.
From here on in, it’s a question of survival. Each of the characters harbors a hidden agenda, usually murderous, in two cases made explicit by means of flashbacks. None of the characters attracts much sympathy, except possibly the small baby. The resourceful Saleh is the nearest to a positive figure, but he’s the strong, silent type whose motives are not made clear until the ending. An hour into the film, by which time two of the characters are dead, one of the vehicles is repaired and the action transfers to the port city of Essaouira, where a new villain, Cyril (Carlos Leal), is encountered.
It’s all pure tosh, of course, and if the gleam of complicity between two of the surviving characters at the close is intended to provide a touch of meaning or warmth, it fails. Given a sufficient lowering of expectations, the film can, at a pinch, be enjoyed, and canny marketing may harvest a commercial return. Full marks to stunt organizer Dominique Fouassierfor the car crash. Otherwise, the movie adds nothing of significance to the CVs of anyone involved.
_Opened: Wednesday, Jan. 30
Production companies: EuropaCorp, Grive Productions
Cast: Roschdy Zem, Frank Grillo, Jaimie Alexander, Marie-Josee Croze, Moussa Maaskri, Charley Bewley, Afif Ben Badra, Carlos Leal, Gabriella Wright
Director: David Marconi
Writer: David Marconi
Photography: Thomas Hardmeier
Producers: Fanny Besson, Gareth Upton
Production Design: Bertrand Seitz
Editor: Julien Rey
Music: Richard Horowitz
Sales: EuropaCorp
No MPAA rating
Running time: 101 minutes_
Read More About:
international
THR Newsletters
Sign up for THR news straight to your inbox every day
Subscribe Sign Up
More from The Hollywood Reporter
Sundance
Quentin Tarantino Pays Tribute to Robert Redford and Sundance at Burbank Fest: “None of Us Independent Filmmakers Would Be Where We Are Without the Sundance Film Institute”
The Intern
Warner Bros. Begins Production on Korean Remake of ‘The Intern’ Starring Choi Min-sik and Han So-hee
international
Park Chan-wook’s Critically Acclaimed Comedy Thriller ‘No Other Choice’ Opens in First Place at Korea’s Box Office
Samantha Morton
‘Anemone’ Review: Daniel Day-Lewis Makes a Commanding Return to the Screen in a Drama That Seldom Approaches His Earth-Shaking Force
J.K. Rowling
J.K. Rowling Applauds Emma Watson Parody Video After ‘Harry Potter’ Star Addresses Their Trans Dispute
Zurich Film Festival
Colin Farrell on His “Extraordinary, Unearned Good Fortune,” From ‘Tigerland’ to ‘The Penguin’
The Definitive Voice of Entertainment News
Subscribe for full access to The Hollywood Reporter
SEE MY OPTIONS Sign Up
THR Newsletters
Sign up for THR news straight to your inbox every day
Email Address
Subscribe Today Sign Up
By providing your information, you agree to our Terms of Use and our Privacy Policy. We use vendors that may also process your information to help provide our services. // This site is protected by reCAPTCHA Enterprise and the Google Privacy Policy and Terms of Service apply.
Most Popular
### Jimmy Kimmel to Return to ABC on Tuesday After Show’s Controversial Suspension
ByJames Hibberd, Alex Weprin
### ‘Dancing With the Stars’ Ends One-Hit Wonders Night With a Four-Way Tie for Top of the Leaderboard: See All the Scores, Who Went Home
ByMcKinley Franklin
### Nexstar and Sinclair Won’t Air Jimmy Kimmel’s Return, Impacting 66 ABC Stations
ByAlex Weprin
### Disney Sued After Enforcing Rights to Mickey Mouse in the Public Domain
ByWinston Cho
### Crunchyroll Boss on the Blockbuster Implications of ‘Demon Slayer: Infinity Castle’: “It’s Now Undeniable How Big Anime Has Become”
ByPatrick Brzeski
Shopping With THR
Article’s Designer-Favorite Furniture Is Up to Half Off — Shop the Top 10 Fall Clearance Finds
By Erin Lassner
The Best Men’s Jeans for Fall, from Classic Fits to Japanese Denim
By Rudie Obias
The Best Gifts for Men, from Sony Headphones, Nintendo Gaming Console, Nike Sneakers and More
By Rudie Obias
The Best Phone Wallets for Every Style, From Stick-On Accessories to MagSafe Favorites
By Natalie Alcala
PMC Logo
Most Popular
### Steve McQueen's Granddaughter Sues Lawyer over $68 M. Jackson Pollock Painting
### Selena Gomez's Wedding Guest List Has Everyone Questioning One Major Absence
### NFL Highest-Paid Coaches 2025: Andy Reid Leads at $20M/Year
### "Rapture" Trends Online As End Of The World Conspiracy Spreads Across Social Media
You may also like
### Invest in Sports: Join Sportico in NYC on Nov. 4! 2 weeks ago
### Dior’s Most In-Demand Aesthetician Now Offers a $1,300 Facial in N.Y.C.—and We Tried It 1 day ago
### Nas-Backed Queens Casino Moves Ahead After JAY-Z’s Times Square Project Is Rejected 2 days ago
### Tame Impala Sinks Teeth In With ‘Dracula,’ Announces U.K. And Europe Tour 3 hours ago
Subscriber Support
Get the Magazine
Customer Service
Back Issues
The Hollywood Reporter
About Us
Careers
Contact Us
Accessibility
Legal
Terms of Use
Privacy Policy
AdChoices
Follow Us
FacebookFacebook
InstagramInstagram
LinkedInLinkedIn
ThreadsThreads
TikTokTikTok
XX
YouTubeYouTube
Newsletter Sign Up
Sign Up
Subscribe Sign Up
By providing your information, you agree to our Terms of Use and our Privacy Policy. We use vendors that may also process your information to help provide our services. // This site is protected by reCAPTCHA Enterprise and the Google Privacy Policy and Terms of Service apply.
Have a Tip?
Send us a tip using our anonymous form.
Send Us a Tip Sign Up
Icon Link Plus Icon
The Hollywood Reporter is a part of Penske Media Corporation. © 2025 The Hollywood Reporter, LLC. All Rights Reserved.
THE HOLLYWOOD REPORTER is a registered trademark of The Hollywood Reporter, LLC.
Powered by WordPress.com VIP
Our Sites
Art in America
Artforum
ARTnews
Billboard
BlogHer
Deadline
Flow Space
Gold Derby
IndieWire
Robb Report
Rolling Stone
SheKnows
Soaps
Sourcing Journal
Sportico
StyleCaster
The Hollywood Reporter
TVLine
Variety
Vibe
WWD
Close the menu
The Hollywood Reporter homepage
Log In
NewsExpand news menu
Latest News
Culture & Politics
LA Wildfires
Obituaries
LA/Local
THR Investigates
FilmExpand film menu
News
Features
Reviews
Box Office
Heat Vision
TVExpand tv menu
News
Features
Reviews
Ratings
Live Feed
Heat Vision
The Fien Print
MusicExpand music menu
Music News
Music Industry News
Music Features
Film and TV Music News
Grammys
K-Pop
Country Music
AwardsExpand awards menu
News
Awards Chatter Podcast
Feinberg Forecast
Behind The Screen
The Race
THR Presents
Oscar Season Hub
LifestyleExpand lifestyle menu
Person of Interest
News
Style
Shopping
Arts
Real Estate
Rambling Reporter
BusinessExpand business menu
THR, Esq
Report Features
Unions / Labor
Film / TV Tax Credits
Signings / Representation
THR ChartsExpand thr-charts menu
Weekly Streaming Ratings Chart
Weekly Broadcast Ratings Chart
Weekend Box Office Chart
Weekly Broadway Grosses Chart
More EssentialsExpand more-essentials menu
International News
Video
THR Podcasts
Featured Voices
Subscribe to Logo text
Follow Us
Follow THR on Facebook
Follow THR on Instagram
Follow THR on LinkedIn
Follow THR on Threads
Follow THR on TikTok
Follow THR on X
Follow THR on YouTube
Alerts & Newsletters
Your Email
Subscribe Sign Up
By providing your information, you agree to our Terms of Use and our Privacy Policy. We use vendors that may also process your information to help provide our services. // This site is protected by reCAPTCHA Enterprise and the Google Privacy Policy and Terms of Service apply.
optional screen reader
About Us
Careers
Contact Us
Icon Link Plus IconThe Hollywood Reporter is a part of Penske Media Corporation. © 2025 The Hollywood Reporter, LLC. All Rights Reserved.
THE HOLLYWOOD REPORTER is a registered trademark of The Hollywood Reporter, LLC.
ad
Your Privacy Rights
Close privacy/terms of use policy
PMC, its service providers, and third-party partners collect information about you and how you use our services (including clicks and screen recordings) using first and third-party cookies, pixels, and similar technologies. Learn about our data practices and your choices in our Privacy Policy. By continuing to use our services, you agree to our Terms of Use (including to waiver and arbitration).
Accept privacy/terms of use policy Accept
Privacy Preference Center
When you visit any website, it may store or retrieve information on your browser, mostly in the form of cookies. This information might be about you, your preferences or your device and is mostly used to make the site work as you expect it to. The information does not usually directly identify you, but it can give you a more personalized web experience. Because we respect your right to privacy, you can choose not to allow some types of cookies. Click on the different category headings to find out more and change our default settings. However, blocking some types of cookies may impact your experience of the site and the services we are able to offer.
More information
Allow All
Manage Consent Preferences
Performance Cookies
[x] Performance Cookies
These cookies allow us to count visits and traffic sources so we can measure and improve the performance of our site. They help us to know which pages are the most and least popular and see how visitors move around the site. All information these cookies collect is aggregated and therefore anonymous. If you do not allow these cookies we will not know when you have visited our site, and will not be able to monitor its performance.
Targeting Cookies
[x] Targeting Cookies
These cookies may be set through our site by our advertising partners. They may be used by those companies to build a profile of your interests and show you relevant adverts on other sites. They do not store directly personal information, but are based on uniquely identifying your browser and internet device. If you do not allow these cookies, you will experience less targeted advertising.
Strictly Necessary Cookies
Always Active
These cookies are necessary for the website to function and cannot be switched off in our systems. They are usually only set in response to actions made by you which amount to a request for services, such as setting your privacy preferences, logging in or filling in forms. You can set your browser to block or alert you about these cookies, but some parts of the site will not then work. These cookies do not store any personally identifiable information.
Functional Cookies
[x] Functional Cookies
These cookies enable the website to provide enhanced functionality and personalisation. They may be set by us or by third party providers whose services we have added to our pages. If you do not allow these cookies then some or all of these services may not function properly.
Cookie List
Clear
[x] checkbox label label
Apply Cancel
Consent Leg.Interest
[x] checkbox label label
[x] checkbox label label
[x] checkbox label label
Confirm My Choices
15 |
8618 | https://pmc.ncbi.nlm.nih.gov/articles/PMC10856212/ | The Iodine/Iodide/Starch Supramolecular Complex - PMC
Skip to main content
An official website of the United States government
Here's how you know
Here's how you know
Official websites use .gov
A .gov website belongs to an official government organization in the United States.
Secure .gov websites use HTTPS
A lock ( ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites.
Search
Log in
Dashboard
Publications
Account settings
Log out
Search… Search NCBI
Primary site navigation
Search
Logged in as:
Dashboard
Publications
Account settings
Log in
Search PMC Full-Text Archive Search in PMC
Advanced Search
Journal List
User Guide
New Try this search in PMC Beta Search
PERMALINK
Copy
As a library, NLM provides access to scientific literature. Inclusion in an NLM database does not imply endorsement of, or agreement with, the contents by NLM or the National Institutes of Health.
Learn more: PMC Disclaimer | PMC Copyright Notice
Molecules
. 2024 Jan 30;29(3):641. doi: 10.3390/molecules29030641
Search in PMC
Search in PubMed
View in NLM Catalog
Add to search
The Iodine/Iodide/Starch Supramolecular Complex
Szilard Pesek
Szilard Pesek
1 Department of Chemistry, Faculty of Chemistry and Chemical Engineering, Babeş-Bolyai University, 11 Arany Janos Street, 400028 Cluj-Napoca, Romania; szilard.pesek@ubbcluj.ro
Find articles by Szilard Pesek
1, Radu Silaghi-Dumitrescu
Radu Silaghi-Dumitrescu
1 Department of Chemistry, Faculty of Chemistry and Chemical Engineering, Babeş-Bolyai University, 11 Arany Janos Street, 400028 Cluj-Napoca, Romania; szilard.pesek@ubbcluj.ro
Find articles by Radu Silaghi-Dumitrescu
1,
Editors: Litao Tong 1, Lili Wang 1
Author information
Article notes
Copyright and License information
1 Department of Chemistry, Faculty of Chemistry and Chemical Engineering, Babeş-Bolyai University, 11 Arany Janos Street, 400028 Cluj-Napoca, Romania; szilard.pesek@ubbcluj.ro
Correspondence: radu.silaghi@ubbcluj.ro; Tel.: +40-264-593833
Roles
Litao Tong: Academic Editor
Lili Wang: Academic Editor
Received 2023 Aug 15; Revised 2024 Jan 21; Accepted 2024 Jan 25; Collection date 2024 Feb.
© 2024 by the authors.
Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (
PMC Copyright notice
PMCID: PMC10856212 PMID: 38338385
Abstract
The nature of the blue color in the iodine–starch reaction (or, in most cases, iodine–iodide-starch reaction, i.e., I 2 as well as I− are typically present) has for decades elicited debate. The intensity of the color suggests a clear charge-transfer nature of the band at ~600 nm, and there is consensus regarding the fact that the hydrophobic interior of the amylose helix is the location where iodine binds. Three types of possible sources of charge transfer have been proposed: (1) chains of neutral I 2 molecules, (2) chains of poly-iodine anions (complicated by the complex speciation of the I 2-I− mixture), or (3) mixtures of I 2 molecules and iodide or polyiodide anions. An extended literature review of the topic is provided here. According to the most recent data, the best candidate for the “blue complex” is an I 2-I 5−-I 2 unit, which is expected to occur in a repetitive manner inside the amylose helix.
Keywords: amylose, iodine, iodide, starch, UV-vis, DFT
1. Introduction
Starch is a mixture of amylose and amylopectin in a ratio of ~20–30% to 70–80%, both of which are polymers of glucopyranose. Amylose is a linear polymer typically containing 300–3000 (or sometimes much more) monomeric units that are interconnected via α(1→4) glycosidic bonds (cf. Figure 1) . Amylopectin has a similar structure, additionally featuring ramifications via α(1→6) bonds. A small number of ramifications (of the same type as amylose) are in fact also found in amylose. The average number of branches in amylose molecules is size-dependent, ranging from zero to 2–4 branches per molecule. The variations have been generally attributed to differences in chain length, with the observation that most branching occurs at the early stage of amylose synthesis . Both polymers are sparsely soluble in water, with amylose more so than amylopectin. Inside living organisms, the enzyme amylase is mainly responsible for hydrolyzing starch. Dextrins, which are polysaccharides with a low degree of polymerization, are produced by partial hydrolysis of starch; with complete hydrolysis, glucose is formed .
Figure 1.
Open in a new tab
Molecular structure of amylose.
Amylose can exist in a disordered amorphous conformation or in two types of helical forms (Figure 2). The first type is a double helix with itself (including forms A and B). The second type is the V form, which consists of a single helix; this is the structure typically discussed in biochemistry textbooks. The V form features an internal cavity that is large enough to accommodate hydrophobic guest molecules such as iodine, fatty acids, or small aromatic compounds. This property has in fact been recently exploited in extensive attempts to design smart drug delivery systems, with amylose/starch serving as the host for inclusion complexes with various molecules of therapeutic potential/use [4,5,6].
Figure 2.
Open in a new tab
Molecular models of helices of A-type, B-type, and V-type cycloamylose.
In 1930, Katz studied the aging and cooking of bread, and with the help of X-ray diffraction, in the analyzed powder he found, in addition to the type A and B models known until then from native starch, he found another crystalline form of starch, which he called model V. To describe the contribution of this form, he used the German word “Verkleitsterung,” which means gelatinization . He found the same V-type pattern when he prepared pasta, which was precipitated with alcohols . Bear identified other different V-type patterns depending on the precipitation agent used . Native starch was fractionated by Meyer et al., separating amylose from amylopectin using hot water .
A- and B-amylose both form parallel-stranded double helices of 6 × 2 glucoses per turn and right-handed [11,12,13,14,15,16] or left-handed turns. Structurally, these two structures differ from each other only in their packing arrangements and water contents. They also differ in biological locations, with A preferentially in grains and B in tubers . A polymorph C of amylose has also been described, consisting of a mixture of A- and B- unit cells .
The V form is also found naturally and is structured as a single left-handed helix with six glucose units per turn and a step height of 7.91 to 8.17 Å [11,12,13]. The V form can be isolated through its precipitation from aqueous solution using alcohols, ketones, fatty acids, iodine, or salts that form inclusion complexes (thus behaving similarly to cyclodextrins, α-cyclodextrin, or cyclohexaamylose) [19,20]. The glucose units in V-amylose (as in cyclodextrins) are all in the syn orientation. This entails hydrogen bonds between the secondary hydroxyl groups O(3)n⋅⋅⋅O(2)n+l, as well as O(6)n⋅⋅⋅O(2)n+6 hydrogen bonds between turns [11,12,13].
Variants of the V-type structure of amylose have been described, each marked with a subscript typically indicating the number of glucose units per shift. The most common variant is the V 6 form , though V 8 and V 7 have also been described. The latter would provide even more space for the guest molecule to bind . Depending on the solvent, slightly different crystallographic structures are formed. The crystals grown with water as the solvent, dubbed amylose V h, feature six glucose units in the unit cell, aligned as a single strand. Within the helix, there are three water molecules, statistically positioned on six positions. In the spaces between helices, there is one water molecule each, on one of three possible positions. The helices point upwards and downwards in a random fashion .
Amylose V is the allomorph known for its deep blue complexes with iodine. The iodine molecules are trapped in the channels within the helices, where molecules of the solvent can also be present. As detailed in the following sections, extensive research has been devoted to understanding the structural features of the amylose–iodine complexes, especially as the blue complexes generally form only when iodide anions are present alongside iodine. This implies the presence of polyiodide species, of which tri-iodide is the most often invoked. Figure 3 illustrates the proposed structures of amylose–I 2 and amylose–tri-iodide species based on electronic structure calculations. Figure 4 illustrates the representative structures proposed for the iodine/iodide arrangement inside the amylose helix .
Figure 3.
Open in a new tab
Structures of AM1-optimized amylose models: A-I 2 and A-I 3− .
Figure 4.
Open in a new tab
Structures of I 2, I 3−, I 5−, I 3−-I 2-I 3−, I 5−-I 2-I 5−, proposed as possible sources of the blue color in the iodine–iodide–amylose complexes.
Cyclodextrins are glucose oligomers featuring the same type of glycosidic bonds as starch, and they can be obtained enzymatically. Depending on their size, some cyclodextrins can arrange themselves in a circular fashion, appearing similar to single turns of the amylose helix, and then forming dimers which generate large cylinders, similar to the amylose helix in its overall structure.
2. General Considerations on the Reaction of Starch with Iodine
Colin and Claubry discovered in 1814 the reaction between starch and iodine (or, more specifically, with iodide–iodine solutions, as molecular I 2 is otherwise very insoluble in water in the absence of I− ions). Since then, the reaction has been found both in organic chemistry classes in school and in qualitative and quantitative analysis courses. Over time, many experiments have shown that the starch–iodine complex shows absorption at ~600 nm: a strong dark blue color . A more detailed description of the reaction was provided by H. H. Landolt in 1886 . The color is mainly due to the complex of iodine with the amylose complex that absorbs at ~620 nm. The affinity of iodine for amylopectin is distinctly smaller (~20 times), and the resulting complex is reddish-violet, with a maximum at ~540 nm. In 1948, Gilbert and Marriott showed that at higher concentrations of iodide, the ratio of iodide ions to iodine molecules increases to at least one, leading to a purple tint in the blue complex.
The nature of the amylose–iodine complex has been debated for many decades, especially in terms of the stoichiometry and charge of the poly-iodine substructures within the helix (see, e.g., Figure 4). It is now generally accepted that iodide ions are also required in this process. Thus, the iodine atoms would align inside the amylose helix as a mixture/combination of I 2 and I− [21,22,24,25,26,27] with an unusual metallic-like structure . This mixture would not entail isolated I− ions, but rather I n− polyiodides (n = 3, 5, 9…) [28,29,30,31,32]. The wavelength of the absorbance maximum in the amylose–iodine complex is known to depend on the chain length. Thus, glucose chains of 4 to 6 units yield no color, while those 8 to 12 yield a red color with a peak at 520 nm reminiscent of amylopectin. Longer chains progressively show a bathochromic shift until a length of 30 to 35 units, when the blue color is reached with a peak at 600–620 nm. Very similar spectra are also obtained through the synthetic action of potato phosphorylase on starch when chains of 50–150 units are obtained.
This relationship between chain length and iodine color has also been applied to branched polysaccharides. Comparing the spectra of the iodine/iodide complexes with dextrin, amylopectin, glycogen, or various synthetic oligosaccharides, estimations of the extent of helical portions available for iodine binding in amylopectin and glycogen were formulated, at 8–18 glucose residues. In line with these observations, hydrolysis of amylose by α-amylase or under acid catalysis gradually changes the absorption maximum from blue to red. On the other hand, hydrolysis with β-amylase leads to a hypochromic but no hypsochromic shift. This can be explained by the fact that β-amylase remains bound to its substrate until it completely degrades it, rather than gradually degrading all the polymer chains at the same time. In this way, little or no red-colored intermediate dextrins remain in the mixture . Variations in color for iodine–amylopectin complexes can be observed due to the difference in structures , branched chain length , and branching points in amylopectin . If the degree of polymerization in amylopectin entails 15 glucose units, complexation of iodine is not observed at any temperature. If it exceeds 30, then iodine binds to amylopectin at either 1.5 °C or 20 °C . Mould and Synge used potentiometric and spectrophotometric titrations for complexation of iodine/iodide with the products of enzymatic hydrolysis of amylose, i.e., dextrins with different molecular masses. Dextrins of less than 10 glucose units were essentially unreactive, those of 10–25 units were orange, those 25–40 were red, and those of 40–130 were blue. Ono showed that the λ max of the amylose–iodine complex shifted to shorter wavelengths with increasing iodide concentrations; this was interpreted as evidence of the breaking the polyiodine chains by the permeated iodide ions.
3. Dependence on the Nature of the Organic (Bio)Polymer
In addition to starch, there is a long list of natural polymers that afford colored complexes with iodine. These include chitosan, glycogen, silk, wool, albumin, cellulose, xylan, and natural rubber. A large number of synthetic polymers have also been described to react with iodine. Examples include poly(vinyl alcohol) (PVA), poly(vinyl pyrrolidone) (PVP), nylons, poly(Schiff base)s, polyaniline, and unsaturated polyhydrocarbons (carbon nanotubes, fullerenes C 60/C 70, and polyacetylene) . It is important to note that most of these polymers do not feature helical structures of the type seen in amylose; binding of the iodine on the outside of the organic polymer, or between polymer chains, is likely occurring in such cases.
Differences in the blue color were reported depending on the size (and, implicitly, biological source) of the polymer, be it amylose or related poly and oligo saccharides or other organic polymers [34,36,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54]. Yu et al. showed that when complexation occurs between I 2/KI and potato amylose, the speciation of iodide varies. Data from Raman and UV-visible spectroscopy were interpreted as evidence that the primary forms of iodine were the monoanions I 3− and I 5−, but larger units were also present such as I 93−, I 113−, I 133−, and I 153− (with bands at 460–480, 560–590, 660–700, and 710–740 nm, respectively), with higher iodide concentrations expectedly favoring shorter polyiodide chains. By adding an iodine–potassium iodide solution to a solution of cellulose, Abe obtained an intense blue solution. At 80 °C, the color disappeared, but upon cooling, it reappeared. Takahashi reported a dark purple adduct upon treating chitin with I 2/KI solution for 24 h at room temperature. Depending on the amount reactant ratios, the iodine content of the adduct was in the range of 9–20%, with one molecular iodine per 6.4 chitin residues. Yajima et al. prepared a purple complex (maximum at 550 nm) by freezing a mixture of chitosan and I 2/KI solution at −20 °C and then thawing it at 4 °C.
Glycogen, as also discussed above, yields a reddish-brown complex with iodine. The largest maximum is at 395 nm, but as shown by Kumari et al. and Lecker et al. , a series of UV absorption bands is also present at 408, 453, 496, 560, 650, and 698 nm. The bands at 408, 453, 560, and 650 nm were comparable to those of amylopectin (412, 458, 550, and 640 nm), and the bands at 496 and 698 nm were assigned to an I 4 species. The bands at 453 and 560 nm were more visible at higher iodine concentrations.
Stromeyer, in 1815, mentions that wool and silk (i.e., the protein structure therein) give yellow colors on exposure to iodine . Amyloid peptides also yield a blue color with molecular iodine. In the presence of sulfuric acid, this shifts to blue–violet, cf. Aterman . Dzwolak described the formation of an insulin–amyloid complex in the presence of I 2/KI that is stable up to 90 °C, interpreted as an inclusion complex between the amyloid fibrils and iodine.
Pritchard and Serra reported that reaction of poly(vinyl acetate) (PVAc) with molecular iodine in methanol in the presence of aqueous KI yields a deep-red precipitate (darker at higher iodine concentrations), interpreted as an I 2-PVAc adduct. Two UV-vis absorbance bands were reported for the complex at 520 and 510 nm. Hughes et al. noted that the red color of the I 2-PVAc complex is independent of the method through which the polymer is prepared. Solutions of the PVA–iodine complex (I 2-PVA) feature intense 600–620 nm, 650–680 nm, 480–500 nm, and 350 nm bands depending on the conditions. The complex of I 2 with poly(vinylpyrrolidone) (povidone or PVP) has a maximum at 361 nm, with the iodine molecule reported to be attached to the PVP matrix through non-covalent interactions with the carbonyl groups. A poly(N-methyl-4-vinyl pyridinium) triiodide was reported to produce a dark-brown powder with molecular iodine in a water–alcohol solution, with bands at 295 nm, 367 nm, and 460 nm .
Charge transfer complexes of iodine with ferrocenyl-bearing Schiff bases have also been described . Liu et al. showed that the brown poly(ferrocenyl-Schiff) bases turn black after treatment with iodine in acetone. In the infrared (IR) spectra, iodine binding resulted in a decrease of the ~1610 cm−1 signal of the Schiff base, accompanied by the appearance of a new wide absorption band at 450–480 nm with a long tail up to ~900 nm in the UV-vis spectra.
Shirakawa examined the reaction of polyacetylene (PAC) with iodine. Strong IR bands were noted at 870 cm−1 and 1390 cm−1. In the UV-vis spectrum analysis, bands at 365 nm and 502 nm were noted to be suggestive of the presence of I 3− and I 2, respectively. A band at 280 nm characteristic of unreacted PAC was noted to decrease in intensity upon doping.
Treatment of poly(p-phenylene vinylene) or phen(p-phenyvinylene) (PPV) with iodine vapor showed a significant effect on its luminescence .
In principle, due to the differences in reactivity between amylose and amylopectin, the iodine reaction can be used for assessing the amylose content of starch. However, a number of other methods may be more appropriate and accurate, such as polarimetry, anthrone, FTIR, penetrometry, Luff school, gravimetry, X-ray diffraction (XRD), and size exclusion chromatography (SEC) techniques including multi-angle laser light scattering–differential refractive index detection (SEC-MALS-DRI) [72,73,74,75,76,77].
Three bands in the UV-visible–NIR spectra, at 688 nm, 1724 nm, and 2292 nm, were interpreted as evidence for a PPV–iodine complex. Polybutadiene, poly(cis-isoprene), and their copolymers were reported to display an intense absorption band at 305 nm in the UV spectrum (with some variations depending on the solvent) upon treatment with iodine . Sreeja et al. found that acrylonitrile butadiene rubber developed two new broad bands appearing between 300 nm and 500 nm, while the 262 nm band attributed to the C=C bonds decreased. Poly(β-pinene) treated with iodine vapor was shown by Vippa et al. to produce bands at 310 nm and 400 nm in the UV-vis spectrum, with the latter interpreted as evidence for a charge transfer between the double bond and I 2.
4. Geometrical Data
Crystalline units in starch and in its partial degradation products have long been explored [81,82,83]. Transmission electron microscopy , light microscopy , the use of atomic force microscopy (AFM) [86,87,88,89,90,91,92], and X-ray and neutron small angle scattering have allowed for some partial visualization of repeating units in amylose. Microstructures were observed, which were called nodules, initially with diameter variations between 150 and 300 nm , and also smaller particles of 20–50 nm [85,87,88,93] (e.g., 130–250 nm for pea starch , 20–50 nm for potato starch , or 10–30 nm for corn granules ). Using combined methods of X-ray diffraction and stereochemical packing analysis of the amylose–iodine complex, Bluhm and Zugenmaier reported that the iodine atoms aligned almost linearly in the center of the amylose chain. Eight hydration water molecules were found per unit cell, located in the amylose helices. The amylose left-handed helix was reported to have an outer diameter of~13 Å and a pitch of 8 Å (six 1,4-glucose units per pitch), hosting an internal cavity of ~5 Å [28,30,31,32,34,36,94,95,96,97]. Some statistical disorder was noted within the polyiodide chain, with an average iodine–iodine distance of ~3.1 Å. This value is larger than the 2.67 Å in molecular I 2 as well as than the 2.90 Å in ionic I 3−, but it is distinctly shorter than the 4.3 Å sum of van der Waals radii for two iodine atoms. With the help of the electron–gas theory, the 620 nm maximum in the UV-vis spectrum was suggested to be due to 14 iodine atoms when assuming an equidistant distance of 3.1 Å. [98,99] However, the X-ray diffraction data so far cannot distinguish between a structure where the iodine atoms are placed equidistantly at 3.1 Å vs. a structure consisting of I 2 and/or I n− units (with internal I-I distances lower than 3.1 Å) placed in non-covalent contact with each other. For instance, I n− units with n = 5–7 (and assuming the same internal I-I distance as in I 3−) placed at 4.3 Å from each other would allow an average distance of 3.1 Å across the crystal structure, in line with experiments. If assuming the intermolecular distance to be shorter than the sum of van der Waals radii (as expected in complexes displaying charge-transfer bands), the experimentally observed iodine–iodine distance of 3.1 Å could be reached with, e.g., I 3− units placed at 3.7 Å from each other or I 5− units placed at 4.1 Å. The same average distance could be obtained by a chain of I 2 molecules placed within 3.6 Å of each other.
5. The I 2-Only Hypothesis
Many textbooks still list the amylose structure as featuring I 2 units aligned inside the helical channel. The driving force for this arrangement would be the hydrophobic character of the I 2 molecules, which thus escape the solvent to align in a more hydrophobic environment inside the helix . The non-covalent interactions between I 2 would then facilitate charge-transfer bands to appear, resulting in the intense blue color. NMR and UV-vis studies have shown that I− ions are not involved in the iodine–amylose helix, but they help to dissolve iodine in water . Simulations of the UV-vis spectra using semiempirical INDO configuration interactions have been reported to support a (C 6 H 10 O 5)16.5···(I 2)3 stoichiometry for the amylose–iodine complex, refuting instead I n− (n = 3, 5, or 7) as possible candidates .
Water molecules were reported to modulate the structure of the iodine–amylose complex . It was also noted that neither the dimensions of the amylose helix nor the rigidity or the helical vs. random coil secondary structure within the amylose polymer were affected by iodine binding—all of which suggest no strong specific inter-molecular bonding between amylose and iodine. This is also consistent with a simple inclusion complex involving neutral molecules. In contrast, the charge in I n− may have been expected to induce local changes in the neighboring polyscachharide units [103,104,105,106,107,108,109,110].
According to semiempirical calculations, the binding of water or of interspersed water and iodine molecules inside the amylose helix results in slight steric distortions of the polymer . The water molecules were found to adhere closely to the walls of the internal channel of amylose and to exclude I 2 from interactions, suggesting that I 2 molecules alone would be unable to dislocate water molecules from inside the amylose helix. In fact, I 2 solutions (in alcohol, so as to not require iodide for solubilization) were only found to be effective in binding to starch at high temperatures, while at lower temperatures, vapor I 2 easily adsorbs/binds to solid/dry amylose in the absence of water .
6. Poly-Iodine Anions as Candidates
It is in fact now generally accepted that the formation of the intense blue color in the amylose–iodine reaction requires iodide ions not only simply as an accessory that allows for the solubilization of I 2 in water, but also because a combination of I 2 and I− chains is present inside the amylose helix, most likely involving I n− units [30,31,32,33,34,35,36]. The (poly) anionic character of these guest ligands may be taken, as discussed above, to be at odds with the hydrophobic nature of the interior cavity of the helix, especially as no counter ions have been discussed or are presumed to be present throughout the cavity. This issue has been addressed by proposing a structure consisting of alternating tri-iodide units and I 2 molecules, where the more hydrophilic ends/entrances of the helix would remain unoccupied by iodine . To explain the fact that at room temperature, molecular iodine can bind to amylose in the solid state but not in solution, it was noted that the secondary structure of amylose, including the internal diameter of the helix, varies depending on the environment (e.g., solvent, ionic strength, pH, temperature, surfactants) [49,50,54,113,114,115,116,117]. Moreover, the solid-state amylose–I 2 complex is not stable in water . To complicate matters, it has also been reported that in the iodine–amylose complex (somewhat similarly to the above-discussed cases of other organic polymers), large amounts of iodine can associate outside the helical cavity, with inter- and intra-chain associations also important . Theoretical studies of the complex have been interpreted as evidence that at low temperatures, an I 6 structure dominates, while at higher temperatures, nonlinear geometries also appear. However, the experimental data have led to descriptions of the amylose-bound iodine chains as featuring 3–4 to 14–15 atoms, and as high as 160 atoms [31,32,33,34,35,36]. Potentiometric titrations at low iodide concentrations have been interpreted as evidence for a 3/2 I 2/I− ratio (hence, formally I 8 2−). However, as the structure appears to be consistently affected by the concentrations of the reactants (especially the iodine–iodide ratios, as followed, e.g., by UV-vis and circular dichroism titrations), I 4−, I 7−, I 9−, I 6 2−, I 8 2−, I 10 2−, I 4 2−, I 6−, and I 24 2− structures have also been proposed [29,30,32,50,52,53,119,120,121,122]. Rawlings and Schneider , using the statistics of binding isotherms of I 2/I 3− and the large variation of the value of the term of the ratio (R), determined the intrinsic binding constant of I 3− to amylose, whose value was much higher than that of I 2. Likewise, the mixed binding energy between I 2-I 3− exceeded those between I 2-I 2 and I 3−-I 3− species . Statements which assume that I− is not required sometimes do not take into account that it is formed through the hydrolysis of I 2 itself. Also, it is known that by adding an acid to an aqueous solution of iodine, the blue color is suppressed [26,124].
I 2 + H 2 O = I− + H+ + HOI
3I 2 + 3H 2 O = IO 3− + 5−+ 6H+
I 2 + H 2 O = H 2 OI+ + I−
In addition to classical short units such as I 3− ions and iodine molecules, the presence of I 5− and I 7− ions was also suggested. These proposals can be based on the fact that the composition of the unit is variable and influenced by the degree of polymerization of the amylose and the concentration of the iodide ion. At the end of the amylose chain, there are 7–8 glucose residues, which do not participate in iodine binding .
Stopped-flow UV-vis and circular dichroism (CD) kinetics have revealed that shorter chains of iodine enter the amylose helix very fast (less than 1 millisecond) and then rearrange rapidly inside the helix without further contributions from the excess iodine/iodide in the solution [28,127]. The optical rotatory dispersion (ORD) spectrum of the amylose–iodine complex was noted to change even when no changes in the UV-vis spectrum were observed; this was interpreted as evidence for complex dynamics of the helix during which the length of the poly-iodine chains remained unaffected . In the structure of the synthetic complex (benzamide)z H+I 3−, a poly-I 3 structure was reported, and a range of kinetic, spectroscopic (UV-vis, CD, Raman, X-ray absorption), and thermodynamic data have also been interpreted to support such a structure in the amylose–iodine complex [122,129,130,131]. On the other hand, the X-ray diffraction data on the amylose–iodine complex have been interpreted to be inconsistent with arrangements consisting of only single I 2 or only I 3− [35,132,133]. An I 5− structure was proposed for the amylose–iodine complex based on the similarities between the Raman and Mössbauer spectra and those of polycrystalline (trimesic acid···H 2 O)10 H+I 5− [129,134]. Of the main bands in the Raman spectra of starch–iodine complexes, at 27, 55, 109, and 160 cm−1, three were assigned to I 5− as the dominant species, while the 109 cm−1 band was assigned to I 3− being present as a minor species or impurity [135,136,137,138,139]. Further resonance Raman, 129 I Mössbauer spectroscopy [129,140] and X-ray diffraction [132,141,142] data have supported I 5− or I 2·I 3− as the dominant species in the amylose–iodine complex. Equilibrium studies in solution have confirmed that I 5− is present as a free species, with I 4 2− and I 6 2− also being present at higher I 2 or I− concentrations [143,144,145]. Teitelbaum et al presented evidence that the I 5− ion is present in the helix. Part of this explanation may be that the formation of the I 5− ion in the solid complex is produced by the hydrolysis or alcoholysis of iodine, and the amylose studied in this case was freed from water and alcohols [118,125]. Analyzing the absorption spectra of an aqueous solution of iodine at pH = 4.8 in acetate buffer, three peaks were found at 286, 350, and 460 nm . The peaks at 286 and 350 nm were attributed to the presence of I 3− ions. If amylose was added to this solution, the peak at 286 nm disappeared, the peak at 350 nm had a slight enhancement, and the peak at 460 nm shifted to 620–650 nm. By adding iodic acid to amylose, the blue color does not appear; this indicates the necessity of the presence of anions for the formation of the colored complex [26,124,147].
7. Structural Role of the Solvent
Benesi and Hildebrand in 1949 showed that iodine is pink–red in benzene, purple in CCl 4 (similarly to vapor iodine), and reddish-brown in alcohol—all of which were interpreted as evidence for charge transfer complexes. Bernal-Uruchurtu et al. provided a microscopic explanation of the interaction between iodine and water. Kereev and Shnyrev studied iodine, iodide , and triiodide in water and noted a band at 203 nm, which they attributed to iodine, and a band at 461 nm, which was then assigned to a H 2 O-I 2 charge–transfer complex. To complicate such studies, I 3− is formed by the oxidation of water with iodine. Analyzing iodine solution in water, the absorption bands of I 3− are at 284 nm and 351 nm. Density functional theory (DFT) calculations showed that two I 2 molecules can bind to two O lone pairs in H 2 O in a nearly tetrahedral geometry with a stabilization energy of 8.3 kcal/mol. This complex was described as involving a charge-transfer character, with a band in the UV-vis spectrum due to the excitation of electrons from the lone pair of oxygen of the molecular complex at 202 nm, where the molecular orbitals of iodine are destabilized and those of water are stabilized. When iodine is bound to water, the lone pairs on oxygen become equivalent due to hybridization . A small effect of water molecules on the structure of the iodine–amylose complex was described, as previously discussed .
8. The I 5−-I 2 Hypothesis
Recent DFT calculations on iodine/iodide chains have been interpreted as evidence that without iodide, the blue color cannot be formed in the starch–iodine system . These simulations propose that the nature of the complex consists of alternating sets of I 2 and I x− units, where the nature of the charge transfer bands responsible for the blue color involves transfer from the I x− σ orbitals (HOMO) to the I 2 σ LUMO orbitals (cf. Figure 5). By analyzing the TD-DFT-computed (time-dependent density functional) UV-vis spectra of various candidates (I 2 chains vs. mixtures of I 2 and I x− with various values of x) and cross-checking with DFT geometry optimizations, a unit of I 2-I 5−-I 2, in a repetitive manner within the amylose helix was the only structure that would fit the experimental data. Poly-I 2 structures were shown to be responsible for the enhanced blue color under certain conditions (e.g., consistent with the experimental observations on dry/solid amylose). Based on semiempirical calculations, poly-I n− structures were found to be unlikely to exist inside the amylose helix (as no distinct local energy minima were identified for such arrangements). Moreover, TD-DFT simulations of the UV-vis spectra of such chains were found to be less consistent with the experiments compared to I 2/I n− pairs. Charge transfer bands from the occupied I n− (n> 3) σ to the empty I 2 σ orbital were instead found to be reasonably responsible for the blue color. Of these, the I 2-I 5−-I 2 trimeric assemblies (i.e., n = 5) were the smallest units that represented the local minima in DFT geometry optimizations. These DFT-optimized units remarkably showed average iodine–iodine distances essentially identical to the 3.1 Å value seen experimentally in the iodine-amylose complex. The distinct charge-transfer character of the UV-vis bands (cf. Figure 5) also brings about a strong dependence on the dielectric constant in the region ε~1–30, which in turn was proposed to explain at least part of the dependence of the UV-vis properties of the amylose–iodine/iodine complexes on various external factors that may subtly affect amylose architecture and hence exposure of the interior cavity to solvent (e.g., temperature, other solutes, solvents, chain length) .
Figure 5.
Open in a new tab
Molecular orbitals in the I 2-I 5−-I 2 that are proposed to be responsible for the ~600 nm band in the iodine/iodide–amylose complex. Iodine atoms are shown in violet, HOMO/LUMO lobes are shown in red/green .
9. Conclusions
The iodine–starch (or, more specifically, iodine–amylose) reaction has a two-century history and a wide range of practical applications. Similar reactions occur with other organic polymers. The nature of the reaction is generally accepted to entail the alignment of iodine atoms inside the amylose helix. However, the structural details are still a source of confusion in many current reference sources, with alternative explanations given such as poly-I 2 (chain of neutral iodine molecules), poly-I 3− (chain of I 3 anions), poly-I x (chains of anionic structures of various lengths), or mixtures of I 2 and I x−. The most recent data suggest that the best explanation is a (probably repetitive) I 2-I 5−-I 2 unit.
Author Contributions
Conceptualization, R.S.-D.; formal analysis, S.P. and R.S.-D.; resources, R.S.-D.; writing—original draft preparation, S.P.; writing—review and editing, R.S.-D. and S.P. All authors have read and agreed to the published version of the manuscript.
Institutional Review Board Statement
Not applicable.
Informed Consent Statement
Not applicable.
Data Availability Statement
Not applicable.
Conflicts of Interest
The authors declare no conflicts of interest.
Funding Statement
Funding from the Romanian Ministry of Education and Research (projects PN-III-P4-ID-PCCF-2016-0142 and PN-III-P1-1.1-PD-2021-0279) is gratefully acknowledged.
Footnotes
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.
References
1.Nelson D.L., Cox M.M. Principles of Biochemistry. 5th ed. W.H. Freeman & Company; New York, NY, USA: 2008. Principles of Biochemistry. [Google Scholar]
2.Wang K., Vilaplana F., Wu A., Hasjim J., Gilbert R.G. The Size Dependence of the Average Number of Branches in Amylose. Carbohydr. Polym. 2019;223:115134. doi: 10.1016/j.carbpol.2019.115134. [DOI] [PubMed] [Google Scholar]
3.Green M.M., Blankenhorn G., Hart H. Which Starch Fraction Is Water-Soluble, Amylose or Amylopectin? J. Chem. Educ. 1975;52:729. doi: 10.1021/ed052p729. [DOI] [Google Scholar]
4.Nouri A., Khoee S. Preparation of Amylose-Poly(Methyl Methacrylate) Inclusion Complex as a Smart Nanocarrier with Switchable Surface Hydrophilicity. Carbohydr. Polym. 2020;246:116662. doi: 10.1016/j.carbpol.2020.116662. [DOI] [PubMed] [Google Scholar]
5.Wang Y., Zhang Y., Guan L., Wang S., Zhang J., Tan L., Kong L., Zhang H. Lipophilization and Amylose Inclusion Complexation Enhance the Stability and Release of Catechin. Carbohydr. Polym. 2021;269:118251. doi: 10.1016/j.carbpol.2021.118251. [DOI] [PubMed] [Google Scholar]
6.Prasher P., Fatima R., Sharma M. Therapeutic Delivery with V-Amylose. Drug Dev. Res. 2021;82:727–729. doi: 10.1002/ddr.21804. [DOI] [PubMed] [Google Scholar]
7.Katz J.R. Abhandlungen Zur Physikalischen Chemie Der Stärke Und Der Brotbereitung. Z. Phys. Chem. 1930;150A:37–59. doi: 10.1515/zpch-1930-15005. [DOI] [Google Scholar]
8.Katz J.R., Derksen J.C. Abhandlungen Zur Physikalischen Chemie Der Stärke Und Der Brotbereitung. Z. Phys. Chem. 1933;167A:129–136. doi: 10.1515/zpch-1933-16714. [DOI] [Google Scholar]
9.Bear R.S. The Significance of the “V” X-Ray Diffraction Patterns of Starches. J. Am. Chem. Soc. 1942;64:1388–1392. doi: 10.1021/ja01258a043. [DOI] [Google Scholar]
10.Meyer K.H., Bernfeld P., Wolf E. Recherches Sur l’amidon III. Fractionnement et Purification de l’amylose de Maïs Naturel. Helv. Chim. Acta. 1940;23:854–864. doi: 10.1002/hlca.194002301110. [DOI] [Google Scholar]
11.Sarko A., Zugenmaier P. Fiber Diffraction Methods. Am. Chem. Soc. 1980;141:459–482. [Google Scholar]
12.Rappenecker G., Zugenmaier P. Detailed refinement of the crystal structure of Vh-amylose. Carbohydr Res. 1981;89:11–19. doi: 10.1016/S0008-6215(00)85225-8. [DOI] [Google Scholar]
13.Murphy V.G., Zaslow B., French A.D. The structure of V amylose dehydrate: A combined X-ray and stereochemical approach. Biopolymers. 1975;14:1487–1501. doi: 10.1002/bip.1975.360140714. [DOI] [Google Scholar]
14.Brisson J., Chanzy H., Winter W.T. The crystal and molecular structure of VH amylose by electron diffraction analysis. Int. J. Biol. Macromol. 1991;13:31–39. doi: 10.1016/0141-8130(91)90007-H. [DOI] [PubMed] [Google Scholar]
15.Veregin R.P., Fyfe C.A., Marchessault R.H. Investigation of the crystalline “V” amylose complexes by high-resolution carbon-13 CP/MAS NMR spectroscopy. Macromolecules. 1987;20:3007–3012. doi: 10.1021/ma00178a010. [DOI] [Google Scholar]
16.Gidley M.J., Bociek S.M. Carbon-13 CP/MAS NMR studies of amylose inclusion complexes, cyclodextrins, and the amorphous phase of starch granules: Relationships between glycosidic linkage conformation and solid-state carbon-13 chemical shifts. J. Am. Chem. Soc. 1988;110:3820–3829. doi: 10.1021/ja00220a016. [DOI] [Google Scholar]
17.Imberty A., Chanzy H., Pérez S., Bulèon A., Tran V. The Double-Helical Nature of the Crystalline Part of A-Starch. J. Mol. Biol. 1988;201:365–378. doi: 10.1016/0022-2836(88)90144-1. [DOI] [PubMed] [Google Scholar]
18.Sarko A., Wu H.-C.H. The Crystal Structures of A-, B- and C-Polymorphs of Amylose and Starch. Starch-Starke. 1978;30:73–78. doi: 10.1002/star.19780300302. [DOI] [Google Scholar]
19.Saenger W. In: Inclusion Compounds. Atwood J.L., Davies J.E.D., MacNicol D.D., editors. Volume 2 Academic Press; London, UK: 1984. [Google Scholar]
20.Harata K. In: Comprehensive Supramolecular Chemistry. Atwood J.L., Davies J.E.D., MacNicol D.D., editors. Volume 3 Pergamon; Oxford, UK: 1996. [Google Scholar]
21.Cohen R., Orlova Y., Kovalev V., Ungar Y., Shimoni E. Structural and Functional Properties of Amylose Complexes with Genistein. J. Agric. Food Chem. 2008;56:4212–4218. doi: 10.1021/jf800255c. [DOI] [PubMed] [Google Scholar]
22.Pesek S., Lehene M., Brânzanic A.M.V., Silaghi-Dumitrescu R. On the Origin of the Blue Color in The Iodine/Iodide/Starch Supramolecular Complex. Molecules. 2022;27:8974. doi: 10.3390/molecules27248974. [DOI] [PMC free article] [PubMed] [Google Scholar]
23.Rani A., Ali U. Degree-Based Topological Indices of Polysaccharides: Amylose and Blue Starch-Iodine Complex. J. Chem. 2021;2021:6652014. doi: 10.1155/2021/6652014. [DOI] [Google Scholar]
24.Landolt H. Uber Die Zeitdauer Der Reaction Zwischen Jodsaure Und Schwefliger Saure. Ber. Dtsch. Chem. Ges. 1886;19:1317–1365. doi: 10.1002/cber.188601901293. [DOI] [Google Scholar]
25.Gilbert G.A., Marriott J.V.R. Starch-Iodine Complexes. Part I. Trans. Faraday Soc. 1948;44:84–93. doi: 10.1039/tf9484400084. [DOI] [Google Scholar]
26.Thoma J.A., French D. The Starch-Iodine-Iodide Interaction. Part I. Spectrophotometric Investigations 1. J. Am. Chem. Soc. 1960;82:4144–4147. doi: 10.1021/ja01501a004. [DOI] [Google Scholar]
27.Stein R.S., Rundle R.E. On the Nature of the Interaction between Starch and Iodine. J. Chem. Phys. 1948;16:195–207. doi: 10.1063/1.1746834. [DOI] [Google Scholar]
28.Hiromi K., Shibaoka T., Ono S. Kinetic Studies of Amylose-Iodine-Iodide Reaction by Stopped-Flow Method. J. Biochem. 1970;68:205–214. doi: 10.1093/oxfordjournals.jbchem.a129348. [DOI] [PubMed] [Google Scholar]
29.Yajima H., Nishimura T., Ishii T., Handa T. Effect of Concentration of Iodide on the Bound Species of I2/I−3 in the Amylose-Iodine Complex. Carbohydr. Res. 1987;163:155–167. doi: 10.1016/0008-6215(87)80179-9. [DOI] [Google Scholar]
30.Cronan C.L., Schneider F.W. Cooperativity and Composition of the Linear Amylose-Iodine-Iodide Complex. J. Phys. Chem. 1969;73:3990–4004. doi: 10.1021/j100845a073. [DOI] [Google Scholar]
31.Cramer F., Herbst W. Die Lichtabsorption von Jodkettenmolekeln. Naturwissenschaften. 1952;39:256. doi: 10.1007/BF00630876. [DOI] [Google Scholar]
32.Bersohn R., Isenberg I. Metallic Nature of the Starch-Iodine Complex. J. Chem. Phys. 1961;35:1640–1643. doi: 10.1063/1.1732123. [DOI] [Google Scholar]
33.Rundle R.E., Baldwin R.R. The Configuration of Starch and Starch-Iodine Complex. I. The Dichroism of Flow of Starch-Iodine Solutions. J. Am. Chem. Soc. 1943;65:554–558. doi: 10.1021/ja01244a017. [DOI] [Google Scholar]
34.Rundle R.E. The Configuration of Starch in the Starch-Iodine Complex. V. Fourier Projections from X-ray Diagrams. J. Am. Chem. Soc. 1947;69:1769–1772. doi: 10.1021/ja01199a054. [DOI] [PubMed] [Google Scholar]
35.Noltemeyer M., Saenger W. X-Ray Studies of Linear Polyiodide Chains in α-Cyclodextrin Channels and a Model for the Starch-Iodine Complex. Nature. 1976;259:629–632. doi: 10.1038/259629a0. [DOI] [Google Scholar]
36.Bluhm T.L., Zugenmaier P. Detailed Structure of the Vh-Amylose-Iodine Complex: A Liner Polyiodine Chain. Carbohydr. Res. 1981;89:1–10. doi: 10.1016/S0008-6215(00)85224-6. [DOI] [Google Scholar]
37.Handa T., Yajima H. Conformation of Amylose-Iodine-Iodide Complex in Aqueous Solution. Biopolymers. 1981;20:2051–2072. doi: 10.1002/bip.1981.360201003. [DOI] [Google Scholar]
38.Mould D.L., Synge R.M. Separations of Polysaccharides Related to Starch by Electrokinetic Ultrafiltration in Collodion Membranes. Biochem. J. 1954;58:571–600. doi: 10.1042/bj0580571. [DOI] [PMC free article] [PubMed] [Google Scholar]
39.Ono S., Tsuchihashi S., Kuge T. On the Starch-Iodine Complex. J. Am. Chem. Soc. 1953;75:3601–3602. doi: 10.1021/ja01110a519. [DOI] [Google Scholar]
40.Moulay S. Molecular Iodine/Polymer Complexes. J. Polym. Eng. 2013;33:389–443. doi: 10.1515/polyeng-2012-0122. [DOI] [Google Scholar]
41.Séne M., Thévenot C., Prioul J.L. Simultaneous Spectrophotometric Determination of Amylose and Amylopectin in Starch from Maize Kernel by Multi-Wavelength Analysis. J. Cereal Sci. 1997;26:211–221. doi: 10.1006/jcrs.1997.0124. [DOI] [Google Scholar]
42.Sashio M., Tanaka M. Thermal Reaction of Poly(Vinyl Alcohol)-Iodine Complex Membranes. J. Polym. Sci. Polym. Chem. Ed. 1985;23:905–909. doi: 10.1002/pol.1985.170230323. [DOI] [Google Scholar]
43.Dintzis F.R. Instability of Solutions of Amylose-Iodine Complex in Concentrated Calcium Chloride. Starch-Stärke. 1974;26:56–58. doi: 10.1002/star.19740260206. [DOI] [Google Scholar]
44.Tashiro K., Gakhutishvili M. Crystal Structure of Cellulose-Iodine Complex. Polymer. 2019;171:140–148. doi: 10.1016/j.polymer.2019.03.034. [DOI] [Google Scholar]
45.Konishi T., Tanaka W., Kawai T., Fujikawa T. Iodine L-Edge XAFS Study of Linear Polyiodide Chains in Amylose and α-Cyclodextrin. J. Synchrotron Radiat. 2001;8:737–739. doi: 10.1107/S0909049500013753. [DOI] [PubMed] [Google Scholar]
46.Knutson C.A. Evaluation of Variations in Amylose–Iodine Absorbance Spectra. Carbohydr. Polym. 2000;42:65–72. doi: 10.1016/S0144-8617(99)00126-5. [DOI] [Google Scholar]
47.Nishimura T., Yajima H., Ishii T., Endo R. Effect of Molecular Weight of Amylose on the Iodine Coloring Species Responsible for the Optical Properties of Amylose-Iodine Complexes. Kobunshi Ronbunshu. 1989;46:537–544. doi: 10.1295/koron.46.537. [DOI] [Google Scholar]
48.SenGupta U.K., MuKherjee A.K., SenGupta K.K. Spectrophotometric Studies on Amylose-Iodine and Amylopectin-Iodine Complexes. Kolloid-Z. Z. Polym. 1966;208:32–34. doi: 10.1007/BF01499864. [DOI] [Google Scholar]
49.Sakajiri T., Kikuchi T., Simon I., Uchida K., Yamamura T., Ishii T., Yajima H. Molecular Dynamics Approach to Study the Discrepancies in the Thermal Behavior of Amylose and Chitosan Conformations. J. Mol. Struct. THEOCHEM. 2006;764:133–140. doi: 10.1016/j.theochem.2006.02.016. [DOI] [Google Scholar]
50.Szejtli J., Augustat S., Richter M. Molecular Configuration of Amylose and Its Complexes in Aqueous Solutions. Part III. Investigation of the DP Distribution of Helical Segments in Amylose-Iodine Complexes. Biopolymers. 1967;5:17–26. doi: 10.1002/bip.1967.360050104. [DOI] [Google Scholar]
51.McMullan R.K., Saenger W., Fayos J., Mootz D. Topography of Cyclodextrin Inclusion Complexes. Carbohydr. Res. 1973;31:211–227. doi: 10.1016/S0008-6215(00)86186-8. [DOI] [Google Scholar]
52.Baldwin R.R., Bear R.S., Rundle R.E. The Relation of Starch—Iodine Absorption Spectra to the Structure of Starch and Starch Components 1. J. Am. Chem. Soc. 1944;66:111–115. doi: 10.1021/ja01229a032. [DOI] [Google Scholar]
53.Davis H., Khan A. Determining the Chromophore in the Amylopectin–Iodine Complex by Theoretical and Experimental Studies. J. Polym. Sci. A Polym. Chem. 1994;32:2257–2265. doi: 10.1002/pola.1994.080321207. [DOI] [Google Scholar]
54.Rendleman J.A. The Reaction of Starch with Iodine Vapor. Determination of Iodide-Ion Content of Starch–Iodine Complexes. Carbohydr. Polym. 2003;51:191–202. doi: 10.1016/S0144-8617(02)00053-X. [DOI] [Google Scholar]
55.Yu X., Houtman C., Atalla R.H. The Complex of Amylose and Iodine. Carbohydr. Res. 1996;292:129–141. doi: 10.1016/0008-6215(96)00159-0. [DOI] [Google Scholar]
56.Abe T. The Visible and Ultraviolet Absorption Spectra of Cellulose- and Amylose-Iodine Complexes. Bull. Chem. Soc. Jpn. 1958;31:661–662. doi: 10.1246/bcsj.31.661. [DOI] [Google Scholar]
57.Takahashi Y.J. Binding Properties of Alginic Acid and Chitin. Inclus. Phenom. 1978;5:525–534. doi: 10.1007/BF00664112. [DOI] [Google Scholar]
58.Yajima H., Morita M., Hashimoto M., Sashiwa H., Kikuchi T., Ishii T. Complex Formation of Chitosan with Iodine and Its Strucutre and Spectroscopic Properties—Molecular Assembly and Thermal Hysteresis Behavior. Int. J. Thermophys. 2001;22:1265–1283. doi: 10.1023/A:1010628712529. [DOI] [Google Scholar]
59.Gunasekaran M. Physiological Studies on Phymatotrichum Omnivorum II. Physiocochemical Properties of Glycogen. Arch. Mikrobiol. 1972;84:69–76. doi: 10.1007/BF00408083. [DOI] [PubMed] [Google Scholar]
60.Kumari S., Roman A., Khan A. Chromophore and Spectrum of the Glycogen-Iodine Complex. J. Polym. Sci. Part A Polym. Chem. 1996;34:2975–2980. doi: 10.1002/(SICI)1099-0518(199610)34:14<2975::AID-POLA15>3.0.CO;2-H. [DOI] [Google Scholar]
61.Lecker D.N., Kumari S., Khan A. Iodine Binding Capacity and Iodine Binding Energy of Glycogen. J. Polym. Sci. Part A Polym. Chem. 1997;35:1409–1412. doi: 10.1002/(SICI)1099-0518(199706)35:8<1409::AID-POLA9>3.0.CO;2-U. [DOI] [Google Scholar]
62.Aterman K. A Historical Note on the Iodine-Sulphuric Acid Reaction of Amyloid. Histochemistry. 1976;49:131–143. doi: 10.1007/BF00495677. [DOI] [PubMed] [Google Scholar]
63.Dzwolak W. Insulin Amyloid Fibrils Form an Inclusion Complex with Molecular Iodine: A Misfolded Protein as a Nanoscale Scaffold. Biochemistry. 2007;46:1568–1572. doi: 10.1021/bi061985l. [DOI] [PubMed] [Google Scholar]
64.Pritchard J.G., Serra F.T. Complexation of Polyvinyl Acetate with Iodine. Talanta. 1973;20:541–546. doi: 10.1016/0039-9140(73)80134-1. [DOI] [PubMed] [Google Scholar]
65.Hughes J. Analytical Behaviour of Poly(Vinyl Acetate) and Its Hydrolysis Products with Iodine. Talanta. 1979;26:1161–1163. doi: 10.1016/0039-9140(79)80032-6. [DOI] [PubMed] [Google Scholar]
66.Schulz R.C., Fleischer D., Henglein A., Bössler H.M., Trisnadi J., Tanaka H. Addition Compounds and Complexes with Polymers and Models. Pure Appl. Chem. 1974;38:227–247. doi: 10.1351/pac197438010227. [DOI] [Google Scholar]
67.Chernov’yants M.S., Burykin I.V., Pisanov R.V., Shalu O.A. Synthesis and Antimicrobial Activity of Poly(N-Methyl-4-Vinylpyridinium Triiodide) Pharm. Chem. J. 2010;44:61–63. doi: 10.1007/s11094-010-0398-5. [DOI] [Google Scholar]
68.Pal S.K., Krishnan A., Das P.K., Samuelson A.G. Schiff Base Linked Ferrocenyl Complexes for Second-Order Nonlinear Optics. J. Organomet. Chem. 2000;604:248–259. doi: 10.1016/S0022-328X(00)00278-3. [DOI] [Google Scholar]
69.Liu W.-J., Xiong G.-X., Zeng D.-H. Synthesis and Electrical Properties of Three Novel Poly (Ferrocenyl-Schiff Bases) and Their Charge Transfer Complexes with Iodine. J. Inorg. Organomet. Polym. 2010;20:97–103. doi: 10.1007/s10904-010-9325-1. [DOI] [Google Scholar]
70.Shirakawa H., Louis E.J., MacDiarmid A.G., Chiang C.F., Heeger A.J. Synthesis of Electrically Conducting Organic Polymers: Halogen Derivatives of Polyacetylene, (CH)x. J. Chem. Soc. Commun. 1977:578–580. doi: 10.1039/c39770000578. [DOI] [Google Scholar]
71.Bakueva L., Matheson D., Musikhin S., Sargent E.H. Luminescence of Pure and Iodine Doped PPV: Internal Energetic Structure Revealed through Spectral Signatures. Synth. Met. 2002;126:207–211. doi: 10.1016/S0379-6779(01)00561-6. [DOI] [Google Scholar]
72.Fasahat P., Rahman S., Ratnam W. Genetic Controls on Starch Amylose Content in Wheat and Rice Grains. J. Genet. 2014;93:279–292. doi: 10.1007/s12041-014-0325-8. [DOI] [PubMed] [Google Scholar]
73.Ashogbon A.O., Akintayo E.T., Oladebeye A.O., Oluwafemi A.D., Akinsola A.F., Imanah O.E. Developments in the Isolation, Composition, and Physicochemical Properties of Legume Starches. Crit. Rev. Food Sci. Nutr. 2021;61:2938–2959. doi: 10.1080/10408398.2020.1791048. [DOI] [PubMed] [Google Scholar]
74.Wang L., Liu L., Zhao J., Li C., Wu H., Zhao H., Wu Q. Granule-Bound Starch Synthase in Plants: Towards an Understanding of Their Evolution, Regulatory Mechanisms, Applications, and Perspectives. Plant Sci. 2023;336:111843. doi: 10.1016/j.plantsci.2023.111843. [DOI] [PubMed] [Google Scholar]
75.Guo K., Liang W., Wang S., Guo D., Liu F., Persson S., Herburger K., Petersen B.L., Liu X., Blennow A., et al. Strategies for Starch Customization: Agricultural Modification. Carbohydr. Polym. 2023;321:121336. doi: 10.1016/j.carbpol.2023.121336. [DOI] [PubMed] [Google Scholar]
76.Chiaramonte E., Rhazi L., Aussenac T., White D.R. Amylose and Amylopectin in Starch by Asymmetric Flow Field-Flow Fractionation with Multi-Angle Light Scattering and Refractive Index Detection (AF4–MALS–RI) J. Cereal Sci. 2012;56:457–463. doi: 10.1016/j.jcs.2012.04.006. [DOI] [Google Scholar]
77.Ulbrich M., Scholz F., Flöter E. Chromatographic Study of High Amylose Corn Starch Genotypes—Investigation of Molecular Properties after Specific Enzymatic Digestion. Starch-Stärke. 2022;74:2100303. doi: 10.1002/star.202100303. [DOI] [Google Scholar]
78.Tutorskii I.A., Sokolova L.V. Mechanism of the Reaction of Polybutadiene with Molecular Iodine. Polym. Sci. U.S.S.R. 1977;19:176–183. doi: 10.1016/0032-3950(77)90164-2. [DOI] [Google Scholar]
79.Sreeja R., Najidha S., Remya Jayan S., Predeep P., Mazur M., Sharma P.D. Electro-Optic Materials from Co-Polymeric Elastomer–Acrylonitrile Butadiene Rubber (NBR) Polymer. 2006;47:617–623. doi: 10.1016/j.polymer.2005.09.024. [DOI] [Google Scholar]
80.Vippa P., Rajagopalan H., Thakur M. Electrical and Optical Properties of a Novel Nonconjugated Conductive Polymer, Poly (β-pinene) J. Polym. Sci. Part B Polym. Phys. 2005;43:3695–3698. doi: 10.1002/polb.20660. [DOI] [Google Scholar]
81.Nagelli W. Beitrage Zur Naheren Kenntniss Der Starkegruppe. Ann. Chem. 1874;173:218–227. doi: 10.1002/jlac.18741730208. [DOI] [Google Scholar]
82.Helbert W., Chanzy H. The Ultrastructure of Starch from Ultrathin Sectioning in Melamine Resin. Starch-Starke. 1996;48:185–188. doi: 10.1002/star.19960480507. [DOI] [Google Scholar]
83.Atkin N.J., Abeysekera R.M., Cheng S.L., Robards A.W. An Experimentally-Based Predictive Model for the Separation of Amylopectin Subunits during Starch Gelatinization. Carbohydr. Polym. 1998;36:173–192. doi: 10.1016/S0144-8617(98)00003-4. [DOI] [Google Scholar]
84.Baker A.A., Miles M.J., Helbert W. Internal Structure of the Starch Resistant Granule Revealed by AFM. Carbohydr. Res. 2001;330:249–256. doi: 10.1016/S0008-6215(00)00275-5. [DOI] [PubMed] [Google Scholar]
85.Baldwin P.M., Adler J., Davies M., Melia D. High Resolution Imaging of Starch Granule Surfaces by Atomic Force Microscopy. J. Cereal Sci. 1998;27:255–265. doi: 10.1006/jcrs.1998.0189. [DOI] [Google Scholar]
86.Dang J.M.C., Copeland L. Imaging Rice Grains Using Atomic Force Microscopy. J. Cereal Sci. 2003;37:165–170. doi: 10.1006/jcrs.2002.0490. [DOI] [Google Scholar]
87.Ohtani T., Yoshimo T., Hagiwara S., Maekawa T. High-Resolution Imaging of Starch Granule Structure Using Atomic Force Microscopy. Starch-Starke. 2000;52:150–153. doi: 10.1002/1521-379X(200006)52:5<150::AID-STAR150>3.0.CO;2-F. [DOI] [Google Scholar]
88.Park H., Xu S., Seetharaman S. A Novel in Situ Atomic Force Microscopy Imaging Technique to Probe Surface Morphological Features of Starch Granules. Carbohydr. Res. 2011;346:847–853. doi: 10.1016/j.carres.2011.01.036. [DOI] [PubMed] [Google Scholar]
89.Ridout M.J., Gunning A.P., Parker M.L., Wilson R.H., Morris V.J. Using AFM to Image the Internal Structure of Starch Grsnules. Carbohydr. Polym. 2002;50:123–132. doi: 10.1016/S0144-8617(02)00021-8. [DOI] [Google Scholar]
90.Szymonska J., Krok F. Potato Starch Granule Nanostructure Studied by Highresolution Non-Contact AFM. Int. J. Biol. Macromol. 2003;33:1–7. doi: 10.1016/S0141-8130(03)00056-4. [DOI] [PubMed] [Google Scholar]
91.Waduge R.N., Xu S., Seetharaman S. Iodine Absorption Properties and Its Effect on the Crystallinity of Developing Wheat Starch Granules. Carbohydr. Polym. 2010;82:786–794. doi: 10.1016/j.carbpol.2010.05.053. [DOI] [Google Scholar]
92.Gallant D.J., Bouchet B., Baldwin P.M. Microscopy of Starch: Evidence of a New Level of Granule Organization. Carbohydr. Polym. 1997;32:177–191. doi: 10.1016/S0144-8617(97)00008-8. [DOI] [Google Scholar]
93.Doutch J., Gilbert E.P. Characterisation of Large Scale Structures in Starch Granules via Small-Angle Neutron and X-ray Scattering Techniques. Carbohydr. Polym. 2013;91:444–451. doi: 10.1016/j.carbpol.2012.08.002. [DOI] [PubMed] [Google Scholar]
94.Barrett A.J., Barrett K.L., Khan A. Effects of Acetone, Ethanol, Isopropanol, and Dimethyl Sulfoxide on Amylose-Iodine Complex. J. Macromol. Sci. Part A. 1998;35:711–722. doi: 10.1080/10601329808002007. [DOI] [Google Scholar]
95.Fonslick J., Khan A. Thermal Stability and Composition of the Amylose–Iodine Complex. J. Polym. Sci. A Polym. Chem. 1989;27:4161–4167. doi: 10.1002/pola.1989.080271222. [DOI] [Google Scholar]
96.Rundle R.E., French D. The Configuration of Starch and the Starch—Iodine Complex. II. Optical Properties of Crystalline Starch Fractions 1. J. Am. Chem. Soc. 1943;65:558–561. doi: 10.1021/ja01244a018. [DOI] [Google Scholar]
97.Rundle R.E., French D. The Configuration of Starch in the Starch—Iodine Complex. III. X-ray Diffraction Studies of the Starch—Iodine Complex 1. J. Am. Chem. Soc. 1943;65:1707–1710. doi: 10.1021/ja01249a016. [DOI] [PubMed] [Google Scholar]
98.Cramer F., Windel H. Über Einschlußverbindungen, X. Mitteil.: Die Blauen Jodverbindungen Der Cumarine Und Anderer Verwandter Verbindungen. Chem. Ber. 1956;89:354–365. doi: 10.1002/cber.19560890227. [DOI] [Google Scholar]
99.Saenger W. The Structure of the Blue Starch-Iodine Complex. Naturwissenschaften. 1984;71:31–36. doi: 10.1007/BF00365977. [DOI] [Google Scholar]
100.Immel S., Lichtenthaler F.W. The Hydrophobic Topographies of Amylose and Its Blue Iodine Complex. Starch-Stärke. 2000;52:1–8. doi: 10.1002/(SICI)1521-379X(200001)52:1<1::AID-STAR1>3.0.CO;2-H. [DOI] [Google Scholar]
101.Minick M., Fotta K., Khan A. Polyiodine Units in Starch-Iodine Complex: INDO CI Study of Spectra and Comparison with Experiments. Biopolymers. 1991;31:57–63. doi: 10.1002/bip.360310106. [DOI] [Google Scholar]
102.Zaslow B., Miller R.L. Hydration of the “V” Amylose Helix 1. J. Am. Chem. Soc. 1961;83:4378–4381. doi: 10.1021/ja01482a020. [DOI] [Google Scholar]
103.Hirai M., Hirai T., Ueki T. Effect of Branching of Amylopectin on Complexation with Iodine as Steric Hindrance. Polymer. 1994;35:2222–2225. doi: 10.1016/0032-3861(94)90255-0. [DOI] [Google Scholar]
104.Dintzis F.R., Beckwith A.C., Babcock G.E., Tobin R. Amylose-Iodine Complex. I. Sedimentation Behavior. Macromolecules. 1976;9:471–478. doi: 10.1021/ma60051a017. [DOI] [PubMed] [Google Scholar]
105.Moulik S.P., Gupta S. Effects of Solvents on the Spectrophotometric and Hydrodynamic Behavior of Amylose and Its Iodine Complex. Carbohydr. Res. 1980;81:131–143. doi: 10.1016/S0008-6215(00)85680-3. [DOI] [Google Scholar]
106.Senior M.B., Hamori E. Investigation of the Effect of Amylose/Iodine Complexation on the Conformation of Amylose in Aqueous Solution. Biopolymers. 1973;12:65–78. doi: 10.1002/bip.1973.360120107. [DOI] [Google Scholar]
107.Vladimirov A.V., Volkova T.V., Agafonov A.V. Temperature Dependence of Stability Constants of the Iodine-Iodide-Amylose Complexes. Russ. J. Phys. Chem. A. 2003;77:612–615. [Google Scholar]
108.Zhang Q., Lu Z., Hu H., Yang W., Marszalek P.E. Direct Detection of the Formation of V-Amylose Helix by Single Molecule Force Spectroscopy. J. Am. Chem. Soc. 2006;128:9387–9393. doi: 10.1021/ja057693+. [DOI] [PMC free article] [PubMed] [Google Scholar]
109.Dintzis F.R., Tobin R., Beckwith A.C. Amylose-Iodine Complex. II. Molecular Weight Estimates. Macromolecules. 1976;9:478–482. doi: 10.1021/ma60051a018. [DOI] [PubMed] [Google Scholar]
110.Mikus F.F., Hixon R.M., Rundle R.E. The Complexes of Fatty Acids with Amylose 1. J. Am. Chem. Soc. 1946;68:1115–1123. doi: 10.1021/ja01210a062. [DOI] [PubMed] [Google Scholar]
111.Calabrese V.T., Khan A. Amylose-Iodine Complex Formation without KI: Evidence for Absence of Iodide Ions within the Complex. J. Polym. Sci. A Polym. Chem. 1999;37:2711–2717. doi: 10.1002/(SICI)1099-0518(19990801)37:15<2711::AID-POLA4>3.0.CO;2-6. [DOI] [Google Scholar]
112.Cesaro A., Benegas J.C., Ripoll D.R. Molecular Model of the Cooperative Amylose-Iodine-Triiodide Complex. J. Phys. Chem. 1986;90:2787–2791. doi: 10.1021/j100403a047. [DOI] [Google Scholar]
113.Kuge T., Ono S. Amylose-Iodine Complex. III. Potentiometric and Spectrophotometric Studies. Bull. Chem. Soc. Jpn. 1960;33:1273–1278. doi: 10.1246/bcsj.33.1273. [DOI] [Google Scholar]
114.Schulz W., Sklenar H., Hinrichs W., Saenger W. The Structure of the Left-Handed Antiparallel Amylose Double Helix: Theoretical Studies. Biopolymers. 1993;33:363–375. doi: 10.1002/bip.360330305. [DOI] [Google Scholar]
115.Moulik S.P., Gupta S. Environment-Induced, Physicochemical Behavior of Amylose-Iodine Complexes. Carbohydr. Res. 1979;71:251–264. doi: 10.1016/S0008-6215(00)86073-5. [DOI] [Google Scholar]
116.Szejtli J., Richter M., Augustat S. Molecular Configuration of Amylose and Its Complexes in Aqueous Solutions. Part IV. Determination OfDP of Amylose by Measuring the Concentration of Free Iodine in Solution of Amylose-Iodine Complex. Biopolymers. 1968;6:27–41. doi: 10.1002/bip.1968.360060103. [DOI] [Google Scholar]
117.Peng Q.-J., Perlin A.S. Observations on N.M.R. Spectra of Starches in Dimethyl Sulfoxide, Iodine-Complexing, and Solvation in Water-Di-Methyl Sulfoxide. Carbohydr. Res. 1987;160:57–72. doi: 10.1016/0008-6215(87)80303-8. [DOI] [Google Scholar]
118.Murdoch K.A. The Amylose-Iodine Complex. Carbohydr. Res. 1992;233:161–174. doi: 10.1016/S0008-6215(00)90928-5. [DOI] [Google Scholar]
119.Knutson C.A., Cluskey J.E., Dintzis F.R. Properties of Amylose-Iodine Complexes Prepared in the Presence of Excess Iodine. Carbohydr. Res. 1982;101:117–128. doi: 10.1016/S0008-6215(00)80800-9. [DOI] [Google Scholar]
120.Murakami H. Electronic Structure of the Amylose-Iodine Complex. J. Chem. Phys. 1954;22:367–374. doi: 10.1063/1.1740076. [DOI] [Google Scholar]
121.Nishimura T., Yajima H., Kubota S., Ishii T., Endo R. Polymer Effect on the Iodine Coloring Species Responsible for the Spectroscopic Properties of Amylose-Iodine Complexes. Kobunshi Ronbunshu. 1990;47:717–725. doi: 10.1295/koron.47.717. [DOI] [Google Scholar]
122.Nishimura T., Yajima H., Kubota S., Ishii T., Endo R. Effect of I− Concentration on the Optical Properties of Amylose-Iodine Complexes. Kobunshi Ronbunshu. 1988;45:945–952. doi: 10.1295/koron.45.945. [DOI] [Google Scholar]
123.Rawlings P.K., Schneider F.W. Models for Competitive Cooperative Linear Adsorption. The Amylose–Iodine–Iodide Complex. J. Chem. Phys. 1970;52:946–952. doi: 10.1063/1.1673080. [DOI] [Google Scholar]
124.Kuge T., Ono S. Advances in Carbohydrate Chemistry and Biochemistry. Bull. Chem. Sot. Jpn. 1960;33:1269–1272. doi: 10.1246/bcsj.33.1269. [DOI] [Google Scholar]
125.Teitelbaum R.C., Ruby S.L., Marks T.J. A Resonance Raman/Iodine Moessbauer Investigation of the Starch-Iodine Structure. Aqueous Solution and Iodine Vapor Preparations. J. Am. Chem. Soc. 1980;102:3322–3328. doi: 10.1021/ja00530a003. [DOI] [Google Scholar]
126.Cesàro A., Jerian E., Saule S. Physicochemical Studies of Amylose and Its Derivatives in Aqueous Solutions: Thermodynamics of the Iodine-Triiodide Complex. Biopolymers. 1980;19:1491–1506. doi: 10.1002/bip.1980.360190806. [DOI] [Google Scholar]
127.Nishimura T., Yajima H., Ishii T., Endo R. Study of the Bluing Mechanism of Amylose-Iodine Complexes by CD Stopped-Flow Method. Kobunshi Ronbunshu. 1991;48:525–528. doi: 10.1295/koron.48.525. [DOI] [Google Scholar]
128.Wolf R., Schulz R.C. Optical Rotatory Dispersion of the Starch Iodine Complex. Part 2. J. Macromol. Sci. Part A Chem. 1968;2:821–832. doi: 10.1080/10601326808051443. [DOI] [Google Scholar]
129.Agafonov A.V., Vladimirov A.V., Volkova T.V. The Concentration Dependences of the Stability Constants of Iodine-Iodide-Amylose Complexes in Aqueous Solutions of Electrolytes. Russ. J. Phys. Chem. A. 2004;78:1584–1587. [Google Scholar]
130.Yamamoto M., Sano T., Harada S., Yasunaga T. Interaction of Amylose with Iodine. II. Kinetic Studies of the Complex Formation by the Temperature-Jump Method. Bull. Chem. Soc. Jpn. 1982;55:3702–3706. doi: 10.1246/bcsj.55.3702. [DOI] [Google Scholar]
131.Foster J.F., Zucker D. Length of the Amylose–Iodine Complex as Determined by Streaming Dichroism. J. Phys. Chem. 1952;56:170–173. doi: 10.1021/j150494a004. [DOI] [Google Scholar]
132.Noltemeyer M., Saenger W. Topography of Cyclodextrin Inclusion Complexes. 12. Structural Chemistry of Linear.Alpha.-Cyclodextrin-Polyiodide Complexes. X-ray Crystal Structures of (.Alpha.-Cyclodextrin)2.LiI3.I2.8H2O and (.Alpha.-Cyclodextrin)2.Cd0.5).I5.27H2O. Models for the Blue. J. Am. Chem. Soc. 1980;102:2710–2722. doi: 10.1021/ja00528a032. [DOI] [Google Scholar.I5.27H2O.%20Models%20for%20the%20Blue&author=M.%20Noltemeyer&author=W.%20Saenger&volume=102&publication_year=1980&pages=2710-2722&doi=10.1021/ja00528a032&)]
133.Betzel C., Hingerty B., Noltemeyer M., Weber G., Saenger W., Hamilton J.A. (β-Cyclodextrin)2 KI7 9 H2O. Spatial Fitting of a Polyiodide Chain to a given Matrix. J. Incl. Phenom. 1983;1:181–191. doi: 10.1007/BF00656821. [DOI] [Google Scholar]
134.Bowmaker G. Bonding and Nuclear Quadrupole Coupling in Linear Pentaiodide Ions. Aust. J. Chem. 1978;31:2713. doi: 10.1071/CH9782713. [DOI] [Google Scholar]
135.Nimz O., Geßler K., Usón I., Laettig S., Welfle H., Sheldrick G.M., Saenger W. X-Ray Structure of the Cyclomaltohexaicosaose Triiodide Inclusion Complex Provides a Model for Amylose–Iodine at Atomic Resolution. Carbohydr. Res. 2003;338:977–986. doi: 10.1016/S0008-6215(03)00016-8. [DOI] [PubMed] [Google Scholar]
136.Ziegast G., Pfannemüller B. Resonance Raman Studies of Amaylose—Iodine Complexes. Int. J. Biol. Macromol. 1982;4:419–424. doi: 10.1016/0141-8130(82)90086-1. [DOI] [Google Scholar]
137.Heyde M.E., Rimai L., Kilponen R.G., Gill D. Resonance-Enhanced Raman Spectra of Iodine Complexes with Amylose and Poly(Vinyl Alcohol), and of Some Iodine-Containing Trihalides. J. Am. Chem. Soc. 1972;94:5222–5227. doi: 10.1021/ja00770a014. [DOI] [Google Scholar]
138.Okuda M., Hiramatsu T., Yasuda M., Ishigaki M., Ozaki Y., Hayashi M., Tominaga K., Chatani E. Theoretical Modeling of Electronic Structures of Polyiodide Species Included in α-Cyclodextrin. J. Phys. Chem. B. 2020;124:4089–4096. doi: 10.1021/acs.jpcb.0c01749. [DOI] [PubMed] [Google Scholar]
139.Mizuno M., Tanaka J., Harada I. Electronic Spectra and Structures of Polyiodide Chain Complexes. J. Phys. Chem. 1981;85:1789–1794. doi: 10.1021/j150613a006. [DOI] [Google Scholar]
140.Teitelbaum R.C., Ruby S.L., Marks T.J. On the Structure of Starch-Iodine. J. Am. Chem. Soc. 1978;100:3215–3217. doi: 10.1021/ja00478a045. [DOI] [Google Scholar]
141.Hach R.J., Rundle R.E. The Structure of Tetramethylammonium Pentaiodide 1,1a. J. Am. Chem. Soc. 1951;73:4321–4324. doi: 10.1021/ja01153a086. [DOI] [Google Scholar]
142.Herbstein F.H., Kapon M. Zigzag Chains of Alternating Molecules and Triiodide Ions in Crystalline (Phenacetin)2.HI5. Nat. Phys. Sci. 1972;239:153–154. doi: 10.1038/physci239153a0. [DOI] [Google Scholar]
143.Haddock A., Steidemann M., Readnour M. Polyiodide Equilibria in Aqueous Solutions of Iodine and Iodide. Synth. React. Inorg. Met. Org. Chem. 1979;9:39–56. doi: 10.1080/00945717908057450. [DOI] [Google Scholar]
144.Ramette R.W., Sandford R.W. Thermodynamics of Iodine Solubility and Triiodide Ion Formation in Water and in Deuterium Oxide. J. Am. Chem. Soc. 1965;87:5001–5005. doi: 10.1021/ja00950a005. [DOI] [Google Scholar]
145.Sekine T. Abstracts. Nippon Kagaku Zassi. 1969;90:951–983. doi: 10.1246/nikkashi1948.90.10_951. [DOI] [Google Scholar]
146.Mould D.L. Potentiometric and Spectrophotometric Studies of Complexes of Hydrolysis Products of Amylose with Iodine and Potassium Iodide. Biochem. J. 1954;58:593–600. doi: 10.1042/bj0580593. [DOI] [PMC free article] [PubMed] [Google Scholar]
147.Bhide S.V., Kale N.R. Ligand-Induced Structural Changes in Amylose Partially Complexed with Iodine. Biochim. Biophys. Acta (BBA)—Gen. Subj. 1976;444:719–726. doi: 10.1016/0304-4165(76)90319-6. [DOI] [PubMed] [Google Scholar]
148.Benesi H.A., Hildebrand J.H. A Spectrophotometric Investigation of the Interaction of Iodine with Aromatic Hydrocarbons. J. Am. Chem. Soc. 1949;71:2703–2707. doi: 10.1021/ja01176a030. [DOI] [Google Scholar]
149.Bernal-Uruchurtu M.I., Kerenskaya G., Janda K.C. Structure, Spectroscopy and Dynamics of Halogen Molecules Interacting with Water. Int. Rev. Phys. Chem. 2009;28:223–265. doi: 10.1080/01442350903017302. [DOI] [Google Scholar]
150.Kireev S.V., Shnyrev S.L. Study of Molecular Iodine, Iodate Ions, Iodide Ions, and Triiodide Ions Solutions Absorption in the UV and Visible Light Spectral Bands. Laser Phys. 2015;25:075602. doi: 10.1088/1054-660X/25/7/075602. [DOI] [Google Scholar]
151.Prasanna, Shrikanth B.K., Hegde M.S. Formation and Structure of Iodine: Water (H2O-I2) Charge-Transfer Complex. J. Chem. Sci. 2021;133:51. doi: 10.1007/s12039-021-01912-7. [DOI] [Google Scholar]
Associated Data
This section collects any data citations, data availability statements, or supplementary materials included in this article.
Data Availability Statement
Not applicable.
Articles from Molecules are provided here courtesy of Multidisciplinary Digital Publishing Institute (MDPI)
ACTIONS
View on publisher site
PDF (3.1 MB)
Cite
Collections
Permalink PERMALINK
Copy
RESOURCES
Similar articles
Cited by other articles
Links to NCBI Databases
Cite
Copy
Download .nbib.nbib
Format:
Add to Collections
Create a new collection
Add to an existing collection
Name your collection
Choose a collection
Unable to load your collection due to an error
Please try again
Add Cancel
Follow NCBI
NCBI on X (formerly known as Twitter)NCBI on FacebookNCBI on LinkedInNCBI on GitHubNCBI RSS feed
Connect with NLM
NLM on X (formerly known as Twitter)NLM on FacebookNLM on YouTube
National Library of Medicine 8600 Rockville Pike Bethesda, MD 20894
Web Policies
FOIA
HHS Vulnerability Disclosure
Help
Accessibility
Careers
NLM
NIH
HHS
USA.gov
Back to Top |
8619 | https://www.horiba.com/usa/process-and-environmental/industries/sulfuric-acid-production/ | Sulfuric Acid Production
Overview about Sulfuric Acid
Sulfuric acid is one of the most highly produced and important chemicals. It finds widespread applications across in various industries and processes such as industrial chemical production (fertilizers, detergents, pigments, dyers), battery manufacturing, metallurgical processes, petroleum refining, dying and textile industry, production of explosives, semiconductor industry, laboratory use, and analytical chemistry, among others. There are different processes of sulfuric acid production: chamber process, single contact, and double contact process. The double contact process is currently the most preferred and widely used due to its high conversion ratio of sulfur trioxide to sulfuric acid. High conversion also means less SO2 residues in the emission gas, helping to reduce SO2 emission into the atmosphere and mitigate ambient air pollution.
Sulfuric Acid Production Process
The production of sulfuric acid involves a multi-step process, widely known as the Contact Process or the sulfuric acid manufacturing process. This process converts sulfur dioxide (SO2) into sulfuric acid (H2SO4). The raw materials used for the sulfuric acid production include sulfur containing gases obtained by burning of elemental sulfur, pyrites or off gases from metallurgical industry.
There are several steps in the production of sulfuric acid explained below:
Step 1: Sulfur Combustion
In cases where sulfuric acid is manufactured from elemental sulfur, pyrite or metal sulfide ore as raw material, the process begins with the combustion of sulfur (S) to produce sulfur dioxide (SO2).
S + O2 → SO2
Step 2: Sulfur Dioxide Conversion
Before sending sulfur dioxide to the converter, it undergoes pretreatment to remove impurities. First, sulfur dioxide is treated in a washing tower to reduce its temperature, then it is passed through a venturi scrubber to remove fine particles. Subsequently, it is routed to the cooling tower to further lower its temperature and remove water. Afterward, it passes through mist precipitators to eliminate any remaining fine particles and mist.
The conversion of sulfur dioxide and oxygen into sulfur trioxide is highly dependent on the amount of excess oxygen. Therefore, sulfuric acid is diluted with dry air to provide the necessary quantity of air for catalytic reaction. The last treatment step before entering the contact process at the converter is the drying tower, where any remaining water is removed from the sulfur dioxide gas.
Finally, the purified sulfur dioxide is sent to the converter for the contact process, during which it is converted into sulfur trioxide (SO3) in the presence of a catalyst. Typically, the catalyst used is either vanadium pentoxide (V2O5) or alumina (Al2O3).
2SO2 + O2 ⇌ 2SO3
The equilibrium mentioned above is highly sensitive to temperature and the excess of oxygen. Therefore, controlling the concentrations of SO2 and O2 after treatment process is crucial to establish optimal conditions before the conversion process.
Step 3: Absorption in Sulfuric Acid
Sulfur trioxide (SO3) is dissolved in concentrated sulfuric acid (H2SO4), typically at a concentration of 98-99%, to produce fuming sulfuric acid, also known as oleum (H2S2O7). Oleum is a mixture of sulfuric acid and sulfur trioxide. It is essential to maintain precise control over the concentration of sulfuric acid in the absorption tower to prevent upset conditions. Under upset conditions, the absorption process may not proceed correctly, underscoring the importance of maintaining optimal operational conditions. Depending on the manufacturing process (single contact single absorption or double contact double absorption), absorption can occur is a single tower or multiple absorption towers.
SO3 + H2SO4 → H2S2O7 (oleum)
Step 4: Dilution
The oleum produced in the absorption tower is diluted with water to achieve the desired concentration of sulfuric acid. Dilution is a critical step in producing commercial-grade sulfuric acid, which typically ranges in concentration from about 93% to 98%.
H2S2O7 + H2O → 2H2SO4
Step 5: Cooling and Storage
The diluted sulfuric acid is then cooled and transferred to storage tanks.
In the case of double contact double absorption process, SO3 is taken into concentrated sulfuric acid and creates oleum in the first (intermediate) absorption tower. Meanwhile, non-oxidized SO2 passes to the converter for the second contact. After contact, SO2 and O2 are converted to SO3 and then directed to the second (final) absorbing tower, where they are absorbed into concentrated sulfuric acid (H2SO4) to create fuming sulfuric acid, or oleum (H2S2O7).
The Contact Process is highly efficient and widely used in industry for the large-scale production of sulfuric acid. The double contact process has largely been replaced by the single contact process, which offers increased production yields of sulfuric acid and is more environmentally friendly due to its higher SO2 absorption ratio and reduced SO2 emissions. Due to the corrosive and potentially hazardous nature of sulfuric acid production, safety measures and environmental controls are crucial.
HORIBA's Solution and Contribution
The sulfuric acid production process must be optimally controlled to attain a production yields of 98% or higher. HORIBA contributes to the sulfuric acid production process through its continuous gas analyzer, the ENDA-5000 series. The custom-designed sample gas pretreatment system, backed by over 50 years of experience and expertise in emission gas monitoring, as well as the in-house development and production of core components, contribute to accurate measurement of high-corrosive SO2 in the sulfuric acid production process.
The ENDA-5000 is offered and adopted in many sulfuric acid plants in Japan and overseas for monitoring of SO2, O2 and other components through the process. Highly important points to monitor for process optimization include (1) SO2 and O2 concentration before converter to maintain optimal conditions for conversion to SO3, (2) non-oxidized SO2 concentration after intermediate absorbing tower, and (3) after final absorbing tower for controlling absorbing efficiency and final regulated by local environmental regulations (4) stack emission monitoring.
Features
■ The ENDA-5000 analyzer contains a custom-made sample gas treatment system which ensures minimal dissolution of SO2 in the drain and provides accurate measurement results.
■ With over 50 years of experience in designing continuous gas analyzers and stack gas analyzers, HORIBA has accumulated know-how in various sampling parts for different sample gas conditions. Among these parts is treatment for the removal of corrosive SO3 mist and other sampling parts for the pretreatment of difficult-to-handle sample gases. Such a pretreatment system allows the detector to serve a longer time without corrosion and major troubles.
■HORIBA's proprietary cross-modulation NDIR technology, applied in this analyzer, ensures the cleaning of the measurement cell with the purified ambient air after each measurement. This contributes to the long lifetime of analyzer and maintains clean conditions inside the sample gas cell over the long time, which is highly important for low zero drift and precise measurement.
Related Brochures
Filter
Segment
Division
Industry
Application
Application Group
other (1)
Product
Media type
There are no results for this filter combination!
Monitoring of SO2 and O2 Concentration in Sulfuric Acid Production (Application Note)
Related Products
Filter
Industry
Electric Utilities (1)Solid Waste Management and Remediation Services (1)Environmental Countermeasures (1)Petroleum and Coal Products Manufacturing (1)
Measurement Object
Gas Component Concentration (1)
Measurement Technique
Non-Dispersive Infrared Spectroscopy (NDIR) (1)Magneto-pneumatic Analysis (1)
Segment
Process And Environmental (1)
Division
Combustion (1)
Name
ENDA-5000 series (1)
There are no results for this filter combination!
ENDA-5000 series
Stack Gas Analysis System
Request for Information
Do you have any questions or requests? Use this form to contact our specialists.
Corporate
By clicking “Accept All Cookies”, you agree to the storing of cookies on your device to enhance site navigation, analyze site usage, and assist in our marketing efforts.
Privacy Preference Center
Privacy Preference Center
Your Privacy
Social Media Cookies
Performance Cookies
Functional Cookies
Targeting Cookies
Strictly Necessary Cookies
Your Privacy
When you visit any website, it may store or retrieve information on your browser, mostly in the form of cookies. This information might be about you, your preferences or your device and is mostly used to make the site work as you expect it to. The information does not usually directly identify you, but it can give you a more personalized web experience. Because we respect your right to privacy, you can choose not to allow some types of cookies. Click on the different category headings to find out more and change our default settings. However, blocking some types of cookies may impact your experience of the site and the services we are able to offer.
More information
Social Media Cookies
These cookies are set by a range of social media services that we have added to the site to enable you to share our content with your friends and networks. They are capable of tracking your browser across other sites and building up a profile of your interests. This may impact the content and messages you see on other websites you visit. If you do not allow these cookies you may not be able to use or see these sharing tools.
Performance Cookies
These cookies allow us to count visits and traffic sources so we can measure and improve the performance of our site. They help us to know which pages are the most and least popular and see how visitors move around the site. All information these cookies collect is aggregated and therefore anonymous. If you do not allow these cookies we will not know when you have visited our site, and will not be able to monitor its performance. |
8620 | https://www.youtube.com/watch?v=m1UgKGDj76A | CrO3+H2O=H2CrO4 balance the chemical equation @mydocumentary838 cro3+h2o=h2cro4 balance the equation
My documentary
65000 subscribers
7 likes
Description
770 views
Posted: 3 Feb 2022
CrO3+H2O=H2CrO4
balance the chemical equation by law of conservation of mass.
cro3+h2o=h2cro4
chromium trioxide+water=chromic acid
chromium trioxide and water are reacts to form chromic acid
balance the chemical equation @mydocumentary838.
Online chemistry classes from my documentary a to z chemistry channel. For more videos please subscribe my channel and press on the bell button.
2 comments
Transcript:
झाल का कर दो अजय को की व्यवस्था केमिकल इक्वेशन बाय लॉ आफ कंजर्वेशन ऑफ मार्क्स फर्म ए ग्रेट प्लेस वाटर इस क्वेश्चसं क्रोमिक एसिड है थे इनिशियल क्वेश्चन रिटर्न सर प्रॉब्लम ट्रैक्शन वाटर एंड प्रोडक्ट्स ग्राफिक एसिड नई कि इसे उखाड़ से सिचुएशन द नंबर आफ आइटम्स रिटर्न्स 68वें नंबर आफ आइटम्स आफ प्रोडक्ट्स एंड न इरॉटिक फ्रॉम बॉटम 1802 हरिद्वार आइटम्स इन थे अस विद सिचुएशन विकास कृषि विश्वविद्यालय रस तो इसे लाइक दिस वीडियो Please subscribe My Channel press The Bell Icon on |
8621 | https://www.youtube.com/watch?v=sFAmnNmrQ-k | TRIGONOMETRIC FUNCTIONS | asinx+bcosx=c | 5. Maximum & Minimum Values
TheGreenReviews
2860 subscribers
4 likes
Description
562 views
Posted: 9 Jun 2021
Transcript:
[Music] so suppose they ask the question what is the maximum value of a sine x plus b cos x or they might also ask what is the minimum value of a sine x plus b cosex now it's quite difficult to find the maximum or minimum value of this single format but if we write it in the format of r sine x plus alpha or r cos x plus alpha then we can find the maximum minimum value because in that case it becomes a sinusoidal now one thing is important to remember it doesn't really matter whether you convert a sine x plus b cosex in the sine format or the cos format because in both the cases they are sinusoidal that means they look like this regardless of whether it is sine x or whether it is cosex they look like this it's known as a sinusoidal curve now in a sinusoidal curve the maximum value is plus one and the minimum value is -1 and there is an axis over here so which is if there is an axis over here so which makes it zero so this is zero so you can see the value fluctuates between 1 0 minus 1 0 1 0 minus 1 it fluctuates between plus 1 and minus 1. so if you're asked what is the maximum value of either sine of any angle or cos of any angle in that case the maximum value of sine theta is one and minimum value of sine theta is minus one similarly the maximum value of cos theta is one and the minimum value of cos theta is -1 now but there is a number over here called r so if there is a number what would be the maximum value well in that case if there is a maximum value for r sine of any angle then if you have r sine theta so the maximum value would be plus r and the minimum value would be minus r again in terms of r cos theta the maximum value would be plus r and the minimum value would be minus r but what about this angle it's not basically x it's x plus alpha or in same in case of a cos it's like x plus alpha theta plus alpha doesn't this have any uh effect on the maximum or minimum value not at all the only thing to remember is that the maximum value of sine of any angle regardless of what the angle is it is plus one and the minimum value of sine or cos of any angle is minus one so what we are doing basically over here is we are just multiplying it with this so this is one so i'm just writing r into one and this is minus 1 i'm just writing r into minus 1 that's where the minimum value minus r is coming from so let me show you visually what this looks like so first of all take a look at this this is the range is from 0 to 360 degree so if i take a look at this is sine x so you can see the maximum value here it is 1 and the minimum value over here is minus 1. so sine x has this maximum value 1 minimum value minus 1. now if i draw 2 sine x meaning r sine x you will see now the maximum value is 2 and the minimum value is minus 2. if i use 3 sine x the value of r has increased the maximum value is three and the minimum value is minus three if i have four sine x the maximum value is four and the minimum value is minus four that means the main sinusoidal is sine x is plus 1 and minus 1 maximum minimum but when you multiply it with numbers by r that becomes the maximum point and that becomes the minimum point now if we do the same thing with course you will see course looks like this from 0 to 360 degree and the maximum value is this 1 and the minimum value is this minus 1 so this is 1 and this is minus 1. if i multiply this with two the maximum value becomes two and the minimum value becomes minus two if i multiply with this three the maximum value becomes three and the minimum value becomes minus three if i multiply this with four the maximum becomes four and the minimum value becomes minus 4. that means basically r and minus r now if you take a look at sine x and cosec you might say but they look different sine x looks like this and cosex looks like this why are they sinusoidal well let me zoom out if i zoom out you will see that basically they are part of the same sinusoidal just the difference is the step so basically course is a 90 degree a little bit 90 degree backwards and sine is 90 degree forward apart from that if you look at the sketch it is identical the sinusoidal is really identical they look more or less the same this is cause and this is sign |
8622 | https://nrich.maths.org/articles/eulers-formula-and-topology | Euler's Formula and Topology | NRICH
Skip to main content
Problem-Solving Schools can now access the Hub!
Contact us if you haven't received login details
Main navigation
Teachersexpand_more
Early years
Primary
Secondary
Post-16
Professional development
Studentsexpand_more
Primary
Secondary
Post-16
Parentsexpand_more
Early years
Primary
Secondary
Post-16
Problem-Solving Schoolsexpand_more
What is the Problem-Solving Schools initiative?
Becoming a Problem-Solving School
Charter
Hub
Resources and PD
Events
About NRICHexpand_more
About us
Impact stories
Support us
Our funders
Contact us
search
menu
search
close
Search NRICH search
Or search by topic
Number and algebra
Properties of numbers
Place value and the number system
Calculations and numerical methods
Fractions, decimals, percentages, ratio and proportion
Patterns, sequences and structure
Coordinates, functions and graphs
Algebraic expressions, equations and formulae
Geometry and measure
Measuring and calculating with units
Angles, polygons, and geometrical proof
3D geometry, shape and space
Transformations and constructions
Pythagoras and trigonometry
Vectors and matrices
Probability and statistics
Handling, processing and representing data
Probability
Working mathematically
Thinking mathematically
Mathematical mindsets
Advanced mathematics
Calculus
Decision mathematics and combinatorics
Advanced probability and statistics
Mechanics
For younger learners
Early years foundation stage
Age
16 to 18
|Article by
Alan Beardon
|PublishedTue, 01/02/2011 - 00:00
Euler's formula and topology
Euler's formula for the sphere
Roughly speaking, a network (or, as mathematicians would say, a graph ) is a collection of points, called vertices , and lines joining them, called edges . Each edge meets only two vertices (one at each of its ends), and two edges must not intersect except at a vertex (which will then be a common endpoint of the two edges). The collection of edges form the boundary of certain areas, called faces . Faces must not have holes in them or handles on them. If two faces have common boundary points, then they must share a common edge (and only this), or a common vertex (and only this).
In Figure 1 there are two examples of networks, and these have
3 faces, 12 edges and 10 vertices,
and
(ii) 2 faces, 6 edges and 5 vertices,
respectively.
Image
Figure 1
It is not necessary to assume that the lines in a network are straight lines'; they can becurves' of any sort except that they must not cross over themselves or each other. Moreover, we can draw such networks on any surface (for example, the plane, a sphere, a cylinder, and so on).
A triangulation of a surface is a network on the surface all of whose faces are triangular (that is, they are bounded by three edges). In fact, surveyors map the countryside by triangulating it with `trig points' (usually on mountain tops) and measuring angles and distances between these trig points. In this case the trig points are the vertices of the triangulation.
The simplest example of a triangulation on the sphere (which we think of as the surface of the earth) is found by drawing the equator and, say, n lines of longitude. In this example, which is illustrated in Figure 2, there are 2 n triangular faces, n+2 vertices (n on the equator, and one at each pole), and 3 n edges, so that (number of faces)−(number of edges)+(number of vertices)=2 n−3 n+(n+2)=2.
Image
Figure 2
It is perhaps surprising that this answer does not depend on the choice of n . Even more remarkable is the fact, known as Euler's Theorem , that this formula holds for ALL triangulations of the sphere. Leonard Euler (1707-1783) was a Swiss mathematician who was, perhaps, the most productive mathematician of all time. Thus, for any triangulation of the sphere with, say, T triangles, E edges and V vertices, Euler's formula for the sphere is that T−E+V=2.
The important thing to realise is that this formula is a topological invariant : this means that if we deform the triangulation and the sphere continuously then the numbers T, E and V will not change and the formula will still be true. For example, as we can deform the sphere to a cube, the formula will still hold for a cube, so let us check this with one particular triangulation of the cube. We divide each face of the cube into two triangles by drawing a diagonal across each face. Then there are six faces of the cube and each gives two triangles; thus T=12. Clearly V=8 (there are no new vertices introduced when we draw the diagonals), and E=18 (twelve from the original cube and the six added diagonals). Thus T−E+V=12−18+8=2.
EXERCISE 1
Check that Euler's formula is true in each of the following cases (and in each case you should try different triangulations):
1. for a tetrahedron (a pyramid with a triangular base);
2. for the surface formed by gluing two tetrahedra together across their base;
3. for a pyramid with a square base as found in Egypt (remember that the base of the pyramid, where the pyramid touches the ground, is part of the surface of the pyramid);
4. for the octahedron (which is formed by gluing two pyramids together across their base);
5. the surface formed by gluing two pyramids, each with a square base, onto two different faces of a cube. You should assume that the faces of the cube are the same size as the bases of the pyramids.
EXERCISE 2
Construct your own surface (for example, by sticking several cubes together), triangulate it, and then check that Euler's formula works for this.
Remark 1:
`Construct' here does not mean that you should physically construct the surface; part of mathematics is learning how to represent three dimensional objects on a sheet of paper!
Remark:
the surface must be a deformed sphere; for example, Euler's formula as given above will not hold on the surface of a doughnut (a torus).
Legendre's proof of Euler's formula on a sphere
The geometry of the sphere is extremely important; for example, when navigators (in ships or planes) work out their course across one of the oceans they must use the geometry of the sphere (and not the geometry of the plane!). In the usual plane Euclidean geometry, the angle sum of a triangle is 180 degrees (or π radians). On the surface of the sphere, the arc of shortest distance between two points is the arc of a great circle on the sphere, and if we form a `spherical' triangle with these arcs as sides then the angle sum of the triangle is NOT 180 degrees. For example, if we look at the triangle on the sphere bounded by the equator, the Greenwich meridian, and the line of longitude given by 90∘ East, we obtain a spherical triangle all three of whose angles are 90∘. Let us call this triangle Δ. You may like to consider now what other angle sums of a triangle are possible on the sphere.
It is a fact, which we shall assume here, that if a spherical triangle is lying on a sphere of unit radius and has, say, angles θ 1, θ 2 and θ 3 (measured in radians), then the area of the triangle is θ 1+θ 2+θ 3−π. Of course, in Euclidean geometry, we would have θ 1+θ 2+θ 3=π, but this formula is no longer valid because our space is curved. At its simplest level, Einstein's theory about space-time is that the space-time we live in is curved and so the `formulae' of physics are different to those that we get if we just use the Euclidean point of view.
Notice that if we accept the above formula for the area of a spherical triangle, then the triangle Δ constructed above has area 3(π/2)−π=π/2. As eight copies of Δ will fill the sphere without overlapping, we see that the area of the sphere (of unit radius) is 4 π.
We can now give Legendre's beautiful proof of Euler's formula that is based on a simple discussion of geometry on the sphere. For any triangle on the sphere, we have angle sum of triangle=area of triangle+π. Suppose that there are T triangles, E edges and V vertices in the triangulation of the sphere. Then, summing all angles in all triangles, the total angle sum is 2 π V (as all of the angles occur at a vertex without overlap, and the angle sum at any one of the V vertices is exactly 2 π). Also, the sum of the areas of the triangles is the area of the sphere, namely 4 π; thus we see that 2 π V=4 π+T π, or V−T/2=2. Now (by counting edges of each triangle and noting that this counts each edge twice), we obtain 3 T=2 E, or E=3 T/2. Thus we have T−E+V=T−3 T/2+(2+T/2)=2 which is Euler's formula.
Euler's formula for a simple closed polygon
Given a polygon that does not cross itself, we can triangulate the inside of the polygon into non-overlapping triangles such that any two triangles meet (if at all) either along a common edge, or at a common vertex. Suppose that there are T triangles, E edges and V vertices; then Euler's formula for a polygon is T−E+V=1. You should be able to confirm this in the polygon illustrated in Figure 3.
Image
Figure 3
EXERCISE 3
Draw different polygons, triangulate them, and check that Euler's formula is true.
Proof of Euler's formula for a simple closed polygon
.
Because the terms T, E and V are unchanged when we apply a continuous change (or deformation) to the picture, we can imagine the picture to be drawn on a flexible rubber sheet. We cut the polygon out of the sheet, and then manipulate it until it fits exactly onto the lower half of a sphere sitting on the table. We can now regard it as a triangulation of the lower hemisphere of this sphere, and we shall use geographical terms, like equator' andnorth pole' to describe the points on the sphere. There will be a certain number of vertices, say N, on the equator' of the sphere (exactly the same number as were on the boundary of the original polygon) and, as the vertices and edges alternate around the equator, there will also be exactly N edges on the equator. Now draw arcs of circles from theNorth pole' of the sphere to each of these N vertices (see Figure 4). This will now give us a triangulation of the sphere with N new triangular faces, N new edges (all from the North pole) and one new vertex (at the North pole). Thus, from Euler's formula for the sphere, (T+N)−(E+N)+(V+1)=2, and this gives Euler's formula for the polygon, namely T−E+V=1.
Image
Figure 4
Euler's formula with polygonal faces on any surface
So far we have considered Euler's formula on a surface with the network only having triangular faces. In fact, the formula also holds when the faces are polygons. You should try some examples of this to persuade yourself that this is so. As one example, the cube has six square faces, 12 edges and 8 vertices and 6 - 12 + 8 = 2.
Here is a sketch of a proof of Euler's formula for a sphere F−E+V=2. Some details are omitted but the general idea should be clear.
We start with a polygon drawn on the sphere. Suppose this has N vertices and hence N sides on the boundary of the polygon. There are two faces (the inside and outside of the polygon) so for this network we have F−E+V=2−N+N=2. Suppose we now join two vertices by a polygonal curve - shown as a red line in the diagram.
Image
For a general curve of this type we add k new edges, k−1 new vertices (in the figure, k=4), and the number of faces is increased by one (one face is divided into two). Then for the new network F n e w−E n e w+V n e w=(F+1)−(E+k)+(V+k−1)=F−E+V=2. In the same way we can add another 'dotted' line without changing F−E+V, and in this way we can build a general network for which we must still have F−E+V=2.
For a polygon the proof is very much like the proof in Section 3.
A Euler's formula for a polygon with holes
Consider two polygons, one inside the other; some examples are given in Figure 5. What is Euler's formula for the region between the two polygons ? You should draw different regions of this type, triangulate them in different ways, and when you have reached an answer experimentally, try and prove it.
Image
Figure 5
Now try to find Euler's formula for regions with two or more holes like those in Figure 6.
Image
Figure 6
A Euler's formula for a square tube
Consider a square tube with no ends as illustrated in Figure 7, and consider triangulations of this that cover the whole tube. What is Euler's formula for a square tube ? You should draw different triangulations of this, and when you have reached an answer experimentally, try and prove it.
Image
Figure 7
When you have done this, try and find Euler's formula for a `doughnut' : see Figure 8.
Image
Figure 8
The angle deficiency of a polyhedron
Here is an attractive application of Euler's Formula. The angle deficiency of a vertex of a polyhedron is 360∘ (or 2 π radians) minus the sum of the angles at the vertex of the faces that meet at the vertex. For example, for a cube, there are three faces, each with a right angle at each vertex so the angle deficiency (at each vertex) is 2 π−3×(π/2)=π/2 (360∘−3×90∘=90∘), or 90 degrees. The total angle deficiency for the cube is 8×90∘=720∘ (or 4 π radians). It is a remarkable fact that the total angle deficiency of any polyhedron that is topologically equivalent to a sphere is 720∘ or 4 π radians .
Suppose that we have a polyhedron which is made up with faces F 1,…F k (each of these is a polygon, which need not be regular). Then we can apply Euler's Theorem to the polyhedron, so let us count the faces, edges and vertices. First, by definition, there are k faces. Suppose that the face F j has N j edges (and hence N j vertices). If we count the total number of edges by looking at them from inside each face we see' N 1+⋯+N k edges. But of course, we see each edge twice (for each edge isseen' from both sides); thus 2 E=N 1+⋯+N k. One interesting fact here is that N 1+⋯+N k must be even ; this is not obviously so. Finally, the number V of vertices is given by Euler's formula for a sphere (or for any `deformed sphere'), so V=E−F+2=(N 1+⋯+N k)/2−k+2, or 2 V=(N 1+⋯+N k)−2 k+4.
The polyhedron has V vertices and, by definition, the total angle deficiency of the polyhedron is 2 π V minus the sum of the angles of all faces at all vertices . As the interior angle sum of a polygon with n sides is (n−2)π, we see that the total angle deficiency of the polyhedron is Θ, where
Θ=2 π V−[(N 1−2)π+(N 2−2)π+⋯+(N k−2)π]=2 π V−(N 1+⋯+N k)π+2 k π=π(2 V−(N 1+⋯+N k)+2 k)=4 π.
We have now proved the following result.
Theorem
.
The total angle deficiency of any polyhedron is 4 π.
An example
Consider the regular tetrahedron. This has three equilateral triangles meeting at each of four vertices, so in this case, Θ=8 π−4×(3×π/3)=4 π. You may like to try this with other examples.
The fact that the total angle deficiency of a polyhedron is 720 degrees, together with Euler's formula, gives the key to finding how many regular polyhedra there are (Platonic Solids) and how many semi-regular polyhedra there are (Archimedean solids) and discovering their properties (the shapes and number of faces etc.).
Adding Euler numbers
We are going to show that if we join two surfaces together across the boundary of a hole in each, then the Euler number of the joined surface is the sum of the Euler numbers of the two separate surfaces .
Suppose that we have two surfaces S 1 and S 2, each with a hole in it. By deforming the surfaces, we may assume that the two holes are circular with the same radius; for example as illustrated below.
Image
Now triangulate both surfaces in such a way that there are exactly k vertices, and hence k edges, on each of the circular boundaries of the holes. Again by deforming the surfaces we may assume that these k vertices and k edges match up exactly when the two surfaces are brought together.
Suppose that the triangulation of S 1 has T 1 triangles, E 1 edges and V 1 vertices, and similarly for S 2. When the surfaces have been joined together at the edges of the two circular holes, we will have a triangulation of the new surface, which we call S, and this triangulation will have exactly T 1+T 2 triangles in it. However, it will not have E 1+E 2 edges because each of the k edges on the boundary of one hole will be joined to one of the other edges on the other boundary; thus the new triangulation will have exactly E 1+E 2−k edges and, similarly, V 1+V 2−k vertices. Thus Euler-number(S)=(T 1+T 2)−(E 1+E 2−k)+(V 1+V 2−k)=(T 1−E 1+V 1)+(T 2−E 2+V 2)=Euler-number(S 1)+Euler-number(S 2).
As an example, a disc is topologically a hemisphere, so that these two surfaces have the same Euler number. If we join two hemispheres across their boundaries (for example, the southern and northern hemishperes are joined across the equator) we see that the Euler number of a sphere is twice the Euler number of a hemisphere. Hence the Euler number of a hemisphere, and also of a disc, is one.
Footer
Sign up to our newsletter
Technical help
Accessibility statement
Contact us
Terms and conditions
Links to the NRICH Twitter account
Links to the NRICH Facebook account
Links to the NRICH Bluesky account
NRICH is part of the family of activities in the Millennium Mathematics Project. |
8623 | https://digitalcommons.ursinus.edu/cgi/viewcontent.cgi?article=1009&context=triumphs_calculus | Ursinus College Ursinus College
Digital Commons @ Ursinus College Digital Commons @ Ursinus College
Calculus Transforming Instruction in Undergraduate
Mathematics via Primary Historical Sources
(TRIUMPHS)
Summer 2019
Euler’s Calculation of the Sum of the Reciprocals of Squares Euler’s Calculation of the Sum of the Reciprocals of Squares
Kenneth M. Monks
kmmonks@gmail.com
Follow this and additional works at:
Part of the Other Mathematics Commons
Click here to let us know how access to this document benefits you. Click here to let us know how access to this document benefits you.
Recommended Citation Recommended Citation
Monks, Kenneth M., "Euler’s Calculation of the Sum of the Reciprocals of Squares" (2019).
Calculus . 9.
This Course Materials is brought to you for free and open access by the Transforming Instruction in Undergraduate
Mathematics via Primary Historical Sources (TRIUMPHS) at Digital Commons @ Ursinus College. It has been
accepted for inclusion in Calculus by an authorized administrator of Digital Commons @ Ursinus College. For more
information, please contact aprock@ursinus.edu .Euler’s Calculation of the Sum of the Reciprocals of the Squares
Kenneth M Monks ∗
January 30, 2022
A central theme of most second-semester calculus courses is that of infinite series . Simply put, to study infinite series is to attempt to answer the following question:
What happens if you add up infinitely many numbers?
How much sense humankind made of this question at different points throughout history depended enormously on what exactly those numbers being summed were. As far back as the third century bce ,Greek mathematician and engineer Archimedes (287 bce –212 bce ) used his method of exhaustion
to carry out computations equivalent to the evaluation of an infinite geometric series in his work
Quadrature of the Parabola (Archimedes, 1897). Far more difficult than geometric series are p-series: series of the form ∞∑
n=1
1
np = 1 + 12p + 13p + 14p + · · ·
for a real number p. Here we show the history of just two such series. In Section 1, we warm up with Nicole Oresme’s treatment of the harmonic series , the p = 1 case. 1 This will lessen the likelihood that we pull a muscle during our more intense Section 3 excursion: Euler’s incredibly clever method for evaluating the p = 2 case.
1 Oresme and the Harmonic Series
In roughly the year 1350 ce , a University of Paris scholar named Nicole Oresme 2 (1323 ce –1382 ce )proved that the harmonic series does not sum to any finite value (Oresme, 1360). Today, we would say the series diverges and that 1 + 12 + 13 + 14 + · · · = ∞.
His argument was brief and beautiful! Let us admire it below.
∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞
∗Department of Mathematics, Front Range Community College – Boulder County Campus, Longmont, CO 80537;
kenneth.monks@frontrange.edu .
1The harmonic mean of two numbers aand bis the quantity 2 /(1 /a + 1 /b ). Thus, in the harmonic series, each term is the harmonic mean of the term to its left and the term to its right, much like the terms of an arithmetic series or geometric series, mutatis mutandis .
2Nicole Oresme was an early Renaissance scholar who had a significant hand in reviving the works of antiquity. For instance, he translated Aristotle’s Ethics ,Politics , and On the Heavens from Latin into French. The work here stems from Oresme’s study of Euclid’s Elements (O’Connor and Robertson, 2003).
1When quantities with unequal proportions (which are less than one) are added to some value, it is still possible that the sum will be infinite. But if it is that each term is proportional to the previous, the sum will be finite, . . . For example: if it we start with a measure of a foot, to which we first add a smaller portion (half a foot), then a third, and then a fourth, then a fifth and so on forever through all the numbers in order, I say that the sum will be infinite. We shall see this as follows: if there exist infinitely many parts, each of which is longer than half a foot, then the total will be infinite. As is well known, 1/4 and 1/3 are more than a half; similarly, [the sum of] 1/5 through 1/8, and 1/9 through 1/16 , etc. to infinity. 3
∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞
Task 1 Pay special attention to Oresme’s claim that “ if it is that each term is proportional to the previous, the sum will be finite, ” What kind of series was he talking about? Why must such a series have a finite sum?
Task 2 Since the question had been settled for series in which pairs of consecutive terms had equal proportions, Oresme then looked at a series with unequal proportions: the sum of reciprocals of the positive natural numbers. Let us examine Oresme’s argument in more modern form. In particular, define
an = 1
n
to be the sequence of reciprocals and
An =
n
∑
k=1
ak
to be the corresponding sequence of partial sums. (a) Explain why 1 /3 + 1 /4 is greater than 1 /2 as claimed without actually calculating the sum of those two fractions. Rather, use the fact that 1 /3 > 1/4 and think about what would happen if you substituted 1 /4 for the 1 /3 in the sum 1 /3 + 1 /4. (b) In a similar manner, explain why 1 /5 + 1 /6 + 1 /7 + 1 /8 is greater than 1 /2. Use the fact that each of the fractions 1 /5, 1 /6, and 1 /7 are greater than 1 /8. (c) Extend this argument to explain why 1 /9 + 1 /10 + · · · + 1 /16 is greater than 1 /2. (d) Explain why
2n+1
∑
k=2 n+1
1
k > 12for all natural numbers n. As a hint, it may be helpful to expand the summation on the left-hand side for a few different particular values of n. For example, try n = 2 and n = 3. This will help build an intuition for what terms exactly are in that sum.
3The original Latin of the excerpt here is taken from pages 105 and 106 of Quaestiones super Geometriam Euclidis
(Oresme, 1360). Many thanks to Dominic Klyve (dominic.klyve@cwu.edu) for providing the translation.
2(e) Use the above inequality to show that
A2n > n
2for all natural numbers n.(f) Consider Oresme’s statement “ if there exist infinitely many parts, each of which is longer than half a foot, then the total will be infinite. ” Why does the inequality A2n > n
2
justify his claim? (g) A standard modern approach for analyzing the convergence/divergence of an infinite series is the integral test . Apply the integral test to the harmonic series. Does the conclusion agree with Oresme’s?
2 The Sum of the Reciprocals of the Squares
A natural follow-up is to consider the sum of the reciprocals of squares! After all, a positive number less than 1 becomes smaller if you square the value; perhaps their sum could converge to a finite value even though the harmonic series diverges. Many mathematicians of the early 18th century attempted to compute this sum, written as 1 + 122 + 132 + 142 + · · · = 1 + 14 + 19 + 116 + · · · .
Task 3 Is the above summation even worth considering? (a) Write the summation above in sigma notation. (b) Use the integral test for convergence of an infinite series to conclude that it does in fact converge to a real number. In the seventeenth century, a pair of mathematical brothers, Jacob (1655–1705) and Johann Bernoulli 4 (1667–1748), began to study this series. In 1689, in his Tractatus de seriebus infinitis ,Jacob was able to prove the above series converged to a number less than 2 (Bernoulli, 1713). However, the exact value of this series eluded the brothers. The solution was presented to the St. Petersburg Academy in 1735 by a brilliant former student of Johann named Leonhard Euler. 5
4
The Bernoulli brothers were born into a successful Calvinist spice merchant family in Basel, Switzerland. The parents pushed Jacob into studying theology and philosophy. He defied them and switched to mathematics and astronomy later in life against their wishes. The parents then tried to push Johann into taking over the spice business; he despised this work and instead decided to follow his older brother into mathematics (O’Connor and Robertson, 1998a).
5
Johann Bernoulli mentored a young Euler in his early days of mathematical studies. In his autobiography, Euler wrote, “ In 1720, I was admitted to the university as a public student, where I soon found the opportunity to become acquainted with the famous professor Johann Bernoulli, who made it a special pleasure for himself to help me along in the mathematical sciences . . . which consisted in myself taking a look at some of the more difficult mathematical books and work through them with great diligence, and should I encounter some objections or difficulties, he offered me free access to him every Saturday afternoon, and he was gracious enough to comment on the collected difficulties, which was done with such a desired advantage that, when he resolved one of my objections, ten others at once disappeared, which certainly is the best method of making auspicious progress in the mathematical sciences. ” (Fellman, 2007, page 5)
33 Euler’s Proof
At age 28, Leonhard Euler 6 (1707–1783) found the exact value of the sum! We now trace through his argument as it first appeared in print, in the 1740 paper De Summis Serierum Reciprocarum (Euler, 1740). It was published in the Commentarii academiae scientiarum imperialis Petropolitanae (Mem-oirs of the Imperial Academy of Sciences in St. Petersburg) , the first journal of the St. Petersburg Academy of Science. It is worth noting, however, that the work was read to the academy five years earlier. Here we use Jordan Bell’s translation (Euler, 2005). 7 As Oresme’s work showed, the sums of reciprocals had been studied for hundreds of years at that point. Thus, Euler opened with the following remark.
∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞§.1. So much work has been done on the series of the reciprocals of powers of the natural numbers, that it seems hardly likely to be able to discover anything new about them. . . .
§.2. I have recently found, quite unexpectedly, an elegant expression for the sum of this series
1 + 14 + 19 + 116 + etc. ,
which depends on the quadrature of the circle. . . .
∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞
Task 4 These days the word “quadrature” is often used interchangeably with the word “area” based on the ancient Greeks’ perspective wherein finding the area of a region was accomplished by constructing a square (quadrilateral) that had the same area. According to Euler’s statements above, on which famous constant will the summation depend? Euler proceeded to state the relationship between the measure of an arc s on the unit circle and the sine of s, which he called y. Note that we typically refer to the input of sine as an angle rather than as an arc. On the unit circle, the radian measure of a central angle equals the length of the corresponding arc, since the unit circle would have a full circumference of 2 π. Thus, radian measure of a central angle and the corresponding arc on the unit circle are in fact interchangeable.
6Leonhard Euler was born in Basel, Switzerland to Marguerite (n´ ee Brucker) and Paul Euler, a Protestant minister who had attended Johann Bernoulli’s lectures at University of Basel. Paul wished for his son to follow him into the ministry, but Johann persuaded Paul to allow Leonhard to study mathematics instead after witnessing his incredible potential for mathematics. The fact that the mathematicians who studied the evaluation of the sum of the reciprocals of the squares came from Basel has led to the problem being referred to as The Basel Problem . (O’Connor and Robertson, 1998b)
7The Mathematical Association of America hosts an incredible digital library known as The Euler Archive .This archive links to not only to the translation we use here, but to a massive quantity of Euler’s writing, translated into many languages. Find it at eulerarchive.maa.org .
4∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞§.4. The first equation
y = s − s3
1 · 2 · 3 + s5
1 · 2 · 3 · 4 · 5 − s7
1 · 2 · 3 · 4 · 5 · 6 · 7 + etc. thus expresses a relation between the arc and sine. For from it, from a given arc its sine can be determined, and likewise for a given sine the arc. . . .
§.16. Now, if we let y = 0 , the fundamental equation becomes
0 = s − s3
1 · 2 · 3 + s5
1 · 2 · 3 · 4 · 5 − s7
1 · 2 · 3 · 4 · 5 · 6 · 7 + etc. ,
and the roots of this equation give all the arcs whose sine is = 0 . There is one minimal root
s = 0 , so that when the equation is divided by s it exhibits all the remaining arcs whose sine is = 0 , so that these arcs are all the roots of the equation
0 = 1 − s2
1 · 2 · 3 + s4
1 · 2 · 3 · 4 · 5 − s6
1 · 2 · 3 · 4 · 5 · 6 · 7 + etc.
∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞
Task 5 Euler discarded the root s = 0 in his work above. Let us examine his decision to do so in a bit more modern notation. (a) Consider the function f (s) = sin( s)
s
. What is the domain of f ?(b) Where is f discontinuous? Does the graph of f have any vertical asymptotes? Why or why not? (c) Recall from your first-semester calculus course your strategies for evaluating limits; often when division by zero was encountered, a good strategy for removing that discontinuity was to factor the numerator and reduce the fraction. Take the function f (s) = sin( s)
s
and rewrite the numerator as a power series centered at zero. Replicate this “factor and cancel” trick to remove the discontinuity. That is, calculate lim
s→0
sin( s)
s
not by using L’Hˆ ospital’s Rule or a special limit identity, but rather by using the power series for sine as Euler did! 5∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞§.16. Of course these arcs whose sine is = 0 are
p, −p, 2p, −2p, 3p, −3p, etc. ,
of which the second of each pair is negative, which the equation itself also tells us, because the dimensions of s are even.
∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞
Task 6 (a) Based on knowing the zeros of the sine function, what number would you guess Euler was talking about when he uses the symbol p above? (b) In §.11. of the same document, Euler introduced the symbol p in the phrase “ where p
denotes the total perimeter of a circle whose diameter is = 1 .” What does this phrase imply the value of p is? Does it agree with your conclusion from part (a)? (c) Why are these the zeros of the function 1 − s2
1·2·3
s4
1·2·3·4·5
− s6
1·2·3·4·5·6·7
· · · , as he claimed? (d) Why is the number 0 not on Euler’s list of arcs? Isn’t the sine of 0 equal to 0? Euler concluded §.16. with a very familiar task, polynomial factorization, presented in a somewhat unfamiliar way.
∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞
Hence the divisors of this equation will be
1 − sp , 1 + sp , 1 − s
2p , 1 + s
2p , etc.
∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞
What is happening here is may look odd at first, but it is really just a slightly rewritten version of the Factor Theorem from any introductory algebra class. We usually state the Factor Theorem as follows:
Familiar Factor Theorem. A polynomial f (x) has a root x = r if and only if it is divisible by
(x − r).
But, one could take the divisor ( x − r) and factor out the constant −r, producing instead the expression (x − r) = −r
( x
−r + 1
)
= −r
(
1 − xr
)
.
6Factoring out the constant −r does not affect the divisibility of those polynomials; the polynomial
f (x) is divisible by ( x − r) if and only if it is divisible by (1 − xr
). Thus, an equivalent but funky restatement of the Familiar Factor Theorem would be this:
Funky Factor Theorem. A polynomial f (x) has a root x = r if and only if it is divisible by
(1 − xr
) .
Why have two versions of the same theorem? Well, the first version is very convenient if the lead-ing coefficient of the polynomial is 1. If f (x) has roots r1, r 2, . . . , r n, then the following factorization will hold:
f (x) = ( x − r1) ( x − r2) · · · (x − rn)since upon expansion, the right-hand side will have leading term xn and thus a leading coefficient of 1. However, instead of having a leading coefficient of 1, suppose f (x) has a constant term of 1. In this case, the Funky Factor Theorem is more convenient! In particular, it tells us
f (x) =
(
1 − xr1
) (
1 − xr2
)
· · ·
(
1 − xrn
)
.
This works because when the above product is multiplied out, the constant term is 1. For fear of a relatively simple idea being obfuscated by formalism, we work out a small concrete example.
Task 7 (a) Factor the polynomial x2 − x − 6. What are the roots? Call these numbers r1 and r2.(b) Show that the polynomial 1 + 16 x − 16 x2 (which is the same quadratic, just divided by −6) has the same roots as the previous polynomial by just plugging in r1 and r2 for x and verifying the output is zero in each case. Use these roots to factor 1 + 16 x − 16 x2 into the form
(
1 − xr1
) (
1 − xr2
)
.(c) In each case, multiply out your factorization to see that it is correct.
Task 8 Let us now revisit Euler’s statement in which he claimed to know “ the divisors of this equation ”. (a) We have the polynomial 8
1 − s2
1 · 2 · 3 + s4
1 · 2 · 3 · 4 · 5 − s6
1 · 2 · 3 · 4 · 5 · 6 · 7 + · · ·
that we wish to factor. Which version of the Factor Theorem should we use, the Familiar or the Funky? Why? (b) Write out the factorization that this produces, using the roots from Task 6. Euler proceeded with a simple but clever simplification.
8Technically this is a power series and not a polynomial, since the degree is infinite. However, Euler had great intuition regarding the generality of the structure of mathematics; he was confident we could factor an infinite degree polynomial in the same way that we could factor a finite degree polynomial, and proceeded accordingly!
7∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞
and by combining each of these pairs of divisors together it will be
1 − s2
1 · 2 · 3 + s4
1 · 2 · 3 · 4 · 5 − s6
1 · 2 · 3 · 4 · 5 · 6 · 7 + etc.
=
(
1 − s2
p2
) (
1 − s2
4p2
) (
1 − s2
9p2
) (
1 − s2
16 p2
)
etc.
∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞
Task 9 (a) Which pairs of divisors did he join together? What famous identity from algebra did Euler apply to produce that right-hand side? (b) When you look at those denominators and think about the question we had set out to answer, do the hairs on the back of your neck tingle a little? If not, remind yourself what it is exactly that we are trying to evaluate, and then see if they do. Euler only stated his next step in words; we will carry out the corresponding algebra.
∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞§.17. Now it is evident from the nature of the equations, that the coefficient of ss, or 11·2·3 ,
is equal to 1
p2 + 14p2 + 19p2 + 116 p2 + etc. . . .
§.18. Hence from these the following sums are thus derived:
1 + 14 + 19 + 116 + etc. = p2
6 .
∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞∞
Task 10 Let us retrace the algebra that Euler glossed over above. (a) Take Euler’s infinite factorization from before, 1 − s2
3! + s4
5! − s6
7! + · · · =
(
1 − s2
p2
) (
1 − s2
4p2
) (
1 − s2
9p2
) (
1 − s2
16 p2
)
· · · ,
and begin to multiply out the right-hand side. Specifically, calculate the constant term as well as the coefficients of s and s2.8(b) Set the degree two coefficient of the right-hand side equal to the degree two coefficient of the left-hand side. 9
(c) Multiply both sides by p2 and declare victory!
4 Epilogue
In the same paper, Euler provided a method to extend this technique to any even power and executed the method up to exponent 12 in §.18. Just for fun, here are the values!
∞
∑
n=1
1
n2 = π2
6
∞
∑
n=1
1
n4 = π4
90
∞
∑
n=1
1
n6 = π6
945
∞
∑
n=1
1
n8 = π8
9450
∞
∑
n=1
1
n10 = π10
93555
∞
∑
n=1
1
n12 = 691 π12
638512875 With slightly different techniques, Euler proceeded to publish the values of the even-powered sums up to exponent 26 in his 1748 Introductio in Analysin Infinitorum , Volume 1 (Euler, 1748, page 131). It is also worth noting that besides the harmonic series, no one has ever computed the sum for a single odd power! For instance, there has been little progress in understanding the sum of reciprocals of the cubes. The number ∞∑
n=1
1
n3
is known as Ap´ ery’s constant . In 1979, Roger Ap´ ery (1916–1994) published a proof that it is irra-tional , (Ap´ ery, 1979) meaning that it cannot be expressed as a ratio of integers. However, it is still unknown whether or not that value is transcendental ! That is to say, in theory, there could be some polynomial equation with integer coefficients which has Ap´ ery’s constant as a root.
9
Euler did not worry about convergence of this power series or of this infinite product. He never tried to plug in any values for s, so issues of convergence are, in a certain sense, irrelevant! Rather, he is interested in extracting information by studying the coefficients of the power series. This approach has been developed into a theory that we today call generating functions . In this discipline, one treats power series as formal algebraic objects (which in this context get called generating functions) in order to extract information about the coefficients. In Herbert Wilf’s classic text generatingfunctionology (Wilf, 2005), he starts the first page of the first chapter by saying “ A generating function is a clothesline on which we hang up a sequence of numbers for display. ” This is exactly the perspective Euler took here. In this work, he never used the power series for sin( s)
s
to calculate values of that function for numerical values of s.Rather, he expressed the degree two coefficient of that power series in two different ways and equated the expressions.
9Task 11 In light of what you just read about Ap´ ery’s constant, what is ironic about Euler’s opening statement?
References
Ap´ ery, R. (1979). Irrationalit´ e de ζ(2) et ζ(3). Ast´ erisque , 61:11–13. Archimedes (1897). Quadrature of the Parabola . Translated and edited by Thomas L Heath. Cam-bridge University Press. Bernoulli, J. (1713). Ars conjectandi, opus posthumum: accedit Tractatus de seriebus infinitis, et epistola Gallice scripta de ludo pil reticularis. Basile: Impensis Thurnisiorum. Euler, L. (1740). De summis serierum reciprocarum. Commentarii academiae scientiarum Petropoli-tanae , 7:123–134. Euler, L. (1748). Introductio in Analysin Infinitorum, Volume 1. Euler, L. (2005). On the sums of series of reciprocals. Translation by Jordan Bell. arXiv Mathematics e-prints , page math/0506415. Fellman, E. A. (2007). Leonhard Euler . Springer Science and Business Media, Leiden. O’Connor, J. J. and Robertson, E. F. (1998a). Johann and Jacob Bernoulli.
MacTutor History of Mathematics , University of St Andrews, Scotland.
Johann.html ,
Jacob.html .O’Connor, J. J. and Robertson, E. F. (1998b). Leonhard Euler. Mac-Tutor History of Mathematics , University of St Andrews, Scotland.
.O’Connor, J. J. and Robertson, E. F. (2003). Nicole Oresme. Mac-Tutor History of Mathematics , University of St Andrews, Scotland.
.Oresme, N. (c. 1360). Quaestiones super Geometriam Euclidis (Questions concerning Euclid’s Ge-ometry) . Edited with English paraphrase by H. L. L. Busard, Leiden (E. J. Brill), 1961. Wilf, H. (2005). generatingfunctionology, Third Edition . CRC Press, A K Peters, Ltd. 10 Notes to Instructors
PSP Content: Topics and Goals
This PSP is intended to enrich a second-semester Calculus student’s grasp on infinite series and power series manipulations. Often in introductory calculus courses, the only means demonstrated for evaluating an infinite series is to take a known power series (perhaps slightly rearranged) and plug in an x-value from the interval of convergence. This project aims to broaden a student’s perspective on the ways infinite series can be handled and the ways power series can be used. The key Calculus II competencies which come up in this project are as follows:
• Definition of the geometric series and geometric series formula
• Integral test for convergence of an infinite series
• Infinite series defined as the limit of a sequence of partial sums
• The harmonic series/ p-series
• Direct Comparison Test
• Power series of sine
• Algebraic manipulations of power series
• Equating coefficients of corresponding degree in two equal power series (similar to how one solves differential equations via power series)
Student Prerequisites
The student should be familiar with (but maybe has not yet completely mastered) the topics listed above. In particular, the standard series convergence tests and the power series for sine are used heav-ily throughout this project. Nothing else in particular is required beyond the standard prerequisite skills for a second-semester calculus course.
PSP Design, and Task Commentary
Tasks 7 and 8 may seem to be belaboring something trivial. However, from personal experience implementing this project, the author can say that the “1 − x/ root” factorization was consistently the part of this project that the students had the hardest time understanding each semester. The students became much more comfortable with Euler’s factorization after seeing it in the easier case laid out in those tasks.
Suggestions for Classroom Implementation
This would certainly pair well with a discussion of Archimedes’ Quadrature of the Parabola . Showing Archimedes’ classic decomposition of a unit square (shown below) and using it to evaluate the geometric series 14 + 116 + 164 + · · · = 1311 could be a great warmup for this PSP, as Oresme begins by referencing geometric series. (Note that the diagram below demonstrates this summation as a result of the black squares collectively occupying exactly one-third of the area of the large square of area 1.)
Sample Implementation Schedule (based on a 50-minute class period)
If an in-class activity was desired, the following would be a reasonable breakdown:
• 10 minutes: Show image of square above; discuss how this relates to Archimedes’ evaluation of a geometric series.
• 30 minutes: Hand out the PSP and allow students to work in groups on the PSP.
• 10 minutes: Point out a few common errors or misconceptions you noticed as you were walking around helping the student groups. Completed writeup of the PSP can be due the following week. The author, however, does not implement this PSP in that manner. Rather, the author uses this as one of a choice of projects. A small group of students (perhaps 2–3) could have time in class one session to prepare solutions and time in class another session to present solutions in a miniconference. Ideally, other groups of students would be following the same process for different projects. To this end, the author has used the TRIUMPHS mini-PSPs M24 Euler’s Rediscovery of
e by David Ruch and M15 Gaussian Guesswork: Infinite Sequences and the Arithmetic-Geometric Mean by Janet H. Barnett. Copies of these PSPs are available at the TRIUMPHS website (see URL in the acknowledgements). Five further projects (without primary sources) appropriate for further study of power series in a second-semester calculus course are available upon request from the author. 12 The author is happy to provide L ATEX code for this project. It was created using Overleaf which makes it convenient to copy and share projects and can allow instructors to adapt this project in whole or in part as they like for their course.
Recommendations for Further Reading
As an epilogue, instructors may choose to share the current state of the Riemann Hypothesis with their students. In the PSP itself, we end with the mystery surrounding Ap´ ery’s constant to show just how much research is still happening regarding sums of powers of reciprocals. However, showing the students that there is a Clay Mathematics Institute Millenium Prize Problem with a one million dollar bounty on it might be even more exciting to the students. Or, perhaps instructors would prefer to not announce this themselves, but rather direct their students to the site
for a short followup reading assignment. If the student wishes to read another treatment of Euler’s work shown here (along with much exciting work from the Bernoullis and others), William Dunham’s Journey Through Genius: The Great Theorems of Mathematics is recommended. A detailed path from Euler’s work to the Riemann zeta function is given in Mathematical Masterpieces: Further Chronicles by the Explorers by Art Knoebel, Reinhard Laubenbacher, Jerry Lodder, and David Pengelley. Students who are excited by Euler’s clever use of power series may enjoy further study in gener-ating functions. This is often a part of junior or senior level combinatorics classes. Also, the author has led independent study classes on Herbert Wilf’s classic generatingfunctionology to sophomore level students. It is the perfect reader course for a student who does not have much experience with proof-writing but who loved Euler’s manipulations in this PSP and wants more!
Acknowledgments
The development of this project has been partially supported by the Transforming Instruction in Undergraduate Mathematics via Primary Historical Sources (TRIUMPHS) Project with funding from the National Science Foundation’s Improving Undergraduate STEM Education Program under Grants No. 1523494, 1523561, 1523747, 1523753, 1523898, 1524065, and 1524098. Any opinions, findings, and conclusions or recommendations expressed in this project are those of the author and do not necessarily reflect the views of the National Science Foundation.
With the exception of excerpts taken from published translations of the primary sources used in this project and any direct quotes from published secondary sources, this work is licensed under aCreative Commons Attribution-ShareAlike 4.0 International License ( ). It allows re-distribution and re-use of a licensed work on the conditions that the creator is appropriately credited and that any derivative work is made available under “the same, similar or a compatible license.” For more information about the NSF-funded project TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources (TRIUMPHS), visit 13 .14 |
8624 | https://artofproblemsolving.com/wiki/index.php/Probability?srsltid=AfmBOoo_BZmU10fwK7vvGq4rys8l_xWBampETRlMH-0Ete-FTthQ8YKC | Art of Problem Solving
Probability - AoPS Wiki
Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical CampusVisit the Virtual Campus
Sign In
Register
online school
Class ScheduleRecommendationsOlympiad CoursesFree Sessions
books tore
AoPS CurriculumBeast AcademyOnline BooksRecommendationsOther Books & GearAll ProductsGift Certificates
community
ForumsContestsSearchHelp
resources
math training & toolsAlcumusVideosFor the Win!MATHCOUNTS TrainerAoPS Practice ContestsAoPS WikiLaTeX TeXeRMIT PRIMES/CrowdMathKeep LearningAll Ten
contests on aopsPractice Math ContestsUSABO
newsAoPS BlogWebinars
view all 0
Sign In
Register
AoPS Wiki
ResourcesAops Wiki Probability
Page
ArticleDiscussionView sourceHistory
Toolbox
Recent changesRandom pageHelpWhat links hereSpecial pages
Search
Probability
Probability is traditionally considered one of the most difficult areas of mathematics, since probabilistic arguments often come up with apparently paradoxical or counterintuitive results. Examples include the Monty Hall paradox and the birthday problem. Probability can be loosely defined as the chance that an event will happen.
Contents
1 Video For Beginners!
2 Introductory Probability
3 Formal Definition of Probability
3.1 Types of Probability
3.2 Example Problems
3.2.1 Introductory
3.2.2 Intermediate
4 Resources
Video For Beginners!
Introductory Probability
Before reading about the following topics, a student learning about probability should learn about introductory counting techniques.
dependent probability
independent probability
Formal Definition of Probability
The foundations of probability reside in an area of analysis known as measure theory. Measure theory in general deals with integration, in particular, how to define and extend the notion of "area" or "volume." Intuitively, therefore, probability could be said to consider how much "volume" an event takes up in a space of outcomes. Measure theory does assume considerable mathematical maturity, so it is usually ignored until one reaches an advanced undergraduate level. Once measure theory is covered, probability does become a lot easier to use and understand.
In the language of measure theory, probability is formally defined as a triple known as a probability space, denoted . Here is a set called the sample space, and is a class of events given by certain subsets of . must satisfy certain properties (it must be a -algebra) to qualify as a class of events. Together, and form what is known as a measurable space, . is an assignment with certain properties (it is a special kind of measure), called the probability function, or probability measure. It assigns a "volume" to each possible event.
As a simple example, consider a single coin-toss. In this case, , , and assigns the following probabilities to events in : ,
,
,
.
We can interpret this as saying that the event of getting Heads, and the event of getting Tails, each take up an equal half of the set of possible outcomes; the event of getting Heads or Tails is certain, and likewise the event of getting neither Heads nor Tails has probability 0.
Of course, to understand this example doesn't need measure theory, but it does show how to translate a very basic situation into measure-theoretic language. Furthermore, if one wanted to determine whether the coin was fair or weighted, it would be difficult to do that without using inferential methods derived from measure theory.
Types of Probability
Part of a comprehensive understanding of basic probability includes an understanding of the differences between different kinds of probability problems.
algebraic probability
combinatorial probability problems involve counting outcomes.
geometric probability
Important subdivisions of probability include
stochastic processes
mathematical statistics
Example Problems
Introductory
2006 AMC 10B Problem 17
2006 AMC 10B Problem 21
Intermediate
2006 AIME II Problem 5
2007 AIME II Problem 10
Resources
Introduction to Counting and Probability by David Patrick
Intermediate Counting and Probability by David Patrick
Retrieved from "
Categories:
Combinatorics
Mathematics
Probability
Art of Problem Solving is an
ACS WASC Accredited School
aops programs
AoPS Online
Beast Academy
AoPS Academy
About
About AoPS
Our Team
Our History
Jobs
AoPS Blog
Site Info
Terms
Privacy
Contact Us
follow us
Subscribe for news and updates
© 2025 AoPS Incorporated
© 2025 Art of Problem Solving
About Us•Contact Us•Terms•Privacy
Copyright © 2025 Art of Problem Solving
Something appears to not have loaded correctly.
Click to refresh. |
8625 | https://study.com/academy/lesson/discriminant-definition-lesson-quiz.html | Discriminant Formula, Rules & Solutions - Lesson | Study.com
Log In
Sign Up
Menu
Plans
Courses
By Subject
College Courses
High School Courses
Middle School Courses
Elementary School Courses
By Subject
Arts
Business
Computer Science
Education & Teaching
English (ELA)
Foreign Language
Health & Medicine
History
Humanities
Math
Psychology
Science
Social Science
Subjects
Art
Business
Computer Science
Education & Teaching
English
Health & Medicine
History
Humanities
Math
Psychology
Science
Social Science
Art
Architecture
Art History
Design
Performing Arts
Visual Arts
Business
Accounting
Business Administration
Business Communication
Business Ethics
Business Intelligence
Business Law
Economics
Finance
Healthcare Administration
Human Resources
Information Technology
International Business
Operations Management
Real Estate
Sales & Marketing
Computer Science
Computer Engineering
Computer Programming
Cybersecurity
Data Science
Software
Education & Teaching
Education Law & Policy
Pedagogy & Teaching Strategies
Special & Specialized Education
Student Support in Education
Teaching English Language Learners
English
Grammar
Literature
Public Speaking
Reading
Vocabulary
Writing & Composition
Health & Medicine
Counseling & Therapy
Health
Medicine
Nursing
Nutrition
History
US History
World History
Humanities
Communication
Ethics
Foreign Languages
Philosophy
Religious Studies
Math
Algebra
Basic Math
Calculus
Geometry
Statistics
Trigonometry
Psychology
Clinical & Abnormal Psychology
Cognitive Science
Developmental Psychology
Educational Psychology
Organizational Psychology
Social Psychology
Science
Anatomy & Physiology
Astronomy
Biology
Chemistry
Earth Science
Engineering
Environmental Science
Physics
Scientific Research
Social Science
Anthropology
Criminal Justice
Geography
Law
Linguistics
Political Science
Sociology
Teachers
Teacher Certification
Teaching Resources and Curriculum
Skills Practice
Lesson Plans
Teacher Professional Development
For schools & districts
Certifications
Teacher Certification Exams
Nursing Exams
Real Estate Exams
Military Exams
Finance Exams
Human Resources Exams
Counseling & Social Work Exams
Allied Health & Medicine Exams
All Test Prep
Teacher Certification Exams
Praxis Test Prep
FTCE Test Prep
TExES Test Prep
CSET & CBEST Test Prep
All Teacher Certification Test Prep
Nursing Exams
NCLEX Test Prep
TEAS Test Prep
HESI Test Prep
All Nursing Test Prep
Real Estate Exams
Real Estate Sales
Real Estate Brokers
Real Estate Appraisals
All Real Estate Test Prep
Military Exams
ASVAB Test Prep
AFOQT Test Prep
All Military Test Prep
Finance Exams
SIE Test Prep
Series 6 Test Prep
Series 65 Test Prep
Series 66 Test Prep
Series 7 Test Prep
CPP Test Prep
CMA Test Prep
All Finance Test Prep
Human Resources Exams
SHRM Test Prep
PHR Test Prep
aPHR Test Prep
PHRi Test Prep
SPHR Test Prep
All HR Test Prep
Counseling & Social Work Exams
NCE Test Prep
NCMHCE Test Prep
CPCE Test Prep
ASWB Test Prep
CRC Test Prep
All Counseling & Social Work Test Prep
Allied Health & Medicine Exams
ASCP Test Prep
CNA Test Prep
CNS Test Prep
All Medical Test Prep
College Degrees
College Credit Courses
Partner Schools
Success Stories
Earn credit
Sign Up
Copyright
Math Courses / SAT Mathematics Level 2: Help and Review Course
Discriminant Formula, Rules & Solutions
Lesson
Transcript
Riley Kench, Yuanxin (Amy) Yang Alcocer
Author Riley Kench
Riley has tutored collegiate mathematics for seven years. They have a Master of Arts degree in Mathematics from Central Michigan University and a Bachelor of Science degree in Mathematics from Central Michigan University.
View bio
Instructor Yuanxin (Amy) Yang Alcocer
Amy has a master's degree in secondary education and has been teaching math for over 9 years. Amy has worked with students at all levels from those with special needs to those that are gifted.
View bio
What is the discriminant? In this lesson, learn the formula for how to find the discriminant of a quadratic equation. Moreover, learn about the quadratic formula, its discriminant, and the different discriminant rules. Updated: 11/21/2023
Table of Contents
What is the Discriminant
Discriminant of a Quadratic Equation
Discriminant solutions
Lesson Summary
Show
Frequently Asked Questions
How do you find discriminant and roots?
To find the discriminant given the quadratic equation f(x)=ax^2+bx+c, simply record the values of a, b, and c and then substitute them into the discriminant formula:
d=b^2-4ac. This will give the value of the discriminant. This also tells the number of roots and whether or not the roots are real or imaginary.
What is the discriminant formula?
The discriminant formula is d=b^2-4ac given the equation of the quadratic is f(x)=ax^2+bx+c. The formula derives from the quadratic formula. Moreover, the discriminant tells how many solutions to the equation there are as well as if the solutions are real or imaginary.
What is the discriminant in a quadratic equation?
The discriminant of a quadratic equation is b^2-4ac if the equation is f(x)=ax^2+bx+c. It tells if the solutions to the quadratic equation are real number or imaginary numbers as well as how many solutions to the equation exist.
Create an account
Table of Contents
What is the Discriminant
Discriminant of a Quadratic Equation
Discriminant solutions
Lesson Summary
Show
What is the Discriminant ------------------------
In mathematics, a quadratic equation is an equation of the form f(x)=a x 2+b x+c. In this lesson, only quadratic equations will be considered. Now, the quadratic formula, which is:
x=−b±b 2−4 a c 2 a
The quadratic formula tells the solutions or roots of a quadratic equation. However, it can also inform on the nature of the solutions themselves. The discriminant is the b 2−4 a c term of the quadratic formula. The discriminant also shows if the solutions are real numbers or complex numbers as well as how many solutions exist. Here are some examples of discriminant calculations:
f(x)=x 2−2 x+3 has discriminant of b 2−4 a c=(−2)2−4(1)(3)=4−12=−8.
f(x)=2 x 2+2 x−4 has discriminant of b 2−4 a c=(2)2−4(2)(−4)=4+32=36.
f(x)=x 2−1 has discriminant of b 2−4 a c=(0)2−4(1)(−1)=4.
Click for sound
4:12
You must c C reate an account to continue watching
Register to view this lesson
Are you a student or a teacher?
I am a student I am a teacher
Create Your Account To Continue Watching
As a member, you'll also get unlimited access to over 88,000 lessons in math, English, science, history, and more. Plus, get practice tests, quizzes, and personalized coaching to help you succeed.
Get unlimited access to over 88,000 lessons.
Try it now
Try it now.
Already registered? Log in here for access
Back
Resources created by teachers for teachers
Over 30,000 video lessons& teaching resources‐all in one place.
Video lessons
Quizzes & Worksheets
Classroom Integration
Lesson Plans
I would definitely recommend Study.com to my colleagues. It’s like a teacher waved a magic wand and did the work for me. I feel like it’s a lifeline.
Jennifer B.
Teacher
Try it now
Back
Coming up next: Principal Square Root | Definition, Calculation & Examples
You're on a roll. Keep up the good work!
Take QuizWatch Next Lesson
Replay
Just checking in. Are you still watching?
Yes! Keep playing.
Your next lesson will play in 10 seconds
0:04 Definition
0:37 Using the Discriminant
1:14 Examples
3:27 Lesson Summary
View Video Only
Save
Timeline
54K views
Video
Quiz
Course
Video Only
54K views
Discriminant of a Quadratic Equation ------------------------------------
A quadratic equation is an equation of the form f(x)=a x 2+b x+c where a, b, and c are real numbers. Here are some examples of quadratic equations:
f(x)=x 2
f(x)=−3 x 2+2 x+3
f(x)=2 x 2+x+1
The discriminant of a quadratic equation is the value b 2−4 a c. This arises from the quadratic formula. It is the term under the square root in the quadratic formula:
x=−b±b 2−4 a c 2 a.
This leads to the quadratic discriminant formula: Δ=b 2−4 a c. Here are some examples of discriminant calculations of the above quadratic equations:
f(x)=x 2 has discriminant of Δ=(0)2−4(1)(0)=0.
f(x)=−3 x 2+2 x+3 has discriminant of Δ=(2)2−4(−3)(3)=4+36=40.
f(x)=2 x 2+x+1 has discriminant of Δ=(1)2−4(2)(1)=1−8=−7.
The discriminant of these quadratic equations tells if the solutions are real numbers or complex numbers and how many solutions exist. However, this formula will only work for quadratic equations. There is a formula for the discriminant of a cubic equation. A cubic equation is an equation of the form f(x)=a x 3+b x 2+c x+d. The formula for the discriminant of a cubic equation is Δ=18 a b c d−4 b 3 d+b 2 c 2−4 a c 3−27 a 2 d 2. Here is an example of this cubic discriminant formula:
f(x)=x 3+x 2+x+1 has discriminant of Δ=18(1)(1)(1)(1)−4(1)3(1)+(1)2(1)2−4(1)(1)3−27(1)2(1)2=18−4+1−4−27=19−35=−16.
How to find the Discriminant
Now, many examples of discriminant calculations were already given but how does one go about calculating the discriminant of a quadratic equation? Here is an example:
Consider the quadratic equation f(x)=x 2−2 x+1. The values of a, b, and c are a=1,b=−2,and c=1. Using the formula for the discriminant Δ=b 2−4 a c, the discriminant of this equation is Δ=(−2)2−4(1)(1)=4−4=0.
The method follows these steps:
Identify the values of a, b, and c.
Substitute these values into the discriminant formula Δ=b 2−4 a c which gives the discriminant.
Here are some more examples:
f(x)=2 x 2−3 has a, b, and c values of a=2,b=0,and c=−3. Substituting these into the discriminant formula gives Δ=(0)2−4(2)(−3)=0−24=−24.
f(x)=−3 x 2+2 x has a, b, and c values of a=−3,b=2,and c=0. Substituting these into the discriminant formula yields Δ=(2)2−4(−3)(0)=4−0=4.
Discriminant solutions ----------------------
The discriminant of a quadratic equation tells if the solutions of the quadratic equation are real numbers or complex numbers as well as how many solutions exist for each. Here are the rules for the discriminant: suppose that f(x) is a quadratic equation, then:
If Δ>0, then f(x) has two real solutions.
If Δ=0, then f(x) has one real solution.
If Δ<0, then f(x) has two complex (imaginary) solutions.
To show how this is true, consider the quadratic formula which tells the exact solutions of a quadratic equation:
f(x)=−b±b 2−4 a c 2 a
If the discriminant is greater than zero, then the square root will exist. This leads to -b adding and subtracting a real value which leads to two solutions. Now, if the discriminant is zero, the square root term is zero which leads to the single solution of −b 2 a. Finally, if the discriminant is less than zero then the square root term becomes the square root of a negative number. This then leads the formula to be -b adding and subtracting a complex number which gives two complex (imaginary) solutions.
Here are some more examples:
f(x)=x 2 has discriminant of zero since a=1, b=0, and c=0. This shows that f(x) has exactly one real solution. Here is a graph to demonstrate this:
Quadratic Equation with one Solution
Notice that in this graph, the curve only "crosses" the x-axis at one point: x=0. This shows that the discriminant gives the correct number of solutions for the equation.
f(x)=x 2+x−2 has discriminant of Δ=(1)2−4(1)(−2)=1+8=9. So, this shows that the quadratic equation has two real solutions. Here is another graph:
Graph of Equation with two Solutions
Notice that the curve crosses the x-axis at exactly two points. This is again verification that the discriminant gives the correct number and type of solutions.
Positive Discriminant
The discriminant is the value Δ=b 2−4 a c of the quadratic formula: x=−b±b 2−4 a c 2 a. If the discriminant is positive, then the quadratic formula gives two distinct real solutions (-b is adding and subtracting a real value which leads to two distinct real solutions). Here is an example:
f(x)=−x 2+2 x+3 has discriminant of Δ=(2)2−4(−1)(3)=4+12=16 and 16>0 which means this equation has exactly two distinct real roots. Here is its graph:
Graph of Equation with Positive Discriminant
Notice that the curve crosses the x-axis at two points which means it has two distinct real solutions which are x=3 and x=-1.
Negative Discriminant
If the discriminant is negative, the square root value in the quadratic formula becomes the square root of a negative number which is imaginary. Therefore, the formula gives two distinct complex solutions (imaginary solutions). The quadratic formula becomes -b adding and subtracting a complex number which gives two complex solutions and zero real solutions. Here is an example:
f(x)=x 2+1 has discriminant of Δ=(0)2−4(1)(1)=−4 and −4<0 which means this equation has two distinct complex solutions and no real solutions. Here is its graph:
Graph of Equation with Complex Solutions
Notice that the curve does not cross the x-axis at any point. Therefore, it has two distinct complex solutions. In this case, the solutions are x=i and x=-i.
Zero Discriminant
If the discriminant is zero, then the quadratic formula becomes −b 2 a which gives one real solution. Graphically, this means the graph of the quadratic equation crosses the x-axis at exactly one point. However, since the graph of a quadratic equation is a parabola, it can only cross the x-axis at its vertex. Here is an example:
f(x)=x 2+2 x+1 has discriminant of Δ=(2)2−4(1)(1)=4−4=0. Since this equation has a discriminant of zero, it has exactly one real solution. Here is its graph:
Graph of Equation with one Real Solution
Notice that the curve only touches the x-axis at one point: x=-1 which means it only has one real solution.
Lesson Summary --------------
In this lesson, the idea of the discriminant was explored in relation to quadratic equations. A quadratic equation is an equation of the form f(x)=a x 2+b x+c. The quadratic formula gives the solutions to a quadratic equation. The portion under the square root of the quadratic formula is called the discriminant. Its formula is Δ=b 2−4 a c. The discriminant formula gives the number of solutions of the quadratic equation as well as tells if the solutions are real numbers or complex numbers.
Finally, the rules for the discriminant were explained and many examples of these rules were shown. There were also many different discriminant calculations shown in this lesson. Here are the highlights from this lesson:
A quadratic equation is any equation of the form f(x)=a x 2+b x+c.
The quadratic formula: f(x)=−b±b 2−4 a c 2 a, gives the number of solutions to a quadratic equation.
The discriminant is the term Δ=b 2−4 a c. It gives the number of solutions as well as if the solutions are real numbers or complex numbers.
If Δ>0, then the quadratic equation has exactly two distinct real solutions.
If Δ=0, then the quadratic equation has exactly one real solution.
If Δ<0, then the quadratic equation has exactly two distinct complex solutions and no real solutions.
Video Transcript
Definition
The discriminant is easy to find when you look at the quadratic formula. The quadratic formula is the equation you use to find the solutions to quadratic equations. It looks like this:
The discriminant is the part inside the square root and tells us how many real solutions a quadratic equation has. Take a moment to look for the square root and find what is inside the square root. Once you see it, you will have found the discriminant. If we isolate that part, we get the formula for finding the discriminant, which is:
Using the Discriminant
The discriminant tells you how many possible solutions a particular quadratic equation has. Before we can use the quadratic equation, though, we first have to change it to standard form. Standard form is when all the variables and constants are on one side of the equation, and the other side is a zero. It looks like this:
Once you have your quadratic equation in standard form, you can label your numbers with the appropriate letters and plug the values into your formula for finding the discriminant. The result of your discriminant tells you how many solutions your quadratic has.
Examples
Let's look at an example:
In our example, our quadratic equation gives us 1 for our letter a, 5 for our letter b, and 4 for our letter c. We take these values and plug them into their appropriate places in the discriminant formula, and we'll find that our discriminant equals 9. That's a positive number, which means that our quadratic equation has two possible real solutions. Real solutions are solutions that can be calculated using the quadratic formula. When you graph this quadratic equation, you will see that the curve crosses the x-axis in two places, exactly where your solutions are:
While the discriminant tells us the number of possible solutions, it does not tell us what those solutions are, but it does give us a clue as to how many solutions we need to look for.
Remember that, when there are no numbers in front of variables, it is assumed that there is a 1 in front. We don't write the 1 because it is a mathematical convention and because it looks neater, especially when you have a lot of letters to work with.
Let's look at another example:
We have labeled our letters with their appropriate values: a = 2, b = -1, and c = 4. After plugging the appropriate values into our discriminant formula, we find that our discriminant is -31, a negative number. Hmm… what could this mean? When the discriminant is negative, it means that there are no real solutions. What this means is that, when you graph the equation, you will see that it never crosses the x-axis and therefore has no real solutions.
There is one other possible situation: when the discriminant equals 0. When you see this, it means that there is only one possible real solution. When graphed, the equation will touch the x-axis only at one point.
Here is a chart to help you remember the possible discriminant situations and what they mean:
| Discriminant | # of Solutions |
--- |
| > 0 | Two real solutions |
| = 0 | One real solution |
| < 0 | No real solutions |
If the discriminant is greater than 0, there are two real solutions. If the discriminant is equal to 0, there is one real solution, and if the discriminant is less than 0, there are no real solutions.
Lesson Summary
To summarize, the discriminant helps you by telling you how many possible solutions a quadratic equation has. The formula can be found by looking for the square root symbol in the quadratic formula, the equation you use to find the solutions to quadratic equations. In order to use a discriminant, we need to first put our equation into standard form, when all the variables and constants are on one side of the equation, and the other side is a zero. There are three possible scenarios. If the discriminant is a positive number, then there are two real solutions. If the discriminant equals 0, then there is only one real solution. If the discriminant is a negative number, then there are no real solutions.
Register to view this lesson
Are you a student or a teacher?
I am a student I am a teacher
Unlock Your Education
See for yourself why 30 million people use Study.com
Become a Study.com member and start learning now.
Become a Member
Already a member? Log In
Back
Resources created by teachers for teachers
Over 30,000 video lessons& teaching resources‐all in one place.
Video lessons
Quizzes & Worksheets
Classroom Integration
Lesson Plans
I would definitely recommend Study.com to my colleagues. It’s like a teacher waved a magic wand and did the work for me. I feel like it’s a lifeline.
Jennifer B.
Teacher
Try it now
Back
Recommended Lessons and Courses for You
Related Lessons
Related Courses
Recommended Lessons for You
Solving Problems using the Quadratic Formula
Using the Quadratic Formula to Solve Equations with Literal Coefficients
Methods to Solve Quadratic Equations | Types & Examples
Finding Complex Roots of Quadratic Equations
Completing the Square Practice Problems
How to Complete the Square | Method & Examples
How to Solve Quadratics That Are Not in Standard Form
How to Solve a Quadratic Equation by Factoring
Quadratic Function | Formula, Equations & Examples
Solving Quadratic Inequalities Using Two Binomials
Zero Product Property | Definition, Formula & Examples
Quadratic Formula Definition, Equation & Proof
B-Value: Definition & Explanation
Solutions (Min & Max) | EP SAT Tutoring
Roots of a Quadratic Equation | Overview, Function & Formula
Complex vs. Irrational Roots | Graph & Formula
How to Solve a Quadratic Equation by Graphing
Zeros, Roots & X-Intercepts | Properties & Examples
Vertex Form | Equation, Formula & Conversion
How to Find the Vertex of a Parabola | Quadratic Equation
Related Courses
ELM: CSU Math Study Guide
SAT Subject Test Mathematics Level 2: Practice and Study Guide
GED Math: Quantitative, Arithmetic & Algebraic Problem Solving
GED Social Studies: Civics & Government, US History, Economics, Geography & World
GED Science: Life, Physical and Chemical
AP Calculus AB & BC: Exam Prep
Study.com SAT Study Guide and Test Prep
Common Core Math - Statistics & Probability: High School Standards
SAT Subject Test US History: Practice and Study Guide
SAT Subject Test Chemistry: Practice and Study Guide
GMAT Study Guide and Test Prep
SAT Subject Test World History: Practice and Study Guide
SAT Subject Test Literature: Practice and Study Guide
SAT Subject Test Biology: Practice and Study Guide
SAT Subject Test Mathematics Level 1: Practice and Study Guide
NY Regents Exam - Integrated Algebra: Test Prep & Practice
Trigonometry: High School
Supplemental Math: Study Aid
Holt Geometry: Online Textbook Help
Glencoe Geometry: Online Textbook Help
Start today. Try it now
SAT Mathematics Level 2: Help and Review
22 chapters | 225 lessons
Ch 1. Basic Math Review: Help and...
Ch 2. Square Roots: Help and Review
Evaluating Square Roots of Perfect Squares 5:12
Estimating Square Roots | Overview & Examples 5:10
Simplifying Square Roots When not a Perfect Square 4:45
Simplifying Square Root Expressions | Steps & Examples 7:03
Discriminant Formula, Rules & Solutions 4:12
5:02 Next Lesson Principal Square Root | Definition, Calculation & Examples
Square Root | Definition, Sign & Problems 10:15
Square Root Property | Overview, Formula & Examples 6:49
Square Root of a Number | Adding, Multiplying & Simplifying
Subtracting Square Roots 3:44
Ch 3. Basic Algebraic Expressions in Math:...
Ch 4. Exponents in Math: Help and...
Ch 5. Linear Equations & Inequalities in...
Ch 6. Absolute Value Equations &...
Ch 7. Polynomials in Algebra: Help and...
Ch 8. Functions in Math: Help and...
Ch 9. Rational Expressions in Algebra: Help...
Ch 10. Matrices: Help and Review
Ch 11. Sequences and Series in Math: Help and...
Ch 12. Exponentials and Logarithms in Math:...
Ch 13. Complex Numbers in Math: Help and...
Ch 14. Coordinate Geometry Basics: Help and...
Ch 15. Graphing Linear Equations &...
Ch 16. Advanced Coordinate Geometry: Help and...
Ch 17. Three-Dimensional Geometry: Help and...
Ch 18. Overview of Trigonometry: Help and...
Ch 19. Probability, Combinations &...
Ch 20. Working with Data & Statistics: Help...
Ch 21. Logic: Help and Review
Ch 22. Limits in Math: Help and Review
Discriminant Formula, Rules & Solutions Related Study Materials
Related Lessons
Solving Problems using the Quadratic Formula
Using the Quadratic Formula to Solve Equations with Literal Coefficients
Methods to Solve Quadratic Equations | Types & Examples
Finding Complex Roots of Quadratic Equations
Completing the Square Practice Problems
How to Complete the Square | Method & Examples
How to Solve Quadratics That Are Not in Standard Form
How to Solve a Quadratic Equation by Factoring
Quadratic Function | Formula, Equations & Examples
Solving Quadratic Inequalities Using Two Binomials
Zero Product Property | Definition, Formula & Examples
Quadratic Formula Definition, Equation & Proof
B-Value: Definition & Explanation
Solutions (Min & Max) | EP SAT Tutoring
Roots of a Quadratic Equation | Overview, Function & Formula
Complex vs. Irrational Roots | Graph & Formula
How to Solve a Quadratic Equation by Graphing
Zeros, Roots & X-Intercepts | Properties & Examples
Vertex Form | Equation, Formula & Conversion
How to Find the Vertex of a Parabola | Quadratic Equation
Related Courses
ELM: CSU Math Study Guide
SAT Subject Test Mathematics Level 2: Practice and Study Guide
GED Math: Quantitative, Arithmetic & Algebraic Problem Solving
GED Social Studies: Civics & Government, US History, Economics, Geography & World
GED Science: Life, Physical and Chemical
AP Calculus AB & BC: Exam Prep
Study.com SAT Study Guide and Test Prep
Common Core Math - Statistics & Probability: High School Standards
SAT Subject Test US History: Practice and Study Guide
SAT Subject Test Chemistry: Practice and Study Guide
GMAT Study Guide and Test Prep
SAT Subject Test World History: Practice and Study Guide
SAT Subject Test Literature: Practice and Study Guide
SAT Subject Test Biology: Practice and Study Guide
SAT Subject Test Mathematics Level 1: Practice and Study Guide
NY Regents Exam - Integrated Algebra: Test Prep & Practice
Trigonometry: High School
Supplemental Math: Study Aid
Holt Geometry: Online Textbook Help
Glencoe Geometry: Online Textbook Help
Related Topics
Browse by Courses
SAT Subject Test Mathematics Level 1: Practice and Study Guide
SAT Subject Test Mathematics Level 2: Practice and Study Guide
NY Regents Exam - Integrated Algebra: Test Prep & Practice
CSET Social Science Subtest I (114) Study Guide and Test Prep
CSET English Subtests I & III (105 & 107): Practice & Study Guide
Glencoe Math Course: Online Textbook Help
Accuplacer Math: Quantitative Reasoning, Algebra, and Statistics Placement Study Guide and Test Prep
ILTS Health Education (211) Study Guide and Test Prep
Study.com ACT® English Test Section: Prep & Practice
McDougal Littell Algebra 2: Online Textbook Help
Glencoe Pre-Algebra: Online Textbook Help
GRE Biology: Study Guide & Test Prep
Study.com ACT® Math Test Section: Review & Practice
Study.com ACT® Reading Test Section: Prep & Practice
Study.com ACT® Science Test Section: Prep & Practice
Browse by Lessons
Identity of Two Quadratic Expressions
Completing the Square Lesson Plan
Solving & Graphing Quadratic Equations Lesson Plan
Lead Coefficients of Completing the Square
How to Identify Lead Coefficients in Quadratic Equations
How to Use the Difference of Two Squares Theorem to Solve Quadratic Equations
What Is a Quadratic Equation? - Lesson Plan
Solving by Completing the Square | Definition, Steps & Examples
Quadratic Equations in Real Life | Overview, Uses & Examples
Create an account to start this course today
Used by over 30 million students worldwide
Create an account
Like this lessonShare
Explore our library of over 88,000 lessons
Search
Browse
Browse by subject
College Courses
Business
English
Foreign Language
History
Humanities
Math
Science
Social Science
See All College Courses
High School Courses
AP
Common Core
GED
High School
See All High School Courses
Other Courses
College & Career Guidance Courses
College Placement Exams
Entrance Exams
General Test Prep
K-8 Courses
Skills Courses
Teacher Certification Exams
See All Other Courses
Upgrade to enroll×
Upgrade to Premium to enroll in SAT Mathematics Level 2: Help and Review
Enrolling in a course lets you earn progress by passing quizzes and exams.
Track course progress
Take quizzes and exams
Earn certificates of completion
You will also be able to:
Create a Goal
Create custom courses
Get your questions answered
Upgrade to Premium to add all these features to your account!
Upgrade Now
Upgrade to Premium to add all these features to your account!
Download the app
Plans
Student Solutions
Teacher Solutions
Study.com for Schools
Working Scholars® Solutions
Online tutoring
About Us
Blog
Careers
Teach For Us
Press Center
Ambassador
Scholarships
Support
FAQ
Site Feedback
Download the app
Working Scholars® Bringing Tuition-Free College to the Community
© Copyright 2025 Study.com. All other trademarks and copyrights are the property of their respective owners. All rights reserved.
Terms of UsePrivacy PolicyDMCA NoticeADA ComplianceHonor Code For Students |
8626 | https://study.com/academy/lesson/y-intercept-definition-lesson-quiz.html | Y-Intercept | Definition, Formula & Examples - Lesson | Study.com
Log In
Sign Up
Menu
Plans
Courses
By Subject
College Courses
High School Courses
Middle School Courses
Elementary School Courses
By Subject
Arts
Business
Computer Science
Education & Teaching
English (ELA)
Foreign Language
Health & Medicine
History
Humanities
Math
Psychology
Science
Social Science
Subjects
Art
Business
Computer Science
Education & Teaching
English
Health & Medicine
History
Humanities
Math
Psychology
Science
Social Science
Art
Architecture
Art History
Design
Performing Arts
Visual Arts
Business
Accounting
Business Administration
Business Communication
Business Ethics
Business Intelligence
Business Law
Economics
Finance
Healthcare Administration
Human Resources
Information Technology
International Business
Operations Management
Real Estate
Sales & Marketing
Computer Science
Computer Engineering
Computer Programming
Cybersecurity
Data Science
Software
Education & Teaching
Education Law & Policy
Pedagogy & Teaching Strategies
Special & Specialized Education
Student Support in Education
Teaching English Language Learners
English
Grammar
Literature
Public Speaking
Reading
Vocabulary
Writing & Composition
Health & Medicine
Counseling & Therapy
Health
Medicine
Nursing
Nutrition
History
US History
World History
Humanities
Communication
Ethics
Foreign Languages
Philosophy
Religious Studies
Math
Algebra
Basic Math
Calculus
Geometry
Statistics
Trigonometry
Psychology
Clinical & Abnormal Psychology
Cognitive Science
Developmental Psychology
Educational Psychology
Organizational Psychology
Social Psychology
Science
Anatomy & Physiology
Astronomy
Biology
Chemistry
Earth Science
Engineering
Environmental Science
Physics
Scientific Research
Social Science
Anthropology
Criminal Justice
Geography
Law
Linguistics
Political Science
Sociology
Teachers
Teacher Certification
Teaching Resources and Curriculum
Skills Practice
Lesson Plans
Teacher Professional Development
For schools & districts
Certifications
Teacher Certification Exams
Nursing Exams
Real Estate Exams
Military Exams
Finance Exams
Human Resources Exams
Counseling & Social Work Exams
Allied Health & Medicine Exams
All Test Prep
Teacher Certification Exams
Praxis Test Prep
FTCE Test Prep
TExES Test Prep
CSET & CBEST Test Prep
All Teacher Certification Test Prep
Nursing Exams
NCLEX Test Prep
TEAS Test Prep
HESI Test Prep
All Nursing Test Prep
Real Estate Exams
Real Estate Sales
Real Estate Brokers
Real Estate Appraisals
All Real Estate Test Prep
Military Exams
ASVAB Test Prep
AFOQT Test Prep
All Military Test Prep
Finance Exams
SIE Test Prep
Series 6 Test Prep
Series 65 Test Prep
Series 66 Test Prep
Series 7 Test Prep
CPP Test Prep
CMA Test Prep
All Finance Test Prep
Human Resources Exams
SHRM Test Prep
PHR Test Prep
aPHR Test Prep
PHRi Test Prep
SPHR Test Prep
All HR Test Prep
Counseling & Social Work Exams
NCE Test Prep
NCMHCE Test Prep
CPCE Test Prep
ASWB Test Prep
CRC Test Prep
All Counseling & Social Work Test Prep
Allied Health & Medicine Exams
ASCP Test Prep
CNA Test Prep
CNS Test Prep
All Medical Test Prep
College Degrees
College Credit Courses
Partner Schools
Success Stories
Earn credit
Sign Up
Business Courses
GMAT Study Guide and Test Prep
Y-Intercept | Definition, Formula & Examples
Contributors: Laura Foist, Kimberlee Davison
Author
Author:
Laura Foist
Show more
Instructor
Instructor:
Kimberlee Davison
Show more
What is the y-intercept? Learn the definition and formula of y-intercept or vertical intercept. Learn how to find the y-intercept of functions and graphs with examples. Updated: 11/21/2023
Table of Contents
What is the Y-Intercept?
How to Find the Y-Intercept
Lesson Summary
Show
FAQ
What is the definition of y-intercept?
The y-intercept definition is where the line of a graph crosses the y-axis, or what y is equal to when x is equal to 0.
What is the y-intercept in an equation?
In an equation, the y-intercept is what y is equal to when x is 0. In a standard equation for a straight line such as y=mx+b, the y-intercept is equal to b.
How do you find the Y-intercept?
The y-intercept can be found by estimating where the graph crosses the y-axis, or by setting x to 0 in the equation and solving for y.
Create an account
Lesson Transcript
VideoQuizCourse
Click for sound
2:58
You must c C reate an account to continue watching
Register to view this lesson
Are you a student or a teacher?
I am a student I am a teacher
Create Your Account To Continue Watching
As a member, you'll also get unlimited access to over 88,000 lessons in math, English, science, history, and more. Plus, get practice tests, quizzes, and personalized coaching to help you succeed.
Get unlimited access to over 88,000 lessons.
Try it now
Already registered? Log in here for access
Go back
Resources created by teachers for teachers
Over 30,000 video lessons & teaching resources—all in one place.
Video lessons
Quizzes and worksheets
Classroom integration
Lesson plans
I would definitely recommend Study.com to my colleagues. It’s like a teacher waved a magic wand and did the work for me. I feel like it’s a lifeline.
Jennifer B.
Teacher
Try it now
Go back
Coming up next: Zero Slope | Graph, Equation & Examples
You're on a roll. Keep up the good work!
Take QuizWatch Next Lesson
Replay
Just checking in. Are you still watching?
Yes! Keep playing.
Your next lesson will play in 10 seconds
0:04 Definition of y-intercept
0:16 y-intercepts & Graphs
1:59 y-intercepts & Curves
2:30 Lesson Summary
QuizCourseView Video Only
Save
Timeline
24K views
Recommended lessons and courses for you
Related Lessons Related Courses
##### Horizontal Line Test | Overview & Function 3:28
##### Linear Model | Equation & Examples 11:00
##### Slope of a Line | Meaning, Calculation & Examples 6:19
##### Direct Variation | Definition, Examples & Graph 7:30
##### Linear Relationship | Definition & Examples 6:29
##### Interest & Rate of Change - Rate of Change: Study.com SAT® Math Exam Prep 10:24
##### Linear Equations | EP SAT Tutoring 25:04
##### Rate of Change vs. Negative Rate of Change 4:22
##### Average Rate of Change | Definition, Formula & Examples 6:59
##### Nonlinear & Linear Graphs Functions | How to Tell if a Function is Linear 4:23
##### ILTS TAP - Test of Academic Proficiency (400): Practice & Study Guide
##### NYSTCE English Language Arts (003) Study Guide and Test Prep
##### CSET English Subtest IV (108) Study Guide and Test Prep
##### FTCE Middle Grades English 5-9 (014) Study Guide and Test Prep
##### SAT Subject Test Literature: Practice and Study Guide
##### CBEST Writing Study Guide and Test Prep
##### Praxis 5195 Study Guide - Spanish: World Language Exam Prep
##### NMTA Essential Academic Skills Subtest Writing (002): Practice & Study Guide
##### OAE Early Childhood Education (012) Study Guide and Test Prep
##### GACE Program Admission Assessment Test I Reading (210): Practice & Study Guide
##### GACE Program Admission Assessment Test III Writing (212): Practice & Study Guide
##### OGET Oklahoma General Education Test (CEOE) (174) Study Guide and Test Prep
##### MTTC Early Childhood Education (106) Study Guide and Test Prep
##### Business 100: Intro to Business
##### Study.com SAT Study Guide and Test Prep
##### GRE Test Study Guide and Test Prep
##### Praxis 5723 Study Guide - Core Academic Skills for Educators - Writing Exam Prep
##### Praxis 5081 Study Guide - Social Studies: Content Knowledge Exam Prep
##### Praxis 5941 Study Guide - World & U.S. History - Content Knowledge Exam Prep
##### Praxis 5952 Study Guide - Sociology Exam Prep
What is the Y-Intercept? ------------------------
The Y-intercept is the point or points where the line in a graph crosses the y-axis. For a straight line, this will only occur once (unless x is a non-zero constant number, in which case there will be no y-intercept), but if a graph does not have a straight line, there can be several y-intercepts in the same graph.
Imagine data was collected about the intensity of sunlight shining on a solar panel each hour of the day from sunrise to sunset, for five days. Then, a graph was created where x was equal to the number of hours after sunrise each day, and y was equal to the intensity of sunlight. The y-intercept would be the intensity of sunlight at sunrise each day, because right at sunrise would be zero hours past sun rise. But since data was collected for five days, there are five sunrises, so there are five y-intercepts.
What is the Vertical Intercept?
The vertical intercept is another way of describing the y-intercept. Since most graphs and equations set y as the vertical line on the graph, the vertical intercept and the y-intercept are the same thing. On an (x, y) coordinate graph, the x-coordinates form the horizontal line and the y-coordinates form the vertical line. The y-intercept is where the line between the coordinates crosses the vertical line.
The red line shows the vertical or y-axis, with the intercept at 3.
What is the Horizontal Intercept?
Since the horizontal line is equal to the x-coordinates, the horizontal intercept is the x-intercept. This is where the coordinate line crosses the horizontal line, or the x-axis. The horizontal intercept is what x is equal to when y is zero.
The blue line shows the horizontal or x-axis, with the horizontal intercepts at about -1.5
Y-Intercept of a Straight Line
When the graph has a straight line, there is only one y-intercept. The equation for a straight line can be in three forms:
General: a x+b y+c=0
Slope-intercept: y=m x+b
Point slope: y−y 1=m(x−x 1)
Y-Intercept Formula
In the general form, when x is equal to 0, the y-intercept can be found by finding what y is equal to:
0 a+b y+c=0
b y+c=0
b y=−c
y=−c b
So, the y-intercept is equal to −c b.
In the slope-intercept form, the y-intercept can be found by setting x to be zero:
y=0 m+b
y=b
The y-intercept is equal to b.
Now let's look at the point slope form; it is important to note that this form is meant to highlight the slope of the line, not the y-intercept (like the other two forms help to highlight). This means that a coordinate is needed to be inputted, as well as setting x to 0 in order to find what the y-intercept is equal to.
Let's say the coordinate (5, 9) is given. This means that x 1=5 and y 1=9
Set x equal to 0
Now the equation looks like this:
y−9=m(0−5)
Isolate y:
y=m(0−5)+9
Simplify:
y=−5 m+9
For the graph with the coordinates (5, 9), the y-intercept is equal to: y=−5 m+9. Or, in more general terms, the y-intercept is equal to: y=−x 1 m+y 1
For a non-straight line, the y-intercept formula can also be found. For example, a parabola, which is a roughly U-shaped line, uses the quadratic formula. This formula is in the form of: y=a x 2+b x+c. If x is set to 0, then the y-intercept can be found:
y=a 0 2+b 0+c=0+0+c=c
The y-intercept of a quadratic formula is equal to c.
To unlock this lesson you must be a Study.com memberCreate an account
How to Find the Y-Intercept ---------------------------
The y-intercept can be found by:
Looking at the equation and calculating the y-intercept
Finding where the line crosses the y-axis
If the equation is given, then the exact y-intercept can be calculated. If the equation is not given, then looking at the graph can typically give a very good estimate of the y-intercept, but it can sometimes be difficult to tell the exact number.
How to Find the Y-Intercept on a Graph
Let's start with a graph that has a straight line, and find the y-intercept:
Since the line crosses the y axis at 6, this is the y-intercept
Take a look at the graph, and notice that the y-axis is the vertical axis, or the axis that is up and down. The line crosses this vertical axis when y is equal to 6. This makes the y-intercept equal to 6.
Now take a look at this graph:
The line crosses the y axis between negative 2 and negative 4
In this graph of a parabola, the line does not cross the y-axis in a location that is clearly identified; instead it is somewhere between -2 and -4. But it can be determined by looking at the tick marks. Each major tick mark (the lines that are darker in color) are spaced 2 units apart. There are also 4 minor tick marks (the lines that are lighter in color) between each major tick mark. The line crosses the y-axis on the third tick mark between -2 and -4.
To determine how far apart the minor tick marks are, calculate the absolute value of units between the two major tick marks divided by the number of tick marks:
Between -4 and -2 there are 2 units
There are 5 tick mark spaces between the two numbers
Space between minor tick marks is: 2 5
This means that each minor tick mark account for 2 5=0.4.
The first minor tick mark below -2 is equal to -2.4, the next is -2.8, the third is -3.2, and the fourth is -3.6, which then brings us to the next major tick mark of -4:
First minor tick mark: −2−0.4=−2.4 (subtract since we are going down, add when going up)
Second minor tick mark: −2.4−0.4=−2.8
Third minor tick mark: −2.8−0.4=−3.2
Fourth minor tick mark: −3.4−0.4=−3.6
Next major tick mark: −3.6−0.4=−4
Since the line crosses the y-axis at the third minor tick mark between -2 and -3, the y-intercept is equal to -3.2.
Y-Intercept Examples
If the equation is given then the y-intercept can be found by inputting x as 0. Find the y-intercept of the following equations:
y=3 x+7
y=5 x 2+7 x+4.6
4 y+5=5 x+43
Solution 1
This problem is in the slope intercept form; this means that the y-intercept is equal to b, which is 7.
Solution 2
This problem is in the quadratic formula form; this means that they y-intercept is equal to c, which is 4.6.
Solution 3
This equation is not in any of the standard forms, so the y-intercept needs to be calculated by setting x to 0 and solving for y:
4 y+5=(5×0)+43
4 y+5=0+43
4 y+5=43
Subtract 5 from both sides: 4 y+5−5=43−5
Divide both sides by 4: 4 y÷4=38÷4
Simplify: y=9.5
The y-intercept is equal to 9.5.
To unlock this lesson you must be a Study.com memberCreate an account
Lesson Summary --------------
The y-intercept is what y equals when x is equal to 0, or it can also be stated as where the line of a graph crosses the y-axis. When the equation is in a standard form, then the y-intercept is equal to:
General: a x+b y+c=0, y intercept is equal to y=−c b
Slope-intercept: y=m x+b y-intercept is equal to b
Point slope: y−y 1=m(x−x 1) y intercept is equal to y=−x 1 m+y 1
Quadratic: y=a x 2+b x+c the y-intercept is equal to c
The y-intercept can also be found by looking at the graph and estimating where the line crosses the y-axis.
To unlock this lesson you must be a Study.com memberCreate an account
Video Transcript
Definition of y-intercept
A y-intercept is the place where a line or curve crosses or touches the y-axis, which is the vertical, often darkened line located in the center of a graph. It is also the point at which x = 0.
y-intercepts and Graphs
Suppose you have two individuals, such as a married couple, and they share the household tasks between them. Let's call them Hope and Stealth. Hope and Stealth have decided to get a certain amount of household work completed each day. If Hope does more, Stealth does less. If Stealth does more, Hope does less.
To keep things simple, assume that there are 5 chores to be done. So, if Hope does 3, Stealth does 2. If Hope does 1, Stealth does 4. Or maybe Hope does 2.63 tasks and Stealth finishes up the other 2.37.
The point here is that there is a relationship between what Hope does and what Stealth does that can be defined precisely by an equation. If you know what one does, finding out what the other does just takes some calculation.
You might write it like this:
Stealth + Hope = 5
or:
x + y = 5
Stealth is the variable x, and Hope is the variable y.
Or, you might even rearrange it into slope-intercept form (y on the left and everything else on the right) like this:
y = -x + 5
If we want to understand the relationship more clearly visually, we might use a graph:
Any point on the graph tells us how many chores Hope does if Stealth does some given amount.
Of course, it's possible that Hope may have to do all of the chores because Stealth snuck off to watch the annual city Groundhog's Day parade. In this case, Hope does 5 chores, and Stealth does zero.
If you look at the graph, you will see that the point where Stealth is zero and Hope is 5 hits right at the central, vertical line—the y-axis. Because this point touches, or intercepts, the y-axis, it is called the y-intercept. The y-intercept is also the point at which the other variable, x, is zero.
y-intercepts and Curves
It's possible for something you graph, such as a curve, to hit the y-axis more than one time. For example, in this graph, the circle hits the y-axis twice.
In this case, there are two y-intercepts.
In the graph of the circle, there is a y-intercept at y= 3 and another y-intercept at y = -2.
Having more than one y-intercept is perfectly fine if it makes sense for the equation you graph. Some equations will have no y-intercepts, some will have one, and others will have several.
Lesson Summary
A y-intercept is the point at which a line or curve hits the y-axis, the vertical line where x = 0. There can, in some cases, be several y-intercepts, or there can be none. Equations with 2 variables (usually x and y) can take a number of forms, such as the slope-intercept form, wherein y is on the left and everything else is on the right.
Register to view this lesson
Are you a student or a teacher?
I am a student I am a teacher
Unlock your education
See for yourself why 30 million people use Study.com
Become a Study.com member and start learning now.
Become a member
Already a member? Log in
Go back
Resources created by teachers for teachers
Over 30,000 video lessons & teaching resources—all in one place.
Video lessons
Quizzes and worksheets
Classroom integration
Lesson plans
I would definitely recommend Study.com to my colleagues. It’s like a teacher waved a magic wand and did the work for me. I feel like it’s a lifeline.
Jennifer B.
Teacher
Try it now
Go back
Create an account to start this course today
Start today. Try it now
GMAT Study Guide and Test Prep
21 chapters
174 lessons
Chapter 1
GMAT: About the Test
What is the GMAT test?
The Difference Between the GMAT & GRE
GMAT Registration Information
How to Prepare for the GMAT
How is the GMAT Scored?
Can You Use a Calculator on the GMAT?
How to Be Ready for Test Day 4:40 min
Chapter 2
GMAT Analytical Writing Assessment Overview & Best Practices
Overview of the Analytical Writing Assessment Section of the GMAT
How to Answer GMAT Analysis of An Argument Questions
How to Focus Your Essay and Respond to the Essay Prompt 7:54 min
How to Structure an Argument in Your Essay 5:45 min
Opposing Viewpoint | Definition & Examples 7:40 min
GMAT Analytical Writing Assessment Prompts: Description & Examples
Chapter 3
Writing Skills for Exam Takers
How to Write a Great Essay 8:08 min
How to Write a Great Argument 7:10 min
How to Engage Readers by Picking and Developing an Appeal 7:41 min
How to Organize an Essay 7:53 min
Supporting Your Writing with Examples and Evidence 4:39 min
How to Write Better by Improving Your Sentence Structure 5:44 min
Writing Revision: How to Fix Mistakes in Your Writing 7:44 min
Proofreading | Components, Importance & Strategies 6:03 min
Chapter 4
GMAT Quantitative Section: About the Quantitative Section
Overview of the Quantitative Section for GMAT
GMAT Quantitative Section Question Types
Quantitative Data Overview, Types & Examples 4:11 min
Chapter 5
GMAT Integrated Reasoning: Types of Questions & Overview
Overview of the Integrated Reasoning Section for GMAT
GMAT Integrated Reasoning Question Types and Directions
How to Answer Graphics Interpretation GMAT Questions
Problem Solving Overview & Strategies 8:42 min
Chapter 6
Problem Solving for Exam Takers
Estimation Problems using Fractions 7:37 min
Solving Problems using Fractions and Mixed Numbers 7:08 min
How to Solve Complex Fractions 5:20 min
Solving Problems Using Decimal Numbers 6:57 min
Solve Problems Using Percents 7:50 min
Solving Linear Equations: Practice Problems 5:49 min
Solving Linear Equations with Literal Coefficients 5:40 min
Problem-Solving using Linear Equations 6:40 min
Solving a System of Equations with Two Unknowns 6:15 min
Solving Problems Involving Systems of Equations 8:07 min
Solving Linear Inequalities: Practice Problems 6:37 min
Solving Absolute Value Functions & Equations | Rules & Examples 5:26 min
Absolute Value | Overview & Practice Problems 7:09 min
Solve & Graph an Absolute Value Inequality | Formula & Examples 8:02 min
Absolute Value Inequalities | Definition, Calculation & Examples 9:06 min
Chapter 7
Fundamentals of Arithmetic for the GMAT
Fundamental Math for the GMAT: Properties of Integers
Fraction | Definition, Concept & Examples 6:20 min
Fractions and Mixed Numbers | Basic Operations & Procedures 3:33 min
Arithmetic with Decimal Numbers 10:40 min
How to Multiply & Divide in Scientific Notation 7:31 min
Real Number | Definition, Types & Examples 4:50 min
Calculations with Ratios and Proportions 5:35 min
Chapter 8
Advanced Arithmetic Key Concepts
Percents: Definition, Application & Examples 6:20 min
Percentage Increase | Definition, Formula & Calculation 5:16 min
Powers & Roots of Numbers | Definition & Examples 5:21 min
Descriptive Statistics Concept & Examples 7:10 min
Frequency Distribution | Definition, Graphs & Examples 5:15 min
Union of Sets in Math | Definition, Symbol & Applications 3:52 min
Permutation Definition, Formula & Examples 6:58 min
Combination in Mathematics | Definition, Formula & Examples 7:14 min
Introduction to Probability: Formula & Examples 4:33 min
Independent Events | Overview, Formula & Examples 5:32 min
Chapter 9
Mathematical Patterns
Recognizing & Generalizing Patterns in Math 5:37 min
Sequences in Math | Overview & Types 5:37 min
How to Find and Classify an Arithmetic Sequence 9:09 min
Finding and Classifying Geometric Sequences 9:17 min
Conjecture in Math | Definition, Uses & Examples 4:11 min
Function Table in Math | Definition, Rules & Examples 5:53 min
Chapter 10
Reviewing Basic Algebra Concepts
Variable in Algebra | Definition, Terms & Examples 5:26 min
Algebra Equations | Formula, Types & Examples 7:28 min
Simultaneous Equations Overview & Examples 11:21 min
Practice Simplifying Algebraic Expressions 8:27 min
Equivalent Expressions | Definition & Examples 4:42 min
How to Solve a System of Linear Equations in Two Variables 4:43 min
How to Solve a Quadratic Equation by Factoring 7:53 min
Quadratic Function | Formula, Equations & Examples 9:20 min
Exponent | Definition, Properties & Examples 8:47 min
Linear Inequalities | Equations & Examples 5:06 min
Estimate in Math | Definition & Examples 7:07 min
Functions: Identification, Notation & Practice Problems 9:24 min
Domain & Range of a Function | Definition, Equation & Examples 8:32 min
Chapter 11
Reviewing Basic Geometry Concepts
Line Segment in Geometry | Definition, Formula & Examples 3:53 min
Angles Formed by Intersecting Lines | Overview, Types & Examples 3:42 min
Perpendicular Sides, Shapes & Line Segments 3:54 min
Parallel Lines | Definition & Examples 4:55 min
What is a Polygon? | Definition, Types & Examples 6:08 min
Convex | Shape, Function & Examples 4:43 min
Types of Triangles & Their Properties 5:48 min
Pythagorean Theorem | Overview, Formula & Examples 7:33 min
Quadrilateral Shape | Properties & Types 6:42 min
Circle in Geometry | Definition, Parts & Examples 4:45 min
Inscribed and Circumscribed Figures: Definition & Construction 6:32 min
Volume of Shapes | Definition, Formula & Examples 4:21 min
Surface Area of a Cube & Rectangular Prism | Definition & Formula 4:08 min
Surface Area of a Cylinder | Formula, Calculation & Examples 4:26 min
Coordinate Geometry Definitions & Formulas 7:02 min
Chapter 12
Strategies for Solving Math Word Problems
Distance Equations | Formula, Calculation & Examples 6:31 min
Unit Rate in Math | Definition, Practice Problems & Solution 6:05 min
Mixture Problems in Algebra | Steps & Practice Problems 6:32 min
Simple Interest Problems | Definition, Formula & Examples 6:05 min
Calculating Percentage Discount | Equation & Formula 6:23 min
Gross Profit | Definition, Formula & Examples 5:04 min
Venn Diagram | Uses, Sets & Symbols 6:01 min
Using Diagrams to Solve Word Problems 5:19 min
Area in Geometry | Overview, Formula & Examples 4:52 min
Metric System | Measurements, Conversion & Units 8:47 min
Chapter 13
Numerical Ability: Reasoning & Data Interpretation
Inductive Reasoning in Math | Definition & Examples 7:03 min
Reasoning in Mathematics: Connective Reasoning 8:16 min
Polya's Problem Solving Process | Overview & Steps 7:52 min
Mathematical Problem | Definition, Principles & Uses 7:50 min
What is the Correct Setup to Solve Math Problems?: Writing Arithmetic Expressions 5:50 min
Solving Word Problems: Steps & Examples 5:54 min
Solving Word Problems with Multiple Steps 5:30 min
Restating Word Problems Using Words or Images 4:32 min
Personalizing a Word Problem to Increase Understanding 4:50 min
How to Simplify Word Problems with Fractions Using Whole Numbers 3:38 min
Distance, Time & Average Speed: Practice Problems 4:54 min
Organizing and Understanding Data with Tables & Schedules 6:33 min
Pie Chart vs. Bar Graph | Overview, Uses & Examples 9:36 min
Chapter 14
Interpreting Data & Statistics
Statistical Analysis with Categorical Data 5:20 min
Cumulative & Relative Frequency | Formula, Table & Calculations 5:47 min
Reading and Interpreting Line Graphs 6:09 min
Summarizing Categorical Data using Tables 4:57 min
Two-Way Table | Definition, Examples & Usage 3:40 min
Histogram in Math | Definition, Uses & Examples 5:12 min
Center, Shape & Spread of a Distribution | Overview & Examples 6:11 min
Measures of Central Tendency | Definition, Formula & Examples 8:30 min
Describing the Relationship between Two Quantitative Variables 4:44 min
Making Estimates and Predictions using Quantitative Data 4:07 min
Chapter 15
Answering Data Sufficiency GMAT Questions
GMAT Data Sufficiency Quantitative Section Overview
Understanding the Directions for the Data Sufficiency Section of GMAT
Test-Taking Strategies for Data Sufficiency Questions on the GMAT
Chapter 16
GMAT Verbal Section
GMAT Verbal Section: Types of Questions & Directions
GMAT Reading Comprehension: Overview & Question Samples
How to Pass a Reading Comprehension Test
Recognizing Details That Elaborate Upon a Main Idea
Logical Appeal Definition, Examples & Strategies 4:38 min
GMAT Sentence Correction: Overview & Directions
Chapter 17
Test-Taking Strategies for the GMAT
Solving Math Problems on the GMAT
Reading Comprehension Techniques for the GMAT
Identifying Grammatical Errors in Sentences 7:03 min
Literary Passage Analysis | Overview, Elements & Steps 10:26 min
How to Answer Multiple Choice Questions About Literature: Test-Taking Strategies 12:20 min
Chapter 18
Critical Reasoning for Test-Taking
How to Evaluate Reasoning 8:32 min
Evaluating Reasoning in an Essay or Article 6:13 min
Analyzing Interactions Among Main Ideas
Opinion vs. Fact | Definition & Examples 4:34 min
Reliable Sources | Definition, Characteristics & Examples 7:49 min
Chapter 19
Sentence Correction Techniques
Identifying Errors in Agreement for Nouns & Pronouns 7:31 min
Pronouns: Relative, Reflexive, Interrogative & Possessive 9:10 min
Independent & Dependent Clauses | Differences & Examples 6:53 min
Commonly Misused Words & Phrases
Parallel Sentence | Structure & Examples 5:47 min
Comma Usage: Avoid Confusion in Clauses & Contrasting Sentence Parts 5:36 min
Sentence Fragment | Overview & Examples 7:33 min
Common English-Language Idioms & Phrases
What Are Misplaced Modifiers and Dangling Modifiers? 6:25 min
Subject-Verb Agreement | Overview, Rules & Examples 6:34 min
Sentence Agreement: Avoiding Faulty Collective Ownership 3:32 min
Chapter 20
English Usage Conventions in Writing
Good Writing Characteristics & Examples 8:20 min
Diction in Writing | Overview, Types & Improvement 7:27 min
Active & Passive Voice | Definition, Usage & Examples 5:45 min
Point of View | Definition, Types & Examples 8:03 min
How to Write Logical Sentences and Avoid Faulty Comparisons 6:09 min
What is Flawed Logic? | Fallacious Reasoning in Arguments 8:41 min
Sentence Clarity: How to Write Clear Sentences 8:13 min
Sentence Structure: Identify and Avoid 'Mixed Structure' Sentences 6:36 min
Chapter 21
GMAT Test Flashcards
Related Study Materials
Y-Intercept | Definition, Formula & Examples
Lessons Courses Topics
##### Horizontal Line Test | Overview & Function 3:28
##### Linear Model | Equation & Examples 11:00
##### Slope of a Line | Meaning, Calculation & Examples 6:19
##### Direct Variation | Definition, Examples & Graph 7:30
##### Linear Relationship | Definition & Examples 6:29
##### Interest & Rate of Change - Rate of Change: Study.com SAT® Math Exam Prep 10:24
##### Linear Equations | EP SAT Tutoring 25:04
##### Rate of Change vs. Negative Rate of Change 4:22
##### Average Rate of Change | Definition, Formula & Examples 6:59
##### Nonlinear & Linear Graphs Functions | How to Tell if a Function is Linear 4:23
##### ILTS TAP - Test of Academic Proficiency (400): Practice & Study Guide
##### NYSTCE English Language Arts (003) Study Guide and Test Prep
##### CSET English Subtest IV (108) Study Guide and Test Prep
##### FTCE Middle Grades English 5-9 (014) Study Guide and Test Prep
##### SAT Subject Test Literature: Practice and Study Guide
##### CBEST Writing Study Guide and Test Prep
##### Praxis 5195 Study Guide - Spanish: World Language Exam Prep
##### NMTA Essential Academic Skills Subtest Writing (002): Practice & Study Guide
##### OAE Early Childhood Education (012) Study Guide and Test Prep
##### GACE Program Admission Assessment Test I Reading (210): Practice & Study Guide
##### GACE Program Admission Assessment Test III Writing (212): Practice & Study Guide
##### OGET Oklahoma General Education Test (CEOE) (174) Study Guide and Test Prep
##### MTTC Early Childhood Education (106) Study Guide and Test Prep
##### Business 100: Intro to Business
##### Study.com SAT Study Guide and Test Prep
##### GRE Test Study Guide and Test Prep
##### Praxis 5723 Study Guide - Core Academic Skills for Educators - Writing Exam Prep
##### Praxis 5081 Study Guide - Social Studies: Content Knowledge Exam Prep
##### Praxis 5941 Study Guide - World & U.S. History - Content Knowledge Exam Prep
##### Praxis 5952 Study Guide - Sociology Exam Prep
Browse by Courses
ILTS Science - Chemistry (240) Study Guide and Test Prep
FTCE Earth/Space Science 6-12 (008) Study Guide and Test Prep
FTCE School Counseling PK-12 (018) Study Guide and Test Prep
DSST Principles of Public Speaking Study Guide and Test Prep
TOEFL iBT Study Guide and Test Prep
DSST Introduction to Business Study Guide and Test Prep
UExcel Organizational Behavior: Study Guide & Test Prep
CLEP Introductory Business Law Study Guide and Exam Prep
UExcel Business Law: Study Guide & Test Prep
Study.com SAT Study Guide and Test Prep
CBEST Study Guide and Test Prep
GRE Test Study Guide and Test Prep
Praxis 5713 Study Guide - Core Academic Skills for Educators - Reading Exam Prep
Praxis 5723 Study Guide - Core Academic Skills for Educators - Writing Exam Prep
Business 111: Principles of Supervision
Browse by Lessons
Slope Project Ideas
Interpreting the Unit Rate as Slope in Proportional Relationshops
Finding Linear Equations to Fit Experimental Data
Finding the Slope of a Perpendicular Line | Formula & Example
Browse by Courses
ILTS Science - Chemistry (240) Study Guide and Test Prep
FTCE Earth/Space Science 6-12 (008) Study Guide and Test Prep
FTCE School Counseling PK-12 (018) Study Guide and Test Prep
DSST Principles of Public Speaking Study Guide and Test Prep
TOEFL iBT Study Guide and Test Prep
DSST Introduction to Business Study Guide and Test Prep
UExcel Organizational Behavior: Study Guide & Test Prep
CLEP Introductory Business Law Study Guide and Exam Prep
UExcel Business Law: Study Guide & Test Prep
Study.com SAT Study Guide and Test Prep
CBEST Study Guide and Test Prep
GRE Test Study Guide and Test Prep
Praxis 5713 Study Guide - Core Academic Skills for Educators - Reading Exam Prep
Praxis 5723 Study Guide - Core Academic Skills for Educators - Writing Exam Prep
Business 111: Principles of Supervision
Browse by Lessons
Slope Project Ideas
Interpreting the Unit Rate as Slope in Proportional Relationshops
Finding Linear Equations to Fit Experimental Data
Finding the Slope of a Perpendicular Line | Formula & Example
Create an account to start this course today
Used by over 30 million students worldwide
Create an account
Explore our library of over 88,000 lessons
Search
Browse
Browse by subject
College Courses
Business
English
Foreign Language
History
Humanities
Math
Science
Social Science
See All College Courses
High School Courses
AP
Common Core
GED
High School
See All High School Courses
Other Courses
College & Career Guidance Courses
College Placement Exams
Entrance Exams
General Test Prep
K-8 Courses
Skills Courses
Teacher Certification Exams
See All Other Courses
Study.com is an online platform offering affordable courses and study materials for K-12, college, and professional development. It enables flexible, self-paced learning.
Plans
Study Help
Test Preparation
College Credit
Teacher Resources
Working Scholars®
School Group
Online Tutoring
About us
Blog
Career
Teach for Us
Press Center
Ambassador
Scholarships
Support
FAQ
Site Feedback
Terms of Use
Privacy Policy
DMCA Notice
ADA Compliance
Honor Code for Students
Mobile Apps
Contact us by phone at (877) 266-4919, or by mail at 100 View Street #202, Mountain View, CA 94041.
© Copyright 2025 Study.com. All other trademarks and copyrights are the property of their respective owners. All rights reserved.
× |
8627 | https://www.sciencedirect.com/topics/engineering/brayton-cycle-efficiency | Brayton Cycle Efficiency - an overview | ScienceDirect Topics
Skip to Main content
Journals & Books
Brayton Cycle Efficiency
In subject area:Engineering
Brayton cycle efficiency refers to the effectiveness of a thermodynamic cycle, specifically in the context of gas turbines, where the performance is influenced by factors such as heat recovery, temperature management, and the design of turbomachinery components. It is crucial for achieving high cycle efficiencies within the operating temperature range of the cycle.
AI generated definition based on:Annals of Nuclear Energy, 2022
How useful is this definition?
Press Enter to select rating, 1 out of 3 stars Press Enter to select rating, 2 out of 3 stars Press Enter to select rating, 3 out of 3 stars
About this page
Add to MendeleySet alert
Discover other topics
1. On this page
On this page
Definition
Chapters and Articles
Related Terms
Recommended Publications
On this page
Definition
Chapters and Articles
Related Terms
Recommended Publications
Chapters and Articles
You might find these chapters and articles relevant to this topic.
Chapter
The gas and steam turbines and combined cycle in IGCC systems
2017, Integrated Gasification Combined Cycle (IGCC) TechnologiesTing Wang
15.3 Thermodynamics of the Brayton Cycle
An ideal Brayton cycle undergoes an isentropic (adiabatic and reversible) compression process, an isobaric (constant pressure) combustion process, an isentropic expansion process to generate power, and finally an isobaric cooling process to dump the waste heat to complete the cycle. These four ideal processes complete the Brayton cycle and are plotted in solid lines in P-v and T-s diagrams in Fig. 15.3, where P is the pressure, v is the specific volume, T is the temperature, and s is the entropy. In the real cycle, irreversibilities occur and entropy is produced due to frictions and heat losses, so the real processes deviate from the ideal process as shown in the dashed lines in Fig. 15.13 later in Section 15.3.
Sign in to download full-size image
Figure 15.3. Brayton cycle shown in P-v and T-s diagrams, in which Ⓟ is isobaric and Ⓢ is isentropic.
1→2: Isentropic compression, W˙in/m˙ (1-b-a-2, P-v diagram), driven by the turbine
2→3: Isobaric combustion (or heat transfer), Q˙in/m˙ (via a combustor or heat exchanger)
3→4: Isentropic expansion, W˙out/m˙ (a-3-4-b, P-v diagram) produced by the turbine
4→1: Isobaric cooling (or heat rejection), Q out/m˙ (via a heat exchanger or to the ambient)
The compressor is driven by the turbine. For industrial gas turbines, the compressor consumes about 40–50% of the power produced by the turbine, while for aircraft engines, the compressor may consume up to 70–80% of the turbine power.
Under the ideal reversible condition without any heat or total pressure losses, the area enclosed by the four processes in the P-v diagram indicates the net specific work, w net=W˙net/m˙(kJ/kg,Btu/lbm), produced by the cycle, while in the T-s diagram, the enclosed area represents the net thermal energy source in terms of heat per unit mass, q net,in=Q˙net,in/m˙(kJ/kg) received by the cycle. The P-v and T-s diagrams only represent the power W˙ (kW or Btu/s) and heat rate Q˙ (kW or Btu/s) per unit mass flow ratem˙ (kg/s or lbm/s), respectively; thus the total heat received and the total power generated by the system can’t be recognized by reading the P-v and T-s diagrams. It can be seen that there is no time scale in the specific work because the time in the power and the mass flow rate of the working fluid cancel each other.
The thermal efficiency of an ideal Brayton cycle can be calculated as
(15.1)η th=W˙turb m˙−W˙comp m˙Q˙in m˙=(h 3−h 4)−(h 2−h 1)(h 3−h 2)
In this equation the turbine power, compressor power, and heat input rate are all calculated as the enthalpy differences. This is the result of employing the energy balance, Eq. (15.2), over flow through the control volume (c v) of a device.
(15.2)d E CV d t=Q˙−W˙cv+∑in m˙i(h i+1 2 V i 2+g Z i)−∑exit m˙e(h e+1 2 V e 2+g Z e)
With the condition of steady state (d E cv/dt=0) and assumptions of negligible differences of kinetic energy and potential energy between inlets and outlets (Δ V 2/2 = 0 and Δ gZ = 0), both the isentropic expansion process in the turbine and isentropic compression in the compressor (without heat losses) can be simplified as W˙=m˙ (h e − h i). Similarly, with the same assumptions but with no work (W˙=0) occurring in the combustor, Eq. (15.2) is simplified to Q˙=m˙ (h e − h i).
For the convenience of a quick and crude analysis, the working fluid is considered only consisting of air without including combustion gases, and the specific heat values (c p and c v) are assumed to be constant, and the thermal efficiency can be expressed as
(15.3)η th=1−T 1 T 2=1−T 4 T 3
By applying the polytropic relationship:
(15.4)T 2 T 1=(p 2 p 1)(k−1)/k
where, k is the specific ratio (c p/c v), the ideal Brayton cycle efficiency can be expressed in terms of the compression pressure ratio (p 2/p 1 or PR) as
(15.5)η th=1−1(p 2 p 1)(k−1)/k=1−1(PR)(k−1)/k
The net specific work output is determined from the difference between the turbine and compressor work as
(15.6)w net=W˙net m˙=C p[(T 3−T 4)−(T 2−T 1)]
which can be mathematically manipulated to be presented in terms of a nondimensional form as specific network output
(15.7)w net C p T 1=r T[1−1 P R(k−1)/k]−[P R(k−1)/k−1]
where r T = (T 3/T 1) is the temperature ratio and PR = (P 2/P 1) is the pressure ratio. It can be seen the specific network output depends on both the pressure ratio (PR) and the temperature ratio (r T), which depends on the turbine inlet temperature (TIT, T 3).
From Eq. (15.3), it can be seen that the Brayton cycle’s thermal efficiency can be augmented by increasing the turbine inlet temperature (TIT, T 3), reducing the turbine exit temperature (T 4), or employing a cooler temperature sink (T 1). However, from Eq. (15.5), it appears that the Brayton cycle’s thermal efficiency has nothing to do with the temperatures, T 1, T 3, and T 4, rather the efficiency can be augmented by increasing the compressor’s pressure ratio (p 2/p 1), or using a working medium with the property of a higher k value. It has been commonly stated by gas turbine designers that increasing PR and TIT (or sometimes being called as the firing temperature) are two of the most important approaches to increasing gas turbine thermal efficiency; however, in contrast, Eq. (15.5) indicates that only increasing PR can augment gas turbine thermal efficiency, but TIT is not involved in affecting efficiency. To further look into this matter, the effects of TIT and PR, respectively, on Brayton cycle efficiency, are analyzed in the following section.
15.3.1 Effect of turbine inlet temperature on Brayton cycle performance
The characteristics of two cycles with different TITs is compared and analyzed in a T-s diagram as shown in Fig. 15.4. First, note that due to the polytropic relationship, two rising isobaric curves (1-4 and 2-3) diverge in the T-S diagram (i.e., the slope of the isobaric curve increases as the temperature increases). Hence, the ratio of any incremental useful energy (exergy) harnessed over the total incremental heat input at any entropy is the same, i.e., 2−1 2−a=3−4 3−b=3′−4′3−b′. Thus, with the same pressure ratio, the area ratio in the T-s diagram of useful energy harnessed over the heat input—i.e., the thermal efficiency—is invariant regardless of the TIT, meaning 1−2−3−4−1 a−2−3−b−a=(1−2−3′−4′−1)(a−2−3′−b′−a). This implies that the thermal efficiency of the two ideal Brayton cycles shown in Fig. 15.3 is the same, and increasing TIT with the same PR value can only increase output power but not alter the ideal Brayton cycle’s efficiency.
Sign in to download full-size image
Figure 15.4. Analysis of the effect of turbine inlet temperature (TIT) on Brayton cycle thermal efficiency.
This also implies that the appearance of T 3 in Eq. (15.3) may give a misleading impression that increasing T 3 alone can result in a higher Brayton cycle thermal efficiency. This is because when T 3 is increased, the turbine exhaust temperature, T 4, will inevitably increase to T′4 according to Eq. (15.4) in such a way that T′4/T′3=T 4/T 3, meaning that as TIT increases, more waste heat will be exhausted to the surroundings at the ambient pressure. Nevertheless, even though an increase of TIT can’t increase a simple Brayton cycle’s thermal efficiency, it can indeed increase output power and thermal efficiency of a combined cycle because more turbine exhaust energy, particularly having it been raised to a higher-grade energy due to higher exhaust temperature (T′4), can be more effectively utilized by the bottom steam cycle.
15.3.2 Effect of pressure ratio on Brayton cycle’s performance
Fig. 15.5 shows a comparison of four Brayton cycles with three different pressure ratios. Let’s first consider two cycles with a low PR (say, PR = 8, 1-2-3-4-1) and a high PR (say, PR = 15, 1-2′-3′-4′-1), respectively. The maximum temperature in these two cycles (T 3 and T′3) are limited by the allowable temperature (T limit,1) set forth by the metallurgical properties of the turbine blade materials and the thermal barrier coatings. Assuming both have the same output power, which can be represented by the useful heat received by the systems as the area of 1-2-3-4-1 and an equal area of 1-2′-3′-4-1, the thermal efficiency of the cycle with a high PR (1-2′-3′-4-1) achieves a higher thermal efficiency. This can be explained by the fact that the high-PR cycle rejects less waste heat (represented by the area of a-1-4′-b′-a) than that rejected by the low PR cycle (represented by the area of a-1-4-b-a). In other words, the useful heat received in the area of 2-2′-3′-c-2 doesn’t add to any additional heat rejection in the high-PR cycle, i.e., 100% of the heat input (2-2′-3′-c′-2) can be effectively converted to the useful power. Whereas, the equivalent useful heat received by the low-PR cycle in the area of c-3-4-4′-c requires a higher heat input represented by the area of b′-c-3-b-b′, i.e., only a portion of the heat input can be effectively converted to the useful power. This above analysis concludes that increasing the PR of a Brayton cycle can indeed increase the thermal efficiency. This above analysis although is based on the assumption that both low PR and high-PR cycles have the same net output power, the conclusion is valid for cycles with different net output powers, meaning that as PR increases, more input heat can be effectively converted to useful work without increasing heat rejection.
Sign in to download full-size image
Figure 15.5. Analysis of the effect of pressure ratio (PR) on Brayton cycle thermal efficiency.
As the PR continues to increase to a very high condition, say PR = 40, with the condition that the maximum temperature is still limited by T limit,1, the cycle efficiency will continue to increase, but eventually, the output power will reduce as shown in the case of cycle 1-2″-3″-4″-1 in Fig. 15.5.
The condition when the maximum net output power is achieved for an ideal Brayton cycle is when
(15.8)T 2,max work=T 1 T 3.
This relation can be derived by making the derivative of the net work with respect to T 2 equal to zero. Then, the optimumcompressor pressure ratio for the condition of maximum work can be obtained by plugging Eq. (15.6) into Eq. (15.4) as
(15.9)P R max work=P 2 P 1=(T 2 T 1)k/(k−1)=(T 3 T 1)k/2(k−1)
Under the condition of very high PR but small net output power, like the cycle 1-2″-3″-4″-1 in Fig. 15.4, if the maximum temperature can be raised to T limit,2, the cycle 1-2″-3-4-1 can achieve significant increases in both the output power and cycle efficiency. Therefore, when the gas turbine designers claim that the gas turbine efficiency can be continuously increased by raising TIT, they don’t mean the scenario as shown in Fig. 15.4 by increasing the TIT alone, rather they mean the scenario of the cycle 1-2″-3-4-1 in Fig. 15.5 by increasing both PR and TIT (which is subjected to the value of T limit). In other words, it can also be said that as TIT increases due to technology advancements in high-temperature resistent materials, higher PR can be implemented.
One drawback of employing high-PR compressions is that the turbine exhaust temperature becomes lower, which will adversely affect the effectiveness for regeneration (see next section) or, in a combined cycle application, reduce the output power and thermal efficiency of the bottom steam cycle.
Raising TIT has been accomplished by using expensive superalloy for turbine airfoils (including vanes and blades), employing internal or external turbine airfoil cooling, and adding thermal barrier coating (TBC) to the airfoils. The direction solidification method has also been used to grow nickel-based superalloys into single-crystal materials for turbine airfoils in some gas turbines.
There are many methods to augment Brayton cycle output power and/or thermal efficiency. The most common methods are briefly described below.
15.3.3 Recovering exhaust energy to increase Brayton cycle efficiency-Regeneration
The word regeneration means “recovering gas turbine exhaust energy for its own usage.” The common approach is to route the exhaust gas through a recuperator or a regenerator to preheat the compressed air prior to entering the combustor, as shown in Fig. 15.6A. The transfer of the waste heat can be illustrated by the two shaded areas as shown in Fig. 15.6B. The area underneath 4-y represents the amount of the recovered waste heat, while the area underneath 2-x represents the theoretically equal amount of heat that is regenerated from the waste heat to preheat the air entering the combustor. Considering a temperature difference is required to transfer heat, T y has to be higher than T 2 and T 4 higher than T x. The regenerator effectiveness is defined as
Sign in to download full-size image
Figure 15.6. Regenerative Brayton cycle.
(15.10)ε reg=m˙2(h x−h 2)m˙4(h 4−h 2)=The actual energy recovered The maximum possible energy can be recovered
If fuel is included, m˙4=m˙2+m˙fuel.
Note that both recuperators and regenerators are types of heat exchangers. In a recuperator, the hot and cold streams don’t mix and the heat is transferred through partitioned walls, while in a regenerator, the heat is alternately stored in and released from a solid, porous (or honeycomb-like) matrix as the hot and cold streams are routed through this matrix alternately. Although the hot and cold streams don’t directly mix, a portion of the mass in one stream inevitably remains in the matrix and is carried away by the other streams during the next cyclic process. An example of a regenerative gas turbine engine is the GT305 automotive engine built by General Motors.
Since the waste heat of the Brayton cycle is recovered and used in the same system, the Brayton simple-cycle efficiency is always increased by using the regeneration scheme. The combustor only needs to provide energy from x to 3 in Fig. 15.6, so the energy from 2 to x is saved. After regeneration, however, little or no exhaust energy can be used for combined cycle applications. Since the gas turbine waste heat is a lower-grade energy, it can be more effectively utilized to generate more power and achieve much higher cycle efficiency (58–60%) in a combined cycle arrangement using steam as the working medium than in a regenerative simple Brayton cycle (42–45%). The efficiency of an ideal Brayton cycle with 100% effective regeneration is
(15.11)η reg=1−T 1 T 3 P R k−1 k
Fig. 15.7 shows the efficiencies of ideal Brayton cycles with and without regeneration at different temperature ratios (T 3/T 1). The efficiency of the ideal Brayton cycle is represented by the heavy solid curve. When regeneration is implemented, a higher PR cycle actually reduces the overall cycle efficiency as represented by the lighter solid lines on the left side in Fig. 15.7. This is because the turbine exhaust temperature (T 4) decreases as PR increases (see T 4>T′4>T″4 in Fig. 15.4) and less energy is available for regeneration, until, eventually, T 4 dips below the compressor exit air temperature (T 2), resulting in an impractical condition for implementing the regeneration scheme. This impractical condition occurs when the regenerative cycle curves are extended below the ideal Brayton efficiency curve in Fig. 15.7. In this region below the ideal Brayton cycle efficiency curve, the regenerator actually adversely cools, instead of heating, the air leaving the compressor.
Sign in to download full-size image
Figure 15.7. The efficiencies of ideal Brayton cycles with and without regeneration at different temperature ratios (T 3/T 1). The dashed curves represent only the qualitative trend of efficiency reduction as PR reduces for nonideal regenerators.
The interception of a regenerative efficiency curve of a specific temperature ratio (T 3/T 1) with the non-regenerative efficiency curve in Fig. 15.7 represents the optimum pressure ratio (PR max work of Eq. (15.9)) for which the Brayton cycle produces the maximum W˙out for that specific temperature ratio. In the ideal condition when the effectiveness of the regenerator is assumed 100%, as PR reduces from the optimum value (PR max work), the efficiency of the regenerative Brayton cycle monotonously increases until it reaches the theoretical maximum efficiency of a Carnot cycle, η = 1 − (T 1/T 2). However, if the regenerator effectiveness is less than 100%, the regenerative Brayton cycle efficiency increases first with reduced PR from the optimum PR, then decreases as PR continuously decreases as shown in the dashed lines on the left side of Fig. 15.6.
15.3.4 Reheating Brayton cycle to augment output power
Although a bigger gas turbine can always provide more power, often the available on-shelf gas turbines are provided in discrete power output. Reheat is a convenient and cost-effective method to notably augment power 20–35% of existing gas turbine systems by using the same compressor, primary combustor, and many other design parameters such as T limit, TIT, mass flow rate, and pressure ratio. Reheat is a process of splitting the original expansion process into a two-stage expansion by adding a reheat combustion process between the high-pressure (HP) turbine and low-pressure (LP) turbine. Fig. 15.8 illustrates the schematic and thermodynamic cycle diagrams of an ideal reheat Brayton cycle.
Sign in to download full-size image
Figure 15.8. Reheat of an ideal Brayton cycle. The shaded areas represent augmented power in the P-v diagram and heat in the T-s diagram, respectively.
The nondimensional specific network and efficiency of an ideal reheating Brayton cycle are
(15.12)w c p T 1=2 r T−X+1−2 r T X
(15.13)η reheat=2 r T−X+1−2 r T/X 2 r T−X−r T/X
where, X=(P 2 P 1)k−1 k and r T=T 3 T 1
The efficiency of a reheat Brayton cycle is lower than the basic Brayton cycle because the heat is added at a lower pressure condition. This can be realized by comparing Fig. 15.8B with Fig. 15.4—the high-grade heat above pressure curve a-b in Fig. 15.8B is not received by the system, which is needed to achieve the same efficiency as the basic Brayton cycle as illustrated in Fig. 15.4. However, reheat does provide a better condition to implement regeneration or a bottom steam cycle to achieve more power output and a better thermal efficiency because the exhaust gas temperature of a LP turbine is higher than that of the basic Brayton cycle (i.e., T 4>T′4). When the split pressure (P a) is set as P a=P b=P 3 P 4, the maximum network output W˙max can be obtained.
15.3.5 Intercooling compressors to save compression power
When the air becomes hot due to compression, the air density reduces and the specific volume expands, so more power is required to compress the air. One approach to mitigate this heat-induced expansion effect is to provide interstage cooling of the compressed air before it is continuously compressed to the final pressure, as shown in Fig. 15.9.
Sign in to download full-size image
Figure 15.9. An intercooled gas turbine system.
The saving of compressor power can be illustrated in the P-v and T-s diagrams in Fig. 15.10A. With the assumptions of isentropic compression and negligible changes in kinetic and potential energy between the inlet and exit of a compressor, the compression power of an insulated compressor can be approximated as
Sign in to download full-size image
Figure 15.10. Illustration of compression power saving due to (A) interstage intercooling and (B) direct cooling over the compressor.
(15.14)W˙comp=m˙∫v d P,
which can be represented by the area enclosed between the compression curve and the P-ordinate. In the P-v diagram in Fig. 15.10A, the compression power of the nonintercooled compressor can be represented by the area 1-c-2′-f-e-1 and the intercooled compression power is represented by the area 1-c-d-2-f-e-1. The difference of these two areas, represented by the shaded area c-d-2-2′-c, indicates the saving of the compression power resulting from the intercooling. Attributed to this saving of compression power, the total net output power of the Brayton cycle increases. Nonetheless, similar to reheating, the cycle efficiency reduces with intercooling because more heat input (represented by the area underneath the curve 2-2′ in the T-s diagram in Fig. 15.10A) is needed to reach the designated TIT (T 3). This added heat is not effectively utilized by the system since more waste energy is rejected (represented by the area underneath the curve c-d in the T-s diagram) than it would have been with an ideal Brayton cycle.
The intercooling process can be also implemented by applying cooling directly over the compressor casing (see Fig. 15.10B) or spraying water mist inside the compressor. The latter involves a wet compression process and may cause potential erosion concerns, which will be discussed later in the section on gas turbine inlet fogging and overspraying.
In the intercooled Brayton cycle, the maximum turbine work can be achieved if the split pressure, P c, is set as P C P 1=P 2 P C or P C=P 1 P 2. Another advantage of using intercooling is attributed to the decreased compressor exit temperature, which can benefit implementation of regeneration.
15.3.6 Reheat and intercooling with regeneration
Employing reheat, intercooling, and regeneration always increases output, but reheat or intercooling cannot improve efficiency if they are operated separately. Combination of reheat and intercooling with regeneration can achieve significant improvements of both net output power and overall cycle thermal efficiency (Fig. 15.11).
Sign in to download full-size image
Figure 15.11. Regenerative gas turbine with intercooling and reheat.
15.3.7 Steam injection gas turbine system (STIG) for augmenting power and reducing NO x
Steam injection was used in the 1940s to boost the power output of military airplane engines (Cohen et al., 1999). Since the 1970s, injecting steam into combustors has also been commonly used as a cost-effective approach to reduce NO x via reducing combustion temperature and smoothing out hot spots. For the purpose of NO x reduction, the steam with approximately 2–3% of the mean air mass flow rate is typically injected into the combustor, while for the purpose of increasing power, steam up to a maximum of 20–25% of the air mass flow has been employed. The steam is typically generated through a heat recovery steam generator (HRSG) using the waste heat from the gas turbine exhaust (see Fig. 15.12). Demineralized water is required.
Sign in to download full-size image
Figure 15.12. A steam injection gas turbine (STIG) system using HRSG to produce the steam.
Injecting steam to augment power can also be used as a fast and dynamic scheme to meet the load fluctuations for medium to small gas turbine systems, particularly during the peak-load periods for distributed generation applications. A variation of STIG is called the Cheng cycle, in which the amount of steam is significantly increased to approach the limit that the waste energy of the exhaust gas can supply. The Cheng cycle literally replaces the entire bottom steam cycle of a combined cycle system by injecting a large amount of steam (25% of the airflow mass) into the combustor, which saves the entire capital and operation and maintenance costs and space of the bottom steam cycle. The Cheng cycle was invented and patented in 1976 by Professor D. Y. Cheng of the University of Santa Clara (Chen and Nelson, 2002). The STIG and Cheng cycles are not applicable in the regions where water resources are scarce.
15.3.8 Evaporative–regenerative gas turbine (ERGT) cycle
Another means to augment the simple Brayton cycle output is to saturate the compressed air before it enters the combustor by spraying water into the air exiting the compressor or using a saturator with a large evaporative surface between the compressor and the combustor. Typically water of approximately 9–10% of the dry air mass flow is required to achieve saturation. Water evaporation lowers the compressed air temperature, which provides an ideal condition to implement regeneration. Thanks to regeneration and added water vapor mass, both the specific power (W˙/m˙) and total power (W˙) are increased.
A variation of the evaporative–regenerative gas turbine cycle, called the humid air turbine (HAT) cycle, further incorporates compressor intercooling and aftercooling, and uses the heat absorbed from the compressed air through the intercooler and aftercooler to preheat the water before it enters the saturator, which is heated by the gas turbine exhaust through a regenerator/recuperator. Since the heat absorbed from cooling the compressed air (typically around 380°C or 716°F for a pressure ratio of 15) is a lower-grade energy than that of the turbine exhaust gas at around 610°C (1100°F), a HAT cycle can more efficiently produce more power than just using the higher-grade energy from the exhaust heat to vaporize the water. Since both intercooling and aftercooling of compressed air are implemented in the HAT cycle, it is more suitable for gas turbines operated with high-pressure ratio above 30, such as for aeroderivative engines used in distributed generation applications. More discussions of HAT cycles can be found in papers of Day and Rao (1993), Gallo (1997) and Rao (2015).
The major differences between the ERGT or HAT cycles and STIG or Cheng cycles are that (1) the STIG or Cheng cycles can use much more steam (up to approximately 25% of the airflow) than ERGT or HAT cycles (up to approximately 10% of the airflow) such that they can literally produce a magnitude of power similar to a combined cycle because they can utilize almost all of the turbine exhaust heat to produce power; (2) STIG or Cheng cycles can adjust the output power more quickly and dynamically with larger turn-down or augmentation ratios to match the load fluctuations; and (3) most STIG applications are primarily used for reducing NO x rather than to augment power.
15.3.9 Gas turbine inlet cooling to augment output power
Gas turbine inlet cooling is extremely effective in counteracting degraded GT performance during hot summers, when the power demand reaches its maximum. The capital cost for an inlet cooling device is cheaper than installing a stand-by unit only for peak-load needs. There are a number of technologies that can be deployed to accomplish the lowering of the air temperature at the compressor inlet, which in turn increases the density of the air and the corresponding mass flow through the GT, allowing increased power output during warm or hot ambient conditions.
The options to improve GT performance/output through inlet cooling are numerous including indirect evaporative “precooling” systems, active “chiller” refrigeration-based systems (both electrically driven and thermally driven), desiccant cooling systems, and a number of water spray/fogging options. Caution needs to be taken to avoid frosting when the static pressure reduces during the flow’s acceleration through the inlet converging duct.
Evaporative cooling and refrigeration are two most common methods for providing inlet cooling. Evaporative cooling can be achieved either by mixed or nonmixed method. In the mixed evaporation, water will be evaporated and mixed with air before entering the compressor. Small water droplets (mist) ranging from 5 to 7 μm can be injected into the GT inlet to achieve the cooling effect as well as increase the mass flow rate. Selections of appropriate cooling technologies depend on site environment and load characteristics.
Turbine inlet cooling (TIC) technology increases the mass flow through the combustion turbine by cooling the inlet air. Cooler air is denser, and since combustion turbines are constant volume flow machines, denser air equates to greater mass flow. Gas turbine output is a strong function of ambient temperature and turbines typically lose between 0.3% and 0.5% of their ISO rated power for every 1°F rise in inlet temperature. Conversely a 1°F drop in temperature by evaporative cooling of the inlet raises the gas turbine power output between 0.3% and 0.5%. Once the specific megawatt-per-degree-cooling number for the turbine in question has been determined (which is available from the OEM’s turbine performance versus ambient temperature charts), this number can then be multiplied by the annual operational hours of effective cooling to arrive at the total annual MW-hr power boost that can be expected from inlet cooling for that turbine at that site.
Inlet cooling technologies can fall into a number of different specific applications (Wang and Braquet, 2008). The two main distinctions will be whether or not active refrigeration or “chilling” is utilized, as is the case with electric or absorption-driven chillers. In either case, the cooling is accomplished by either evaporation—whether through fogging, evaporative cooling pads, or spray injection—or through active/mechanical refrigeration.
1.
Active refrigeration systems can chill inlet air to just about any temperature that does not result in ice buildup on the cooling coils from the ambient relative humidity. For optimization and practical reasons, systems are usually designed to operate above 9°C (48°F). Refrigeration, whether by direct chilling or ice storage, is relatively costly, requiring considerable maintenance, has a fairly high parasitic load, and requires a fairly large area to accommodate the refrigeration equipment. The basic difference between direct chilling and ice storage is that direct chilling has a sustained and relatively constant parasitic load while the combustion turbine is operating, whereas ice storage has a high parasitic load that is used to make ice during off-peak hours that is then melted to provide refrigeration during peak hours. The required installation space for ice storage is also considerably larger than the space required for direct chilling and can often eliminate the viability of that option.
Typically, active chilling will be applied around five primary systems:
a.
electric-driven, compressor-based refrigeration chillers;
b.
waste heat-driven, absorption refrigeration-based chillers;
c.
a thermal storage system where low-temperature ice or solution is made through active chilling using power and/or waste heat available during off-peak times so that this low-temperature storage can be used during on-peak times to increase gas turbine power output without significant parasitic power used by refrigeration equipment;
d.
a “combination” system that may include both electric and absorption chilling as well as partial thermal storage;
e.
indirect chilling systems where inlet air passes over a cooling coil containing a fluid that is being circulated to a separate cooling source, which could include a heat exchange system using a cool primary source such as a river or well water.
2.
Evaporative systems do not utilize any active refrigeration equipment. Instead, the system deployed acts as more of a “cooling” process where inlet temperatures can be reduced to approach the coincident wet-bulb temperature. These evaporative systems generally fall into three categories:
a.
direct “wet” systems, which use water sprayed onto the air–steam or onto a medium in the main inlet air stream;
b.
fogging systems, where water is carefully atomized so as to achieve full evaporation either before entering the compressor section or to provide some degree of compressor intercooling by completing the vaporization within the initial sections of the compressor;
c.
indirect “dry” systems, similar to the indirect chilling system except that the separated primary source is provided by evaporative cooling.
Evaporation cooling has an advantage in that the mass of the water used for cooling also contributes to the turbine’s performance (i.e., by increasing mass flow through the turbine). The major disadvantage of evaporation is that its effects diminish with rising wet-bulb temperatures, associated with hot and humid days when power (electricity) is usually most valuable, and that it is rarely possible to cool the inlet air below about 42°F (5.5°C), which sets a limit on the potential performance improvement.
Evaporation also uses a considerable amount of water and can lead to compressor section and turbine section maintenance problems, if operating parameters are not carefully observed. While evaporation can be relatively inexpensive to install, maintenance costs can be high and performance can suffer if water of the required quality is not used, which is an area where operators often try to reduce costs. For applications in which water will be ingested into the gas turbine compressor such as sprays and fogging technologies, the best approach is to utilize high quality, demineralized water.
Bhargava and Meher-Homji (2005) presented the results of a comprehensive parametric analysis on the effect of inlet fogging on a wide range of existing gas turbines. Both evaporative and overspray fogging conditions were analyzed. It shows that the performance parameters indicative of inlet fogging effects have a definitive correlation with the key gas turbine design parameters. In addition, they indicated that aeroderivative gas turbines, in comparison to the industrial machines, have higher performance improvement from inlet fogging.
Chaker et al. (2004a,b,c) presented the results of extensive experimental and theoretical studies conducted over several years and coupled with practical aspects learned in the implementation of nearly 500 inlet fogging systems on gas turbines ranging from 5 to 250 MW. Their studies covered the underlying theory of droplet thermodynamics and heat transfer and provided practical points relating to the implementation and application of inlet fogging to gas turbine engines. They also described the different measurement techniques available to design nozzles. Their papers collectively provided experimental data on different nozzles and recommended a standardized nozzle testing method for gas turbine inlet air fogging. The complex behavior of fog droplets in the inlet duct was addressed and experimental results from several wind tunnel studies were documented.
Any cooling schemes that may affect the flow field inside the compressors have not been favorably considered by the industry due to concerns of any disturbance that might damage compressor airfoils or adversely affect the compressor’s performance stability. However, as the inlet fog cooling scheme has become popular as an economic and effective means to augment gas turbine output power on hot or dry days (Meher-Homji and Mee, 1999), consideration has been given to applying overspray (high) fog cooling or injecting fine water droplets between stages (i.e., interstage fogging). There are several concerns associated with overspray or interstage water spray cooling: (1) the potential erosion of compressor blades caused by tiny water droplets, (2) drilling holes through the compressor casings to install fogging devices may cause a compressor integrity problem, and (3) the manufacturer’s warranty could be voided. Furthermore, Bagnoli et al. (2004) showed that more power could be obtained by applying fogging upstream of the compressor than between the stages. Therefore, overspray or interstage fogging has not been widely applied and is still subject to further research and investigation.
Ingistov (2001) pointed out that a water injection system was installed in a 7EA gas turbine to wash the compressor blades and preserve the gas turbine output power rather than augment power output. Shepherd and Fraser (2006) presented field data from a Siemens gas turbine to compare the performance between applying inlet and interstage fogging. The data showed that output power augmentation obtained by applying the interstage fogging is almost insensitive to the ambient temperature variation. They treated this feature as a merit of interstage fogging. The data further showed NO x reduction was more pronounced by applying interstage fogging than inlet cooling. More detailed discussion on the theory and fundamental physics of inlet fogging will be presented in Section 15.5.
A comparison of the results of applying various gas turbine performance enhancement schemes conducted by the Electric Power and Research Institute (EPRI 2003) is summarized in Table 15.1. It can be seen that inlet fogging and evaporative cooling are the most economic means to augment gas turbine power up to about 15% with a cost less than $50/kW, whereas in the same performance enhancement range, the conventional steam injection scheme would cost about $250/kW. By adding 1% overspray, inlet fogging can further augment power to 20% in hot days with a cost of $75–$100/kW. Inlet chilling using refrigeration or ice storage can enhance power about 22.5% with a much higher cost about $200–$400/kW. Advanced Cheng system steam injection can significantly boost power up to 68% and increase cycle efficiency up to almost 30% at a cost of $310–$548/kW, which is approaching the cost of purchasing stand-by gas turbine units.
Table 15.1. Performance enhancements for 50––100 MW combustion turbines in a simple-cycle plant (EPRI, 2003)
| Method | Δ Capacity | Δ Heat rate | $ Per KW |
--- --- |
| Water injection to combustor or interstage | +10% | +4% | <$50 a |
| Inlet fogging and evaporative cooling b | +5% to +15% | −2.5% | ~$50 a,b |
| Fogger with 1% inlet overspray | +10% to +20% | −5% | $75 to $100 a |
| Conventional steam injection to combustor or compressor discharged air | +10% to +15% | −5% to −7% | $200 c |
| Conventional steam injection with supplemental firing | +10% to +15% | +2% to 0% | ∼$250 d |
| Inlet chilling: refrigeration or thermal (ice) storage b | +22.5% | −5% | $200 to $400 b |
| Supercharging | +15% to +20% | −4% | ∼$200 |
| “Power-cool” supercharging with inlet fogging | + 15% to +30% | −5% to −6% | ∼$250 a |
| Compressor coatings and enhanced wash systems | +0.5% to +3% | −0.5% to −3% | <$60 |
| Thermal barrier coatings | +5% to +15% | −0.5% to −1% | <$50 |
| High-flow inlet guide vanes | +4.5% | −1% | $20 to $100 |
| Compressor aerodynamic upgrades | +10% to +20% | −1.5% to −5% | $150 to $250 |
| Turbine hot-parts upgrades | +5% to + 10% | −2% to −5% | $150 to $250 |
| Wet compression | +10% to +20% | −1.5% | ∼$100 e |
| Cheng boost steam injection f | +15% to +30% | −5% to −8% | $149 to $349 c |
| Advanced Cheng system steam injection f,g | +21% to +68% | −20% to −29% | $310 to $548 h |
All water-use technologies assume a viable source of suitable, fresh water is available.
a Add approximately $30 per kW if treated water system is not already available.b See additional performance and cost data specific to inlet cooling and chilling technologies within this report.c Assumes that suitable steam source is available.d Assumes that HRSG is already installed, but includes duct burner installation.e Approximate cost including treated water system; current market prices are targeted at $200 per kW or more.f Cheng data at 85°F, 60% R.H. and standard pressure with natural gas.g For the advanced Cheng system, the cost per kW is calculated on the total output of a retrofitted plant, not just on the incremental output over that of the simple-cycle engine, because the cost and complexity of the advanced Cheng system retrofit and the resulting performance improvements are so great that it would not be undertaken unless there was a requirement for an entire plant’s worth of capacity with the attendant performance characteristics.h Assumes “new” plant installation with used and rebuilt combustion turbine; performance improvements are as compared to underlying simple-cycle engine performance; cost per kW is for total plant output and includes new HRSG and balance of plant improvements.
15.3.10 Irreversibilities
The previous sections lay out the fundamental thermodynamics theories for gas turbines operated in the ideal Brayton cycle. For real gas turbines, entropy is produced through irreversibilities such as total pressure losses through frictions and heat losses to the environment. The major irreversibilities occur during compression in the compressors and expansion in the turbines as well as pressure losses through combustors or heat exchangers implemented for compressor intercooling or regeneration. Fig. 15.13 illustrates a nonideal Brayton cycle, in which friction losses in the heat exchanger air side are referred to as Δ P a and the gas side as Δ P g. Aerodynamic resistance in a combustor and the momentum changes induced pressure losses by the exothermic reaction are included as Δ P c.
Sign in to download full-size image
Figure 15.13. Nonideal Brayton cycle showing irreversibilities in compressors, combustors, and turbines. The states labeled with “s” are the theoretical end of an isentropic process.
In Fig. 15.13 the states labeled with “s” are the ends of theoretical isentropic processes. The dashed lines represent the paths of real processes. The difference between the isentropic and real states is associated with the isentropic efficiency of the component. In Fig. 15.2, the isentropic efficiency of the compressor is
(15.15)η c=h 2 s−h 1 h 2−h 1
and the isentropic turbine efficiency is
(15.16)η t=h 1−h 4 h 1−h 4 s
15.3.11 Gas turbine heat rates and efficiency
In the electric power industry, the gas turbine cycle efficiency is also represented in terms of the heat rate (HR), which indicates the amount of energy (kJ, Btu, or hp-h) used by a gas turbine system or a power plant to generate one kilowatt hour (kWh) of electricity or horsepower-hour of mechanical energy (hp-h). Heat rate is defined as:
(15.17)HR≡Q in W net(kJ/kWh, Btu/kWh, or Btu/hp-hr)
Heat rate happens to be in an inverse relationship with the cycle efficiency as shown below:
Cycle efficiency,η=3413(Btu/kWh)HR(Btu/kWh)=3600(kJ/kWh)HR(kJ/kWh)
With the following unit conversions: 1 hp = 2545 Btu/h, 1 hp = 0.7457 kW, and 1 kW = 3413 Btu/h, an efficiency of 100% converts to a heat rate of 3600 kJ/kWh or 3413 Btu/kWh.
Example:
HR=8950 Btu/kWh⇒η=3413 8950=38.13%HR=6820 kJ/kWh⇒η=3600 6820=52.79%
It can be seen that the lower the heat rate, the higher the thermal efficiency.
15.3.12 Polytropic efficiency versus isentropic efficiency for compressors and turbines
The efficiency of a compressor is usually quoted as the isentropic efficiency (η c) of the entire compressor; often another efficiency, referred to as polytropic efficiency (η c,p), is also used in turbomachinery to describe the elemental efficiency of a differential piece of a stage. The concept of using the polytropic efficiency lies on the fact that the increase in temperature due to irreversible friction in one stage results in more work being required in the next stage. The effect is termed the preheat effect. The polytropic efficiency—also called “small-stage efficiency”—is defined as the isentropic efficiency of an elemental (or differential) stage in the process such that it is constant throughout the whole process. The relationship between the isentropic and polytropic compressor efficiency is given by Cohen et al. (1999) as:
(15.18)η c=PR(k−1)/k−1 PR(k−1)/(k η c,p)−1
The difference between η c,p and η c increases as compression ratio increases. Isentropic compressor efficiency is usually degraded from the polytropic compressor efficiency (i.e., η c,p>η c).
Similar, but opposite, to the polytropic compressor efficiency, the irreversible friction heat of the prior stage is recovered as work in the next stage, resulting in a polytropic turbine efficiency (η t,p) being lower than the isentropic turbine efficiency (η t), i.e., η t,p<η t.
The relation between η t,p and η t is given by Cohen et al. (1999) as:
(15.19)η t=1−(P 4/P 3)η t,p(k−1)/k 1−(P 4/P 3)(k−1)/k
where P 4/P 3 is the turbine expansion ratio, which is the inverse of the compression pressure ratio (1/PR). The foregoing relations cannot be applied to a steam turbine as vapors generally do not obey the gas law. The reheat factor (R H) is used to measure the hidden inefficiency of the complete expansion (from state 1 to state 2) due to the irreversibility.
(15.20)R H=∑Δ h s,p/(h 1−h 2 S)
and
(15.21)η ST=R H(η ST,p)−−−(Steam Turbine)
where, Δ h s,p is the isentropic small-stage enthalpy of an elemental stage of the steam turbine blades and ηST,p is the polytropic steam turbine efficiency. R H is usually between 1.03 and 1.08.
Show more
View chapterExplore book
Read full chapter
URL:
Book 2017, Integrated Gasification Combined Cycle (IGCC) TechnologiesTing Wang
Chapter
Heat engines
2022, Machinery and Energy Systems for the Hydrogen EconomyJoshua Schmitt, ... David Sánchez
4.2 Conventional engine cycles
4.2.1 The Brayton cycle
4.2.1.1 Cycle overview
The Brayton cycle is a thermodynamic working cycle that was originally proposed by George Brayton in 1872.
The ideal Brayton cycle consists of:
•
Isentropic compression
•
Isobaric combustion
•
Isentropic expansion
•
Isobaric heat rejection
A real Brayton cycle (Figs. 4.10 and 4.11) would consist of the following:
Sign in to download full-size image
Fig. 4.10. Pressure-volume diagram of the Brayton cycle.
Sign in to download full-size image
Fig. 4.11. Enthalpy-entropy diagram of a real Brayton cycle for a two-shaft gas turbine.
•
Adiabatic compression (1–2)
•
Isobaric combustion (2–3)
•
Adiabatic expansion (3–7)
•
Isobaric heat rejection (5–1)
The Brayton cycle describes the working concept of gas turbines (Fig. 4.11), where the necessary steps are created by a compressor (1–2), a combustor (2–3) for heat addition, a turbine section that drives the compressor (3–5), and a second turbine section (5–7) that provides the useful power output. The working fluid is generally air (in the compressor) and air mixed with combustion products in the turbine section (in an open cycle). Brayton cycles can also use other working fluids, like CO 2 or helium. In these cases, the cycle is often a closed cycle.
Fig. 4.10 shows that in this cycle, the highest allowable temperature limits the heat input, thus the maximum cycle power. Component efficiencies influence the amount of power output.
Other key features found in gas turbines are the requirement that the expansion process provides the power for the compression process and the fact that the combustion process is at a constant pressure (Fig. 4.11). The constant-pressure combustion process typically uses natural gas as a fuel but could use hydrogen as a fuel in pure form or blends. It also can be seen that the Brayton process will leave significant amounts of heat in the exhaust gas that cannot be used beneficially within this cycle.
Modifications of the Brayton cycle include intercooling of the air in the compressor, recirculation of exhaust gases, or the addition of topping or bottoming cycles.
4.2.2 The Rankine cycle
4.2.2.1 Cycle overview
The Rankine cycle is a regenerative process that William John Macquorn Rankine laid the basis for in 1854. In a Rankine thermodynamic cycle for a simple steam power plant, the working fluid undergoes a series of processes and returns to its initial state. In this form, the Rankine cycle is a closed cycle. Unlike the Rankine closed cycle, open-cycle working fluids will have a different composition or will be at a different state at the conclusion of the mechanical cycle. For the Rankine cycle, we consider four steady-state points in the closed-loop. Each point is defined by four processes:
•
(1–2) Reversible adiabatic pumping process in the pump (isentropic compression)
•
(2–3) Constant-pressure transfer of the heat in the boiler (heat addition)
•
(3–4) Reversible adiabatic expansion in the turbine or expander (isentropic expansion)
•
(4–1) Constant-pressure transfer of the heat in a condenser (heat rejection)
These states and processes can be shown on a p-v diagram (Fig. 4.12). The Rankine cycle may also include a superheater that will superheat the vapor (Fig. 4.13).
Sign in to download full-size image
Fig. 4.12. Process power cycle (p-v diagram).
Sign in to download full-size image
Fig. 4.13. Superheated process power cycle (p-v diagram).
•
Process from 1 to 2: Taking a liquid and pumping it from low to high pressure. This is one of the benefits of the Rankine cycle over Brayton cycles since pumping a liquid is much easier than compressing a fluid that contains vapor.
•
Process from 2 to 3: This high-pressure liquid from the pump will enter a boiler and be held at constant pressure and heated with some variation of an external heat source until it becomes a dry, saturated vapor. The required input energy can have a large influence on the efficiency of the process. Often, an additional heat source is added to superheat the steam to provide more available energy to the system for the next process.
•
Process from 3 to 4: During this process, the dry, saturated vapor will expand through either single or multiple stages of a turbine or expander to produce power. During the process, the temperature and pressure will be decreased, converting the potential energy of the fluid into kinetic energy for the system. The efficiency in which the energy is extracted from the fluid and the state point at which the fluid exits this process can have a large influence on the efficiency of the process.
•
Process from 4 to 1: The fluid typically will enter a condenser, where it will be condensed at a constant pressure to become a saturated liquid before returning to the pumping process.
The Rankine cycle is traditionally used in large coal or nuclear boilers but recently has been applied as the bottoming cycle to a Brayton cycle to recover the rejected heat in combined-cycle gas turbine plants. The ideal Rankine cycle processes for the turbine and pump would be isentropic such that they would have no entropy change and maximize the workout produce for each process. In reality, each process has heat losses, fluid friction, and parasitic pressure losses, as well as others that reduce the amount of work in and work out.
4.2.2.2 Typical types of units applied in the Rankine cycle
There are two primary types of turbines or expanders that are used in the Rankine cycle: radial-inflow and axial flow. Each is designed to take the available energy or the process fluid and convert this into kinetic energy to produce a torsional force to the rotor. As fluid flows through a turbine or expander, the potential energy is subsequently reduced to lower potential energy through stationary nozzles (stators) producing a high-velocity jet as the fluid expands. High-velocity fluid exiting the stationary nozzle pushes against the rotating blades (buckets) mounted onto a shaft or disks to develop a force that causes the shaft to rotate. This in turn produces useful mechanical work. A turbine or expander stage consists of stationary nozzles and at least one set of rotating blades mounted on a disk that is part of a shaft or rotor assembly.
Radial-inflow turbines allow flow to enter a volute in the circumferential direction. The sectional area of the volute, as shown in Fig. 4.14, is reduced as the fluid radially exits through a series of nozzles. The stationary nozzles are designed to accelerate the flow as it passes through each subsequent nozzle. This nozzle acceleration produces angular momentum into the rotor or rotating portion of the turbine. This rotor is designed to convert some or all of the kinetic energy of the fluid. Turbines are often times designed with several stages that sequentially break this available energy into stages. Applications with large amounts of available energy are typically used in a series of stages to produce the overall power of the unit. Once the fluid has reached the final stage, any remaining kinetic energy passes through a diffuser that is designed to gradually expand the volumetric area and convert any remaining kinetic energy of the fluid back into an increase in the static pressure, ultimately improving overall efficiency. The radial-inflow turbine designs are not well suited to multistage applications due to high losses between stages, and they are conventionally coupled with axial-flow stages after the first radial-inflow stage.
Sign in to download full-size image
Fig. 4.14. Cross-section of a radial-inflow turbine.
Courtesy of Elliott Group.
Axial-flow turbines are much more suited to handle the series of sequential pressure drops and much easier to design for multistage configurations. The cross-section of these types of turbines is shown in Fig. 4.15. While radial-inflow turbines are better suited for micro-turbines or turbochargers, the axial-flow designs provide added flexibility for higher efficiency applications. Axial-flow turbines consist of similar stationery and rotating airfoil; however, the working fluid, as the name eludes to, passes through the unit axially versus radially. For the first stage of any turbine or expander, the stationary nozzles are built into the nozzle ring. For all subsequent stages, the nozzles are part of the diaphragm. Diaphragms divide the turbine or expander into stages, each operating at a successively lower pressure and higher volume flows. Successive stages thus require larger and larger nozzles and rotor blades, or more generally, provide more through-flow area.
Sign in to download full-size image
Fig. 4.15. Cross-section of an axial inflow turbine.
Courtesy of Elliott Group.
When dealing with stage designs, the two primary parameters each stage is designed for are the pressure ratio and velocity ratio. The pressure ratio is the ratio of the inlet steam pressure before entering a stationary nozzle over the pressure leaving the rotating blade (Eq. 4.1). The steam will expand as it makes its way through a series of stages that provide this sequential pressure drop, thus converting the stored potential energy of the steam into shaft rotational power along the way. The velocity ratio (Eq. 4.2) for a stage is the ratio of the blade velocity (Eq. 4.3) to the steam jet velocity (Eq. 4.4). The peak efficiency of both pressure ratio and velocity ratio can vary based upon the airfoil designs. It is important that the design of each airfoil be made to produce a high level of efficiency based upon the specific operating conditions required by the turbine.
(4.1)PR=P in P exit
(4.2)VRO=V b V j
(4.3)v b=π·N·D 720
(4.4)v j=224·Δ h is
Here, PR is pressure ratio, VRO is velocity ratio, v b is blade velocity (ft/s), v j is jet velocity (ft/s), N is rotor speed (RPM), D is wheel diameter, and Δ h is is the isentropic heat drop (ft 2/s 2).
4.2.2.3 Rankine turbine/expander stage designs
Looking deeper into types of turbine or expander axial-flow designs, the control stage is where the unit begins to convert this available fluid energy into power for the driven equipment. For turbines or expanders with throttling valves or partial admission, the control stages are predominately impulse-style stage designs that are velocity compounded (Curtis stage), whereas pressure compounded stages (Rateau stage) are typically applied as full admission stages that may have varying amounts of a reaction built into the design.
A velocity-compounded stage design consists of stationary nozzles with two rows of moving blades and a reversing bucket assembly between the rotating blade rows. Stationary nozzles direct the working fluid against the first row of rotating blades. The working fluid then leaves the first row of blades still having a high velocity and enters the reversing buckets, which then redirect fluid into the second row of rotating blades. The large pressure drop through the nozzle produces a high-velocity jet. The high velocity of the nozzle jet is absorbed in two constant pressure steps. The two rotating rows of blades make effective use of the high-speed jet, resulting in small wheel diameters and tip speeds, fewer stages, and a shorter turbine or expander. The high-velocity jet, which is as much as four times the blade velocity, produces high bending stresses in blades, thus limiting blade heights.
For pressure compounded staging, the heat energy of the working fluid is converted into work by stationary nozzles directing the fluid against a single row of moving blades, in addition to a reaction force generated by the rotating bucket. Unlike a Curtis stage, where the stage pressure drop occurs entirely across the stationary nozzles, the Rateau stage can be designed to produce a certain amount of pressure drop that provides a reactionary force. The nozzle exit velocity for a Rateau stage is approximately two times that of the blade velocity. Blade bending stresses are lower allowing for longer blades than can be used in Curtis stages. In an impulse stage, the smallest flow area is the throat at the stationary nozzle exit.
4.2.3 The Otto and Diesel cycle
The Otto and Diesel cycles are idealized thermodynamic cycles, which are commonly used to describe the operation of spark- and compression-ignited internal combustion engines, respectively. They are both air-standard cycles that capture the fundamental factors that impact internal combustion engine efficiency and are widely used to evaluate engine concepts.
4.2.3.1 The Otto cycle
The Otto cycle is named after Nicolaus Otto, a German engineer who is regarded as the first person to build a working, four-stroke engine as patented by Alphonse Beau de Rochas. In general terms, the thermodynamic Otto cycle describes the processes acting upon a trapped mass of gas over the four strokes of a typical internal combustion engine. The Otto cycle describes a closed system that consists solely of the trapped gas within the heat engine. The four components of the Otto cycle are described below and plotted on a P-v diagram in Fig. 4.16:
Sign in to download full-size image
Fig. 4.16. P-v diagram of the Otto cycle.
•
State 1 ➔ 2: Isentropic compression of a trapped mass of gas inside the heat engine
•
State 2 ➔ 3: Constant volume heat transfer to the gas from an external source while the piston remains at top dead center (TDC, where the piston is at the top of the stroke and the cylinder is at its minimum volume)
•
State 3 ➔ 4: Isentropic expansion of the gas
•
State 4 ➔ 1: Constant volume heat rejection to an external sink while the piston remains at bottom dead center (BDC, where the piston is at the bottom of the stroke and the cylinder is at its maximum volume)
The canonical Otto cycle presumes that the trapped mass of gas is not exchanged with the environment, though a zero-state can be considered where a gas exchange process with the environment can happen losslessly. This gas exchange process is shown in Fig. 4.16 as the line between State 1 and State 0. This gas exchange process does not change the thermodynamics of the basic Otto cycle and will not be considered further in this discussion.
The Otto cycle can be expressed by the change in internal energy of the gas through each of the four processes:
W 1→2=U 2−U 1
Q 2→3=U 3−U 2
W 3→4=U 4−U 3
Q 4→1=U 1−U 4
The thermal efficiency of the Otto cycle can then be calculated as the ratio of the net work from the cycle to the heat added:
η=Q 2→3+Q 4→1 Q 2→3=1+U 1−U 4 U 3−U 2
Then, considering the specific heat equation for constant volume:
c v=dU dT V
Using the specific heat equation to replace the internal energy terms ultimately yields:
η=1−T 1 T 2
The ideal gas law can then be used to relate temperature, pressure, and volume. Considering that the compression ratio of an internal combustion engine is defined as r=(V 1/V 2), we can ultimately obtain the equation describing the efficiency of the Otto cycle:
η=1−1 r γ−1
This equation reveals the factors which ultimately impact the efficiency of the Otto cycle: the compression ratio, r, and the ratio of specific heats, γ. For an Otto cycle analysis, γ is generally a fixed value, typically 1.4 for air. This leaves the compression ratio as the parameter which can be changed. Fig. 4.17 shows the cycle efficiency as calculated using the above equation for pure air with a specific heat ratio of 1.4.
Sign in to download full-size image
Fig. 4.17. Otto cycle efficiency as a function of compression ratio with air as the working fluid (γ=1.4).
Even without considering the real engine cycle that includes such factors as heat transfer, friction, and finite combustion speeds, it is evident that the basic Otto cycle calculation yields an unrealistically high efficiency (the blue curve) due to the use of a fixed and optimistic value for the specific heat ratio. While a value of 1.4 is correct for pure air at normal atmospheric conditions, the gas composition in a real engine is more complex as it contains fuel, water vapor, carbon dioxide, and other species with larger molecular structures than O 2 and N 2. This brings the specific heat ratio down to a value around 1.3 on a cycle-average basis and yields a more reasonable, though still high efficiency, prediction (the red curve). While optimistic, this cycle analysis is still useful for showing the impact of compression ratio on the cycle efficiency and for providing an indication of a practical maximum efficiency for the Otto cycle.
A practical realization of the Otto cycle is generally considered to be a spark-ignition engine. While a wide variety of mechanical systems have been developed which can produce the four strokes of the Otto cycle, the most common and practical implementation is a crank-slider piston engine. Fig. 4.18 shows a basic configuration for this machine. The piston rides in a cylinder with minimal clearance and with one or more rings secured into grooves in the piston, which act as a seal to prevent combustion gases from leaking past the piston and to remove oil from the cylinder wall. The piston is attached to a connecting rod with a pin (termed a wrist pin), and the other end of the connecting rod is attached to a rotational joint on a crankshaft. The linear motion of the piston is thus converted to the rotational motion of the crankshaft. Air and fuel are drawn into the combustion chamber through one or more valves which are mechanically linked to the piston motion to open when the piston is moving downwards and to close when the piston reaches the bottom of its stroke. The mixture is then compressed by the upward motion of the piston, and the spark plug fires near the top of the piston stroke to initiate combustion. A quasi-spherical flame front proceeds outward from the spark plug and consumes the air/fuel mixture as the piston starts to move downwards. The pressure rise from the combustion energy release performs work on the piston and pushes it back down to the bottom of its stroke. One or more exhaust valves then open, and the burned gases are pushed out of the cylinder by a combination of the still-elevated pressure and the upward motion of the cylinder. As the piston nears the top of its stroke, the exhaust valve closes and the cycle starts again.
Sign in to download full-size image
Fig. 4.18. Basic crank-slider piston engine.
The above components and processes describe the essential spark-ignition/Otto cycle operation. There are several key differences between the ideal Otto cycle and the real cycle as occurs in an internal combustion engine:
•
The gas exchange between the cylinder and the intake/exhaust system does require work to be performed by the piston and is subject to losses due to flow through ducts and restrictions
•
The compression process is subject to heat transfer between the cylinder wall and the gas
•
Combustion occurs over a finite time and not under constant volume conditions
•
The expansion process sees significant heat transfer through the cylinder walls, the cylinder head, and the piston
•
The heat rejection process occurs as the exhaust stroke over the entire up-stroke of the piston
•
Friction losses occur throughout the cycle
Despite these significant differences, the overall cycle still behaves quite similarly to an Otto cycle as can be seen in Fig. 4.19. These P-v diagrams are for a series of cases on a spark-ignited gasoline engine running at a high load with a range of exhaust gas recirculation (EGR) ratios. Note that the axes are logarithmic, so the exponential curves of the expansion and compression strokes appear linear. For the engine cases represented by these diagrams, the higher EGR ratios allowed for more optimum combustion phasing and a more constant volume combustion event. The curvature in the traces at minimum volume is due to the finite combustion duration. The work required for gas exchange can be seen in the area between the intake and exhaust processes at low pressure. But other than these differences, the P-v diagrams are quite similar to those for the ideal cycle.
Sign in to download full-size image
Fig. 4.19. Spark-ignited engine P-v diagram for a gasoline engine running with a range of exhaust gas recirculation ratios.
4.2.3.2 The Diesel cycle
The Diesel cycle is named after Rudolf Diesel and his 1895 patent for a compression-ignition cycle. From a thermodynamic cycle perspective, the Diesel cycle looks near-identical to the Otto cycle but for one crucial difference; the constant volume heat addition of the Otto cycle (State point 2 ➔ 3) is replaced with constant-pressure heat addition. The equation for the efficiency of the Diesel cycle can be derived in a similar fashion to that of the Otto cycle to yield:
η=1−1 r γ−1 α γ−1 γ α−1
The first portion of the efficiency equation is the same as for the Otto cycle, but the term in parentheses accounts for the volume change over the combustion process at constant pressure. The ratio of the start and end volume for combustion is denoted by α:
α=V 3 V 2
Using a similar analytical approach with air as the working fluid, the efficiency of the Otto and Diesel cycles can be compared as shown in Fig. 4.20. For this figure, air was used for the ratio of specific heats, and the Diesel cutoff ratio was specified as 1.3 (V 3 was 30% greater than V 2). The lower cycle efficiency of the Diesel cycle relative to the Otto cycle for a given compression ratio is immediately evident.
Sign in to download full-size image
Fig. 4.20. Comparison of Otto and diesel cycle efficiency for air and a diesel cutoff ratio of 1.3.
The results shown in Fig. 4.20 are counter-intuitive, as it is well understood that in practice, a compression-ignited engine (which is what the Diesel cycle is intended to represent) is much more efficient than a spark-ignited engine (which is what the Otto cycle is considered to represent). The key is that the real implementation of these two combustion approaches does not occur at the same compression ratio. Because a spark-ignited engine relies on a flame propagating through premixed air and fuel, there is a limit in how high the compression ratio can be. As the fuel burns, the unburned mixture sees higher pressures and temperatures as the flame progresses. If the pressure and temperature get too high, then the remaining unburned mixture will homogenously ignite and result in a strong pressure wave, termed “knock,” that is extremely damaging to the engine structure. For any given engine, this knock phenomenon limits how high the compression ratio can be. Generally, the compression ratio is limited to a value of 10–12 in practice. But since the compression-ignition engine does not have premixing of the air and fuel, but relies on diffusion combustion, there is no danger of knock. This allows for much higher compression ratios in the range of 16–22 for modern engines. Comparing the two cycles at their respective practical compression ratios shows why a compression-ignited engine is more efficient than a spark-ignited engine.
The Diesel cycle was defined with constant pressure combustion based on limitations of the original concept of the engine by Rudolph Diesel. His original idea was to inject and burn a coal slurry, which would have had a quite slow combustion process and which would have yielded a cycle that looked much like the thermodynamic definition of the Diesel cycle. However, modern engines use very high injection pressures (in excess of 2000 bar) to atomize diesel fuel and yield a much faster combustion event. As such, a modern compression-ignition engine is more accurately described by the Otto cycle despite the differences in combustion systems.
The modern compression-ignition engine looks quite similar to the spark-ignited engine illustrated in Fig. 4.18. The only difference of note as it relates to the discussion in this chapter is that the spark plug is replaced by a fuel injector. For the spark-ignited engine, the fuel can be introduced either in the intake or directly into the combustion chamber, but the fuel delivery is designed so that the fuel and air can be fully premixed by the time that the spark plug is fired, which initiates combustion. In the compression-ignition engine, the fuel is injected directly into the cylinder as the piston approaches the top of its stroke when the compressed air is hot enough to cause autoignition of the fuel without any external energy input. Beyond that difference, the arrangement of the piston and valves is functionally identical between the spark- and compression-ignition engines. Fig. 4.21 shows the combustion portion of a P-v diagram for a modern diesel engine. There is more curvature than in the spark-ignited engine case in Fig. 4.19, but the combustion event is far closer to constant volume than constant-pressure. This indicates that the Otto cycle can be used as a thermodynamic model for either spark- or compression-ignited engines at this point in engine development.
Sign in to download full-size image
Fig. 4.21. Combustion portion of a P-v diagram for a modern diesel engine.
4.2.3.3 Common fuels and opportunities for hydrogen
Through a century of development, modern internal combustion engines have largely settled onto near-universally used fuels for spark- and compression-ignited variants, but each also has a range of fuels that have been or are currently used in some applications. This section will briefly review the key characteristics that fuel must display, commonly used fuels, and how hydrogen as a fuel fits into the landscape for internal combustion engines.
Spark-ignited engines rely on positive ignition from a spark plug to initiate combustion as a propagating flame through a premixed blend of air and fuel. Because the fuel should not ignite before the spark plug fires, it needs to have a reasonably high autoignition temperature (or alternately minimum ignition energy) to prevent preignition before spark timing.
This autoignition temperature also needs to be high enough to prevent the premixed fuel and air to ignite before the flame front consumes all of the mixtures; the rising temperature and pressure in the combustion chamber during flame propagation can lead to end-gas autoignition, commonly termed knock. In practical implementations of a spark-ignited engine, the necessary autoignition resistance of the fuel is driven by this limit, as it is a more severe thermal condition than the compression conditions when the spark timing is set.
The resistance of a given fuel to knock is described by the Research Octane Number (RON) and Motor Octane Number (MON) of the fuel. These numbers are determined in two ASTM tests: D2699 defines the method to determine RON for fuel and D2700 defines the method to determine MON. The two tests were developed for historical reasons, but for modern engines, the history is not essential to understand. The RON value is the best indicator of the resistance of a fuel to end-gas autoignition as experienced in modern engine designs, while the MON value is useful as an indicator of the potential of the fuel to experience hot-spot preignition (where a hot surface such as the spark plug or exhaust valve causes a local autoignition event). In general, a high RON is desired to allow for higher compression ratios and higher compression pressures, while a high MON has been attractive to provide resistance to hot-spot preignition.i
While gasoline is generally the first spark-ignited fuel that is thought of, there are several common fuels that can be of interest in modern engines:
•
Gasoline: 91–97 RON, 87–93 MON typical in the US
•
Ethanol: 109 RON, 100 MON
•
Methanol: 109 RON, 99 MON
•
Propane: 112 RON, 105 MON
•
Methane: 120 RON, 120 MON
Methane, as the largest component of natural gas, is common in stationary engines as well as in some mobile applications (city buses, refuse trucks, and taxis are frequently fueled with natural gas). The high RON and MON of methane allow for the use of high compression ratios, though the high RON also correlates with high ignition energy demands which make the ignition system more expensive than for other fuels.
Hydrogen can be an excellent alternative to natural gas with an easy conversion since both are gaseous fuels. It can also be a good alternative to liquid fuels like gasoline, though with more hardware changes required. However, hydrogen brings a significant challenge; the RON of hydrogen is quite high, estimated at 130, but the MON is difficult to define, with most estimates being in the range of 0–10. This results from the unique-to-hydrogen combination of very low ignition energy and very high resistance to bulk autoignition. In practice, this both simplifies and complicates the design of a spark-ignited hydrogen engine. The ignition system demands are minor, so low-cost components can be used, but preignition from hot spots in the combustion chamber can occur at quite low loads. Careful design of cooling circuits and heat transfer paths is required to maintain low wall temperatures, especially near the exhaust valves and the spark plug. When done well, efficiencies on par with natural gas engines can be achieved (~42% typically) with a conventional engine design.
Compression-ignited engines rely on the compression heating of the air in the combustion chamber to ignite the fuel, which is injected near the end of the compression stroke. Some of the injected fuel premixes and autoignites, which then start a diffusion flame around the plume of fuel (Dec, 1997). This requires fuel with relatively low ignition energy demands in order to minimize the amount of compression heating required to ignite the fuel. Positive ignition timing is controlled by the timing of the fuel injection and the resulting ignition delay that is a function of the gas temperature in the combustion chamber.
Compression-ignition engines have been made to operate on many fuel types, but diesel fuel is near-ubiquitous. Properties vary across regions and fuel types, but a Cetane number of 40–50 is a typical value. Cetane number is measured in an engine test that is analogous to the Octane rating methods (ATM Method D613). It is difficult to precisely correlate Cetane and Octane numbers, but diesel fuel is considered to have a RON of 15–25, so it is convenient to think of Cetane and Octane ratings to be roughly inverse of each other (high Octane implies low Cetane and vice versa).
Hydrogen has not been directly measured for a Cetane number, presumably due to the incompatibility of hydrogen fuel with any of the accepted methods for determining Cetane (which requires a liquid fuel that can be injected into a combustion volume). Using the typical relationship between Octane and Cetane numbers, it would be reasonable to assume that hydrogen has a Cetane number near zero. However, the very low MON of hydrogen and the correlated low ignition energy suggest that hydrogen may have a non-zero Cetane number. Without data, it is difficult to estimate a precise number, but it is reasonable to assume that hydrogen could function in a compression-ignition engine, should a suitable hardware system be developed. Examples of such engines can be found in the literature (Dimitriou and Tsujimura, 2017), though none appear to be suitable for series production at present.
4.2.4 The combined cycle
4.2.4.1 Cycle overview
Traditionally, heat engines are utilized a single cycle, such as the steam Rankine cycle or an air-Brayton gas turbine. However, the combustion gases being exhausted by some cycles remains hot. These cycles are especially inefficient when they are run in an open fashion (such as a simple gas turbine or a diesel generator), and the exhaust gas heat is lost to the atmosphere. In order to recover this heat, engineers began introducing combined cycles that feature a topping cycle and a bottoming cycle. The bottoming cycle utilizes the waste heat generated by the topping cycle and converts that waste heat into additional electricity. A common example of a combined cycle pairs a Brayton topping cycle with a Ranking bottoming cycle.
The available pressure ratio of the gas turbine limits the amount of heat that can be converted into work. Brayton cycle efficiencies improve as the hottest turbine inlet temperatures are increased with new technological innovation. However, since the pressure ratio of a gas turbine is limited, this typically results in hotter exhaust gas temperatures.
If the available exhaust heat can be used, significant improvements in the overall efficiency are possible. The exhaust gas heat, and potentially another heat source (e.g., a duct burner) to increase the exhaust temperature, are used to generate steam (Fig. 4.22). The steam is generated in a large heat exchanger known as a heat recovery steam generator (HRSG). In this case, the topping heat engine is a gas turbine, and the bottoming heat engine is a steam cycle, which drives a steam turbine. This configuration is typically known as a combined cycle gas turbine (CCGT).
Sign in to download full-size image
Fig. 4.22. Gas turbine and steam turbine driving a generator (Brun and Kurz, 2019).
Fig. 4.23 shows the temperature/entropy diagram for the CCGT. Cycle 1-2-3-4 is the gas turbine Brayton cycle. The cycle A-B-C-D-E-A is the Rankine steam cycle. The exhaust heat from Point 4 is subsequently used to heat the feedwater (A-B), evaporate the water (B-C), and superheat the steam (C-D). The steam is expanded through a steam turbine (D-E) and afterward condensed in a condenser (E-A).
Sign in to download full-size image
Fig. 4.23. Combined cycle gas turbines—temperature/entropy diagram (Brun and Kurz, 2019).
The steam is used to operate a steam turbine that either drives a separate generator or the same generator as the gas turbine, as in a Steam Turbine Assisted Cogeneration (STAC) system (Fig. 4.22). Flexibility is gained by adding a clutch between the generator and the steam turbine. The STAC system is optimized around flexibility, i.e., the system can be used to vary the electrical power output based on the needs of the process steam load.
The amount and quality of steam produced do not only depend on the available exhaust heat (Q ex):
Q ex=c p·ṁexh·T 7−T sink
but also, on the temperature (T 7) at which the exhaust gases are available. Heat transfer in a heat exchanger follows:
Q=k·A·Δ T
with k being the heat transfer coefficient, A the surface area, and Δ T the average temperature difference between the exhaust gas and the coolant. The term kA can be considered constant once the boiler geometry is defined. The average temperature difference Δ T depends also on the heat exchanger geometry. In a counterflow arrangement, Δ T becomes:
Δ T=T exh,in−T w,out−T exh,out−T w,in ln T exh,in−T w,out T exh,out−T w,in
Assuming we know the steam temperature (T w,out), the feedwater temperature (T w,in) and the exhaust temperature (T 7=T exh,in), as well as the exhaust flow (ṁexh) and additionally kA from the boiler geometry, then we can calculate the stack temperature (T exh,out) from:
ṁexh·c p,exh kA=T exh,in−T w,out−T exh,out−T w,in ln T exh,in−T w,out T exh,out−T w,in
Knowing the stack temperature, we can now calculate the average temperature difference between the exhaust gas and the water/steam (Δ T). Then, we can calculate the transferred heat (Q), as well as the resulting steam flow (ṁw) (if the steam enthalpy is defined) or the steam enthalpy (Δ h w) (if the steam flow is defined) from:
ṁw·Δ h w=ṁexh·c p,exh·T exh,in−T exh,out=c p·ṁexh·T 4−T sink=Q
If the temperature difference drops (as T 7 drops), less heat can be transferred to the steam. On single-shaft engines without variable inlet geometry, turbine exhaust temperature (T 4) will drop at part load while ṁexh will stay roughly constant. With variable inlet guide vanes, T 4 will stay roughly constant, but ṁexh will drop (Fig. 4.24).
Sign in to download full-size image
Fig. 4.24. Steam power at part load for three different engine concepts: Single shaft with VIGVs (for dry low NO x control), conventional single shaft, two shaft (Brun and Kurz, 2019).
4.2.4.2 Hydrogen fuel for combustion gas turbines and combined cycles
Hydrogen in general is a suitable fuel for combustion gas turbines. Though pure hydrogen has not been available as a fuel in the industry, fuel gas mixtures with 60% and higher fractions of hydrogen have been burned in gas turbines in the past from, for example, coke oven gas or refinery byproducts (European Turbine Network, 2020). The gas turbine models used for this mainly include industrial E-class turbines with diffusion flame combustors and water or steam injection for NOx control. In fact, tests with up to 100% hydrogen have been conducted successfully (Cocchi and Sigali, 2010) in a 10 MW E-class gas turbine with steam injection. Similarly, aero-derivative gas turbines, used in the process industry for mechanical drive or by utilities as small peaking power plants, can burn up to 100% hydrogen at acceptable NOx levels when equipped with wet, low-emission diffusion combustors (European Turbine Network, 2020).
In high-efficiency F- and H-class utility gas turbines for large power generation, dry premix combustors dominate the market today and enable significantly lower NOx concentrations in line with the latest emission regulations. In such gas turbines, three major issues are encountered with fractions of hydrogen increasing to 100% compared to natural gas:
•
The fuel gas volume flow increases by a factor of 3 at the same pressure
•
The flame temperature increases, leading to thermal NO x concentrations beyond current emission limits reached with dry, low-NOx premix combustors
•
A flashback risk along the fuel/air mixture path due to higher flame velocity, leading to a danger of overheating of components upstream of the flame
Thus, modifications are needed on lean, premixed, dry, low-emission combustors to tailor velocities and cooling flow to deal with a much higher flame velocity and mitigate the flame temperature increase and associated NOx rise and material issues. These are engineering tasks currently underway at all major gas turbine manufacturers (European Turbine Network, 2020). Gas turbine combustors themselves account for a very small cost fraction of a power plant and are routinely replaced during major overhauls. No significant plant cost increases or changes in performance are expected once the combustors for 100% hydrogen are developed and on the market.
Dealing with the increase in-fuel gas volume flow requires changes and uprates in the fueling system such as bigger fuel compressors and larger pipes and valves to reduce pressure losses. Again, at least for new installations, no significant cost impact on the overall plant is expected. Besides the uprate to accommodate higher volume flows, safety systems must be upgraded to deal with the increased risk from leakages and the wider flammability range of hydrogen. Gas turbine startup may require a secondary fuel such as distillate or natural gas with safer characteristics than hydrogen. These safety measures will lead to small increases in cost and operational complexity but appear also to be very small compared to an entire combined cycle plant.
One gas turbine manufacturer that is building a combined heat and power plant, planned to be operating by 2023 for the municipal utility in Leipzig, Germany, announced that two, 62 MW industrial F-class units with dry low-emission combustors could run on 100% hydrogen and switch from natural gas to hydrogen as soon as this becomes available (Leipziger, 2020).
4.2.5 Stirling cycle
4.2.5.1 Cycle overview
The Stirling engine is named after Rev. Robert Stirling (1790–1878) of Scotland, who in 1816, patented a hot air engine comprised of all the elements that are also found in contemporary engines of this type (Walker, 1980; Stirling, 1816). The engine was aimed at harvesting the energy carried by a stream of hot gases to then convert it into motive power in the form of a rotating crankshaft, as shown in Fig. 4.25.
Sign in to download full-size image
Fig. 4.25. Artist view of the hot air engine originally created by Rev. R. Stirling (Finkelstein and Organ, 2001).
The cylinder holds two elements inside a displacer and power piston. These have coordinated, reciprocating displacement through coordinated crankshaft mechanisms actuated by the same wheel. The space between the displacer and power piston is where the cold gas is compressed, whereas the upper volume in Fig. 4.25, between displacer and cylinder head, is where the hot gas is expanded. The difference between the work produced by the hot gas during expansion and the work needed to compress the gas when it is cold yields the net work output of the engine.
The annulus between the displacer and cylinder walls enables the flow of gas between cavities, as dictated by the pressure gradient between cavities. This pressure difference is in practice determined mostly by the variable volumes of the cavities as piston and displacer move axially. Interestingly, halfway across the annulus, a regenerator exchanges energy with the gas in order to enhance the performance of the engine:
•
When hot gas coming from the upper cavity flows downwards, this heat exchanger harvests thermal energy from the stream which, in turn, becomes colder.
•
When cold gas coming from the compression space flows upwards, the stream absorbs energy from the regenerator, thereby increasing its temperature and raising thermal efficiency.
The timing mechanisms ensuring the relative displacement between power piston and displacer enables the variation of total volume inside the cylinder. Moreover, the timing is designed so as to make sure that most of the fluid is confined in the cold space during the compression phase, whereas most of it is in the hot space when expansion takes place.
4.2.5.2 Operating principle of an ideal Stirling cycle
The working cycle of an ideal Stirling engine is broken down into a number of processes as shown in Fig. 4.26 below. The schematic on the right shows the variation of the compression and expansion volumes (right and left respectively) as the piston and displacer move:
Sign in to download full-size image
Fig. 4.26. Breakdown of processes in an ideal Stirling engine (Finkelstein and Organ, 2001; Muñoz de Escalona et al., 2012).
•
1–2: Isothermal compression, driven by the decreasing volume within the compression space. In order to enable constant temperature during the process, heat rejection across the cylinder walls is needed.
•
2–3: Regenerator discharged. The relative motion of the piston and displacer drives the gas from the compression space into the expansion volume, even though the cumulative volume of both cavities remains unaltered. As the gas flows across the regenerator, its temperature increases thanks to the thermal energy discharged by the regenerator.
•
3–4: Isothermal expansion. The volume within the expansion space increases, while that of the compression space remains constant. Heat is added to the cylinder head to keep temperature constant. Pressure builds up due to the increasing temperature and not because the internal volume of the engine gets smaller.
•
4–1: Regenerator charged. At constant engine volume, the volume within the hot space decreases at the same rate as the volume of the cold space increases, thanks to the relative motion of displacer and power piston. Hot gases flow across the regenerator, which absorbs thermal energy from the gas flow and are cooled down in the compression space. Pressure decreases due to the temperature drop and not because of reduced internal volume.
The resulting working cycle is shown in Fig. 4.27: indicator (pressure vs. volume) diagram on the left and temperature vs. entropy on the right. The combination of both plots makes it clear that the cycle is comprised of isochoric heat addition (2–3) and rejection (4–1) and isothermal compression (1–2) and expansion (3–4). The volume within the cycle envelope of the indicator represents the work done per cycle, as it is also the case for an Otto or Diesel internal combustion engine.
Sign in to download full-size image
Fig. 4.27. Breakdown of processes in an ideal Stirling engine (Muñoz de Escalona et al., 2012).
In the case that the engine incorporates a perfect regenerator, the external heat addition to the engine would take place between 3 and 4, while heat rejection will only happen between 1 and 2. This is just a particular case of the generalized Reitlinger cycle. This means that, under such circumstances, the thermal efficiency of the engine would be the same as that of a Carnot cycle working between the same (hot) source and (cold) sink temperatures. However, the adoption of two isochoric processes in lieu of two isentropic process in the Stirling cycle yields a larger cycle work per unit displacement, which is a significant advantage of the Stirling cycle over the Carnot cycle.
From the considerations above, the indicated (cycle) work of an ideal Stirling engine can be expressed as follows:
W 12=−∫1 2 p·dV=−∫1 2 m·R·T c V·dV=−m·R·T c·ln V 2 V 1
W 34=∫3 4 p·dV=∫3 4 m·R·T H V·dV=m·R·T H·ln V 4 V 3
W ind=W 34+W 12=m·R·T H−T c·ln CR V
where:
•
m: mass stored in the engine
•
R: gas constant
•
T c: cooler temperature
•
T H: heater temperature
•
CR V: volumetric compression ratio
For an ideal Stirling engine, the heat absorbed by the regenerator in the process between 4 and 1 equals the amount of heat released between 1 and 2. Therefore, the amount of heat added to the engine can be expressed as follows:
Q 34=m·c v·T 4−T 1=m·c v·T H−T L=−Q 12
Using this information, the indicated efficiency of the ideal Stirling engine is:
η ind=W ind Q 34=m·R·T H−T c·ln CR V m·c v·T H−T L=m·R·T H−T c·ln CR V m·R γ−1·T H−T L=γ−1·ln CR V
where γ is the ratio of specific heat (isentropic exponent) of the working fluid. This efficiency can of course be expressed in an equivalent form by realizing the amount of heat added to the cycle during the expansion process (3 to 4) equals the amount of work developed during the same process, in order to keep temperature constant: Q 34=W 34.
η ind=W ind Q 34=m·R·T H−T c·ln CR V m·R·T H·ln CR V=1−T c T H
4.2.5.3 Non-ideality of a practical Stirling cycle
The performance of the ideal Stirling engine described in the foregoing section is nevertheless not achievable in practice. The first departure from the ideal performance is found in the variation of volume in the compression and expansion spaces reported in Fig. 4.26, which cannot be replicated in practice if conventional mechanical drives are used.
Indeed, various engine layouts have been used in the past to resemble the variations of volume in Figs. 4.26 and 4.27 as much as possible. These are shown in Fig. 4.28:
Sign in to download full-size image
Fig. 4.28. Configurations of kinematic Stirling engines (Egas and Clucas, 2018).
A.
Alpha-type with crank drive: the engine makes use of two cylinders, hot and cold, each one of which incorporates a piston that varies the internal volume independently from the other cylinder. Both pistons are driven by the same crankshaft, but the corresponding rods are 90° out of phase.
B.
Beta-type with crank drive: the actual variation of engine volume is enabled by a power piston and displacer, as shown in Fig. 4.26 above, which are driven by a common crankshaft. The embodiment of the concept shown in Fig. 4.28 is not practical given that the displacer and piston are coaxial, and therefore, the need to accommodate two rods brings about eccentric forces on at least one of the elements inside the cylinder.
C.
Gamma-type with crank drive: in gamma-type engines, the piston and displacer are held within different cylinders, connected by a transfer pipe housing the regenerator. Both elements (piston and displacer) are driven by a common crankshaft with the rods phased 90°.
D.
Beta-type with rhombic drive: this configuration is also of the beta-type, but it incorporates a rhombic drive to ensure the relative linear motion and displacer within the cylinder. With respect to the crank drive in a beta engine, the rhombic drive enables purely linear motion of the rods, balanced forces exerted on the displacer and piston, and easier sealing of the (high-pressure) cylinder. On the negative side, the complexity and more demanding manufacturing tolerances compromise the cost-effectiveness of this solution.
E.
Alpha-type with Ross yoke drive: akin to the rhombic drive, the Ross yoke drive is aimed at eliminating lateral forces on the piston while still enabling a conventional crank drive. Even if lateral forces are not null, they are largely reduced with respect to the drive in layout A. The added complexity of the Ross yoke drive with respect to the standard crank drive with two-phased rods is evident in Fig. 4.28.
The utilization of a particular configuration in Fig. 4.28 has different implications. First and foremost, the particular variation of volume within the hot and cold spaces is different, so the internal pressure distribution varies with respect to the ideal p-V diagram in Fig. 4.26. As discussed by Egas and Clucas (2018), some configurations yield higher engine pressure ratio than others even if the bore and stroke of piston and displacer are set to the same values. This is a very relevant feature in as much as pressure ratio has a very strong influence on engine efficiency, although this does not mean that higher pressure ratios are always preferred. The pressure ratio of interest is in practice linked to the temperature gap between the energy source and sink.
Internal pressures along the cycle are also affected by the fact that the engine is comprised of voids and clearances in addition to the compression and expansion spaces. Thus, while the working fluid is confined either in the hot space or in the cold space (or partly) at any crank angle, a real engine sees a fraction of the working fluid stored in the regenerator and connecting pipes and in the dead volumes of the compression/expansion spaces.
The second departure from the ideal cycle is brought about by the limited heat transfer rates across the cold and hot cylinder walls. Indeed, isothermal expansion and compression require an infinite heat transfer rate which is not possible in a real engine. As a consequence of this, isothermal processes cannot be realized in practice. This heat transfer-related non-ideality is aggravated further by the adoption of a single-cylinder configuration, which makes it more difficult to keep the temperature difference between the hot and cold spaces for given set temperatures of the energy source and sink. Heat conduction along the cylinder wall is high for the expected materials used to manufacture the engine block, and this tends to decrease temperature in the expansion volume and increase it within the compression volume. As opposed to this, the adoption of two cylinders makes it much easier to build a large temperature difference between the walls of the compression and expansion spaces, hence enabling better engine performance.
The regenerator is also affected by the limited heat transfer rate between wire mesh (typical regenerator material) and working fluid. This means that there is always a temperature gap between these elements, regardless of flow direction, which limits the temperature change of the working fluid across the entire regeneration, and in turn, brings about a higher duty of the heater and cooler sections of the engine; the gas flows into the compression and expansion spaces at higher and lower temperatures, respectively. The consequence of this is a lower engine efficiency brought about by lower expansion work and higher compression work, also affected by the friction losses experienced by the flow of gas in a real engine, originally dismissed in the ideal cycle.
Last, crank-driven configurations are usually preferred over more complex drives, despite the potential benefits that the latter could bring. This is based on the lower costs, longer durability, and simplicity. Moreover, slight modifications of the conventional crank drive can help to offset some of the shortcomings of this solution. For instance, the adoption of intermediate pushrods between the crank-rod and piston head facilitates cylinder sealing and eliminates lateral forces on the pistons (i.e., mechanical losses due to rubbing between piston and cylinder wall), thanks to the linear motion of the former element (very much like in the Ross yoke drive).
The cumulative effect of the foregoing non-idealities is visible in the p-V diagram of a real two-cylinder (alpha-type) engine shown in Fig. 4.29; the right plot shows pressure and volume variations in the expansion and compression cylinders, and the resulting indicator diagram of the working space is shown on the left chart. All the features above are observed:
Sign in to download full-size image
Fig. 4.29. Indicator diagram of a practical two-cylinder (alpha-type) Stirling cycle. Left: indicator diagram of the working space (Martini, 1983). Right: p-V variations in the compression and expansion cylinders (Escalona, 2013).
•
Higher pressure during compression and higher minimum cycle pressure.
•
Lower pressure during expansion and lower maximum cycle pressure.
•
Constant volume processes are not visible anymore.
•
Lower area enclosed by the cycle (∮p dV) on the left, representing a lower cycle work. It is noted that the compound indicator diagram on the left plot of the figure is the result of the individual p-V, and this is accompanied by a substantial efficiency drop with respect to an ideal Stirling cycle working with the same boundary conditions and design specifications. In practice, a 50% efficiency drop with respect to the reference ideal Stirling cycle should be expected, and even for very well-designed engines, it is difficult to reduce this performance drop to values lower than 30%–35% (Walker, 1980).
Even if not strictly a departure of the cycle itself, it is worth noting that leakage of fluid across the seals and piston rings of the engine brings about additional pressure drops in the diagram in Fig. 4.29. This is particularly relevant when the working fluid is hydrogen, given (1) the low size of this molecule, (2) the need to find a compromise between the tightness of clearances and the rubbing of piston rings and rod seals, (3) leakage across pressure-control valves in certain engines, and (4) wear of the engine. Though this is not visible in the indicator diagram of a clean-and-new, refilled engine, the performance drop of the engine is experienced with time.
4.2.5.4 Performance of a Stirling engine
There are several approaches to estimating the performance of Stirling engines. A commonly agreed classification follows:
•
Zeroth-order models based on curve fitting of experimental engine data. The accuracy of these models relies on the similarity of the actual boundary conditions to the reference data (West, 1987).
•
First-order: these models rely on a number of assumptions simplifying the modeling of pressure and temperature at each crank angle, which hence correspond to those of the ideal cycle without losses. The most common 1st order model is the isothermal model, enabling the direct calculation of pressure for a known set of temperatures in the compression and expansion cylinders (Schimdt, 1871).
•
Second-order: these models are based on a first-order guess of pressure along the cycle which is later updated with a pressure correction factor. This pressure correction factor accounting for energy losses is uncoupled, meaning that it is not incorporated into the initial p-V calculation and yields pressure distributions that are closer to the real working cycle. The most common second-order model is the adiabatic model developed by Urieli (1977).
•
Third-order model: pressure and temperature calculations are coupled, and they are resolved simultaneously through conservation of mass, momentum, and energy at every node of the engine. NASA developed a number of these models in the past (Sullivan, 1989).
•
Fourth-order models rely on Computational Fluid Dynamics (Pistek and Novotny, 2007).
The accuracy of these models varies largely, yielding more accurate results in increasing order of complexity. Zeroth-order models are capable of yielding accurate performance estimates, but this is only the case when working with engines for which a wide experimental database exists. This is confirmed by the plot in Fig. 4.30, where several models are used to guess the power output of a four-cylinder Stirling engine at different shaft speeds. Second-order models exhibit superior performance prediction capabilities, though all models fail to account for the complex, unsteady aerodynamic effects at high shaft speed. Despite this, first- and second-order models are still very commonly used to have first estimates of engine performance.
Sign in to download full-size image
Fig. 4.30. Power output of a four-cylinder Stirling engine. Models of different order are used: West (0th), Isothermal (1th), Adiabatic, and modified adiabatic (2nd). Experimental data is also provided for reference (Muñoz de Escalona et al., 2012).
The performance of a Stirling engine (work output and efficiency) is mostly affected by swept volume, compression ratio, (average) internal pressure, maximum and minimum cycle temperatures, and working fluid characteristics. Estimates of output for given operating conditions and engine size can be obtained from Beale’s number, a non-dimensional number with the following definition:
Be=P p·f·V d
where:
•
P: power output [W].
•
p: mean cycle pressure [bar].
•
f: shaft speed [Hz]
•
V d: swept volume of the power piston [cm 3].
Beale’s number can be represented against maximum cycle temperature, yielding the plot presented in Fig. 4.31, where average and lower/upper confidence limits are provided. From this information, the output of a well-designed engine of given swept volume, shaft speed, and peak temperature can be estimated. Conversely, one could have a first estimate of the engine size that is needed to produce a certain amount of power.
Sign in to download full-size image
Fig. 4.31. Beale number as a function of peak cycle (heater) temperature (Walker, 1980).
Beale’s number does nevertheless not capture all the functional dependences of a Stirling engine performance. For instance, the lack of sensitivity to cooler temperature is noteworthy since this has a strong influence on engine efficiency, according to the Second Law of Thermodynamics. This is why the chart provides upper and lower bounds of the power output estimate for lower and higher cooler temperatures, respectively. According to Walker (1980), most of the engine data used to correlate Beale’s number to heater temperature in Fig. 4.31 correspond to 340 K cooler temperatures, so deviations from this value must be compensated for when using the chart.
The efficiency of a Stirling engine is affected by heater and cooler temperatures mostly, as already discussed in earlier sections. In addition to this, specific work and efficiency are also strongly influenced by compression ratio, but this is more of theoretical consideration, given that Stirling engines do not have the capability to achieve high volumetric compression ratios, as the void volume of the engine brought about by the intermediate heat exchangers is very large compared to dead volumes in reciprocating internal combustion engines. Therefore, in practice, compression ratio cannot be used to increase the output and efficiency of a Stirling engine, and hence, designers usually resort to increasing mean pressure to obtain higher specific outputs. Higher mean pressures imply higher mass stored in the engine, and therefore, higher output according to the earlier discussion (but it has a much weaker effect on efficiency). This can be observed in Fig. 4.32 where the influence of pressure on output and efficiency is shown for a United Stirling AB engine; the same chart shows the capacity of improved second-order models to replicate the performance of the engine at high speed.
Sign in to download full-size image
Fig. 4.32. Power output and efficiency of a United Stirling AB engine at different mean operating pressure and comparison against the 2nd order model developed by Muñoz de Escalona et al. (2012).
The expressions obtained for specific output and efficiency of the ideal cycle evidence the importance of the working fluid characteristics through γ and R. This is important for the ideal engine, but it becomes critical when real Stirling engines are considered since the nature of the working fluid has a dramatic effect on the heat transfer characteristics of heat exchangers as well as on friction losses across the ducts and void volume. This is shown in Fig. 4.33 below where the superior performance of hydrogen is observed. At low speed, air still enables a high enough transfer rate, and thus it can be used as an inexpensive working fluid. At higher speeds though, the difference in performance when hydrogen is used becomes very large.
Sign in to download full-size image
Fig. 4.33. Influence of working fluid on engine efficiency: hydrogen, helium, and air (Walker, 1980).
This superior performance of hydrogen does not come without certain challenges. First and foremost, the cost of hydrogen is higher than that of air. Additionally, the flammability range of hydrogen is very wide, as the mixture of hydrogen and air is combustible from H 2 concentrations between 4% and 74%. This poses safety concerns in applications where combustion or even high-temperature waste heat is involved. Finally, hydrogen is a very small molecule, and it is therefore very difficult to keep it confined within the engine. Tradeoffs between tightness and wear rate of seals and clearances, mechanical losses, and leakage flow are needed during the design phase of the engine, and in any case, these incur higher operating costs.
4.2.5.5 Final remarks
Stirling engines have been on the market for almost two centuries, but they seem to have never reached critical mass for wide adoption. The fundamentals of the working cycle announce excellent performance, but the practical limitations brought about by crank and rod mechanisms, void engine volume, and very importantly, heat transfer rates, limit the performance of these engines. Small-scale applications like domestic combined heat and power units or portable backup power in recreational yachts can still benefit from the smooth operation and low noise of the technology, while achieving fairly good efficiency. Nevertheless, in larger applications, the low specific output (kW/cm 3) translates into higher capital costs that compromise the economics of the technology. In order to overcome this problem, double-acting Stirling engines make use of an arrangement whereby adjacent cylinders are connected through a reciprocating element and heat exchanger. This is shown in Fig. 4.34 for a four, double-acting cylinder concept developed by Siemens (Martini, 1983). The reciprocating engine is connecting the hot expansion volume of one “unit engine” (above) and the cold compression volume of the adjacent one. The most interesting feature of this configuration is the 50% reduction of moving parts, hence lower mechanical losses, and the more compact configuration that enables significantly lower cost and footprint for a given output.
Sign in to download full-size image
Fig. 4.34. Four double-acting cylinder concepts (Martini, 1983).
Free-piston engines differ from kinematic engines in the lack of a driving piston/displacer mechanism. Rather, the reciprocating motion of the piston/displacer is a consequence of the pressure exerted by the gas and the action of gas/mechanical springs as shown in Fig. 4.35. The electric generator is of the linear type, and the overall concept enables a hermetic design with hardly any loss of working fluid. Additionally, the lack of a driving mechanism ensures that there are no lateral forces on the rods and pistons, and accordingly, this drastically reduces the wear rate of the moving elements. The cumulative effect of these features is a highly efficient engine with higher specific output, very low noise and vibrations, and little to no maintenance. On the negative side, controllability of output frequency response and stability (amplitude of the oscillations) are more complex than in kinematic engines for variable operating conditions (Walker, 1980; Zheng et al., 2017; Shahryar and Tavakolpour-Saleh, 2020).
Sign in to download full-size image
Fig. 4.35. Free-piston Stirling engine concept (top) (Shahryar and Tavakolpour-Saleh, 2020) and embodiment showing linear alternator (bottom) (Zheng et al., 2017).
Show more
View chapterExplore book
Read full chapter
URL:
Book 2022, Machinery and Energy Systems for the Hydrogen EconomyJoshua Schmitt, ... David Sánchez
Chapter
Microturbine Generation Power Systems
2017, Distributed Generation SystemsR. Noroozian, P. Asgharian
4.4.1 The Brayton Cycle
Large gas turbines operations are based on the Brayton thermodynamic cycle. Understanding of the Brayton cycle concept is an essential issue because MTs are a subset of gas turbines and they have a similar cycle.
This cycle was proposed by George Brayton (1830–92), who was an American engineer, in 1872. He applied this cycle for his patent that was for a reciprocating oil-burning engine. Today, gas turbines and jet engines based on the Brayton cycle are widely used in industry.
The Brayton cycle or Joule cycle is a thermodynamic cycle and needs to have a working fluid. Working fluid in a thermodynamic system is a liquid or gas that absorbs or transmits. Air is the working fluid of the Brayton cycle. This system is based on four steps: compression, heat addition, expansion, and heat rejection, and each cycle consists of two isobaric processes and two adiabatic processes. An isobaric process is a thermodynamic process in which pressure stays constant and an adiabatic process is a thermodynamic process without heat transferring between two systems.
In a Brayton open cycle, inlet fresh air at ambient conditions is running through a compressor and after that, the resulting compressed air enters the combustion chamber. In the combustor, fuel is added to air and ignited, which causes an increase of gases volume. Expanded gases run through a turbine and provide mechanical force for driving. Finally, combustion products are discharged to the atmosphere. Fig. 4.9 diagrams the open cycle that discharges air flow to the atmosphere and occurs in each cycle. The compressor power requirement is about 40%–80% of turbine power output, which is a large amount, so power loss reduction is necessary.
Sign in to download full-size image
Fig. 4.9. An open cycle gas turbine engine.
A typical close cycle and T-S and P-V diagram of an ideal Brayton cycle are represented in Fig. 4.10.
Sign in to download full-size image
Fig. 4.10. Brayton cycle: (A) A close cycle gas turbine engine and (B) P-V and T-S diagram.
A close cycle consists of four main stages, as follows:
•
Stage 1 to 2: Air compression through adiabatic process— fresh air enters the compressor in stage 1. Through stage 1 to 2, air is compressing and increasing temperature whereas entropy is constant and volume of the air is decreasing. At stage 2, high pressure and high temperature air is discharged from the compressor.
•
Stage 2 to 3: Air combustion through isobaric processes— in this stage, compressed air enters the combustion chamber. Fuel and air are mixed and burned at constant pressure. After the combustion process, gas volumetric expansion occurs and at stage 3, there is a very high temperature gas.
•
Stage 3 to 4: Air expansion through adiabatic process— at stage 3, working fluid enters the turbine to drive it. When hot gas runs through the turbine, volume is larger than before and gas pressure is decreasing, so the turbine is called an expander.
•
Stage 4 to 1: Air heat transfer through isobaric process— in stage 4, turbine exhaust gas is not proper for stage 1, thus heat transfer to the surroundingis necessary. After the heat transfer, there is a gas at atmospheric pressure as well as appropriate volume and temperature, which is ready to enter the compressor again.
The Brayton cycle network with respect to Fig. 4.10B is equal to area of a 1-2-3-4 close loop. Output efficiency of this cycle is an important parameter because it represents a reasonable operation or inappropriate performance of the gas turbine. The following equations are according to Fig. 4.10.
Added heat to cycle is equal to :
(4.1)Q in=h 3−h 2=C p T 3−T 2
where C p is heat capacity in constant pressure, T 2 and T 3 are temperatures in stages 2 and 3, respectively, and h is enthalpy. Heat rejection from turbine is as follows:
(4.2)Q out=h 4−h 1=C p T 4−T 1
From Eqs. (4.1) (4.2), net work is
(4.3)W n e t=Q in−Q out=C p T 3−T 2−T 4−T 1
Brayton cycle efficiency is equal to the ratio of net work to added heat. Efficiency is as follows:
(4.4)η=W n e t Q in=C p T 3−T 2−T 4−T 1 C p T 3−T 2=1−T 4−T 1 T 3−T 2
The isentropic process is an idealized thermodynamic process in which there is no heat transfer from the system to its surrounding. In the isentropic process, the following equation is true.
(4.5)T 2 T 1=P 2 P 1 γ−1/γ=V 1 V 2 γ−1,γ=C p C v=c p c v
where γ is heat capacity ratio, C is heat capacity, and c is specific heat capacity that index p and v are related to constant pressure and constant volume, respectively. Similarly,
(4.6)T 3 T 4=P 3 P 4 γ−1/y
According to Fig. 4.3, because P 2=P 3 and P 1=P 4, thus
(4.7)T 2 T 1=T 3 T 4=P 2 P 1 γ−1/γ
(4.8)T 4−T 1 T 3−T 2=T 1 T 2=T 3 T 4
From Eqs. (4.8) (4.4), efficiency is as follows:
(4.9)η=1−T 1 T 2=1−T 3 T 4
Pressure ratio is called r p and Eq. (4.10) is rewritten Eq. (4.7).
(4.10)T 2 T 1=P 2 P 1 γ−1/γ=r p γ−1/γ
Finally, the efficiency of the Brayton cycle is as follows:
(4.11)η=1−1 r p γ−1 γ=1−r p 1−γ γ
Note, all mentioned equations are based on ignorance of kinetic energy and potential energy changes. According to Eq. (4.11), efficiency depends on temperature and pressure ratio but maximum and minimum temperatures are limited. Maximum temperature is limited by materials because turbine blades cannot withstand high temperatures and minimum temperature is defined by the temperature of inlet air to the engine.
Example 4.1
A gas turbine power plant operating on an ideal Brayton cycle has a pressure ratio of 6. The gas temperature is 350°K at the compressor inlet and 1500°K at the turbine inlet. Utilizing the air-standard assumptions, determine (a) the gas temperature at the exits of the compressor and the turbine, (b) the back work ratio, and (c) the thermal efficiency.
Solution
The T-S diagram of the ideal Brayton cycle described is shown in Fig. 4.11. Note that the components involved in the Brayton cycle are steady flow devices.
(a)
The air temperatures at the compressor and turbine exits are determined from isentropic relations in Appendix B:
Process 1-2:T 1=350 K→h 1=350.49 kJ/kg,P r 1=2.379 A t compressor exit P r 2=P 2 P 1 P r 1=6×2.379=14.274→T 2=580 K,h 2=586.04 kJ/kg Process 3-4:T 3=1500 K→h 3=1635.97 kJ/kg,P r 3=601.9 A t turbine exit P r 4=P 4 P 3 P r 3=1 6×601.9=100.32→T 4=970 K,h 4=1011.9 kJ/kg
(b)
To find the back work ratio, we need to find the work input to the compressor and the work output of the turbine:
w comp,in=h 2−h 1=586.04−350.49=235.55 kJ/kg w turb,out=h 3−h 4=1635.97−1011.9=624.07 kJ/kg→r b w=w comp,in w turb,out=0.377
That is, 33% of the turbine work output is used just to drive the compressor.
(c)
The thermal efficiency of the cycle is the ratio of the net power output to the total heat input:
Q in=h 3−h 2=1635.97−586.04=1049.93 w n e t=w out−w in=624.04−235.55=388.49→η t h=w n e t Q in=0.37 or 37%
Sign in to download full-size image
Fig. 4.11. Brayton cycle T-S diagram of Example 4.1.
Example 4.2
In a Brayton cycle-based power plant, the air at the inlet is at 27°C, 0.1 MPa. The pressure ratio is 6.25 and the maximum temperature is 800°C. Find (a) the compressor work per kilograms of air, (b) the turbine work per kilograms of air, (c) the heat supplied per kilograms of air, and (d) the cycle efficiency. (Consider γ=1.4, which is the specific heat ratio value of air at room temperature)
Solution
Based on mentioned values: T 1=27°C=300 K, P 1=100 kPa, r p=6.25, T 3=800°C=1073 K, and thus
(a)
Process 1–2:
T 2 T 1=r p γ−1 γ=6.25 1.4−1 1.4=1.689→T 2=506.69 K,W comp=c p T 2−T 1=1.005 506.69−300=207.72 kJ/kg
(b)
Process 3–4:
T 3 T 4=r p γ−1 γ=6.25 1.4−1 1.4=1.689→T 4=635.29 K,W turb=c p T 3−T 4=1.005 1073−635.29=439.89 kJ/kg
(c)
Heat input,
Q in=c p T 3−T 2=1.005 1073−506.69=569.14 kJ/kg
(d)
Cycle efficiency,
η t h=W turb−W comp Q in=439.89−207.72 569.14=0.408 or 40.8%
There are three general methods for improving Brayton cycle efficiency: regeneration, reheat. and intercooling. Fig. 4.12 shows a Brayton cycle that simultaneously uses reheating, intercooling, and regeneration methods.
Sign in to download full-size image
Fig. 4.12. Improved Brayton cycle.
T-S and P-V diagrams for improved Brayton cycle are given in Fig. 4.13. According to these diagrams, surface increasing causes more net work and higher efficiency.
Sign in to download full-size image
Fig. 4.13. Diagrams of improved Brayton cycle.
Example 4.3
Solve Example 4.2 if a regenerator of 80% effectiveness is added to the plant.
Solution
With respect to Fig. 4.14,
ɛ=T 5−T 2 T 4−T 2=0.8→0.8=T 5−506.69 635.29−506.69→T 5=603.14 K
Sign in to download full-size image
Fig. 4.14. T-S diagram with a regenerator of Example 4.3.
and T 4, W comp, W turb remain unchanged.
The new input heat is equal to:
Q in=c p T 3−T 5=472.2 kJ/kg
so,
η t h=W trub−W comp Q in=439.89−207.72 472.2=0.492%or 49.2%
Show more
View chapterExplore book
Read full chapter
URL:
Book 2017, Distributed Generation SystemsR. Noroozian, P. Asgharian
Chapter
The gas and steam turbines and combined cycle in IGCC systems
2017, Integrated Gasification Combined Cycle (IGCC) TechnologiesTing Wang
15.3.3 Recovering exhaust energy to increase Brayton cycle efficiency-Regeneration
The word regeneration means “recovering gas turbine exhaust energy for its own usage.” The common approach is to route the exhaust gas through a recuperator or a regenerator to preheat the compressed air prior to entering the combustor, as shown in Fig. 15.6A. The transfer of the waste heat can be illustrated by the two shaded areas as shown in Fig. 15.6B. The area underneath 4-y represents the amount of the recovered waste heat, while the area underneath 2-x represents the theoretically equal amount of heat that is regenerated from the waste heat to preheat the air entering the combustor. Considering a temperature difference is required to transfer heat, T y has to be higher than T 2 and T 4 higher than T x. The regenerator effectiveness is defined as
Sign in to download full-size image
Figure 15.6. Regenerative Brayton cycle.
(15.10)ε reg=m˙2(h x−h 2)m˙4(h 4−h 2)=The actual energy recovered The maximum possible energy can be recovered
If fuel is included, m˙4=m˙2+m˙fuel.
Note that both recuperators and regenerators are types of heat exchangers. In a recuperator, the hot and cold streams don’t mix and the heat is transferred through partitioned walls, while in a regenerator, the heat is alternately stored in and released from a solid, porous (or honeycomb-like) matrix as the hot and cold streams are routed through this matrix alternately. Although the hot and cold streams don’t directly mix, a portion of the mass in one stream inevitably remains in the matrix and is carried away by the other streams during the next cyclic process. An example of a regenerative gas turbine engine is the GT305 automotive engine built by General Motors.
Since the waste heat of the Brayton cycle is recovered and used in the same system, the Brayton simple-cycle efficiency is always increased by using the regeneration scheme. The combustor only needs to provide energy from x to 3 in Fig. 15.6, so the energy from 2 to x is saved. After regeneration, however, little or no exhaust energy can be used for combined cycle applications. Since the gas turbine waste heat is a lower-grade energy, it can be more effectively utilized to generate more power and achieve much higher cycle efficiency (58–60%) in a combined cycle arrangement using steam as the working medium than in a regenerative simple Brayton cycle (42–45%). The efficiency of an ideal Brayton cycle with 100% effective regeneration is
(15.11)η reg=1−T 1 T 3 P R k−1 k
Fig. 15.7 shows the efficiencies of ideal Brayton cycles with and without regeneration at different temperature ratios (T 3/T 1). The efficiency of the ideal Brayton cycle is represented by the heavy solid curve. When regeneration is implemented, a higher PR cycle actually reduces the overall cycle efficiency as represented by the lighter solid lines on the left side in Fig. 15.7. This is because the turbine exhaust temperature (T 4) decreases as PR increases (see T 4>T′4>T″4 in Fig. 15.4) and less energy is available for regeneration, until, eventually, T 4 dips below the compressor exit air temperature (T 2), resulting in an impractical condition for implementing the regeneration scheme. This impractical condition occurs when the regenerative cycle curves are extended below the ideal Brayton efficiency curve in Fig. 15.7. In this region below the ideal Brayton cycle efficiency curve, the regenerator actually adversely cools, instead of heating, the air leaving the compressor.
Sign in to download full-size image
Figure 15.7. The efficiencies of ideal Brayton cycles with and without regeneration at different temperature ratios (T 3/T 1). The dashed curves represent only the qualitative trend of efficiency reduction as PR reduces for nonideal regenerators.
The interception of a regenerative efficiency curve of a specific temperature ratio (T 3/T 1) with the non-regenerative efficiency curve in Fig. 15.7 represents the optimum pressure ratio (PR max work of Eq. (15.9)) for which the Brayton cycle produces the maximum W˙out for that specific temperature ratio. In the ideal condition when the effectiveness of the regenerator is assumed 100%, as PR reduces from the optimum value (PR max work), the efficiency of the regenerative Brayton cycle monotonously increases until it reaches the theoretical maximum efficiency of a Carnot cycle, η = 1 − (T 1/T 2). However, if the regenerator effectiveness is less than 100%, the regenerative Brayton cycle efficiency increases first with reduced PR from the optimum PR, then decreases as PR continuously decreases as shown in the dashed lines on the left side of Fig. 15.6.
Show more
View chapterExplore book
Read full chapter
URL:
Book 2017, Integrated Gasification Combined Cycle (IGCC) TechnologiesTing Wang
Review article
Molten salt for advanced energy applications: A review
2022, Annals of Nuclear EnergyRobin Roper, ... Paul Marotta
4.2.5 Supercritical CO 2 Brayton cycle
As suggested by the name, supercritical CO 2 is used as the working fluid in the SCO 2Brayton cycle. SCO 2 is non-toxic, non-corrosive, non-flammable, and non-explosive. (Liu et al., 2019) Similar to the He Brayton cycle, the addition of inert gases (including N, O 2, He, Ar, Xe, Kr, H 2 S, etc.) to improve cycle efficiency has been investigated extensively in recent years. (Guo et al., 2019)
As seen in Fig. 12, SCO 2 cycles are capable of achieving high cycle efficiencies for temperatures in the range of 450–700°C (Liu et al., 2019) than the He Brayton cycle. The SCO 2 Brayton cycle offers the added benefit of featuring a more compact turbine than those used for the He Brayton cycle. (Dolan, 2017) Mechanical methods of increasing SCO 2 Brayton efficiency include: (1) recovering heat from a recuperator or adding reheat; (2) reducing the inlet temperature of the compressor and adding intercooling; or (3) adding an appropriate bottoming cycle. (Liu et al., 2019)
The SCO 2 Brayton cycle is still experimental, indicating a TRL range of 4–5. Test loops in the United States, at facilities such as Sandia National Laboratory, (Wright et al., 2010) the Southwest Research Institute, (Turchi, 2014) and in South Korea, including at the Korea Advanced Institute of Science and Technology (Baik et al., 2016) and the Korea Institute of Energy Research (Shin et al., 2017), have been created for evaluating the performance of SCO 2 Brayton cycle components. Obtaining accurate measurements of the efficiencies of radial and axial gas turbines, centrifugal and pump-type compressors, and printed circuit heat exchangers (PCHEs) is an essential area of research for the SCO 2 Brayton cycle because the efficiencies of this turbomachinery can significantly impact the SCO 2 Brayton cycle efficiency. (Liu et al., 2019)
Interest in SCO 2 Brayton cycles for MSR coupling is limited at best. In recent years, molten salt applications of SCO 2 have focused on improving power production efficiencies of concentrating solar power (CSP). (Guo et al., 2019; Liu et al., 2019; Wang and He, 2017) Furthermore, Liang et al (Liang et al., 2015) suggests that SCO 2 Brayton cycles are most appropriate for coupling with other advanced reactor concepts, including the gas-cooled fast reactor, sodium fast reactor, and lead-cooled fast reactor.
View article
Read full article
URL:
Journal2022, Annals of Nuclear EnergyRobin Roper, ... Paul Marotta
Chapter
Vulnerability of Energy to Climate
2013, Climate VulnerabilityC.A. Miller
3.02.2.1 Thermoelectric Cooling: Reduction in Efficiency
Increasing ambient temperatures adversely affect the thermal efficiency of thermoelectric generating systems. This means that there would be an increase in fossil fuels consumption (almost entirely natural gas and coal) if average temperatures increase. The temperature effect impacts Rankine (steam) cycle plants and Brayton (combustion turbine) cycle plants. Rankine cycle efficiency is most sensitive to higher condenser temperatures (and pressures) that result from increases in ambient air and water temperatures. Brayton cycle efficiency is most sensitive to higher inlet air temperatures, which require increased work to compress the air prior to fuel injection and combustion. In addition, increases in ambient temperature can also reduce the power output of combustion turbines (Daycock et al. 2004; Wilbanks et al. 2008). Wilbanks et al. (2008) suggest that increases in ambient temperature from summer to winter could lead to reductions in power output of about 1–2% for coal and nuclear plants, and Daycock et al. (2004) suggest that power output could be reduced as much as 3–4% for each 10°F in ambient temperature for combustion turbines.
In the near term, thresholds for Rankine plants would most likely be in the form of discharge water temperature limits, which are in place to prevent significant changes in ambient water temperatures that can harm local ecosystems. The probabilities of reaching these limits is high in some areas, illustrated by the fact that there have already been instances where plants were required to reduce production to avoid exceeding discharge temperature requirements and more are expected (Averyt et al. 2011; Beshears 2007; O’Grady 2011). Over the longer term, measures to adapt thermoelectric power generationcooling systems to increasing ambient temperatures would likely be coupled with measures to adapt to changes in water availability, which will be discussed in more detail below.
View chapterExplore book
Read full chapter
URL:
Reference work 2013, Climate VulnerabilityC.A. Miller
Chapter
Advanced Nuclear OpenAir-Brayton Cycles for Highly Efficient Power Conversion
2021, Molten Salt Reactors and Integrated Molten Salt ReactorsBahman Zohuri
5.2 Background
The major growth in the electricity production industry in the last 30 years has centered on the expansion of natural gas power plants based on gas turbine cycles. The most popular extension of the simple Brayton gas turbine has been the combined cycle power plant with the Air-Brayton cycle serves as the topping cycle and the steam-Rankine cycle serves as the bottoming cycle. The air-Brayton cycle is an open air cycle and the steam-Rankine cycle is a closed cycle. The air-Brayton cycle for a natural gas–driven power plant must be an open cycle, where the air is drawn in from the environment and exhausted with the products of combustion to the environment. The hot exhaust from the air-Brayton cycle passes through a heat recovery steam generator (HRSG) prior to exhausting to the environment in a combined cycle [1,2]. The HRSG serves the same purpose as a boiler in the conventional steam-Rankine cycle.
In 2007, gas turbine with combined cycle plants had a total capacity of 800 GW and represented 20% of the installed capacity worldwide. They have far exceeded the installed capacity of nuclear plants, though in the late 1990s they had less than 5% of the installed capacity worldwide (Ref. 3, Table 2.1, p. 7). There are a number of reasons for this. First the natural gas is abundant and cheap. Second combined cycle plants achieve the greatest efficiency of any thermal plant (Ref. 3, Table 3.2, p. 21). And third, they require the least amount of waste heat cooling water of any thermal plant.
A typical gas turbine plant consists of a compressor, combustion chamber, turbine, and an electrical generator. The working fluid is air [4-6]. The gas turbine exhaust is ducted to the HRSG.
The HRSG serves as the boiler in a typical closed-cycle steam plant. A steam plant consists of a steam turbine, a condenser, a water pump, an evaporator (boiler), and an electrical generator. The working fluid is water . In a combined cycle plant, the gas turbine and steam turbine can be on the same shaft to eliminate the need for one of the electrical generators. However, two-shaft two-generator systems provide a great deal of more flexibility at a slightly higher cost .
In addition to the closed loop for the steam, an open-loop circulating water system is required to extract the waste heat from the condenser. The waste heat extracted by this circulating water system is significantly less than that in a conventional thermal plant as the open air-Brayton cycle exhausts its waste heat directly to the air.
The layout for a typical combined cycle power plant is given in Fig. 5.1.
Sign in to download full-size image
Fig. 5.1. Typical gas turbine combined cycle power plant.
The approximate efficiency can be calculated for a combined cycle power plant by the following simple argument (Ref. 2, p. 38):
(5.1)Brayton cycle efficiency=W B Q i n=η B
(5.2)Heat to Rankine cycle=Q R=(1−η B)Q i n
(5.3)Rankine cycle efficiency=W R Q R=η R
(5.4)Overall efficiency=W B+W R Q i n=η T=η B Q i n+η R Q R Q i n=η B Q i n+η R(1−η B)Q i n Q i n=η B+η R−η B η R η T=η B+η R−η B η R
This efficiency has to be corrected for pressure losses and assumes that all of the heat in the Brayton exhaust is used in the HRSG. For a combustion gas turbine, this is not usually possible if condensation of the water in the exhaust products is to be avoided. The detailed models developed in this effort give a more accurate answer.
For a nuclear system to take advantage of combined cycle technology, there are a number of changes to the plant components that have to be made. The most significant, of course, is that the combustion chamber has to be replaced by a heat exchanger in which the working fluid from the nuclear reactor secondary loop is used to heat the air. The normal Brayton cycle is an internal combustion one. The working fluid is heated by the combustion of the fuel-air mixture in the combustion chamber. Four-fifths of the air (nitrogen) does not participate in the combustion. The walls of the combustion chamber can be cooled and peak temperatures in the working fluid can be significantly above the temperature that the walls of the chamber can tolerate for any length of time [6,9,10].
For a nuclear reactor system, the heat transfer is in the opposite direction. All reactor components and fluids in the primary and secondary loops must be at a higher temperature than the peak temperature of the gas exiting the heat exchanger. This severely restricts the peak temperature that can be achieved for the air entering the turbine. However, all is not lost.
In a typical combustion system, there are pressure losses approaching 5% of the total pressure needed to complete the combustion process (Ref. 9, Table 6.2, p. 363 and Ref. 10, Fig. 6.20, p. 220). Heat exchangers can be built with significantly lower pressure drops than 5% of the entering pressure, possibly approaching 1% (Ref. 11). The most straightforward method to overcome this severe temperature limitation is to borrow a technique from steam power plants and implement multiple reheat cycles. That is, the first heat exchanger heats the air to its peak temperature. Then the air is expanded through the first turbine. The air is then reheated to the same peak temperature and expanded through the second turbine. Based on the relative pressure losses that appear possible, up to five turbines might be considered. All five turbines would drive the same compressor .
Multiple compressors on concentric shafts driven by different sets of turbines might be possible, but that has not been considered here. Multiple reheat cycles allow more heat to be put into the working fluid at a higher temperature. This improves the efficiency of the overall cycle. Interestingly enough, this does not improve the efficiency of the Brayton cycle, but because the exit temperature from the last turbine is higher, it does improve the efficiency of the Rankine part of the cycle with a net gain for the overall cycle. For this to work, the reactor coolant temperature must reach temperatures significantly higher than current light water reactor (LWR) temperatures. Even sodium-cooled reactor exit temperatures in the 550 °C range are not quite high enough to get an NACC to compete with the efficiency of a pure steam-Rankine cycle. But when the coolant exit temperatures reach the 650–700 °C range, the combined cycle systems, with multiple turbines, surpass the performance of steam-Rankine systems. Therefore, the analysis that follows is targeted at a fluoride salt reactor and a lead coolant reactor . A pressurized sodium reactor that could reach these temperatures is another possibility, but no one is proposing such a system at this time. It could also apply to a high-temperature gas reactor, but the heat exchangers would be quite different. Gas-to-gas heat exchangers have not been considered as the primary heat exchangers at this point but will be addressed in the recuperated systems.
Liquid metal and molten salt heat exchangers were developed and tested successfully [11,15-17] during the Aircraft Nuclear Propulsion program in the late 1950s. They were conventional tube and plate exchangers and were tested for over 1000 h at temperatures up to 1100 K. The largest size tested transferred 55 MW of heat in a package of approximately 1.2 m 3. The heat transfer area on the air side had a surface area per unit volume of1200 m 2/m 3. Certainly, some development will be needed to bring this technology up to modern standards, but the tasks involved do not appear insurmountable.
A number of additional heat exchangers were designed in this work in an attempt to estimate sizes of components and validate that pressure drop criteria could be met. The heat from the exhaust of the air-Brayton cycle transfers heat to vaporize the steam in the steam-Rankine cycle in an HRSG of a fairly conventional design. This includes air-to-steam superheaters as well as an economizer and evaporator section. A condenser of conventional design is included. For this work, all heat exchangers were considered to be counterflow designs .
The heat exchanger design procedures and experimental data were taken from the text by Kays and London . All of the data presented in this text were developed from steam-to-air heat exchangers and should be particularly applicable to the types of heat exchangers used in this analysis. The only ones not using these two fluids are the molten salt (or liquid metal)-to-air heat exchangers referenced above. Thus, this is a real basis for all of the design calculations used to estimate power conversion system performance and sizing.
One of the significant advantages of the combined cycle power system over current LWR power systems is its reduced requirement for circulating water in the waste heat rejection loop for the steam-Rankine cycle. The typical combined cycle plant considered here produces approximately 50% of its power from the steam-Rankine cycle and 50% of its power from the air-Brayton cycle.
This automatically reduces the cooling water requirement by half. In addition, the combined cycle plant achieves 45% efficiency so that only 55% of the heat generated has to be released as waste. A typical 25-MW(electric) system will only need to get rid of 6.9 MW of heat via a circulating water system. A current LWR plant generating 25 MW (electric) at an efficiency of 33% would need to dump 16.8 MW of heat. This represents a major saving in circulating water requirements.
As the combined cycle reduces the circulating water requirement so significantly, the natural question arose as to whether it could be eliminated completely. With a recuperated air-Brayton cycle, it can be. All of the waste heat can be rejected directly to the atmosphere. At first it was thought that the efficiency of a multiturbine recuperated cycle could not compete with a combined cycle plant. However, after performing the detailed analysis, the efficiencies of a recuperated cycle system come within 1% or 2% of the predicted combined cycle efficiencies. This would seem to be a minor penalty to pay for being free of a circulating water requirement. However, achieving these high efficiencies requires a very effective recuperator, which can become quite large.
Show more
View chapterExplore book
Read full chapter
URL:
Book 2021, Molten Salt Reactors and Integrated Molten Salt ReactorsBahman Zohuri
Review article
Molten salt for advanced energy applications: A review
2022, Annals of Nuclear EnergyRobin Roper, ... Paul Marotta
4.2.7 Air Brayton combined cycles
Air Brayton combined cycles coupled to nuclear reactors (also known as Nuclear Air Brayton combined cycles or NACCs) are capable of accepting heat from a nuclear and non-nuclear source(s) to facilitate operation in two modes: base-load and peak electricity. (Dolan, 2017) Both liquid-fueled MSRs and fluoride high-temperature reactors (FHRs) have been considered for NACC systems. Such cycles have the potential of achieving higher efficiencies than standalone natural gas plants. (Forsberg and Peterson, 2016) The concept of a NACC is similar to a commercial gas turbine combined cycle (GTCC), in which natural gas provides baseload energy, and a Rankine bottoming cycle is driven by a heat recovery steam generator. In a NACC, the gas turbine combustion chamber is replaced by a nuclear-driven heat exchanger. (Fathi et al., 2018)
Nuclear heat is used to provide base-load power. During peak electricity times, a topping cycle, such as natural gas or hydrogen firing, can be used to input additional heat, because the temperatures within an MSR do not approach maximum allowable gas turbine temperatures. As seen in Fig. 13, traditional NACC efficiencies are well below SCO 2Brayton cycle efficiencies at the lower end of MSR operating temperatures; however, the addition of a recuperator and intercooler (RIC) results in efficiencies near those of an SCO 2 Brayton cycle. (Fathi et al., 2018) Another option is the use of a nuclear air Brayton recuperated cycle (NARC) with a RIC, which can achieve efficiencies greater than that of an SCO 2 Brayton cycle over the range of MSR operating temperatures.
Sign in to download hi-res image
Fig. 13. Cycle efficiencies for SC CO2 Brayton, NACC, NACC&RIC, and NARC&RIC (Fathi et al., 2018).
View article
Read full article
URL:
Journal2022, Annals of Nuclear EnergyRobin Roper, ... Paul Marotta
Chapter
Nuclear power
2017, Fundamentals and Applications of Supercritical Carbon Dioxide (sCO₂) Based Power CyclesJ.J. Sienicki, A. Moisseytsev
13.1 Benefits of supercritical CO 2 cycles for nuclear power
A number of benefits have been envisioned from the utilization of supercritical CO 2 (sCO 2) Brayton cycle power conversion for nuclear power applications. For terrestrial nuclear power plants (NPPs), those benefits have generally been envisioned with applications to advanced nuclear reactor types that offer performance enhancements beyond the light water reactor (LWR) NPPs currently being deployed. Current LWR NPPs have nuclear reactor core outlet temperatures that are too low (e.g.,≈287–327°C) to realize a greater efficiency by switching to an sCO 2 Brayton cycle than is already achieved with a Rankine saturated steam cycle (e.g., a net plant efficiency of 33–36%, depending on the design). However, advanced nuclear reactor concepts and designs generally raise the core outlet temperature significantly above current LWR core outlet temperatures such that the performance of sCO 2Brayton cycles is comparable to or exceeds Rankine steam cycles or gas Brayton cycles in efficiency. For an advanced NPP, a goal is generally to reduce the NPP capital cost per unit output electrical power (i.e., $/kWe) or the levelized cost of electricity(LCOE). sCO 2 Brayton cycles have been viewed as an approach to reducing the NPP capital cost per unit output electrical power or the LCOE below that with a Rankine steam cycle or a gas Brayton cycle. This might make the NPP more attractive to potential owners and operators.
Applications of the sCO 2 Brayton cycle to sodium-cooled fast reactor (SFR) designs have revealed and confirmed multiple benefits (Sienicki et al., 2015). SFRs are under development in the Russian Federation, India, China, France, Japan, and South Korea. These nations currently have or have had plans to transition to a fast reactor economy based upon SFRs and a closed fuel cycle in the future, recognizing the resource limitations of continuing the use of LWRs. The use of SFRs together with a closed fuel cycle can provide a factor of 50 greater utilization of the energy content of uranium resources than LWRs. The US. Department of Energy (US. DOE) Office of Nuclear Energy (NE) is conducting research and development on SFRs. An SFR utilizes low-pressure sodium liquid metal coolant that has a high atmospheric boiling point (881.55°C) such that it is a low-pressure coolant that does not flash following a postulated boundary rupture. It has a low freezing temperature (97.85°C) such that it is practical to perform refueling and maintenance operations with the coolant molten. It has a high thermal conductivity resulting in a low Prandtl number (0.005) making it an outstanding liquid metal heat transfer fluid permitting heat removal from high power density fast neutron spectrum reactor cores. It has only one disadvantage, namely, its combustibility/chemical reactivity with air and water/steam. However, this is not a safety problem. Design approaches have been successfully developed to accommodate the possible reaction of sodium with water following the postulated failure of one or more steam generator tubes as well as the possible burning of sodium following the postulated release of sodium into an air-filled compartment. However, those design approaches add to the capital cost of the NPP.
SFRs typically have a core outlet temperature in the range of 510–550°C. At the higher temperature, a Rankine superheated steam cycle provides a gross cycle efficiency (i.e., the generator output electrical power divided by the rate of heat input to the cycle) of about 43%; the net NPP efficiency is less than this due to the electrical power requirements of the sodium pumps and other electrical power consumers. Replacing the superheated steam cycle with a sCO 2 Brayton cycle provides a similar gross efficiency of about 43% or slightly higher. Fig.13.1 compares the general cycle efficiencies of sCO 2 Brayton, helium gas Brayton, superheated steam, and supercritical water cycles coupled to an SFR as the cycle turbine inlet temperature increases. In deriving the efficiencies, limitations on SFR temperatures due to material loss of strength and other phenomena are neglected. For cycles coupled to other reactor types, the curves could be different. As the turbine inlet temperature increases, the sCO 2 Brayton cycle efficiency increases at a greater rate than either the superheated steam cycle or supercritical water cycle.
Sign in to download full-size image
Figure 13.1. Comparison of cycle efficiency of supercritical CO 2 Brayton cycle with cycle efficiencies of other power conversion cycles coupled to an sodium-cooled fast reactor.
The sCO 2 cycle is well matched to the SFR system. As the CO 2 expands inside of the turbine from an inlet pressure of about 20 MPa to an outlet pressure of about 7.8 MPa, the CO 2 temperature decreases by about 113°C. The primary sodium coolant temperature rises by about the same increment of 150°C inside of the reactor core. By effectively recuperating the sCO 2 cycle, it is possible to deliver CO 2 to the sodium-to-CO 2 heat exchanger at a temperature that is close to, although less than, the sodium core inlet temperature. This contributes to enhancing the cycle efficiency.
It is expected that the capital cost of an sCO 2 Brayton cycle for an SFR will be reduced below that for a superheated steam cycle. First, the cost of the sCO 2turbomachinery, which is relatively small, is expected to be significantly less than that for a superheated steam cycle with its separate high-pressure, larger intermediate pressure, and still larger low-pressure turbines. However, the sCO 2 Brayton cycle requires large heat exchange areas for the high-temperature recuperator and low-temperature recuperator (LTR). The cost associated with the recuperators, the sodium-to-CO 2 heat exchangers, and the cooler could tend to offset the cost savings associated with the sCO 2 turbomachinery.
Use of an sCO 2 Brayton cycle eliminates the need to detect and accommodate sodium–water reactions. It is necessary to incorporate instrumentation to detect small sodium leaks in sodium–water steam generator tubes before they can grow to large leaks as well as a system to accommodate the postulated double-ended guillotine rupture of a steam generator tube. Those systems add to the capital cost. However, sodium does react with CO 2. Use of an sCO 2 Brayton cycle and compact sodium-to-CO 2 heat exchangers requires an understanding and accommodation of the potential effects of sodium–CO 2 interactions. At this time, experimental investigations of sodium–CO 2 interactions are ongoing (Miyahara et al., 2009; Ishikawa et al., 2005; Eoh et al., 2010, 2011a,b; Gerardi et al., 2016, 2015). It is expected that sodium–CO 2 interactions will be found to be relatively benign compared with sodium–water reactions such that the cost of systems to detect and accommodate sodium–CO 2 interactions will be less than the cost of systems to detect and accommodate sodium–water reactions. However, sodium and CO 2 can form solid reaction products that do not readily dissolve in sodium. It will be necessary to incorporate systems to clean up solid reaction products from postulated sodium–CO 2 interactions.
The small turbomachinery and compact heat exchangers enable the footprint of the sCO 2 Brayton cycle to be reduced relative to that of a superheated steam cycle power converter. This can also contribute to a lower NPP cost by reducing the size and, therefore, the cost of the turbine generator building.
Using cost information available in 2010 and 2011, initial unreported estimates of the capital costs of an SFR sCO 2 Brayton cycle and superheated steam cycle power converters yielded comparable costs. Subsequently, there have been efforts to reduce the cost of compact diffusion-bonded heat exchangers suitable for use with sCO 2 cycles. If significant reductions in heat exchanger cost can be demonstrated, then it is expected that the cost of an sCO 2 Brayton cycle power converter can be made significantly lower than that of a superheated steam cycle power converter to produce a significant reduction in the SFR NPP capital cost per unit output electrical power or the LCOE. The potential reduction in cost is the main driver for interest in sCO 2 power conversion for SFRs. The potential for reducing the power converter cost is the main driver for development of less costly compact heat exchanger technologies suitable for use with sCO 2 cycles.
There are advanced nuclear reactor designs with alternate primary coolants that may allow higher core outlet temperatures, for example, up to 700°C. Examples are lead-cooled fast reactor (LFR) concepts and designs that use lead (Pb) or lead–bismuth eutectic (LBE, 44.5 wt% Pb – 55.5 wt% Bi) liquid metal coolants. Both have high boiling temperatures and are therefore low-pressure coolants. Both are calculated to have little or no chemical reaction with CO 2. Extension of the reactor core outlet temperature to values above about 550°C will require the development and demonstration of an approach to protect fuel cladding and reactor structural materials from attack by lead or LBE. Structural materials and their protection approach will need to be codified for use with the appropriate American Society of Mechanical Engineers design codes before a reactor can be licensed. By raising the turbine inlet temperature, one can take advantage of the rise in sCO 2 cycle efficiency with increasing temperature, as is illustrated in Fig.13.1. Increasing turbine inlet temperatures reduces the NPP capital cost per unit output electrical power or the LCOE, for a fixed reactor output thermal power. From Fig.13.1, it is observed that the sCO 2 Brayton cycle efficiency increases at a greater rate with temperature than the superheated steam cycle or at a comparable rate with the gas Brayton cycle. This makes the sCO 2 cycle especially attractive for application to advanced nuclear reactor concepts and designs that seek to raise the core outlet temperature. A consideration is selecting the primary coolant temperature rise through the core to be comparable to the CO 2 temperature drop in the turbine; that is, to attempt to match the heat source and the power converter, similar to the situation realized for SFRs.
Another nuclear application is for shipboard propulsion for which the nuclear heat source is a water-cooled reactor with water at elevated pressure. The efficiency of an sCO 2 cycle is low for the core outlet temperature of a water-cooled reactor. However, efficiency is not the main consideration in propulsion. It is the compactness of an sCO 2 Brayton cycle that makes it attractive. The relatively small turbomachinery sizes together with the use of compact heat exchanger technologies enable the power converter mass and volume to be decreased relative to a steam cycle. The reduced mass and volume are the main benefits for shipboard propulsion applications.
When the turbine is utilized to drive the compressors with a common shaft or the compressors are driven by a separate turbine on a common shaft instead of by electric motors, then the compressors can continue to circulate CO 2 through the cycle, in the event that electrical power is not available. This feature enables the cycle to continue to remove heat from a reactor and reject heat to a heat sink when electric power is unavailable.
A universal benefit of the sCO 2 Brayton cycle is that it can be controlled over a wide range of electrical grid loads to enable load following. This is especially beneficial for small modular reactors (SMRs) deployed on small or remote electrical power grids. On a small or remote power grid, an SMR might be the largest producer of electricity such that a need to follow changes in load demand can be an essential attribute.
Show more
View chapterExplore book
Read full chapter
URL:
Book 2017, Fundamentals and Applications of Supercritical Carbon Dioxide (sCO₂) Based Power CyclesJ.J. Sienicki, A. Moisseytsev
Chapter
Comparative Assessment of Three Integrated Trigeneration Systems for Dairy Farms
2018, Exergetic, Energetic and Environmental DimensionsEren Sevinchan, Ibrahim Dincer
3 Thermodynamic Analysis
Energy balance equations are examined under the first law of thermodynamics for each unit of the systems as a control volume under the steady-state operations, then, results and some significant inlet parameters are tabulated in Table 1. Some of the inlet parameters are the same for all of them as well as some output values, but the overall efficiencies and the Brayton cycle efficiencies are different as listed in Tables 2–4.
Table 1. Input Data
| Systems | System 1 | System 2 | System 3 |
--- --- |
| Mass flow rate | 0.1531 kg/s | 0.3788 kg/s | 0.1871 kg/s |
| Electrical generator efficiency | 95% | 95% | 95% |
| Conversion ratio | 4 | 3 | 3 |
| Turbine inlet pressure | 405.3 kPa | 303.9 kPa | 20000 kPa |
| Organic Rankine Cycle |
| Organic cycle pump isentropic efficiency | 80% |
| Organic cycle turbine isentropic efficiency | 80% |
| Electrical generator efficiency | 80% |
| Electrical motor efficiency | 95% |
| Mass flow rate | 0.24 kg/s |
| Single-Effect Absorption Chiller |
| Overall heat transfer coefficient of the absorber | 75 kW/K |
| Overall heat transfer coefficient of the condenser | 80 kW/K |
| Overall heat transfer coefficient of the generator | 70 kW/K |
| Overall heat transfer coefficient of the evaporator | 95 kW/K |
| Effectiveness of solution heat exchanger | 70% |
| Heating |
| Water heating inlet temperature | 283K |
| Water heating outlet temperature | 338K |
| Air heating inlet temperature | 283K |
| Air heating outlet temperature | 298K |
| Ambient Conditions |
| Ambient pressure | 101.3 kPa |
| Ambient temperature | 293K |
Table 2. All Outputs of the Three Systems
| Empty Cell | System 1 | System 2 | System 3 |
---
| η Brayton | 27.37% | 25.61% | 24.31% |
| η ORC | 15.61% | 17.19% | 13.93% |
| η overall | 84.1% | 63.27% | 84.4% |
| W˙C,b(kW) | 21.87 | 40.97 | 65.73 |
| W˙T,b(kW) | 86.73 | 136.6 | 101 |
| W˙e(kW) | 58.38 | 86.09 | 31.78 |
| W˙T,ORC(kW) | 16.72 | 20.06 | 20.06 |
| W˙e,ORC(kW) | 15.08 | 18.06 | 18.06 |
| Coefficient of performance | 0.70 | 0.70 | 0.70 |
| Q˙heat(kW) | 30.07 | 30.07 | 30.07 |
Table 3. Thermodynamic Properties of the Stations of System 1
| State | m˙ (kg/s) | T (K) | h (kJ/kg) | P (kPa) |
--- ---
| 1 | 0.1531 | 293 | 293.4 | 101.3 |
| 2 | 0.1531 | 434.5 | 436.2 | 405.3 |
| 3 | 0.03403 | 983 | – | – |
| 4 | 0.1531 | 1473 | 1603 | 405.3 |
| 5 | 0.1531 | 992 | 1073 | 101.3 |
| 6 | 0.1531 | 473 | 475.7 | 101.3 |
| 7 | 0.1531 | 422.3 | 423.9 | 101.3 |
| 8 | 0.1531 | 314.7 | 315 | 101.3 |
| 9 | 0.2 | 365.9 | 161.3 | 2000 |
| 10 | 0.2 | 549 | 708.5 | 2000 |
| 11 | 0.2 | 432.9 | 624.9 | 35.7 |
| 12 | 0.2 | 365 | 157.5 | 35.7 |
| 13 | 0.037 | 336.5 | 2618 | 5.4 |
| 14 | 0.037 | 307.5 | 144 | 5.4 |
| 15 | 0.037 | 278.2 | 144 | 0.87 |
| 16 | 0.037 | 494.2 | 4460 | 0.87 |
| 17 | 0.173 | 303.4 | 67.3 | 0.87 |
| 18 | 0.173 | 303.4 | 67.3 | 5.4 |
| 19 | 0.173 | 332.2 | 128.3 | 5.4 |
| 20 | 0.037 | 365 | 244.5 | 5.4 |
| 21 | 0.037 | 321.9 | 168.5 | 5.4 |
| 22 | 0.037 | 329 | 168.5 | 0.87 |
Table 4. Thermodynamic Properties of the Stations of System 2
| State | m˙ (kg/s) | T (K) | h (kJ/kg) | P (kPa) |
--- ---
| 1 | 0.3788 | 293 | 293.4 | 101.3 |
| 2 | 0.3788 | 434.5 | 436.2 | 405.3 |
| 3 | 0.02196 | – | – | – |
| 4 | 0.3788 | 1173 | 1246 | 405.3 |
| 5 | 0.3788 | 857.5 | 885.6 | 101.3 |
| 6 | 0.3788 | 614 | 623.5 | 101.3 |
| 7 | 0.3788 | 598.7 | 599.4 | 101.3 |
| 8 | 0.3788 | 546.3 | 548.5 | 101.3 |
| 9 | 0.24 | 365.9 | 161.3 | 2000 |
| 10 | 0.24 | 549 | 708.5 | 2000 |
| 11 | 0.24 | 432.9 | 624.9 | 35.7 |
| 12 | 0.24 | 365 | 157.5 | 35.7 |
| 13 | 0.037 | 336.5 | 2618 | 5.4 |
| 14 | 0.037 | 307.5 | 144 | 5.4 |
| 15 | 0.037 | 278.2 | 144 | 0.87 |
| 16 | 0.037 | 494.2 | 4460 | 0.87 |
| 17 | 0.173 | 303.4 | 67.3 | 0.87 |
| 18 | 0.173 | 303.4 | 67.3 | 5.4 |
| 19 | 0.173 | 332.2 | 128.3 | 5.4 |
| 20 | 0.037 | 365 | 244.5 | 5.4 |
| 21 | 0.037 | 321.9 | 168.5 | 5.4 |
| 22 | 0.037 | 329 | 168.5 | 0.87 |
Before writing of the energy balance equations, the net electrical power of each system should be defined as:
(2)W˙n e t b r a y t o n=W˙T,b−W˙C,b
where W˙ is the power, and subscripts T,b and C,b indicate the turbine of the closed and open Brayton cycle and the compressor of the closed and open Brayton cycle. The net electricity generation per second is defined as:
(3)W˙n e t e=η g×W˙n e t b r a y t o n
where W˙n e t e is the net electrical output for Brayton cycle, and η g is the efficiency of the generator.
After the calculations, the electrical efficiency of the Brayton cycle is determined as:
(4)η b r a y t o n=W˙n e t e/Q˙i n
where Q˙i n is the total heat rate that is produced in the biomass burner by utilization of the fuel. Q˙i n is calculated by:
(5)Q˙i n=m˙f×L H V f
LHV f is the lower heating value of the fuel.
The net electrical power of the ORC is defined as:
(6)W˙n e t,ORC=W˙T,ORC−W˙p
where W˙ is the power, and subscripts, T,ORC and p indicate turbine and pump of the ORC unit. Then, the efficiency of the ORC can be defined as:
(7)η ORC=W˙n e t,ORC/Q˙i n,ORC
where Q˙i n,ORC determines the amount of heat that is transferred to ORC from the Brayton cycle. Q˙i n,ORC can be defined as:
(8)Q˙i n,ORC=−(m˙6×h 6−m˙5×h 5)
The mass balance and energy balance equations are defined for each system. Due to similarities of the systems, most of the balance equations are the same for all systems with different input and outputs. Balance equations of system elements that are only special and different from others will be examined in this section. The equations are given with state points that are estimated in the diagrams of the systems. The energy balance equations of the compressor and turbine of the Brayton cycles are written as:
(9)W˙C,b+m˙1×h 1=m˙2×h 2
(10)W˙T,b+m˙4×h 4=m˙5×h 5
where m˙1 equals to m˙2 and m˙4 equals to m˙5 because of the mass balance. The other energy and balance equations are given for the heat exchanger between the Brayton and organic Rankine cycle as:
(11)m˙9=m˙10
(12)m˙5=m˙6
(13)m˙5×h 5+m˙9×h 9=m˙6×h 6+m˙10×h 10
Another significant balance equation is written for heating processes. The equations of the heating processes are the same for all systems, and they are also similar for water heating and air heating. The mass and energy balance equations of the heat exchanger for air heating are given as:
(14)m˙a i r−i n=m˙a i r−o u t
(15)m˙7=m˙8
(16)m˙7×h 7+m˙a i r×h a i r−i n=m˙8×h 8+m˙a i r×h a i r−o u t
The next energy balance equation is for the turbine of the ORC unit, and it is also the same for all systems.
(17)W˙T,ORC+m˙10×h 10=m˙11×h 11
Finally, the energy and mass balance equations are written for the generator of the single-effect absorption chiller, which has one of the complicated balance equations:
(18)m˙19=m˙20+m˙13
(19)h 11×m˙11+h 19×m˙19=h 13×m˙13+h 20×m˙20+h 12×m˙12
The overall efficiency is another important performance criterion for the systems to compare them with each other, and the overall energy efficiency of the trigeneration systems become:
(20)η o v e r a l l=W˙n e t e+W˙n e t,ORC+Q˙h e a t i n g+Q˙e v p Q˙i n
where Q˙h e a t i n g is the total heating load of water and air, and Q˙e v p is the cooling load of the system.
Show more
View chapterExplore book
Read full chapter
URL:
Book 2018, Exergetic, Energetic and Environmental DimensionsEren Sevinchan, Ibrahim Dincer
Related terms:
Gas Turbine
Pressure Ratio
Solar Chimneys
Energy Storage
Combustor
Rankine
Turbine Inlet Temperature
Cycle Efficiency
Output Power
Power Output
View all Topics
Recommended publications
Applied Thermal EngineeringJournal
EnergyJournal
Energy Conversion and ManagementJournal
International Journal of Hydrogen EnergyJournal
Browse books and journals
About ScienceDirect
Remote access
Advertise
Contact and support
Terms and conditions
Privacy policy
Cookies are used by this site. Cookie settings
All content on this site: Copyright © 2025 or its licensors and contributors. All rights are reserved, including those for text and data mining, AI training, and similar technologies. For all open access content, the relevant licensing terms apply.
We use cookies that are necessary to make our site work. We may also use additional cookies to analyze, improve, and personalize our content and your digital experience. You can manage your cookie preferences using the “Cookie Settings” link. For more information, see ourCookie Policy
Cookie Settings Accept all cookies
Cookie Preference Center
We use cookies which are necessary to make our site work. We may also use additional cookies to analyse, improve and personalise our content and your digital experience. For more information, see our Cookie Policy and the list of Google Ad-Tech Vendors.
You may choose not to allow some types of cookies. However, blocking some types may impact your experience of our site and the services we are able to offer. See the different category headings below to find out more or change your settings.
You may also be able to exercise your privacy choices as described in our Privacy Policy
Allow all
Manage Consent Preferences
Strictly Necessary Cookies
Always active
These cookies are necessary for the website to function and cannot be switched off in our systems. They are usually only set in response to actions made by you which amount to a request for services, such as setting your privacy preferences, logging in or filling in forms. You can set your browser to block or alert you about these cookies, but some parts of the site will not then work.
Cookie Details List
Performance Cookies
[x] Performance Cookies
These cookies allow us to count visits and traffic sources so we can measure and improve the performance of our site. They help us to know which pages are the most and least popular and see how visitors move around the site.
Cookie Details List
Contextual Advertising Cookies
[x] Contextual Advertising Cookies
These cookies are used for properly showing banner advertisements on our site and associated functions such as limiting the number of times ads are shown to each user.
Cookie Details List
Cookie List
Clear
[x] checkbox label label
Apply Cancel
Consent Leg.Interest
[x] checkbox label label
[x] checkbox label label
[x] checkbox label label
Confirm my choices |
8628 | https://pangea.stanford.edu/ERE/pdf/IGAstandard/SGW/2003/Verma.pdf | PROCEEDINGS, Twenty-Eighth Workshop on Geothermal Reservoir Engineering Stanford University, Stanford, California, January 27-29, 2003 SGP-TR-173 A THERMODYNAMIC ASSESSMENT OF DISSOCIATION CONSTANT OF WATER Mahendra P. Verma Geotermia, Instituto de Investigaciones Electricas Av. Reforma 113, Col. Palmira, Apdo. 1-475 Cuernavaca, Morelos, 62000, México e-mail: mahendra@iie.org.mx ABSTRACT A thermodynamically consistent formulation for the dissociation constant of water (KW) is presented. Based on experimental data along the vapor-water saturation, the following equation for KW is derived for temperature 0 to 380ºC and pressure 1 to 250 bar 2 3 2 3 10 805175 .
1 T P 176057 .
0 T 5189145 10 678282 .
9 9581 559 02620 .
10 log P P T KW − − × + + + × − − + − = INTRODUCTION Knowledge of the dissociation constant of water (KW) at a specified temperature and pressure is fundamental in dealing the chemical modeling of aqueous systems in the laboratory or in nature. Working Group III of the International Association for Properties of Water-Steam (IAPWS) has compiled all the existing experimental values and established the representative values of KW for 0-1000ºC and 1-10,000 bar, which were published by Marshall and Franck (1981). The values are widely accepted and reproduction in handbooks on physical-chemical properties of water (e.g. Meyer et al., 1993; Lide and Frederikes, 1998). The present study scrutinizes the values of KW on the basis of basic laws of thermodynamics. Similarly, a formulation is presented to derive the consistent values of KW in the range approximately 0-380ºC and 1-250 bar. A STATE FUNCTION A thermodynamic variable (for example, pressure, temperature, volume, Gibbs free energy, internal energy, enthalpy or entropy represented by P, T, V, G, U, H or S, respectively), which does not depend on the past history of the substance or on the path it has followed in reaching a given state, is known as state function or point function (Smith and van Ness, 1975). A state function should be single valued and continuously differentiable unless there is a phase transition (Smith and van Ness, 1975; Chatterjee, 1991). On fixing the values of any two state functions (for example T and P), the values of all the other state functions are uniquely defined. Similarly, the equilibrium constant (K) for a chemical reaction is also a state function and is expressed as ∆ − + ∆ − = ∆ − = RT S RT H RT G K P T P T F P T F , , , 303 .
2 1 303 .
2 1 303 .
2 1 log …1 where R is gas constant, P is pressure and T is absolute temperature. F stands for formation. At low temperatures, the values of Gibbs free energy (∆GF T,P) or enthalpy (∆HF T,P) and entropy (∆ST,P) for the first order of approximation are constant and the variation of log K with 1/T is a straight line. There could be positive or negative deviation from the linear trend at higher temperatures, but the function should always be single valued (i.e. the trend of log K may be asymptotic at higher temperatures). Figure 1(a) shows a schematic diagram for the variation of log K with the inverse of T at constant P. Let us consider two other behaviors of log K as shown with curves I and II in Figure 1(b). In case I there are two values of log K at a given T, whereas in case II there are two values of T for a value of log K. Thus K(T) in case I and T(K) in case II are not single valued functions. Similarly, ∞ = dK dT in case I and ∞ = dT dK in case II. It means that T or K are not continuous differentiable. In other word either T or K is not thermodynamic state function in the respective cases. Figure 1(c) presents the behavior of Log K with 1/T at two pressures P1 and P2. The functions are crossing at temperature T1. Then at T=T1, ∞ = dK dP and ∞ = dT dP . It means that P is not a state function. It is well known that T, P and K are state functions. Therefore, the behaviors of log K presented in Figure 1(b) and 1(c) are against the basic laws of thermodynamics. Thus the permissible behaviors of log K with 1/T are as given in Figure 1(a). This is valid for any property (for example, viscosity, thermal conductivity, etc.) of a substance or system, which does not depend on the past history or on the path to reach a given state (Verma, 2002a). ∆G=0 ∆G=+ve constant ∆G=-ve constant 1/T log K ∆G=-∞ ∆G=+∞ May be asymptotic at high T +ve deviation -ve deviation 0 +ve -ve 0 log K 1/T 1/T 0 log K 2 T 3 1 2 K 1 3 I II P2 P1 K K T T 1 T a b c ∆G=0 ∆G=+ve constant ∆G=-ve constant 1/T log K ∆G=-∞ ∆G=+∞ May be asymptotic at high T +ve deviation -ve deviation 0 +ve -ve 0 log K 1/T 1/T 0 log K 2 T 3 1 2 K 1 3 I II P2 P1 K K T T 1 T a b c Figure. 1. A schematic diagram for the variation of log K with inverse of absolute temperature (after Verma, 2002a). DISSOCIATION CONSTANT OF WATER The most accepted values of dissociation constant of water (KW) from Marshall and Franck (1981) are plotted in Figure 2. Figure 2(a) shows the temperature behavior of KW. It can be observed that KW increases with T and P. It is well known that T and P produce, in general, adverse effect on the physical and chemical properties of a substance. The increase in KW with T and P does not seem reasonable. Accordingly, there will be an increase in the concentration of ionized water as the water is buried deeper into the Earth. The chemical reactivity of water should increase with P. It looks quite unreasonable. Additionally, KW has irregular behavior with T. For example, there is a drastic drop around 400ºC for the curve corresponding to 250 bar. Similarly, there are two temperatures for a given values of KW. It means that Kw or T is not a state function. Figure 2(b) shows the pressure dependence of KW. The curves corresponding to different temperatures are crossing each other. This is again against the basic laws of thermodynamics. Thus the values of KW reported by Marshall and Franck (1981) are violating fundamental laws of thermodynamics and are unreliable. Clever (1968) reviewed the earlier experimental data and techniques for the determination of KW. The mostly used techniques are electrical conductivity, emf of cells without transference and thermal measurements. In all the methods, a salt solution like of KCl is used and the measured properties are extrapolated to the zero salt effect, as it is difficult to measure the properties of pure water. Sweeton et al. (1974) performed a comprehensive study for the determination of KW along the water-vapor saturation and their values have been used for geochemical modeling at along the water-vapor saturation up to 300ºC. Since an aqueous solution in lab or in nature is not always along the saturation curve (Verma, 2002), a numerical formulation is carried out to derive the pressure and temperature dependence of KW. The equation for log KW as a function of temperature and pressure is written as 2 2 / / / log P f T P e T d P c T b a KW + + + + + = …2 where the values for the constants are a=-10.02620, b=559.9581, c=-9.678282 10-3, d=518914.5 and f=1.805175 10-5. Figure 3 shows the values of log KW calculated by eq. (2) together with the experimental values (Sweeton et al., 1974). The calculated and experimental values of KW are in good agreement. Mathematically, a quadratic function (for example Y= a+bX+CX2) must always have one value for two values of independent variable (X). Thus Y cannot be single valued for all the range of X. Since KW is a state function, the extrapolation of KW is only permissible up to approximately 380ºC and 250 bar. The calculated values of KW by eq. (2) are tabulated in Table 1. At P=1 bar, the feasible temperature range is 0 to 100ºC. If T is higher than 100ºC, the water -25 -20 -15 -10 -5 0 500 1000 T (ºC) log KW 250 bar 500 750 1000 1500 2000 3000 5000 10000 Anomalous Increasing with T & P -25 -20 -15 -10 -5 0 2000 4000 6000 8000 10000 P (bar) log KW T= 0 ºC T= 25 T= 100 T= 300 T= 500 T= 800 T=1000 Crossing -25 -20 -15 -10 -5 0 500 1000 T (ºC) log KW 250 bar 500 750 1000 1500 2000 3000 5000 10000 Anomalous Increasing with T & P -25 -20 -15 -10 -5 0 2000 4000 6000 8000 10000 P (bar) log KW T= 0 ºC T= 25 T= 100 T= 300 T= 500 T= 800 T=1000 -25 -20 -15 -10 -5 0 500 1000 T (ºC) log KW 250 bar 500 750 1000 1500 2000 3000 5000 10000 Anomalous Increasing with T & P -25 -20 -15 -10 -5 0 500 1000 T (ºC) log KW 250 bar 500 750 1000 1500 2000 3000 5000 10000 Anomalous Increasing with T & P -25 -20 -15 -10 -5 0 2000 4000 6000 8000 10000 P (bar) log KW T= 0 ºC T= 25 T= 100 T= 300 T= 500 T= 800 T=1000 Crossing Figure. 2. Temperature and pressure dependence of log KW for the values reported by Marshall and Franck (1981). -15 -14 -13 -12 -11 1.5 2 2.5 3 3.5 1000/T (in K) log K Experimental Calculated Figure. 3. A comparison between the experimental (sweeton et al, 1974) and theoretical values of log KW along the water-vapor saturation curve. a 200 100 300 0 -16 -15 -14 -13 -12 -11 1.5 2 2.5 3 3.5 1000/T (in K) log KW P= 1 bar P= 10 P= 50 P=100 P=150 P=200 P=250 a 200 100 300 0 -16 -15 -14 -13 -12 -11 1.5 2 2.5 3 3.5 1000/T (in K) log KW P= 1 bar P= 10 P= 50 P=100 P=150 P=200 P=250 -16 -15 -14 -13 -12 -11 0 50 100 150 200 250 P (bar) Log KW T= 25ºC 50 100 150 200 250 300 350 b -16 -15 -14 -13 -12 -11 0 50 100 150 200 250 P (bar) Log KW T= 25ºC 50 100 150 200 250 300 350 b Figure. 4. Temperature and pressure dependence of log KW for the values derived in the present study. Table 1: Logarithm (base 10) of the dissociation constant of water (KW). P (bar) T (ºC) 1 10 50 100 150 200 250 0 -14.940 -15.020 -15.338 -15.654 -15.880 -16.016 -16.061 25 -13.995 -14.075 -14.395 -14.714 -14.943 -15.081 -15.129 50 -13.272 -13.352 -13.674 -13.995 -14.226 -14.367 -14.418 75 -12.708 -12.789 -13.112 -13.436 -13.669 -13.811 -13.864 100 -12.262 -12.343 -12.668 -12.992 -13.227 -13.372 -13.426 125 -11.984 -12.310 -12.636 -12.872 -13.018 -13.074 150 -11.692 -12.019 -12.347 -12.584 -12.731 -12.788 175 -11.452 -11.780 -12.108 -12.347 -12.495 -12.554 200 -11.581 -11.911 -12.150 -12.300 -12.359 225 -11.414 -11.745 -11.986 -12.136 -12.196 250 -11.274 -11.606 -11.847 -11.998 -12.059 275 -11.487 -11.729 -11.881 -11.943 300 -11.385 -11.628 -11.781 -11.843 325 -11.542 -11.695 -11.758 350 -11.621 -11.685 375 -11.621 380 -11.609 will be in vapor phase at P=1 bar. Similarly, at T=125ºC, the minimum pressure for the existence of water is 4.76 bar. Figure 4 shows the temperature and pressure dependence of KW for all the possible values from the vapor-water saturation to the maximum values of T and P. The behavior is consistent with the basic laws of thermodynamics. CONCLUSIONS The values of the dissociation constant of water for temperature 0 to 380ºC and pressure 1 to 250 bar, presented here are thermodynamically consistent and are in good agreement with the experimental values reported in the literature7. There are practically very little efforts to improve the analytical quality of experimental values of KW since 1974. Therefore the values of KW at higher temperatures and pressures need an evaluation for analytical uncertainty and its propagation in different formulation to study the temperature and pressure dependence of KW. REFERENCES Chatterjee, N.D. (1991). “Applied mineralogical thermodynamics”, Springer-Verlag, New York, 320pp. Clever, H.L. (1968). “The Ion Product Constant of Water”, Journal of Chemical Education, 45, pp. 231-235. Lide, D.R. and Frederikse, H.P.R. (eds.) (1998). “CRC Handbook of Chemistry and Physics”, CRC Press, London, pp. 8.56-8.57. Marshall, W.L. and Franck, E.U. 1981. “Ion Product of Water Substance, 0-1000ºC, 1-10,000 Bars New International Formulation and Its Background”, Journal of Physical Chemical Reference Data, 10, pp. 295-304. Meyer, C.A., McClintock, R.B., Silvestri, G.J. and Spencer, R.C. (1983). “Steam Tables: Thermodynamic and Transport Properties of Steam “, ASME Press, New York, pp. 147-152. Smith, J.M. and van Ness, H.C. (1974). “Introduction to chemical thermodynamics” McGraw-Hill Kogakusha, ltd., Tokyo, 632pp. Sweeton, F.H., Mesmer, R.E. and Baes, C.F. (1974). “Acidity Measurements at Elevated Temperatures. VII. Dissociation of Water”, Journal of Solution Chemistry, 3, pp. 191-214. Verma, M.P. (2002a). “Geochemical techniques in geothermal development”, In: D. Chandrasekharam and J. Bundschuh (eds.) Geothermal Energy Resources for Developing Countries, Swets & Zeitlinger Publishers, The Netherlands, pp. 225-251. Verma, M.P. (2002b). “A numerical simulation of H2O-CO2 heating in a geothermal reservoir”. Proceedings XXVII Workshop Geothermal Reservoir Engineering, Stanford University, Stanford, CA, pp.217-222. |
8629 | https://t5k.org/glossary/xpage/divisor.html | The Prime Glossary: divisor
Prime Glossary
Home
Index
Search
Contact
FAQ
PrimePagesPrimePages' Home5000 LargestTop 20 PrimesProver BiosPrime Curios!
• Previous• Next• Random
divisor
The divisors (or factors) of a positive integer are the integers that evenly divide it. For example, the divisors of 28 are 1, 2, 4, 7, 14 and 28. Of course 28 is also divisible by the negative of each of these, but by "divisors" we usually mean the positive divisors.
The proper divisors of the integer n are the positive divisors of n other than n itself. The proper divisors of 27 are 1, 3 and 9.
Prime numbers are those integers greater than one whose only divisors are one and themselves (so whose only proper positive divisor is one).
Several number theoretic functions are related to the divisors of n. For example, tau(n) (or ) is the number of divisors of n, and sigma(n) (or ) is their sum.
There is another use of the word divisor: when we divide an integer a by a non-zero integer b, to get a quotient and remainder (see the division algorithm), b is the divisor and a is the dividend.
See Also:Divides, EulersPhi, AbundantNumber
Printed from the PrimePages © Reginald McLean.
Collections
Prime Glossary
Prime Curios!
Proving Primality
Frequently Asked Questions
Proofs
Prime Lists
Largest Known Primes
Top 5000 List
Lists of Small Primes
Records for Different Types
Mersenne Primes
Pages
Donate
Largest Known by Year
Prover Bios
Top 20 Provers
Contact Us
Copyright © 2025PrimePages. (Privacy notice) |
8630 | https://www.geeksforgeeks.org/dsa/closest-pair-of-points-using-divide-and-conquer-algorithm/ | Minimum Distance between Two Points
Last Updated :
23 Jul, 2025
Suggest changes
110 Likes
You are given an array arr[] of n distinct points in a 2D plane, where each point is represented by its (x, y) coordinates. Find the minimum Euclidean distance between two distinct points.
Note: For two points A(px,qx) and B(py,qy) the distance Euclidean between them is:
Distance = (px−qx)2+(py−qy)2
Examples:
Input: arr[] =
Output: 1.414214
Explanation: The smallest distance is between points (1, 2) and (2, 3), which is 1.414214.
Input: arr[] =
Output: 2.236068
Explanation: The smallest distance is between points (-2, -2) and (-1, 0), which is 2.236068.
Table of Content
[Naive Approach] Checking All Pairs - O(n^2) Time and O(1) Space
[Expected Approach] Using Divide and Conquer - O(n log(n)) Time and O(n) Space
[Naive Approach] Checking All Pairs - O(n^2) Time and O(1) Space
The idea is to check the distance between all pairs of points and store the minimum distance found.
C++
````
// C++ program to find closet point
include
include
include
include
using namespace std;
// Function to compute Euclidean distance between two points
double distance(const vector& p1, const vector& p2) {
return sqrt((p1 - p2) (p1 - p2) +
(p1 - p2) (p1 - p2));
}
// Function that returns the smallest distance
// between any pair of points
double minDistance(const vector>& points) {
int n = points.size();
double minDist = 1e9;
// Brute force to check all pairs
for (int i = 0; i < n; ++i) {
for (int j = i + 1; j < n; ++j) {
double dist = distance(points[i], points[j]);
if (dist < minDist) {
minDist = dist;
}
}
}
// Return the smallest distance
return minDist;
}
int main() {
vector> points = {{-1, -2}, {0, 0}, {1, 2}, {2, 3}};
double res = minDistance(points);
cout << fixed << setprecision(6) << res << endl;
return 0;
}
````
// C++ program to find closet point
// C++ program to find closet point
```
include #include
```
```
include #include
```
```
include #include
```
```
include #include
```
using namespace std; using namespace std
// Function to compute Euclidean distance between two points
// Function to compute Euclidean distance between two points
double distance(const vector<double>& p1, const vector<double>& p2) {double distance const vector< double>& p1 const vector< double>& p2
return sqrt((p1 - p2) (p1 - p2) + return sqrt p1 0 - p2 0 p1 0 - p2 0 +
(p1 - p2) (p1 - p2)); p1 1 - p2 1 p1 1 - p2 1
}
// Function that returns the smallest distance
// Function that returns the smallest distance
// between any pair of points
// between any pair of points
double minDistance(const vector<vector<double>>& points) {double minDistance const vector< vector< double>>& points
int n = points.size(); int n = points size
```
```
double minDist = 1e9; double minDist = 1e9
// Brute force to check all pairs
// Brute force to check all pairs
for (int i = 0; i < n; ++i) {for int i = 0 i< n ++ i
for (int j = i + 1; j < n; ++j) {for int j = i + 1 j< n ++ j
double dist = distance(points[i], points[j]); double dist = distance points i points j
if (dist < minDist) {if dist< minDist
minDist = dist; minDist = dist
}
}
}
```
```
// Return the smallest distance
// Return the smallest distance
return minDist; return minDist
}
int main() {int main
vector<vector<double>> points = {{-1, -2}, {0, 0}, {1, 2}, {2, 3}}; vector< vector< double>> points = - 1 - 2 0 0 1 2 2 3
double res = minDistance(points); double res = minDistance points
cout << fixed << setprecision(6) << res << endl; cout<< fixed<< setprecision 6<< res<< endl
return 0; return 0
}
Java
````
// Java program to find closest point
import java.util.ArrayList;
import java.util.List;
import java.lang.Math;
class GfG {
// Function to compute Euclidean distance between two points
static double distance(double[] p1, double[] p2) {
return Math.sqrt((p1 - p2) (p1 - p2) +
(p1 - p2) (p1 - p2));
}
// Function that returns the smallest distance
// between any pair of points
static double minDistance(List<double[]> points) {
int n = points.size();
double minDist = Double.MAX_VALUE;
// Brute force to check all pairs
for (int i = 0; i < n; ++i) {
for (int j = i + 1; j < n; ++j) {
double dist = distance(points.get(i), points.get(j));
if (dist < minDist) {
minDist = dist;
}
}
}
// Return the smallest distance
return minDist;
}
public static void main(String[] args) {
List<double[]> points = new ArrayList<>();
points.add(new double[]{-1, -2});
points.add(new double[]{0, 0});
points.add(new double[]{1, 2});
points.add(new double[]{2, 3});
double res = minDistance(points);
System.out.printf("%.6f\n", res);
}
}
````
Python
````
Python program to find closet point
import math
Function to compute Euclidean distance between two points
def distance(p1, p2):
return math.sqrt((p1 - p2) 2 + (p1 - p2) 2)
Function that returns the smallest distance
between any pair of points
def minDistance(points):
n = len(points)
minDist = float('inf')
# Brute force to check all pairs
for i in range(n):
for j in range(i + 1, n):
dist = distance(points[i], points[j])
if dist < minDist:
minDist = dist
# Return the smallest distance
return minDist
if name == "main":
points =
res = minDistance(points)
print(f"{res:.6f}")
````
C#
````
// C# program to find closest point
using System;
using System.Collections.Generic;
class GfG {
// Function to compute Euclidean distance between two points
static double distance(double[] p1, double[] p2) {
return Math.Sqrt((p1 - p2) (p1 - p2) +
(p1 - p2) (p1 - p2));
}
// Function that returns the smallest distance
// between any pair of points
static double minDistance(List<double[]> points) {
int n = points.Count;
double minDist = double.MaxValue;
for (int i = 0; i < n; ++i) {
for (int j = i + 1; j < n; ++j) {
double dist = distance(points[i], points[j]);
if (dist < minDist) {
minDist = dist;
}
}
}
// Return the smallest distance
return minDist;
}
static void Main() {
List<double[]> points = new List<double[]> {
new double[] {-1, -2},
new double[] {0, 0},
new double[] {1, 2},
new double[] {2, 3}
};
double res = minDistance(points);
Console.WriteLine(res.ToString("F6"));
}
}
````
JavaScript
````
// JavaScript program to find closest point
// Function to compute Euclidean distance between two points
function distance(p1, p2) {
return Math.sqrt((p1 - p2) 2 + (p1 - p2) 2);
}
// Function that returns the smallest distance
// between any pair of points
function minDistance(points) {
let n = points.length;
let minDist = Infinity;
for (let i = 0; i < n; ++i) {
for (let j = i + 1; j < n; ++j) {
let dist = distance(points[i], points[j]);
if (dist < minDist) {
minDist = dist;
}
}
}
// Return the smallest distance
return minDist;
}
// Driver Code
const points = ;
const res = minDistance(points);
console.log(res.toFixed(6));
````
Output
```
1.414214
```
[Expected Approach] Using Divide and Conquer - O(n log(n)) Time and O(n) Space
The main idea is to use the divide and conquer algorithm, where the points are recursively divided into smaller groups. The minimum distance is calculated within each groups, and during the merging step, we will check for possible closer pairs across the dividing point. This approach reduces the number of comparisons, as it only focuses on relevant points near the divide.
Step By Step Implementation:
Sort the points by x-coordinate.
Recursively divide the points into left and right subarrays until each subarrays has one or two points:
If one point, return infinity.
If two points, return their distance.
Find the minimum distances dl and dr in the left and right subarrays, and set das the smaller value between dl and dr.
Build a strip: collect points whose x-distance from the midline is ≤ d.
Sort the strip by y-coordinate.
For each point in the strip, compare it with the next up to 7 points (having y-distance ≤ d) to check for closer pairs.
Update d if a smaller distance is found in the strip.
Return the smallest distance found from the left half, right half, or across the strip.
Key Insights:
Let the dividing point be at coordinate (x, y). After recursively finding the minimum distance d from the left and right halves, we focus on points near the dividing point that could potentially form a closer pair.
We stores all points whose x-distance from the dividing point is ≤ d, i.e., points between x - d and x + d. So, all these points lie inside a vertical strip of width 2d along the x-axis.
Why only consider points within x - d to x + d?
If two points lie farther than d apart in the x-direction, their distance is already greater than d, so they can’t be closer.
Now, to reduce unnecessary comparisons, we sort these points by their y-coordinate. For each point in this strip, we only compare it with points that are within d units vertically (i.e., y-distance ≤ d).
Why compare points with y distance ≤ d?
Because any pair with a y-distance > d will have a total distance > d, so we can safely skip those.
This means for any point in the strip[], we’re only checking points inside a rectangle of dimensions 2d × d (width 2d along x, height d along y).
For each point in the strip[], we check at most 7 points, as there are at most 7 points within this rectangle that could potentially have a smaller distance.
Why at most 7 comparisons per point in strip[]?
Let’s look at this 2d × d rectangle more closely:
Split it into two d × d squares: one for points from the left half, one for the right half.
Each square is then divided into 4 smaller squares of size (d/2 × d/2).
The diagonal of these smaller squares is less than d/√2 which is less than d, so no two points can occupy the same small square (otherwise, they’d be closer than d, violating the earlier recursive result).
So, each d × d square can have at most 4 points, totaling at most 8 points in the full 2d × d rectangle.
Since we’re only comparing the current point with the others, that means at most 7 valid comparisons.
C++
````
// C++ program to find minimum distance between points
include
include
include
include
include
using namespace std;
// Function to compute Euclidean distance between two points
double distance(const vector& p1, const vector& p2) {
return sqrt((p1 - p2) (p1 - p2) +
(p1 - p2) (p1 - p2));
}
// Comparison function to sort points by x-coordinate
bool compareX(const vector& p1, const vector& p2) {
return p1 < p2;
}
// Comparison function to sort points by y-coordinate
bool compareY(const vector& p1, const vector& p2) {
return p1 < p2;
}
// Function to find the minimum distance in the strip
double stripClosest(vector>& strip, double d) {
double minDist = d;
// Sort points in the strip by their y-coordinate
sort(strip.begin(), strip.end(), compareY);
// Compare each point in the strip
for (int i = 0; i < strip.size(); ++i) {
// At most 7 times this will run
for (int j = i + 1; j < strip.size() &&
(strip[j] - strip[i]) < minDist; ++j) {
minDist = min(minDist, distance(strip[i], strip[j]));
}
}
return minDist;
}
// Divide and conquer function to find the minimum distance
double minDistUtil(vector>& points,
int left, int right) {
// Base case brute force for 2 or fewer points
if (right - left <= 2) {
double minDist = 1e9;
for (int i = left; i < right; ++i) {
for (int j = i + 1; j < right; ++j) {
minDist = min(minDist, distance(points[i], points[j]));
}
}
return minDist;
}
// Find the midpoint
int mid = (left + right) / 2;
double midX = points[mid];
// Recursively find the minimum distances in
// the left and right halves
double dl = minDistUtil(points, left, mid);
double dr = minDistUtil(points, mid, right);
double d = min(dl, dr);
// Build the strip of points within distance d from the midline
int k = 0;
vector<vector<double>> strip;
for (int i = left; i < right; ++i) {
if (abs(points[i] - midX) < d) {
strip.push_back(points[i]);
}
}
// Find the minimum distance in the strip
double stripDist = stripClosest(strip, d);
return min(d, stripDist);
}
// Function to find the closest pair of points
double minDistance(vector>& points) {
int n = points.size();
// Sort points by x-coordinate
sort(points.begin(), points.end(), compareX);
return minDistUtil(points, 0, n);
}
int main() {
vector> points = {{-1, -2}, {0, 0}, {1, 2}, {2, 3}};
double res = minDistance(points);
// Output the result with 6 decimal places
cout << fixed << setprecision(6) << res << endl;
return 0;
}
````
// C++ program to find minimum distance between points
// C++ program to find minimum distance between points
```
include #include
```
```
include #include
```
```
include #include
```
```
include #include
```
```
include #include
```
using namespace std; using namespace std
// Function to compute Euclidean distance between two points
// Function to compute Euclidean distance between two points
double distance(const vector<double>& p1, const vector<double>& p2) {double distance const vector< double>& p1 const vector< double>& p2
return sqrt((p1 - p2) (p1 - p2) + return sqrt p1 0 - p2 0 p1 0 - p2 0 +
(p1 - p2) (p1 - p2)); p1 1 - p2 1 p1 1 - p2 1
}
// Comparison function to sort points by x-coordinate
// Comparison function to sort points by x-coordinate
bool compareX(const vector<double>& p1, const vector<double>& p2) {bool compareX const vector< double>& p1 const vector< double>& p2
return p1 < p2; return p1 0< p2 0
}
// Comparison function to sort points by y-coordinate
// Comparison function to sort points by y-coordinate
bool compareY(const vector<double>& p1, const vector<double>& p2) {bool compareY const vector< double>& p1 const vector< double>& p2
return p1 < p2; return p1 1< p2 1
}
// Function to find the minimum distance in the strip
// Function to find the minimum distance in the strip
double stripClosest(vector<vector<double>>& strip, double d) {double stripClosest vector< vector< double>>& strip double d
double minDist = d; double minDist = d
// Sort points in the strip by their y-coordinate
// Sort points in the strip by their y-coordinate
sort(strip.begin(), strip.end(), compareY); sort strip begin strip end compareY
// Compare each point in the strip
// Compare each point in the strip
for (int i = 0; i < strip.size(); ++i) {for int i = 0 i< strip size ++ i
```
```
// At most 7 times this will run
// At most 7 times this will run
for (int j = i + 1; j < strip.size() && for int j = i + 1 j< strip size&&
(strip[j] - strip[i]) < minDist; ++j) {strip j 1 - strip i 1< minDist ++ j
minDist = min(minDist, distance(strip[i], strip[j])); minDist = min minDist distance strip i strip j
}
}
return minDist; return minDist
}
// Divide and conquer function to find the minimum distance
// Divide and conquer function to find the minimum distance
double minDistUtil(vector<vector<double>>& points, double minDistUtil vector< vector< double>>& points
int left, int right) {int left int right
```
```
// Base case brute force for 2 or fewer points
// Base case brute force for 2 or fewer points
if (right - left <= 2) {if right - left<= 2
double minDist = 1e9; double minDist = 1e9
for (int i = left; i < right; ++i) {for int i = left i< right ++ i
for (int j = i + 1; j < right; ++j) {for int j = i + 1 j< right ++ j
minDist = min(minDist, distance(points[i], points[j])); minDist = min minDist distance points i points j
}
}
return minDist; return minDist
}
// Find the midpoint
// Find the midpoint
int mid = (left + right) / 2; int mid = left + right/ 2
double midX = points[mid]; double midX = points mid 0
// Recursively find the minimum distances in
// Recursively find the minimum distances in
// the left and right halves
// the left and right halves
double dl = minDistUtil(points, left, mid); double dl = minDistUtil points left mid
double dr = minDistUtil(points, mid, right); double dr = minDistUtil points mid right
double d = min(dl, dr); double d = min dl dr
// Build the strip of points within distance d from the midline
// Build the strip of points within distance d from the midline
int k = 0; int k = 0
vector<vector<double>> strip; vector< vector< double>> strip
for (int i = left; i < right; ++i) {for int i = left i< right ++ i
if (abs(points[i] - midX) < d) {if abs points i 0 - midX< d
strip.push_back(points[i]); strip push_back points i
}
}
// Find the minimum distance in the strip
// Find the minimum distance in the strip
double stripDist = stripClosest(strip, d); double stripDist = stripClosest strip d
return min(d, stripDist); return min d stripDist
}
// Function to find the closest pair of points
// Function to find the closest pair of points
double minDistance(vector<vector<double>>& points) {double minDistance vector< vector< double>>& points
int n = points.size(); int n = points size
// Sort points by x-coordinate
// Sort points by x-coordinate
sort(points.begin(), points.end(), compareX); sort points begin points end compareX
return minDistUtil(points, 0, n); return minDistUtil points 0 n
}
int main() {int main
vector<vector<double>> points = {{-1, -2}, {0, 0}, {1, 2}, {2, 3}}; vector< vector< double>> points = - 1 - 2 0 0 1 2 2 3
double res = minDistance(points); double res = minDistance points
// Output the result with 6 decimal places
// Output the result with 6 decimal places
cout << fixed << setprecision(6) << res << endl; cout<< fixed<< setprecision 6<< res<< endl
return 0; return 0
}
Java
````
// Java program to find minimum distance between points
import java.util.;
import java.lang.Math;
public class GfG{
// Function to compute Euclidean distance between two points
static double distance(double[] p1, double[] p2) {
return Math.sqrt((p1 - p2) (p1 - p2) +
(p1 - p2) (p1 - p2));
}
// Comparison function to sort points by x-coordinate
static Comparator<double[]> compareX = new Comparator<double[]>() {
public int compare(double[] p1, double[] p2) {
return Double.compare(p1, p2);
}
};
// Comparison function to sort points by y-coordinate
static Comparator<double[]> compareY = new Comparator<double[]>() {
public int compare(double[] p1, double[] p2) {
return Double.compare(p1, p2);
}
};
// Function to find the minimum distance in the strip
static double stripClosest(double[][] strip, double d) {
double minDist = d;
// Sort points in the strip by their y-coordinate
Arrays.sort(strip, compareY);
// Compare each point in the strip
for (int i = 0; i < strip.length; i++) {
for (int j = i + 1; j < strip.length && (strip[j] - strip[i]) < minDist; j++) {
minDist = Math.min(minDist, distance(strip[i], strip[j]));
}
}
return minDist;
}
// Divide and conquer function to find the minimum distance
static double minDistUtil(double[][] points, int left, int right) {
// Base case brute force for 2 or fewer points
if (right - left <= 2) {
double minDist = Double.MAX_VALUE;
for (int i = left; i < right; i++) {
for (int j = i + 1; j < right; j++) {
minDist = Math.min(minDist, distance(points[i], points[j]));
}
}
return minDist;
}
// Find the midpoint
int mid = (left + right) / 2;
double midX = points[mid];
// Recursively find the minimum distances in
// the left and right halves
double dl = minDistUtil(points, left, mid);
double dr = minDistUtil(points, mid, right);
double d = Math.min(dl, dr);
// Build the strip of points within distance d from the midline
List<double[]> strip = new ArrayList<>();
for (int i = left; i < right; i++) {
if (Math.abs(points[i] - midX) < d) {
strip.add(points[i]);
}
}
// Find the minimum distance in the strip
double stripDist = stripClosest(strip.toArray(new double[strip.size()][]), d);
return Math.min(d, stripDist);
}
// Function to find the closest pair of points
static double minDistance(double[][] points) {
int n = points.length;
// Sort points by x-coordinate
Arrays.sort(points, compareX);
return minDistUtil(points, 0, n);
}
public static void main(String[] args) {
double[][] points = {{-1, -2}, {0, 0}, {1, 2}, {2, 3}};
double res = minDistance(points);
// Output the result with 6 decimal places
System.out.printf("%.6f\n", res);
}
}
````
Python
````
Python program to find minimum distance between points
import math
Function to compute Euclidean distance between two points
def distance(p1, p2):
return math.sqrt((p1 - p2) 2 + (p1 - p2) 2)
Function to find the minimum distance in the strip
def stripClosest(strip, d):
min_dist = d
# Sort points in the strip by their y-coordinate
strip.sort(key=lambda point: point)
# Compare each point in the strip
for i in range(len(strip)):
for j in range(i + 1, len(strip)):
if (strip[j] - strip[i]) < min_dist:
min_dist = min(min_dist, distance(strip[i], strip[j]))
else:
break
return min_dist
Divide and conquer function to find the minimum distance
def minDistUtil(points, left, right):
# Base case brute force for 2 or fewer points
if right - left <= 2:
min_dist = float('inf')
for i in range(left, right):
for j in range(i + 1, right):
min_dist = min(min_dist, distance(points[i], points[j]))
return min_dist
# Find the midpoint
mid = (left + right) // 2
mid_x = points[mid]
# Recursively find the minimum distances
# in the left and right halves
dl = minDistUtil(points, left, mid)
dr = minDistUtil(points, mid, right)
d = min(dl, dr)
# Build the strip of points within distance d from the midl
strip = []
for i in range(left, right):
if abs(points[i] - mid_x) < d:
strip.append(points[i])
# Find the minimum distance in the strip
stripDist = stripClosest(strip, d)
return min(d, stripDist)
Function to find the closest pair of points
def minDistance(points):
n = len(points)
# Sort points by x-coordinate
points.sort(key=lambda point: point)
return minDistUtil(points, 0, n)
if name == 'main':
points =
res = minDistance(points)
# Output the result with 6 decimal places
print(f'{res:.6f}')
````
C#
````
// C# program to find minimum distance between points
using System;
using System.Linq;
using static System.Math;
using System.Collections.Generic;
class GfG {
// Function to compute Euclidean distance between two points
static double distance(double[] p1, double[] p2){
double dx = p1 - p2;
double dy = p1 - p2;
return Sqrt(dx dx + dy dy);
}
// Function to find the minimum distance in the strip
static double stripClosest(double[][] strip, double d){
double minDist = d;
// Sort points in the strip by their y-coordinate
Array.Sort(strip, (p1, p2) => p1.CompareTo(p2));
for (int i = 0; i < strip.Length; ++i){
// The inner loop runs for at most 7 points
for (int j = i + 1; j < strip.Length &&
(strip[j] - strip[i]) < minDist; ++j){
minDist = Min(minDist, distance(strip[i], strip[j]));
}
}
return minDist;
}
// Divide and conquer function to find the minimum distance
static double minDistUtil(double[][] points, int left, int right){
// Base case there are 2 or fewer points
if (right - left <= 2){
if (right - left <= 0) return double.MaxValue;
return distance(points[left], points[right - 1]);
}
// Find the midpoint index
int midIndex = left + (right - left) / 2;
double midX = points[midIndex];
// Recursively find the minimum distances
// in the left and right halves
double dl = minDistUtil(points, left, midIndex);
double dr = minDistUtil(points, midIndex, right);
double d = Min(dl, dr);
// Build the strip of points whose x-coordinate
// is within distance 'd' from the mid
List<double[]> stripList = new List<double[]>();
for (int i = left; i < right; ++i){
if (Abs(points[i] - midX) < d){
stripList.Add(points[i]);
}
}
double[][] stripArray = stripList.ToArray();
// Find the minimum distance in the strip
double stripDist = stripClosest(stripArray, d);
return Min(d, stripDist);
}
// Function to find the closest pair of points
static double minDistance(double[,] points2D){
int n = points2D.GetLength(0);
double[][] points = new double[n][];
for (int i = 0; i < n; i++){
points[i] = new double[] { points2D[i, 0], points2D[i, 1] };
}
// Sort points by x-coordinate
Array.Sort(points, (p1, p2) => p1.CompareTo(p2));
return minDistUtil(points, 0, n);
}
static void Main(){
double[,] points = {
{ -1, -2 },
{ 0, 0 },
{ 1, 2 },
{ 2, 3 },
{ 5, 1 },
{ 6, 3 },
{ 8, 0 },
{ 9, 2 }
};
double res = minDistance(points);
// Output the result with 6 decimal places
Console.WriteLine(res.ToString("F6"));
}
}
````
JavaScript
````
// JavaScript program to find minimum distance between points
// Function to compute Euclidean distance between two points
function distance(p1, p2) {
return Math.sqrt((p1 - p2) 2 + (p1 - p2) 2);
}
// Comparison function to sort points by x-coordinate
function compareX(p1, p2) {
return p1 - p2;
}
// Comparison function to sort points by y-coordinate
function compareY(p1, p2) {
return p1 - p2;
}
// Function to find the minimum distance in the strip
function stripClosest(strip, d) {
let minDist = d;
// Sort points in the strip by their y-coordinate
strip.sort(compareY);
// Compare each point in the strip
for (let i = 0; i < strip.length; i++) {
// At most 7 times this will run
for (let j = i + 1; j < strip.length &&
(strip[j] - strip[i]) < minDist; j++) {
minDist = Math.min(minDist, distance(strip[i], strip[j]));
}
}
return minDist;
}
// Divide and conquer function to find the minimum distance
function minDistUtil(points, left, right) {
// Base case brute force for 2 or fewer points
if (right - left <= 2) {
let minDist = Infinity;
for (let i = left; i < right; i++) {
for (let j = i + 1; j < right; j++) {
minDist = Math.min(minDist, distance(points[i], points[j]));
}
}
return minDist;
}
// Find the midpoint
let mid = Math.floor((left + right) / 2);
let midX = points[mid];
// Recursively find the minimum distances in the left and right halves
let dl = minDistUtil(points, left, mid);
let dr = minDistUtil(points, mid, right);
let d = Math.min(dl, dr);
// Build the strip of points within distance d from the midline
let strip = [];
for (let i = left; i < right; i++) {
if (Math.abs(points[i] - midX) < d) {
strip.push(points[i]);
}
}
// Find the minimum distance in the strip
let stripDist = stripClosest(strip, d);
return Math.min(d, stripDist);
}
// Function to find the closest pair of points
function minDistance(points) {
let n = points.length;
// Sort points by x-coordinate
points.sort(compareX);
return minDistUtil(points, 0, n);
}
// Driver Code
let points = ;
let res = minDistance(points);
// Output the result with 6 decimal places
console.log(res.toFixed(6));
````
Output
```
1.414214
```
Applications:
Used to detect planes that are too close to each other, preventing potential collisions.
Used in motion planning to avoid obstacles by determining the closest obstacles or points in the robot's path.
Helps in data classification and pattern recognition by finding nearest data points for training models.
Optimizing the placement of devices by finding nearest neighbors for improved communication.
Closest Pair of Points
K
kartik
Improve
Article Tags :
Divide and Conquer
Geometric
DSA
Closest Pair of Points
Explore
DSA Fundamentals
Logic Building Problems
2 min readAnalysis of Algorithms
1 min read
Data Structures
Array Data Structure
3 min readString in Data Structure
2 min readHashing in Data Structure
2 min readLinked List Data Structure
2 min readStack Data Structure
2 min readQueue Data Structure
2 min readTree Data Structure
2 min readGraph Data Structure
3 min readTrie Data Structure
15+ min read
Algorithms
Searching Algorithms
2 min readSorting Algorithms
3 min readIntroduction to Recursion
14 min readGreedy Algorithms
3 min readGraph Algorithms
3 min readDynamic Programming or DP
3 min readBitwise Algorithms
4 min read
Advanced
Segment Tree
2 min readBinary Indexed Tree or Fenwick Tree
15 min readSquare Root (Sqrt) Decomposition Algorithm
15+ min readBinary Lifting
15+ min readGeometry
2 min read
Interview Preparation
Interview Corner
3 min readGfG160
3 min read
Practice Problem
GeeksforGeeks Practice - Leading Online Coding Platform
6 min readProblem of The Day - Develop the Habit of Coding
5 min read
Improvement
Suggest Changes
Help us improve. Share your suggestions to enhance the article. Contribute your expertise and make a difference in the GeeksforGeeks portal.
Create Improvement
Enhance the article with your expertise. Contribute to the GeeksforGeeks community and help create better learning resources for all.
Suggest Changes
min 4 words, max Words Limit:1000
Thank You!
Your suggestions are valuable to us.
What kind of Experience do you want to share?
Interview Experiences
Admission Experiences
Career Journeys
Work Experiences
Campus Experiences
Competitive Exam Experiences |
8631 | http://cimss.ssec.wisc.edu/wxwise/class/tempscal.html | Temperature Scales
Temperature is a measure of the average kinetic energy of a substance. There
are three commonly used scales for measuring the temperature of an object.
Fahrenheit (named after the German instrument maker G. D. Fahrenheit) is
commonly used in the United States and is used to report temperatures near the
surface. At
standard pressure, water freezes at 32F and boils at 212F. The Celsius (or
centigrade) temperature scale (named after the Swedish astronomer A. Celsius)
is based on the freezing and boiling points of water. This scientific
temperature scale is used to report upper air temperature, and is the common
temperature scale used in Europe and Canada. The formula to convert from
Farenheit to Celsius is C=5/9(F-32).
The Celsisus
temperature scale is not an absolute scale in that 0C is not the lowest
possible temperature. The Kelvin scale (named after Lord Kelvin) is an
absolute scale in that 0K is the lowest possible temperature. As the volume
of an ideal gas, such as air, is proportional to its temperature, a gas with a
temperature of absolute zero has no volume. Negative Kelvin temperatures are
impossible. As temperature is a measure of the random motions of molecules
and atoms, 0K physically means that the molecules and atoms are not moving.
The Kelvin scale is used in thermodynamics and radiation.
The freezing and boiling points
of substances are a function of pressure.
At standard temperature (0C) and sea-level pressure (1013 mb) water freezes at 0C (273.15K) and boils at
100C (373.15K). |
8632 | https://math.stackexchange.com/questions/3098962/discrete-probability-flipping-a-coin-5-times-in-a-row | Discrete Probability - Flipping a coin 5 times in a row. - Mathematics Stack Exchange
Join Mathematics
By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy.
Sign up with Google
OR
Email
Password
Sign up
Already have an account? Log in
Skip to main content
Stack Exchange Network
Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
Visit Stack Exchange
Loading…
Tour Start here for a quick overview of the site
Help Center Detailed answers to any questions you might have
Meta Discuss the workings and policies of this site
About Us Learn more about Stack Overflow the company, and our products
current community
Mathematics helpchat
Mathematics Meta
your communities
Sign up or log in to customize your list.
more stack exchange communities
company blog
Log in
Sign up
Home
Questions
Unanswered
AI Assist Labs
Tags
Chat
Users
Teams
Ask questions, find answers and collaborate at work with Stack Overflow for Teams.
Try Teams for freeExplore Teams
3. Teams
4. Ask questions, find answers and collaborate at work with Stack Overflow for Teams. Explore Teams
Teams
Q&A for work
Connect and share knowledge within a single location that is structured and easy to search.
Learn more about Teams
Hang on, you can't upvote just yet.
You'll need to complete a few actions and gain 15 reputation points before being able to upvote. Upvoting indicates when questions and answers are useful. What's reputation and how do I get it?
Instead, you can save this post to reference later.
Save this post for later Not now
Thanks for your vote!
You now have 5 free votes weekly.
Free votes
count toward the total vote score
does not give reputation to the author
Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation.
Got it!Go to help center to learn more
Discrete Probability - Flipping a coin 5 times in a row.
Ask Question
Asked 6 years, 7 months ago
Modified3 years, 3 months ago
Viewed 19k times
This question shows research effort; it is useful and clear
0
Save this question.
Show activity on this post.
Hey guys, so I think the first thing I'm going to do is write out all of the possibilities (Sample Space). But basically since we're flipping the coin 5 times, and there are only 2 possibilities, there should be 2^5 = 32 outcomes (size of Sample Space)?
Also, how can I write out all the possibilities to ensure that I am not missing any? Since they said "By counting" so I'm assuming they want us to write everything out in the Sample Space, and then count it out.
probability
discrete-mathematics
Share
Share a link to this question
Copy linkCC BY-SA 4.0
Cite
Follow
Follow this question to receive notifications
edited Aug 3, 2021 at 10:17
N. F. Taussig
79.3k 14 14 gold badges 62 62 silver badges 77 77 bronze badges
asked Feb 3, 2019 at 19:17
StuyStuy
1,149 2 2 gold badges 12 12 silver badges 22 22 bronze badges
2
Yes, there are 32 cases. I'd advise you to write out the numbers from 0 thru 31, in base 2 notation: there is an easy pattern to follow, making checking easy.kimchi lover –kimchi lover 2019-02-03 19:21:04 +00:00 Commented Feb 3, 2019 at 19:21
1 "...is write out all of the possibilities..." It is...fine...to do this for such a small introductory problem, but I strongly discourage this as it is prone to error and not generalizeable to larger problems. Part of the point of a course on counting is to learn how to count things without using your fingers and toes to do so. If the problem were mostly the same but instead of just flipping the coin five times you flipped it a hundred times then your approach is infeasible.JMoravitz –JMoravitz 2019-02-03 19:49:05 +00:00 Commented Feb 3, 2019 at 19:49
Add a comment|
4 Answers 4
Sorted by: Reset to default
This answer is useful
1
Save this answer.
Show activity on this post.
There are two possibilities for each of the five tosses of the coin, so there are 2 5=32 2 5=32 possible outcomes in your sample space, as you found.
What is the probability that heads never occurs twice in a row?
Your proposed answer of 13/32 13/32 is correct.
If there are four or five heads in the sequence of five coin tosses, at least two heads must be consecutive.
If there are three heads in the sequence of five coin tosses, the only possibility is that the sequence is HTHTH.
There are (5 2)=10(5 2)=10 sequences of five coin tosses with exactly two heads, of which four have consecutive heads (since the first of these consecutive heads must appear in one of the first four positions). Hence, there are 10−4=6 10−4=6 sequences of five coin tosses with exactly two heads in which no two heads are consecutive.
In each of the five sequences of coin tosses in which exactly one head appears, no two heads are consecutive.
In the only sequence of five coin tosses in which no heads appear, no two heads are consecutive.
Hence, the number of sequences of five coin tosses in which no two heads are consecutive is 0+0+1+6+5+1=13 0+0+1+6+5+1=13, as you found.
What is the probability that neither heads nor tails occurs twice in a row?
Your proposed answer of 1/16 1/16 is correct since there are only two favorable cases: HTHTH and THTHT, which gives the probability 2 32=1 16 2 32=1 16.
What is the probability that both heads and tails occur at least twice in a row?
Your proposed answer of 15/16 15/16 is incorrect.
Since 1−1 16=15 16 1−1 16=15 16, your answer suggests you mistakenly believed that the negation of the statement that neither heads nor tails occurs twice in a row is that both heads and tails occur at least twice in a row. The negation of the statement that neither heads nor tails occurs twice in a row is that at least two heads or at least two tails are consecutive. For instance, the sequences HHTHT and TTTTH both violate the restriction that neither heads nor tails occur twice in a row without satisfying the stronger requirement that both heads and tails occur at least twice in a row.
If both heads and tails occur at least twice in a row, then there are four possibilities:
there is a block of three consecutive heads and a block of two consecutive tails
there is a block of three consecutive tails and a block of two consecutive heads
there is a block of two consecutive heads and a single head that are separated by a block of two consecutive tails
there is a block of two consecutive tails and a single tail that are separated by a block of two consecutive heads
A block of three consecutive heads and a block of two consecutive tails can occur in two ways, HHHTT and TTHHH. By symmetry, a block of three consecutive tails and two consecutive heads can occur in two ways. A block of two consecutive heads and a single head that are separated by a block of two consecutive tails can occur in two ways, HHTTH and HTTHH. By symmetry, a block of two consecutive tails and a single tail that are separated by a block of two consecutive heads can occur in two ways. Hence, there are 2+2+2+2=8 2+2+2+2=8 favorable cases, giving a probability of 8 32=1 4 8 32=1 4.
Share
Share a link to this answer
Copy linkCC BY-SA 4.0
Cite
Follow
Follow this answer to receive notifications
answered Feb 4, 2019 at 12:24
N. F. TaussigN. F. Taussig
79.3k 14 14 gold badges 62 62 silver badges 77 77 bronze badges
Add a comment|
This answer is useful
1
Save this answer.
Show activity on this post.
"By counting the elements in the following events" doesn't necessarily imply that you must write out all the possibilities and check every single one to identify which of them satisfy the given conditions. More generally, "counting" could also mean finding a formula which expresses the numbers of possibilities as a function of the number of tosses. As JMoravitz pointed out in his comment, the first approach is infeasible if the number of tosses were as large as 100 100. As it turns out, however, it's not too difficult to derive formulae for the numbers of elements in each of the three types of events as a function of the number of tosses. The answers for the special cases of 5 5 tosses and 100 100 tosses can then be obtained by substituting those numbers into the formulae.
Let h n h n be the number of sequences of n n tosses that have no consecutive heads and end with a head, and t n t n the number of sequences of n n tosses that have no consecutive heads and end with a tail. Then
h n t n t 1 t 2=t n−1,=t n−1+h n−1,=t n−1+t n−2,=1, and=2.h n=t n−1,t n=t n−1+h n−1,=t n−1+t n−2,t 1=1, and t 2=2.
It follows that t n=F n+1 t n=F n+1, the (n+1)th(n+1)th Fibonacci number. So the number of sequences of n n tosses with no consecutive heads is t n+h n=F n+2 t n+h n=F n+2. For n=5 n=5 this gives F 7=13 F 7=13 as the number of such sequences, and for n=100 n=100Wolfram alpha gives
F 102=927 372 692 193 078 999 176 F 102=927 372 692 193 078 999 176
as the number of such sequences.
The number of sequences of tosses with no consecutive heads or tails is always 2 2, because heads and tails must alternate, so the whole sequence is completely determined by the result of the first toss.
The easiest way of getting the number of sequences of n n tosses in which both heads and tail occur at least twice in a row seems to be to use the principle of inclusion-exclusion to obtain the number of elements in its complement. The complement is the union of the set A A of sequences containing no successive heads with the set B B of sequences containing no successive tails. The intersection A∩B A∩B of these sets is the set of sequences containing no successive heads or tails. Therefore, by the principle of inclusion-exclusion,
|A∪B|=|A|+|B|−|A∩B|=F n+2+F n+2−2,|A∪B|=|A|+|B|−|A∩B|=F n+2+F n+2−2,
and the number of sequences of n n tosses in which both heads and tail occur at least twice in a row is therefore
=2 n−2 F n+2+2 8 for n=5.2 n−2 F n+2+2=8 for n=5.
For n=100 n=100Wolfram alpha gives the number of such sequences as
2 100−2 F 102+2=5 070 602 399 058 172 221 600 654 823 154.2 100−2 F 102+2=5 070 602 399 058 172 221 600 654 823 154.
Share
Share a link to this answer
Copy linkCC BY-SA 4.0
Cite
Follow
Follow this answer to receive notifications
edited Jun 14, 2022 at 12:52
answered Aug 3, 2021 at 12:26
lonza leggieralonza leggiera
32.7k 2 2 gold badges 13 13 silver badges 40 40 bronze badges
Add a comment|
This answer is useful
0
Save this answer.
Show activity on this post.
Hint:
These are the possible outcomes when a coin is flipped 5 5 times in a row.
Now you can easily find the solutions for part a),b)a),b) and c)c)
H,H,H,H,H H,H,H,H,H
H,H,H,H,T H,H,H,H,T
H,H,H,T,H H,H,H,T,H
H,H,H,T,T H,H,H,T,T
H,H,T,H,H H,H,T,H,H
H,H,T,H,T H,H,T,H,T
H,H,T,T,H H,H,T,T,H
H,H,T,T,T H,H,T,T,T
H,T,H,H,H H,T,H,H,H
H,T,H,H,T H,T,H,H,T
H,T,H,T,H H,T,H,T,H
H,T,H,T,T H,T,H,T,T
H,T,T,H,H H,T,T,H,H
H,T,T,H,T H,T,T,H,T
H,T,T,T,H H,T,T,T,H
H,T,T,T,T H,T,T,T,T
T,H,H,H,H T,H,H,H,H
T,H,H,H,T T,H,H,H,T
T,H,H,T,H T,H,H,T,H
T,H,H,T,T T,H,H,T,T
T,H,T,H,H T,H,T,H,H
T,H,T,H,T T,H,T,H,T
T,H,T,T,H T,H,T,T,H
T,H,T,T,T T,H,T,T,T
T,T,H,H,H T,T,H,H,H
T,T,H,H,T T,T,H,H,T
T,T,H,T,H T,T,H,T,H
T,T,H,T,T T,T,H,T,T
T,T,T,H,H T,T,T,H,H
T,T,T,H,T T,T,T,H,T
T,T,T,T,H T,T,T,T,H
T,T,T,T,T T,T,T,T,T
Share
Share a link to this answer
Copy linkCC BY-SA 4.0
Cite
Follow
Follow this answer to receive notifications
answered Feb 3, 2019 at 19:23
Key FlexKey Flex
9,673 7 7 gold badges 20 20 silver badges 36 36 bronze badges
3
a) 13/32, b) 1/16, c) 15/16 ? Are my answers correct?Stuy –Stuy 2019-02-03 19:46:33 +00:00 Commented Feb 3, 2019 at 19:46
how is it 1/16?Stuy –Stuy 2019-02-04 19:05:59 +00:00 Commented Feb 4, 2019 at 19:05
@Stuy Your answer is correct only.Key Flex –Key Flex 2019-02-04 19:06:54 +00:00 Commented Feb 4, 2019 at 19:06
Add a comment|
This answer is useful
0
Save this answer.
Show activity on this post.
I'm dealing with similiar problems at the moment and yes, my general experience is that when it's written 'by couting' it literally means by counting. So just, write all of the possible combinations occuring in the sample space and calculate the probability of each of the stated events, by counting how many times the event occurs and dividing by the size of the sample space.
Your sample space size is correct :)
Share
Share a link to this answer
Copy linkCC BY-SA 4.0
Cite
Follow
Follow this answer to receive notifications
answered Feb 3, 2019 at 19:25
VRTVRT
185 9 9 bronze badges
Add a comment|
You must log in to answer this question.
Start asking to get answers
Find the answer to your question by asking.
Ask question
Explore related questions
probability
discrete-mathematics
See similar questions with these tags.
Featured on Meta
Introducing a new proactive anti-spam measure
Spevacus has joined us as a Community Manager
stackoverflow.ai - rebuilt for attribution
Community Asks Sprint Announcement - September 2025
Report this ad
Related
0Probability of a run of heads of length at least 2 2 when flipping a coin n n times.
3Probability with flipping a coin 6 times
0If a fair coin is tossed until heads appears twice in a row...
4Coin flipping probability taking turns between players
0Coin flipping: 4 4 players 3 3 flips - whose result is more probable and what's the probability?
1Counting number of possibilities on flipping two coins. Why don't we double count 2 tails(TT) or 2 heads(HH)?
5Flip a coin 5 times. What is the probability that heads never occurs twice in a row?
0Discrete Probability: Four dice are thrown, what's the probability that...
Hot Network Questions
What's the expectation around asking to be invited to invitation-only workshops?
On being a Maître de conférence (France): Importance of Postdoc
Does "An Annotated Asimov Biography" exist?
An odd question
How to locate a leak in an irrigation system?
My dissertation is wrong, but I already defended. How to remedy?
Is it ok to place components "inside" the PCB
Do we need the author's permission for reference
Bypassing C64's PETSCII to screen code mapping
How to convert this extremely large group in GAP into a permutation group.
Exchange a file in a zip file quickly
Is existence always locational?
The rule of necessitation seems utterly unreasonable
Why is a DC bias voltage (V_BB) needed in a BJT amplifier, and how does the coupling capacitor make this possible?
A time-travel short fiction where a graphologist falls in love with a girl for having read letters she has not yet written… to another man
"Unexpected"-type comic story. Aboard a space ark/colony ship. Everyone's a vampire/werewolf
Countable and uncountable "flavour": chocolate-flavoured protein is protein with chocolate flavour or protein has chocolate flavour
Numbers Interpreted in Smallest Valid Base
Does the mind blank spell prevent someone from creating a simulacrum of a creature using wish?
Weird utility function
I have a lot of PTO to take, which will make the deadline impossible
Can peaty/boggy/wet/soggy/marshy ground be solid enough to support several tonnes of foot traffic per minute but NOT support a road?
How to start explorer with C: drive selected and shown in folder list?
Is encrypting the login keyring necessary if you have full disk encryption?
Question feed
Subscribe to RSS
Question feed
To subscribe to this RSS feed, copy and paste this URL into your RSS reader.
Why are you flagging this comment?
It contains harassment, bigotry or abuse.
This comment attacks a person or group. Learn more in our Code of Conduct.
It's unfriendly or unkind.
This comment is rude or condescending. Learn more in our Code of Conduct.
Not needed.
This comment is not relevant to the post.
Enter at least 6 characters
Something else.
A problem not listed above. Try to be as specific as possible.
Enter at least 6 characters
Flag comment Cancel
You have 0 flags left today
Mathematics
Tour
Help
Chat
Contact
Feedback
Company
Stack Overflow
Teams
Advertising
Talent
About
Press
Legal
Privacy Policy
Terms of Service
Your Privacy Choices
Cookie Policy
Stack Exchange Network
Technology
Culture & recreation
Life & arts
Science
Professional
Business
API
Data
Blog
Facebook
Twitter
LinkedIn
Instagram
Site design / logo © 2025 Stack Exchange Inc; user contributions licensed under CC BY-SA. rev 2025.9.29.34589
By clicking “Accept all cookies”, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy.
Accept all cookies Necessary cookies only
Customize settings
Cookie Consent Preference Center
When you visit any of our websites, it may store or retrieve information on your browser, mostly in the form of cookies. This information might be about you, your preferences, or your device and is mostly used to make the site work as you expect it to. The information does not usually directly identify you, but it can give you a more personalized experience. Because we respect your right to privacy, you can choose not to allow some types of cookies. Click on the different category headings to find out more and manage your preferences. Please note, blocking some types of cookies may impact your experience of the site and the services we are able to offer.
Cookie Policy
Accept all cookies
Manage Consent Preferences
Strictly Necessary Cookies
Always Active
These cookies are necessary for the website to function and cannot be switched off in our systems. They are usually only set in response to actions made by you which amount to a request for services, such as setting your privacy preferences, logging in or filling in forms. You can set your browser to block or alert you about these cookies, but some parts of the site will not then work. These cookies do not store any personally identifiable information.
Cookies Details
Performance Cookies
[x] Performance Cookies
These cookies allow us to count visits and traffic sources so we can measure and improve the performance of our site. They help us to know which pages are the most and least popular and see how visitors move around the site. All information these cookies collect is aggregated and therefore anonymous. If you do not allow these cookies we will not know when you have visited our site, and will not be able to monitor its performance.
Cookies Details
Functional Cookies
[x] Functional Cookies
These cookies enable the website to provide enhanced functionality and personalisation. They may be set by us or by third party providers whose services we have added to our pages. If you do not allow these cookies then some or all of these services may not function properly.
Cookies Details
Targeting Cookies
[x] Targeting Cookies
These cookies are used to make advertising messages more relevant to you and may be set through our site by us or by our advertising partners. They may be used to build a profile of your interests and show you relevant advertising on our site or on other sites. They do not store directly personal information, but are based on uniquely identifying your browser and internet device.
Cookies Details
Cookie List
Clear
[x] checkbox label label
Apply Cancel
Consent Leg.Interest
[x] checkbox label label
[x] checkbox label label
[x] checkbox label label
Necessary cookies only Confirm my choices |
8633 | https://www.ahapuzzles.com/brain-teasers/riddles/wgWyn6M6/there-is-a-kind-of-fish-that-can-never-swim-what-is-that/ | There is a kind of fish that can never swim. What is that? - Riddle & Answer - Aha! Puzzles
×
Custom Search
Sort by:
Relevance
Relevance
Date
Home
Logic Games
Featured Games
#### Zebra Puzzles #### Einstein's Puzzle
Logic Games
Einstein's Puzzle
Greekest
Logic Equations
Logic Puzzles
Zebra Puzzles
Word Games
Featured Games
#### Mixed Words #### Word Search
Word Games
Hangman
Mixed Words
Mysterious Words
Word Search
Word Snake
Wordzilla
Daily Word Search
Easy
Medium
Hard
One-Word
One-Clue
Puzzles
Puzzles
2048
Jigsaw Puzzles
Triplet
Daily Jigsaw Puzzles
Daily Jigsaw - 29/09 - 00h
Daily Jigsaw - 28/09 - 20h
Daily Jigsaw - 28/09 - 16h
Daily Jigsaw - 28/09 - 12h
Solitaire
Featured Solitaire Games on Solitairey.com
#### Spider Solitaire #### Klondike
Solitaire Games on Solitairey.com
Baker's Dozen
FreeCell
Klondike
Spider Solitaire
Tri Peaks Solitaire
Triple Klondike
Fun
Featured Fun Games
#### Memory Game #### 4 Colors
Fun Games
4 Colors
Brick Out
Memory Game
Pinball
Snakes and Ladders
Games##### Games
#### Candy Bubble #### Hex Puzzle #### 1010 Deluxe #### Onet Connect
Riddles
Aha! Puzzles
Brain Teasers
Riddles with Answers
Riddle
There is a kind of fish that can never swim. What is that? Riddle
There is a kind of fish that can never swim. What is that?
Show Answer
Answer: Dead fish.
🤣🤪🤯🙄😑🤡
Next Riddle in 6...cancel
#Animal#Funny#Medium#Short
Random RiddleNext Riddle
Printable Riddles
This riddle appears in the following downloadable PDF files:
Animal Riddles.pdf
Funny Riddles.pdf
Medium Riddles.pdf
Short Riddles.pdf
More riddles about...
Animal Riddles
Funny Riddles
Medium Riddles
Short Riddles
More Riddles
Categories
4th of July
Animal
Astronomy
Best
Bible
Biology
Chemistry
Christmas
Clothes and Accessories
Cool
Easter
Food and Drink
French
Fruit
Funny
Games and Toys
Geography
Geometry
German
Good
Halloween
History
Italian
Jobs and Occupations
Kids
Logic
Math
Money
Music
Nature
Physics
Political
Portuguese
Rhyming
Seasons
Spanish
Sports
St. Patrick's Day
Teens
Thanksgiving
Tricky
Valentines
What am I?
What is it?
Who am I?
Difficulty
Easy
Medium
Hard
Size
Short
Long
Before you go
Einstein's Puzzle
Can you solve one of the most famous logic puzzle ever?
Memory Game
Memory Matching game for kids, adults and seniors.
Onet Connect
Connect the matching tiles and pop them all.
Math Riddles 🔢 90 Math Riddles Puzzles ❤️ Teachers, Parents & Brainiacs Download Now
Logic Puzzles For Kids 🧠 15 Logic Puzzles ✅ Answers included Download Now
Logic Puzzles 🧠 50 Logic Puzzles ✅ Answers included Download Now
Impossible Word Search 🔍 50 Puzzles ❤️ Teachers, Parents & Brainiacs Download Now
Number Pyramid 🔢 360 Number Pyramid Puzzles ❤️ Teachers, Parents & Brainiacs Download Now
Zebra Puzzles 🦓 60 Zebra Puzzles ✅ Answers included Download Now
View slide 1 View slide 2 View slide 3 View slide 4 View slide 5 View slide 6
View slide 1 View slide 2 View slide 3
Get notified about new brain games
Relax, we'll let you know about new brain games
Go!- [x]
One email every two weeks. No spam, ever!
Daily Games
Greekest
Hangman
Jigsaw Puzzles
Logic Equations
Mixed Words
Mysterious Words
Triplet
Word Search
Easy / Medium / Hard
One-Clue
One-Word
Word Snake
Wordzilla
Logic Games
Einstein's Puzzle
Greekest
Logic Equations
Logic Puzzles
Zebra Puzzles
Puzzles
2048
Jigsaw Puzzles
Triplet
Fun Games
4 Colors
Brick Out
Memory Game
Pinball
Snakes and Ladders
Word Games
Hangman
Mixed Words
Mysterious Words
Word Search
Word Snake
Wordzilla
Riddles
Riddles with Answers
Best Riddles
Funny Riddles
Hard Riddles
Math Riddles
Riddles in Spanish
Tricky Riddles
Printable Games
Printable Logic Puzzles
Printable Riddles
Printable Word Searches
Printable Zebra Puzzles
Aha! Printables
Printable Kyudoku
Printable Logic Equations
Printable Logic Games
Printable Math Games
Printable Sudoku
Printable Word Games
Privacy Manager
WARNING: This site is famous for stimulating cognitive explosions. We advise you to buckle up mentally, and preferably, find a comfortable seat. Side effects may include: addiction to brain teasing and spontaneous eruptions of "Aha!" moments. Consume responsibly!
Aha! Puzzles - Puzzles for your brain.
© 2020-2025 Aha! Puzzles | Links | Privacy Policy | About
4
Aha! Puzzles+
6 hours ago
New
Daily Jigsaw - 29/09 - 00h
Daily Jigsaw - 29/09 - 00h
Play Jigsaw
Jigsaw
😀
😐
😕
7 hours ago
New
Play Word Master
Play Word Master
Play Word Master
Word Games
😀
😐
😕
8 hours ago
New
Daily Word-Search
Find out the hidden words about: Citrus Fruits, Root Vegetables, Superheroes.
Daily Word-Search
Word Games
😀
😐
😕
9 hours ago
New
Daily Mix - 28/09 - 21h
Daily Mix - 28/09 - 21h
Play Broken Words
Word Games
😀
😐
😕 |
8634 | https://www.youtube.com/watch?v=mCmALs0QJ3A | Lecture 5 : Linearly Independent Set in R2 ( with Animations )
Pratul@Maths
1230 subscribers
6 likes
Description
327 views
Posted: 25 Jan 2022
Linear Independence in R2 by Prof. Pratul Gadagkar, is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
1 comments
Transcript:
[Music] in this video we are going to discuss about linearly independent vectors in r2 so now what we will do is we will take two vectors v 1 bar and v 2 bar to be 2 vectors in r 2 and we will write a definition of what is meant by the linear independence of these two vectors in r2 so if the combination c 1 v 1 bar plus c 2 v 2 bar is equal to 0 and if this implies that c 1 and c 2 both are 0 then we say that the two vectors v1 bar and p2 bar are linearly independent so let us take one simple example suppose i write the two vectors v1 bar is suppose i am taking 2 comma 3 and the second vector v2 bar i'm going to take say 3 comma 2 these two vectors are vectors in r2 and i'm going to show that these two vectors v1 bar and v2 bar the set v1 bar v2 bar which i'm going to call s is a linearly independent set means the two vectors inside this set s are linearly independent vectors in r2 so how am i going to show that so i will consider c 1 v 1 bar plus c 2 v 2 bar and i will equate it to 0 and i will try to substitute the values v1 bar is 2 comma 3 plus c2 v2 bar is 3 comma 2 and that is equal to 0 bar this is a vector 0 bar and this gives me comma 3 c 1 plus 3 c 2 plus 2 c sorry comma 2 c 2 is equal to 0 so that will give me 2 c 1 plus 3 c 2 comma 3 c 1 plus 2 c 2 is 0 0 and if i get 2 equations the first equation is 2 c 1 plus 3 c 2 is 0 and second equation is 3 c 1 plus 2 c 2 is also equal to 0 when we solve them simultaneously we will get c1 is also equal to 0 and c2 is also equal to 0 this means that the set s that we were talking about s was the first vector was 2 comma 3 and the second vector was 3 comma 2 this set is a linearly independent set in the set r2 so when we plot the two vectors in the in the space r2 the first vector which is 2 comma 3 and the second vector is 3 comma 2 we see that the first vector is u bar and the second vector is v bar we understand that these two vectors turn out to be linearly independent in r 2 because both of them have different directions it does not means that the second vector can be the opposite of the first vector that is not allowed so these two vectors will have different directions is somewhat a geometrical meaning of the two vectors being linearly independent so now what we will do is we will consider only one vector which is u bar equal to two comma three and we will try to multiply this vector u by a scalar i am going to call that scalar a which is a real number and i am going to look at all such possibilities a u bar which is a times 2 comma 3. where this a is going to be a real number so this is nothing but scalar multiplication in the vector space so let us see how does the scalar multiplication can be geometrically visualized now as you can see in the figure the vector u is 2 comma 3 now what we will do is we will multiply this vector by 2 so that i will get the vector 4 comma 6 so when i join that vector you will see that this w vector is a vector 4 comma 6 you can even multiply this vector u by minus 1 so when i multiply it by minus 1 i will get minus 2 minus 3 which is a point in the third quadrant when i join that i will get this vector so this vector is this minus 2 minus 3 is a scalar multiple of the vector u so that is minus 1 times the vector 2 comma 3 i can even multiply the vector u but by minus 2 so when i multiply by minus 2 i will get minus 4 minus 6 so this is that vector which is minus 4 comma minus 6 this is the vector right so now it means that when i take all the scalar multiples of that vector 2 comma 3 it is understood that all these vectors lie on this particular line and the equation of that line is 3x minus 2y is equal to 0 and all the points on this vector space i'm sorry on the all the points on this line are basically nothing but the scalar multiples of the point 2 comma 3. you can now see the point varying on that line when i multiply it by scalar a real number that point starts varying on this particular line so this is nothing but the span of that vector u you can see that the point is moving on the line this scalar is a real number so it will continuously vary from minus infinity to plus infinity and accordingly the point will move on the line and the equation of that line is 3x minus 2y is equal to 0. so this 3x minus 2y line equal to 0 is this line is nothing but the span of the vector u 2 comma 3. |
8635 | https://ocw.mit.edu/courses/5-60-thermodynamics-kinetics-spring-2008/resources/5_60_lecture23/ | Colligative Properties | Thermodynamics & Kinetics | Chemistry | MIT OpenCourseWare
Browse Course Material
Syllabus
Lecture Notes
Readings
Exams
Video Lectures
Course Info
Instructors
Prof. Keith A. Nelson
Prof. Moungi Bawendi
Departments
Chemistry
As Taught In
Spring 2008
Level
Undergraduate
Topics
Science
Chemistry
Physical Chemistry
Physics
Classical Mechanics
Thermodynamics
Learning Resource Types
theaters Lecture Videos
grading Exams
notes Lecture Notes
Download Course
menu
search
Give Now
About OCW
Help & Faqs
Contact Us
searchGIVE NOWabout ocwhelp & faqscontact us
5.60 | Spring 2008 | Undergraduate
Thermodynamics & Kinetics
Menu
More Info
Syllabus
Lecture Notes
Readings
Exams
Video Lectures
Lecture Notes
Colligative Properties
Description:
Lecture notes on colligative properties, vapor pressure lowering, boiling point elevation, freezing point depression, and osmotic pressure.
Resource Type:
Lecture Notes
pdf 112 kB
Colligative Properties
Download File
Course Info
Instructors
Prof. Keith A. Nelson
Prof. Moungi Bawendi
Departments
Chemistry
As Taught In
Spring 2008
Level
Undergraduate
Topics
Science
Chemistry
Physical Chemistry
Physics
Classical Mechanics
Thermodynamics
Learning Resource Types
theaters Lecture Videos
grading Exams
notes Lecture Notes
Download Course
Over 2,500 courses & materials
Freely sharing knowledge with learners and educators around the world. Learn more
© 2001–2025 Massachusetts Institute of Technology
Accessibility
Creative Commons License
Terms and Conditions
Proud member of:
© 2001–2025 Massachusetts Institute of Technology
You are leaving MIT OpenCourseWare
close
Please be advised that external sites may have terms and conditions, including license rights, that differ from ours. MIT OCW is not responsible for any content on third party sites, nor does a link suggest an endorsement of those sites and/or their content.
Stay Here Continue |
8636 | https://wisc.pb.unizin.org/minimisgenchem/chapter/enthalpy-of-formation-reactions-m6q8/ | Standard Enthalpy of Formation (M6Q8) – UW-Madison Chemistry 103/104 Resource Book
Skip to content
Menu
Primary Navigation
Home
Read
Sign in
Search in book: Search
Book Contents Navigation
Contents
Introduction
I. Module 1: Introduction to Chemistry Concepts
1.Measurements, Units, Conversions, Density (M1Q1)
2.States of Matter (M1Q2)
3.Classification of Matter (M1Q3)
II. Module 2: Atoms, Molecules, and Ions
4.Protons, Neutrons, and Electrons (M2Q1)
5.Rutherford's Experiments (M2Q2)
6.Isotopes, Atomic Mass, and Mass Spectrometry (M2Q3)
7.Periodic Trends (M2Q4)
8.Monatomic Ions (M2Q5)
9.Polyatomic Ions (M2Q6)
III. Module 3: Qualitative Analysis of Chemical Reactions
10.Solutions and Solubility (part 1) (M3Q1)
11.Solutions and Solubility (part 2) (M3Q2)
12.Acids, Bases, Neutralization, and Gas-Forming Reactions (M3Q3-4)
13.Oxidation-Reduction Reactions (M3Q5-6)
IV. Module 4. Quantitative Analysis of Chemical Reactions
14.Moles, AMUs, and Conversions (M4Q1)
15.Stoichiometry (M4Q2)
16.Limiting Reagents (M4Q3)
17.Empirical Formula (M4Q4)
18.Heat and Stoichiometry (M4Q5)
19.Molarity, Solutions, and Dilutions (M4Q6)
V. Module 5. Gases
20.Pressure and Ideal Gas Laws (M5Q1)
21.Stoichiometry Involving Gases (M5Q2)
22.Gas Density (M5Q3)
23.Gas Mixtures and Partial Pressure (M5Q4)
24.Gas Behavior, Kinetic Molecular Theory, and Temperature (M5Q5)
25.Chemistry of the Atmosphere (M5Q6)
VI. Module 6. Thermochemistry
26.Energy Forms & Global Relevance (M6Q1)
27.Direction of Heat Flow and System vs. Surroundings (M6Q2)
28.First Law of Thermodynamics and Work (M6Q3)
29.Heat Capacity (M6Q4)
30.Calorimetry continued: Types of Calorimeters and Analyzing Heat Flow (M6Q5)
31.Calorimetry continued: Phase Changes and Heating Curves (M6Q6)
32.Hess's Law (M6Q7)
33.Standard Enthalpy of Formation (M6Q8)
VII. Module 7. Light, Matter, and Atomic Structure
34.Waves and the Electromagnetic Spectrum (M7Q1)
35.Photoelectric Effect (M7Q2)
36.Emission Spectra and H Atom Levels (M7Q3)
37.Wave Interference, Diffraction (M7Q4)
38.DeBroglie, Intro to Quantum Mechanics, Quantum Numbers 1-3 (M7Q5)
39.Orbitals and the 4th Quantum Number, (M7Q6)
40.Electron Configurations, Orbital Box Notation (M7Q7)
41.Core and Valence Electrons, Shielding, Zeff (M7Q8)
42.Periodic Trends (M7Q9)
43.Electron Configurations for Ions (M7Q10)
VIII. Module 8: Chemical Bonding
44.Bonding and Electronegativity (M8Q1)
45.Lewis Structures (M8Q2)
46.Resonance Structures and Formal Charge (M8Q3)
47.Bond Length and Strength (M8Q4)
IX. Module 9: Molecular Geometry
48.Predicting Molecular Shapes: VSEPR Model (M9Q1)
49.Molecule Polarity (M9Q2)
50.Valence Bond Theory and Hybridization (M9Q3)
51.Valence Bond Theory and Resonance (M9Q4)
52.Molecular Orbital Theory (M9Q5-6)
X. Module 10: Intermolecular Forces
53.An Introduction to Intermolecular Forces (M10Q1)
54.Melting and Boiling Point Comparisons (M10Q2)
55.Vapor Pressure and Boiling Point Correlations (M10Q3)
56.Explaining Solubility and Surface Tension through IMFs (M10Q4)
XI. Module 11: Phase Diagrams and Solids
57.Features of Phase Diagrams (M11Q1)
58.Heating Curves and Phase Diagrams (M11Q2)
59.Types of Solids (M11Q3)
60.Types of Unit Cells: Primitive Cubic Cell (M11Q4)
61.Types of Unit Cells: Body-Centered Cubic and Face-Centered Cubic (M11Q5)
62.Ionic Crystals and Unit Cell Stoichiometry (M11Q6)
Appendix A: Periodic Table
Appendix B: Essential Mathematics
Appendix C: Units and Conversion Factors
Appendix D: Solubility Flow Chart
Appendix E: Specific Heat Capacities for Common Substances (M6Q5)
Appendix F: Standard Thermodynamic Properties (M6)
Appendix G: Bond Enthalpy, Bond Length, Atomic Radii, and Ionic Radii
Appendix H: Electronegativity
UW-Madison Chemistry 103/104 Resource Book
33 Standard Enthalpy of Formation (M6Q8)
Introduction
Concluding Module 6, this section introduces the enthalpy of formation and how to perform calculations for enthalpy of formation reactions run under standard state conditions. How to determine the enthalpy change of a reaction utilizing enthalpies of formation and Hess’s Law is also included. This section includes worked examples, sample problems, and a glossary.
Learning Objectives for Enthalpy of Formation Reactions
Use standard enthalpies of formation to calculate a reaction enthalpy change.
|Standard Enthalpy of Formation | Hess’s Law with Standard Enthalpy of Formation |
| Key Concepts and Summary | Key Equations | Glossary | End of Section Exercises |
Standard Enthalpy of Formation
A standard enthalpy of formation (ΔH°f)is an enthalpy change for a reaction in which exactly one 1 mole of a pure substance is formed from free elements in their most stable states under standard state conditions. These values are especially useful for computing or predicting enthalpy changes for chemical reactions that are impractical or dangerous to carry out, or for processes for which it is difficult to make measurements. If we have values for the appropriate standard enthalpies of formation, we can determine the enthalpy change for any reaction using Hess’s law.
The standard enthalpy of formation of CO 2(g) is −393.5 kJ/mol. This is the enthalpy change for the exothermic reaction:
C(s, graphite) + O 2(g) → CO 2(g)ΔH°f= ΔH°298= -393.509 kJ
starting with the reactants at a pressure of 1 atm and 25 °C (with the carbon present as graphite, the most stable form of carbon under these conditions) and ending with one mole of CO 2, also at 1 atm and 25 °C. For nitrogen dioxide, NO 2(g), ΔH°f is 33.2 kJ/mol. This is the enthalpy change for the reaction:
½ N 2(g) + O 2(g) → NO 2(g)ΔH°f= ΔH°298= +33.18 kJ
A reaction equation with ½ mole of N 2 and 1 mole of O 2 is correct in this case because the standard enthalpy of formation always refers to 1 mole of product, NO 2(g).
You will find a table of standard enthalpies of formation of many common substances in Appendix F. These values indicate that formation reactions range from highly exothermic (such as −2984.0 kJ/mol for the formation of P 4 O 10) to strongly endothermic (such as +226.73 kJ/mol for the formation of acetylene, C 2 H 2). By definition, the standard enthalpy of formation of an element in its most stable form (most stable allotrope) is equal to zero under standard conditions, which is 1 atm for gases and 1 M for solutions. For example, carbon exists as several allotropes, two of them being graphite and diamond. Graphite is the more thermodynamically stable (ΔH°f= 0 kJ/mole) whereas for diamond, ΔH°f= 2.4 kJ/mole. While this table does not list the complete reaction that is represented by the enthalpy change, you can use the definition above to write these reactions when needed.
Example 1
Evaluating an Enthalpy of Formation
Ozone, O 3(g), forms from oxygen, O 2(g), by an endothermic process. Ultraviolet radiation is the source of the energy that drives this reaction in the upper atmosphere. Assuming that both the reactants and products of the reaction are in their standard states, determine the standard enthalpy of formation, ΔH°f, of ozone from the following information:
3 O 2(g) → 2 O 3(g)ΔH°298= +285.4 kJ
Solution
ΔH°f is the enthalpy change for the formation of one mole of a substance in its standard state from the elements in their standard states. Thus, ΔH°f for O 3(g) is the enthalpy change for the reaction:
3 2 O 2(g) → O 3(g)
For the formation of 2 mol of O 3(g), ΔH°298= +285.4 kJ. This ratio, (285.4 kJ 2 mol O 3), can be used as a conversion factor to find the heat produced when 1 mole of O 3(g) is formed, which is the enthalpy of formation for O 3(g):
ΔH° for 1 mol of O 3(g) = 1 ~~mol O 3~~ × 285.4 kJ 2 mol O 3= +142.7 kJ
Therefore,ΔH°f [O 3(g)] = +142.7 kJ/mol
Check Your Learning
Hydrogen gas, H 2, reacts explosively with gaseous chlorine, Cl 2, to form hydrogen chloride, HCl(g). What is the enthalpy change for the reaction of 1 mole of H 2(g) with 1 mole of Cl 2(g) if both the reactants and products are at standard state conditions? The standard enthalpy of formation of HCl(g) is −92.307 kJ/mol.
Answer:
For the reaction:
H 2(g) + Cl 2(g) → 2 HCl(g)ΔH°298= -184.61 kJ
Example 2
Writing Reaction Equations for ΔH°f
Write the enthalpy of formation reaction equations for:
(a) C 2 H 5 OH(l)
(b) Ca 3(PO 4)2(s)
Solution
Remembering that ΔH°f reaction equations are for forming 1 mole of the compound from its constituent elements under standard conditions, we have:
(a)
2 C(s, graphite) + 3 H 2(g) + ½ O 2(g) → C 2 H 5 OH(l)
(b)
3 Ca(s) + ½ P 4(s) + 4 O 2(g) →Ca 3(PO 4)2(s)
Note: The standard state of carbon is graphite, and phosphorus exists as P 4.
Check Your Learning
Write the enthalpy of formation reaction equations for:
(a) C 2 H 5 OC 2 H 5(l)
(b) Na 2 CO 3(s)
Answer:
(a)
4 C(s, graphite) + 5 H 2(g) + 1 2 O 2(g) → C 2 H 5 OC 2 H 5(l)
(b)
2 Na(s) + C(s, graphite) + 3 2 O 2(g) → Na 2 CO 3(s)
Hess’s Law with Standard Enthalpy of Formation
In the previous section, we learned about Hess’s law and how it can be useful for determining the ΔH of a given reaction since we can use other reactions as “steps” to achieve the overall reaction of interest and the ΔH values for the “steps” would sum to give the overall ΔH. We also can use Hess’s law to determine the enthalpy change of any reaction if the corresponding enthalpies of formation of the reactants and products are available. In this case, the stepwise reactions we consider are:
(i) Decompositions of the reactants into their component elements. If you were to write the reaction for the decomposition of a reactant (where you start with a compound and end with component elements), this would be related to the reverse reaction of enthalpy of formation (where you start with component elements and form a compound). So the enthalpy for these reactions would be proportional to the negative of the enthalpies of formation of the reactants.
This step would be followed by:
(ii) Recombinations of the elements to give the products. The enthalpy change of this step would then be proportional to the enthalpy of formation of the products since we are taking component elements and forming a compound.
The standard enthalpy change of the overall reaction is therefore equal to:
(ii) The sum of the standard enthalpies of formation of all the products plus (i) the sum of the negatives of the standard enthalpies of formation of the reactants. This is usually rearranged slightly to be written as follows, with ∑ representing “the sum of” and n standing for the stoichiometric coefficients:
ΔH°reaction= ∑n × ΔH°f(products) – ∑n × ΔH°f(reactants)
The following example shows in detail why this equation is valid, and how to use it to calculate the enthalpy change for a reaction of interest.
Example 3
Using Hess’s Law
Find the standard enthalpy change for the following reaction:
3 NO 2(g) + H 2 O(l) → 2 HNO 3(aq) + NO(g)ΔH° = ?
Solution: Using the Equation
Use the special form of Hess’s law given previously:
ΔH°reaction= ∑n× ΔH°f(products) – ∑n× ΔH°f(reactants)
= [2 mol HNO 3 × −207.36 kJ mol HNO 3+ 1 mol NO × +90.25 kJ mol NO] - [3 mol NO 2 × +33.18 kJ mol NO 2+ 1 mol H 2 O × −285.83 kJ mol H 2 O]
= [2(-207.36 kJ) + 1(+90.25 kJ)] - [3(+33.18 kJ) + 1(-285.83 kJ)]
= -138.18 kJ
Solution: Supporting Why the General Equation Is Valid
Alternatively, we can write this reaction as the sum of the decompositions of 3 NO 2(g) and 1 H 2 O(l) into their constituent elements, and the formation of 2 HNO 3(aq) and 1 NO(g) from their constituent elements. Writing out these reactions, and noting their relationships to the ΔH°f values for these compounds (from Appendix F), we have:
3 NO 2(g) → 3 2 N 2(g) + 3 O 2(g)
⇒ ΔH°1= -3 × ΔH°f(NO 2(g)) = -99.54 kJ
and
H 2 O(l) → H 2(g) + 1 2 O 2(g)
⇒ ΔH°2= -ΔH°f (H 2 O(l)) = +285.83 kJ
and
H 2(g) + N 2(g) + 3 O 2(g) → 2 HNO 3(aq)
⇒ ΔH°3 = 2 × ΔH°f(HNO 3(aq)) = -414.72 kJ
and
1 2 N 2(g) + 1 2 O 2(g) → NO(g)
⇒ ΔH°4 = ΔH°f(NO(g)) = +90.25 kJ
Summing these reaction equations gives the reaction we are interested in:
3 NO 2(g) + H 2 O(l) → 2 HNO 3(aq) + NO(g)
Summing their enthalpy changes gives the value we want to determine:
ΔH°rxn= ΔH°1+ΔH°2+ΔH°3+ΔH°4= (-99.54 kJ) + (+285.83 kJ) + (-414.72 kJ) + (+90.25 kJ) = -138.18 kJ
So the standard enthalpy change for this reaction is ΔH° = −138.18 kJ.
Note that this result was obtained by (1) multiplying the ΔH°f of each product by its stoichiometric coefficient and summing those values, (2) multiplying the ΔH°f of each reactant by its stoichiometric coefficient and summing those values, and then (3) subtracting the result found in (2) from the result found in (1). This is also the procedure in using the general equation, as shown.
Check Your Learning
Calculate the heat of combustion of 1 mole of ethanol, C 2 H 5 OH(l), when H 2 O(l) and CO 2(g) are formed. Use the following enthalpies of formation: C 2 H 5 OH(l), −277.69 kJ/mol; H 2 O(l), −285.83 kJ/mol; and CO 2(g), −393.509 kJ/mol.
Answer:
The combustion reaction is: C 2 H 5 OH(l) + 3 O 2(g) → 3 H 2 O(l) + 2 CO 2(g)
−1366.8 kJ/mol
Key Concepts and Summary
The standard enthalpy of formation, ΔH°f, is the enthalpy change accompanying the formation of 1 mole of a substance from the elements in their most stable states at 1 bar (standard state). Many of the calculations are carried out at 298.15 K. If the enthalpies of formation are available for the reactants and products of a reaction, the enthalpy change can be calculated using Hess’s law: If a process can be written as the sum of several stepwise processes, the enthalpy change of the total process equals the sum of the enthalpy changes of the various steps.
Key Equations
ΔH°reaction= ∑n × ΔH°f(products) - ∑n × ΔH°f(reactants)
Glossary
standard enthalpy of formation (ΔH f°)enthalpy change of a chemical reaction in which 1 mole of a pure substance is formed from its elements in their most stable states under standard state conditions
Chemistry End of Section Exercises
Calculate the standard molar enthalpy of formation of NO(g) from the following data:
N 2(g) + 2O 2(g)⟶2NO 2(g)Δ H°298 = 66.4 kJ
2NO(g) + O 2(g)⟶2NO 2(g)Δ H°298 = -114.1 kJ
2. Calculate the enthalpy change for the following reaction: 2C 4 H 10(g) + 13O 2(g) → 10H 2 O(g) + 8CO 2(g). Determine the enthalpy change for the complete combustion of 22.3 g butane, C 4 H 10(g).
3. Propane, C 3 H 8, is a hydrocarbon that is commonly used as a fuel.
1. Write a balanced equation for the complete combustion of propane gas.
2. Calculate the volume of air at 25 °C and 1.00 atmosphere that is needed to completely combust 25.0 grams of propane. Assume that air is 21.0 percent O 2 by volume. (Hint: Use the information that 1.00 L of air at 25 °C and 1.00 atm contains 0.275 g of O 2 per liter.)
3. The heat of combustion of propane is −2,219.2 kJ/mol. Calculate the heat of formation, Δ H f∘, of propane given that Δ H f∘ of H 2 O(l) = −285.8 kJ/mol and Δ H f∘ of CO 2(g) = −393.5 kJ/mol.
4. Assuming that all of the heat released in burning 25.0 grams of propane is transferred to 4.00 kilograms of water, calculate the increase in temperature of the water.
Answers to Chemistry End of Section Exercises
90.3 kJ/mol of NO
-5313.8 kJ/mol; -1020 kJ
(a) C 3 H 8(g) + 5O 2(g) ⟶ 3CO 2(g) + 4H 2 O(l); (b) 330 L; (c) −104.5 kJ/mol; (d) 75.4 °C
Previous/next navigation
Previous: Hess’s Law (M6Q7)
Next: Module 7. Light, Matter, and Atomic Structure
Back to top
License
UW-Madison Chemistry 103/104 Resource Book Copyright © by crlandis is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.
Share This Book
Pressbooks
Powered by Pressbooks
Pressbooks User Guide
|Pressbooks Directory
|Contact
Pressbooks on YouTubePressbooks on LinkedIn |
8637 | https://www.youtube.com/watch?v=PJOdnPtOMKo | Real World System of Linear Equations to Determine Cost | 8.EE.C.8 💗💙
The Magic Of Math
20200 subscribers
6 likes
Description
1250 views
Posted: 3 Dec 2023
In this math video we will write and solve a real world system of linear equations to determine cost. We will be presented with a real world situation where two friends are shopping the same store for a game system and video games. We will define variable terms to represent the cost of each. We will write two linear equations, one for each of the friends that represent their individual purchases. We will solve the system of linear equations using elimination since one pair of like terms as the same coefficient. We will subtract each pair of like terms. We will then evaluate one of the original equations using the value for the variable by substituting. We will then solve for the second unknown variable. We will answer the question using the information. An exemplar solution is modeled and explained.
Free Edpuzzle -
Free Pear Assessment -
Website:
Instagram:
TPT:
math
maths
mathematics
00:00 Introduction
00:15 The Question
00:52 Understanding the Problem
01:01 Understanding the First Situation
01:26 Understanding the Second Situation
01:53 Create the First Linear Equation
02:30 Create the Second Linear Equation
02:27 Solve the System Using Elimination
03:49 Substitute to Find the Second Variable
04:47 The Solution
Expressions and Equations
Analyze and solve linear equations and pairs of simultaneous linear equations.
8.EE.C.8 Analyze and solve pairs of simultaneous linear equations.
8.EE.C.8.a Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.
8.EE.C.8.b Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6.
8.EE.C.8.c Solve real-world and mathematical problems leading to two linear equations in two variables. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair.
8.EE.C.8 Create two linear equations, each involving the same two variables, to solve a real-world problem.
2 comments
Transcript:
Introduction it's the magic of math here and today we're talking real world system of linear equations where two friends go to the store to buy a game system and a video game and we're going to write a system of linear equations to determine the cost of the game system here's our The Question question today Glenn bought a game system and three video games for $375 Jim bought a game system and four video games at the same store for $410 we're told that each game system costs the same and each video game costs the same we're asked what is the cost in dollars of one game system and then we're going to enter your answer in the box so here's where I'd like you to pause the video do your best work and then come back to see mine good luck welcome back again we're going to Understanding the Problem find the cost in dollars of one game system to do this we're going to write an equation for for Glenn and then we'll Understanding the First Situation write an equation for Jim so we're going to start with Glenn and what did Glenn do he bought a game system one game system and three video games so three video games for a total amount of dollars of $375 so Glenn bought a game system and three video games for a total cost of $375 now let's go to Jim and we're going Understanding the Second Situation to come down and record what Jim did Jim bought a game system and he went and bought four video games and he paid a total of $412 so now we know that we are wanting to find what one game system cost this is our unknown that we're trying to find and we're going to go on to the next slide here bringing forward what we started with we're going to start with Create the First Linear Equation the game system and we're going to let the variable G represent the cost for one game system now we need a variable to represent the cost of one video game we'll use V so three times V right here represents the total cost that Glenn spent for three video games and V is that unknown cost for one video game and then he spent a total of $375 so a game system plus three videos equals 375 we'll do and write an equation for JY Jim also bought a game system and he Create the Second Linear Equation bought four video games so four V the cost of one video game and that equals $410 here is my system of linear equations seeing as the variable G has the same coefficient I'm going to subtract one equation from the other if I subtract these terms I must also do the same and subtract these these like terms as well so G subtract G is zero there are zero pair 3v subtract 4 V is -1 V or negative V and 375 subtract 410 is -35 I don't want to know what negative 1 V equals 2 I want to know what Nega positive 1 V is equal to so I'm going to divide both sides by the coefficient 1-1 divided by itself is pos1 leaving me V or 1 V and that is equal to 35 -35 / -1 is 35 so that tells me that the cost of one video game is $35 but remember we want to know the cost of a game system so let's bring down you could do either equation I'm Substitute to Find the Second Variable going to start with the first one G plus 3 times the cost of a video game equals 375 well now we know that V has a cost of $35 so let's rewrite this G + 3 35 we know that V is equal to 35 and that equals 375 we're going to multiply here so bring down G add 3 35 is 105 and that still equals 375 to solve for G that cost for one game system we're going to undo add 105 by subtracting 105 from each side this gives me a zero pair left with G on the left is equal to 375 subtract 105 which is 270 bring this back to our problem over here and that tells me that that game system the cost of one game system is The Solution what I was solving for G our game system is equal to 270 so to answer our question the cost in dollars of one game system is $270 and there you have it that is a real world system of linear equations that we wrote two equations to represent each friend to determine the cost of a game system and that's the magic of math where we continue to master math one video at a time I hope you have a great day and that you'll come back soon |
8638 | https://blog.chip-chip.com/why-is-it-called-dc-gain/ | Understanding DC Gain in Electrical Circuits and Control Systems
chip-chip
Home
Product
Manufacturers
BOM
About US
Sign in
Get started
chip-chip
Home
Product
Manufacturers
BOM
About US
Sign in
Get started
CONTENTS
Key Takeaways
What is DC Gain?
Why is it Called DC Gain?
Importance of DC Gain in Electrical Circuits
Importance of DC Gain in Control Systems
Examples and Calculations
Related Concepts
FAQ
See Also
Understanding DC Gain in Electrical Circuits and Control Systems
·15 min read
DC gain represents how a system responds to constant signals, measuring the output magnitude when the input remains unchanged. Engineers rely on DC gain to analyze the behavior of circuits and systems over time.
The term "DC gain" is derived from evaluating it at s=0 in the Laplace transform, which explains why it is called DC gain. This corresponds to the system's reaction to steady signals, such as direct current (DC). For instance:
If G(s) = (s+1)/(s²+s+1), the DC gain is 1.
The steady-state error for a step input is 0.5.
A higher DC gain reduces steady-state error and enhances system performance.
Key Takeaways
DC gain shows how a system reacts to steady signals. It compares the output to the input when the input stays constant.
To find DC gain, set s=0 in the transfer function. This shows how the system works after temporary effects are gone.
Higher DC gain lowers steady-state error. This makes systems better at following the desired output.
In amplifiers, DC gain shows how much the output grows. For example, a DC gain of 10 means the output is 10 times bigger than the input.
DC gain affects system stability. Too much gain can make the system unstable, so balance is very important.
Using DC gain in circuits keeps the output voltage steady. This protects delicate electronic parts.
DC gain matters in control systems. It helps the output match the target input closely.
Always check the transfer function gain at s=0. This ensures systems handle steady inputs well and avoids design errors.
What is DC Gain?
Definition of DC Gain
DC gain shows how a system reacts to steady input signals. It compares the constant output to the constant input when the input stays the same. For example, if you give an amplifier a steady voltage, DC gain tells how much the output voltage changes.
You find DC gain using the system's transfer function. Mathematically, it is the transfer function's value when (s = 0). This helps explain how the system behaves after all temporary effects are gone.
Transfer Function Gain and Its Role
Transfer function gain is important for studying circuits and control systems. A transfer function links the input and output in the Laplace domain. It is written as a fraction, with the top part showing the output and the bottom part showing the input.
For example, take (G(s) = \frac{s + 2}{s^2 + 3s + 5}). To find the gain at (s = 0), replace (s) with 0 in (G(s)). This gives (G(0) = \frac{2}{5}), which is the DC gain.
Transfer function gain helps predict how a system acts in different situations. It shows if the system makes signals stronger or weaker and by how much. This is useful for designing systems to meet specific needs.
Physical Interpretation of DC Gain
DC gain has a simple meaning. It shows how well a system boosts or lowers a steady input signal. For example, in an audio amplifier, DC gain tells how much louder the output sound is compared to the input sound.
In control systems, DC gain shows how closely the system follows a steady target signal. Higher DC gain means better tracking and less error. But too much gain can make the system unstable, so designers must balance performance and stability.
Think of DC gain as a way to measure how efficiently a system turns input energy into output energy under steady conditions. It helps understand the system's performance and guides design choices.
Why is it Called DC Gain?
Connection to Direct Current (DC) Signals
The name DC gain comes from its link to DC signals. DC signals stay constant and do not change over time. For example, a battery gives steady voltage, which is a DC signal. When this signal is used, the system's output becomes steady after initial changes stop.
DC gain shows how much the system increases or decreases this steady input. If the input is constant voltage, DC gain tells the ratio of output to input voltage. For example, if an amplifier has a DC gain of 10, the output voltage is 10 times the input voltage. This connection explains why it is called DC gain.
Role of s=0 in the Laplace Transform
The Laplace transform is a math tool for studying systems. It describes system behavior using (s), which includes frequency and growth or decay rates. To find DC gain, you check the transfer function at (s = 0).
Why (s = 0)? Because (s = 0) means a constant input signal in the Laplace domain. Substituting (s = 0) removes all terms related to time changes. This leaves only the steady-state behavior of the system.
For example, take (G(s) = \frac{s + 3}{s^2 + 4s + 5}). To find DC gain, set (s = 0):
[ G(0) = \frac{3}{5} ]
This shows the system's output will be 3/5 of the input for a steady signal. The role of (s = 0) explains why it is called DC gain.
Steady-State Response and DC Gain
DC gain connects to the steady-state response of a system. Steady-state response is the output after all temporary effects are gone. When a constant input is applied, the system settles into this steady state.
DC gain shows how close the output matches the input in steady state. Higher DC gain means the system boosts the input more. Lower DC gain means the output is smaller than the input. In control systems, DC gain affects steady-state error, which is the gap between desired and actual output.
For example, if a control system has a DC gain of 1, the output equals the input in steady state. If DC gain is less than 1, the output is smaller, causing steady-state error. This link between steady-state response and DC gain explains its name.
Tip: Knowing steady-state response helps design systems to meet goals like reducing error or improving control.
Importance of DC Gain in Electrical Circuits
Image Source: pexels
How DC Gain Affects Circuits
DC gain helps decide how circuits work with steady signals. It shows if a circuit makes signals stronger or weaker. For example, in a voltage amplifier, high DC gain makes the output closer to the needed level. But too much gain can cause problems like instability or distortion.
Here are some key circuit performance numbers:
| Statistic | Value |
--- |
| Gain Margin | 3.15 dB |
| Phase Margin | 115 - 131 degrees |
These numbers show why balance is important. Enough gain margin keeps circuits stable. Good phase margin stops unwanted signal changes. By studying these values, you can make circuits work better and more efficiently.
DC Gain in Amplifiers and Filters
In amplifiers, DC gain shows how much the input signal grows. For example, if an amplifier has a DC gain of 10, the output is 10 times bigger than the input for steady signals. This is very important for audio amplifiers to keep sound quality high.
Filters also use DC gain to control signal levels. A low-pass filter uses DC gain to manage low-frequency signals. Changing the transfer function gain lets you adjust the filter to do what you need. This helps pass wanted signals and block unwanted ones.
Using DC Gain in Circuit Design
Knowing DC gain helps you make circuits that meet your goals. For example, in power supplies, DC gain keeps the output voltage steady even if the input or load changes. This protects delicate electronic parts.
In control systems, DC gain affects steady-state error. This error is the difference between the wanted and actual output. Picking the right transfer function gain reduces this error and makes systems more accurate. Whether you're working on amplifiers, filters, or control systems, understanding DC gain helps you design better circuits.
Tip: Always check the transfer function gain at (s = 0). This shows how your circuit works with steady signals and avoids design mistakes.
Importance of DC Gain in Control Systems
Role in System Performance
DC gain is important for how a system works steadily. When a steady input is given, the output depends on the transfer function gain. Higher gain helps the system reach the desired output better. For example, in a temperature control system, DC gain helps keep the temperature close to the target.
Think of transfer function gain as how much the system boosts or reduces signals. If the gain is too low, the system may not reach the needed output. But if the gain is too high, the system can become unstable. By studying the transfer function, you can adjust the system for better performance.
Influence on Steady-State Error
Steady-state error is the gap between the wanted output and the actual output. DC gain affects this error directly. A higher DC gain lowers the error, making the system follow the input more closely. For example, in a motor speed control system, high gain helps the motor match the set speed.
To find steady-state error, use the final value theorem in the Laplace domain. This means checking the transfer function gain at (s = 0). Raising the DC gain reduces the error and improves accuracy. But too much gain can cause instability, so balance is key.
Connection to System Stability
DC gain also impacts how stable a system is. While higher gain improves steady output, it can make the system react too strongly to changes. This might cause shaking or instability. To keep the system stable, check factors like gain margin and phase margin.
For example, in a feedback control system, gain affects how the system handles input changes or disturbances. Adjusting the gain carefully can make the system steady and reliable. Tools like Bode plots or Nyquist diagrams help find the right balance between stability and performance.
Tip: Always check the transfer function gain at (s = 0). This shows how the system behaves with steady inputs and helps avoid problems.
Examples and Calculations
Example of DC Gain in a Simple Circuit
Think of a circuit with resistors and a voltage source. You can find the DC gain by using the transfer function. Imagine a voltage divider with two resistors, (R_1) and (R_2). The input voltage is (V_{in}), and the output voltage is (V_{out}).
The transfer function is:
[ G(s) = \frac{V_{out}}{V_{in}} = \frac{R_2}{R_1 + R_2} ]
For steady signals, (s = 0). The DC gain is the ratio of (R_2) to (R_1 + R_2). For example, if (R_1 = 1 , \text{k}\Omega) and (R_2 = 2 , \text{k}\Omega), the DC gain is:
[ G(0) = \frac{2}{1 + 2} = \frac{2}{3} ]
This means the output voltage is two-thirds of the input voltage.
Tip: Use this method to quickly find DC gain in simple circuits. It helps predict how steady signals behave.
DC Gain in a First-Order System
First-order systems are common in circuits and control systems. These systems have one pole in their transfer function. For example, think of a low-pass RC filter with resistance (R) and capacitance (C). The transfer function is:
[ G(s) = \frac{1}{RCs + 1} ]
To find the DC gain, set (s = 0):
[ G(0) = \frac{1}{1} = 1 ]
This shows the output equals the input for steady signals. The filter lets constant signals pass without reducing them.
First-order systems smooth signals and remove noise. Knowing the DC gain helps design filters that work as needed.
Note: The DC gain of a first-order system is always 1 unless extra parts change the transfer function.
DC Gain in a Second-Order System
Second-order systems have two poles and are more complex. These systems are used in oscillators and control systems. For example, consider a system with this transfer function:
[ G(s) = \frac{\omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2} ]
Here, (\omega_n) is the natural frequency, and (\zeta) is the damping ratio. To find the DC gain, set (s = 0):
[ G(0) = \frac{\omega_n^2}{\omega_n^2} = 1 ]
This means the output matches the input for steady signals. The damping ratio (\zeta) affects how the system reacts to changes.
Second-order systems are useful in control designs. Knowing the DC gain helps predict steady-state behavior and improve accuracy.
Tip: Check the damping ratio when studying second-order systems. It impacts both steady and temporary responses.
Related Concepts
Steady-State Error and DC Gain
Steady-state error is the gap between desired and actual output. DC gain helps reduce this error by improving system accuracy. Higher DC gain makes the system follow the input better, lowering the error. For example, in motor speed control, more gain helps the motor reach the target speed accurately.
Controllers like PID and FOPID improve steady-state performance. Advanced controllers, like multi-stage PIDn(1+PD), work better than basic ones. They reduce steady-state error and respond quickly. These controllers also handle complex systems, like DC-DC converters, effectively.
Tip: Adjust the transfer function gain to lower steady-state error. This improves system performance for steady inputs.
System Stability and DC Gain
System stability shows how well a system handles changes or disturbances. DC gain affects this stability. Higher gain improves steady output but can make the system unstable if too high. Balancing gain is important for both accuracy and stability.
Tools like Bode plots and Nyquist diagrams help check stability. These tools show how the system behaves at different frequencies. Metrics like gain margin and phase margin ensure stability. A system with enough gain margin stays stable even with input changes.
Note: Always check the transfer function gain at (s = 0). This ensures the system stays stable with steady inputs.
Frequency Response and Its Relation to DC Gain
Frequency response shows how a system reacts to signals of different frequencies. DC gain explains the system's behavior at zero frequency, where the input is constant. This helps connect steady-state and dynamic responses.
For example, in filters, DC gain controls low-frequency signals. A low-pass filter with high DC gain passes steady signals but blocks high-frequency noise. In amplifiers, DC gain ensures the output matches the input for constant signals.
Studying the transfer function helps predict frequency response. This lets you design systems to reduce noise or improve signal quality.
Tip: Use frequency response analysis to adjust transfer function gain. This ensures the system works well across different signal ranges.
DC gain shows how a system reacts to steady signals. It helps understand how systems work with constant inputs. Engineers use DC gain to design circuits that make signals stronger or filters that control frequencies better. In control systems, DC gain lowers steady-state error and improves accuracy.
Knowing DC gain helps make systems work well and stay stable. Whether you design amplifiers, filters, or controllers, learning about DC gain helps meet goals and keep systems reliable.
FAQ
What does DC gain do in a system?
DC gain shows how a system handles steady signals. It compares the output to the input when the input stays constant. This helps understand steady behavior and design systems with less error.
How can you find DC gain?
To find DC gain, set (s = 0) in the transfer function. For example, if (G(s) = \frac{s + 3}{s^2 + 4s + 5}), then (G(0) = \frac{3}{5}). This gives the steady output-to-input ratio.
Why is DC gain useful in control systems?
DC gain shows how well a control system follows steady inputs. Higher DC gain lowers steady-state error, making the system more accurate. But too much gain can cause instability, so balance is needed.
Can DC gain change system stability?
Yes, DC gain affects stability. High gain improves accuracy but can make the system too sensitive to changes. Tools like Bode plots help find the right gain for stability.
How is DC gain different from frequency response?
DC gain looks at how a system works with steady signals (zero frequency). Frequency response checks how it reacts to signals of all frequencies. DC gain is part of frequency response, focusing on constant inputs.
What does DC gain do in amplifiers?
DC gain shows how much an amplifier increases steady signals. For example, if an amplifier has a DC gain of 10, it makes the input voltage 10 times bigger. This keeps signals strong in audio and communication systems.
Is DC gain always 1 in filters?
No, DC gain depends on the filter type. A low-pass RC filter usually has a DC gain of 1, letting steady signals pass unchanged. Other filters may have different DC gain values based on their design.
How does DC gain affect steady-state error?
DC gain reduces steady-state error. Higher DC gain makes the output closer to the desired input. Adjusting DC gain helps improve accuracy in control systems.
Tip: Always check DC gain when designing systems to ensure they work well and stay stable.
See Also
Enhancing Amplifier Performance Through Gain Type Knowledge
Grasping Amplifier Gain Concepts Through Examples and Equations
A Guide to Selecting the Right DC-DC Converter
Investigating Thyristor Roles in Power Electronics Applications
Simplifying Inverting Versus Non-Inverting Amplifiers Explained
Short Discontinued Model Inquiry, Quick Quote
Consult Now
chip-chip.com
Home
Product
Manufacturers
BOM
About US
Menu
© Copyright 2024 chip-chip.com - All Rights Reserved. |
8639 | https://www.aaidd.org/docs/default-source/prepressarticles/vineland-3-measurement-non-invariance-in-children-with-and-without-intellectual-and-developmental-disabilities.pdf?sfvrsn=d2a63c21_0 | %PDF-1.6 %ãÏÓ 307 0 obj <>stream hÞäYÝnÛ: ~¾Eê nK0 §hzQ ØEC,)wèÞþPí´É·Ó¦mQ-ä÷Q"%!(Àôï�4 XÏï¬)8ph§}0à,B ÎóàHà5¢ ÖAäçÑóïø7mµåímH£(éñt2DE0V³Ftã'ÄF¡çQÜÏÏO´fÛBj°5 Ql]H 6Ëb#¸ÀO¿ã)° ¬Ô;~üî]ñ~1¯ÊyµÈÿßfÓÅÍXAú{éKq6YrPês^®×Ëi¹áV¿ðÔ8á¿÷UÉc¿''µ¦ëúÅçÙ·ÕS¯v²¨&°kQ[ ®
ÎfQwi K%íUñù|QÆFå¯cÓx<Öù[ÚeµêEÌËÅõU2~T|(ͦåùð´ËÉ|uOguGöù¡�~¼©£jRÜ<GÝe8 uFòÈØ¢6·8�kËÇIl¹Êj\}åMU|úÉfñ>O_Ëÿ¾þ(§IÏåÓ´,ªåõ´Ê¬@¯«Á¹FÆeûÛ.L O y¿å&v¸iÀÍú/aO©Ëáðü÷à¦îpÓ©§qóqÓ¾�L�ôÊÛv9l^ÅåºÜtÇØÛPô½ÿ3(R¸p· �_Yì ö�G÷AQ ¸ÅÜò¬>·¿oÿks»zõ¹¤¹f¹RME}Z¤éEJ?Ì'ær«]³?qÔ]õRxX¨zXR¥ îtϼ Öû�ÃtáÞ°°çýòïv1+ ^)3)iÍFøBòþºêcKï;,U²lMj>÷9лA#12"íXG½6õ Vè�K"$²´L\ éì3Ǧ]·ãbÙ©%ÐäP7r×2oê3îû £££n´-Fb¤c¥4h m¢$F¼ íFÉK,ùX7 ½,¾aFïrùpÑ
GÌÇ!w4¶ÍÑÍ&´Å\_´¦½@ÚëtúÑ÷õ. 0ïºëû Ìÿû\_P+!^'6ÕT³((Dêçb
úY!å·Y ªe%èuéäZVBl# 4Zß)ÜëTx!Vàµz¹¤h,
+Yìò0BY§ÄB(×"ÕB¤BÜúÒN5 :îÇB¾~{Ƚ]J²ÔdY9Öj²¡Pr;yþ/À�¥÷W endstream endobj 308 0 obj <>stream hÞ¼WÛNÛ@ýúHwgïBÊÒiÅ!V±8È1ªøûÎ:)øª (xwvÎÙ J4aÌDZA$ZXZ@³Ñv?Ñ^(ÜÊpÖDãe1hg¨ÄE|[Oh Ô-KÞ¸/ ¿àÄi¦8¥¹ÛH·±Ò9S3´% Ò±@¸ÈÜãÄ(#\#a@?Þêk'D¹-Ù¬ç|¾E½ tì§Á,ºõ{Åx½<£döo"sã%ÑcºHÉ�NµÓKòÿÊö¡çiúÁ˧-\_ c£uܪ«EùÚ¿Ù)7½a^(p6n £Ç´ÈlܧÅlqhªµ½ÀK;Ç^ñ0ÌqÏz1JéíûØ£y¸üô,üãåbÄÍÌøûPÞY0DzòHcºÝ¥Ô¾v³iC~\ÀÖü ?½É¯Óíõ½Å]5deß³vMW e]ÔÐÿ1òÏËHjÄñøÝïú¸á~$mtJmiRA°P°+¥qþ^0ìjk²èZFÏó»ÅlGCmm^ÛÐÚئ0å^êoålÚxVÔ=k =yAbºêj©ÖÍ£Vó6Ê$l+U^#ߤ¯+ó
TWìV7i~^ûWeêúÈÚêÊåüÉð¨3¹)EV¢ ²þÄj3,,ÔÔ©¶¬ÜYµ¨SmFVoZUÑÛ$«RÅyǪiê³lW}ßBKV¢õAçÉ帴4nÐêQZTÝÑð¬?)íõ(4A' ÀLQUó¾#Q�mD�v\_l�yábµþ¿¬Å1ßáèÚ-p^ØSþù¾\_þ÷I6Hm^ØI÷núý£RdA 3ÿ�Zö endstream endobj 309 0 obj <>stream hÞÔmoÛF ǿʽ¶w"ä= mvÀ{1´V É[¥iäÙNÓöÓ'9ÃêZë&3ßYÖÃïHþI FMHR0DÊ&¢èèMJe$@I'lпDõjÔ£z³Áõbè'ã¼NØx e"ÆI2^Ʋñ!êÀøDzBãsÖGr |ÒçP4Äã9Ê@ÏQ~±L¼¡Ì £.ª°ÄdÜ+3\¸9d} xÃIá½Ð>|X=Ñ÷óê×g?÷7¯·ÛÕ滪º¸¸p]Ûou_¾ÏêWÝÚ-·U}Ò·mÕg§Cݺz³zÿ¸o!è²¼:×i#)Ï#wK«ÏK6K̶e-yéÛø³xݽíá·GGÕÓaÝvëÅ-p}?Nð¸zÖ-¶/%'�äXgá¸z~Þl?¬ºê§þìÍÑѽ_A01j|e21Ã?òÿR?Ylûáìµ ¥ëÊoW©zü®ï.~ÎN?<¥P ®i8r½ ÐJS×v 5×#Ûs½ýotúá)d§±£ÑGXRãæ¬hÛõùbûs½îζ&_
´z7¬_UN¥ªzµê]3 o6.«Sö"9Ì¡xõ³øàƪ:]¢ã¦+)kûÍÖö;ßk¸¡:qk çÁ1Í2ǤI$9Õ83;ðû¤×6äDv²Y)&ÞðÀ4ìð8hà ´æávÖúRRM\b},6¥j¥÷sÀLÙë4# ûbIdp,ûÀx/'õ$QN5pw¤ßçõ×ö<Äj¬zÞRhgµC &I.¤Ñ²/Ï\_l2ØNÔÓºxPÔ^®ÍE±ÒÒ6~DNSù´2ªCk-8)ÿL+áúÙ0ê ³Æ hÉN\_Ô°÷TL¥8)Õ\}®xÓipTD¿Ö,�³~ti è¼v"K¨Ò>\_¾qÉ~§Õ4 Q8]]¼½KM=;&T¢uØy-ÌJ,NÜû¤7?3[ï%ØÙüçÿF\£ÒM¾§Ú9X°è-EÒÖÌF÷BcíÙLè yW/½Î]øïpçì©{вô9»ºvJ$ñîuØF¤ÁiVÁ0ï]r.ûí+nöt¥Xúòµ]}Ú9ïE)õ,@vu+£Yw)g$]äd³·a9¼ãTõàç4Öu vYa!9©°tky¯Þ´S86\³ndu£nw%d'%áQ'-´Øÿò{o´[¬»Í;Ö¤ã¬Y99ú«ðíÌz¨ð1Ý9ëd×|u§Ëù7Õ¦MuÛ.ûF¬Ô,· l2Zê8wóýü[àVVð_þ¸.øëå2,ê¥wìjËÍmh)ç¦ Ë%Fîî©nc_ÓiïX¤¤vÑÆo£M¢L°XDíºZù×ü1&µ>¨íÿÝ (ën5ë "yàO�æ;(ý endstream endobj 3 0 obj <>/CIDToGIDMap/Identity/DW 1000/FontDescriptor 4 0 R/Subtype/CIDFontType2/Type/Font/W[310[190 333 333]15[277 333 277 277 556 556 556 556]24[556 556 556]28[556 277 277]3236[666 666 722 722 666 610 777 722 277 500 666]48[833 722 777]53[722 666 610 722 666]68[556 556 500 556 556 277 556 556 222]78[500 222 833 556 556 556]85[333 500 277 556 500 722 500 500 500]98]>> endobj 4 0 obj <> endobj 5 0 obj <>stream xì½y|UÚ6|Zº«÷%éÎFÒ4!ÄlLË0È@Dd ,"""d1 BdQds¢®¨"(ã K:é~¯SÝ Aç}Þïû}ß_o·wªS§Îr¯×}ªB42prû¤1Sh×é¨ÙBñÛï¹;¥Ç ø¾h0óÌñS+&=Ýä Ä#DP1ñÞñ _ì@H»ä¾[9nÌØÖäB\>r+Qá|äÝq>ç'Ý=3üÒ®ý8_@¸«ÃÄ)·éì¾Kxå§O3sêÑGeÖ?Ñ>eê]ã¦þðíQ8?Û_FÅ<ÅÍD2¾FÈ1äú+ó°Zö%û÷ácøt~/ógø»ü#þ)ÿÉ¿æÿÌÒ¤Ò_¥ýÒaééGéWé²l|ÔXk¬3n7¾bÜcÜom}Öúuõyë+ÖWû ÖO¬_ØmwØÛVØVÙ^°í²½b?¿:;Ö;.·{²Ýåää¤ä>É%OþKrYòìäÉû?Lþ,ùçäóÉ¡KJ§¾)SSIÙrÜï{Ðwºý¿üÄEúu8º"õç®SÒ¤4cZZZÿ´iÒ¦§U¤=öbÚÞ®y]º>Þõ©®¡n×t{¤ÛÓYY½²feÕe½Ê¾'ûìÐÓÏ{ö|Ê{jì©SÚt]¿ÔØ%¨=ÁÜaðàÁÞÁmÍEÍK5CþPïÐ ÐøС»Bwåá½áCá÷Ç¿ÿn
fu´ duìkNøh~·®mAÛáÏò#ÚöD%«ôg©\zT:$½-~.ÉÝäáòhÛ2ùYùïòù\TÛ|J@LUv+¯)GsFj
ÆÆRãÑõnr·nÝÍ"Y)Y£³æd½u$dÏÊþú¸ÍS´UÛ;G/I]ÁvÁ|h[ Ø+Ø7øaóÀæÇÏ
RB]C7
\CS¡m÷ixLx\_øðáOÃ\_O
áð·¨ÙþkxNøpeøpa8?îÎ \_î¾&Ü)N w §½aO866µ°!,yÞ ý#´7ôjh{è
ÐÊÐPuè¡ÐhøÔÐäФÐÄЦÛnkÝTÞô¦þMýJú6õiêÝtcSQÓ M
Mݺ5uiº¦©s¿É×ÔÐß×ämò4Å4¹ìMÖ&ÞÄÁàÏÁgg?O ~<ü.øMðÃà»ÁW{»õÁ]ÁÁÁíÁ/ >\\\\¬ þ58?øp^ðà¬àÌàà´àÔààøà°àMÁ1øÖ¶yÁk^Xql0&èmAkÐTáÆÆÆS'¿o<Ñø]ã×\_6~ÑøÏÆw7æ5f7Z-¦FÑÐÈi#¹¾ºÜt9x¹ñò¥Ë/\_¸|þò/\~ïòáËû/¿qyßå\Þ{éãKï^zëÒÁK¯\_Úsfçíg¶Ù|¦öÌÆ3Î<{fýgά;óÔÿ:³öÌ3OYufågYrfì1gérÆöã¯?úñÃü¸ÿÇÍ?~LýÑ~:xú\_§þìt¿Ó7.>Ý÷tÓ½Oßx:púÓ]Ow9}ÍñwüÛã\_ÿø±ãÿàø]ǧ¿õøÀã7ïq¼ðxþñÜþðr뺴¿æØ5G¯yá'Rízµë8:áîOp\_½ 3f$ MèÐ;!1þRüøSñ'ã¿ÿ&þxü×ñÇ,þhüñïÇ¿ßÿWÿïçÿ~þüÈ»I~7߶)5R>yu}¨RÔßƾ$N:I,ìõð¿Ñåÿ|lCþ\_®O6ÑÄ^þ7=Ä Ú6¦IÓÈÈðò&ùÙGêɲ¿%!#+Û? d;W×8\_N6ê׶ç"½Ñr:>E§Ð¡ôÆßÌg.¨ßJr3õýÁ|kñ]GîÅÑÃd¹ßwh
ïRò6ÌUm=,ü7½DÄg(¨\¼-È;øò f?ªWRCÊI5FZJÆ·Öu"[Ø ì>6>BÊÙ\²¾JÞa[H#ÛL&³?'#ÍäI$Uä»,#%y5I #»qÞ¼AH®ÀÕZ3ôÓÓÏÈZd4±$ ¯%YáwÉ-ú÷q|·÷ëñCæðu¬Ïa}òTÈghØ % ß¡.är7\M"V6/FóI4²xYäFÌé,õ?AÖ©äý¬¡u=P»¿ãôã.RK?ÑëHRBÆêçSð]©¾"¶·©Cfã÷hTëøHR@ÛÈ,ò(´íêOwÒ�Çý©?IvBê;¡UëÁ«øþñç8ù<ÌǼäÐë1·t¶Î7ðÞd\ÝHúb~ô^rÆ òa1a®÷b{ßw¡IÉPÛÏZÌ~9yL\_ËH/ «þ£Ïmø H;ÑNàLßÀã»ANn£ÉäK¾\_ÃGkX[Ë¡Ð÷øvþ7z~J¿#9$zÓíeÙȪk¸tæ©Êve;}T©ëÓÉ\_È}èiTN^aåd>õtTtVGR=éϧÒÝü ©UÐÅdBph,tAHmݨûRþé+%È=ümpàMÈô«Ï~²Þ®¼|<÷GuÑr^$¯ÀÈ?¨»ô!¿R#Ík- Ýñmºsðýÿêó¼ËPò'2òëú""©ävèUÛ½øu}ÁàBK9<ùm±æßóüaÝÝûGuOò¤ÅÒbþ´¢39Lª À¿%øn§ûIäKC¥¡ÿc¾ÜïR iöÇJFc¼Á~¨¹úSö³ùmøŽÈ@:º=¯-"ÚG¤/¾DRÉC�Ù¢\_:´é©¦oW¿$ŨjøÆC»+àqçpXäzØÑtHj fø�YEæaÉc k¡S7;y=m]c¬/ÿà£ìT'nÅ)ßFìÒÓÒr>#Ãÿ ÿØüÓU
}´ÈVhúýàËÃðPë0ðåºn&{×næUíNÓ!U-ú«ÿ£Ùü?üäßÊG7|Æ7±´ìj]óQEbìÑìõ¦e¡OÀ3B0»;Êéu~³[§Ïôsº,"V""næ\_¿½B&È×Óeä#e7À}·öt-WÁ\KI/Éãì"3/ýKã|¥ÃÞeø4þ ?~eÚÐmðáC1AäUØûdÄÑßp°N~J´ J¨ÜÿZ8î§2Vô0ëÊzÃkï%ýX ÌJ¡ï Ý<<ª5:Òðe<é DÇH2ZE+¤/0kBðµ³0Ò4ë ¡ÿȧ§G£Å$ åø
"Ü?ñá;|#HåVq÷Õ9':îJ̸¼Ê|Ð8\»ïw E]$ܪiÑq¡Ï1FkÉÐð%Q$D"ÑÖØÈsfæFk1 ÒÈS|ÊÈuäpøþ>ú/½¶lùÖ¬§×Ð=àeùA÷ ÑëÄËX¬{ V6ÓÒ¾;|Ô>ô{u줳>Rð4ùÐö 8.Gão/Öö,/d÷am§pÃ@\©®íIþb]thÖð îFëùzHM$¯Óÿü~] eæ@¶éX×TjZéK¤×K8ïèå}ï6¼sÚ�®oAA,lf«Á·GÈ&ØÌ�ÑL½%\ß©XûxøáY¸s~ü\ë,ï< ¨>\ÊÅÚEÇÈGßSØà!Ìj¼ãcüW2ÌeÀ¯À Jz\ß½ ?/7çºì¬n]3¯írMFçNéiS;øÛûRÛ%%&ÄÇy=±1nÓa·Y-ff4¨,qFÉ5Ô[çíUÚgB]\¯Ñu}ý½ýöº¾·YG >¿#%;sxh«:9£¸ú×¹n#üáuJÆoÜRÇSí¿øpó>uR\þó÷3¶.}P©Ïoÿ(¡õúpÜSß«ÔçK¨c©øï&\ýƤ³D½/!RsSX\¨>üM>\I¾o8~Öµk9>ü&ù 8úZë4³ÑäºÈ¾o\¯ÞuĽôý¦ÄFçòO ëÒ30 ;ô¾HfuÿRG]u4f�&|õ�¶¯óÿ}ÆNð÷{ø9vôðÓ²(eÑ RG6õ)÷¯;üçÒm&¿×8 D¯ Û4jL¢]LÝFûÞ@õÖ·O÷m,SL· uÅ£qàï ®áëÊúðk·½Dp[Ë+rDÒ«NL"åºÀ:²8eÛ5¯-z¸ÞNnaë;fdiÛOíS9¸.±ÿÀ¨P Ñ)Bؽõ!º>)p.ÚƯ¿·ùUõc+ÇJBGû{ã±Wiïµ:'Ê>uºb4+¾ï»¾¨÷qºhQMJÝ:L·ÍUø
x1õE}ü õÐS$³Ulº.Þ4VNñº¹·MhÞ[´ß·È^×÷Ò|p§~ccGOS0F,³ÏEÇéK}X\_´5¥ÏÞÄÐ}2w(íSéïse@,<õ·÷ú|uqÆEú)ÙG¦ Wæ/,"!b>½êõ ÖeczVE·+£{îÈMëÔÔùZÊ"Ñ£ZçΰûöãÚk]®é?¨´Oï}õu¬Wi³Þ³8î?°µzÑfQæÙúßêïÿçT¶ü1\_Ö\y4¶×{mð&4ฯ¿ïèEúúSú.½hL}xîmþ»Ñ¶þýMí3:E·{úÝêú><¼Î>ºv
¾õ$$Ó7¥rLÄKù}ù >ÇðËÿÓå¨AÙ¡òÄÙÏZf¸¢¾¯ÔÃ!$ÔÙó
bCJa·ëêªÿÀ4nEç HøðÔ>wÜå 1ª+áý9ZN|>a>ëä6ÔÍýsiä<
Ü°23 ¶ÑÊk-Wb+s[®´Þ>Ú1yûßú¿Q綪¼Èáw¦dê¬×ýìغ×còë ùQI»zò=b \ið\
u ýFÁ8ÈEvÊ=£NîUúZBáð»¶êA´G¡¡ö#þ7©ðÄm¯£
u4VÔøSÝ©sO>.¶ÞÒgÑè¨åAv:#ëNÀl;Az÷ýfåÑà0¶ò÷Ë7cùhc÷×/$DÚ;~Áwtøbüvöý· pßp IJJWs J%JdnÁþeþ#٠ݺú>G~(7ÎIPd®�rd$ßN;É»ê6¸Lø, ¨ìű]3h¼×>àlA)B/.Ãtkè[~þ9tRÜ?+4?$f2
¥bü³¤7øÄ óôIø=V�áä@Ç :2¶0îfs)s³ÆJ3BQÃ2=Ùee¤¨°¨¾>£fö~L¢ÌHýJÑZywcü#ô <òëàì <Øö£ìfí;.õrÞîd
¦McÌlNy ,¡;£¬ÄÒ@]f¿9ÛÜË,¹bý±Ù±½b¥DÛ5¶BÛÍ6)Ñu«Ðu³KªUèj®ÌTæ;hF jb'ª=!», {¡c÷3¯:eD-Á.XQt¼ÇgTYX)´É(½¦8ì̧+Ëé¸îZêoÏng,+ _^<ò_ü~Ùã7ο}Ùc=_ovªCKÙ:NkÜMãhÑáÙ¡oBû? M-J¾FÈÍ&=YWÊ3å²$«0&²¸R)-ê%I¦Ãh]E7QYèY-tjè.´,˪Þ-k_(ú¨¬ù©CQs:äes²£æW5]5wøÖ1/i3\<ÆQ 1Ø(óÊ\^ëuÇÆzkciZ,¥ªÃãv¦Ùꦆ >39{¥8Å¢=áPã P§!Ù¦XJ^¥3bWÅnåLͽËþ1gIÛcãr5N!ÐR¾3c4ñX}Á ºÕ8O\^ÜÅ8É+©[ZîûYõu(/³ÝïôàX¤\C8.f±¤³Ù4\Ë z»²ëµÞ5V]/q\_yY'z]nÞ 4çºþöÌ:dg!ka²¢ú$)ç
³Ë¾û¯÷n ¯8(?ôñ¿Þ½ÿôuìýûCÿ¸;jöùÚ=ªz4J4ÍòD@q©ér÷äáÕçñåùЬU
¥1;%é11í¦ÒVgnº}!÷-n¯-6®á÷³ ã>SØ=vÀ©è<,/ÓYap6´M0¹¼ÌeÍGEx]®Û ²~QD%ÚƸ¿}Ç´¼v4;+Wg½uk TÊ ¾x{\_SÌ¥Óxñé¹×õ)\Ù{Ù¨I ;ü?oÜõë<.þÃûßì[ö趹³Ö»²æë?i梹¾ìýb]Ã!¡OÞH©H:æ CãåYè0K
S\V\UJ&Ö\Y×åpOp³|;©ÑCeDÔFýIBûçH=øzÄyÝqqÞ>68@æýØsÊsÉÃ=1kwÁY1CqJ¾³ÙöÁõkeÇNÆAêqÞ8fLóæy½Ã¼Áéu23K±!.6)7Á¹Pö¸DÉk^h ´ïgI¦+9èɲ½?̬¨?¤
.eYÎQefÛ÷«¡îUeº×$¢æfg]®EË㣽ÿúè.jùzWè¼{gåä
·.|¡lóRv¬¹W 5üiè½÷Þ¶.^øÞª5Û¸Ù/[áק§%ã)½«i
qtNtQ^CÉíøv¥UݾWû[ÛóØc?òq Ob«±±Zs½AÇó5Ê:·§oµÿ´=ãi1yb(-uÑUvÊkðó@ô²0´[«z1nô v@V§ÓýUõ>0¸ ÞôØüZÛlé:ÿ_ÊÈÎe CjZ.ò \ìHX(«¤ÔUD21¤B$ÚdðGÿ<+¬ C·ÝÚz¡YDu!Ý2t¹EÃIDRÕ"VL¤�xI9áüíÃNµ£54\2%ÆKÍ÷ØÑ<òaèµû.nG·CÒ¾÷¯Yóà-cç÷
.¼÷~è3OÒtqýËöB?
>i®9üÈ#»\_\úØ»PëÛ<°Ô¦ÝªPG¡Æ¬,¡ÌC)TR>4\Ó«æ« àZvÉËò\�|#ìò5h- Y:Æ+¡¥T1"dÕ 3îT Ð
äHèÎLîg&ÎO°Ê ös¿E=|ËílpêðåW<9¸DýNæ<ö'Ö«4îëælþ²p¶¹oèaÁÞ¦LèhiÔK{Iê¼Zá¡I´ÒÑÆ ]M+}´±lDi¢·zè Ͷô²Üjá<ÏSìæádºòövÆj,C7&HbÜËè÷87ã3 «<Ï f¨Ô S\\|YbÄ·¶SK97ÎmÉOU+æ KmïiÖ~W{ÉNò KLlo ÔÝ3»;Û/TÍ÷³T¥MT·«°(Ð9ÝqÒµ°±-Ou,3JG9SjP&ç:¨P8ÝYÇBr
ãcs^ør3eo¿C;Î5?÷¥wK§·6Ö
zòéûgCå¶@}JcCûB÷¦
f¾\o=I3©Áÿ÷¥ËÞxoùm ÖÃVK©zp ³)u²H/ÒbÉ §°¹PeXªC@© ¼´!øYúÐ1¸Þ
ér } @á²G ]¶T¯ $®ëÉi6n7´mCþ&1Öù(/ÈÀ0®ß74D!¹>V\ÎH]']/)úª® x½2Ý@Jd5I¦FékÇWB|öB{!iQù¢°Ý!Ôøú¦\þ ^^Û¼¦°pø¾\%6h×ÝHáó4Ù ßgYX10J6.IVS}xnXzFî8¥Ó.Ó!ÓId²óêO$i¬¸eY1ZÆ �ü 6�''bi2¾o9D.ÅHìÀX�9ÐgÈé\UUzH=¦Jy MSòd^å°ò±"ÙÕ¨ÀÜ Y»©§H8ó%ËÓ¤!zM2 Á¢Bèm¡àF3 D³"¡ôêÄ ¡ÊöÙûU{aa·®\ ª|~îã~íbÓüÊåÒ϶4¯]ßÀ²ß|ǵrbæº7ÔSÞÝôý|ÞÄÇÝ$÷jx¾3ଠXØO6¦¡5¦UÙEÄî÷Q^Ï6a¾ øÛF}y\×øåF°Üà4$S«Çfa©±¬²È@k©È· ª W1,^Kº
Û%y6·'Ì1RùÖÁpg7Ú#:3Ì¥ã-]ó3H$M(¨+V\¥ã)Ø+T&¨IIõe <{^1WÿdÑû¿{£ù=d¦ÏÆ5'Ðn«_±tÁ¤ý«×ÚÍ_5ëÝ1òîS~Ö[~{åí±Áõ½gF5^>)Ï@ÆGjåé)±t µXìË°'.-nX\Eܪ8ÄÅÅ.ã®qµqõqÇI×0[¡yæbó 3\_ ÐI¡>ù¤ÛSOC÷ªWQû$"Ù¯k@a
/7Ù�)àÛeÎ!ª4GXÈdfÛºÚLøàVÖûÀt0>¿Ù²ÿfeß>ÖøÛÞ|¼»y#ö¸9úNQrÀ.væ('Ϩõ½PsY¤wMç¼ñFDfyá|$íäÎKìzåtH]{ªg±]lÎ\WÕbuº !F¶«ª¶»¬XÀJÀ @
Å«8±1V«E¶rDvZñÍ»@£g<ðl>ò}le'Þ>:v=ú÷ñ/¾ô>öXù¬üg1Ïàä ÁI: ð¹ÓK" (kÒLDãª^4ÐcøÕè)£�É@Lµ&ZAãÚ<£Ám4 ¿&}áK$f&îá«8ã5Ò4^ÌÐ Q4#lÕHg/Y?±µjL72º]d Å£611²@$9ÜhШW£+´ç4ÆT-OÛ¥Ò$zT;¡ýªq^!ìÖ&¢ºn¨Ì·84A×%ø(\_�Ç�±\Ë]
¯Ô¢B(ÖOv¤Ô·þ¤Ù]c½ª¾¬ºK5Ø
pÚIl, Õ¢>\ æK¿@¹>¡Jóvëls«ÁZB+dàîyu1ÑîNc¢«Ltl#éaJNóA&¹Ä?ÍxPݪ\Ç n×ÚÛ+µUÚfÚØ$j°ymé¶|DÆ©ÓUÆÕqÒFK¦³ pVaÔ&©ÌiB%§$ÛlÄè4©l.Qçãa#3xyÆbcqññ¤QUiÆ]ÆCÆFYðÚJb bq[ü¿¿G5Õ?-§lv¶�:emQ»Ø0n¬ÖÞFöWŧ ¸Çá§ø 'Ic=Ù¹yÔ§Xë¥Ð¡J=mOÿÒÝ(%dÓrʤ·9ÒWA¼;øË × YÉ"#ê/®È1rÊç©FðÏC P:رx>÷wÞãlýlkE:+¨iÙUæÌ,f /r¨IõYHzED¤Îæ úÿ >Üåܸ÷P,=ì¦íôI½²C=ÌèÇÎSNÆ£19ôù¤fæ̼ìBW)²+=e»dc¼ÔãÄm6WcèÉ1LìJ05Æ¥ÆÌ6żÆm³z==·X�f£ª¢vKþD#XDÿ¥YÄ%&øséutAè 7B ÎýÆÐÐ>}GcåCå¡Í¡çCcèVKï O7îF¾ Lw]èÐápè�¢:H&¤æ@ÁI5;3Ä{ã1·KlǬÜKýã ¡QÎçÅwL áÔíY® ¶$¢g\7× ª¬R=¥^Ry
JóÄ3¦<¦qDdçL;Uí3ì5öUv ¡3fX £Í<]Ý vÀg«vjSíiö<{±]r{{a1à3¶q$\_êj.@[~d·¶JÏèËtß¡'¨k¾;¨³üPdö2>ÁR÷Úå¬[5b×¢s¡ãYíЮë¡Ä +×ÿlô!_ïc «w,¸é7ÏkÚI×=°¼ô±ÚP7:©ªvÍ÷Ê»ÛOÚ8rã[ ÒoÑ÷û°þ,|À$Ò@]u½jHO¢,-&/¦8ÃåG638Ö¦T¦ÌLá3ÛÑR'ü~y40Ñ æ%%ºgÖ$\¥&å$1}× ÃY±6L«+Îç%bÿ;IKìí+ ÕÓ|$"ËÛKnªº=î4w»Ø=ÌhÄ !Ú]IDM²$#©ñG,Êd÷W Í2
utM¼jÓVOA1{:d.sD6SZ¿WmD@6ì9 fS¨Eï\HûÊa9×é EÌÛ7®Ü³¬aç÷Bþz;Ô¬·\ØúìR÷Oçi'£d~xÉÔ;G]>xÛW©ûs{Ý36sòû¬ûþÄó]wêYmv ÈËÎ#lN�I¦ÈÜäsDòèE ÒsÃ!6êØЯ§Ë;C o¥
²¹ñ¼lÖ£ÆzxáÓã"ããf£oìlìnìga¹f¤@ Ô¤æ à\_¯Jó8ssÎ�ª1I{ÁÀ VGÔËôd{3³FELi¸#ûêH ~ר»îº NOW¢?×˹¤yßÁæÃïÓÙ>©Ýµt¼»±/û±ñ §ß'u^ì±/õ >A.6èLù±TZZÊÙêJkM§LlØç-ÑJµzí°&{t%\_>d¶"ò¼ÄÄc'%¦&zÓ¹¡Ò8Xìíéa²xÞ@ÊÁvÉËþ«µÖzëa«ÄJH¬V9vkE¼½Ó d Î!±[yÏöt>i\_[mðPx<ç]+Ù5ñò×ÍIÏKí' #;Eg¿ðhy²TVÞ¢µnöúüÓtðy]nÍÍ»îZÖúÀ£M¢îX£H°:ä{5Þôòµûþ~²¶¼Öm\_K:÷Á?^êä8°$T9¹¢úí¬?¯]ôÐóq¶Ø?÷\¹fètϳªÖ=ðË«è3óÝ4êiÁù
à¼\_ÇÙIäµÀÍïe¯±GOB½´>îpóxiBó%ê¡@å\W«Ì«}¬1I#yä¼tO¾ýó<XH<+]s¬0¸,N n¯{µ»Ö-YL8»çËñ¶æjK Y¾µ³kÍ9¾Û¯§Ä-s±½+þ\ü¬"Ñ�y"Ý ñ5Í£Ç+{í(í9vwÃ
ÐÙú¿WJµ;¨eö³+zðéÕ|ëÒÐGçS}ò2ÕæóM Çöýý³o^øÛÒJá'"ù ºOæÍpÒUæ]æCf¾ÊD Òécé$ ÌP |î¡éj}ƱB³Ç¯÷¬à®ôøüxqÙÜb°zÌÆ«µÚ¯-w°eüo76ô xö;üY(Ö^FÛì½êOÁ";JD%òøOþƧN6-ûtæ¶&Ü[¾jíÞ]í SÞ~f6¾wyÜæçîøÑkûVWCîý°ªo£rß\_HYÒ<c>FXïì¤<Í,¶f8= Ûé,v8b=±rÉ%O!f]fèÆ.åò=4ÏAÙQ åsg
Éκ)©ñÔ¬Æ{Ó7Åï-ÌmëI«Ir!ЩØ!wi¿xpºÉ±Ë!Y
BÅv¿ÓWÊxÎeï-|ðk\·y
Ín$Nx³±Z Ä#H\ÄkÉ[v¿EòÝ/ÒsUBûÞF£oä ¥ág^:eò ;vÌ}bÏÍì©|@ó¸÷\_ Þú¤ù0ñúÎJÒ1²ª¤: Pj¬¬6è¸ÈU¡xÚ@2ÄKpb\G<Ýé±)õÍw°ú¬þ¾t]à¯þ[j ½V(ñ."÷:aô/ù)±[s^¢µnEvj¢»AFcÆÝ.N±à¿rFû ±àbWhì1ã¦p±;Ä9Ñ6±T¿S §Êk |©e§Ø%ÆW1zxçU¤3´m¶Ißÿ9¦]ÓTîbRA!|$ª´îü»° t_^-v|ÊÄOôtd»&îÉ.Òµ±JßZÍÖí]G n¼«¼"ëCþÝêɦüÒ¡§çìØA7MÉ÷FÈ»^¥[C3"ZXJÀÁ8uBXÊÔ"+±XÔ²¡5%Õ(iðíÉx© ^£)õáS£Ë+Ò99ÍUp澨M6r½IĶ"^YW=ÉiÉÜÌí16·8>çºrÕ$ORZÈ$Ú몸OLãüÝýLÕc»¦w(éÀÔTíס¢f÷¹ÛoÞoo>p^ oõ¤BkþDJ}|KYÙîíÛÁµÅ¦Á5y00ø@ e8»ý±T{½Ëîw\²ÕM ^©H%ú»2Z¸Í$¯pÚÍ6F5±H\ÅûN.5]mÜybkYXwËc¬ªr G|]ÔÄ4º83aåÓ+.-Ä¢ 6M¼¡Ì˪½ïGlÞ#¥®6¬pæ»
g-b÷hb ÿ0s
xàTê¬tÎtrç b7¯P\%6ZCá{bfÄ\áûl\lÐÍäTOämóyÌ|±tóÇ_ýä¦,
Ë"ûûº«>$¿®6N£ ¡w_{3ô1m÷áu/W/¬{Iê:
©F}ÔK]M7|·ûÐ[ o¿¥?7^ªR± ;bÝÀ£ÕUëmpús®]×ÇQ² nu\m§ÞÚú±¯1Ý(Þñ(iJ
2CÒ% b¡^Oä5>àqÆQ»ÀëV{�=!>PpCn·SÝÓeÙÌÞ=é÷½¢Ä1V$zé0>ºm#w zð-v×e|Õ ,ÙãÞzÿÐù7¯wl <÷ôýs{¯}&ôþ¿CÍ¡÷4__]ØôÕûï½ñÁéÝW�·ldRÀÞ'IG�dêáÇ(£<&¤&ÕJùkÀÄ%$çZíFêT <(qhjµ!7´úÁýº)<i>l£Ç#¢ktÊ\1Bvà^q;6HC¤i/?Ûü¡º®^¼c ^9¦Qê§dU]
lªGT^¼0³øtWÇCëÒ/uv²>fÊr̨!É jÜÉÉIÈßÙäûß$2YÑhb§_¶ñ $é[uÏñì0q?ëY°» >¯/Ýïôw ]Ìg7sõ }pÉ$M²øHuZZü|»ÚåRª}NHcÒ}mQ ì?bc)òì.ú<ó;Ì|WöÛ·wZ·Hy ¾iÍ£èæw;OJÛ÷¯¥i×R¤Îç±OÌÓ++æ[Q»©ìÔ¾·¾ÉºpÖ=t¶úeÿ|iï1¬iØ°¾½rÓ2zÎðȾçO2ùö®ÝRs×ÜY½eÅãà?ü K×�÷ äJ]\»ºB5Óë¨1Kffu¯d.[ Y ãT,jI
eíR8QÈ[xó²H·¨°¡¬LßÊHÅô1cu# Ø+ò¸<60ïÞùó.|½Ò:1ûsïb·/¦I¡Åìex«zsBJ%1äþÀÍãT¶¦Z;RLÜB$jBÙÅ[En¦zZ®jèb)�\Lvk6»QWµ1à>Ùk¼Ê'
î×÷1×[�WFôIi=zÎ%ÖÝÊÏÌN<#ß÷áWv$Üõ§ñ;v¬}¹åGÙcÍ;Ü÷§%\_²<Ì?±å$æo"ì,;ÎÖß tÌÊÏeYø©7Ñ]X¥ÿ¦T¼ÑÌÌ61ZJbTÑttç¬\cVAa®ØϪ¹²Usõ~Íd %ôÃDÙØÔòªÈWJÔªðòaÖT$§9À-àh&²ª»ÄæÕ]b¬¬\ë?íe½KüpÅÕ·À\ú M© MÚûÙ fÉvAJ ~&¥6¯ÙzC"ÛyiÈXÄMïª×(÷ÈpÒÔü± P¤QÉÕ!I;»]îèmLa \6·Ëe³Ã³Õm6[MfSf4¢²Iù9§I³YÝĶËuÈÅJ]3¿§»ò]%.N5W¼í2ÓRs¥y¦3¯Ydq%fjÈt?¯q�²N·x:8 HqC¹h©«ÒÅ\&æÕJµJV5ý m²f4hú£f#f«Íì%%&Vé9¶k},"¯üÄ´GðXKÁqý¸,[çwvö¨èYt³LT¢:úQ¾ÚyûÅË1uú?§Ñ¿©_ÃñïÓ®?4ú1¿ý[kûß4D«ÄßÒVv ¦è5¬R¨ÊH¶ÌõãI¤·´
T¡í÷D¹SÜ+þÎè[Ph(44J·EËR´÷>Zi)7$iòÐpãJЯ¾%2Aã¾=!y¨_{Ö(ÉZÔ¯ÇõѨ{J/#÷Ä}]Ä5Õ%$¥&õ©ègqt½>þ:ÉHø<Ö"æw h!Ʋh ÚØQöõô ð\¯Ç¿õA%Ñòô3×pç58NÀ<4VÔm!ÌMö ìõêsÙL¾Aûª(ßÄ<h#økEò ´¹@2p¾Dç?x/êÄÜy6æ©ù@ßêë¸9Òå³LÐé é"îW¶9QÊ?ðý÷$4MÅP>Ûgáã(U´a-rø-a^¢~°.¶ÇSyt½¿#è_ߨ,®"ßf|S´}®&¡_ßyº,Ú
¸¥X«èãw%Ö.ÆÿoÊ ²DúÉôöýt}Åüþw¥Ðg¡Sÿ©ý ùEû°Îw¡wËÅzQnFyån¬}7ôp¤° Ð>
=|®¿y¸¾^·è£¯y ÚVDËÒ´ð´yN³Å¸g\FT/¿+Y8(¿I6c]®àíoKèË<ØÚ-º¢åhY Û%lã?fu»AÉ"¥8ΡæZêö~(jïº|#v/lï·¥°cð[ù t½Î÷jÝvtþCV>Èz½.k¦ ÔÛ½,M'%èw|s? ßñmøäò2¾ª!ðÈB¶÷Då
?~²=ÚK%'Ê¿4÷D¿Çuº8êË&è6ð>Í~"}uþ,#-|½ë¥ X{æ _§H¤§tº]¬A r) åÇt&üsÎo~W#ôï½M²E Ð6 FmNzÀÏ5B¯P·¼ö¼¸ÅF°æ/@·èÀÿTF{¸ÚÞ¿6ÿ;{ò/:FïRê¸?ÕvÎWîû]¿
/ø½½_mXÏ£ Ç
~ýGõ91Zøµ Æ¥:!·)ýZüGóã©ÇÑçlèUéÇÜáÕÑxÓ:ÖOèN~u½¤çM¬'tëÒdåã#z,»õ4?JÐãðbÝÇÍøF¹>j,úq6O!2_§9ßJ=ÃzDÝÊrËE >éÔuï�lh¾¶1ÎtûêxÓä@·| ~¢ÔùðîË!«Åzõ5b~¢o\3ðÁúûµÜ£îô°<øùWð{%ãÑðʵá±)d<0$6xRÎI9Êù iÑr 0é i ¾:Ebh]{ÝLD,Ø ÍùP½n<Ù,¨µ¿zøaA=@'H5Út]¬ê¹Yiæß ëèF l ìî8ÚÿiFÕ|#!MA3AK þåÓ¨¿å6ÐO8v¡\i×¼å9иH;½íPÐøHIlWúm¾%BM_ÎƱ¸çP!¦ püÁñÄXÍ8>rlt¼½ ºè<_·íõy·Qb.ãþùûS¢}ìÀñHûÖõÏù« ¹ø!Äß+í¡S ·D°høMa×X^®UáZV² |^¾Ò>][Ò5¥ñd£<£X3UØ´°sá
ß±£gÊÙdRùðÉ:ÎmfèTS¦rŬU@=î¬%%ü^r½°C=ÖTQ'ÍØ#:¯nÅ#E;Þû×'EÚñé<#O -CL?óó¸×sôS"u!VģĬ¬±AK¢Nó')# ß oÖË4øµ"¦cuKI¹¼d«ø°¾ò^
¨«µëI6æ[.ið»MD>V=Næ%Òbux¤Øý2}ÎV|UÎà¾
àÙG-x
b8F'Æ:§cá ðapÆ]>=t>¯Öï�l¾ÔÛ0îQ=ë8R¿gqFe¤Ë.r?yJ_[ü=UÄA9ÏubMêH²Aö¥ v´ï¡Ïïå¡Hþ#Dl/¯íXÏf=&GùfÌõ^ÔÝÅVËGrî£K¤¡ºØr Ú8ìSV"Þ@´ï �¾XþØãB]&ÈMºÉçQÕ±©>¶[¿·Hèøf|UG'ésxãªïC@Ö²ø;á&u ôx¯$XµúÙ£,Äùõ¸Þ@:=rÞ¹ýÕu¨%ÖSèþ=Zj×ÑYWbaËxã 'O³lÁܬðo(õãY°Ó«x.n7a:Î2|rÕy êkºR>'ܤ$áÚÍà×O¸o%p^6d5q9¢4 ±OÌiVt"Î|K2ÿ£80¡Mi²PöÃu¡ãIRàÜ
ÊzØÓPbm)
Þ¶Î5:75a35÷ >ÿ¦iÅGõ<¹µüÝÚE [ÐuûÞ(¦¸÷<~EÜQûÛ2:]·¡\_¨Ï.×C@}qÜ4O y(w\¼6å8îÆ )@nòJØ@é6(Å}¢í<´ËÁµ÷PöõÔû-"KP·eêêDÿ²ÐùÍä=~lGyu¨û$æÝ1ײ ²´f#öÇDÏøH½¼@&Diln×ë¶èç@#«Üã IѲ2ÚÖ®·É fõXßÍz=o#cT®ö\_º¡-c¢ßzBF&àxú èòqrt3è;NEÌþtÇvµ ã£:½~3Ê/@q8^rhè\´Í6ÇqÑþÏEÚè÷þHt¬I %ÒÏ¢t>B-mDIû"wè&êiYëBOFî"³náýà»ã ¯£M ñ\_cÐbq.Yq=í¾öÔ÷ÐÛí¬B¿ÆW ¡×K ãÍhûä5d¹l'R=úÝ Ý¬Ç±'T}3
þ>B!åü°\_ò'¤'ÊõJbM#Jýxä%^
y \W 2.!i#¤ì%÷ ' b×Cj-ú¼ÀJæÈ7Ã&v}úbl1® "þJgä·Ûs,þ©Q ¹ÕÛ Ð@Ð<ÐHÐõ ( FCW µòyb?JXëqÐ:Уè«kd½[º-ìWº-kýOÔ¿¥Vüwå×U4êó^þD:\_#s 3pRð¹Öý1 9´dqml$'r( ^ýw$ä§ì\_j¥ èé2ABDò{ðk3H[FJØb¢±=ðEM$[Þ 5kxñë²NO�¶lÀÜpf¿e=ÃÓú tïKß�þhûú!Ê ×Ûêð-pqÈÃF�£,&CÕbèL¯p³N¯åáfu$Ê;qþíÕ$Ú¶%Äfùs\Û2 e3ʹ>ôr;èqÔËÑþ72Q·å(ã#cËÉ(\_ÔIÕ8\_J{PÎ'í¨k7óo#$OÞæ^2@úå58/7||íÎ 7G¨ù"êúÓ÷�¦p|&/d£Ý'ìеh{åa\oV]/·p :}Õzç¶Yoþ¡ÏZAl#U¤xªÈI´ h²DÞBIPzA@ÿ�½�ú7r:ä)ÍÏð©dèUå&8¡Ðp03Å¿¶ÙÉdY#½åû.¶éJèz}ïh²ô !§ã¸¹Cj¥8>ûdàTøQ#FúÇÒ§SqN¤¯p~�>uÎÝä/°1.Þ'£<¬·%²72GÌÝI5#Ô|+Î;| ÙàÇ Kð Nàïç\ºß{êÝt«;2í¿§ûí¯Ñb»î#t?öÏ;·bÍðy<ãz¨z>ög %cM\ïS½BõÎ÷iýk¬Ï¦¢Ï¾t¹x.!ëòF¬ ¾¾ g'äw Èw¾5×X-ïæ¯�/Àÿ9÷ÓÀfÄlÍï[£½§°ÎMÇh/ððC H¼ d¿>ülëÌˤEXÜéý5ÝéÞùÜL ÛÑî·CuѨzY}üGvVðcð=ú ÁS�æEô¥Î+ ¡÷Sqïÿañsÿ¨ÃÜtñG©Õù hä)e)|o ¾¸ÎúúJd?} ú8bÝcý´Éx¾×~
1üj@Gïü,²ÿÆAyÑGë9ícíûcìÎcB×9GÑ7oûstÖßrþuzÿ«ráçî/é6ߥëý·éúèoèZ÷´ÚH÷'C¿HóÝÐ|ï,}ÌDuÑï£:÷Nô¿GiµómúdÔ¢¿¤¿ñæÓêäðçÆ~ëÅÇ|ôÕîaZíýmuªàÞO×ü~¸Côì{çû;4?ùÈx7Ü=LO]ÜïL¥yݱ4÷uºPkt3Í}uc~ÃW^|ʲ°¼üOA:ZgÊ'Yv®× ÷>Úê>M=^Fì·Ö)Zq¿Ðú}Ñ=@ 0÷/ÀXþdÒØ#-¥
~ -L®£
ÑbºËi¸;1WÞL Þϱw¾yÞ}éGéöèÃØÿpÜb¤gTѤGáÿ)d²~÷6 oýû8Üè¿ /Ýå\ ê¹9û®¤:-ÆB7îz¼wP-ê\¹.#ZH¹îµb}>·^mpÛb¬}5nßÀÎEÈ Do¤ïjD_¤÷.ºü5tæü£)gQß\<ÌõÇÜó85ý�wp îõ KÙmæYÍøªæA¶Û¢¤[°N\Aó0ÝænÜgd9¨F±ýúÏEyü>Bïw îÏÝp 0繦 ²éN¢¬¶XµûÙXµýX¬::7Vlýxµ#vÄú>ÒfÓn-ÆÃjùtwùÀL (nÊßÕ×AwoÓgû©.éëTü"棩nÔ·!×#ì¿b½NÐe]ÒªLú;ªLݸ©¨Ó?#l5OÒÑPeò+T¼þú@2~Z9êàÛ:.Áø\_JI\_F>gÀ²¢À^ú:Æb]ÔW¨ôÿñõÇþÛîÝk¨'ºúùü-e}m|tW
xWú¿C/þsÍ t§]AdkÇ©a¤4ôÏ:uKÎÞÆ¢køßD{&ì9POÿÈþ ò\@ð^A Ü[PÎiAÒwàÎDùa̯¨ë·rW¡\_Ýö³xεb 2q%@+åxËÐQ3ÆKBü\_OÑ?ôGÎ3SÇszä©uÑF
Up¯<6»¨×±é°/®TáÓhÖ)n>ܬ·lI³ í½:©qUò|e£Ùö¹^K¤Å�ëåS O¡o³QNõ9:sM0Ób.ÇÚúzðm?V±~ÈÔëmÎwàn´Yê×ý×c¿Ó9×Í1Ò½Ýð¯À¸ZK9Ñ3Têeb]º
\ÖYöCK©ÈmNv�OCwÛÙT-ßT8ãeç<Ü5@/öúÔ®Òõªð&w.5ÛmXåSýÛ/åeE&
ÕzÌÄm±«°wk ea2ÕzkÕþhÏ, î\ÐÞ}V˹~9êQévß)ò¦h9£¹Ô<ZX4 &øêqM:ý)poò}Öx/O¤ hÆþä(fÀý&àiÐ!Ð|á? ÌÖ À2¸}áµà/æHys Øü;® ÃZÚr9ß ,LZÓ"g:ܳr¦ú]§ÝÏ íÊ8 ¬?ä6ÈË+éiwÞ¨ålÕõPàp®OçPD&jÐr Ö·_¡aµé:EfÁ¿úÀ1ØO{uº]éÔ 42NüÇÅ¿ l¢ ¸=T>{ãNIûº|R¹|·Ê±g©xªr{ÐN'¨Ê+Å3¸ã§�ë¬=ÆÑ2/ñë1}Zæ²]æػ8~{ }ß· ¼ãÀ° ÐïÌ& ¼éàe¾"<{8¬&¶Ø} ëþõà=v8t ÔîcÙÕ(ó�h (ôk5"¾t (×»�5�GÛÐ X¬º.G>hCê®úǸÐu ¬´ o_(Ì Þ'X:ô�CË· pµè úÝß3}f±ô¿ø¾Æ¥úf~©ÊéMÐ?C}Öw3 C
meêPж¸°ø¾~|xß5}Û5ÜÏݦ/H[È®g²ã§!û=e·w9z
v}l«Êv'ÃíøÑ9üÝÖø¬Ø#Ïž̶lÏhì©Øî-×4öm¡ÿ ô»ü í~é6}BGú¶ú¼Añ:Ô¾ÏÐÉBÇ^ÎÎ/½\_@coôÛýÅ}£6ö£ò=85NÉVP·Ùbë4EÛ¿aß2ý¨¢±g.%ô²ý/\_82ÎìSvËlïØ3#ÆϾ2ß>ÆFñrt½fªìf.AÛåX¯ Øéîma(;ÀZg °Ù[Û"hºÄgD¸çî|lsÔm4]ÑZ¸èz1бùà5ÁnÕÛl¶³l Í@ý2ÜÈYµGì�°3: åL= ìv+{KÀþt<À6QS×¥rÒb;½ð2Þ~
´
FïFF/¨sÌSÆþÈØ );¤÷Øïµ]þPõ¾¤ì!Üí�Nj;ËH(÷ ÈÝ\ÐvkræaFì,Û|ÛUöïe: ZP?Y±WÙ:zíéÄÞíïv9²¦3Ͼ¤"åÊ~J3Qÿà þÐÏ ðY�óÏ>ÓàN7ͶøÌÛì Úp6±¼|EPé³ýÔ�°<»áßÍk^vràOOÞDéÆ6ÔIÕÍcÃ~3´ @~�¯NÛÏbÇvÒ!º<óx«0BÞHù@'ásWä Æ8í@Ó"Èv¤ÛN»AÒ4În\qzÈÅϤG½áµÄ$9çDfø]õH¾:c¡ìܱ?Ìj±Ìp{±Né
Þuxö3м¿fÒ?0ÝàÏN¦júN?ú?Ieð71¿?cÿVðñ±þh.; Þç°G' w ¥9M4{ÓZ«=¶|.ö¦¿úø\{Jb»Ä¢h#õxUª¬i¼'Jªã}õg<ÊàS½NY w-eY=Xc)ÎØ�ò8êÎlìñX{¢^ÈPt5þ)\´s©þn¹õ4e®ñ¶Q#01D§�Ä?%DkÙ¯Î(µÐ>o í¶+ÒèbJñsÉIN'kã5H²_çÖn漯×íçJ¬Ë^ucè¤h MJ÷N |SÕ<+¶åÜöÑWµ}ª¢»°WÞYözìæðA$Ng£RoÖÁ´ DÓnÔmê½6«4õÔÁò±¶Í{¿T^Uðhïy§ê½9èç:Ïô'}ýzK/ü{³Ï¨±¤ÞÝ NÕÎX2ê¨B°h< ôÖR&ùM ¹
øÌñRç¼ú©luÔ LrÏWçöÙèNðhÔ+¤Ð¯ûÕc´î¹pÏ
{.»ÑIå »&R¨ÜqeÀg¬JÁ7ù<#¥1s»Ìó²>ál¤ÿ¡óVr7]Û ¯Õ0áè?¯�a ?0Ò ì êH¸[Ðá>/a»PÏútÝãûX]$Ïýð¸êûHv<$]:à VÁÊãp ×ä5.«ÿÚ¡¸?¯ ä½èc3kÐFÈ2{ é³,§.¢Tl5m#ßöæ�³ÝLJ÷rP/ÍS0ÿIä;w¦DZxlò<ÀàµÕuRåø¤¾¶~ùvÈØÌï7zé'~·µ¦Ãð²ÐmÀÖñüÞ¿2ÿùò»D\¤àiÒG²4Ëã÷éõ%ùöKX8ß +wñ³ãÊ-ïõ;°æ{nÜ8àVç
È8Áá?!ô_A߯Á<ÃÊï:I#þÁô&_G5ðøQ¯é°Pã>ìÑP ûÀæ?mX À,à3)@% äù)9üÐñǯÏÿ½púcoiXi{.ïgjÜáø0ß»G<ÿ\úÔ¿Vcß»¿Ò²Ãnõ}Y0¨ãýj·ý!yãè{ÒæTzahùN¦þþy%,×æïìÖÜNìÐÌ+ºç+C˲ãÊäe¨:S$\Å]ª¾+FÛtO}#¢v> ãéo½eT$çÅXK¬ éíw¦ '°^x#~t;Í £Ý´-Úm1EÞü=»´¨v6Q½2°O(ÀÚd'Sì=¶%í§Un=my»Ø÷¯ºÀÓax½À6m2OE2±ït¸Y(ûÖD©T
5Ñ,§{ß]TìñßT¨ËüMT½ ¨×LµÞÛ}p¡ÛCxÐV Ï
lq¯×ßAcO\¼ë\¬W]¼ß{ZíZªuö¡Ò°»é¸ ;zZ£;¨XѣР¯r¼2¬Õª5^·wG © ûþEW¡!ÿd¬ý¥|o"Õx Ûgz óñêSöN#ÖEXÇm@=S¶W¾P F)öøðWBWè!×Û ÙÙ½:ZÍ(»õ:5öA¬«w)PN5î\ðÍVíÌþ.×ÝB¹îVÊwÛh»fxë¾(Q¨D|ÛEyÐy¶Ûx!êÌv»P^Ú¼f{¨ u¥0NO
ImÍ¡l» ýNUëûb
ã´Ó©Ä¬²¢êS1t± Ôµàè4§ t,ZGsAÏE+ [#Ú¼¶0<ìYQ×Bo#MOB9¶ ¶b¯¼ª¢iÐáðìN~E è®á ªWG{ £Ðá):=:öG÷QèAnCÃeo�oÆ~wù4°ë¦\_C}EÇ|røw¨ZºXó8¼nZzRÒÔêo¼.I<;_ ?Ç#è9 îIÓ } ^ Ôï´Y òmºCóð7fzÃWv ®ô|Ìv+ÀÏë¸u@àÝ�dçoúÓò½úMÐ}òzø±^äýZöÕ÷fj|æ;r¯í øÜ%>¨uÙ"ißÒõPákByÑáÞkñ·rèÎë¼ÌÞKS5$Å×sõ.à®/㦾¶wà2�knÝÜn \§×t[)Ù,ÉãþþÍúã6ºñ÷íÈx¹®Ò Ûlò¹îR·%ÿS"3Êw±ÖvÑtmÒm¦°NÊÞ/åöË7õn©÷g¥~»¤=ÀYÔ©[ô¸YêÞ-íÜ'òqÙ©ãY_Jg¾C7Hy¬«¤ ÀB_·z%ó}©èÛ¬FCµG¦Ù¾¶yà¼ÚBXú½,ºzUäÚ¨Ã"<ØbèøÞ]'28¤Çy¦¨v>®¡út³¯m_Öì^¨ù<éÏl#1Sô×/òó}\iº=XG<6¬Tñ; 'ÅæäS,<çtßUãoÔyFʤ½Kt{;g§´qiûcu
6(ïÆFêßÅþô¼ÏEJwÛýĶ!»c{5hk5¿7b[5^Ãö)a;§°-Qc�ýÎ|ï×uVc¦Bú8óVIÝ\CýªDtÁãßKÝÿ\_GÎÿÇï~öñøçgù¹ÈϹÏÕðÇ÷½çhW<°QcÍü¼áqÔªg{4$@±äÁî¹j))?[JÅÍàç Y§Uv¢¹Ëó[QZEÞc~ó¼gs%@éh<ÄÝ&ewJºå!,^ægò3øÈk>q¿Ûþt)Ô&�Ï«KEç¼öX"y>³D^^oòú$¿Q°DÚ´+Ä·IÀ¼{Îó ÏÏ»Cd¸àë:~æ¬|·ÞÈófK¯ñd¥ÈÒ"y½ïöIûD·û×oëE¦!öÌ;
²¬ÛýÝV/rv¯n#~ϪtÉúõZôÉbä¹2TöJ)×2ü³ä¼I]VH^e¢|á; é;\wÞÿ¬òÌú¡JêR.}©GÜ"oÝäsaÐVI¿FÊMÒt¾D?ÛÅÍ#û#3Eæ i_þ9[üu¢Ç"ÉÛì% E>ÓO%¾Fê^.òxÞÛy÷¦íRÏ6-Gˤܹ±èH^+Efk DÎuRfÔº¤ê¶ÔHxÈY,æy~Dä+þÆP:¤möKþܶÙWÈaú¿ég!yyüð3Îô{Óþ~)w¶ÈÇ:á}îá-ùM©|{¤\^ènıÇü/I[Ê éU»Cí+|UgÔ§Xø9|ÈjÚt±¤+UúÁZæiѵA©S¤î¤Tó¦EöøºÿèÎ ¸ß½_Ýñ³UîúIDùN¹ëïà.GùmÈ«ò¾K¿º§ç jê©îT:¬ïV{{ Û»ü)[¡nïå±wG¨ºéUÊ7ù«{}öÆ&'¤#Äy¹4>ÚNs,2ÕP3ú¼[ÈñúÕÔØ¿cï[Ý1TAùmÚFÍ %·Ì·ò£ð#|7!ÜðÆóÇù[¨9¹ÚPqj¦æAÿèV¡ì þ¤ÎRsÒáÅÜfàÏx9òãýh«>ÿ |kþ%ð¾>r}®FÞÿ<òËD~#ó'¬ óQÊujd>s·UM¾UO©A·Ûx úgÔ}óÚÍ¿uù
'g�ùõz¨/é¨óÔs/ Ncø½VÄóàùq_).Zþ%ðZÑ÷þË
1MÎ $5§£æHnݤܸ?ù0éa ¨ÿrè@®àÞ°Ç˧ò0'yÊ{ZÏ ëøQÐ)×-©! ^ÊÚ¢ããy-äia²jc)s(?m0¸: )ÝÀºZ ô§)'s³I[b d_Ã|ø§+¯øÒpúnËoÖÒÓ¡}cß kõñý|:kÉõôu5§Ü¾G#õQjNýrlÌ ?æ{6/6Àñ<æ1jó%Ä1¡öÔï"î©9í)øOS{Ú=ÔÎqc ñ ßØäõ,°{wÊ i«)?õ¿SûèCÜhÊg÷!¸S1ÆFãY9ú³Ô<ú&jO¹þÁs/twÔaJþ¡?®ÃðL×Á¸Ã4õ§«ë«òôÊýM�2ú4æ#ËòËÎ?í'Öϸ?}LSæe>Ý©ÿ |Y&êòU='èºp~Jº\·´GQÆ£º,vÔE]1cµ®Uþ³¨+©È§ÆäÄÒ^§qc©},ÆBÊ|êJvÀóL Ö¤«hÝK-�ê5¯½C#Ækdõh¬·Í»4~¹÷½ü¿Ø
°?RNõÖ° Mìùß Nì\¤ ÀT2PLOªI³#¯Òt«é{iS>ËYÎ\ªtN¨{KÇó¡?µ6èçg?VÍó GS"ÙäEøîA}×åa¬
Ô}GÕfÍbüñk g<Þ×f}u¹õæ{^?ÆͽÞtÄûï\ÿòá~¹sû\Ê{u¸\_Ý×ÉÿEÙwsì|¼?~½¬î\K%þ¸õ£ºW²ú%úà;?¯¦vûÜyÁú5^¯ü¼±sõ§iEöqbQçH, ôyÐçJs�BÑ÷¿õ ìOþ'\_ÇÞÙþ\0ÀÙÚLÖaÖiPþ.ú:ðAàÀ$àC@Tça}�¸V(xlÉOñLÓû;~yY¾éõÉwÆ4ùF½ý¤Õ¬ã¬Ùú 'æo
êÛãlIËßùöJ\ÞGò>\_aòaûB¹Ì3ú¼»5Al?3²ÿ |#ÅÕÞ djÛv³}U¢¿
\~Æ o HÜDýRñÀmM×<üPñòy´&ïûÅ~hv(ÒÊw¬/qØJÇuÅ@¸9¬ÕaûIÌM@9Àï Ë|áåðRu°?Õ<üß, ¨?§©z^øÊf:¤þóQ�Pg½ÿK}3Q½J\_#el|Fù$Àõêø$\uYË äh]¨¼,%!zæ.E~àÜS r¶kè¬KêTêtØ#´Thè/#4GsE/¹ÏÒqºïD~¥ûµêW|g¿3B~-¿?DY¯ò¼Ù>ø}æÿË0YþeþÙ¹»JþÝ¥ù"ÌÛemí¾qÿhÌ7zpWáÙAHëI£(uZ-Ëiµç §Ë:)gwKy|Þ/ÇüYäV~ûtÓó=!é·K|¾òMó¯+)ËÓ4öÇÙûh"0ËÑÿS\à3ÙÐÃà- VöVLë6à0ÓaíôRbõÜ»ùù£æhÌÇwß{ñïì»èfkÆÚ-ôMëZº&òü¦zÁsûh]´k´<öH_ YìCÚ¯®}¿¾óÖ¾SËfg ¶#ì!> þ$àqºÝFÞ6êi-ý¢¾KWÅ OÙc¥LÈk¿B÷Ùï? yÜ¿;ch;y£÷Ãý»îþE¨ÍVË}Ã|×ðîáw ;Uø"ÌÜO eDÐ=Yn{¯²$l~Ý.Nºng|v±3Cí2^ÚåïÿéØ;CÚåê¸6YGªM~¢e¸dyw²Üö^e!º°LµÃ®zgºô×ÀJ Já¢Ó@÷TaÜq5æêÁûª/_Þ¯à½W-8%¨Q0þûÀ÷bï¢ÚÅúÎlèt/i¨;³× ýs'SîÍ=ñ^=2°Z0 ¸VðÉdð\H4ðUU¿KÌËVçêàÎpWjÌ ¼oß1ÈcÖÿÏz ó\<a á޿ļüÝl¨é~nÿ@Ð&Ƽÿ5Ý~ ÒßügG.ð% gwز²5ü·D®ÏðQp_«ç #î"=÷ªgÏkp;2G¼¦icÒëÆ|ñMÊðZÔw ?Ô×Üè$ð#·IÐfîOWý!¸ÈïèÛÎê®ð_��{øíawkÇÝ«}Ù;µÜ§îÑ®;¼Í=ÞK¯ðï½WPæ7G¾Ç{ðï:ú¿C<\¶èyèú?¦,Üîʽ7A¹ïÞç~Ý«îNßu¥÷Àÿ¾q¯r\x#òECe¿ó÷©tú}4ù ïRn®³[è\ûUº+éÉÐ]êÙqw©ßà.õKÝ¥î¼MmÎiêÐv'Ç9Ç"wZ§±ïC~S(ÓÚ¯e³s©PÉC÷B¶|ûÉá÷¼Ûýô¼µ:¬×ô=ïÖNÈ0>$#}½ý4ÊïÍÄ~£^ßónçS}{£To]G½à«pfÑJäÛf·#¿Ó§¯LO(]ÕÓXÖäy2$Ov<·'§ðòl Üm¦åà{¡>Nùî¿÷¡çQÎt§ r¾Ãmö½á÷ߣ½ZL{¹Óéz´×ê½ÞOÍnn/ÈrãÊ Y¦,,Od¹5Naý²|Fd'²Ü!²\>;B»4T»lTïsÿÜDÏÛQ®Úå0öÈávéHÐOBmb®G<¬Úú° 4C÷Ñq²\PAø@=]å$ÓUåÉ,×ÅÉ2¼ä#NË2Od¹Cd¹ëÏåÅ×ëz/·5æ ¬».¶Å
£pc¶¼X|¢JÎ3 {cì7ª ÞØ UA=ÒsÙGÀÏr¦O+ XW! +uÆqX®õiÀz õöù¥Þ ô³ÐêZèÍÞz^á;BÐ ßzxÔó¥ó¡¸ýçÑ[Ñ^L'ÿPÿXeþö¨1ÿàpá9(é^ NôÄhÐHy²L7»Ípn?£á£¬7Ø/Rzì<çàgÌwBçwsÞôaÈi?Ap¶ér9OÅËÿCø)Mu è>ïF´1ÿ á:Ìç7ë! û3ðüÿAèJðBªFþ ÿà}|ÓTïÝç¿NÉ×Ómþ5Hÿc=M·¥¸´ %a©´(z -ðG EÞ¯i?nAtü¾Ò�ÉíT7ê Èk-HAuÑóT²îô¸¯BÙ) Nþ|ªãxç ÿs!îCÿq¿èäþ¸CÜ ]4@'wCõ üoèäþþ AÐ ]¦üámrþÄÿ¦¸Ga¤6á¿´OØg.>¼ºûox5ä9cì6;kc °|H
±Á}ë1ÊD[f²Á}}è´á Ê È5~©A·¯íÃùûaùæºOÀç ü! Û&6lóÈvºÛ%-Ûáo{Äm~£Î¶¤ë
w¹Ø06}$ÛBöKÛ~.¼ºÄÆq
Îì¹ÚÎ12¸ g{D69o¨ìÙþñ¿'ñy?UËÁ¶ê
²½u:ùÃ9ß·LòùÈß+éØn²YtÐ-å²)Û ²mpijíæN?°Ñ4öØ|.òGÌÚæ\BøÙõY}þQËç5ù#Ûº²Í(Ûî¸i> ?Cd(]nïè"w³íè«R¿z)¹ØÚw^¸
º.V©ØúWúÝu¿ïös=ÒÊE¿KüÀ®¶GÚè«Ûή󳵦ýWI]9mf¨.ÅÝÒýѳ9´Ólv£´ßßÍý&OÚíróýÀNó«:Ìò»å:©÷\ɧUô.<æL±fWK9l÷Û"ù¶H>&q;xCýàlÑz?°í#nÎ7Cd/råüªkÐ9KÊo<æ ï,?8%yÕùý¹±=6çòEe¢ß:ɧFÜ "¿±moñûus®ªDô\%e;ëúP~9+íoì¡ó$¿úP»¹¢Kò7¶ôåÒsEù!ü¢ÏJ?°A© !×ÎûüB?°Ùæ¸PK
§@Ê56.Þ)mX#<ˤ|Ö×á+¸J½ØìiJBí[&|Mq²3/Ï Í!9$Î
[î<3$Oc»nlvÒ
.ÙkE÷y~p&Ã}-ô^rÝêhæs+Kd©2XöñkdɹνWPîçJ ¥Ó üÀn¿EôÒ·@ê:]Ò;K¨±ë7ca±øÍÜbô>¯W#rä ofHß"§gòDLÁlË^&þrq»þriש¢sFÃ)ô36æÛJÄmú¥ ÏóñÏru ¥«ºMöhøb)·\tR :gËà¹
?87búB¶ðs!æTa¨<æèý°RÚc±ÌA5~pvĸÍ9ks¾0ÛÎq4ÇÎw¹Î¿0sG¡ä9×æ�s¦Çè>WÐ"yud3óY¥ÔÉ)¹ýÀ.+OèØkæ¡Ò¿f¬5IY¦Íùˤ®3üàUç箸LóËñ³ÊPU~0Wò8¬Cô-mÀ4ÓæÓ̼jú9å}>Ç|?tìò¤-ýʽ!KôgÆ 7ç¡gK:æªgQ(ÏLѵ9T%yÕÚ[ì¨TÝîÍ3£EÚ±KôV!õZáýª$9³Ä<"·y¦ä\Óýà9Wág²ÏÈiê\Ã\yL¿0ç³òýN4mWÊÛÌéæ9Ñàë"s&ÉU/~Ö±6çkü o¹ÆsJzy±ãþ¬ÿR?Ò 1Æþ³lßËÙó±ô|¯=Teæ²Ùè¹ó²m¸.Çi.fÎï65ËÊ|2Ä|øsþþ5óÙÏ磻ù^ûùîµÌ{UË÷Gý.û6ÊcúRÖ/ÊN¬ïÉz"7J?,d?îÈãh%»"ë0ÄȺbOÖçGY#Ç?1®'ó7s+ú÷¬/Æñ0ÿÏeÛ±çD£ý7í8u½DÆÛϱÇFÞG\_Ï}u¸íeøF^ÿÐ#½¿dñݯ\_eÝdËò»Ù?èﲬOäõ8ò!¶ýû|ý?¶²¯/ªÁÃæ'nxöPû §Ð éÌCV¡ Ífÿ \_#oƲíÀ æ³ìQîo²Ø×+ÆÒmµê%ÈÇ ÙÞÌ¿%ùi;²,\_åiÊãOÀ5Ʋö¯jæøè ?BF51Èóh÷óø)ÏÏów_÷þÛ¯MõãÁ>WÍ5+÷ýn¾k¼þåA¿6÷ýúÙ_Ñ/ëY÷_òú÷YßwòXzÇøù¢õlÙÑóÝÙ÷uÝI ³ÙVÙÁ£oëhÃ8øÍ2 äßÄ8ÿ}ÿö >;àÞ7Ó~!ËÜé^ýÇý¨Ø &îuæÙδc¼L9öjÕayBÅcÏø ÉD1Ôõ¡¼ºó©Ì{LÆv°IÙõþ®VÛühæÙ®VÝú¬¿Õ}$&2ï¿eùÀ^÷^þ>þúàw³ÿÁêCèá³ÝÑíØ?ùq&æótq4û>î}XïÛ>õhó#²mk~s,Äãým£þ-Óñã0dÒÓ<þû,ÛzæóéLçm>÷CîãÏÆg±<é|çh«Øè«ü?tPæs1ö~õºÍL+Úp+ïGLÊ@þþ&Ûù»<ØÇc6Û/ÊÈ8ËYÐÌvgïæùa¶}¤û,ÓÈôdy·²þßdg^g½#ýyýy<ÛÜÃ#ðãLk&Û=ãr^mú>ó~÷åûA6ËùÜζ{«ÖïÆ»oJùåuöµÌ=/cüu9^å½WN¿ÙìËÃý(ÇÏÆlïó^o¶ã¼ÿ÷Ì÷V¾úáùÌr¼Êº¾Íríeqo^þ^èÃ|÷qþýøm)'c.Ú9Æƹ¬Ï~õ<_ÈvùÚï=ÌòÅ3ù-Ô{=Ë6A÷Z÷ºÏ}mÁz÷_¶|ÿ#Ò^wä1MÄó÷òÚÓ¼ö(ëü] [çy#Þ½å\Ë6üMÞûév¾ó:ßúÆü¿mïh?ó<ù¾Ê¶ºy?Îgïe9Kq¾(A¾½ù<e\ßÏ4¢=~̼Þå9ÊxVÞÏeK³¼73ÿF¾Ç\Yɶü²åu^»mÍ8zõû&¯ïdÙÙylýís³ó]ûz¶ã|ïìË3zsý_Ùo%ÆúnÞegBå~³Íÿî|Ö'Þ¹ôÌ3Æ̵¼Æ¾£ñ\_¶ÛLûV^gñ½ß³üød£^ìÏʾe´ßÏY§»²ø2Óû)Ëð&óº½Êûy¾íxWÞ«,Ê7yýõÞ¿ü&óaLÝɶ¿Ï=Ìô¾÷ýZÞn6Ë÷&ת½|>Ês%äÀtg.Û6¢ÿ·òÙÇòþM_dy÷óÚù,¸'óñ±äïwyùú,Û±î§|-ûômÖó]Öñ{-ËýNç¬Ì;Ùn¬O´Ç#YǼ)ïEÈü¾+ïMÏ
|¹.ï§÷÷¬küfc|«ûj/²=X3wòwõãy¶ÅË|çZ
~@úYÇ{ùî,ËÅ<#CðÿÇïʹç«ò>/d¹ît;¯
íc1ÆkèY¡wöçû7K»ÐÇp¾ÌòÊkUõñlù|óõÇüý¼üÿ6ß}wPæfÚ_åõ÷òþìøQÙ·e réЯ Ì}ñQà§bïÈñ,Øü¤¼÷ 6ò¬cÝk\yUÞS_夼§aÝG2æB[0ÑŬ¾l||mw+ym¡en8äJæÁ^ìÿ0¼È}ï®Ê¼i0\ÚqHöiã¤\_WJ}É?0ùlï³À%Æe-xÐ ÉRøà}cY5ÙÇ×#ïË O\_;ÜxñümÙ.g̬ßW²ÎK2Î~>ó÷1YȲïÊ(Úüó¼ cÇÝêÃý{Ñî½¥ïÀ~eüeiÈþ¬Qµò$ðoòîlÖ÷¼¼?siå m çÎÌyé2d^Ü2ðÌÅ|9ýÜgêé±íìÀ_zákß'å}mOñOæ¼07ÏÉr.ÓÜ¡~F>±÷êJæËÚï=gáIÀýøý³)ïëù,YG¢lp$GJ~+²ó,«XK×ò7{ëF"7äùe¹Q8ÁÈÈþL¾DÔ1°ÛXÓC_OáÐ Gs)L?ò:.Ûì¥:UóÝÛYx6äË] W©?ïÓ¾qÿX;ò[çkckE1k²qL^§%?ö?ÇãHæeß¹BC2ïb&©¼,üÇu×¾|¶S^çGä}ºá' Ç3¡c],÷©{VÃÅwÚ-ï\_ i7ßß2±
ó ¼6Le>Ëåú%Gg?àWÄÿÇ\÷=Ço\_dõPúè\øWdÝj.ϲµ¿÷ÙgÈTxJk²þtI^SøÆ�\¡OÔò!²\çe Z|ÁWÔºÆ\_Î~½×ÏÊ{þßÊ{ÈOx² ¡ç]ý/õú¶¬ï©ÁÇC;£áÓ¼P~G}«ñ7wK¶Xç{²ìNÛÒõû#2gþ$ü+¸e~ ÛçÌ}|G�ýäRæ5íT9{óÙÆèQÉóoY³«ÁZÏ:;+Ïx¬û¬Qp²~XÓÆfƶ!]üùøòixØè!ØÅphç3å÷FÑÝ)ßRI¹´\ë#ÙOÈD|÷èc rüÁ²l\ç=~môÚ3%oÖÖÑ,keåÝ£7e×çjå±G<ßá¹(ëúð¿Ñ³òX²ÜT«Éÿ ^ºzã"å
þ°Ï Óió2¿Û6æ ú|©;8ì1γg²à&¡§°vuɺý®ÔyùØÕÖ¿ýM[ó5d½ ÔÙß&í'¶?ëuJ¶ñhô¸LÝ ýr9/ɼiì@ôßÄÚyý±Ï?mZ¾W°i¶gº±fÃ1/?ÏÞ-e$äó¾nÎë¦_1y\ÈgÐc6³Ñ&l1¦^Ëß.ÙËgCØÍw°åo¸/¹é/ð¸|ó Ü$d÷KyuKW÷ÊûÜ?ͶµóVæÍÚ÷¦ôû¢·Æï2.AÌ vç~¶Ý¥|önog{.ß=\_ò÷e½oe{Æûpú.Ê::}M7'óµÎå³OQÞÆÒÿèq˲O{>G[¾Ñ.DZVÁg
{-0UòȾÿW²¢¬y}6ÛE8'ó>Þgtg»1µ@fÜTë·Q6dN÷Ù¼N 7¶×ÍuôÅëþºl«³.¢'G{Á9g}ÿDÉqY¿Äî\_ÿu5 ^T¬[о!3OÈã2êÓïç=rï|ÜÑG¼;øOý8¹|>úà]é;k~+ügü¯ñ\Øտ̶¿+sv³!ÿg;D>ÊøË{¥ÍNîQü¯~ÎwY×ÙÄ2£ïbÞŸ¹!óI¾Îg'd?Ö£<£WöÉò½óyòb/gú'î½Í>úá íßaÄÿ<ëöBæGGÝ\_e¹6óýgÙf¬÷;}ØÌ5p/Ës Å£l¯ð+\cp®g+9÷±Óóÿ¥ì¯Hëû¼~)óZ¿9õ¨ê3#î¼u»¡V¿¾ú²ø}\\_ÄÖßèÁ½å:ôÛx¯®4:+±ø.bLÅX yã;²m$uÉ7ÉGÀ¿óîlÏĸeïgky=Æ+1ky-æ1NeýmW·¼UòY+ï]¿ã¹çåÞ¿Tò»ZÊô0ÙÈ3ûLe;¤íÐVcÿYN|6ÒpÊñáßÌ4>Á.ÇGHÿUÞ�±é2¦=:Itð4LÎȾN|Ú+Ä2}½Ø}àÙà8gËÑ'cð ÙD f~®ä3%c)Ò[êç >Bìíz\_J\«~öøÏ&÷#߯àÜé\_ü?öõ¸Ì8/û5°ñÀåÁÀÁ&ñÝF×Ë3üøMñsÒÑ~ø)À+/a 9!óºJÛð½$¨>;Úo\ÆõKyÀ®£Èü«¡Õq<'¾8oä8ã±3-Ïò/ÐSÒ^ùÄÁ£÷Ê~Ùxÿ�²ù|I'Ú«Wã38Hg>í}Jö9NåõÉò¬u$¯¯Ë6ÿ|^ÃG=ËXúXz08Ä¿:igzÈÖ¸"'ÔºwÃXûIùÒ´>^y̨õ»]Æ{4ÓÇÇd¾ 8+û0 áÏcJ¶à1:Ë;u-Äfa¾°nD¹ÉñÈü%øÍ1ýO~Ï>î'Ô¬g·ìÛ!~1¯94!ïÁí|/ý}Jºå½×'øNèCÈÊAXž¢ Ðñ¿¬\_pP.ʺüqÙÏÊ|ìÊçOËö7ó¡;ÛÄ)Ù/AÐ ãK27i)ë¼,ï1+ϳ¥OÖäñÏõu |}ºÔcZ^çÍÌ9¸!´ñ¤ìcûk¶ßÙ×ÈØg¯þO¬¡¬Oè²ÿ¶'ßÈ5ùÎ<
;×/ûÒ6eý¤ÖB)g´%çÌIFË:>rf¾ 4àñ\8qìÿB;Ðöè«È ,û§¡OÿðÆ5ÌÆR¯<¯±ÄMR\_äÓçj#¹øöù®to)kì÷¯y17ÉqÍÿ÷"«Ðý:å¹,Å2çÌ$Mö¾À¨2Û
ñÕ},q{9ôÞó;òwÈyq»ñt'˽e»ô[Ù>'ûì{Ã&aí2¢_æ/²NSÆïv³¿ù0ï<}Ôy4Ë\_ýÄ|WÙø¡n|\s&à}ÌÉþ~öö J»2àDnÊ6zô[u=='Çj3WÕðÑiø X£ð}ÕdýoÞS×åyßbIÖ'&eú,kí%yoxþì×k1®CÎ÷ádà²ÃÈØõãåãüø�xØu¬¼Ëé¯Èû cã«
s.¾XÚ5äZÈ
]Ù>\9)e^gtÇîlÊÛ×ÔS²\_w yìì¹|.êÞ�¦}BÞ§Ëòz oyH:Ìãɼ¶'Ͻ5Ù'?w[üß©CÿÑíà6áka~3dy,Àö¿ù"·Èïx¾ý{aE¬
øç\·®cý6ýíAÀ\WàAÕþD¯G¶º$ó$À èstsjÝ/è´¼ÖTÝ| ~?)yÑè£så)ýÎÜë§d?"ûËàÓGyáÁ,rá2Wå8 ú¹ÈÜdî¡[÷ËxËrɵ±5Rò!ì®xõ }83ø|¬=È08zË;ÔYÞ½>'óÁZ%}°¸¨Qü]¥àeÐo|®[¶À!àU²mË{Y±× x=zÈlûny-c]gf\_õ<ðã² á½
,ïBI5-ÖÉyídîÄ9¸>ù÷ÿæ> xb8¤AY®E{^Íkà¦èØ U Ç çªýßUú=?ú6æ\\ÚýöGïiÈûõËz4óo¸ä;\óÄÀà·\_Bp¾lpà\'#-ÖÅl ä4\vlNì´ÙlwÚ »z!ëq^æÁïÉ|nϲÌ
6a¿ñÍ|.üÅpã+9Òý)c1Ãv¸'sàÁÚª2uDF ïáè³ÊWðXø>FpnÈëÎÅÒô3¼rÚ{ýÏá É2¿õ¿å<÷ó
&¸(ø׬#�~Í×íßYþS®QcoåûcÇmä«Ù6ð á!Ïe½»ðVáýò^Þ/åµ±7\ò}]ØÙøÊ&eÎ+ë#z 6Gʲ?ùÃ8m˽߶!Ǹ?Pë~ q?íN¶!ûËü®1ãù¼gk|¼³Ï?ÆÊzÖ9lõðYßÌ6b+mÓe~/dz¿ùXÑw{Yçjå ÅñµÌ͹ðo\_ |gøTe^ÛY¾cÝ÷×ÚȾþí ÝõÁöFgÁßmBÞ¿ã¼O ûßle:a×ýPsç&§"êöEùBÖ?ÏìEiíľo².ßf»Gùf]2ý¯åý§¾Í²¼ÉÿÑ1>þVêû&Ó¥¯#ÿ{yü ËÆeü
Ì«6ø]Þ£áçl»³¬kùLæõJÞÇèfiÇÈk5Û$ÆøkyoãyÆÞÿ:Ë o1ôüùî¼ç{qÄýäõ}üŬ×ëü%Ïßä½çGüÞȾÇmüUiß²ÿ~í h¾|¯;Û+Æ ¨çYÇ2·$æÊ£L+ÎóYò~ 1/Þg?LæyJÞåÖAû6¿?üDÖ»8ãÝÑ3;oúF¥À2ôÊ{ EyOdýÐ×.æ5tVÖ[|"ø°å¥QöCèñ,õól«ËtE»&óÁM¯Êú,±+øñÇK õdNú-X!øúü¹¼¶ ã øJ±ç'ÔêAoGyÁÃ'<ð³D=C7>&c¾U¤-Àð1Ö2MÉ1·Ä÷bseYĄ̀?6k4z%úÇáò ã»ÆÖ®º÷Úe< »ôh^ Ü?käó×ÒvØ_èßä]£r{ÐÿMÖÿ©3:ëáwáO´Y×J»Mfõd}á]Tt²,'ó?~éÒîð¦e»«ílW5ÔÊÉ ½YOÁNå]>þÖmãî³ìÏòú±rôäq4Ë &u^õ@·ÄÞ".¸nlCÆg9ô¾W þ3±½ý%þ,ïg² §Ý =n¾äMð¿à#0ñMÐöøºçJ±8¦å±Ò/cCñ;æöç¥ã9ö
èÈ#ú©]1['e_tí,?cÖp;Ày©åGWèm#¸hØåØhÈìy/jæ\_ôÓá8iÆ^C c2¯yżD¾POü|ÌIæøîL¦"õÿÆ϶
?øs7)s·2=±fdû¿5² Ù îíný¦b´)?cõ´'ËÅ»&;.Ïið|³7 #ÝÙ'Ýå ¼´167ch&Û|³Oæ)Õ:ÓVG½²ísyëÏó2ìT^ãuÀ6dg\åïRe kr Þmõ¹ì©ã YÓ'ïÓ.ÏCÆMõçÒ&ãüÔ¡úwg¹±c£ï>q]δS¯üvYö3üò/ï·ëí%}d.mlgÎåµFéc°uü�´aå6±fÕ>óÒFèyBÞÌxÉuo^ÏyaqýÏ¥¬×Ñø}4ó? Íþ'eLyòç<ºdß×v÷£\_ÐiÝZ-å,iö÷ñÙu: ÑpGõÇ8cÜqG»ýwãg^u>ì-óþ¸Z¿üK, Ï¿µ9tLæ°1å7é±îp¯¿\-ïåêD©m ﾺ4vx5äA»ý#ÓhûQy=qôWÇÛÄ^R¾òzÏÿÇd.üÿù[sôk¬ãË|ºs�O#vûxéðöÒCÜ66¸ÜÙìcbðã7{Ðá kÁx&ûÙ·!ÿÇË;Oeú˾¬OVµÏb;,Ë}tʹ¾(c ç³}ðy±>T|9ø´+ýV°CÁp3ôØ~W|Oô#¸;>Bt\pþ§¥~ð{o¨[©òh[Þ×ØÈþØB¶]¤Ø[ààÚd_=>àÙELò°üýÚ ~ c|&Ëü'³YÞ}Ù¦ëʼàÿE7òwW>óE¦Sç<}¿;ºzûpRÆ{ =Ë2nɾðÌë~µÚÔu2ÒýwÙnsî¿æ¼þ±ök ÿá~xæ /óLw]öó2fd| büdzMáµÁ ¿áÑϹÇÎu1îÓ27¼(ÆqÁøûÖåïþ0/e5FÛ}Jxæ{7Àï;å'.==àZá»DþNÉØÿ5y?°Õùo\Ìw£>ðäeßÞÌ3wtZ¡£ðÕ¢¾Û(?þ"Ú}[ëò~c2wñ5%Ïü´ÈwdÍ×%md>obߢ®Ë>Pú� /ê÷&£'ú10+û?ã@[9¯Kj( rñÌzmÊû7UNeÝ Y ÿv. ?¾û±ö{xØ.²p]æºL¾þÏ| lÅ &ÇyÌXód¾�ÜPÖÚ~y^²·º züàhvG7¼'
7 \_ùÞYÊne øyð¯Óï¤ @]wÐïù 2/½ ù|¶ïTyG¾bªÀ=[È·ªwD9«u;ÖoøñLÌ[bEå½geîu<ãå<®¶äøÚxîfi·8®Ëû ¡LæÿÛòÎp2¸y<ë÷[uylÃ/·ÿÁз{òxÇsü5ÆÑ<±çÑÐ
éKdý>%Ïû 9Ù~¶¼SÏ+òÜ@u·ÇðíÍd;S&ƬÈUÊϺ|³!ûdº¤9%ëpVà\sÿ´ÌÉQkì:ØYy&puä 2î²t²¤CD9BoZ/k�e
W9XËUã\wÄÕàûɺ\áêÇ^iÚ
ñÜcDj>1ÕA9èty0,øIĪU®!¯¢¢³ã#«Tæ;q´GåÅ"Ñ£YçðQ.|aô~ì.ÖLôTÚ¡|# Î?rß5îePÆäÀÐð/V å!~5 ¿;~8üQµ\_X/Áh#ÍòZ° Ç!úÊy;.ëoÈzì¾ã¿òøYñëá «.iÀgÿG^Ìpê?¯c|rd¿yØWÖU}0ÈqÎèêpndy\Ã'Qç'Ø<¾á©RÖÍl/ìÙuµÎã)YWFîs £Øå=åë!ø vÎHùÍZKósZåëcháSOÉ�0Ep+Ê\ ùȾ<¬/¯K²k÷ä½OÖ²Èûá÷Ô.Ûögc쪬W-fÙ6åµ|{GÆÀÛ°Å®e7ò9¾Ymìuëdþ£#áSc~¯Èë<^è/ÆxaÍ@¬vaý mãòþ·»²mVu]Æ<:,ssEqeæ#ó"ã9nÃü£MFÔcßù6YaücPÖsäë¼×.z0¶m¯¹@;3Á=i¿ñò¶ø¤Ì_gíàïFßFÇ[%&'Þ»#¯#Òù0ÛÉØSÔ 7½r ͤ.ñ\èhÛYCÇ¥òßæBùþ¿çgjõYÙ÷ø;ÿçKË[U«~î\!=düÚxîsaÏ" à³ö^±apnl°¥»¼ËXZ×'ô/¸f6¸þaòÆ0"c:Ìi¸Pô9c ýµ¹°·wü^\ç9Y/o|Êsñ²Þ&kÉX¹ÏzÅXª:çò;tÌuµò©Æv«m»Rò£×åõu>ÞKÅzã »]ue¡¤Ëz
ôð#O[¾ 5¯ú"íËrÌyU|qÄúpRß¼Áá>85ö6òµî aO£sÐØ=ä¿\ò>3LLJ¬¿Ð¯¹vøÝ«¬å¹«´ ¶h]«áÉà/Gÿ'O©¿¼ ^2VÒª9}Æw]Xy/øm#ÍãòÜGç'¯(Ëm9.¹¼ú/ñí§gð,°AúµvÇØç0Cæxu¯E¹ î
^Þ\_ÎÈ"üÖèóÕg>QtNô£(ÿL®ê¢EDùb^^ó8¸×Ä·Vûxú®®y¬áp@Y¿Ñk¬|"xHðCNÈ\|¢+Yÿëi>IÚùMÆÑõ+²üCwåjÃßykÈ b£îèxWdÝa,Ûy~.Û¬C¶[¢ !kO«um¬83q
KåZäÆÇ¿.NøìÒÏ|ÐÁ?ù7ÙWÝúÀäñA"cá ¡wÿáFµËñ¬Æ¶Ìög|ëôȺ0¼ ô¸ðÈüRÌI°ôPt 2FÚßn�ÜƼvT»cmÅæé.ýD<>¨ø=/ËgÆë ߣǿg~Lf?>e]@#§Ð×±uð%õÉ1[ò^àSÙ?uï8ö¢÷ØG¤_ÖØK¼+Û)êÀÞ»U¹ÈÓå@16a̹#òüàXeù µúX'±wgKþÔrTädÞ×FiÇ%ë|ÆÇMÉÜÿÜu¸Îw¡ÏÊßlùüÔò76eûì £ÍçèÈ^Öûʱ;fU}ÎÏÃI9Üo=QÆßo«vXþtÉs®Ü§jÜß(ìågáø>á#gÃZ;Tòs £Óòc¿ ®q°^ý±ï±ÁØ?ÈFÁ¯]÷@aÎ{1®«²n´üæÝú î Ö= æí|>ïïÊs«ö-:\>yýÃþê=Tf0ôOtUtð±'1rùQíºVsT¾¶íÌ5a¿Ið,Ǻ¼ÞkÔýÀ³CU®"[ñ£Q«ÿ[l¦\Çîª>ÿÃñÔuV/g|
@kcuéïj¿×¼±ãÐÅ°÷èêÜ^ËòwrUÆið³Ì¨Uo¥ ±åÐéNË<ìÊCÎ2á cÙÁ¾D÷e<Ó>Q[²^ ¦<ë
ZuhpMlß«røTIçº,OèGd¼öÈëkèËkÁ {CÞcïVþ,1w1×ve9ÛSÒÚ9x¿{6Ï'È÷[+ßHùÐüÆòd¬ôüÛ>´cÛßÍ5k ßÿ!ÛãéA{´}¢¹×¿øpýôÁ¹Ù\_ßä{Ä)ïfÆÿàý]ÈòbóF{<Ê}xcÓ½Ì7ô·F<{í ßæÉw³í£m^ÊßÝ@¾|%cJ6ßÚÊg{³l73Ïï2¯³ó2Ç(îý,s:vò~¤8]Ìë÷ä5 <þ\_e9\)æuýþr{ù\õøîð X·YÁ³Y˱9ã8)ÛÌH£CÆ 'd¿iɼhðalsüvÌå¸wDÞWÌß/éVL Ϲµ(û¥ÈN qຽyoPÓ1~Ê:$ó,ûóLÉ·¿¤Í¾sø£X )ï;Ç\_dÿNóèÿè^äp~Ðñè#ÒÉv¤X_=jõyýOqíìO|}4NÈqz³å¸øà5pßOÈßáé-ÿá$Tþ|{d5>±x»¢rx=eãlÀÆò]ôgìPtb8#¬èòèZÄó»óx\_械}h¶ï!ïÞ}ø}5Ëòøã~ò]öø5ÛçY~×%dä[Y?CnÄc6dÚ£üå¹ñó!ÛBn
Mxð×y/±{öMù[£YÞ{yíMæ vÀmì!¸)pßÑÀ\_6Ôª¯Oʶ¶\ò={Þ¾ð ; 7)Ò/ÿo¨áo#\_ü3ky>)Ïgø¹=úÈm»mýõK¦ýö[¬;WåïÚD\_¼ÏûÏò+Ï´ýk¦ýk\_e\_üýñ(ëïN½ÏÜÉsôéË|þ¬ãûÞH/ÚáyÄÿÐA.db¼5ò½(ÌÝìgÖÞûyï~¦çý<³fFýÞf¾³¢î±Þ½(é}í~?ºÅüüZþ¦ÓÓ|¼ÈûÏ2hÛËÿ\_ò÷÷yýÇüM=¾Îwò{ÍkôÕ½¼¾$ä«wßö¢½¾ÌzÅ^¯²\\_3Çå÷ÛLûu¹¾~ð6ÛüYö÷^>eþ=¯÷y=tùìYù\ä÷SöçÌû,ó×Ù.Qwt×YÇùÿ[YPÏ»æYO÷m'ò7^åï§YhÏ+çlöÁ^û&ÓÜ1¾_Ãÿ³¥¾Í´Ð·We|(æ(}ùø6yìS°e.çsì?Äúþw¢ÿÁ¦¬»ã!sª½f]1$°CìxîãÆNÀXñéTN¸ñ\_à¶=èDð fd{~<ëÿ½_Æ:å8.pÓ¸GL5qàrøyXÏûäøWl®Ñqhè<®ãË.yʳWö3ãçÎë'K¨;¾=Þ¡øÝWþKöÉüSt)tuìÍêï;áÌnwI¿þFÏúÉ?¨7<8töëb\à=]Új£äÁªåí}àøºKþõÜ£Ö»£GóÚÉ8&ß<Ã) ²Ú)øòx«Çg�^îNL !ýþ<\hÀØsð<vdl¨rfè?Æ8c~ c¹óPS¿Ê1¤\ü?.ã³¥ßÀ8ÀsªO7þ¯ÈßW\ù'¿Ù£öù ÛÞd½Á0C®]Ç1ííÊü® aÙ櫬bÞÀÝ{ø2¨ãB½P«yý·-ûùOßÓ2H=ÁðÀ4"MöÁàÀ°Àãhç^µêÊèÎÌ=0+ÒemËçàÑw÷Kù¨Ã¼_íD¾¼#ƾå>¼QÙvÿ-õÜõüÞ%6,ú{åÛ3¿»¬gÌGb
À{Á÷#Ý?¸>åwÐçynàÐ{øðc+#'å}aVdid#:uÏkìñ ÖÙmO®]¶ÁzyfXÞ'Øà Ùöoze¾awŤÐV_ZÞ8zì?æÕ#8åã¸íTN7õÀ¤/ðÝÎzÞß#ïpFæ »P/ ËßE×lÆ(¸:{cTïÙ0×¢ïC7ůï£æ÷ÁPã£zÊoü¶øùßX¹¥C?#ãL¶ÁF÷mßn©òIYG£ðUæöår^åM¯áÓĵÊëDo"þ¹r¼Ñ§ð=Î6êÇüÌG¿ílÚ¹IóYø u=ÃÄÎÆ×0'Ç$bð-UY×>Ñ1Ywd~/Ë{WÒ¯Äü&6ùÐ)ËtÑÅìëËe×5pÛY¿½ÝH\_d{×\_0dø5ìç¿c|²ÞÏÊólÉùn8÷Ntªý~½Eù\_ø ÐÏÀ{ûJº¡4d.Ü9y] ¾8=ú¢ÚØ�Ì£IÙïÇüÑ#ëqÀKècÙêþ3d5m&1.ÇÁ �êã.ݲÇ<Øù$¤Q×98ß³%\_xø\ÑoÑ[ÑY)/~¹HëXþ>/¯ ø0éòBGïc|h[dø¬¼¶71¯ÙÄ>u˶¾óUµîqÔiÞ}ÍsìÑÁz]÷!϶çò~îÈNt¾<ã-»^~Ã;i:°Þ³Þác¯iÕcGæÏfÇuöð#0Úí,Ó¶lKÃ9:'ó5rÜX{OLveN1&3ýËå±°·/<&0ò{ý.2¶Ê9yÃ&þ4î#GÉïp[¬á¼ö.ï²æSßaye09öRß$8ôز¥pS°õÏÈßÉó6UÎè°¬àÔß´×'Ëot8ô ôô:KøÌ{Ê=)ÿ§ÕZ&Ödørp÷iKdã¹GsÒÞä9WþÓ·ÑÎe,r¾<[ßc CC9ñùÓ×=¥ÝÛåqÏ)êº&û¯ãè+é÷~hcAû;kÜC¯ä÷@Iú-y¶ß2oz#Æ 29Þ ®Ì/©å9ô{FÖ=j5ÔºçAôþÊËü?[Ò#j¢¤ÇÃôP¸ùïwúЯr¬cpRèÈç®7åjÈãµ =ü\¼?£Ä)}Nú1¬SC¥Gôq=ü$ÖÝ,Ëogö¯'>>zNØéøïù.¸XüÕ|>úuo§äÅ:ÅÑWÒf¿ºßüXìBÞÉ<.twå¸8¶Ô3Ò»q(op^t¯Ù.yGOË<åÞt^£ôóZ¿7×·«¼[ùÌèägt®ñüJùM|öÉ8éCy [2ÞÀtzÕZªS¡g,©áÙ¥ò ñ ¬ð³Ã¿¿ \_�:-z&:g´é,ûÄø[rÃW>êÏð\À=ãdOÞKLý�óÜû6Þ÷aì ã¹ðÇOü]¶{ó$>1ÃèøDßd¹~cN°Á\ø&ètÖ9QmÝCOU¿ }s¼ô>^ôµ)yU?:¸ ú}¤Ý+¯E¬-øãü©l»Ì<yø¥Ègä%<ny¿íÖw·;¼ Ær2Û¢'ÛjG±|L}ãyÆ úðýÒoèðñNÈ6lÞÇnkÆ\ÇwÊ^X]ùì%Ù6<&Ç
W Ò .º'¸|(°$ð4|fØÐØ.ø8§2\_äïÃa@cT{ýÎ8 {Áq¨ûÀÃæ:6úìνç¶JÛvÊqéÌo|pëè-÷§e¹ò«¬ö|Ëoð¶ª1·°1á¥ù÷ÄYý®ü\_Î÷¶óSk½éæÚ/2Ümñ^ÆvÚ¯íßõË^öGþÿ þm{] ß@C¶ÍÚå9÷c<)cøNÙî~pàwþS£éWmûÐm ÑUÏZÊüÐ
Àµ±ëÐ ð»ÔñÊ+z ¯û bk}.Ï%l>9.ö´8Q&üZî·EC¯û-û²;x¯-dì=Y'Ç:ù:˺-X {¿ð^Û4Ûþ,cV·dLä¼/ílý¼qÌÿ]y}� "FuüyZ}Qw£6eÝë²,+çËoôºµ�U§%¯Ð±àT!³Ñã£ïMÌåÙxïÌé©{! T+�Ý\£\_^#é¨>ËFI¼ÀѦäuãGy\_cú½¥Ð3à9ò-Æ ®{Õ:}tC¶1Wò géJæu]Ö%Ù´ã¼QÞcï
,zálw§0éÑû¥"OÖ"ôH0~ìº(ÛÍ,ç²¼7x²¿"{µàÃÎýØÜòñÔµ�øòÖåµv#ëÏage3Æà^Kùî¾¼¯4þÿ=k<õZ/ý ·=bF¿ðvå½}Ùn' eÿ¤KòÞèúðeKÝ®Êö�ñùä1µe¦ØoäOÆô1û±§ã\¶U;ò~»@Ø£øxÀ°¯Ðøî.+1mø=°Ø³>OJ:ÕöÏ:)bã7wñm-fÿ_çÿuyymËßJǵYÒªö4øÞÌ7ç:Ü/ôYäËt¦¹ óØã [ôR¾³+c<¬ Qî£ò(é-å5üù¬3ôߺÌ3!> 2zMûH /_9%Ó¥,ÌÍh«;²ü#6<¸[0GäQÔã~>-}¹}×!ï7~WÆWQ+ÖÁÀo\±§ ò~ô?wößú>É2l˲þE7}Ik¥Íñ³'�¾U¸Iä3í1"¯eÎáµü?-O6½uí/í�FmÃ"f.ÝH¦ <-Û|ìI1.ïÌ·é'òÚz¾Ï\sÄ'³)ã1àHèpJÀ{äµ¹~YWè5ìO NÒ(}~KæÎ4²ýÏeã½òÜ<'ãðð-¢ç"/É87ýd=r~ØcY7«Ã²/yÌ\×¹ÙÈ^·¤Móý%ôäv¹ìÅ~d}'nl¦¤ ÜyVÆGð× g+Ýy®¿þ ^ØT¬¬ùàø³°yÀð笯·dì>+¯oìKÉ=ÆÏc##¯«²o _m9®È:ýg2ÿÊa,,ÝßöMµîu
Mÿ }v¾Ð~ÌUx&´iüù
?îTåûÓ'ðØXs«½ò ¯÷¼+ßàÖØüÄ~vËã.þÚaLöÓ&ô3u&>5\YGuÍliãz6 1~eÝ;Æ[Èúm&Ö\_|WðJ9Ìkl3ìó²ÎÑÈúÍÛ!cíÔ¡[ÆÀ,+÷ñ\k/íE gåøÃÌX§O>í$xO¬5èx¬;u÷�S¯ºX}ߴ̶ͪüT¨z£ë#H¾v/zr ²1y?(ä3><ê6eFÿ$¢QÚ&gd1L3ç{JÞø¸ò=!û¸ÿpw!'äqÇ?þµÎ5Ê8#ù?d;Øä YG?i|&Ç"MÉoð'ÊÙ+ûõN>
Ó×OÊqÞà³è^Ô
kìÐS²oóè¡öAÎgßÑïpÀW"Î<«óÀoî<µ/ô0t+°p 0«Xã?qÆøÌ#Ìç¿uYÊ=-ÛØÞÈKêÅXڹɲõ©§ùûBæU½Èô.d|.ûhïö¼Þ!ï y>eÖF¾?\ÇJÍæ³ü³pÏÀë÷5'?äÒP¹\_÷ }g YÇ8"¯©èÚøðW/ ]åö5çO²}ß ¬|F0ûÇoldZ§½CïDÛ_l¹\Cg«ÿñ¥×Ƨ´[ÛÒ.OýÝRÓ^+×çJ=½´Xo¡~Økðnå«Ë^?TÆòÞÖµRwl?0¥ÙCéÉNÉ>"tÊ?-þ£ï ¯ÿ jxÒkå}bª]®Ü?ïóMlÞÇwõæ@Çgo}H:Õþ+é ,J½qîлqíWÙÇ|NÞÛùVO]æJ^øJ±Ý#«å}|õ}|Uè¼Ø ðv° Y³À'BÐ>ßY#cßAê·æÊA_GY±åàLöÉ«Ù®Á}c_ã5kÄ"àà[TKÚòQewýåˤm±Ej»Òö7d|²m¾(ó(óå?s°Q~S¦«²>G=QßmcÆÇ\f¾bïð»d<p:¿jß~µÊ 0Òç¹[yt˲.êéîʼ5Ú»áRÉýÞ9>Ø©ÒÈ#æ :>Ái²×k[²_mJæ¹Gþòÿî º²rS9+ë©Ø¥L|Çýºì/_ñVÚüð0ÏK gpoϸßeßÅÿÎÜ÷FfóÎ_×/üÌpÁ¸¶RAÞág¹\ê»$Ë#pwd>}Ê|Þç(}?¥Ö¹lZþ.újö1cþ^ÌgømêúÆ÷X+À ¨kCíu{°ìjKþLæÐÞÙ³¥M£íùß¿¿¢[æÝ÷Êø ~÷ÿØ÷eÞ ¾¸~Mæå¼íÆÁ»mÿE+¢ØóØ
ýÙ7øÑáwUN/ë¶.tôÒë G}« ï>Pëz/Þ)ì0Uôp1°¿Èþ+æØR¦×4#Ʊ ·y¶Ç1ùÛGÀùÿÒÌO_.Ç,È08äû¦?c¾mLÿ²¦Gë2V& ¾¾8�ÝrÌ>¹8¿ËßWÔºW ë�8#ÛoU+àûàÞ ò^.\ó.Y¸&Ç7±f²ö!¿£M°ÀIàãÆïÓ²ª+óGWñí[nÅÀ¥°uà÷5á9bÁÌ Oêvþýu~séod½ û g nßPÖµCÆ;d~Û¸¼÷ý{"û<À¯\FÍÿhÏËö·/ñD'å}3ãðêOÉØx,q ·è+é²ü'ì|°xÔ®T¯¼q è½ÄÖñ}qb÷Íóåwì;o7ÃN"hc2V¬/é}!³ÁÛV²þø«ºËûààÖø<ñq ûã˱Wd2tfeÜx^/ä]yà7c%?þ#;öå}©Ð/»äo2UÜ6¬¾,¸bÊCÙÀà!Lôá°àó¨ ²vKÆ9Ñ»ÛÄuå}æxÝË+êxT¶Ïò^3r ehȾùSjÕ¡ÁüJ>ý²Ly¤àqGòü%&ªK §7Ïè<Ük6ç |5çÅÁöZËúRÚ}Ìëäýý5d= ùVÆÈÚ´#¯mð
£}ÑéÁÁeý>¾6"Ç1?Oʾ©º_ y<± %d~.|ãó¥Àà7c+£çcÏó¶3ús¬5øUà@Ù§ÊùîÊz3üôbô¸«ð§ÐÉ&JÚ·§ýöoÕÍ);l0ì"ð©¸~MÆÀÀtfK°÷Ñ /zwBÞ£ýN$,1ËØô¶æ¤Ì Së£}ÐÁ»±Á[V3ð$ìMÊÿ ÿ<ý|¹0Çz¥äÏÄÝ.}2[úð|þïÇ(¾ËOß2k#ëübÍ°sá! ²FG½Á(Såh_ùtñ9sïì_É2ödyYCáXÑÌt7trôìôÇiµîSÎyúе(ßïé9Ô½ýõy~Ì7¯}í2FöÙGuk%>¤î¬;¸ÜVI[ ýZ÷¡g(3º-¿áq½ÿÐõÓjý>þÁhã²HÌuI >í<.ûDÈú²&ÀmÆO/j´<_9aøNàªÕ<Ö²?ɺåAµÝH¿®iÝå8ôëЬlãk®{{rDßvÈëv·=7UòAßé-Ê<¸ú|ñáò,m8QZVÆÁçå>Æeà9± µ(wø
B^-É8�:ÝçgtW^°W1ïäÿ«²\f¾ÌÈ#¾ÅüHæ{ã?OÇôoÊ6pòù{LÉñÄ(G{+Ë]89QÖßeܸ¡e! Èö>kÓWØ'¯ÇÈßíü]m4øGèI±èÇÄ<Ü×ê}÷�/½S^c]j¨u¯ÓxöD;äu!ã 8fÖôöÜ»ÉþWþsÙ7òDÆÄ+G©/ï¿×Ïe¾ü¥óYîãåvñÏqÚÜ\_ßµú\ÁRÑsÀtø"ëlo²Ör?Põ»S¿C»2Ç} yÿ:¶º\ÝeUæ6a\_ro¹ü^ü7>¾ ÇR\_pAö¶¡¬%]ò¸{(ýÚèRÕ÷
ÎUuKü ´ëåL9À ñb'Àçf¾P\_ðpäã?Kô < lrì©>SM<ælËDϪջ[Úw©3ï±!GÊûý2>0¨V» ùuÖ¸0¬Äa1²þ<$6~¿;PÞ
·�ï
5kø'Y£¬Ñã%ÏqüàµôåA9'KRíNx\ø0§ì]þÕ^Òwƶ!s÷k%]üüGëõ#ú®Oyû¼¼~3ÔýÀÛè´]²í=Wò ðg}cíï�·eß8¬}òG]2ßz^Oø\ೡDZW\eÀ¯X%/äÜ/t^Öi9Ö:újUÆ<åïSaçà oDçý@ÇïqO¸jöqwÁÐáÕ1Ó«V,¹ÉtÈX4}¾ÇxC'#a^¬ÉøÐiã·¸GlwõÓ~µîÔ%sÐà(í©8·~Të©äoÞ)íЮֵµòÓ¯íòÚîZcB&dß5s¼S . oÑEFJ:Øq}é¿ýã£ÿ9¥blf~cSQGt d8¸Ì,_5®a ¤;WÚÙ7NúÑwçãåõãÆOWI3î#ñ¹±3c ̹ÿá£+À'þñázÛ?åuµ³¿\C.ÔçªîÃ8d]ü£ÿÐ{ÿ§ØßÑr£ü>£V?,ºsö¡ =ù,ò/y?ö©u³Nµ¶xë|Ó²ÜgSFÚGJÿþ_Y¯£ò¾pçÏdÿTöWܧNC¤ÿ6ß:"ãµÞ©ú»;´rkV^WÎeÚGò̺ϸbÐøÛ¬Ið_À¯:å¹
¾Äü?«VݪKCjÄÇ<¯±¶ô6ýe¦Gäx@dóyÜ%óÃÑ;úÕ¨sw¶Øa£¤ßû]Ù~íò:
ï²Å½jåÿÃI LèLøràÿøa¾ÿüì&WèÝÁ¹yü[î#üZûÇñ"ÿ¿ÌóóÏèÃñIy/ß hÛÍtÚ¿è/Ñxï÷÷ë<¿ËkF3¿\_®5ó{×þùþ§Ã{Í´ã¹ôUóÚË|æ·25Ó>zȹJÿµäû2ýé ¯æ>Éï2ï2þ3õ|μe½~Í4~Êöû6Ëý>ïÅõ¯óÿ/îfúï[Ókþ§>O3¿×YÖùÌY¦Èãû¼qoóük^{éÅ{OòÝWyþ]þvô7¥¿ãÞwY×ù.mõ\ÿè£|îQÖíQùlûäUÖéy¶Åü¦ük÷åÇã×<"»Y·×ϯYÆß3¯o²ÞûYO7\_|Oû¯ûÔ6Çõ£ÌÿÛ|÷}>û ëyÜÏ4þϼÈr¿Ï¼ù¦ý«,mTÛ÷FiãW>côF>ó,Ëñ6ù¦Ï² £;åù:'¯g;þKæÃØú.óúúàóþÒ7oJ»}íûMé·/ä9ðe9åsïäqûüýøoóý»Ùòxç¸v+ßõ®uó/³^æñ6Ï\_f3Íé×ò=°d0|y¡Ã
¼DG$ ýh¡¼çXPð<®±¦¡Å{p«ÏnxCäÜÅÅdÿ\_O^u?ü«ÙáëÝ\_>¸»YôÛH+æþ¶¬ßýÞø÷dþ{�ÅXªëðFæ½u]ÌvÆÿ ÇyPÞ³0ÚåÌ¥'s##ã¥Þ@Ô÷ñE÷¦ì¯ ÿ>Êx_÷[òÕøÁ±û¢ecKö]9'{YßìÏ'ò:ý"ðf|¼àã2VÆý_Î5þHʳí_ì{SÞã_D]×á<¬ÉûR±¿íc]æÀ=,²Ï8;}°ZîW¬Ø feþ8|(ü\_¹fp78¢ýà¼àc¨±KáøÎÊØ>XÍÌÑà'¯Ps²½\÷ÙÊ2ã+wb÷üËgáÌÉöûGÒÈ5×w. ܶ-ÄÔÐpëÀì·JÞ[ò¾íøàA³c{÷@ûhS¦ÙÇ _ @üþg®1Øýq< |ìÚ?æTè_e9ñqxríÙÇ=2¾?.tß®/e
ÛÞÞ+ÛpR®Éã\9
} ~á\¦ùðbð xéÑ'ëjµÙe\Lñ¤Z÷pÀ¿?û'úçt^ë˺_Ïÿ÷eÌ6Áî{Ü"mÙ}ßå;u#Ü£/Nä{Öñ¼þ7OííiÙîVmÈãþ¸ìÏë¿]Ä:Þß·ñÜgò>ðuرàu£åÑrû§=lóO³ÿ#£Y·£²#~_÷¨Hù'a ëÁtñáÉãCg?#ü!r¿qè6שּׁ£GqæÇСg¨oåµaÒæø(i0ßùÏ÷ðÖçy¦]Æ{Ñ×°ÛkløCï»y½F#ï1£ß>/å¡7ÌÓz½=û|l;ÚmÑoø7sðf°Ï+6\¨¶¿É¸¶GæU÷q]tK8øúeùÑ£V<¾úøM[U¾¼©ÊYCV¿Nwr?ÇúС¬5äOèF³¥'³?ÀK!@®û-ÿ««Ò%óòÁÙ{ð¤Íè såùSòº;X®ÃÕg\À¯~:¸¬ø ÀÁ/2¶þµ¶Q~ã÷oÏ~LpÀߪa§ìc/ø8ðÐÁ1{d:>ʳå?å±ÑÕ|Ìt8Ûý�9±¦Ë=ÖxÆ9zÝ)[T«|g?ªX?fÛMªu¯´eµÊ/à!Çqµtè¬|s]9RåV¤óËþmd~ dé´GUùB'åï5D_ÀAÙ÷x©~Zÿr׬SѶQÛ²ß:®£¿Ö¸ tÎè"áä3¿£Ê=ÖZò©qèÃ'eÛ(Ú1lü½,Kè5ÄÁnd½à¶ÉEY¯ßcj×d½h _cóÙs¼»)ëì±È¾¾Äåµý¼¾'ÇÎF/mȶt¤m3¨Vû©[æj³þaë¢'cÓ\,ý?ÿáUNg9Îʱñõû ØÌÕ9öu ~U¶¹kO¾Çï å´¼¯xµ©Ï£]æ°/ù¼/GJ?g>Ñßey¿%Ç¢ck ¯+
ícá¤,Swd ÷-ó´úNÊ>®8ЧÏÈßDã¾¼b¯,9àEã\_;|¨üƧy¦\C¦4dÞ8~Á%Y¦àÝÙ¼ö©,ßá àǪå§Tý·ø«Ês¹öqüEëÕ§WÝ«Ö÷lIßSÒÄÞ'm�Ç;ò¨ßÒêÌ{q\_ýHèUøÚбNz.åÏoúëC9ÿ×y=ìÇqo=y|i±.Ã=!&¾.{áó)ÿëZ ¿&ý¿Û¿)6 }Y´ÛK]àiqæIù#uë5 ¾r2ïü.üæsò°%Mt]úó2Ò=õ ÝàÀ¹ÀÉø=%¯éèÈYöRæ8/Ëã¡3ϲµ³ãùãå=ô"ì;ôÖPôtæïñ<°µ:êµ<3ohoÚç¸ÇÈzî|Ö>Yø¹Bî)iádzL)×ðeÓ.Üïy�~Gæß¡ÿ\ø¡+!µëüïË3e(ïëëõܹ?åå¾¼6\ÒçdÝO>XÛ¬¼>²>.˺ÿk#KbÜÿì+
OP¿kÉÞ^uÝÃ~þ\û=FÐË« º\óLXc×2O°JböÁ'±Ëyí¼Õ»,\_è�jýæñrL(:gè¬Qf°dOå¬ £0ïd=¦õ�'tb¬¿î´¼W]Ü!Ë2ìÀÏeyØ·u[øDþûú·Ëjti8IØp°Ëdö@ÁÉvÀĦDfÎÊ{pR'xèïËò~Wè×ý²ú©7^ÈôÃß7Þ!Xì|þ×{ ÁVcüÞ÷ÁÀÿÕuçßɼÙcXEü!NÈC8ðàºèWkrì \æË0 aKÎf/UyÈIµî/>)Ûc¬cÓùņò6ø
Ãp ÿ\_9îä¤á®Éñ¨``ø¢X®ÈñÚ¬Åd=ØÖvÙ[-¹/sw³ìÑ¿![NȾtæ^ø|·äñ±!ËærL6<Þ¨÷\_¥Ã>à´ÍZæ^c2lJ¾Ï7+ï#rCöý¡ßc·òì§r×æÇú¶![r ¯[ÙÞGå}ð#ÃU"ñÉ>ÉyGØÄáß »c=Û-Ê÷eö ¸c¥=ºÊÁw0ãË2¾ZxºøxfôíȾîÈ6«×ásÍܪ1ðó5ÆücgÿÞ{ìo쫼ÊA,7Ü=t_úFô8 ¸aü_ýËå ýl]zXötÈ8(}xf!;õ:Fù~û0Fþ"ójñÁ^÷V�a/«xö¶>)û¸Á9[åö860¾UÙ×à Y¼+û0ñ¿²÷²= ÁPäïL.ÉqjSùÌ=ÙoO9mèãAèÞØ,sòíW¬_û Oø¾Bp²ÕRoLnļ§&{G°w~hbǶeÜq &WcÁɯÊqúãå7þtt³²Ü¿§ïEpëÙß1ßåïOå}NË1íØLÝò·6vòÀ' ÿ ¼T8¥È0ì|7XQþûr æWåq}#ÛᢼÐóv3¿Ìܱr3ÓØmÓ(Ë~´9±±æ<È8p¶9Ü{ï!×>m Ödë(íN¿÷Èx ~!Æ0ý¡ÆçrÐQÏÐV3ýl×k2Þy¦ü<ÿÇ<yc7Æ\ÌÏ'Ùò^<s¾m¤üÉß÷å=MÁÞûå=ÁÀ ÑéñÉUÀl¹Gü<6Äì go®¼K;¹äÃZq]öuÃÉ þ»}5óz)/k3ØãxÖ.N¼GÜ&s<¾üüUÈ#ä+vW¤ÅwBYãÀôçeLñ<65eË@1¶"ì+ϺϻÈd&8(¼¼xïìßÅWëÞøöéçNO4!ï±u
¸²<ìqQ9DÄOÍrms[M^,y2Ö¯gBöÆ\x's³ð
Ðc§åoPbS2áÔÿ²ï ½Îã»1êwJæ^Rk<þeô>êTã}#2µqèÖkSðöJ#˲}[æGàkÀ;-cbp)#Í\_>¬)Q§ ¿ÁîùX Ø$òùL©÷ ùÛÃÙ'¬÷ð3«îÎÏ^T+o�èÏà7ïßfNæ³neÚ%mð
ù»Røèó:ÐécÄ£4"ëÙ'³}ÿ$s±ÁïWåýËÐÁnXGçÊp 8ÉìQÞ¸&ëkjÝëùºm:Ä}ÏÊý¯dr»ßv¿L˶ þC²ñðõÃzB½6V~zöò"¿?ýoò:ÏÀ¡óÿéæ mvM{17d[pVþ.|ä \_¶Oþø¤ü½tßÜÕù¯ú¿Ôä;Æ ½àEö5ñ2Ïd¬½{Àô®È\-y¯8!µhQæóÁ¹cnþ=åÇ×0!ó[Àcà³à7èÎ2nÈ8J<Î\êÈ÷bTÞ×r¾Þ:UàO±¦°Ó¥ì»²miëÐj¶Ý,Ç7ÙN7óË6{·eÛ ½$Úk^Þóv}øýÐ'dÞç2>®Ñ¿#°mWd\_ º9y#«±K±àtáK{¬Q&xFøIßèIJÎNàÊ1/gþØ<³ûwnä{×óÇÙ¦·ó¹í¼>éîg:ûùþBááÁ¸jÊT1¿å,W¸�aýÒÿ32v]̺ V|KÆfN³C5Ö=#À¢,Øð(ÁÁ¬ÁÚ°«nËU\ß÷«Aw2¢Vîzx19ð¡gpæƲlð/Ðiåo½ ?²Ç6:¾ôôòs26òHþÞØöÏl¦kg31îb·îmòeö=¾®(ÇwÇó¬ów^¼³y\ýç\ÖÌ(Ò¿ï ×ã>É2ìËû®Þ̶fÝ^¿sæìßcO1ÖËqßÌ/d+XÆÍ1±ÄDƽ;ò\z¿C¶ÝÎóf>ÃZüÛA¿6q¬S/³úåøïXG#à÷ò>ºísñ¬óÕ¼v3Û$;#Ûèá²=yMg{GïæµüYÊ<÷ò¸ís=ågÓ2/Ø9lÖ#ðFîéclO˺:=6 ¸ k3s¾�ó?Nèày¡Ï·Ê:þ"ÇWÞÊzîg½ñ½Ìg»OÆ=t¦}µî-_®êíØà P¹èæàkçe¼Ù>=)cks²D|%rø:tiöw:-ë. w¼cVæ¹¢Ï2ä9ÞæçýÌ v�ï.Á:� F�§ý Ì\Î=ºã>IMß vì¨üÝx<øûfùÁQ×d|qQÈ^Ú5Ò:'Çwaw¢Ò¾øéÚ°ªEW£ìà/X̸ ÄÌ8ZÒý6Õ/Æcãx~\ÏæØÆQ®-×<)ÇWÎÉ24èOÖéCÕ1?òwÈÔdoMæy¢îö¥"Kô3¶Àì_¤_£ý^Ês1Î2^Õ+cË5îgTÆ>£¬1gy>èmÇôQ·lî_ñ<ó_õî^ߺ²DÈ0øßffÚìUzìvÖïmöçÏy=òñkNȿзc}¹$¯k)å{ÌÐweuº ¯ÑàáS0[Æ|Klªv9þ¢hOü'ØO3òpØ4ïIY7#~n/¼ÆëÇh¹Vû5qÞ'ÇùL¨u5ö£(iç7Ôêûý:¬}ØÌ]|ð1}ÃüÃ6Æf®{˱Z×ø|ÑÑá³gdÃæf'þþuá;�ø¨zåo,ÊñààÑð×ñÞ¶ìCæaþfìs³1bÌÆ8½òþï2ÌÚ5"°k8dè'øXKñã»ý¢\_ãήÞ2+Û¦¬óøù#¿ËòÚ\_ù!¬ûò< f½mkðwú칸VG?«29Múúºó¡ÞC¯KU1¶ªK0ÎðY ʾ.ü·Ä1ÇÐ\VÇ®ë/xã¹Ë¼Ä§Wq:üS´7óNKÜ?'ëZÏ uµ³úiiÇÙLÛÁïcg &\_bfJWÊ}Æ:r¹ÎµÍ¼FlÑ+ùÛô뮼Çw´sìëX×ämN</{íj~/ïÏ8AÏ
\_7Ö=C#]ðªiYAváC-×ÑÁÏåØ5Æ6òþÒû¨}°$ñª»ëí£¬oÝ#Êí¿yªVîì;
cS}\_Óò~è?ÄcÔ8 x¶Ãò|CVòm=ô;ÒÿÿZ±ïx·ÆïUþ=pÖGôñ)yÖª"OKø´ðg¢þ'V;÷,ÎÉv�<¹+Ùî¡]Î~HܵÆ#C6epÖDêì»(ë«àfø4à°Á7ÀeM³º'Ƕ/Ê:Ó@É y\ß¿ /ïç¢ûW"߯
3>2SÚÎÊR¶;k$²õʪ̤îèeü¶íǼPxç²ß~ëàÛª¬õèÈáEy¯IÖ}ìHø´ì¹íNßy [nȲõü¨î1µ¦ÖµþÃ&C=�{ºöɼFú½ûì
kàÿ3ë|Nö ¡lðÕy k¡KK7Sò«Üp7òÃöÙsû¶?)í@,N¤ Ǹup ô/8ðÖñïÁýfÐ3ð}t˺mÎø£ÝåBå[5d+:Y¤þö²Ý½í ó䥬#±Þ/É6<6v+Ò9ɺ�Ç·+÷Wº�Ø ¾¾ W±!û^mÈ2ûa<ûø¬ìßé׸U¬Ä6UOlÿ£{ ¿[ÿªõ?ÞßÏ6Þå%ú/}|?ßã»xçÕú}>pW0L°uÆÔ§m]ÖÅáÏóMEæ qdè¬Ä FÛÞÌû½ò>s7ó}x;¿UYïÜõª|æ3ëDØù[Ù^Qν|[^Þh¶ç¹5c¾ Ç2¬ËߺܹVKåë<ºõºC!ÿð©0¾°ÙÀ4gJ»jËËÿó÷z¶7!{G\Ì~ÛȶØmOæÙ¦¼Òðiäù6(Kx-xÜȶËùή¼Ïcwþã9%Ë\Êo\_&>YÖ^ø&Ý¡sä³×J0ö«¾É7ÑÕNçµ[¥ ³ò~Ä2&ðnÛ+ïûÜ[¿óEÛŬ¹®Ýó¥/£<1wÌÝ yܱßcä[mËúS}Ä0ÃGELÀ)G·&ï7]òß×<ô"lyÌ%YfÃáGoGV3ù· NÉúþ®n?(ëسàI3²®±ÚõøGfäýAeû4>#Ö_æ-åA¶Óþè:=2?4þzÒBg@¿EOaßîCïa û¹bð)ñ
D[öÊûÓ]¿ÅÚ[ý§è¢èag$îLÿ(ÏÏüzä¸P8K¬yøÛð©b{ ?tÈ4üýò7:å}PÀ=Ñç£Õ÷ªuUäàùrÞ¿wcº&óéÜ«x,~¼9N<á?À>'ëxkò7bÑ/§Kz¬Ùk¥ èô5F=Y/¼XßX?×ùi.Èñy¿Èë.óþ¥.èq2Ü'ôR&d{ Éim©U^³¯ê´ÎBàÍøáËzð\µ«ÈmæùÍ´fe.2í}¿:Ðð&ãqÕ±~Äü;ï^³%ó±%¦Ôê÷?O>~tò¨SGéæ):0ò(öCöoÀW9¦Výµ´ÁÀE«¾³[ò+í |2qýKÙÝ+ÛØf¬ ¤µ!ãÙk2uõ¡òÜÓA¯GÏëU«?µ¿6vþK²ïñCºØÒ´;vÊûÎÈë(ýM{W ÆÆ`¡£cãËë4HbÁ³¨ãm¦ô;1_è¿øð@OÉ:fä:;Üŧ²-½üÖñßbAS<üû?ºåÐoªÏº.{èuàßAW5îJi/°Å§dÜÎî´¼FîÈkÄç±Þªõûkòÿ©Sö á2÷Gåx}Ê~ZÆI+N2(ïvxƶÆÝYêÈxþÇ7éã㥮þSæå6pÓÀÓÃ/ó÷<åû á§A¬OÁ© ¿ÑÃ|.äjømß8üãL÷}òàÄ^Ï4'®kèÿa½¿Ñz1¾jb·+Ç9É8ƦþamKÛ¢"³and[³ãûÃ{Tå7ã½ûß«Yoð[r¬ZæO¤OÖãüD:º�ò ]>ÆZ×µÆáp(0ĵrý¼ñZs7û©z÷)#ÁÜd?S2ü¾ÄoMi'dÄ
Ì£GÞ [Øè;8
à²ÿ_{ggUqåÿº[S¬Ý,Íb ½°ûfÇ'ú5 !ÒQt@ÓQ ò'? $ "iuDbÄóqÔ¨ÉdüÕÑlc1jòIbbv5ò?ç¾úQÕ/Ç~Ï<|ø¾º÷ö½·öªS§NÕåk ÀùõX\_}ú7÷ag9dô]ho¹n]=8u¨¶ëÙµý¾$ävèhÿëÿ¡rí« ïÄZbØOaÌv´N·n# V¿C¯þk¡Çà´Ú¢m]¸ÉS¤
« «íx í?l´8]Úîo6ZÛ±$Çã¾û'»v#B[}Ñfb¬;?8ñæ¼Åþ7"ý!O�6Xз¬Ò¶ïC[8l¬f9ls/Ö(¹¥x½úæ &øÞʧCÚvU#²éåq~Ô¶BÏùsL ÿÌÕÇ\_ªí·ÎÕ¶\_FTj»¯'ÆÑÐ;öÕ¶-@4VÛ1/Æ
k£MAÛ \½¶vhG° ù¾á~8iÓ\_·^#6¤ZÛùê
ÆÖvN6bõÚîº97øôCÛ¸I[Ý[µÝkc¾¶¶[&´sf=k¼ný ׯmÙ4×ñ'Ìßï7é{ÈÎSÍßa{ùå·²6V^Îê³Óí:t]e¹r½¶ß|ÕÜ»MÛz¿×VÛÔ &=¸gc»¶sÐÕÀözCÚÚ¶ðûXþ¹kj¸M±ÊVó]N³×³á¿OµYÛõFø6Ï6óì&ý×{ºAxÔv¬úÀØÆ| yæMmç8oïÓvÊ+&=9xþåaûÇvýÜ@¾yÝ?nWnòäyïk\_¸ö¶²=÷#i/~÷·ÍµoÒ&ìôïÜ·ÀÎáaf.C,çL5×ww2çp"½±¦ä±o}Äø±OÛ±õvLü^1{Óøõm¿ô9çp,3ázÓ¤Çv÷´k6ñxJÛu[»LØ~jÒÆßCæo/ç°Öa§ó.¬¹ú¡Iã7L|ÒNø¶i»¦ãu-æïXoôÙi6Çü̬߷0þÿ.»ßLÖï¼ãt1û·ønëÑGsýå29cÈ ×/Øܯí¶Qfì#ÃíÅ\míï7ù?®Xà Ý-µ&o±.:dÌ C§x¶:oØÀÏa¼»}è/ ×Xåñ@8Vj;®,ù»¹ÚÊ¿§Á¸ì¶î^cµ¿o2RÛùG¬mo½¶m!æ\Ðwa|®¶ûn@9¸Áälgi»Þw¾¶v£ßC!W¯mÿàÚ�B�ÛyÚöÐÃ>a¹¶6h!+ ,_.ï<Ú§íÚµíÚî²ÞÜÃqzN[]úò©ÚöEAÐ7bü²Ù¤ÏvÝZ7PgüÅØc®3ÐáÀæóÔS´]';BØʮնº6ÞfXÜð»¸Ý2ó¦ñç[&~ìr{ïñÍ1aç{¸mà6ë?·ë\¹m¸ßÄkÓ¿¹ÚdxÕÜó¼yf ïË&]^ÒV¶xEÛoKòýÐÞgÞxbóq§y×æ¾;-æÝ{´µÿç¶ðÛÚê·hk?p¾Ö¬aÉ& Ì=ì2÷¢]2{7x±¸p[ÊzM8iÛ&½ª¼ [®]Æ]kÐ}Öãôá6c×ÆÖµUÅ\;¸D·¶ëû0sëøoÜs¹iþ9è¡?X©í|�5îoúëc2ÇÇÐíÃæý}F[[ªFw(#kûMµµÚÎé_èÀáf9úæéÚîñ|¶ºXÌr{ÏûUA×»:m×8Luî½T[«Ç¼ {!Ø¡ ç²ÓK[4? ~¶uûtñ{Úö%a û§eÚö¼ER«í~vX]/ôlе#>(ô°ef¼¶cM¾>ÁÔV û s´ÝöBØ#sðÐÍð}µíêtë½±^o¾ô^&jku!Ðkblyú+[í¼c¹nÝnoþ]5tÑ;ì6¡»Æ8¿®°qá<
Ýk¯}æ,ÖiûMìK´ïc|ö;¡my¨mßõÓ«´µmE¿±+Æ_Ð÷¢ìºóäîÜ8Æ5è×FjÛAÇ=Ê¡k·½ë0·�;R~m ìåP±vãu´MX]ì·\là ÷-ì 7BçÏw®Cï{u¯H{¬k:[·os9ø¾¶s,X'Ëï«íúÒñÚÚªáX?t3EÛ²=ÝÜÃýë®Yö~ÑøóS¾/7~ì0÷d¤¶ßqíàfÈ ¨+hk0?õ 'Cx¡ó<ÆÐwÀî6,اêUme=ØõpxÝùÈÍhÿj[ÞQOÐGî4þr;Ãõåç-&üfím\Ƕh«W\¬ÖaòûÞÔvÌö~
¶eÕµÚ í<;7þöHGÔyØa Ù6 XÃ>Æî#µoÃFusé°é:ªÍÚêñÞõÚöEX66m°/@½ãw¬Õ¶| ®°À 2?dE¡§ÃºÌ=+ä-è'1Çû@Æý?l· cÓ¶Ü0Þ¹üÆøéÛ¦¹ÚÚ¡ýA9nòlaýæ a§�Û-׶ s_èóÑߣnCFe°n{ð7aîýòyù³Ú®ç¼+Ë ÚÊ+XO2KÛ¹ ô1S´óÞ²9Æ°¡Í æÅÜw#ÏPn°||®ö6ÚîÇ~áÄÚØé?£íøçjmÛTèËöeuòîãWèXø½N¾Í×v>²Êà.meóZmÇ\ÐéÁ}?Òº¨:mmØrmmWjk³ÛE´è¿¡Ä\6Æ6#ðÀÆûé#¯ë\ÐÏö}Ãhm×#Àfc´¶ã�èé0>r×B~.Æ)°Å¼ÆuÐòë¦!wOwäçmuq[÷¡¯ÑVVÃØãy¬½«-ímÛ:ÌC_ªm=¾h¶²;h0áE~òñrm¿÷YãA´'k«oqëkÆÏn6ñ\¥m»Ó¬[ÛYaìì®
þ¹óáifµýÆÍ#Ú®C¢í>^µÆÌþûró§¿Ðç³ÌÏú2#Ü¡[¯×|1Þç7ÚªjÏ;¸Ì÷Õv=ôÀs´BWq®¶å³Ö¼o¬yôÚµÚ'ô¯h.Ѷ½¬¶óÇÖå½¢í\+À¶eö·°Eçcñøê;&Üo´»ÕäÖ
ðÛÁæùí&ݱ¾ÃrÈ<ïÚ3÷s¾ì1ai1÷ì1ÜñkÛDsÏ.ãÏ
&\õºõ^¸CµýVÆN°cÁXúÈOÃ~}3òòô9ÚÎsûÈõf´n×Ἤí~ h¿0W2KÛú;sÔi·]®ÓÖöïí«íØzPè½êÖ¶7\^{h;Dß ;"ä½xÞu½¶åó2ÖAG±Ê#ì+ ·¡ocèëÑãYw¹ ôMhË0æá2ò¸¶kQÖ;y½d5æØþñøÛåæz£¹öóNÐ\_@GÀaÛæÄóu:ï¹Üäõ6õºõX ö ÐÓAv}Ä\óNÔQ´IÍæ^Ì ¢í½Ü6¦ÁÚ®óìu¢hÑ×b\Ù|¶vwhcP>ë¿AO°Ìñ£N[]$ëúZã×ýÚî?²úHèP\_'hÛ£¾Bç&\C}-ÇxÝÚV c ×þ´V[èÑê´]ÿ:=ÏÙÿ®NËfÝú;Þ°)å6ë0ÃwÑ»c£eÚ®+׶\_¨ÕÖc¹Úê¦j;¶íòm÷ììë1¿¶LÛ¾mä5Ø%¡\_Azîálm÷6puBÐW×;þ¸¶"7BØ¡ý0ìyaã9U[½ímË-ûñÊ9ÆrÐÁÖö¶ÐëA÷óÒó´-KÁ¼ëùÚîOþãóùÚÚ£®ºv»·j»öy£ïjû½#W¸ß{B[{;è60÷1L[r ÖþmÊ°«g ÚZ´#Ð\_BÇ�»Äk²?'B{̤¶ýú%¼÷%µµµÞk/ uÞíÎÒsì÷ÝÚ6¨+'cmÊ/új´#îòmæîweýó §b½t3'°GÑvm tIСb_éó~È+î>MGÀcJÎ;ê¿ #lt\ôÐ =hÔÖ2Ò ãntÒîn®¶ö æãBØÉcî±Î'´-³Lø&l|>T·cð³,Ó¬t®Á r2æ0¦ÎÏàþç´]W²8ÌøáÊG]j;·äþò t´Ð@çË~Ñ?ê¾}îÀçЯÌ3aC_c#G¾[{ËjmåD®?{´Ýá|mmRFkÛ"0W1dy§ð¡GúCÖãsôßïa~2tshs¹<ÔÛAù¨ÓLÔÖ6ò#ôMÚÚ&s9^ë<綿ü^ì»Ü<ÛnnG0ßýæ!Ï×ÖÖ I[»xôs´ÝöÅX?ù ëÕÐfÀnwñ§I[ýì³whÛÇ/6aãûî0ïÁØéÝÇ ûOmÖ¶C¹+äa~Ç&mײû´/BÏÙ íZ#ÃmõСڮë¯íÚ/¬t×r¸¶j+Ãþüjmíó1?»EÛ=0\Ý!>V[½<Ún¬#| Öz.Ôö» ³¿Á.ÏÁ.w»qa:¶ÐIßyÚî5½B[]GIÓ&ïMºõ| õþrºÁ.�Ï¡ìÌÕv¬½Bsõ°\W0×X.I:é}Öøcý!Ö]¢íÚxÌMÑ×ÊB¿Éß¡í¸kÔ¶óæÝÌ+óµÍÚ~¿d¹yÿmu¸Ðg\¼v:X ]ÞTm¿/Ì÷ñX ²ÇÚCÛ6ãÔÆKLC\_¹J[Y±Áüm¾ÉÿÍÚ®«eÐvýd'Ô þ;ÆXÛ½§9<\/1¼ ôðOÆ7)oÚ~×a
ùÛVm÷ÀÚ\èСÓF[ÁáÞ æý.ѶÅßæi»Oþmm¡'X¨íº[Í3µÝÿá>mm\ݼkÐÖó%Ý0¶3yu¼h°ËÚNm×X¢üb ÌõwY´Y´?bmö2óÞ Ú®khÔv~s¥¶ûbÌú¸\Ûò9k çaßÔãéÚeô!¿:´JÛqÖ»'uëõÜóx¡íúsöo®¶{· öÁø :è1xíÛ9ÚÛµÚÚ-a^v8УOã0= íxã~«¶ó'Xß½RÛ1�æ2ݹ\¤òÝ]Û³][d\_~Ïhmç ¡ÿ
ìû UƯMÆØÃ2úf¬ tË1æ¬Ðb/wïMÚÎ/×xã]µÝãg±¶û<¸òZ¶òt¢«À¸w¢¶sr° ^ú=ä],æ #rÜîÓÖvìRÝzI¼z#°Æ:'Ø&ïCçµHÈuÚê\_H?�tð]k«ÀX´Î¸i+¯Np1Óà\ü ÖÓaþr/Ê ÚNÌá{th÷0?vûlmmWÖHáÃ;cÜyúhµ]ÛZÄë4j«+A~æÔ?Ã|"¿:ÊsµÕM£ß¥$ìmkk'û×æíPµý9tt:aÁ¼tPÿBmmS°>cZÝZ]äÚ79̵Õ_¹aÀsB[ÛØ\a-òz¬aÚÊÎÿÃùλùúmmó\ ûgA?0Î_eÅÞ-cÄ1Ê;§Æ+!sbOìÃö
7ö:ß
Úî¯6{4:ïÙäø ÛOìÕ
~ ,ûkȦÉù|>ìÛ Ø¸m1Ò y»LÛo¡b r¦Ûþ{4èÖk|ÐÆ×vý\ÖÍúãºõ~%¯j»Ïôà\°6/h§¡Fßù ØÍcÌwñýµCÅý®koq;äAèźµ .tj¨Ð?£ çê¥1¯ ½kn½g\ê1ô\îÜæaСîc~i±¶z¾ósß »b?!KÁ¦ãôoÐ;«[Û3ó;P\_ sBÄþ&ïíÚîáüþØÕ9bLÜè<»Ì\ÇPÈê±ÐAG}^cw¢sþ÷ÜWësûßë]Õ9ïp¢ã:>?È°å\>x¥ù;l&ÑÕ¶\_®öns@îþ»û¸×¦:Çðólãº@÷=®í>d,èæÜ{®¹9hn]1¿±ÜøÃã.¼6Û1¶itÇÍpaÇå×/Bܤí°I@½æ²õ¶ue÷ay¶{+£Î ~¦ ãFè¶hÛÏ£ýþãEs
¶2,öÀ<Ì Ç
®©Q[[ÀK´ÃÜï0ajqÞÝP£¶òlÊ°'×Bmu÷ÐUݧí8ʵÞÔ Y«íܧi³·HÈÆ°\9rƳ÷0×rí®íÒãaèk»Nó
è Ãåsh¯Q§øÌïBWÇSµÝ3suÚê Ý1æZîж^ÑÙ2Çq
¼æ¶û°½å¼Ì
út%䢱ÚÚI£n£aÐ]ûÆÇX·}
n½fÉ]ü®[ïwä®)X¥,;+è]¿098ÔGÈHÌÕj[ö0>Å~I¨ç Ú¶ß(Ç°¬NÛ¶òÛöð³ëãw!>XË6T·ÞÓ½¶ö\_#µÕÛ@gR¯=le0îþ¶Ñ9F°ÿø¢Z[Y¡^[Äuö µÚöÍËÁú ¬ Å::ÈHþÏM±þë'÷ÛsµÕ¹c8ûÉòñsÌãvó=(c\¹Üc^2?»Óß±IÛ}[ðrè(ë´Õï¦v$ÜÞ¾¢íwl 3ÝÚù¥&ï°fe«ç°{¾Ü9ªí>uIçò6-è¡+B½A>z¢¶sÖÐ3ÃÕ%´Õo¡»Úäê&ǶRXs6E[m´óçhkO9õ:)u«Ö¹ß°¯í&Ûb°p¥ñ÷EmmþèÃ]öõ~ÐáÃÆòæÍ1ßr¿Òø7Ðú£ûM8÷h«g^Ú]Kiù3' ÓÐv}ks4Á|òx¾¶ßº^m Ó´9} ˽°
ý2lÐ7@ÿô%î:Ķ¢Î3ñü¶zì«Ý¢íü êôßèëѦ³ õóJ«cðÄ Ç.mç´í9ì¨ÓbÜ>_Û9,Û®', íshÉ9ok¾[ÿ¾mü('{üX^Þê°9ç¼)tü¸vçè7ç·5«òüþ\¸Xë°4ç¼A[®Î9okVäøßçø\0
,+ ÷]Ûvå·5ÜovØsÞÖ,þùÿ~m]¡ÀtGÎy[3?'¾u~ÓÛì5wä·5X VèüöïÓµÝà ß"ÞG$~m æ·ÁÔó¶ædǧí'·grÎÛ.£nÿ´Tç·ÿã6ÛmO7éü¶×
οBsnÝÞìÔùmÏǶå·5®¼}ί<ÿ@NúîÊsþ-ÍóûsY¯[Ë¿[u~åkÖ½ãðýó¶æåÞO\_ƯۼÆopý\¥ Ûa=تó+OÌÈñ¿)Ïñ[çø|Ôä¥MöÇðM-9çmÍæü\÷>óÿýòaè é˼îxiÎëx¿RH>~éwD'ñû¿]^Zs~uýy©íû¿íÛrÎÛÖnqØsÞÖ¼Ô¶BÁmè»äñótøIüÞoü é_¡ûÛÜñßR×ñ_ܦr>î~]Øözyó¯Ð-°Fçw~,w>ncýãø¼½p[¾'Û|§©kºNç×¾í
ݱîÓ:¿ciÞgûyrÎÛ\ÿòÍG-~
ð¯ÐÌñ+´~Ç﮽ÈV\_{\ýÒ2\_ù¥Ðú¥Bç\_¡a7ú¤ãé´Ìǵ¼TÀúç ,ð\¨³û´o¾÷ßÛÜ¿¿ÁG.~¯ÿ/KÏxü>yéõ9çmÍÊÿò¿\ù,ßýßÉ>÷aÛ/=¤ó+/å®ßÌ7"/µ¼4¨µÄmxíà{ÿ½ÍÛÿ¿!O|àøHüZSèþÝÈ
ädÈKm §§Û?Õùíÿ> ú¥mÛu~×;åÊKÛu~å¥Í0ÿE^jͦg®¼äϳ<ñήÁ+ßÿnÿ ¹RËG ~
ÖOà8
²_Ê}¾ý[¦ :¾=éõK,S¸ö>ët~íæåçÛ~)W^¯óÛÇç¦ïÉf¿9þBÙ/½¢³{d¼ÏÛ+Le9çm ¯/vÇ«t~Ç+¹ñ[uÅoyãwyÿËò¿ËsüÏwùÄ7tÀó¶¦Ðú îß]ýOί~÷Ñuû§¥:¿ýëÏD¿Ôv° z«Ãöó¶õK®ýþßõk>þ¿\_Ò9é{²ÉKùÞ¿\sþÐá»9çmÍPÝzÿØ&ýþö}¿$´ýÞö\_G>ø¹ß:sÞÖ ÕùÕWçþÍusÞÖ ÕvúB¿ ó¿B÷·Óuëú±Vç·þñ~k®ÐùÛ{áñm!ÛëBçß}öûsw<¿^çW\_Àú¥Bë Y^róïdxÌw×éü§¹=Éçz\x½ØO¾sÞÖðÇ!9çmÍ;rüöä9~»?õßý¯ùýnyÍ·=ÃÉ¿Û~)ßú%|\_ªPä~!ßìú%Öù²>ðþ\®þg¥Î¯~)×~)ßú¦ý¹û}·1ñþÉî~v/éüîÇõá~-9çm ©pØsÞÖìñÛø!ÄÏí/u~û#sõu~õ¬qûß«uþûøB¶g¦
ì\_î|Üɦ\_\4\ß>Ìù¸í:ÿò»ÁVݶû#ä"óqm˯ta¿ßËZî07ç¼áïT\_ê0?ç¼9ÙãÇó+®|ïïG~ù·ÐaqÎy[ÃñkpsÞÖp| ÙqùwÛ&ßöýs÷S^¯ó³O³ûþY^:Ùç㸼r~ådkÌóûs)´=ØvÝÚ>e©Î¯ýK³nÝÞ¬Õùmϸ|äîÿÏõ2?·>®Õùï» ?·þ¯×ùm\_ØO·¼æ»jÎ9Ï]/ÞÖäÆ/ßìú¥¥º°öu^ÏüÐ/¹s
ØO Ðú¥B¶g'»~)7=óÍ:»<sÞÖð»\_û&Ýv{¿h?øBú·µÀþ:~ß-°ßúÅomý+tkoýûWHùì<ÇçÃrûÛBô
ríò-/Z¾ÎÍ¿§òòüþ\8î|Kîy[óÝz}ìã:¿ëoß8ÉùÙÿ0(tËöXùWrQÿ8~'³¼Tèù8Øôm:¿úy?ä÷z -/}ÖÇ0==Ýsé¹çmÌ_éîÒùOçÚ_ߥók,ñk[rç§ï×ùhCÁÆó¶¦Ðñû0ôK
_çoȳ
Nöý9¿ÜùÎm:¿ó©¹û-å[¿Tèù÷ÜôyéÁë ÈãÆÏB±£Àþìñ+´<æî´Bçg_"Àßrõõ{t~çò{ÿ¥BÏ,°,_²½û¥Ë~)ßz>u~ý+´¼ÄúgÊ9ok÷ñÖ>ÑIÇ{fÆçÄNÇÙïÛ×0èã?b=çï¬yÔ6zçw"HFóvdýô;Ö?ß´ç~¢=?~,¥÷ùöËKÕ;úæ{Üo]ìÓ¿IÁmìfçí¹÷iʦM|¾Æ|³Ó
ÚJÆÏ^Ñ×ýMzðy%1ßP ï§I/Bã5æìÕ¶Ô
«Í¹»Ç&Ê%·-»l8_Ãßïr®sÉíÐ&î+̵¹ïqÎ~O=Î9Þ¿IÛï/-ÕÖnÿxý3ï±
}9Ö9asÇVÐócMè˼öý;LxsÓ)&SÖêã7e¯ã{ÕÇüÃþï÷<7k|?^8Ñ_mq;^8ñü»Ç OþÞsàÆiN¾/ßk´ý¦ô¹ùùæeãïó:»Êó¹çÌÜs·Ü!=Ýo¤/=ñ²÷Êwóü±|ëïàõ÷8Ç{ïpÒ¥é=üwËçCúÄú ³¶µß¹ßÓü«óo5;?Q8× =!S5÷Dá4×ñͯ÷ý½¢ÜóÜò ÷½Êç:mËç:ýÞåßÚà¿á«´ý¦ñÎ9÷ã[nzb}Úûã
{8lÖ׺ä×±Ç5öfæõ·ósì¡ }Þ\Ýz¿^ìÑ÷b½5ö=Q~Né}¢p¢À|èOuvOD ~=ç±±»'Þöãç¦'úÉå;ãô\i²æ÷q|°§öêyÀ<¢söóȳ«N¸'JO7ßݽ.Nïè7±À¥&?sϱç Í9Ö»ïmÒ½æøDùüü[åkVXó¶×7:þÍ-ÿnz
ÓÍ÷íúÄõÏcíÀzmm¦ß¯»Vÿuzþ|?^8qßSÎ}° ÍÞµ rÓÓå½ò}¶ýìßÈwÌÅÜëÎ×kSÄùñ¹A89^z/ß¡Gþ{õÐnsÎçæóæûqúÍìØPeÿÕý\¯×Øañ+sÏ?Pø¹RáåJE{j7%nVªÃ5Ju¥Óê<hÉÒ¥Kkç)UòE¥ºÒ3ÝÎ ö)Õ}R=Î$v?RªBC Þó7:%´ÑµS_SªïÇÃJõû÷¿ kå\¥ªÎRª?Åe@Rô%¥ÓûßeÈ¥R¸N£{§t>¬áJUÏ"ÞRjÄÃJ¤÷¦0¹Q©q½ÿ' ÿP<¥ü¶·X V?P¯JTµºH©D»öR^MOyý¼)jï=f ¿¡j¼uneÉo]ÍEº}òåÃ={úwégÅzÍ^±pŪÁĬðW|¡ÏsÿIׯ]F?W/¢\H? ®éYvÁ5³¯YxÍ×jÁª«4/Yà-¸fÕçN¹ UæçpÑo QA&%T á«Þ ÊUOÞolºGé©û½QÞèqt´d)ý|v>ý̽~®×£lÕ¼Íó÷ò¼pÄ^ùÞóÖ->¥OcÏÏOêS¹ð÷=\Té\ÑRTéÚ=9.5¼èT ÊeEgª_u¤óÎEDµêKn_r¢qjcÑéj+ñÝÓA%FЧ Q÷ V»él¹È}(:£¨OÚOÔ´¥;'EïVÔ-$Zz¦{ô¦ëãz¿¢ÞE½Òabzª={jýÞÿ¥¢¨Wfø$=Ð+Ó·"ëvëLÐcÕ§»+¯¨½ØO\êSÔÎúõ+Wº;´è´¢E«Ñ¢ª³ª+ (À{ýÝO2ÕÈÞU!¥}ô3¨Dôkã¾eÜ·£×ÉèõLϲä'D¯SH+¢£é^}£#Ñkñ]¿^ËÞõZzØdªgQQt8¡&S ¹|ã_È%ï¢w²?z8úq¦Sat8ÓpÖíÖ+Ù1Õ/ú¹ºðÕÑKj6áG¿Þ~©:E?^þJvuô#åEþýYu~½ýÜ¢}é(ñdª{´O½@øê´è.5 z@!ê£;ÕbQ¤j¢ýÒSe©Ñ651ºCív«ß¡mËö¡bíL¯¡¤£[8ÌÑNãÞfÜƽ%ºrhJ%©àEM®=ø ÿ)é¬;ý3¥Ýòè~ ôýÑÓR£[ÕeÄ"P·Ó¯wôÙèæLqW\¨£4=ðEþöD7Å xoì ¥ÓMñÉdìVT±[Òä{º
K{ÇÔ)Ñ×¹óot{´=úJ¸6E7Run§«\_î¾J vw´#ºGu>úXt]fp"¥:E×Ñ£¿;DKÔ]}"ݯ²,Õ+ºJ}X@¬ 6}VM®VÓ+éø"¢ôéX\q O.ê³¥dffÔ8ûÌ4îÑ?FõqÖGçdpvºKWº>;ºÞ..¤jrûÁèBµà» S9ß° Ó'» éá£û£+£óé \_<@r)½<Ýw]=':üíÝBΨÉÔÑ5ÑBÕ%Z-VÅÎÔmÄWcQÎóo ùê ô»Î6»ÍÜQ¼ ä5�רÏÄOÓQ9 &|åãjW4ÞQ}vÏ42:Cî rwÇ\¬ÚG®>CùyY4+úU÷²è2\ú©$1©:ÔuÑÙTµÎV7¥Ò¤4óýÑYQª7§eúU$)µÊäÔG£!uC£qV fo®H ó¸8Ì$'pF LWôORuUªQª2qÌIyØñ4Òm$§ªØ»©QuáGâáQ5¥O¿¨<ª 7M øÆDc)>¢aúöáõùðwêKÄO¢öêwjcél2±øÝñ]ýsÔªxEøûÌ)§&£CáèéÉáïãQ>2Ù>ufÔ]õ!|umT¬6D¥ttføKÊÄbJèbÊøîTÿJ©t ÊÙ]µºoÄeµ³q;ËõR·È¸¹ÜÐÙû_g¯o?¡»6UÄÁyG]LøQIø:OD!¹üO.ßÿsº_Q2 ÿ¾!þÝF¿÷~øðÍð·ªSøãðÕð5R¬f~øðÝð¨êþ1üSøgN¼ðõPøÊ?z8üôÜTÐÁ©}ÍA§dê´ðGáÜ\/ßÝÿ Ýï_Ýÿ ÷rèçû0ÇîøTìþkÈ-V"ü7ú;>~=R§?P^ø C;ºúÃðÉø¯ß ÿ=~Ë¿ÓÝT¸o§!ßØ=?]ÑBÕöTðQº¡þðñ~¿q[½T¸ÎHu¥s/ÌûTULM91oOP]/ ÛgúJ©îáݪñ4ñ"ñ m¢H
ô;ð>ÞéÖ;Y\ w¨zb5q;ªÇè÷YwDÞÞ©ú\_wo§»¯JÞ¡6wÏEtÏvººj@ø/êz"8úl¸5Ó¾Kò2zt+]ÞJáÙª¡ênSe5záªC,"VQxKØ.=²Gª¸YU D@ÝL÷oV#ÌÏ«»f¢=E¦Ií&|µ+¼î¦°\}ZyÇTyøeòóË°\_VÛÝDQ««®l +è
\_¢wl »¤ûÿî@øO\ÏM^})ën¤;o¤;o¤goT+D¥ò Ý\UÞÀa¸^ÕÓ×(üjx\_z@ù¢T·ð>ºç¦øwLxÝuZBÜLì#"ðÊôÔO&+Ã\Õ|eøôÐòTIøºõ Îëé÷æøèæp ¥Æ8m¯O÷>»>ì?¶¢1´¼858¼»ü¼rüZu¨l-¥P.¥¿,¥üß.óÿ^ã® ·¹7îrãþpYº\_y¾eòeqPQL~.¤ßô[F$ªä¢LûÎÉ©áçÔ WÓFJ³FõKm"¤ÜH/j¤x4R\_þ£Z@øTª¯¡RÍ PYX@G ág©¸~^ ßף˹ôÄ\º>^ÅC8O=rÇuAøEµ¸ .~«Gà3ôÌíô»à¶åLñ)ɳSCÙCs(г)¥æ&¯fSTfS$fÓ#³©�á,Ä,ª ³ÔÎðS¢ÀÏ¢ÀÏ¢T¥4ôKãr43Ó¾SòöGÃäÑL\z3) ¤ MRÓØ2»R¸Ürr+È&·ÜFr{¹ýÉJî r;;\α!Y\_&éóXEÅ .<ö0^t +ìEGrÙNä ·³qÉý ¹%äInWrÙ«nä²WÝÉe¯zPÁ-ïxN=òo8ËõÔuISS³?øsð'"Å©kߪàÄT9WÇüø#ñ'J¨¯P~
$Á[Ê ~üN¢¿ªôwOm¸§H¿³
ÄíÄn¢
þÞ3Ø«¾ú"ý>Ý|Þøjð5nÿ»¯óÿ6îw=Ü/÷ã>|#v¿nο<»û³çG{Òݺ'{èEEñ
×ÒcÜ ÑAÕ:øQðZ¦¸;¥Jð\_áµì>9µ\Ùê¼N¡}]ùÁSÁ zætY¿øá'Óatð,]éÔ:àEÒË!xÞ¸Ï\_¥Pr( æ8ÕTqzÒ ÊT{ÿ|¿;~ìNÍLêWYêèOeù~§7T éí;&êäO¡7ÔûuÜÇÑê¸O;ú¬\_îÝæ§Òí)~ÊÀ})]¨Iÿ¥&ݳo²
I+[ür\Ø¥&é�
f"yºÏÿzðéö¥{öûXjÅAÿLöù§óéþEj ±üóøVJó2ÅÝÓR½üóXñÏ¡ßñíè÷ïÒÌåÏ¿Ü0ÎÉQýqþ85Çúc(5;R¾¡�!ÇÏcȧ1\òÏöÏ"¹ìíþYJ£ýQé²ÔpùñXü;~§KÕD3©»ÌÑkÄÛ5ô[A4+oû#éùú]Bì#U4¼l0yÙ¦¼l .Ëò)~\_ d\_îÊüÞDÊ¥ÞDUOn=¹sÈCî"rÛAýÎ{ü¹Ó{UyÞ«Þ+énåwð^¡¯y{HFÝ|ÈÛú¥¬ÞÑÉ\_ÿ@¦} ýqIh©9BÍUoÞ'Ô½otKÞ»3L¬¾Ç»ggX½ÃÛqw¸ïòî"Gí,Ù9gç¢ajÿ®ÿVC!\«ÿ¹\áÞ6î[þØ}×ÿu\yÏ ÆñýÁrùüLréïAÒ¸gK¹7îXã ÆQüÔ)Á©Aßøβ oüîA׸èF.\_/1n±¹Þ%èJÍ
êç7ûÆayÐß=þ×ó¯ù»c÷rùúýÆýªqïówgÈoêì¯S%D1¨!ê"}¦)L¨THøª$¸H æ~êzb'Ð/ë¯zÐoÑ@\OÞÞ/¹É >Ǭ\ÁÆ=߸çwZpNÓOó©ýïáô0ѵ¤Ø9^ës0½éU9ûÓ+£Dª·É[C%)áý·:voðÖ$>û·ÊÑzo%½pö!üKåh~º¬/ ½¹Þ<®fÞíôäDËÑÇ2[®JÆîÌOeÝó.f÷\_·¤>ÜÒÔï©Þ4v\²é/qã-QbûmQ¢æö¾åÉÛèçvºrñÏÄVVésËðêdÍ-ÃGÐOEýP\.hò.¸Í£í\_íq&ÜN.gʶ{`»·MqvÞL.ÿåËÆÝ4qvôß4uä7þ-uGÒ4¯òEÿ°Ë¿ßàßK.ï4î=äRñw÷.ãÞiî¿Ã¿ .½ñîôød2Õ/W¿ärFËa©6îpã#b¸§§slö=B]¹ûïôïìôÉöÆ}2§ôKú©nAû ]\¾£È¸¹íbê¯Í¬ë@ë_ÉýïC~ÚL4A0'}Jk0;ë\9ÈJ ï¥tÿ!ÜqzßË&<ê}OM'^#ï9 µèR§ú©R ¤j60®zΣºûª¸×© ~¸ÕÆô;¸¼ýÊXwåý(Ó¡S²cªÄ{{'ï;ªðÕÞPG¡¼wÕxUîýû9öÕx?W7>RíJU¦Ä 6%Gv²qëÈåD¨%>eÜãN4îÇIiMo+ ½Õä²àõ÷
t帪|!ݽgr¿·ÅcUctïMÔEüëý_o%ûã̬Z¼ÆôJr>u³ó·$ÞÊ£½Å=ê ·X$ý{Ò]ùÍK½ÏP(¨_Wü+ÿ U|ªs2CÉÙ©.Þ¸÷¡_¯Ú�öõS¦-^·.ô.daÊûwjqÜ;3}n}3Ó©Iæä¨øà¬ôyIÈdNÉ>Ö¦{õ/¤ÒÉ3ÍÁÐ9ès9 Q&¦'N4É æìÁ°jsPQe8%ù Ó¾C²æ¦ØTyý9½þuEG»xÜîÜ9Tïá¡ÌTu°C-"V7wÍÄcij¦>à^zî^êîU¿&EôôÎà~/ýýî¡ ×Wý§ù6jl°²Ì¨ñɨ8r¡¼2ʪ2ÎʨÜQ¡_βô©Ìý]{Óhü s' !½îtÔä»îôLwuÑLÉÔéta±¸h&"ïRï2ÕÅî]Ì2ÿQJ9þ-¥ßCô¯+]¯öJ(¯J(¯J(uK¨\uñù~ÿÖe¼UòÞ¤x¿¡Ë¿ò~ýõ.5ÝfÓ¤dèÐ8MªÃ¡ÃÅ%];uêÜ¥Sû;µÓ0êDí§
¼ªg«üªúªÇªW©øAUÔ' 'Å^qðëÀ/óúvîÝîÎ¥%½:w {t®í5w¦¦M¯mîî{QmóèÄ´ æQiͺþ²{=ïÆtµÙ¿¡
Záæðn.½lF×ÿ¼®$XO5O³nÓ̽¾ªmönhîÑ vj>9£¹5}Æ^ß«-k7Í9³yü´ú|çÌDßæitëê¾3GñÁM}gªýklä%ükÿ5&« 8üoïAOüéæÓ'Ï©soöZ?kÿ-qßD56.É^'ïèÊ¥Kédi|Nà-ñ =;fì§ê´§Þ¨+©ü~êR84Nt©[<"®¨H&3[
«À!j4ïm4o ÚeÆçÇÚe ͺ¥§$oßO=yϲì=Ýz&¿\_[b^I¼UÐÔgQÆóÛÔ%'((æ)Gq¹æco4ãpN¯)ç¥ûUÆá9/=ÈÐhúè[y<¿ ,Õ^ѨÃôÍÙ'cñzôɦMSº&ÍxÒ~w¬ÙçÀîLUò»ò»9åùoºgY2[ÆËóÌ)qÚïÎvzÖÍæÅn638а/îBù ÿÀd\^ 1¡- ÙÛÆe#Ìý\èà :è]f®tí-Eqéêlcñ<{е÷\_®=K¸Z-AçtoS�)±ªí±º¶$[ûøM'SÁãvèXs6Jlã¡s1¾Ô:(^ãýÁ¾ïäy-Á×&_õéºØi Ò¯j®Ùøéæ9-A¦y© áµ]èæ°KÿºÖP«Çôjþ¥æËk¤#>àÿÔ´l¦5Uñ)_ôfJÄQ\Kìo oeOÿiöȯ£¤[B~7&ZôF74¹ß¾4¦öïq²ßKöoGØ«Ä ÁNKøSá[ýëTK%»Êf"]-5ÙdJóã³Ic)ÎÑnmòÞ#¡H»¹È bnkÕm($¼èRJ4¾a ®Ç¾$nh©ð5÷nA½(]^Úý:v;{Ûs÷Za-õ±Æýló)ýéäßèdt £þqL×gì-òj÷¶Sµ{;ÛQÕy{êY²wZ´WMùXKøðdÕÜÜ1ÑÜëØ¿VMØ;Qrwmõ½:5ÑÕvýkg¥þ?UÍ! endstream endobj 6 0 obj <>stream x]Í@ï<ÇD9zzflɪK¢H{Ȳ(W e)Æ{ûöÁ]^)H|ÃtUCo>>|zN·zó}¹ôåV§iXÊõò¼ô¥>ãiªZ©S{½2öçn®6ëÇë¦ñRí÷õæÇzóz[^êwOO¿?4ï«Í·e(Ëi:®ÊÏ_«òø<ÏʹL·º©z(ãúª/Ýüµ;zc ßħ¹Ôb×-+è/C¹Î]_n:j߬ö×Uÿn§ÀUñíñ�§40©S2¥²¥ÔÃ);J#ÒÔ6pÊ@©
S %SFJVXW«pR3¥g¶pHigHÌ0h»5wd Çv3t á4pÚ«TV/V7©¬^¬nRYýÚ§\¥-JC²SiHØA£Ò°Fe¥SéQ8¥À©ô8C §²µAàT¶6D8#mg¤íá´ØAc¤í°3Ò¶\Ak3qGMp&î¨ÎÄu gº3½îØÁ´àL Z±11h-p&¦ª#ÿP´ÉÄ £EL&ÎÄ c3óÎÌ$b33ÈÔID¦nÌL"2ucfÑ2 3D´ ÈÌ$¢}ldæ'ή±qöonÝ'Û}èú ìe¡6mRÞgäi\>¼çË|\_U¯gõ~dD endstream endobj 8 0 obj <> endobj 9 0 obj <>stream xµYÛrÛ6Å ßäÛ½dÚNÛác'i¤ðNê±n¾´ºY¢:ÎØθS[N&?ÕoìÙ É4ÁRjDpö],ÔÛÊ/§?²ã0²O/+AP»êò?/Z®Mw×Ï>UZèãÏþ;ÏÆÛuã×À´4ÍÞT=Ñ1'¢%¢/,Q1AKÿwñÿ[ã3EÏ.^ôMÑG~O£Ú¸ ý{ Äg¥²Ïæã¾kïï?=ý³Øoð&ÛívÅ
åUIÐ!z^T¯ù®] 5רæ±ôdU5'kõZs%(ZÕauJÎ;¸9hiiÖó¶ËÊí°rgèAÓ¸z¼æ,±/Y6¥¬Îñ®º
¼~Úù²¬Äµz¨5YSÏæket¶ÔysoyFwïàå0ã ÐZ¬îõ9À}C}A!'WQÔü@}2¸3t쪰nÍÓJ¿òcëë"X3WÌ(ví( m7öj .h½ª\ ñÜÄ A~èÐí $W\_J±[Ä ¼¸< Êà<ûã ¹#¿ä×wGñ>¤zRÞ©q\î,ósðîb1#(¤+×À¨Þ¤!ê9PĵÐk�«Õe¸eà "¶ÁkÈB±ñy¶3´êh³rëyùͱÆnçW·j\» .6,ÒNvi½ÏtGl$m&wöø@qÌ=ʸ cqéÈËZÃ>{aÈ>LgJÊî³ú0IEsBBËy®è4ø¬jWéñ0Ha@°äÜ!§o ë.w±ÍG2á]ÜaÅÒùÈn÷ä¿QÃxc'?
ED9]ß«Y3>³WóɦÄ=N änzLqa\_\±dù\_ùkÌ
-)b̧hÅåëÜWÅÞ [ASµÄùlµ¾JÀ,ÚY!4£|¬ %ùiSáÄA²B$JU6¯X¶VfñôÑ ~,ÝLvÿÄT~Rµ7\Áô¬èH BÎ| éÊÐ)X¿qa_K
lÏ\¹þþˆ“\‰”µ1׊õRdàn\sìaÛ¥ü8XOKsWtÓC@)iãAA2¿s@¾·¸0 Ïž.ÀØY©42D,ÎhUB4àšzyÔöR±2¹û":T¥M_íXÛÙYªoÜý9d-Aööå÷ÁѼª[6ÍÅ~ä¹F
o¶Z_pñ~~-v ýºg¤Ã¼µÃ}²D§Å- 6ßbÃ:]»JøÉR¹ß[öÍ>ÞVx#vôØÄg",Kùt@ÊLªßºbW=})ûÙ}mÊۨб8eÒ¢ßUûÔX9TFFÁÖVò7ØÌGð¯-,Åç Õdã{qq¤x×ÄÐKFeSiHñƦË!U¿¾YÖé£üiäV°ÉºëWæcNñ¢-@ ×/ëRÊÏ5 ®)M ÈnÍÌ)^º,¼^MÄ\óAòþTÆ 9ähtô¾:f!#´©µ´øÒ4m9¸Wg±ÙËn"ç Kmà¤][/'ÕoôàÉîw?y¶i)vøÉ5GÛveó·»K¹ºçÑ÷¯ ìE˳\½4k×ókS÷'½é«ÙÕ¥ýú£Ý¼¼y?Ý\üeÿvqwñæjö¯}qwi÷fÓË&7Ó»Eóõlzkÿ<»¹zg?¾{uûÎþu:û{:»x~dÌúxkg endstream endobj 13 0 obj <> endobj 14 0 obj [250 0 0 0 0 833 778 0 333 333 0 564 250 333 250 278 500 500 500 500 500 500 500 500 500 500 278 278 564 564 564 444 0 722 667 667 722 611 556 722 722 333 389 722 611 889 722 722 556 722 667 556 611 722 722 944 722 722 611 0 0 0 0 500 0 444 500 444 500 444 333 500 500 278 278 500 278 778 500 500 500 500 333 389 278 500 500 722 500 500 444 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 444 0 0 0 0 0 0 0 500 0 0 0 0 500] endobj 16 0 obj <> endobj 17 0 obj [250 0 0 0 0 0 0 0 0 0 0 0 0 333 0 0 0 0 0 500 0 0 0 0 0 0 0 0 0 0 0 0 0 722 667 722 722 0 611 0 0 389 0 0 667 944 722 0 611 0 722 556 0 0 722 0 0 0 0 0 0 0 0 0 0 500 556 444 556 444 333 0 556 278 0 0 278 833 556 500 556 0 444 389 333 556 500 722 0 500] endobj 19 0 obj <> endobj 20 0 obj <>stream xì}|TUúö9÷N/Lf&df2IH!!¡$d ÞBMHAA ¢(hÅÞXŵ e2 ± bûZÖÊZVWq¬Òò=÷¾s0ð©ÿÝýÜuÿ97Ï<ÏyO¹ï=í¾#ÉOÆc.|hØøÕ~ɤ Ï1æ;¿rXŤ ÔÇØf;cÒÌÊa£ËÞ9ÑÄØÍQäë«+«þòØ7táÛÉ_TW3·iÐ9ݾñêÈ°Çßý I+«1®¦ è߬c¿»Ö7ÎoXÐRtòõ!+}Û¸dq zÝ®L:{Áùûö±0Ö÷[Æ)s-©,ûoBûy§ÏþfÈC3ÙçZ>Î:|úòþ-0Xï´ÈÈ_|fËüÅËj6Á½J+xêäæ
§ð ¾±+Ç£Ü3ïÔÆ]'<¼±eÆÒ«æ7,[,ÈÜö(Ò0¿9{Éøre c É N]´¸ÓÇÐþæUJù
Í N¾[:ÌØW¦)c«½¾cÓoiüó¢[¤xNáÇ3®^~ÿ¡uÆÏõ÷"kd£v:vñ]¦Möïßdü\í©K2<ªXllÓ²Áг³ÖÌcz\_ÎdMßRöm1ºL'\_dk$fM+IF4{Ôfwtª © Xs6ù ¿^Ê0~R&ߧMP½'üè ý¯Oº×ÙÇÛÿ¤if7oþ¤Óý{ü÷þ¶Æáß4}Yýñö¡;ý¿'évÍñöá·¤Øð¥ßÇæýÚ¾t§îÔºSwú×t7ýlY=ÛûÿÓßJû±uÇÛ®IzVùoÿsw»þ¡þÖ3ï?RO¾ÿúûýãßÿcýýêm¿c}~ÍzÝ©;u§ÿ]I3]t¼}èNÝ©;u§îÔºSwêNÝ©;u§îôߺ¿gv§îÔºSwêNÝ©;u§îÔ~I#5þW"%É4l òfEùÓ+Ë}ØÖIJMl{ZiàìÎNµ~¢w~®èü;»§t6ÊC>[ûÁyñ»$ý'òHù\¦ã«¹¯ý«õïèo$$öËwéïý\_ýý¦2Où
² ÿEWÕÞË W7Í1}ÚÔ)uµI5'7vÌèQ#G ¯®ª¬(64\6äÁ è߯ w~¯ì¬ÌPßä°Û¬fÑ ×i5²ÄY¯ÊPU} ]ÕdÏWò¡ºê£ª® Ô«ÕG× £æìcj©føHMn fó{\CèîP OP ½¾"TîUõUk²Õ-@×£UKZÚë+Ð_»ÙTo6å÷bí&3¤ZÐÎspUH9Û%f°·ÊY MÑñj++|Ájcåj\_Q]yT¯ö«øÌÖÚ{íl»°ÃÎfÕçYBM Ój£rµÉmmk£¼hn¨"»üC¹9Ú+TQÍ ¡³QÜGµYöP í;çC{??ÚÒ·è²ìß1E\xdP.4oðÏ \¾¬ë³YÈDWM¨¥|ÍòÅX¸ ¯.\Õ+%;E+¢¬%G×ÊTUÖÇ´x¢«fò{aôÕ,ü <³ëg5¶(ÜÐܪ¨ qT W@ÏZÙ^Xú õx¹Ê0L¨DBè eæÖÔªMÍ¢IåQVßo-¨¬Pü T¶ÕWJ\_¡ µÛYqçûí}¾Å¬/«Sü&cR²+ÛjfGýõ¾&¬ÏÙZ\_0®ÃðÕ
jëY Ù£¹ïãvAõj+<Û1µEeåÉõY@äëÙ!P
аÁ(°cºÔ¬2£Ãj¹j¸K¼¢ê9«|¸R$+MËûuAJ¿à/î6+jèÒ#>Ñ}~Ö5ª8¨l®èàQjãÆ{ûi?%e,7F 2ÃE
nT2@ÔCu!¬¡ðøZåÙ±VçwTMhÔ)µêlÇWɤ£rT^B¹( ¢Xd¤r¬Áª<V5\_ædSNKQVº ttvNª îöí b©M¦ÔFy8ûµY#Q¯ZA=ÌÕÑU ,R«´Õgh¬Ã²¢Ê¨=ã= FÚFYhÔ¹ÁªíW!]UËSnZ;·N]Îö(i§>µÙÊ êÚ¡"uob+²Ö\do¬¦,>dq³:$½7PÔXÀhkXc :¥&Yq$j²U|ñB¦<e¶¢ÆÞè?6÷V¶¤6K\_WGΫ¹µñ ¸·=jGÙ]2Þ�£¢/øY Wª(ÝLèCËp²(N«=éQµfhÀáOíÍ°JDcrFã}ì"«^yr Æ]ÎÔѹ9tz°KÊïR^Êd¾íXج®íXCtj^~/ñV«jnk3XºÁzc o ÆbF9Ð!»Íèá#!V qg ±J³8SB¬ ! qºËXÄ!Z
X,Ä"!Nb§ qó
'ÄÉB$Ä!Z#Äl!
h¢QYB4Q/ÄL!f1]iBLbuBÔ q¢
1I!& 1AñBb¬c-Ä(!F 1BáBT Q%D¥B 1L¡B
(b'1XAB ¢T!Ñ_~Bô¢X"!úQ(D½
È¢yBô"W!z-DBÈ"(D@¿éB¤ O!¼Bxp ,K$!
p á.M!¬BX0 a(A½:!´Bh
àB°¸àB
8 Ä~!~{!þ.Ä>!¾[!¾k!¾K!þ&ÄBìs!>¯B|Ä'BüE
øH
ø@? ±G÷
xOw
xG·
xK? ñ¦oñº¯ ñG!^!^%!^!b·Ï ñ¬Ïñ´O ñ¤Oñ¸»xLG
xDB<,ÄCB<(ÄBì~!¶ Ñ!Ä}BÜ+Ä=Blb«1!Ú
q·w q§w±E?q»· ±Y[
¸E
¸Iß q£¸Aë
ø
¸Nk
¸F«
¸J+
¸BË
¸LK
¸D B\,ÄEB¬B!Ö Ñ&ÄB/ÄZ!Öq"ìá"ìá"ìá"ìá"ìá"ìá"ìá"ìá"ìá"ìá"ìá"ìá"ìá"ìá"ìá"ìá"ìá"ìá
ññññññññññññññññññññññññññññññññaaaÑÑÑÑÑÑÑÑÑ/ߪDͱô!~Ä̱tèÊKZE¹³Î¥[@+)·è ¢åD§ÇÒÅÒÊAKµRÙbÊ-"ZHÆÓbiÃ@ N%:
ªÌ'Gtr,µtÑ\¢¢9D³c© fÊ55Í"j ª'I4ÚM§Ü4¢©DSêjN$L!DTC4hÑx¢qDcÆ&E42æA4<æ ª&ªùFc¾Ñ ¢r¢aT6Ú
ʨݢSÍAD©y)Q Ñ�¢þDý¨³¾DÅÔKQ¢Bꬨ7µË'êEGÔ((¨uME}f
2¨ë QÚùÒÒR|D)±± /'2ä&J&£(DN"ÙldL ²Y¨ÌLd"2RHO¤yÇ´1ïH&£D9NÄTDÕüå ÚOe?Pî{¢¿í#ú.æú6æ©}C¹¯¾"úÊþF¹/ö}Neý}Bô¢©ÊGûrPîÏD{Þ§²÷Þ%ã;Do½Eô'ªò&åÞ z=æ>ôZÌ=ôG¢WÉø ÑËD/½HU^ z»#zèªò4ÑSd|è ¢Çv=F5¥Ü#D;¦²$ãD;î'ÚNÔA5ï£Ü½D÷m#ÚK.ÅbÉSAíDQ¢»î"ºè¢-D%ã¼æ·S/·m¦²[n!ºè&¢ßÝH´èêìzêåwD©ì:¢k®!º\E¹+® ºÊ.£^.%ºÊ6]LtÑz¢ ©æ:ʵ]@t>ÑZ¢51Wè¼kè\¢Õ1×lÐ9DgÇ\Ъ 1?+æê:h%5_AíÎ Zs5N§æË-!j%ZL´º^HÍO#Zs5N¥ÎN¡óæLtÑ\j×B4<Mͨf#Ñ,¢¢z¢D3衧gÓ¦ÒCO¡®ëèFµD'»éFêeQ ÑD¢ ±¤0h|,I¹Ã¸X²¼ÇÆVÆÄòA£©Ê(¢±$Ä|åU±t&¨2´TK: TKZsV
ÊÄx¿ó(78æ¨ "s(K£¨$æ¨ 9jAýc) ~TÖ¨8æè¢}båÁ ceoõ¦æùt^DyÔYO¢\ê,¨Q6QVÌ¡R&QúÌ >ÔYzñ¥S»4¢T"Q 7fòÄì3@î}&(ÈEDHä¤j'£(ÈJd¡fªi"£È@¤'ÒQM-ÕÔQ&8 wÚfù¶5úÙü¡�ûû¶¿ûïoaÿøe_!ÿ%ð7à /ì¡ì¯È |üø8aÿ£ÿÀÀ=°½~xxù·ÁoÞÞ°ìÝÚÇÿøÖyþWÙþW¡\_²æù\_^�GùnسÎ÷? ý ôÓÐOYOò?iëÚÜ:Ç¿ mC�áÎø|xxÐrÿËBÿË"ÿýÅþí@pì÷÷ lʶÚ(p·ùtÿ]æåþ;Í+üwWú·Ïôÿ¸¸ Ø Ü ÜbÎ÷ß ¾ ø=ÚÜÞd>Ùôõп6B\_¾®E\_× ¯«a» ¸¸¸¸ ¸í.ALcýÆù/2Íñ¯7ݿдÙå?W.ñ¯æ%þs"«"goY9+²2ræóJn^é[9jå+·¬|keØ©3,±eyäôÈÒȲ-K#÷KkØlé¼ðàÈ-MkRëVùÛV¾¥W´òV.±V{k U¶,,,Ú²0\_¸jat¡fPtáû %¶::wn]èK¯W,´Ú«NY°åÔÈ)³çGNsKæDZ¶ÌÌ.i4oi4Ì4ÔGfLÌØ2=2dJdê)ºÚȨ?¹dR$²eR¤¦dBd q%c#caS2\2z˨ÈÈá[GªKª"xxjO ¤Êvű©ðùø°B\_Ø÷¾ïKù¢¾>ÙiKñ§H¹6//çå§zÏò^ìmÇÞçØû{cïïsø7ê·¿ñä91ýõ¾ì¨ß¢ÏNbØ\\kØzv{½Åf±ÕP×°MìVv;²GØÓìõ\_é·ÄÕtøtí|fïc:ÈXçþνo:´ ],!¨ ühé´w~qíÃuÚwè̤¶µJ/Ãú ?Ô¹ïWä;û+yi-´MmñþúÃwÞ|ÌLSØT6Mgõ¬ÏßÄZØ\ÌÉlÏNQs§ l>g#7µp¨úÇZ§²ÀB¶µ²%¸@/ç²ÓÔ|+[k;-gg°leüs©jYåj~p&; 3s6;GUɲËÎìeç³ ~1wÁÕÆÖ± 1ϱV¯?·×%ìR¬ËÙìJv5ÖÅulã1Ö«TûµìzvÖRv,7¨J)}=Áîaw±»Ù½êX6bÔhDĸÌVÇpÆpuiü3ñìʳµÅtìçti±$>JÍÕ¨I½Ð<(½¬ ä÷Ü㪨0äë«\bÆyyئ¬÷¥¤
îë§[/;Ftðümeúõs˽{èùCïîuìåïìywý«ç¥Å{^ÝÓ§;I ^¤ eôúõÈî\_\\4Dê×7; ©¶¾ý Ò%9IXHJË/";¤Î M.Ö¦§Ø¬:êqæβ×LÍÜ;M/ëu²Ö Ï0,cÔ¼Ê?éi®ä4§ÁàLKv¥9ôÞÒ&ìÿZp \3ïÀå²nд²LùjAÒètéoÏAÁmv9ÑîH6èKNÅ´Ck\©J©.õuh ãìÎýº<àvGØ^?dÁÉZXè.(0õöxR::?ÙjçcÀ_nµÅÙªò¾?ÙjVXrÓ3ûX,&ªì6åM&Ô2yPÅt?¾°Îa/2,³ÿ³Çm-ðôéóçLðGm!9Ý¥2^ðjÞõXä(¶QÒ Å} §gux¬¨<ä8bì«ÌIºäæÅ¡H.Ïä÷ºép±lv¥%¹ÒÌÒájnH x=D}/_K 0ÓcäKµ|9Åíoó%ZR ½V«·4s\®7éeÞ¤ÃÀ_sÄ~kÏLKJïàòé=½fcb îFÆäxï9e<Àñ%±))l4z~Hhòý ÃÊöaÑRed«Îá±¾ooÊÚh{jý¤ÌÌ$îh{duE4'²vÞ%f¯©ë%ù/|nÍд |s0òÜÏxá¿èÓ|òWx7vî×6ÃV½+¿§wÖS~~F_s°~MùÉf9-»)ÅÞ¢m¡LÃ"'ÝYZjßSä(-íSØuàuºÿqà]Úf}bÀí 8õÒáuP¡Q>|¤w¼^¿SíçïĨçjxÅÌMíÍtëÍzòÒçZ,²Î¨W¼àõÉ2úJO¥÷L12ñîÜ/oij³0;a;3I®m}ìy¾Ê/drtäm©y rî 4¹ãOªîõRLBÑ«{ð¯íùñ{ôè-~8eb\ÊöOÝîäd¹Ëm4¸²R}AIlË,Úwº®ILZJýyS ÓúîãËÏ ÚëLúÏ]
£W\4dl7Q æ¯{V¤wä! ¦eWÍÚwreÝ, ç|Þ Îó¾Ë[Vþ ³¾s¯¼qY6N¿uaÙ nö\;¯TÙy¥v»òÝXªìÁÒøÎXAçûÊ&.oîøæVÙ·LaSb°Ê\ÚçIè©ü3gdß®Ù0F;CµK«6è«ñ}ZÚu{ö£5Bç\_²Û?]r¶zZºÒ%eðÈõÔ$媾fjã
'æͺdæ¸Õa}ßµc¼µ|eEYí�¯«ïä¡ÁU=¼Ø}&aéÉcV·ÏZ¼ãÜêÊrɬ·\Òª?TYsàY+ç4àìYÞGÙ × Ü,?Ò¸mA?mc¶ø#¿Üf³óѶøAgëàß,3+ìÀG�F}6æ̶¹#\ÊPá(y_e x| çÔwYDñp©o´YÒ wZ¦Ë[ØoÈà¤ÃHçLu'§ÙõYC¦YiÌåYÉé£ÑhHê=zÀ¡¨ÁlÐhð!k0±ÌÕý+zØdÉdLð)ëc¸ô¸´\ç¬+½ývà+2cùü°Ýáï5Ê9ÑäÓ®³,Åg´TQlüRåLÒýã³Ø´Üt$Ût M-M YÖgbÞb×SÙ×o³¤egè-}h±j°¯ô)7wpÕ¢qyÙÙ¼·Ö eA{¸¦wï@ßòPfU¿^?eöæIÏò´>ϲ·f¤0ÌÍaKiWÓ2l®ô®E?ÎÃW»¥ñõ'& ¿r,ÑðkøKF¯5m.-5JÖÚÉYo(äöôÌ%&õ®yÙáIÐkuZ³''íðmp[£ø.y,HÕþ·AcÐ%¸ÄMûøÛÚÌÅrY=Ú,ß{{çùýé'gÇÇ/ñØðàA½òzNuêÜà ¥úB.CÑã÷çzðÖÈõûs¼FÞj°(ón1È÷[ΰ( æùÌf\_^0ï5½ùû½{ùݪ'X²ÔÄÌ%Þk¶÷\_s²ïêúöQ<¹ïÏùv
Þæs%ûì:îÐ%f¦ú2pp3ÓR³ÝF£;;5-3ÙÈû)/HR§ÅnÒjÍ6ËÁ@ZÙìéã5¼9ðm<[ºVÛ\FÉ]m¯Æ(í.ê:Jñë±$»¤Õ:»ÛéôØtnSRÐ3Öȯ=ÊV-¯ÃÄ\_êp£mvåÿÕ© Óä° ó·»ÍRÊVæ°;ø°©S1eev«/®ï>½NÌ, ô\Îä ¸.óþDwVÖ¼§Æð8×hQÅlë.7k¬é×aÑ=,ib°àÅ]Êÿo@Ëz³í,WJ Ûé3.æIt¦æ::xR»n:"¢²wàЮÒÒ%8»Sü çj,zÎsqÞbÖéVO8ÊS§Çé 8uÅÓV$¦Øõz{J¢3Åa>S]v;<³n§pùà £#~{;÷ó·5=^Ö :zC·¦6«/ØCO(ôÏ^ gf¦óÌ4ÊC>3½<ÛóÝ<×ÉsÜ@¨P¨ü©Õ&©/ä, ò¹ñ "7DäÆ\_¹ñ7+ø{åÍ«|CHH÷(èø÷cõ²1É»Is 84Z26}¤³ùjcãBز¤ä÷´Ïìdñ<ÄJÚó½Ê/n
M ±P¿é¼{z»Írz¢Ò9iuý³·È¾·ëÅ!Ç¿2Ê¡ÄÿëKbq¢øÊ(¿Í§JÐʶd=9ÁÈßæ\o÷ÀjÓ§'V¹^»îiù½Óåu4%ZÒZ¦MµRøÐ2¾ÜËú±#ö×R\èqèkÉêL±é´ÏxãÆÅÇþ&® ÿKÒýתã}Éîÿ×ÿÀkßÏ_ÝW÷õ_r]tÔõöÎ¥Íø®ÿÃÞ@GUÝÿ¾ÉÌdfHØ@�"ÙdQAÙ "VÍ a $( Rª¸áZEªKÑÖÖÖN£Rd1ÄP@Pé°HaÔ÷û¼û^°?ú?ÿsþçùÏ{÷½wï}ßí.½|t¡X-;/"/_(¶é»ÿ£¬ùrè"rô²\f _ËrY.Ëe¹,åRÄ>û²\ËrY.Ëe¹,å²\ËrYþÿùÍí¯/´å¨«¼\¡tTiÿÓ©«húû¹òhµåV6D³E4ýEþæd£l©'RÌ7ekÌ}8cií¢·y§QnËmFÙaZÝ\ß)¦[FÙ%z[¶åÓ³SF9QäÛV4ÿ¶¨QVÍÞÛ(-~QÓß)ñ÷eKL8áÎ([cîÛÄøײ]´ßoãEkGW£ìP²ë;EÇ�£ìíwåeÃoÅgPû .xÃÏzY÷³^Öý¬u?ëeKLÝÏzÙs_÷³^Öý¬u?ëeÝÏzY÷³^Öý¬u?ëeÝÏo· ú#Ã(Ý"¿ñãE¢f�÷nßÒ¿/åáR¡HçÉõ"q)ÜËsxV"¯¼½Ô^À17Ò.:³¸ç£OÖó@}åʺ
\p¯P>ÓÛûÐÀ êùè¡«
¼Ë-¿5r>uÝRçRZçÊïåÉ^^Ô(0Þ©Õpcc|§W~ÏK³e¼´u6w<òûG~i
[=ÒJí½º9<é+{.wòe|¤ßozKýäKZr§@¾UïS³3£öÆbiKÓ÷Ótoëºkonùͬ<éü.ö·¼Ò,4ÇC÷þ·Ô½Ð°«Húv¬yNãX4¯ÉvºÕó¸NùÍ«eo²réR#ò±þÖ"¦ÛïúköëqñËlÐÎúµX»é£¸Ù]Ç<£N WÞX¡GhAs<2G<Ü-haWS6ç G¾?ÇxºÌØ<+íÉ
c ã«3GÍ1ÝÈ"oéqO/õ^#uk<þyò©®×ð¦c®Ì\M«y2fMm.þtö5Ïei\ù¤Úû§Êl´c?C¢ rqVze.ONè)cÜ:¹²ÿ¥VzÛ�Rû! ¥¤Ë1ÞRótÙ{u䦴 ʹ«Ep¶´E9-{mº?[~KÔ/ó·©¿Û¥ÎzÖËl+ä¸\óÞÚ-mÐƤWfO¾C÷Ð,Ù¶É{cñß$fD½?æ>s¥OÎÑ
Æ·+çüÈ{õknYT\}Ûó¹òy±ÌØòÒ¬ñ6÷£ÕÌsÚÐÿqqnLôÚhc@\_Òe¬EÙîýûsßËñ͸o,òù=\_Qaºûúü|÷\_Þ@{·Äë\_àÍM¿Ñïå÷¹}%n» (×ë/tx KÜ<÷ÍvÏöøòËÝ }9îÒY|¯Û\_TZë+Ì+qQ5à- ea®;§È\_èõ¤»Çܳ½@©ß[ö{=ùn\_wäôuxÐ ÇSLYkRPðÓeai×OÍo@vP.ö¡·¦6½çç-tÏAq·¯ Øpû ÝÍ4£;ßWÈ»f»gùòdÇúÞ²�}ó¼énÃÌ«KÜrwN)Æëzæð~ïB·ß-~fÓÐSà.-Ö^CyÜ)ñ-¢z h&yÜ =þý]sæxü(æõ§Oñææ{üÍÈhzuÁÓqF¹§ãz/þå5úÏóizxQÌïÉõxüóÜEÚËÙ°t ÖL+ôh?5à è6ö£"ùbðû¼%éJszzJz¹s½îýE< 3úõ[¸pazASçé9EýåÅEy~Oñò~9ÙE
£ªVíÁyZ½ÛJqm¹»´Ä¤=v{¤×\_à h Í\ê6ézúåqÎ-Õ#ºp/gNL[ξüÒ\ÍEî\\_Iq>/Ð|^ì÷Q!ZÞ@º»éÝE
$DO\_/··Öè\W
M/ª¬®¥4î/Á=9zÞ5¿]úÕèk¸T §·úëýÚ�É-ZX_ä}):{tMq|sJťܾÀãÕêÌñægÐ¥ÄBF¢_®w¶Aî)).kþ<(Ô±üüñÁø¬eæ³
C´6U¿ü¨}é¢'çAB4>»øÏKK¡ò³K 민ÔúZÉú.µ~ëÖ²~ôRë'%iõM}/µ~6Ôc>-´OY_ûTmÓþj¥ÒO+DGóqù1À|TdPóf¡ý-ÛÜÓ¦mºÑ&6×ÑF«y+Ïï8¯Í;1m:ЦmÐæzÚL ¦öå¿Y-Û(I1mRiÓ6i36Ô¼ç¾óÚ,iÓ6}ióÚL ÍTjæð¼è¼6ßÆ´¹6ýh36is;5çò|_v»bw|òÉkü<÷=N±Ûìö²ø)³«e
öcW»ED
Ù¬ØãV¯^mvköj·öµÝ<ºzôh{¼bw~TñQÅä)ä!$>N§ß¦-5îÝj·xE7:Ö{×{F¦ãL!{ïP]Õü¼2úÑOHY8¬Ãn±X+-[¶"³(6£÷ brÄ5w\_a±(ëJ~¤>Æ äùSq$TgWg£ÄêÇÝ»A!N«¢}ýó¢/q\&gÓK·8å[q&ÝÝüY¸,Æ3Aq¶ªN©NYÝsuÏãVÓóýûývMqÅøɸé~~nÊ°[»ÕxW
K1¹¬-ßæ²iosÚg|ÿ¬õx]BÓë\«õþÎû;¸ngߺüºüM¶mÛ°Ó¸>q%Øáyh?yÃ¥{êöWë? &SµºùGTWÇYû6íÇeW\aeu)Æj³Ä¸wûk?úèu5¦SîÏmóüÞy"#ß(d×îÊÔ)7¸Eо¢Z«HÐþ©¼RÈßDÑNÞ×ïbÌíóø¬¬q¢ûÉ·¸EÿÛ¦LtFmþl-:È+3oHjîÝ""Y¤WqÅ,ËxÊ).)¯Êãòø®<¾/ÊãÇóØtMò¸Ckäq<îÇÃòÒÖqR;\Vyì(éòx\«äñ%y\+ëgÁÿí¨\Ñ'ÍøÀíBûÕÿ»{&ð\_Åò÷Ú'ÌûÅbxO|,vqR1xi©Ý°6$´ßÝ1öȬÎò¯\ %C??´\?ÿ2Ó|;¶¦Åµ:Ûò:±GËë¤ä×moy}Õ-¯{÷¼wÇ×úxSìõ©çV¡Ü|]ËëIpvÓ=Y×LxÇW©¿)K,5½júR¬6ÿÒüKQc X^»ã¾°>¤SåÇNEÙäjíkºÑuë%SyBn\Ó&¬0mH4%ÚM»O'6ýM(Í7ÖÚ÷/\;= bä!;/"§»6KO$Ìòìù°3qMo[¯2du¼©I¸¨8²å¤§%¢KrçH:2¨íó1òª.òÉyÒö½¶eG»ýÈaMÚ[.&ÉéíÛ÷ìðH<%åãÊÎg$¥mJÇfcÈJéƹ¥TGÞF)5Í¢·Þr"µwjnêK©ë49¿÷Ôõ½÷Ô?¤6rêhoI=#ßU¡qŤnÍ2©ÛfÉ5d.RÑmn÷Èè«Ò¯Óm.Çô«>î±éêZ)§zÎD{õ@úöjè
^?ôÞÔç%Mz5ôù°Ï>GúZú&ömÛ÷HMúH$+}f¿ ^[1°ÇÀo=1d2rhÊÐC˽gÈÃ6«Éè ËX>|ï«#>rväoòþ³\¿=ò¼:1Ê4Ê4òíQ}G?:úÃëÓÇÎ@öݸ¦õªÍs&u"Ýrµ¶ ï'>{nîÔ½ÄìM«Jrçös+ÚÚ&k[´´ý¸}²öD»£ÕÒî'wNØÙ©);³&kíµ²~÷Ü:»jºÈUÓX7cVÎdz8|ªÅê¸ÓXÛ6iÏó3úÛµ÷OÌj·?e ú´ð¾æ5ÍÇD\fÄ6ùX7õLé¿'hÑÔ<ÕöyïuZlbFuFêzlmZakô^+B)!]´7hçnS´¨h%=Ó´sEèªôîtô®û�¹\Ŷé«\ÿE®©1ra ¹ÒƱ6Ë
-´ö¿¹\_²4¯Ø?"ç{JæuüGD®ì,r·qr¾wä%F.ôÜ»Ä÷z¤ÿ;¹°çÿ]»KÝÏÚÞ%qÍë®#Î&ìÑv=RVÊ;Vm§#¯VNèªíg;¨aÚ®I¿«ÍýZI¹;!wVÚêÄÈrÄîÒÇ#VÊÝIEó.Fµ+2÷N®Ðv0òj±ÏÑËkÙ5hw´Ö.Ó¹ã ȽuåÓµÚ1u=µ×j»)f{徫Ì,y§¶ëWY{µyÉx°sëÏ^MÛ¡iíË"÷iÅr?G]¹SkÞ¯MÌe9«ùÖîViëNÜ"ûÖÞ´\ö%ûm9/hl\]«\_ «Rî1ߢ~h&ZgÙ¯~g¡Ä\5ÊRÈ1YÙ(º±Ï<]\g¾CØÍwC¾xȼT\i¾O¸ÍJËËê.Ëjxv^/ vC-| uð7Øõ°þ.zŵV?Û¯îûpÅ
(
ê.kíÅy èe9\_ýÜZ�
P¥ê×Öo¬øÆo¬�ßXßÃïïá´në-ºØúÀÝ¢-fÁ|ðC9TÀ}l+áqx^7ØÞ|Á øNiÀöÈ/.ñB o+ºÈÜ=L^;dé[¢~Z´#k«ÈÚ\²Ùv=Ùv?Ù6lE¶'ÛFSûUò%Ý<]}ÔüSu4¼y²ÍAuù yÖ(ÌæÃäà·g¨µmfÓ¨¸Kôéý/ ÿ±ô?Ú3éû)úþ=Ò÷\ú~þ>¤¿é"^ÓËqziM/WÓK!½ô£~ôÒ^®FË}ôÔrée�=¬n¦ôH¡¿ÐÇ\_裧r·úGúéG?wÓÏ úJ?£ú}õSUÿ@Ë?Ñ
þ Ùlúlfôössz í¶¿a´~+®11Fl½ö¦W½¥×±ôÚ{ÒÛ´üw VNNcù736³<'\ÕXÀrxáø9¬-jTl
m°vÀg°>]ðÔÀn¨¿«ªØ\_Á~8� pPÝ\A#TëÅ?ç§ 8 Qf·ñü |gáßðº¨jH ÈYñ y&ö3õ¸ù.ÎÙêqË.5dùj7ÔP=P{áïðµ| Gà£p à øN?á åPÕqÉêVÛh5j "dª\_Ûnã< fòü¸ îVC¶lóx6³Bs}/ç Î÷ÁrÊq°=Æy%çÇáIÊOÁÓ° ¡ÿ¹¿ò«ß¢üå?1²#1²#[½ªÚö1²#1²í§Íh�bdûV·Kª;mÇà8ÏNÐ÷wpNqMìlΧ¹&FöÈ/ñ2GE[¹rÅ£äî4rX[½¸ú5W¸Oo0&úhÿºcÈÌz2³Ì¬'3ëÉÌz2³Ì¬'3ëÉÌz2³Ú_iQ2-J¦EÉ´(%Ó¢dQ12&BÆDx_5ï«7ß)ÌEå¨Éz²¦¬©'kêÉz²¦¬©'kêÉz²¦¬©'kêdHFd(ÖÅz"!jõDhETHÕz¢Q×£x=×£x=×£x5WCx4G#x4ëñb/ÖãÅz¼X/GìaÃ×3í¬½fíýy'kíç¬B¬6Ò¿G°ðs,< ý{/W)\uÆ¿÷ÓÃbëdëdëdëdëdëdëdëdëdëdoZÙµ²;c¶1[ÃaÌ̳aÆl1fÌYO³ÙFÆl#c¶1K¼ÅDÖÍAÓÓ¯§§\_gæÈËXG»°vaíÄÚÆÚÆÚÆÚÆÚÆÚÆÚÆÚÆÚÆÚÆÚÆÚÆXld,625½0c®1WÃkdKcKc}Kc}Kc]Kc¬4²¶¥±¶ug¬4²¾¥ÿ5ä ù\_Cþ×ÿÈÿäü³þ%³þ%ÿä| 9&çYÓXÿÒXÿÒXÿÒ´|WOëìÏU ãÏ0qD5® ÛÇw±ªQ0ï³dôꩽZu¬ªK¸EÛ]´ý»£iû(m?¥íÚÖÐîva5ÆÑO©¹5Ô ÷WZμ.{òò|Ïwð¼çÃééa¾KO7ÐÓzê/ëÿMî÷ÉcD8V¢2ò¡� æð+}R-xËýô^F?åÞhµèþlþø7n¬ÚSÙ%&³rwdØÍü 3÷hp{ÿYÏýêG´hϲ«¶¦Ó>_gIÎß!Æﻯñ"Í:¡Y'4ëfЬuB³NhÖ Í:¡Y'Z¶¥e!-ÛÒ²P¶L e-h@ËZ&Ð2 ´L e-{ÐòZZö åµ²¥.Zºh颥.Zºh颥.£å £å ,¹Cô¦Ô[ú¸JîNãzíÿ©[a L
Û½½½½#^ûï´<Ü6YÆNcÑQ£ôT^Ðú@_¸Ò¡ôka� A0ÀP? ×Á £4\7À0ÆMp3ñ0&$¸2a2</Àð¼ «áX¿Wá5X ¯Ã:xÞ·àmø5¬wà]ø ¼Uð[ø»µjΩ{áØ�
ÜÿTÝlÍ°¶6õ°²vÀgì fòiå.u§å¯ì$6§° 6ÃØ Û»ºÛ²>SwÇ%© qm¡´©jõ1xðõ%õ°uzÜú:¬7àMø-÷?áÌnÓúWÊ;ÕÝÖ/¨_G9¢6Ø®+¡ ¸!M=në Ý ;\=Ôݶ«¡§ºÇÖ È¹#î¶\äÙpõ°í:ÎSÔãvÚ7À 6°C<8À .HDh{íÉаÛÝvì¶c·»íØmï 3 ¿ýíèoG{t
nЮè4@=l?QwÛ38÷FÃMp3ÜM½Ygó,zsÀs¡ga ,
xû¿¢þëÔ\_§î±¿Áõp{aµ!^lo£îÇøvêáx79tw¼£àï(xGÁ; ÞQh¡àï(xFi~$A2´¶ÐÚCHTèÈõJènH®Ð ºÃUЮ|Êô
k úA¸À@a
aðÈáp0 FÃõpÜc,Ü7Ã8"LÚ¿pëT2a2d©[a L
ÛzO ¸«G%°à>¸a<�ËáAxø¼¡¬TO+Ãð$<OÃxg|^àeX ¯Àø¼ ¯ÁZTÖÁð&¼oïa=0×\̵Êoà=¨ßB5sùGð1|à¯ð)lÍ°¶ù³È4ÕÃ,=u 3ÿu¬ý¯cÖþÜgaƳ0ãYñ,Ìxf< 3
ÏgaƳ0ãYñ,ëùò¼ ¿÷ ~ ¿?¨G-ÀáOð!üð¨àcø6Àvá²ìÏ+.I8Ú g\;h R
ÓºB=jý
²>Fyågկϱ&9æ¶X\_ã:[ÑÙÎVfië;ê!ë»ðϪ@åÞ§þï¹÷ÏÿúC@O+zÊÙïS®·ðl+çmÜÛ;à3Ø)\Ö/x7í¬|¶³ÖrïKõ´)÷ ç¬\_ÓÏ,Öev×Vv×ÖãÀg+Y¬|f±þNA"ØvZ=dKTÚZAkHõ´-:B'è WíJènè!\¶«¡'ôk¹7ó@µ±ºê³®pÙMi7À 6°C<8À .HDh! ¡ ´{;h R¡#tÎvô´£§=íiкAw¸ ®VÚûð/\é\³S°_K¹i&Dy
aðìÈIo>çÚ'Ó.KÝ¿¦ÀíêiûÝè9zçÏÒ|Þµóy×¾£ÃX Ôw3þ嬽ó³ôû</Àëô·fñ·¸G íaÚ~¯ê¡x
½] ÅãÏxç$î·.9³³BÅwà^ ¤óq|gí÷ÚH7öU¡»åíãæû
Ü/¿GÑö[ÇDiú3ó-ê'ìNÚï¶xvTô5õWÁPãÔÏMãÕ¦p »òiê>v{Ù]ìuÌP·:fêÇCð0<?ð à³ã1X Ãð$<OÃ\x
çàyx^\_Kð2¬W ü ^U¸ú¨GM#¦|&öóz8úÑ?lÊPÑ?lºóÃêÓ#|v¹C\Ãüu 5·:¦ªÛ`:ü rÔ¹
P xP c[ÛØƶ0¶
±-mal c[ÛØƶ0¶
±-mal c[ÛØƶ0¶
±-mal c[ÛØƶ0¶
ÔÎ0 nL Yp«z�ÛÃÄp¨ú%ÚfqT7ÉßvÁöuؽÎtºÞ ð°ZªµÏßؾÛ×aû:l_íÕØ^íÕØ^íÕØ^í(S×;Êá¸P×£W5zU£W5zU£W5zU£W5zUëøÐí ð¡ßi2èt =¿B:4©3OûáyÆaV"ÓÕ%èô3>ão »N]§Ð®íêЮíêЮíêÈøÈøÈøÈøÈøÈøÈøÈøÈøÈøÈøÈøÈøÈøÈøð@¨Ãux Ôá:
<áÃcøð><áÃcÎ u§s8#$Ñp=Ü�7 7ÁÍ0ÆC.vxa6äÁðÁ\ùP�
PÅ0üP(
°Ê © ³/~V©ßKËÔäÉiE¢D!JÎ-ðDXq"Ôàå(^²DXa"¬0V+L&÷£x?÷£x?÷£x?÷£x?÷£x?÷£x?÷£x?÷£x?÷£x?÷£x?÷£x?÷£x?÷£xÿ Þ?÷Ïàý3xÿ Þ?÷Ïàý3¬rV¹«\U.\a°ÊEXå"x7w£x7w£x7w£x7w£x7w£x7w£x7w£x7w£x7w£x7w£x7w£x7Ê[@vkcq1>]Bv/xû ÞnÀÛÇE1>ã þ 57áëøú i×ÕoiuÌù!2?DæÿC8Ãw¦GÕO\_2¾d|Éø±´¹a#1ÚMv£ 1 £ 1 £ 1 £ 1 £ 1 £ 1 £ 1 £ 1 £ 1 £ 1 £ 1 £ 1 £ 1 £ 1 £ 1 £ 1:H£Äè 1:H£Äè #$Ä 1BB#$Ä 1BB#$Ä 1BB#$Ä 1BB1ã 1ã 1ãàÿwçñU×w¾Ç9'9'¸kkÖnÓZ«¶Sí>c§S»ïS;3j-TZ©¢¢ÐZmë; àR)-KIiEÁ+Æ8ÃI I'? P¾÷y2´×ν÷qï?÷Þ?^ßYßï÷ýÙclü?ÅÆÍlÜÌÆÍlÜÌÆÍlÜÌÆÍlÜÌÆÍlÜÌÆÍlÜÌÆÍlÜÌÆÍlÜÌÆÍlÜÌÆÍlÜÌÆÍlÜÌÆÍÑxìÁÜÃÞϱn+°Ü.Y.f¹åböϲÿ Ö+±^)q³×neéYa v±v± ö°?YÃm¬ØÆ%V,±bK¬XbÅ+v°b+v°b+v°b+v°b+v°b+v°b+v°b+v°b+v°b+v°b+v°b+v°b+v°RÌJ1+Ŭ³RÌJ1+Ŭ³RÌJ1+Ŭ³RÌJ1+Ŭ³RJ¬Tb¥+X©ÄJ%V\±R+µ±R+µ±R+µ±R+µ±R+µ±R+µ±R+µ±R+µ±R+µ±R+µ±R+µ±R[ôVb¥¡ÑhüO+ °B+ô±À ç¦>êöQ·º}Ôí£nu¨;DÝ!êQwºCÔ¢îu¨;DÝ!êQwºCÔ¢îu¨;DÝ!êQwºCÔ¢îu¨ÓG>êôQ§:}Ôé£Nuú¢÷È ¯Ë ¯þzIÜì·8Åèî=¾sÔû¹êö ººñ¼'ám8oÇÅ>s ¾Kñ}è i=LëaZÓzÖô¦õ0i=LëaZÓzÖô¦õ0i=}Ö]´î²ãDAQEAQGõÿKÐýð||¢üÿµ·w±G{t±G{t±G{t±G{t±G{t±G{t±G{t±G{t±G{t±G{t±G{t±G{tQ°DÁK,Q°DÁK,¢h(¢h(¢h(¢h(¢h(¢h(¢h(¢h(þDC
,Td¡" Y¨ÈBE\²P
,Td¡" Y¨ÈBE\²P
,Td¡" Y¨ÈBÅÑß;ú\_!?ÄV%¶\É6%Ù¦ö%Ú5.ѸDãK4.ѸDãK4.ѸDãK4.ѸDãK4.ѸDãK4.ѸDãK4.±ä%g,9cÉKÎXrÆ3±ä%g,9cÉKÎXrÆ3jʾ0Wájð7g,9c):L.üÛái7Fú:ô¿½ûUzT©h˶h{U¤-Ò2ÑÍ(Uãé¸Î\~µ~zyv¯OÍ^ÕyÀ·ÞOá! ¼©kêåݽ¼»w÷òî^ÞÝûÿ(Ûôò¾^Þ×Ëûzy\_/ïëå}½¼¯÷ÿjWTVF(õ\_ç(yðµVÚ}¶õ´g¿öë¡my²)°D};éÛ9ÿfy~·áÒ¯Í tí¤k'];éÚI×NºvÒµ®õt§k=]ëéZO×zºÖÓµ®õt§k=]ëéZO×zºÖÓµ®õt§k=]ëéZO×zºÖÓµOõð©>ÕçzøTêáS=|ªîtï¤{'Ý;éÞI÷NºwÒ½îtï¤{'Ý;éÞI÷NºwÒ½îtï¤{'Ý;éÞI÷NºwÒ½³¦|Îɸ WãLÁÔÐ9ªñÞ0XM<«ã|\_>f$^ {ôaVbohLÊÉ÷^OËg
¿þ¶ò×¢Ã\_²§°+ÛÖ³Ø÷]çDÀó!XËÓ\_ÀÖ\çúrhI¬7éæ¬Öìº ]ÑD·HÔãé±/ô%£ÐL£ÇþO;g=ÉLçíÙ ¥ì%¡!ûÈÙ¹^Z² 'd§¹Nw½zèìO¡bfo
¨ÌÎòþ]^û²³=yî± ìÍ.vÿeXöd^«ó|«3e½Ö Øìy-·¢ÝçzB{vC{íQ!®=ca:¬5Öêõ ¡¡VO\_k\_µ7
ÚÃÚ{08úçªØiª©ÚCÕª¾NÕTÍSu3U÷Pu3U7SsýÔì§d?%û)ÙOŽT¤ )ØCÁ7Sp3 ÜLÁ<ó,P0ÿ\_,P°=ì¡ (ØCÁ n¦^õz¨7H½AÊõPlb¤Ô ¥)ÕC©~JõSªRýê§T?¥ú)ÕO©~Jm>¨TR=¤Ô ¥)Õ½=ñhXSê)>¸B ©²+±-\ÊÏ&'ºÃ¼ûköÞðq~ö§d2¬M¦mÉlø!ooNNN}/ùp%Ï{òýáÓT{÷ÇçîO~<\üTøöÁßÎjK~=<üFÖÉ©U%Ça«_cmVì°B·»öºãvwÜ-ÎK3>ÊbÏ&ßÇËGc¤+z«ooðÍ|s§½uØ[;äFãá¬óÍgÃK¾õoýÁ7ôW×6¿¦êÑ>I¾ÏóÓÃ6ßj·ËµÑ[xÖÑo®åY/yÙ·×óª.²ÙuSØÉ;vòÌK£ÙÏ·³ô²Ï²Ï²Ï¬þaþHC¤¿!ªßNd²-hßKû^ßJÉQ}rTÕçìÎ>àìÎÝëܽÎÝ묽ÎÚ+¿ôÉ/}rKÜÒ'·ôñï>¹¥Ï^ì³W®è+úä¾gò{YÿÖ¿õïL¬aѧðlx1±VU|/GxÁþįçøV>LNl «´ [±-ÜhsÝî¹ÓµèfòºDÑãy\f$k<ΣùìP×Ãp8ÀQá\Þz>o=·¯¶Þ<>\<Á{'¤èÉ]ßSä¼Sñð¯ÉÓ<'Þåù»ñÿï ÿ Gþ»Ì²Õf²ÚLVÉÛ?'\_ÞüÏ|~üë987\ü¨ëÇðñðQq~ò\üXd|íàoÌ.!W'¿¼ãÃ+òëo³ãCcv.ûEÉ~r§ÙÏKfò¼dfv¦÷ãø%nÆfoÅmåó÷xí^Ìö|æºÏý?àú}A¸1ûëpµjv}öQÏÃo±$'ªÎSá®ç3yàLýÁªÜõÙßdW>·ÊkOúÜj×à)¯¯õüE¯¯sßz¯½?{ëÑè^MØ>¿ÙgóؽdoÞ=SÔÝVÜóTÑëEïù¢÷¼ì¯ñÁ,̾~íBwx&˳ü0[ÌîF/úd~ y<Öd÷bÇoÏeù¬0£ßÕò»ÚdXS[éZ&ËeɵÕ=2àµÙðLm-ñøPæõÃqôúQ!¯ÒçUú|í1îw¬ÏãqNÄ[|ö$ï¿ '[ÿí^ae£µ×F>³ö¦hl-[ײu-[×Þ[p«÷î Wü2Õy2Õy2Õy²ÀLÙê¼ÚûÝg¾}?è¸ÿÏ
øM¸":Yø±,ñ»ÑÊüÜh=A&èñ³DöwDöJQ»LÔ¾¤æاEìQÙ$ëEáQ¸QÔý£ÈºP$-1·DL§(¹GlOñþ\_óþÏóþgxù\_\|Ç¿ý|µØN~«bmH,S¥VÊ {mSç÷ÞÚ°Iöܤr=#gõ¨\+ÕÀ»íV½Vª^+å¯vþ<Õmçë墵vo¶Ë7Ûí¼S¾ÎÙùn9;'gçäµv¿D.X",±Ëývù¥rÏ£zmÈþ»L{IX©TÁ6¨+ÅfØìQÁ6ÏųG|.ÅçblCößûnÆ-a¬¾IVß$6{T³ ªÙ~ ¿Il.VÍVÍÅbi ¿_Ïðénõ$§äøm·ã«Ýüt-¿\À/ðË|±¯mçkÛùÚv¾ÕÍ·ºùÕv~µ_Ur|j ·O-Vá6¨øÇþÑÍ?¶ë ×ð§ð¬íÅð8¥wªM|áÓ²y«lÞÊ^¦j;U©ÚÈ'þ(so£ì:º²ë(»oì¯ÉÆeã²ñF>òw|dX-Ȳ¾²
tȬ 2kÌÚÀgeÓ-²h^æÜ(#6ÉMTßIõÔÞ)6ÉM2 Ø$6Qv§¬×$ë5ÉtM2Z^+ÈbY,/5Èb 2X^Û"m¶ÈVÙ© ;d§ìÔ ;5ÈN ²ÓY© +f¥Ù¨ åe£¬³NfiYZYi ]¶É.Ûdm²E«lÑ\3´Ê 2C+K5²T#K5Ê ÛdVjd©FßÊRëD~oñM"¾IÄ7ø&ß ÚD{A´D{A´7öhoeÅFQÞ\Ê[Ey«(o5wéË}õYáõèlQV³~ ¢æ¨9"ê9v!jö²ëBvc×:ÑRd×ìºM²éR1" FØb[Ì#ì1Çðò9¼|/Ã3xù/áåsxùÞ¼^Ké´7ï¥ÕRZí Õ^½^;xò^úÔѧ>uôÙÁ÷òæ½4ª£Q}òÞÞ;çîuæ:g|>ÜÊcg{ì}0<Ê7·EÇ;ÙÏ:¬Ûɺ¬×©ä¢58YÝí±»»k°»=v×W{ìhuÛQ·uÛÍ»Ùc7ÝvÓm7 vQe»£¬4h¥-Vê°RºhXQ6µF«5ZmÐjVk´Ú ÕiÑO~«Ò¢ßÊVî°r;hÑoõA«Z½ÃêVo´zy>ì0#l/÷Wú+X±U.[%ãnqËóÁG3nʧÎPÅÿéôä7¢×îVï´>+ÏvûGu¬:ø~ÏJî¿ÉýûtÃy=mû3CUzÒÒ8ÙóÓ0?ôºÇ¶QË4ùtRÞã@t{¼àÇé×ï^OøÄkïGëM$¿¤QLx©¾è4ѱÛè¸åùzýúíá {xÁ^°hù·s÷ 8ñMó÷É>ªX<Íu¾Ï?èµòÌ]áÌqtýõÙS=í²§]³Ûî»ík·}í¶Ýö±Ûv[»ÏÚ}Öî³î.ëî²î.ëí²Þ.kí¶N5vE§ºûNÿ''_÷¦,£ó+ fÕÌèoüì -·8ýøòoôü%û8ñ:«>iÕ'úäÿ4ó3ÍÉ>WÎ2§¹3Æ|ý¯cÌhÝ£Øk¶N±ëWÃå»ã+sô7F?hßÛ|ò¬Ö.ØdÿOSiÙ2H¹2ä)5Ëu÷5jͧÖ|çyÚ]ov·¥¬Ø wÛDÁùÏ T/"ò""Ï¢ Î÷´¨È;ã6gÜæÛXµA¶I¶I¿µé¿d<+7°rÃ\_3ÇÉîqjïìO;÷6VnÍ'P½
ê-£?Eöçíºò-vÜcÇåáôP»
Ú-vÙc=Tn¡r [¨ÜBå\·P¸ÅJ=n¡n u[¨ÛBÝQ5(ëîSýx OG UpNioÔ¼èYgÑÉÅfýI¬?UÊarX¥>ø3¢¥W?¢Uº¢J7¬Ò ë×GT»¢}D\_ëÉGT·aÕmXuÖwè»GT¶amXß«lE½G¬Ò «4êËp4F-ßk'óÔîXÍ.÷u¯Y5fÁGXðѬ2FµH%¼/ Û§Jɳ£Ce3OtuòQ¥ûìtòÏ\GÊ'pìèOåÏS(ñtvñzù§²>á{Û££=+~Àé~ôäßÒ+\ßtò'=u£k6 p:'p²'Þfµõô¤ïfún~ódníU:h;h
+tüu_1ú¿ÚÒv3mÿfBßìy~ô§£:m7[½¶ß<GN>¬õè¨ð n)Ö-ź¥ØþO Ö ©[ÇTþéZvébxcÇÌG#Ë¿Yîzºu=ÝöõÝM·î¦[wÓ»éÖÍtëfºíç:n]LlOÐQtë(ºuݺî(m7¿³ò+XqÕöZíe«½ÝWéÖi[ìqOüö·ÐÙ:»sùõ§è° tÒp ÷ýÕJ+¼Vçù\×'uZ/º¾Ùj=Ïã/ÖÛê3í>¿=lù+¥Z;ÕÚ©ÖN©vJµÛwÛÁIµS¤"íÔh§F;5Ú©ÑNvj´S¢íTh§B;Ú©Ðï[q«3nuÆÝÎsÆθÑ7êTË^·Ñy6ê\ºÊ¢³lÕY=p£³lt:É¢sltαն:ÃFgØè Gÿå)ÉïF§Ds¢ÃÜè|W¢©áh®Åt\aN´è÷½áöhöãu¼n¯xWh¬x7Þ÷ïð>¼§ã8Ä8 gãCø0þÁ98ÅÇðq|ħðiüÆáqþ çã3øg|ÿÏá|ã£c\ OW<þXñÇZ¼Ãux õx9¬©|0ÜQùFçëñ µò�B¸½ê°0·ê0§J]¥Ë®ÒeWcqÚÃU%éAo¸#õn|
¹©ø!~Éá¡ÔU {jVhL55)Oú´°&ýN¼+ü1ýn|gzþQ|+ÌIÛÓ³±�í¿í³twx(]Änï x>n¯NÆê$Q
tÕ:Åê1È YÔÃp8ÀøHXS}¾ëñ÷\g¸þÆuQøcõhã^cÔ':"¬ì±8ïÄ»ðn¼ïÅgñ/ø.Àçñ|\_ñ5|y¯jsX\_ÕhC»×\_síDÉû=èõÚa}\
4Æ cqÞÓ@xÇ¢ÔY®r=×õð\ïßpYÇsæñy4'>8#8Ëõ8?LK|&\ø,¾èùø031!\ø. éÙ>üv¸RßöÙäw]¯ õÉÉ¡)ÙU%¢£u½Í¦òMQ&¹#,JîÔtDïJ¾æÚYþÛ@®»¢#\¯¯«p5®ÁLÅ4\é¸×ãÁ0Q¾(\_L¬ÜZ¹94c6#-( Ø zòöé¼}º\3êðÐÌë§Ê1«vEùeü2M~Xµ?:<ßJ#q Þ&¦ÞãzÎÆÊ)Söø²0Mþ&L?¦ÉåÉòÇùcB/¥¦/¥ææÔ}£ÿ¾9ý¼'ám8E"mªH\Ò¦§'E¦;0Ûëº>½U4MO?æq»Ï¿íàs"çns·ÈY$r¥{¢1é»}~ÀûüOMOGV«ÆXcqÇ 8öZm¯ÕöZm¯Õ'ãí8§¸È½.Æ%îùu¸>4©Ío+2ßôpYæz¸É¸É¸ÉÜÛ0 ·Ãy3w.Ü{p/fcæ>Üy@ÌCx¿#XZ3 ×b:®Ãõ m mk~ñ]#¾kÄwø®±Ïû¬±Ïû¬±Ïû¬±Ïû¬±Ïû¬±Ç{¬±Ç{¬±Ç{¬±Ç{̾7:ô1È ¦ügI¾"RvÈFåGå¿=rLjÙ,;úH!jÿoÔ ;úì³²YVPÐt�@APÐt�@APÐt�ïHïH@Q'PÔ uE@Q'PÔ uE@Q'PÔ eÉKeÉKeÉK£ï8 ø.ÃDü?åã0^F½\F½\F½\F½\F½\6'MÇɦãdÓq²iF6ÍȦÙ4#fdÓlM3²iF6ͨ»ên«ºÛªî¶ª»ên«ºÛÞ±ñ(VEÇɼǩ¿±ú«¿±ú«¿±ú«¿±ú«¿±ú«¿±ú«¿±l=I¶$[O:Ͳ]èF»PBbìF/úÐfËì eö
2ûB}¡Ì¾PV"«OÕ§ÈêSdõ)zú¼>¯§Ïëéózú¼>¯§Ïëéózú¼>¯§Ïëéózú¼>¯§Ïëéózú¼>¯§Ïëéózú¼>¯§Ïëéózú¼>¯§Ïëéózú¼>¯§Ïëéózú¼>¯§Ïëéózú¼>¯§Ïëéó\_ÆV|\_ñÜr\QN%Ê©D9(§åT¢JSr\QN%Ê©D9(§åT¢JSr\QN%Ê©D9(§åT¢JSr\QÎ,QgXmXmXmXmXm¨3KÔ%êÌufº?G¬Ç+QF˪bYU,8§üoT]ÿÁõüp½jvjvÁh5ûv(%.ÆxÕíMU-11T¶©lT¶©l̳W%É'Ãsɧ¢CϪ~¯çÌé£cT¹¢\Ln6ßÿg¥«RéNýE¯ïRy®²ª\V˪rYU.«ÊeU¹¬\U岪\V˪rYtQ']ÔIuÒEtQ']ÔIuÒEtQ']ÔIuÒEt±rv+ç.îÃýùx�qç8s¹«ÎÜUgîªSE3ªhFͨ¢U4£fTÑQE3ªhFͨ¢}f¬Ïõ±>3ÖgÆúÌXë3c}f¬Ïõ±>3ÖgÆ¡T9a/öa?^P§¨ÌSTæKUæÊ<Éü7ÿåÍyó\_Þü7ÿåM SBÁP4%TðqU;ClR( \ù¥\ù¥UöTeO\ú8=kj(Tð<8¡ $£¬J5QLEÁDQPù³\ÖdQ0YR'úì[p×ÞáùikMÁ8A6õïóAÝÁ¦a!kò(< &É£ò(< :Kuê.Õ9\GSòhJM]+19×M×M\®¸\1Î<×Iät¹Ô£ilj9~?úWƦ^pm uº\-ͽùÔp4VÇÓqät9GÎ,\g®3 ¯6 ¯ÖäÌëÍÃués£¸Î\ bsAl.ͺ
æØ\ëV&éV&¥ÿ5ÒßÁ
aù N_æ±Jÿ?åä?sZͱÙ!6;Ä:'cÍqú>óè_u=óDlͱy"ÖMÑetAÇ+bÐPÆl-b³El¶Í±Ù"Ö!MÒ!MÒ!MÒ!MJïtï¼¹>-×ëfëfëêê¦è&éê¦è2fý¼Y?oÖÏõófý¼Y?oÖÏõófý¼Y?oÖÏõófý¼Y?oÖÏõófý¼Y?¯ëÊéºrº®®+§ëÊéºrº®®+§ëÊéºrº®®+§ëÊéºrº®®+§ëÊéºrÕ´§3ñPW}¾ëÞy~1.Á÷¼v©ë÷1ð£PÔ¡åth9Z®zïÌòúo|vQX]½ØãG1òc¢h¬.7ÆÙÆêÆe2_;2_Á×ðpÎîÌ¿z|M(e¦þÒéÍôøg¸)Êêø²:¾¬/«ãËêø²:¾¬/«ãËêø²:¾¬/«ãËêø²:¾¬/«ãËêø²:¾¬/«ãËêø²:¾¬/«ãËêø²:¾¬/«ãËêø²ÿ;¾ìßt|GG·
V\}®ß¢/Wü{tMÅDÿXqQôÑ£¯'ξ}-ùÕðéä7§O
ɧçÛÃKzã2\òµpG²+¼ìNHÍ[»PtRtÛç£ÇhmØàî?ø×?äîïu÷÷ºû'+Æ!µµÃ¦9SÙWÃ9VùU&'W'kðÔRò°RÛ|.¼|>Üfõ¬<ìV?Ç곬´úV>ªN® ödOn%7UÉom -ªV}êcáOöö'ü¦Ú¹Þ§gûô´ä|úaþ:ºÒ7®öûFÿ¶ãév;]5êýÄçTòña|Q2ñ¨>ùùðÃĶèìÄ |Tthòôðëäê(«JÒåÿãzrY³9ü^®r÷NS©§¬ÔÉ3iÒÉ:ÝNUôú®ÐSñõ¨2¬ªBÕ jE-Á¡áÉè0Z¢sñÓ°<º?ø ?Ç/ðKÜ[p W
¦èÐT-IT¢ )¤Q1È µ8 ã£p4ÆãðV·ád¼§àT¼§áøBØZñE| _ÆW0×ázÌÀLü?Å ønÄMø9n[îÀ¸ wãÜÙaKay,|_ '~ _/ÿ«øÙë|l9KøØçùØëÉ¡]Éa1Òɽû´$÷TòõÉ7'¼qUºSáÓ鮬>0\9æ@Ke&¤ktVfÃ'k½~Ï]VUNÆU¸×¦b®Åt\ëñ«ÐRùà×XßãQ<ßb bãwX:ü+ñxØZ¹ OàI¬Æ<
§ñ Åsxk±!,¯Ü± ÇÐVl Ë«öU©$øoª<:õH÷à ZRv½%lMÝ9;gê×;OÊyRÎrÔ2¯-Ç ÔáXåõ'ð$VÃÞSöª÷øeüÙã¬Ç+ØÍaKªà½NìBú±ÄpØ>0cÃôq8'àDZÒƤ°<ýcÌÀLÜñphJ?æ:W¿3l~oh©~¿ë\/Àç=þfØR}÷/Æ%ø
×çx}.îÃýx ûÃ1QØ:æpWñ5F\9'ÌE¡ËðC\+!Þ3=#Þ3=#Þ3=s+nÃ,ÜûÍÜ»p7îÁ½9ûp?æa>3fÃøÁ°¼æC¡æ³ø|àóø¾iáñk1×ázÌÀLü?Å ønÄMø9~_fÜ[qfávÜ»p7îÁ½9Ͼ7,?dLxü jãQ¥Z±\æ/&£÷Ë˯G÷DSÃýÑ4\é¸{CÁü\0?ÌÏósÁücósl~Íϱù96?ÇæçØücósl~Íϱù96?ÇæçØücósl~Íϱù96?ÇæçØücósl~Íϱù96?ÇæçØüÿu_Eýÿê®$B¸7°è¢®¢«xp«( .«"¢ àº?t=@AÁ 9 r%@ G8¦þߪÄrÀêþ»?ÕS]çëªWßz¯ªûMúsúsúsúsúsúsØXábßÎoutÖè¬9ÐYs ³æ@ =t2ôÎåÐ;Cï\Îgêlû~dô£MüÞÙl f±)b)ÕÇ|¹3Ø8èpS ÃM7:\t¸èpFJþý):S:S:S:S:S:S:ÒèAS §LN2:Äèaè9Ð Ðr ä¨Suº:ÍÚãÌìodùtÈÙéÓ! §CNüüüüüüüüüüüüüüüüüü¼y5òj2j;e?ÿÆjCÞ CÞÌñ'a<ݨ'CÆ r9dÊåÞHí³CIzS¨\2}¸4¸>Co"YåÌëãÄ<:O̧b!µS Úw®øÔWÔL,¡«ÐÖWA¯÷Ab¸º}¢XAÐî 9ÔÐL:òUlº å~[Ë> 5}¡g!ýs¶Î÷w¤ù°Å¸[jìR¶¥Ëî¤veÛÓ=-1:.@0^¢!-1[@è%-çc¶Ünmï0ÿFÐ:¸»È®)ÖDÚ& ÁüÁV:)ÎÄÝRj'LB\=<«±úv£þI ¦¶ ÿKy!ä5ïp÷RcnL»u¸ëO!ÜÆÝwÔ$µ#§à\8?\�.çÁ
àPcWª!ºAÆë×Ï4ràç3¿ÐËäj'À
»nÜpÃá7nÜhj]¾tövÐÙÛAGo½tòvпÛA÷n}»ýÿ¤Û<Ô´O±U,DO3ùBÏt»Ï>m2t}TxZ<{ÙÏÔ-£h^h¿nHÕº^ÖÆ\wÑ\_a¬¡:C¼HmÄ$:õÑÓM ÉÌçQ+ÙZ µºS=ä¨zZ£7SjÚiê·5
bÿkòèÜ=¾~oÆïpØÏúWÈÈ9YþYE.r rÌ?¡ u2R¥)ÃHKÉ
E[ 7 DM¦Oêå»sÐëñ@Üe¶¼èÁÈ
2DìKÔÐá Ã@G.\�¹�:rtßÔÙUg/P©)ʶRçQÍuö�fõg I|©Þ êrñap\ Ô½¹¡Þ ê=Xn½AÔaþ¥%¢^JÜsPbJô£´Ý±§(À8ëPc/°$ù>p3j!§;ȹ9 3Z"¦Õ3£".§Íp[à³ÃåÃÀ:t
ær£n!z�-zRoÑ¿7ãw�t g¨!F/^¤sí¿~/.´oúæý²m
^
1-çpGZI-#pùérÕ ®;\/j¦&ÁÍÛûMpp Så",¿ûA±ÿ Êá²SñÜ@Ù©xîT<·A ÏÀ³fÕ¹îSäø96#G\rlFTä8©@óVËy¿è|Ð}97Û\+ìÿtC}ÝÁɽðÛ¿CÔ �k«ï>µÿ¨cú/©BrÑ]á»Ñ c /Y WÝùn+èÎFÛtØòÛFäÛ|î¢dtªE}õnº îv¸Aèý®èÏn «Üp¦I .ÙÎMÛ \_nG);0O^H5} z·/n§Þíô�wܽpCà¢Ü¸ØAÉé(9] S æg 3ÁE1ìÓ³ÑFÛôV¯ úòA\_>èË=½YS^RÖ£RN (å�J ciÞE Ìÿ¾|ÐúòA\_>èË}ù /Î ¾Ôn»n8u áFÀEPcZå^ 3"ýä o©VB e;àRÿÈVÔÊrÄ!üæ£Öpf4gj@;ÍÅ9ÒÅo�T-©�O°\×zíb(%¥DPF Ù¶nrç"w¹5rfÇh8Å´SdhÈ@îFȽ¹÷ñúðñp\rÖG@KJkÒÖ¢´ýÒ¯Wا ¢=J¦¼%MÿgfQÍQAбND#×AÔ½Nào®¥H
úLK¥#EÊ42v¡uê/ô~¬»þ±im¿ m9ýg<Á~�V²ý2'¹ÝñÇhoSf;SL"¿¬úR( SQZmä©¡.üõWq ×÷M×qÍ0Hj#6 ¿MÐ'LtYõ§¢Ú¨ÉUáõÞ�áÞá(½RkÇRLY #vLFHM¸ªúr ʬú8èãȵE¦!¾\C7F&k 3ó¯ä(eh5OÈe-ÐJ¾X)&÷:ÐoËFk¸hnçM«ÞKÍ)(7ÏR½_uÕ5Ï
øúOC|CÄ7FXÄ7E|3<}Så´&\^"h Y}YÏ\iê#MÀ5DFHÓi"M3Ìl¦<Û®):LI #:<Û¶ qßضàAЦWHØgOµszÓz>w4GnjNñUå Ú0Úï(¾Àh?Bå äjAêXüØ&TýdñJ;O]E>AîæTíDy¥gèäð z{ÛU;7Ë7Õým@Ò>@:@µÎp$¨v©(lúôª¥ÕÚJ_dµÐ¨P³ôGrjÊd;©/P- ¨ÖV&Eö£EÎ@9E¦à¾>-ªZ¢U¢UÈz¯tiHÓ�®!î!]c¤ktM®¸ÆÍÍÎÕNÿõùªCÚM¤ÛRŹAÚ·ÿ-4õ¢óYºÝLO±[ð{+4÷®zª¸ºÈ z$©öêN9NªE6ù¤Õ6´ðî½¢;M~û\¿g}æßí2à|µ
Nz\]³u¤k©%]O7 ô&ÈrÐ4®¤gèmºæÑÜ}óYúVÑDZsv2²P[¬6«M¿°zì ZÎ:±Îɺ°ëh ëÆzÐÖõ¦0»õ¥\ÖÝC{Ù6ö³áLeSqÖf¯à¬ÃÞbo³ºìs¶Õç-x+voÍÏax[ÞµáñvìþWÞÇ/ã±óùßxGvïÌ;³öü~-»\_ÏodxwÞ]Î{óÞìo¼/¿]Áûñ~¬#¿ßÃ:ñ|(û;Æ7ð'ùÓ¬Ï_dýùdþÌgòÿ°¡ü¾ýËW±I| Ïdoòm|ûçò]lßðù!Ïp-}!¸ì+¡D-ñ"ý(DûY$T¶L4 ÙÑX4akD3q K§3Ø:q8m-E+¶Q´mXh+Îg[Ä
"%Úöl¸D\¶¢Û!:.,G\'nd¹¢¸åþb�~NbÁ1JJ¼(&qW̳y@|(>äA1WÌåøD|ÅCbXÍSDØÁýBóÓ¥OÆñ62I6çíå
òBÞUOðëåXù¿K~,ðåOr)Yþ"·ðé2[jþ¡/à ð}Ïã?ù||o¹ïW¾Ì÷o#_ãËôeòu¾¾|½/Û·oðíðí|{|{xoï�Ïöò;|ù¾|ã;øøNG9q|¿à$ðèÔàÚIqê á4pþ"ÎÙÎÙ¢®ss¹¨çtqº³Î#¢óOçqÑÃyÒyJôvÆ;ãÅ-γÎDq«óó¸ÍäL·;Ó颿3Ã!8¯;¯{YÎ^gó©æ,t¾£ooÅ£Îbg¥xÌYí¬t']<ï¬w6,g»äìv ÄE7RimÕTµ_«óÔ
b¹j¯Ú5ê¯êrñ«ºR]%Ö«kÔ5"S]§®Õõêz±EuS½ÅVu«ê+rÔêNVw«a"W W£ÄõzXrõ¸zBJ5V=%5^M®úúLTSÕTY]½¢¦É$5SÍÉj/kª¯ÔbÙ-S«äYjÚ#ÏVyê°ì¬ ×¹MݦòF·¹{ª¼É=Ó=Köp[»e/÷<·ìí^à^(ovÛ»íåîßÜ+e_·ÛIös¯r»È;ÜkÝ®ò.÷&÷&9À½Õí'ïqïuÿ!¹ÃÝár¨;Ò)ïwrÃÜ'Ü'åîSî89Êï¹Ýòa÷Ew|Ä}Óý·ãÎrgɱîlw¶|ÊÝãîãÜ}î>ù{Ð=(Çû|r_ú¥èWþ|ÎïùkÊIþZþZr¿¶¿éOó§É® toúúÈÿúúÊ÷wîîÜ-? Ü#çî Ü'? ËyÑr~àÀ;raàóÀwrKeà7¬lû©2làK N ¾ê{&8'¸À÷JpipïMOy)¾¼Ó¼K}ë¼½;|½»½û¿7ÐìÄ{C½aN¢7ÜîÔðFx9ÉÞï'ÍàMpy½çæÞÞtç4ï5ï5§7Ó{Ç9Ç{×ûÐiïÍõæ;yy9½
ÞB§÷
÷ÓÙûÑûÅéêðV8=¼UÞ§§împúx¼]ÎíÞ^ï 3Ô;ì8#¼HÑ!Î#!rGCn(ä<J%;ãB)¡ç¹Pj¨ó|¨^¨±3)Ô4ÔÔy%4:4Úz8ô3=4&ô´ózèÙÐsάР¡Ù¡B/9ï
¦¦8ÿ ½zÕy?4#ô¦37ÇÅ9Æ%ÆÕtÇÕ«ë,;wØù
x�ò;wIµ«©9¥ÑI:ô<©·R ÿÚ2SDôý.Î\=wWëîȳ¾ìX|¶Þë¦ØÝþRùMìvó÷8UF={á/Þá>+²5$Zy@óBº\_u>üfò}fI ¦:ÔuXÿ2ð´YåÑXÃE©/ÆJ߬sô"½%v·§Tí;àÖé z¹>¨¯$?ÚîTjP,>R^ezú.%üN9ÚK4öuý:ypE}xTîp[t:ÊX[䬦t|õmì×z^þï@o/»þ·õkúük§ÏÔCôøµcáÓÃSwD£³ÀAßè@úÁ´^É\Ei,§)zQõ= £ì y³8WÄBòðä{Ðökõ^Èûñj^(ª]ï°=´£0u©ü9zÆX¸°ÅÍʨýýxòè¥K/q÷wßU¬ -mú§éÕè?W¯.§æÅÆvK:·Ôïè¿©0M%óo5Üax¶TÌÊ äÆéÇoÎÑãYßRüàý¡Åõ¦ß{è·,¾
v-}¸!Wϳ¨YA¾(£=çª2rÇVÿR¥ÜïÙëj'ýøKêßËt>øho¥kðÛ îï¶oSôÅ×/#Ï)8ëã<¥oÄ~FÏãäoYfþXëKööàçN½¶Ñ)ÃÕmøs6ºþ\/Ð+Ì~üÅüOQ-àÿ ÔÅXØ:Ì óKcqQübþ yé êÿìXX&ZoÙ±gÕú-G¿ü~ ÏÀð÷õ»$ôÜcæ? }ú!üéXüwú[´ÿ÷±»Òø}¸,r×¢Îd$¡v±°Ïô'(áÿYÿæ²Ã#è1ú}î«»ÄRO+ÿ ØëúÿôÏzE±N=éQß34Þ|3CïsgÓ\Hóiµ²« mè+ZEçЯ´
:Rct#ëÃúÐ hô§ÁF§¡F§ûù]|�=�}| äky&Ù<àÛùctsË÷ó4çó|zÆèæ4Þèæô,tó ='êú4Yô=é%ÑGÜLSä9V«é_¢/~t>r>¢Ï´ÄYëüF?;ÚÑôÑéh¹ÑéhºZ]CëNG ÓÝ@NGF§£l£ÓÑv£ÓÑ£ÓÑ!£ÓQ:ÝS Í=ËõÌüF§cñF§c F§cÕÔ 5U7:«at:Ö:Ýv´9ͺ¸õ±î®ëX/×sãØÍn5·:ëëÖpk²~nª[ÝåÖsÓØ�·ÛÝç^ä¶c µÝÆ@;ÃA;{ 7ú{ÐèDlÑØÈàÁ ìa£é°I^æ{ïxï°¯½Lo[dt ¶ÜèìW£k°ß®Á6]m4ºË4ºÛft ¶Ëèl·Ñ5XÑ5X¾Ñ#XÑ#Ø£Gpç rW#®&Ä;ÌÍjË1Ìr Ǽbý <=
f"äuÞ ·1KÍ?9ðÓ§u««àªÅÿVPVäà²Uª¥ß ]£ ±Lð\Ê¢Ýñ{p6¤½tÑAé¡&GV³Y×r¤°éYôÀý)�_z/Áë(¯çë©:ßÀ7QMÁ3(
g_ëX~mù5Åòk ˯©_«sÍ5Uÿ) \ËqÅA5À» ~t>Õ~ðqåãÚàãÔTô777÷ÿfðt3ËÓuÁÓëÉõr q¹Uf#³e2WæQ=¹Oî§xy@P}yÜßÄrËýu-÷×µÜ_×r]pÿ_)IuP(¨.UTa<ø0®DHGÕ!T'Rª³êL®º ã¤ÆÉÕÈ{ Fß Y¡ºc&c¦;5P=TOW½T/j¢zcU³£¨E £ènäê¯îC¨¤WÕÔ2T EÉ÷c¤1ÒD®jGªH? c/dÇ3ë)H3F=zǪ§;^GÈ5¹UÏ"ÍsêELR@Éd5!0ãå¼¢^A®ijg¨(g¦³Ô,¼£f#ï»ê]´Ã{êC´ÌGêÐ9OÍCÌWóAÕWj¨ýF-FË8STàIµZ¥£´µj¥©m²Ye£®mj;5T;TZr§ Sc«rQã.µ4ç©<¤Ü§ö!v¿Úðê�(9¨¡üÃê0JÎWù(¹@PuuDAíA^´ùU×Gu à 4Áh+ÐW ®@\&¸Mp<ëw q)$ ¦3B0e®#£)Á Ëò«k(ü5¸ Ê0(Cµ2TÝÛìm¦$o·
BÞVo+%{Y^b³½lJñ¶yÛ¨·ÝÛ Ø #}®4»¼]H³×Û ·R½ýÞ~¤9àDÃÞaÄæ{ô"¦Q«üU$®¾C@1jü¡�ÕCA¤ôB!ª\«¤P2¥t£d [®µCu¦^¨>%
ÒBi(§A¨!üB¾q¨1üÀ>ûòrèÔ2-4¹^ ½gf¢Ì×CoR $ RACJ�bý'p > ádø§�
ÅA(øü³éc\?!pÐðsø¿ ZÀÁ@ÌUÀWa×ï]ÉkZL±8X˪ÅAųx ±n¬®ýÙ�\ïeqÌã:¥Pò%ý@ɾ¸ Zô[³ÄsxU³8hq°:?P¼EÀ!
¤D @¨è&ºQÑݾÉf°¯®Å¾ú¢è
ðÞöí6u-Ö·[©vf�æ ì+ E½TzÉfÕãóbu1Fï%êã\u90Nã:oÐMXts,º¥¨.ª B º uº×ëTW¤4'-º%[t XtKºõ!OÝ¢nÁõVu+Òߦnõê«A:×"] tÕ Ò9ã\õz�y«áH_t£ábÜ#êQø Ò¹éEº§Æ!×ÓêÔs-êy1Ô¨&"Ük±/Õ¢°¨'ÕË@=C½éj:ü¯ªWh¯©×Þà °8Z
ÅA88þ(ö}ª¾ÿ+õ3®û\_:üõjXÔK¶¨°¨WÓ¢^E½ZõR-êyj¯Ú\û-ö¥XìKa_0NXó\æ2Q´ <@þÀqIÁÀhS0ðpàa<xü§xpbð%q¼Ào··-¾$XdI²ÿ w)s)ÕB"$(h¢(ÎH¢Å$ H"üAªjj"Á¤PÝP]×aG°#ÑbGÅj;/£Ìi¡iÈ5#4ég5-jpv×s¶þµ ]I7KÎÿÿãÐÙzq±»eé]fÇ®õU¶ìÍf
ËjÞÛûµ
uÚëÏ1í3ÇèVM×:«äNùõ®ÐéûOáÉ=tGhæ÷ºw©Ùд¿úºLQ99GßéÝö ®ÍÐa¸¢½bhR±ÜéHµÌºGMøb+
Úõt¨)^¯G7Ù°e.èí¥×æô½IÿR»U= WÉKÞñãêbë ]ùsÕËzCéUÍu½Sn®úUû[Wÿ3άé·à[KSÈYfïÓK Ã+UÏfË£¿ßU0½®X§ízY+ß}AMqµoEû×®Zg®ò8X¹z¿.;lÖºôé·/õ?vüÁc¾z¯.£¼ j¬w¥ÿhN[ ZL-ó�6TxñÄç£Ê+AUñ±WÁüïëú½Øþ@¦ØÐL3»½«$?¬6n´òCM,9IoÄï¬Xª°Ýoûnά+×ÉjQáÚì× ëepSz¥^®°á+¢RÝѾ©ò¢|[;;êÿ ¹KÏÐôf_, =aqWz×Ìké½Ðíúsz¥}{%3Dß±£³¿¹>0¾ë9Q+Ïì~'zêP÷¾þùQìîký6÷æ?(ö5P¢P ÎúþÃèþY\©2¿Õé
±2Ì],¼:h]yjm>RýkÑ]¡î²Éø õʨ$nm±áè;"±ñ³Ç"rO}µ½[Hf7ï>¸ûá 'c®»?VJ±w[ÐóõðP{³©_Óàû£ú5ÝÏÃÓ^C;/ÔSõ[sÍ }²yz¶96k¤ê/3K¯V¹gùbr§>uKgÇ{Ñ[A%g);Oi¾VòÝhß{(þÆÅ%ßXù£»¸ö ¦åSb¨ÔûWÄQR5 Þ[~ÚÞ9ineòFѲVã÷;ÝE)·8½úe=BÿSO²þ¥à÷WÍ2±y(/îÓ-8±zlI-¢o²Pz+fB;?¢O·dîh¯ë]9v%Vº®ÈÜÅrÿíUÐbpð§ØÝØøQýçç²}»¾Mªç·w#õP u¨D ç긧ÿ¡ÏÓ£õýúΨ+?¦½1Lê´Eï¾Z2ödzæI(Ãpïª(ªC¾-Õû6>Cÿòû,üç f-Æ]óM±HSJºýîïªþÑè}¦øÈ
|5ïϤçØFÛ#;EßtÕ ÀèÆ-´×µúÝ]? ßxý[4¬u}{ôV²Æ¼ïyýïE2¬wÝOæ!oÁ¬wV,Ê{Gù¸y+ÈQú]»¶¿£ê5;jR\t@:aÉU?{2()§ÒAº=áuùÔKåÕ Éö¿½û$µb+Êe~qtýÊÉì /-#¶¼²ÍwT súì ÿ¦XHaçÛº¢«ØÝ¿YHù^«U櫬f¦\Z»ªßÐó¾ùD[Ó\ZDGËRô¾QùúJä¯Bú»+ñ}ѽ}ò¦Sá¾ |½wºËü6¹<[íªÉ-Ø»¯1ö¢È8|igxº¨bßk¿\ï?,7ß[Z·?zo¯±Uóã£CìYê|ßüµ[/³n\ÕLº-¶´1:¦-¯ÝUyJËyè[1m]÷Ñ÷ë7õ+Ön@Ñ;=º£~¿%ýÇHÌÆc×£#eí\Gw Û]þ.NUûL õÈ{ Ñé¿#ÎAÙ3>W\_oï?�¬Ò=õ"s¯êçõ7fÅÜÆ=W¢ìu
ᢨ ÖWÆî¬ØÏúßÐ3ô@ðÁTHkó0ósôGúÃجmVç©
Ýs¦ûÛ°èû¯@®~Ùô±PôPµ }¨ðkþJÑû~ ºÚ¿bwKlÝS-Î/±mv_ßÓyú úÕ~ì ]ùZÿ¬ã¿ò5véZ6"VtßùÏ:ª²O
ÞIÅV,$Tdî©Næýk¿µîfónÔ±ÅÎ&µé/z%F¨9×éõú<~äéè¼ÓS1:£:UÍØýû± NE_LÛðwóöÝ =ó\lR_¬{ÃuÔ·Su mh»T¯»êØ ú;ý}[Øí6Åô×Ó¨¹9O³©¿ºQ6]¯ê¸¾Ut?Ïèr%Þ¬¸.æéN§s©µÓÄÆö@ä°3å§únýÃô(ý¨ñ¡Ô±%ª¾vwèí¯ïÅóßko\øú[Ü|ÔÎÔËÐYèôsUö¬+£:^uo+?M©<9ö�#'Xn²Üü5î¥ö+ï\ñt¨ç´¼;vÝbvì¡+g5¨¯µN7ÌZ§cÓeÝXOÀîdwÒóÖ.Ý lKÙ86fët4ÏX§£ùÆ:}j¬ÓÑgì ¶ò¼%-áyúÙX§£å¼oG+u:Zɯài5ÈQ:Æ ßøþç3ùLÊàoòÙÉçð¹´?¡üS¾ük¾vóÅ|1íå?ñ%ÇæËh?_ÎÓA¾¯¢C!:,D" s¤
9²æ|¢±h̵0çZ«rAÑF´a!kU.ÎZK°Vå=¹ê¢èÎD/Ñ%oåX±úÆRÕ7v¦+°nÆê»ÅXzc·Koìv_¯ëçKòÕbw{oì^ßo¾Ml¨±÷ÆF{ol¤±÷ÆF{oì!cï=îÛçËgOoìcãM26ÞØ4cãM76ÞØLcãÍ26ÞØcã-46ÞØÏNOçq¶ÚXwãÌXwãÒXwã>cÝ+cÝ»Îtg3vÝx¢±ëÆ«»n¼±ëÆ»n¼³ØYÃO1ÝøyÆ¢oëd9;øÆ¢¿ØXtãE7~µ±èÆï2Ýøæû8>Êå.ç£]ÇUü!7èù#n¼ÀuÜ$þÖ»uݺ|ÛÀmÈ4×øSÆg,®ññnK·%ÖØ]ãÝ5þ±»Æ_pÛ»óIÆîÉØ]ãSÝ5þ²±»Æ§»kü5÷v·aì®ñ×ÝÁîþoc}¿e¬¯ñ·õ5>Ë}Ò}Ïvǹãø»îxwÏX\_ãïëküc}b¬¯ñùîîþ©û¹»ç®rWóßÜ\_ݵ|½»ÎÍÜmî^c¬²ñÆ\?èj?ãU6^¬²ñ#ÆþZþz"dì±êþþæ"ÉÿLQÛßÊßJÔ÷í?[¤ùÏñ/ø/ô\_"ú;ø;Óýùÿ&Îð\_éï(Zø;û»Vþü7³ý÷øsiÆcÝM\l¬»+µ6q¥±Ö&î3ÖÚÄÆZxØXkO¯ Þ\f¯öÄ|cM|å)/^ühì´^wï±ËØic§MJc§M\c§M64vÚd c§MÖ1vÚd]c§M¦;mò4o¦7Knì´ÉÖÆNlkì´ÉvÆNloì´É6y
±Ó&¯6vÚä5ÆN¼ÎÛäeÈnÆÊìa¬¬ÉÆʼÅXYw+kònceMãq®¼'ÎCãä0cYM>w îOñL&Î2zqÐø)UÃ)(ó°¤ÌÝ>ÌêMÞ§¢f]:(éOðÐüÏÃEö0 bÆYÄb^\7à¬Üì{ÑÔúC/ä0ç%4Q z�g2 §Q¨y46ëQ-bqj¿®Í¹g�s!¤9kN-Ø)ìTÆNÿtq-Å-Å]p½|©µZõ.·²¸ÜÊò_Ë#>=AÙ6e> ¤þ TT×½ö÷ sÎàð!E$ÄBH%hB%ãµÖafaùbÎ|2Zc±kµÖk1ÖZk¯µ^c|¯åk¬×j½ÖZ¯µ^kµæ>û?ÄÚ®õ®õ¾ÉÚÏìõ?ûìsæÌ°÷ïa
'Ó±RU¨Ö¨¾ÉV aÕE«ö,ZµgѪ]Uûûèí.ÅÚý>ö#ª#ìÕªYµê8Vó9´«±C.ÒNkºÖôtZÓ3iMÖô'iM¯¤5=kú÷ÙCêõ{Pýú]ö°zVùÖú±Öÿè ¬ø3°Bÿë~.û¹´î?u\_f&¤õ/ Õ¿VÿÇ°úg±¢i ÓØÌìlVÇwô±°Ç±<-Lxga?Å|?ÀYU UÐgÁÑ9 s sæb ö(öTøßZ¿@kÝ@\_ýý}uýMu=ö�« j3v5,Móf}Aó¦fMÖ¼¥ÙȪ4ok¾Ã¦j6kÞeÓ4;4?bÙØQ~ÌÊx(+çû «æû ù¾MÒÙÍó0Ã##/æcü£b!Æ?.>ΦEbê3ÅL#Å,U|B,ÁüOOÜR±³ÍgaLXsg³Ìw\\ëiñiÔ+Å\|F|3T5LçÏcd½XÏ´ ¸çÅ/á}5¯bþ¯z\½E4à\¢ óÅNV#ZÅn6O´N\Ñ%ºYè±z}¢MûÅ~Ü\_Tð^bóÄfaÌ#,YQ\ePÄÃU@�l:'�V xkŵl6ç�6 ð&Ã,[|KÄ: ~Kü«7ð´7 ß7³2ñÌðøt»o©¸CÜsßw²çÅ?ÀÌ»ÄèqÎý±øcÔ÷û0ò§~üxG.b #¨ ~ÀJÀÿãÇPùPü#¿ÀÈQq÷óïIùHüwxJü%îù´x=!~,~ÌÏgp.gcùSñSõ[ñ·íxã/þãÿ(þ cn7ñ4þ,þ÷vK¼Ã¦qa³Á1)è§j'±rmv2®ÍÔNeÚ,m{Zû vÊyUk µ³µEÚìm±¶'´O²9ÚRm)f¥
eÚ2£åZxG°ÑØSÚm®õö¯ÖVãèí\g ¨83±2ÎLP0Ì3AÁLP0Ì3±lÎLl:g&(=Á }0«æÌĦñ¬ZV"Íæá, '9AAN¬{ä' %3~êfi]êÁPÎE¡È ÄÏ=HôyúQ¥å'$'$£ú/!î/#î/'îoN:§ÿG@ÿO&þKþKÐ/N(ÎJÍfà <~eB%{"á ð%äf%TÃ$<ð,æç~ À«äxÀ"ò�õ ÿ! îßÈ2ø³ø§ñWjöøñfs4ïk³Zþºû2ÊdJ£L¦ÉÉÔDN>9yÏôù\øH@+ü @$ %Jô¯%úÏ.@ù
ߢ¹\_$JÄ???pC¸åL\_OL¯%¦Ï ¦¯'¦W"^K4¯%FÔ^O¼®%RÏ RFt^O\®%.Ï".¯Ã÷% rX<X¼~ËÅr¯+0³x=Qx¹µÄÙZbëbëùÄÖÄÖ §[O%¶Î"¶FôÆsV® JN\I<(¹N< J®&JG\A<(¹(ùy¢ä§ÄOÄOpóq¯×Pá|\I|\E|üxW¼ Båd\Md<d<}ÎÄ5ÄÄó´keµDÆuDƯ?G<8ø5à:àéÚ§µOC9?O\§}Vû,æäbi%&PX¥¥Q@)bI"ÖH)b¥ Úfm3®Î³ÄÊK£±(El2¥5QX6¥eS@)b¥ "F)bïKK£±$JK£±lJ(E,RÄûRÄJK£1RÄ&SX6¥ "F)bÙ÷¥ "F)bM"&P~p\_~@ùa)Fùaå5Ý&P~Xå Fùaå Fùaå½Hùa/Q~ØdÊû"å5R~ØËÖDùaÙ&P~ØKÖHùaM÷å Mùa<ÌdVÇò(Gþ¤VzLz Þ P\ëÏf²J©Xz~£D\A½T\ð-R4=Oî¥Bª\¡ÜÃÔIÏHÏîaj¥zéhôf[ }c¥Föô2Ì©IjCxMz G¹©t÷c 8+ÄÈNN®ÅNªÔ#90O¯Ô³\='y$R�ïûò6Ó)¹±Nµ´ZZ å>çyò9ÕÒ7%¬äsÈáÌÞÞFå»ÒwquîvêÈí¼&}_ÁYÜóÌÞÞÅ÷¤Ðù$KcÒo ÿ ÏLçò<µÒ éfæ§Jú«ôW¼;îyÉó¼Bgyjr;ävªÈíT$¦ÀáTÃáLb5äpêÈá¶ê4äv4p83á"¸Iæq¹ æa¹]î�©sßò09p,ÝèÛå8·ìF
{Gä~¹9�ýÉ£äOf?yþd%¯Ã¥Ï , ¼À<Ê ,¦¼À<Ê |òHså¿ÉòìÀbù3H ù�s å>DÙ¤¹&ø eæR¦e S6iQJjJê<\_° erÊdTxÊ!¥ >£O., ä)¹° )¹° 5L5ýzNð+'¬d2¦_¦G3¡YÑhÞ{¯û^´(ÚJ´5hChÐ6£mCÛ¶mÚA´#hÇÑN¢Acêà1jL:8výËh×Ðn¢Ýa¬E&¡¥¢e¢e£ÍßCKÁÿåµ8>WKÙDãçT¢Í¥c¬¥m~ü~éÍñ÷ØÒ¶mi¼>ñª£¦²ïDÛþù{µx»vu¢íÆDÿv¼
ØDÑd´ ´,´ÜøØP>g-4Kü9µØî=óøØ"ÇZh>´ Zl=¬_/T:ñ^×¢ £m8¾exÅD«F c ?ûÑÝ{/ñ÷¼m?Ú!´£h'ÐN¡EG»8ñzå¾×ÏÇ_G»5ñzv¼[÷¿ËA6-çï¯üó3ä¡þ?¿ªCµÿ¬ø{3L|Öÿ¿-û}¿WƯCß«ìø8ºîýêï¯÷æÏ«5 ^V?ñýÃ1ÿ¿Ñk&-ï?0ªv3RT®ìήéÎuçB7tçC7w ò³KõÛºKe»N/»Òµhà~Gwiõ½þîîÚsühÀ²ìz×Òóú}Ý çãý ½Õe¸¤?ØÝHºzúG¨¼{ ôd·z¦Ûë¶\glP úw»lWõºÐËÝ^èµneà¯:Msàþfwz§{eÀ§Kêò ÜnQw¯!"Ý�Zê ©Ý¡ÝÛ ÙÝ; 3ºwÜæg-Ýû ºô® 'Û}Paº)]1EäérºV)rKY÷he÷qEæÀªx}BóºÖº®a%«en÷É{Z×}FÉõÀÚ -éÚ¨ä¶Ìï#½�m¢þ¢îËÐ¥Ý× îPK÷{j³«Ã-N»Ø¨+ïÚ¢ä·øì©J>ÍV4Q Ú3?W^ lÑUu(¥-1{6éÏû¼ÑÕtíTZVÙ ÞìÔÕØѯïÚ£T·¬µVÞëÛçB7Úë [ìó¡#ö&èNû"ê/Uªù¹=º]ûZ]s×!¥¡eÝpO÷Û ý-ì¥A·¸ë¨Ò¨[ÖuîÁFê¼×?j÷áN]§
-'ìÁ{zÊSê:ºÎKÚöIc¤« GúÖB÷ COöméÛëQð³}íúvuö®qE¯sw]TLíûö@¯õí'åý}?:Óù»®(bû¾£Ø¡îº2¸®ºp×uÅÚ!õ =M¥~õ3ûÎB³ûÆ¡3ú.B ú®(V~ÖàZè-ôWtÝUÅ}סe}· }¨ðúà°nµM£x;æú¸Öù7êÖÙ¥c¾/kGúS M¾è"_t©¯jð@-¾rEág né°ùªGtëuçhÓW£DuléÊJ®¡|ÝVÛeMÏW ú(kxepg¼>¡Ûm9Ên-OÙÐó5ßÓU¾ÅøÙA}pÏîµ;Öúïõ}Ð>;tÏ ñù¡;}aèßÁýû}«Ý²ãoÝà!mÇDå¨o=ôW^<ª;l+Wvwòm"Ýúy×OèÙª}g}Û}¼?xªcÜ·kð¬nÔV£ì¸'õí½×¿;�½î; ½å;½ëUvj|§¡I¾sÊA~îà¸î´^9¢;g[ ïL÷ÿ'»¤×·5+'ul39¾«¤7îõó|·3º«¶eÊXga?»§%ý¢2¦»a3ZÎÚW®
Sÿ¢}zžzݾzË>½kß©\àg4ö=£ºÛ¶å²ÙìÊ5C}?4t iýr Ð6·rS/Úrå}CýD U/ÛüÊC¡ýéÙêØÇ¡åöÐûhýºr8¥Ï°
j}mE@2ÔÛoAØïB{4ÐÅ=IIk[H5,#5ö¤Îêómë)¤9¤yL}~O!úö¨»§êï©u7{jPYÑS¸¨/²dV÷,®ëidëKm\W ë{®ë+l[1~SÏ2ÌPÑcäÊx¼>¡Õ¶íúZÛ.ÜÛÖèvÒ]=v<^¿eØÛãÆîI}}mo Àp ÇO¾§{V@õ¬ö¬îY=׳ z¾gkà®áRÏö óës{vAkm¡¶c¸Ï«={¡7¸Re\¿Ð6(3Üî9ðÊëAØÖÃV±çX0]¿Äv:PÙ÷y?8E¿¤½ÞvÞW\ÏÞoÍè¹Íê¹ Íí¹Íï¹ -r0h©CÄ{ççÞÒlçsõVÛ¥@]k
Cþ'vdêôÛÕÀ|½×v#ÐÔZk_ËÕuO¹&½b»XÔÚèÈ.$
£«ÏjëkëãsÚFÚFø¼¶©¶)¾ m¡m/j}µ¾ø¶¶¾¬íªí¯h{iýñ/Ú@m ¾i�a?NýD|K àÛbï#Åø%ðìRü xöþ�< $"u#Tê!%}äXù ×ÍÒÅc_îoht+»Ò÷+Ú³·Øj"Mö¨�Ư¤æ[A¬a¥ºX´[w ©wÙ8a'5Up ;ìm6Æ\ãæÇñbvM ê#X ¥k¡´#l;¡¬PN°Ê à(Tg¡²PEpô«à&TÅ¿à«ø¾oà8ßiø6¾ït|ßÇðCü?ÆOðSü ?Ç/ðKüçxËæ>r9\.÷Ëã>sù\WøßÙÇ)ª²õ&w6óª3MÝ:õd«N½ÙZU±µ°Êìn\/Òz¬4q%® «êêoÄHC8Ò4åØ´åØtBè�¤å3ã4ñ'¤gs$®×!#q¸×ÅÈ\Lo#K1]|:LìE{!#0ÄJä¤9r<¿\Åï jäé¨:ùCTðü1òë)òLj®¿D^í¯ÁW¿µ¥ ³¥) [l¾³ÅxÕ"´g Ì" è¼®2»´ "fÈìÒ1»L]fVq;X´SÜJ1meŧsTA|)¾»TK«0K]¥Õ¥Àk NXÑFfµ³º1ðS6jìÊ«¯ê¯û2\TÕ'í¡E6º}Æfï¼ðÛ>7í°eöís0~еôë ùVý¡aýAYhYîíDÖ+ó¶ÌúF'v» "ó È¢¯¹àóùRd 1Ø^d/î ê½EÞb@CLEA-ÌAcAlFÀý{ÐàúT´ù|óù~ðûè�óüAæùCÌóÌóç0Ï'³¿EÉÀîè(0|>:|®A@ãX¢ë kìÐ]Ð2ÐP%zêe�Ç[A�HÒPÔÕWWP[õnÔ^'û ðÒxÉ_þ{Úåßôéoãù1¯êÙoSl<èÈyÛGP#víÞìÛçÄejøIb ñ\I9°Eù_Îľè,¿kM@³Ýá%"
a!Ç°gX(0,Ô0,¤ µ %
2ÃB
a¡!ÃB#
& M1,,Á°Ða¡úÛæ£LpPÝ?½D°Mà,ËJØ ×Äõq3ÜÎÎá<ôS$gãùðWãðZ¼ïÄûða|Á onC?<Ãoð{ü HCdbB, ±' «ãJ½ôEeVvVËØ=qMVö^¬ìk±²®ÍJ?ìÍJ\}q]Vàz¬ì²27bå uµ ÆY#X¨%¿W°dePR-<¤©VVKÍjZ
µ¬Ì×±²@kÌÊBZ2eeCÌþNv42AËò® 8TýVH]!ïÝ ï«BîAËmÕ ï«Ck@Þ×Wï?Á }
UM ÁM!A> ·<·|nùRÁ °·ä ºú§Ç¥0ªÁNòD-h°Q£ÞQ¥¥hµj Û@ië G[ÝóסÉh&¡Õh3Ú¡ãèºn£GèàKÑ5EI0Öía,équ\ FSÜ ûBoô«áÐ[1ÐCXÙofeO¼
½ðVVöÆÛXéè®þx+ûà¬ìw±2�ïfe?miµmÔ¬,ÃÊD-+kíX¯-ËÊm9VjíÕ,.ÏÊ:xó_,ó\óÜJæ¹xæ¹UÌg«ÏÖ0/®e[Ç<·ynêëñ¬ÇÍY[°·d=^õ¸ëñR¬ÇYcÄ"vg9Ç°± Õ¨OnÅîëw@nL°Õ0lÎÆ#êßV[Á%¿Õ¨#IÅ^À
l¬°\½J�¡.qfHD¾¨¼j¦ã¸î;ãxØ°ëµi2L ÓÈ.ÛÀíT>+ùJR(»\!ÆqJ1^\%®ÄMÇÄã ñ¤xJLQ>Dá^B+~óÄÏb¾X J�{ÒÏÒ\i4_Z -I¥%Ò^)AÚ'íH¥CR¢tXº%ÝîJ÷¤Ò#éôLz!½ÞHï¤LÊZÙ@eIe¬È²ì,W«È.²^vÝ䪲»\M®.×=dO¹¦ì%×kËÞr¹®\O®/7Ê>¬èE1QL3%GÉU>)¥kE½ZEE¨®æÀiAd(Pe2¢J»oVa1¤! Øú¯1·ÛL4Û4Û©&AJh>j>fx Y¨ñh«»cä¨FM ¤¦~¨)måÐ�"þTÔ¢þ4ÔéVL?´fú¡ Óm~hÇôC{¦|~èÀôCG¦:1ýÐY�åÐE6µàÇÔx¦&%@-L; ®Å£ÿÿ?}õÈz±Þ4ýhú±ëG{fyefyufy[f¹/ÓI¾D{Û Ô!um¹>²)>þ;ÿx<~;Ð1)yXÃü©0213021þ4gþ´þ´dþ,Éüi~³@¥Î^bg¯æ-±êgã±qÙ8%lrEßÃbßµUò ¾Îtl°,°LÙHÖ~¤qÎÆyEjÀR¤qä þBÐ\_ #QRN)¯TT'¥²¢¸\îJuÅC©©ÔR¼ºJ}¥¡ÒDé©ôUú)ÁJ°2T¡RÆ(áJ¥LS~Rf)ÑÊ\e¾²PY¬Ä(ËJ¯¬VÖ\ëÊfe«²CÙ¥ìQýÊAå°¬SN(§ÓÊYå¼rQ¹¬ü¢\Sn(©J®¼VÞ)Ê{%û?¿ôøÏ}ÿc¿ô0ÍßO0Uòóëü¥ûÚa& Ííbw!kÕ»t¾Ýãó¸OçÛ>ЩMz[éP÷4ú¶^ߣ Ñ«øDØ×´%IÒô¬ Ô¯^WûQR¯¥OÐÊ÷Éã÷I½òV<©×é~ü&5R¯}Zÿ>©Wô'°åðÁw lþ>uùQþø.A/}z²ôëvßߤþþ
ü(Iß'ïSÉߤ²ß§"û¾/ká?ë#°>Ñ]àÏZÀõM@eû²g±|}ú4д¢x´m
øç�JB'!ºnBÿéÙõæÿÛÜã_Ê[ÿ+ùWA¾¬ÈP,TãTOëÌYô ^gÁØhl¿�ê ñ"¨/ÆêÄW@äEðüV} -Îx%½ãÎúGË83êqÔ úBxsÑ@õÉøèØ;EÄØÄA½1º
úàÕRP·&vP/K r#öêÛGc¡^Tºqº3qFê[UC½ Qß´,
ú2² êËÉr¨¯à³'É6E×L0UU'½à£>]Qh8¡ÐG}V¸õ õÍÄÀÕ£ >Z}j%DA}Ô·,'Cý¨YK $Ú 6d�JÏ°nº:zutÉP?ª;õ T±b:5YÈ"<@eCbh÷åwÖÌ3ùý:øW ÁLàb¿bÅL¦A0Ó iÌ~{ÁL¦A0Ó iÌ4fä˦D0S")ÌfJ3%ÁL¦D0S")ÌfJ3%ÁL¦D0S")ÌfJ3%ÁL¦D0S")ÌfJ3%ÁL¦D0S")ÌfJ3%ÁL¦D0S")ÌfJ3%ÁL¦D0S")ÌfJ3%ÁL¦D0S")ÌfJ3%ÁL¦D0S")ÌfJ3%ÁL¦D0S")ÌfJ3%òõ%ßXRj8fl/5PYª¿Æ Ò&S>ê0%q¥:Ã._±«¤7ÐN G¬¤ï£4Ç5æãÚëÛêí±·hÍ.)ÕBDP(üW/1yëí5ÆU¤«Z{dUþ¼>¶úò§[ϼi^IÉè#ɧ8àfÔª5Íø²w¶ÿ«ôºzÝ·3Å<S«ÞQÃuà%Ó² CÆ ì? ÔÖÁßÑÖÕÓ³mË@ÿaÁÃû
Ú6RÅÕFoýåÃ%¾?<¬OhðW;}õ8gjùëñvÁÁ¡¶õF:Foc¡ó¬¡wuÕëkèá_W ÞÕªkÑæ?pF¸lñnQßT °ûE1Ú@Cxe¶å»htOýø ³Ê÷Ê)XÐbUBÁòx[ï°¶ñKã£{» ¼\¿ï7Gö½ùrÙëèØÉýv8Ö¯ÜõÒµî¹Ï?R¹_LÌ K.Õt>"ïé!¹ÑSÑÛc¡óÏõ¯Nªÿp²áÁAúl [Ù»ò¨ÏìîëÓÆÚUko»áéÏNOj/ö7ëÝY-]£ÝÔëÞÎ''Kýr¤Ï®éÔ|å;¿ÕÖüuc¶Úfyn¡ê4§wÍMh
ÝòV÷µk¯Dtìôv¯WOóQüìÃ['.(Ø~>üú:«aÝk9ôN»ª¬~&êô.ÛQ¦Qé¿\b½>b>"z³4æ#bô&u»ò6pØrm'íl9»ðìÊaÿ~ÿEþÉçT.x&%ÍÊZdYíõ>lsqV÷Þn±+¤³ÞÏÓ¢O×|bù®Ó<ç=qSüÞ~¾qÎË«ëê¾öë>·ñ®vÇuVíX£ &->\_jðи«më~ã¶m,T£|åÃ+M~\oè¿ê£¯u®Ýéë%²ÚmÒÀæGZä<î?H×6;1£Ý©Ä§Çõm] ¦^àhÕòZi²&c=nw·÷;î¤tzÐôT;ß½»99×ßi£'ì[tbS çGcõpdºT'ùJõîÕ3Y\_-¨TPZµûWùGë}ø®U=´´Öù%ñ3¹æ[§ÑyëkCÒLjN7"vÝ8@
ÞúH®ÅT«l2¾Ý¦°ûò³I\_1¥ô?0ï=Üà «lVû cB#SÒ¡½«©ÞXÝÐú 8¤(ü#½¢î¤¦´]@ßÁÁCú~=1ñN¬ÞîËY?Þ7À¶}ÿ!ЪmõþÆ¿Þcçz÷Í®·rËWk:)¯ÌS>Cß^nôìêÌc[´ó{¿ky³é {ï#Ê%HMGÜñIÜ´9QÞ)3î©®\Ëõì?ù-¹XÒgͼfeßåRöX³ÊaÁ©%l¼fzyÞIt|ßÏ«2v+,¨ØdíAxê²¼;ýÃ#s»ÇEL½=sßüU=Ö¶²¨8µÕ}6ªýþdníÃS^ò\WÅ={wmx¿G÷[¶x¸nʶÌãY¶û[Ìò?ëêæSòÍÁf ½Ú´·¼Ð¯í[¦¦tôl3m°£Zò8ûÄvýj/iuÎiBÕ!k.¯¸Ôl 2NÞ¾>é#>êMUP(ÏËzQ£BÊqÿ; P=GSõµB_ZÝ¡ðæ¼Ù¹ÒF¢nÛ2noÓ¶aU ýßé%õ°!ÏÃ4Rlê0·iëf2/jß¹bh¥»¦äoj14jùüÌKËÛ'ø°,Òàä©çrÚ;Ø1øà Ñ
)1׬÷I±%uóoܲÙ8þíëµÃ7Gßõ]{qÐ!ÁW¦m+þüz ÁÏÓ î£îYÃrLª/Ϋ?ÐahGô=ª;ÝcÀùÄõö[0áàl÷3QØØWîÕOWpÿþæìôkº]!×ç>l½×#>¬òÕÚiî_ Tnzvwÿèí]zÞè=³Ãd«ª¼ÇEÊñ½fìrNX¹æì¦[¶{èKFÙéj÷¾Þ½ús§&<ÈZ·éÄúÃF1A1í0¦áèL!qÅç�8óÎê¯SU¯Ä© £÷Ô»©UÕM}èßrjEǹ?8þ§X&κx4¹éÒókºo)×eÚ Ãve槼Øzääµ GÝgºÕÃ9¯zGN[£uwÌV qhn^§ÞæYuw4¦K¿eæR§#»¿Èø¬<]Uõlèã·û¬À%ø^ó6¹¶ýLOÝ¥q ¦ºÏ½¢FÌLØr(êÅî9?ïõëñÚ8½æ»n3¶M~Ìçáé£z/}ºeTrYUÍ\LÓüNoµÚÐzqÿ-Wm=õCïÍêßèÁIë÷º6¡õ\ öAvn{|§ç©úkw·l¶)úÆìIÞ£ÅÆ7Wï\îØÌqýv4 M¬P¯ù²>f½[éS"³.I!ao:´uEÛadDÖäè#>°¾/m¨ÎX¤b6Ë®îì°¶9¾Í?¶¸4É]¨RáÙ¡IÅÒåxK½ùÄOóêÊðµõ^zϸqÕ¦TRÓÅÅØ \¿ú°ðê^aÁ}GøwiÐZØ¥oòõO©¥¯©÷øº'S5jÔ V¬¥ÐßL 6u;·ï¿v;VX4¯µååÍð7º1¡£Z/jl
JNHóßå²G:ÜXRÐæHO]û×¾ÌZlÜåÓûò/3´Þæ¶öø4ÖVèÝÉ ùüwÚsZy÷ CµvÃÒ{íÝhb?ÿÍsw´ CçíÎTjÑt£óg+Ïõ¨pèP{ÝvNT³n=Ù§qáÁù+»Ð ïNì¾f]«s[ÅÔ{p¶»½÷íp÷Ʋ/¦[þrïéeþfí·my{ãÈŸë4Õ9éTêçAÜ#[2.w/iaôñÌĵFZ«;sÊ=ݾ²
÷íÆF+ÉÎûW<] Ðf9 MÔW´iö¡ðÏ¡oààá¡}GêzO×êz×jÕܼqenzuS±öo9·úò_ÒfHÀÃl¶÷±õiߪ¦«¾¡Gåjî57¨ßÈãë9S?0¢}À°þ P/þ)©c¶Nnè½f×ñ×-Vا{´1¸îÖ¬óè+N©kè·Okç%V[÷xü·©µgxÖÈ̹éån~undû«Qì¢ïíkqo_TVU$Ç^EûÍÆÞ7tZ¡MTú^¯ØÉäò¤Ö^?ÝÍñºzxín\YÍWGÔúX÷ÅýéGhë¡ãË¿Ø4(ãZÿmùñ¦?0hñÉ/ïugLÍÆ)}lü:ß}']ójÞüADÞV³ç nõx)[d'¸Ì×ʦ]üÜ9ù> }«íð©±%pC@®{G½<ïÍÌ´úзM¯å®[Ô¯4aØ»u::Þ+ÿqÀ>ü¹ùý z=ÁÏZÕY´ß}Só)ѽØìU¯ÁÉKÿ-ì ßç{¾¶ú#Õþ
�Pc# dóËw/6^åÈe÷±áêUÊúÅn®²hK¯ö=s_'û6[?þ£é%É,·eæhÈÃI¥|Ö9{ºÝ ©ÑõM¹vѾܬ:ëõõÈ®bÖoMïŧuÇF8dõ[çú {èÜvíîw9oÎò@Ó/\_Ù]t?¬á:§n|Ã}ìK?þS£åèhmqò]YçF=Þç®=9Ê»\pîÚ¾Q³ãýt\Û¬<Ç;¼pûìÏ^eäóÛÎ7½Ð5tó§,Ó2¥p´ùåp!çíÃ3N¿¶50¤ßèýíMxµÿüù&«¥.e].ÖMíȹgpßÁÍ|ÓîdÜ=;éÔÄôð¤ÆüI+´m4>4²»Ëø½åùÝ»YûN «õæ\_¾¥ç¨PÚãh÷Sj88¼°ñ4Y°|б÷^yG/Z1TéU©µ¥\_o·W&µr\Û}Ï».ÚìÌÝc5kÐòKMsÎÝRÞG^ýä±ñçzk^ Û§TMÈßmr½É£æÙ^ƹɹàe î= ¯0{à¼ÄÐ#ö?¡ÿ+/#AÑh ¬ÆT¨zþK~ûM
G!¡!ÁÁµkѯÃR@Mñ4 ßqbê¤,ûåÎk)½7³«²¢ yÛ¬C;ýÜ´Vü{uܾ¤éÀºUEÁ±;2×½6´Í¥Þðy Ûçwê4]ÊÙ¿þ~á¦ÍI+¶1ÜsÞÍúùû³ÖEÑît·<¬ÜHá ¼-¼ �ì ¥ endstream endobj 21 0 obj [226 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 507] endobj 23 0 obj <>/CIDToGIDMap/Identity/DW 1000/FontDescriptor 24 0 R/Subtype/CIDFontType2/Type/Font/W[48687076818385[333 500 277 556]]>> endobj 24 0 obj <> endobj 25 0 obj <>stream xì½ \UÖ?¾ë¹r8 DFJ¨¤ã
SBBDC$LÑÔÐÈÌ!5of¯cfÆËX]MͨÌ1Ç)+3»9Mãrà<¿ïÞÏ>p@´æ÷öÿü_àËÚµ÷^{íµoçÙÏ>$î'
¢n6qFÔwB¶ÙÞ»ýîÙQ)£ÃÒÁ0È9·xÆi¼vú&¢n?YJ§Ü9¯øã½Ï]þK&O,zÇEÔëZä1 ÁËÞ?ÿð_Q2mö\ã
Þ)]®¸³ìö_+%eð6ø{M8wÆá¶IÏúüQ3î<ã»5àoFò&AN&±TÛëDò2î¤û¤È¡rüü2B¨ÌQîQîS++O(ï++Q>U>S¾Pþ®:ÕÕ|õwêÛê>õ=õ{õêy-G Uk´'´µ·µSÚÝ©ëúuúh}þ¾[_ÿA?oSlɶlcl3l¶m+lµ¶zÛVÛ˶]¶·mûl ¶#¶c¶¶&»l·ÛÝö^ö¾öDû�{²}}¨=Ã~=ß^m¯´/±/µ¯´×Úëí/Ø\_¶ï±ï·7Ø?´aÿ«Cq8nG/G\_G¢#Ù1Ìá¸Éq«c¼c¢c£Ò±Ø±±ÚQãXçøc£Îñ¢ãeÇ}ÇÇONÅétº½Îdçpgó&gs¢³ÌYé|عYë|Áù²ómç>gócçgÎãο(Î�w@¯äa7Ü01 Ìuë6×®]¹»ºsýÁõ¤ë)×Ó®?º^v½êzÛõ«ÁuÔu,ðÀ;®|,ðÀ\_vïqu{Æm :Ùï/;Ù52"rDäM·F¼-r|äüÈmoGü$ò¯g"½QQ½£Ò£fD-ÚuÚw}\_o¿«û-ë÷þ=ûë\_Ñ¿¾ÿþÞÄ»%zÿðÔ'¿Õ¾íþmÑ·S¾]ñíÓ?Ê?ýÑÔÇ£{B=·} kÓìö.ö«ìýíIöA°¶ëí7ØGÚÇÁÚí íU°¶åöUö:ûVûö]ö½öÜÚ¾vHÍèè¸ÊÑß1À1Äqc¤#Çïà(v<à¨r,s<xÖ¶Þñ¤£Öñ¬ã%ÇnÇ;wamß8%§æt9W9×;ÓαÎBg±ógsógóEçnç;Îwam\_8¿´�W@pÀUפdÚ ]+ÕµÖVÍíQ×ã®'¸µÕ¹q½zÖ¶Ïõ¡ëÓ@ ¼1pfà XÛãÏÃÚ6¸?µQÐ ÚËè² Ù=2\2#2[XÛÈû#·G¾ùqä±È"ÏFQ;ª/¬mvÔ¨£)úè¬MµÑiWÆRì+éÊ°8)NsÆõµÝ kËwwÜÚ¸çû¬í7}ïûD?\_¿~+úSÿ¨þ
ýô½ÿXøEÍo¥Vk{êGégú¹çyò§ç2Ï X[g'Ýs¨%»å½QÞ¾Þ½c¼%Þ°¶» Éh¼i¼k|düÅøøÖh6 ãBv¿3÷%F1Äd$}ã£qµÑÛèeÄW±Fw#Ôèft5Ý°¡÷]ï>ï^ï;Þ×¼»½¯z·zñ®ñ>äò>è]� áîæ½Ó[Ú|{ó¤æææÛG6g6g4§7hÞüæÔæë4\_×ܯ¹OóÕÍW5Ç4G7G475÷hîÞÚܵ9¤ÙÝìjVeÇóWÏiÏFÏ÷Sï<ßz¾öô|åùÒsÈó¾çUÏ.ÏNÏÏKímç<Ïxò<áÙàYçYYì©öüÎSéyÀ³Ðs§3×Sîåá)ó{nõÜèÉðÜ�½@\_Mó¤B=×x®òtG/îæéê ~=.OGo2¼MMMß6}ÓôuÓɦ¯¾hú¬éXÓÞkÚ×4°)±ÉÕÐäh²7Yä&©Îç½çÏ{Î7?wþçógÏ9ÿÓùÏp~ßù·Ï¿uþÍó¯ß}îÏçÞ?÷î¹wνqnWãöÆ[ëk77nj|²qcã×7þ[ãºÆÆÇk\Ó¸ºqUãÒÆ¢ÆÙ}¿ÿÇ÷ß~èû=ß¿ý}Ý÷ißÇ~ï>å9õ·SÇO}r\óÔ§n8~jÄ©á§~s\íÔõ§úêsêêãßÿêøã?vüÈñCÇ?:~×ñ²ã·Ï>þã)Çt|ÀGË¿{±Ïµ}.¿úÈÕ¯~æêÇcÓ.vÙÐ
áË¿7¼"üðyásÃËÃsÓÃ÷ ;ÖömØ7a\_}v<ì°OÃþv$ìpØ¡°aï5½ÎçÕÿûù¿ÿÆm'õÀÿÍì¿ÿkß´÷°ö¿)z¨ßPüñwäðÏmøQ
åÆãkÚDä¿K{åd6u:òM´^£7im£:ü'P¢§5~î±²-GìzþU´Ç¥§Ìܤ©TZ/I9Òo:Ès?Ðß%Ew"o-~7Ð<¸¦
t/~ßzÐü.§×äT¡XQÖÁ}«ñïN£Ñ�ûÉfËÁÑ@ïáèH?G²´+¥ ¨ %-§Ö°Þ´E~F¾G.-£ù~Z'½JïÉ[¨I®£éòÍô{MF=ä²¢}·ÑJº¢äµküH;áïGoQ¥I«[rr(»ÖnéZM7 £"cõ7Þ§,þû(~·BL÷ñ»(ä\eÞrXEûäVuÉø]ãíCwÐlÊQ§]Ñ7g½
Ê4)mñ"r3dúf¡þS Í eÜ×ÐZ×Öà1v'JªòðßcÀZy,q~¡U_V7¨[ý'Ò|ü?Lhu
R4õ¡dʦITA+mí®£tÊÆì¤ÕOÛÑêÛaU¡«5øíüç8}F+Å4S4²õë¤
ÐÆXe8Í6ÓHËäæÑ©ÅÓ!¿2VBÖyÆqã´¼ñû>,i:íüÖAúUô¯KZ¯?jÝÙÏ$üfPE¢Ô[ê ÍôW6)«ð»IC¤HúLy[í:Oq¨ÏEÒQée«òïÒÒ_¤¯(7×É»åwä]h«&Ôa¼³fè[õÒ m®ÎúX1çÐmtrÚ¤ÐËrUJÝéYiªÕ¥î Ê i§rLm§HféOC ¡"Økµ ûL{ÒÕôÝö£Mïpé§s¾jèò÷ôTganë9z½¡Ê; »FÐ?$4°6À¯Ão3êßÿª1ºäÐÍßiX:Ý)fÐí°+ÿ=×iX:´-øèXè¤#«ó
: NÃ:KÛYØïÔ%êåXÅU´fFCKñ»UzÒh£æüj½Üß2ÊEkDM QÞÐG&BÚÿïD!ì§iK)[ ÛNlÔf3Ú¿Ñ2õ3 ^ XzHêJae÷ô9Õg¨¥}yõ3ºá÷IvüÁº§´nAÊqèÑæ ¥Bûè1ª¤[Ñ&�×À¦Fþ6£ FGèw'?úvýQ ѵIäVÿ ®RJQ²Çøñ}Ëév¬øÚYú½ÐËá6@mˤRí¾on;¾SÒXHUüvt&Íðgñ.jgl&¯v#6Õ4%o¦"iúÕÃÔ_Z² å7WzBgDánf ñ²M!õ ¥Òmà~í¤Åä"R§òÆ·©T,¤õҤˢ˥u:°ÓÏÐo. Sàþ;ËÏ&wÊa÷Ó:®l nÒÞoRf)(ÇÛÄ<1mIºåÞxVc!Yò[iJÕÑìg<]Kûp½.óþMA_¾[A¥«¥]Ðe}A3äZù²ê]åÀrF¢5ï+iB¼ÐÜT7]ÅeÉ
Þ2aɵo»�óe?uGÝTÈ÷ nß"A6b¦ºý^9¹®6´ux�©gI;-Ò^KOzCz®ÓuÁ?ù£7°ÙÕÊVIhÛ^¨× É
ÕJ:ý^¬~JlÅy7FëîûéþQÈ´ ZßÑrOzm4\ ½-£§Ñgnbl,Ѿ3P÷bÃH¹»\_¯Òv1ÐÓR(Æ(¡¥¿|(cò~TÒ±6xRÍèøò2î±~MK}cFÊàë tmbÿ~}®ésuüU½{Å]{EÌåÑQEô ëÑ=´[×.ÁAî@WÓa·Y-º¦\²DWKÝë»ËQZßcXa}zÌðwT}zÖ7%ÔSpxtLPTb¸>«^¯§.#ëC²s§´Aãêõø,YõJ¬û§h$¾)Rß+b¸áyQB½òS½Ô¥^êzn\_�KöÅ N40¢¨4fDÑÐgQaF4õµ8jñèÜ D8¹È#ë÷ý6÷y}XÌ°Év çí8X�²ñ¼~½ÄrúëÉ�å3qG0Ö§-)#f8´.m1;×ö"$ó¹º.Sz}X½Å"êú´õ´$êù«\_\_üð7M\wÅMÌÏW&áyRbG©ï92;A( (,b=ÿcM5¢$j1ü·ÿc³&o^T2¹T3q¶a¹Õѯר¯¿l7ÜóU¸²xD÷;¢wñê¨ú ×/6ý tèGÄ 4d6¢t(kÖfã¶xco´%£êïTjZÞÄ}Ö½Ø]~6öAJP¨²¨°\:UsDiÔ%yUæUµF(ÎÀöi,Rçå(ÑV \Û1mtt}xpñLÄEÞmò³/Aaõic8¡1¼ PbÚÄáãDÈcÉXLáðqã¢Ívk½%¶Z»&&j1ËÑ[ï~q¯÷¹zäèÜÃÃyíëåa¹)?tÿîÙÁRwð,Nø!ÜÔÑÈ[bFþÖ´ß¿1f÷R©@À$ máþÑ@ÌÊJ¥,¸ói^7ç''èÔo'èÖ,c&0Xä°_Èw(ç!JáÞ Dx í=b)~2>~¢å-#Qí( ݨÇc® rwÚC@ ¹ ÛRÊf°1¡ï¦y y<ØðZjO1ö.Z BØNåûx¬lV.±Ójæ înv\OWHvì�@6°Èñ9j!l%Ù8Êêª,¤ÞØq«¨ëq°y(vÚÈÊV'1zê«ëÅà¯hÕÉ¥ ôÕyíý>]vÔ'Ó#ÓagàzeØ@ ¬åð2=·bCçíÐ ´E;lÆ: m£c!Àtu)°öÓwç|mÉ�;]ÉÀlå§}qÊ+)C^BvyÆ¢fJÔvb¼Z:ÄüEp¬- ¤xã<Ô¨ÃÉ æ6.½
õ�ãíÄ>MÞð·¬o±.öÊyX£,¡Ë °aF ÇËF¶Íh±äN
ÿD{0^þiÑ>EÜnÐÐÐûÍ<8Ý <pMä_$ l ès afÙZ$èsfZÿZãî¤57å mpÄçG7©^ ÑöÐëuò2£ÅDËÏ5} 0(à÷ ÷a¤Ó°NÅ8µ&ù1·ºóé ø{Ñõsø÷L}þºMgåcçµ1©ç?tç%»)#d°Ö4Ñr üW�ÑÀ|èã¬ÖÐïBé¬WrÜiæå«øÿ.?×|@ýîÃÚ¤�òÜzÞ ÿ"Ôk0áKAgby'ãºQ]dd&iké6ÄOcè#¤+U°§DØÛy\_× ÷¢¼;MhêlÐû/®# µÐîûwÀ.�»Uu'0¸lÞ6uÃõÃîE<£w�÷�ØÙbL m èhSÇ\Ïáüí!ïX º7á}º9)ÍCí�b½qY«DÙ Oq:x¸ ,æ�kLàA�53ú Ù\¨¿%ï#[uÓ6ìán´æA¿ÛÑ\_\¨íU¡ßE½YÇ%nа^<¹»mp ´nÔªi~/k}hDÊhò8ÍSá>D3°Ø§ÀáNÌq+)z§®";ÖÄÅZëN\V\_§Êß°>Í~t©uy)Ö/WÝJKÙg Èwª@ûLxWϦïÁ;úûi3"äö|Ë ¿un,ÖD;ãÀ;ÀVà0ü ¸ù)ò]£h.Ö!cõ»i¬6 ãÒÍh÷¡Ç"ËT½Ìµìÿ!ø3ø^�}Bà5�ã"liºú1èèÆ÷©poþÏòøÈ Ób
ßB³Ô¿&XK7daòê òoMÀþcº6ee=côû;áÏr\_A)6þ$ï4NbsRþqVÛnUºQ-úËsm¼Q ¼XvîyÞr½qH~iÓµÚ$ô(ó#ùc,ôJ,ªì³ú"èî'/Ï "Ûï©Èþ>Æ£ë©Èñ äê°Õèc\_QtYd«§<Û¿Qk;bP§S[ø4Öò)¨òìS6ãµFÓev�vçx|;PÇ
èÿUk[|Î7eYô{ôÅ"#ÏøbÄcÿB÷Ú´Ò²ú¹þe9ým|tgËó¬çæw \_c¬yÆ\¹DÊb¬9^¦le?¹aE\êf ü¡«Ø;º-5ê4ì9POë{ýMä9.Ó?Fw¡Üa(çn{îDı2¯Æøúaý¢¦õô 0·²óÜ,¬A".E)z5lø�¡¿¤p?áë¨ÔÒáûþÀòL4ãYzäÉuÑåp/yl¶Q\#=Òa\_ÇÃã(ëR-n¦·Á"M5Ú{-ÍWc ãzÊcãrs^¶pµéå&Ð×Ã[äR³j0ÄX°´Ë±¶n^¾:ðc%Øõ7Ñ\o³|½Ñf¨_ô_ýN Æ´m(Ü !«o úuè$J±¡Ñz"Ö¥µ4Mc:ëýPÒ¦A'õÀÐÝ ò-cÀwa �g8,ÀòAÚ {}:çéjxø4m8)s°&J§9Ê·? YO"ÚC.mz}ÐggùØ»ÍÿLJÆ:%FÞµêJöP>� îTÐ,½¶Çcý"Ô#Oÿ|!¯ÓÓJeö>4kRPöê'&õîW¤h¸7éÀ.r«@Á#-�Ê°?9¸x¸¿tt�4]ðLP3P ·UðÊðöEyÃ~À ùWü:HMh3Êeùne²©@80 8-ä
;Èú¬P¬ëL÷AÊp°°?·)òbHéüÓÖ ×!¼ÔOª3ëÁÁY}æ·ÔÃ-2Áôôó_bBcÖIB: þgA¡ ~ôÁÚ¬5ÛEÝ@¥áÿDø×õama¬=x>;;5öouóåãbåk£({|ô§|m%ÚésÊ×GcÎFÿÉÀú¡kªÖ"~ú°ª5v®aáÕÃnakìóã/cqàEX3ôÛñ¥qSÞ¼oæVhÌÓcÝ¿¼GÑG®ÊõôÇ þÜxæ½ n'õVó0î§~úOÈ'qfJá®B~[A µ4Z«�-Üì@õѯ"ÿùàï1i0öU«KP~)k(òHgÏg´Óp¡éè3ó(cWò /#W_8i«$«è;ÐÀ< 0 Ø!Ðñ�Çá^�ú9pĤÒf Ò R¸è×ÈG´w{Ð à· )àãcéVv?Ph îþ8Ó>c'wYIdæ-»;á¹>1ó¥½f>24 ¿¤ëán�ã»ÆJJÊ| Ô ýÊ¥Z Êê]Jù¶¡%À49°JÔuòAÒÁ³ÀϽР.%sïF¿0üÃ7 T¡+ùíËW2ÚÀä:cݵÚÏfæ ûëh;þ¸mvb¼NìÓÏFiÝ
nFýÁÃ6u%¦=hK°¶þÉ
¶ë³AÖ®þvÖêöÙÒéöæøëYËÏqu¤~ç÷ø¹½\_¢¿ò\;«ÊÎ\xï:=·õùådãi¯±¥dç}ç:RqvoIgýwAgÏÀÙ¹ô¢gú½Ø³E¿gÀë9¿¶?ßç£ýÒ9¿ÎÎûµQã;áøÕçþ:<£öÿû%\»:Ðè4ÛqÖ)Ú<ÿ}ËÅÎrj¼yéø\_CÑþÒÍr:ÃvñsËloãÍÆ'ý:Ú±}|g¶;¯Ù åçf.Ö³Ë3ÀVm·QËÀÏÙt¬ujÍú£V X Îã\ZÒ5-ºQ+P,°L®çÓÐ7O ØÎ O3jÍÊ<&G°@9 7 vv·ñ°Õr7mÀv~ÞçÐ?بhäu)¸QÛت{ÁËðð#ÒfðéݧG^Pç 1NeøÎùÎ ñsHÿb;þ«íòUïKÊîô»zàI³ ʽroÍskhÆYvæØ,þÝ"ÏrìÁ/Ív°Wªöù}ïEh¨;ÇdgèLÐòNõlcçé £cì<8ïÆÎßmSÅÎG1Ì^GÉáç§ø{&ü^¨¶êÌwPØ»$�Æ3ì+ VR2½óÁÎìµáÆÜàÍeï£ä©¾w¨òl;[ó2·¯øcí(Öw6TíCÒfJb¿é]� ?N+{°¿cç¤!e9é G_ñòJ9Þdñ> {ï"ïÙØ9ÐXÒ³H÷,EÁ]"Òú½»ñ«ÓC.6'¨æ»7l-.ÞIaïÄ[ûQ¬¥ ëtþ?çýa20 û×ëèÝ ÏǾ1¾ tíï-´Ø 37ø-TÀ�Ûeíý 1ðOcPeöüû· TJ°|޽أ m!¹Õi½é$y Ùuv^r¯ñu mdïµhü!v.q¥Vêù¼¬8ØÈVÄöÔ�ªæP±KÉ ÚJWbÅÏp^äqDl¬=©2d ]¡µ2T%°H~AWé[¨èáG£Páö£ k¿£4véµ´ËRÎÏ:-óÈiM%ÕKÖ,AìÖ ÌkX»ùÞwbë5~ös)Öcþ.G8h¸¥mqpoåa.PòuñqV-gmo9fOåtöÊÛ¤uÐPÝs³ðVdwÎÿðq4Z_hqká? áG÷nÔmê½6ó4ÅTÁÐ\×>ïvþ|¿òòýh¶/ÜRÓNÞhÕ¿iõ³:' 4t"ûºèÓ||½)gx_Ý NêI¬I(W;P"°Ø¯¯¡Däãÿ^[½sÇÞñïy5ÐVQ лݻtì=°þÐ.%¶xÅNZ¨Ï~õ(-{8ÜÃáÎÜO9 J¡ÉÝÊð{Çj4ø".Ás±tÇÀñOæù°¼aBÝ"ÅþgçÍåöëÐÚ@kø$¾pØϼ\_ Àb}PúQçÐ ;õÇvaÛPÏb?uïhcER\æýàÏÖúñç#;B¤eÐriÀBÇÁ6½ÚøEýOj\_gkñ¹Ð ú¦d5ALHÕÊZ:äµHÕ"ÃRPçkñlo ¤%R¬z<ù�ã?|·R§4õM60°µÀäX'm¤+ñç
sp A<;dØÌ>ß0ÇbíGôt P×I<ûÜ?¯þ¦äw8)ãi:¤M´Ç>O/FXÈç°[ùç["²ÄgñIÊOëW¨ïaÍ÷¸ÅÁÀõ=)X þÏýhwç¤øå×K¤þÖô¾|}y�oÃoW£^é >÷AÚaÌ6ÿï\$�èDy@"0ÔJRP ¤¤\_ÿóõùÿ.Ôã;²ýÇ/ûõ¹ýãýùþøèÈÿ¿þú& ÿ$ðýῶl7¾,ЪãÝm|¼ÝvûÉÛþKzÀJïµ/_M4þHL\àß(°ÝDëþ3T ÞDk^ø¨R IÀR?øËRÿëäeàu¦á<îRõ]r¸í~èógDTÎÞ
ÑÍgõj%ÞûÈÀZby¤Ðí êÃñ9Ö zÉjyJÚb©³@Ç0¼ÙóìbÐY@ºòg©ö Xleûæ2%m åZ1mñV2ß¿üR�O
W Ø¢³3R"öó Z2ÊnÄÈEùX%¨Øûn£,½ãû9eXzðð1ÖMcÇÞT/£IúO4ûàLm%a> : óá^g>í<{"ä]¤a=¼:úõã4KDÔ](L(ôÀËe©§,NSϦ%úhJÑÇVÅ×xúgɤØ÷gpºíp�ùÛ±öÛCéz\ÔCùv8QÎC j¥ ì3ÔR¬=Fa·õ\DõLØBxFc tû ©úNÈÎÜuÐQ5x6£ìI¨×¬±÷c]íEÞ£\Ô/·3óg°rµZJÕê(]CáÚ:××!¼ ÈæÈC|²¶BçµJL¨3;W° åAgPÞLcP·tÊi°<+£è7¯ï³8>¡jÖylÉàϲ u�¡®ÿ°Ä¡?5ÓQK =gÉ
l¥hóiTË cϺfê(<9tÈ2v: {åmoqCõà©N~¤ù ÐÝ(Ý:zy=#-+¡£xè°ö¶ÓnË.ºµ!Êae×YÞo.úùÙå@=ÖM3æ3ô�ÖWtÔJ\{æ½T-gò¨lÝ´ôH3É|Æ«gÏÎÏãzáß�zÈ;C¤}^TfÏiA3ijé =c¦V~nASÝ!ó¬èKÀÏÖqk"@ç ;{¦/Ï«¿Ý%CW ?ÖlÿÃ×üy3|¾çÈ£¬æ9öÞ%{¼ßÔT+Ò~gÖ¯òËë.±½{VýQoïÅשY&¿ä´cõ6 Ѭ/{ÆMoYÍó¬CÀ ÒlÖn¬NÇͶ²É"sæóo¦?ÖF¬nìù¶,2ëÃuÎÔòYÝ3D~ù7 Q¾µ¶öT¡k¥Ñl3µ¢ìÝ¢ÜñL½RÔÙÎQ¿m¢+³ÀzQ§J¡ÇÍ¢î¢7 ùÍl5㾸ÎXú ¾^Çtõ(aërÈÃ÷|¿JèµY¡ Þ¢$YÍ3,¯9~Xîغ:&äÚI¬?±³^Y«÷G ýÃ7§ðvþÄ·é2«yöe
gür¼ i¨°gvF¢Ð_ÝÇå6Ûéõ Ù%üGÄ9 ÖO3¾|²Ï9Óvyÿ«á'Ì<¥1¢½³ÍöäçL¼B¢+¬>;õ+Ï ü3g¤þKÎÿ9nøE\wÏZ;?«w.Åwöªõ¬ÕEÎY±ÏØYÞ_ýϧøsò?KTÚv'c¶~['dÚê'ÃN!óVA¬ÏZÛÖmÅBÎù¿Ól#ö9+×%ÓXOðµÉF±Ø&ò\êWöRQ>[Ë°ù¸ä¿Á:k<¹ÚXÉï©�Ï ìå 娣,e§áU^K2-Pzr¡2"Oÿöh&·ô}Ðs½GÃÞBÜ}æWd×ÞföNf Ü·.ô_ß6?ßY dgmÁÆ
"aS¬½|}]ülÍc5ÇÖ~l¼-ìæ<ÙÚahJ7ëlÝ0MØÅV,x3ßh¡ûá_ d)²òl)2Íe¶X"Òg ÙÒEý|coì$Êʲð#"O&K¶àóY^D]¼Æ¤ ¾·Dú Vw¶ÿYÊóòE]r-î !o({µm,,tH¿J\¤©ú&ôó¬p3 ûÖGúsEû²gI_$ô8JäíÛKf ù|vR EÝs¼¾x¶·õKloZ.ê9GÈbÊÄÝ©"ÿ,¡ÿM"<[äµTÈÌÆÚ !çZQæQoèFrm/ð1BÎ,Á3F´QÈ·Xȳ_È-ê2Ú¯}jºÈÇ7rþR¿:T¶Ù-ògm;XäU"äðÙ¿ÏÎrýäeýÍq>»÷µ?C(7IÈÇtö¹o ÞQB~½Lù®udåMº[.}}ñ ÚïÜ|?}²¹j»_;§ ¾|çLQ,ÁÏ×Y}m:O¤%tgm[˼!tíÃhu¿z2ݪ³{êÒ"ÝM©z:åù!×ËÏÒõáwþÓPm ký¨vVi©¥¡áÜí¦ÞzÅñgøì® ?ðû|èô"ãÜQÒz7ï^¢Nî$é/KÇyÖÓpé'ãK9Áøß³E ä$Ã0ï¶ßy¨uNêàg÷ø°÷µxëõøçýü:q×OgÝñ3_ÜõÊîà±,BùsWäý'ýüýí£¾zò;w+±{{ [¯R þÍ4c>Ü�»G© ün¦cîËßë³Óè^$NO¥K9 ÷ÉÒS!mTch£>T]'UûÞÐ.Bmµ4Ü6vcxÐØQpó;r)½µMçPG6ú-ã[Oé ø}ÈAøroB¸·#¿}&ÙGQy;Ü2kE&üÖ êóÇß´í,ÙúQm>X?OÊìMéýh«ÖWæÃQÈ·é?�ïW¯-y X_"ò˸8¿ý-Ⱥ2¡tGãÅù|wX±6 ÈK ÕÍD<éBåW1ޡݬQÿ¤p{? ½ê+i£ímÔy!ê¹aÐ Kãã×g!þvÉÙÒÅ,°/}lozÿµ§^[Ù
Ô1¼ öi LØÚÜ>¿ýtÐδQëá6X¼~tE´;örçòñ<|èǾå½aÖq�h¿SV7[IÕsQVßWF2Æ(yáu¤t'ÚÀa"Û¨³XÛFU@Í
Ôy8×F;öY[Q·ðgß|ÖF/(oðÐ
ô-¯cû3Ýp=E··v|Ñ&äÚÈîçSù»æRcôu 9'oÙ
p=Ie®Å7ð=J|ÒÝ /Cúsà TøQ¹ëyÄ¢2÷kð¦r÷dgqA/øñ ¾ /òÚøo6qCPdp¯ tש<àOÀ ÄP:s^<àv¡® OeC¨Üy ü÷g t7YÔa\þ{¡?V@ð2ëàsûSקfÖç#Òs÷S�drÜñák±²Dyå ¿{©'¦à. /ñ2>\_zævýX,zåózuaùq9öõdus?24Ëò¥u´õ 2uÍóO ¶B Jí1JÜ\_QVp¡/8§Ò÷´æÀX.§µJ ÍÈG¤Í«Ô0ØÄz´P¬·}kîÑ~á¿ôy@ ö}F³¿_Ê¡bkXÐiÌ ±ïÞ�-QsR e} DÇ¥$RtzË H_CqúÀÞeO:òÔÏù½¥!làóÏ$r·úÙÄÇ°ñóÆi0é»{мëò ÖBü¾Kô£ßÅç︶ýõ·ëHßúêÖÿòú±ÃzÑw/¤Þeý÷E.Ñ
~q/ç\íÊ;v¡ß×ɾ¥î2£©£¿ãzß9¹²[ýÖü^ÉIÐgлó3Ê£pm[¿vÔ+o¶õ§¯]8ÖÖ>jaQ6Ð} ûÔr«�AaûXIþ{ÏAìæspìG@�{fyaɧAÙsѯ(à¯@8p91ó/®<ÈóùgîïØçxÒañLo£xÎèÏÈ°×S°ËÌ89ÉüL='fÏ ù³Ç$=çÛ)&ûH¶ÏçØ
|Øù²2Ïï»Ë¡ìg?A¿xF=ª²H4Ï07;ß gÏB9?Ãçf¼ìq=Ìçn¹·ÉÃr^ö>Ú4Ý÷ý2P¦´Ý±¾Peç¡Ì¸ù@P"Ü,lÊÎOblr�öá ]ð²ðÑüþÔäaßÛ äEÏÒä Ú$øƨÿ à-þ®7±ï%¾¨Xì¾PP&c9pï(�X½\E|¼HK"Tþ³)Ó ÅÔÏ{%Û¾zº%À>P
Ü1@¨x·kÐÙQÿ#¢N4[è¯0PÐ\ô\üÉf´Õ´éGÓ®¹]±;;ØgFÈOªü»ý(Ókomýó;ö½ ; |ßÙ±;\_|wC²É'1ÞrQ/øútø^Yù¦P °î|Ì´ºcX Vgɲ:KØZYÖ ñ.àvQ{ß/Å÷¾rýers¿r½'hÐÜÄòý\¤VäÁÞ¯üÆ÷]W¢,ݤÆw,NÙE=Õü>¥ öN6ô ¬ø$öVs£@=^n¯»ä,>ï»ÙüÃÇhÇl\¾Øݶ¿x§mÇ{l;ÜaÛéݵU.Zq\_-»£¶³»iµ§ém »¶åßÛèzy.úÚ0zJ¾®<ð\_M׫ôyCím´Èë¥ÐGyé³A!rÀôw¼WWaÞy«5eSú¡ EØOß¼L7+È[A=åjÐßwéò¸%t$ʼÊÇtRÊóþ\_!O»ûwãÛ·�tûüEîþÖ¯ÍVûÙ]ÃÛ/¼kØwÇ0ñµá«ð7¢ m¸ò?Eÿª,¶KÙ.j¬Ù.jHçí¢$úµKh? þVÃÓ®]";´ÉZêÇÛä/¦ ´Nÿª,DÍÕ¼¶u¢w¦Cîù-¦>ðþî?u@]$´ïêÖûª¹¼¿µÃ¿Z^@£@qÃç¿|ï·Åk°A%˼3Ûèt' àwf¯xÅï;wŽ¹§Ñß3äeÞ À7úá à^6yåõ»Ä¸üKñÔ¹ íÎp6®0ð>+ ?z|GYD~è³Ö«Ì>ÌÆ2{0fÃ?¼iÑõ>ê³ë³]¬oµµ>o}ÜlëL¤¿ù=)æTà!jµ ý,P²°~'亳 zàûJs¼}Dk½|î9·Æã&µ Oê/¢~úLþ£?øÒÁ44üH«oã»?Û¥Íýñ·A:OϨ¯ò»¿;ØÛÜÝá^í_¼S»û´ù=ÚEoß=ÞU¿ò￢̧.~wëÞEôß~¸ÙBÏ;®_ù
µ;wïl»ï¼ÿÅv Oáw§ûßükïÿÆI²6h>)õåÔ¯lÏ{½Oýbw©ßA¿ò.õÞÊãÔKI=C§-¥WýîRÜá.õ;¹K=óRw©«?Ñõ4UçNZÕ£ÒXù4Öò_4%Ê»MÙTÊä²¥ÐÈ®¼zá=ïJíwP
|ܼç]Þ iGúbå ¿�4ûbów%f\§±9BÅr#èZª\_®@Kï¥ùÆ8ýëä¹ëª® Ï«~ò î ÏÍÈyIyÖSj'mfÊÁî
ú ¥wvÿ½þ&íC9½Õiã ôíßf/]xÿ=Úk¦¯½´Þt ÚëaÞ^Ý©Lj¶d¹öWÊYz Yn,¯úÉ2¢,Ød'd d¹EÈÒ 6zv)óÉÀÛeÿ<·íû�Ð>ÅB½y»ÄÙ¿]\:±¿6Q£kÐ&ó6>·#eÐÇtmYzvË÷? SOÕN=!Ë«~²ôê Ë
68S B[,½XýYyëËê:
µ5Æ ¬»ZfµGóÃÀ¸1Z¶ä! ×¥�ËÓÊãï¼ jÏyy>#=+ÛCàgrøpnå2ùÀtå Êc:óÇÉXdêÓ¦W0½úûüÑúç°3?ÀKÔýüû´eú!ÚÇñ èUçý
xÔó ®óö¸öËÆÑh/F#Ú¾CÄrßwF = }ßÁÏ~ÁÀ ôAü[@5ÿkxØ8Û¾LÚ±9báç~ØÙÌÖ:à ¥u~úªAöTþDÝ4;ýFÙäÓ]a~©¢ì}L¦«)ÐC ç½e=}n´L¢ìû,i®öüóþ")ö-¤°ïiàøÔéAÇ6R¶b%ÅjI¤8Æ$ßã gÑdÌßX#ß�ªcßéqQþ=°%Hó�Òÿ¦Z^ ©Ö"7aÓB;ä´T í@ð½JSÙ÷>X^ãßù0ÕöWêX9ÿ#ß)1ñ¿SéEÙLóåØ÷[¢Ì¦tõ úÝ\_¨¿\Ô¹µéz îÓ�7M¨ÁÔ\_îãí¯¼Om(Æ~øgäÁp~=4L~úKÏ·üVÙ
ñî&±üÌ;IWCÖDåºLÝbËòä¼ìû>¥5îЯE³ïBè
ñüzó».ø7áÿÍ%¾aA'ßIÈ¿Óï@Ðoßã£;¬_Qýi½éÿL,§i¤S£éÎ~sÑ\Ë4Ýê¸æê£ Ö ¸,ñð[°Sã>äCÓíñTdi¢5ÖªÝe;A»Ru±xu?ûÎ
ßÑþû!fÌèðÝ¡i>@'¡böÝwµÓÉ|/A'ô å_Ø&àïü»)&s\¬M.ò½»ååLË+öÐÝcËJ½CÕ Æ\å}ïfu1{Éxs30ÛzÑÌÍÀl6ÔmØ»qK Õm5ϳÏÜg®»Øû¾3ðDØqÆydçtiÙ9üM<kÛuvôÿQöþ~¦Û¶ïIßca;¶ãÚ,Û¶lE´ºQRJR"¢Ô-uKQ¢DD)Q"""r#BDDDDDDÑZ -Zk-¢µ¶´Ö¢µÖÖeYÖ9ßù½®ëüñz¿ß÷ÇóûÏcù¼·üìEsÌ,.ä ß+î秵nã¥4þ7Ç;ã<Ç-¿{süÍ1W]6ÇøÖÅüz£ åOªx¿A~Uªß÷¹øÖ[ßܯ8kï»àtþäòßô{Å<á68ç|cZÔÏû~q7?T8ð±+.äMù·°§6q ëþ_|ÔÏ6ùV¼fÅ8×µ8£ÅýØ÷ÖÜ/ ¿Ýe8䶼m?ú{Ý×;ú½ë·àü/\ûsnªÇÔF]Þ9l®ï.ãe
wý×qW=¦Ý¾g^íU÷ÁY·×9%öô[Kÿ_q]ëÝVûþ½ÜÚþ¶Û8 N0\í%÷Oíß]ãfÌ}R¼Ü
§[é͸;Þò¼ë=îtNº]·úbàWw>Åû]qº+N4ý<çJlÑ-
۽ǿ+Ý].ûçëø£A]ç|>æ2ïuþ§Æ¸Ý©Ä#íuZó ÿî1qFn#nßy§3ëß.?Üö
¿N\Õ!·ó1ç Ïz¡¥·ßϺÿáC9½
Ö/Èu§~Ú}1îöhmrØåq{U8(ÇÚ1ªÄÁÏR8Ûuo²Õñ°t¾p\êú÷ᬹàü«½Þõs}ï¨Ë~PáÓjý{ÄÏ-o{=[2áD+Dzï Q\ÿ5w9M¸ëpv¶ú|ÖesÛ)1ľNù8ÔÚ}¦ý¦í¶:"gÃÚë²Ì:û¿úYÊ2ærðyÛýh;¦¸)çÃ8Txû+n¥VÞI×õ}¿3íß{}×Ï\Xõd íÞãõf]Q?;ÒÚ{ÒåDι#>vûÞûÛüÚ¿áõO»_ßsû£AÜÌcC²í3.¹>¦Ìÿ×ºÏ í½y×í÷îeß?è|§Ý&nC¯-oã&¸Æ>?G\qRS-¿zþßqxÔý±ªÈ Y%vßÄY_¸OÁfíQ;uÄ_ ;¦æ¸"é¡íG}¬83lȳ£®ñ!]®£ noKgÖmL¾È¡ÃíyæsmÙy1ÎvºüG\×íÊ5¯Ä]U¬sû5<϶4)²üêÚÛwû Î#¬a !WÄÄíÕðúÐçü~ Ïýn1÷EW̽C.Ã^·sëÄCïö{õÌ{Z#nk9ÙÖßæQ êNÛ³f¬ø ×Ýn3®×%e\jé³TÏìr¹YSö»\ï+ëÜkµ×ÏQNê3Ñò ^hÊåa\5¡ÈDúîKÎ:±¨èEÄ$×ÿó.1ÖÄ7Î\cYCÓ?»È ÖÃÊú4æ3uÚïöb Q^æÿ¿\_ú¶Ëu\_+=NÇà êyI×+}úÓ<ër {÷3й¦\.ô½e¨=lêÛ(2¬¹ÎG8Ì\wÙ¥Oò¢3,+ßÃA÷f\/ø:øÎMUöU=?ø]m[ºøwz»^ì®Y׷줿zÿs¾ÍRc¯ì¦.såõ}kE%î{Ñå\_sûpúÈ¿/Ý?k|ÊÁ½3nO,gå³¼\ñóJÜÿAgm÷Î)6Ѫ¿´æÝ~GüÜ)%&}q|BÃúÀÓÄxWY ªÏûþèª[PæºìX{oUl¡¼¯É?y½³Á«´ónwìýïolwþ¨Ä0^q;V[Üv±{¯ùÚUEv+\mSñSn²Ïk¬×¸ÿÈõ?åòWZe3ÖØáö½è¾8á|¥ç{M졲ÝeXv9gÝè¹'Ûð²§yÒe&>ØßmùXe¨ùðïþýó¹çç«·ü^'~÷¶ó^Sòð}Q¿vßVYjL\_pýªLèDÈú1×Ù|¥õÃûñ2VýÜ%E!æ|ZÑÇ\9«ÿĸîöoæVõïAE\_¬ãcÿ\_vÛ±çDcý7ãv<î:°^"cÝïçØccÊ÷Ñ×½ï çÜÎU/õ=°Òû½ûï~ÕøûÌu{ê4O+{hÜrÿ s¾vY+ëqåClûü<û\s\_×0=ÍOÜðÒ¦öAN¡Ò/ĬBZrÿ \_#o¹íÀ N¸ìUîoÏÜGìëUcéõäã9·7óïò´,\_Uä©åñ;àÇÜ?'7ö¯äøl£?JF 0ÈýLµûO>þèóKÿ.ìëÃÿöÛ@fý´±ÏÕÍò¾ß÷kMüÀ×?Ùè×Á¾_ÿÃýUýrÙuÿÕ×ÿàú¾VÆjÕ»ÆÏGm|\v|쾨þüÔï>pß÷uÇæ ÜÅ£ëjÃï68ø2ÚÈqþÎ}ÿãF½³¼Á½¤ý"s6xõo÷£b/º·ÏcæºÓ®ñ2»QæÚ k0V(ëº(6{ÎÔolHÆ$ºH¡ E¿9ì{¥;ïqÞÇlýX¯êÿV ë¼Øæï;Ï]Öwú[ßGbÖyÿÉå{©ßwý¿úøóßþkè{ ¡êvG·cÿ äÇAÌçqÆ}_÷Þ¬÷[ÞUôjómmç50jͬqÄþ6
QÿÙi×ø¬qX2é uÿí²óùôÓùÑÏ}ç}êÙÚã¬0g-¯|®¶ª½>óÿÃõì1WcïWE¯»ê´ª ¯ù~Ťðï/ÝÎ_û%·\_=qN» ¥'þävªgûü±Û¾Ò}á´gþs÷ëÿ¥ëð©ëðÊõ®ôú>ûó|±ñìo8E·'zÆE_ÿÅmúó~éû/|x?ÈAy+Gn»5üÐz÷Vþ¾Î¾Þór0Æ_µã{ßû>éÚ}y¸_åøÙiPÆ4}PÏèk_øÚ3×ùk [ç¥ËQïÞt9×ÝVö~zäw^ùýªoÍÿ;n|G÷çYçû½ÛêóþÔÏ~èr2ê|^|{ó¥e\?qÕ?9¯×>WûYY¶ ÊûÀùOø=æʪÛòVW¾öØmÍ8úÁõûÒ×o¸lõì´2¶~vûßTöaüF;àN|l5^KϽsÊï?híBÃ9øÄåQÖ80ªëÇÝü|=[så¢ëÌ7¯+ìß{~ÿC×û3FØgunEÙñ¦¯±6Ýq76ôÁ^̳<ûÞî7=¸~c§{»çsõÿþ_ãïÏz»'ô�zé2=ßèëíüGßûkKç[ÿÿ«²üÜòùÌÏÖ|ýÉ¿_¶ÿ_ùÝ×e¤ý¯ÿ¢Ì£}9S÷ÐÉ~òÿgݼ¨¶òþöµ³æ r|½Æ?û±WéuÅ/½ë=p8°\ü<ìZçÃ?ÿÙ¢ <Øîã¿ÁQ|ûUÒ!Om:ð5y²Ï"> üTìyÜeSöÄF>è:ö½æÀ×=µñUÎ){ö}$k.Ô¸=ézà{ÁÆÇ×v·êk+í|Rá&C®:ödÿCx(øÁàuTÞì{wSá ,(<#+>müôëj«/ùצumø¹û\¸ãà³:¤S|7p]f¿ÄJëö ÄVõ)|d·ZVø"üsOrºÕþ�xß1c]ññ)ûrÓÀ×wX=ÿH±Ë3kí÷%×ù³u¾Ç}ÁAV\ö[ 6?§Ie®Á{:¡Ik¾?¦ÿ ¾Ïû¼3ß¿ñ¹¦o|©ÿ=Üò<¦pªàñác¨6Üë÷Áº}>©øðíãË=¢¬+ÊܯwyS°cÆn÷áÎþ½j÷ñÖw¿'üSePüY3æIàßäÝ%×÷¬²?sRioSä m Xç}ÎyVé2+¼ ¸eà'ý>s¾úyÌgêQé±í¬À\_zák®ß»}mOõÿ®yanQä6\¦åMý|bïÕUçËÚï=gáIÀýøêþ¹ªìëùµ"YGªlp$§[~«ªó5EV±®û7{ëV"7(ó¹ÊrBá2V}à##§|ªca·µ¦¾À¥æU§\_yíPlöR·N5x÷ËQÏ|y¬ð3á\Mù>í[÷·»Üßeen°f0¶V³®èÛµyz¨åÇþÇËÊxvþU60ø+ Wè°»Xô3
ÿ¸ïó:ég÷)ëü´²O7ü0áz¦t¬óí>õbÏjx³øNGýká/Ýòûõû¤Çe,ÁGá|HYæÏévýÏ#»ð+ÿc CNVö=Ço\_dmSúè\øWÝjÙç«uÒ¿÷ÜgÈTxJëþtAYSøÆ�\Ã-ª[çCr¹Î\>8µø/ix¿è~=çëG=ÿúòç9¡ô¼;ÿ¥\_¿®è»pjðñÐÎèCø4ϵßU\_s5ÞrÀáæ^Sl&ÖyÆÞ×ýÒi[Ú¢aZásÀ༢Ópû\R¸ ß'ß@?¹à¼»:gïÛ=Ê<ÿ¡5»ë¬õ¬³KÊ|gÁºÏçpBÑ{ÚØÌøÒ®(c>¾|Ú6zv1 Úùû}ÅgtwÊwª¥Ç\ZSôY÷2ß=úØpZ±.ó¿6zíÁ7ë ëÎËÁZÙy÷èÍÙµUÃü!öç;<ç]þ2zI U¶#YÃàÁËBAÏc\S¤¼ðàÑ.ûd:mxTá·aÛÖ\'ÞGß/uC�Ç=Æyö û nz k×~E÷£_Ð&^>vk·µð/cÓØ|^êhËïRKö;åzíQl<Ú=nÑi ;¡¿u.ç
7þklhKá"º?nøùOï¬9Í]N·Öl8æðåOøÙÇä¼bþ×õ¼øuGÇ9?sÕu¨6c«1õJùvÉ]?[ºÀ-¿ pZù ø">N|oÝÀ5à_à&%»XÊ3×,\ÞÿXÙçþ ·U7kßÿؽµ~ßWp bV°;ï¹í.øÙÇ.ßÎ(öß=Ûò¨÷_¸ÞÝõ>¾óGaÓ-+|3~ì©ÊûÀiûüÎNeÞ#÷øÎÇzbÿ®psÑ¡?õ±ìç«^û°ïlðLü Ï}¯Îø_ë¹²; «ÿÖmÿXá"ÝrKþ½t;T>süåÍ6÷¨þw?çkáÛ8@æcõ]Í»7÷>Éç~vVñc=ó½rRèè/ͽèô)OÝûÑ}ôÝFÛ¾Ãþ¥ëö®ºïr]õû/Üf¬÷76úp'kà]·æ>gn¯ò+ÜVbpî¸<àûØiüÿû«Òú¯_p^oN½4o{ÅõæPu|àvúD×¼àüÏ~²n\¥^{ÜmT¿Ç[à«ýØϾr{!ïºÝ°õëÙ_5&ϸͦÌìö½®ø=ÆÝöKNë{èuõÜÛúÕFßÎ3~µõ§{WY¦ü»æß/ÆÝ65ú}d6ñkN]ç[§ùË]}Sóî¾Þ~gsæ}
;ݧà~§Ò»ãr\ÞèßAýî::Ãá ÆúÓ\wÿ¹}ß³9·5ëë¢û¤Îýû¾Û9ÖÚø÷a¥¿ËgqYo9¾ÿÃX¹è¶{áô·>øÚç£~\¬zý¬àL7ïûwÉyßpø>=8®ø«pË{9ágî(sÿ;í:ÜÐÛom9Ò]ñ½Z[°ßÿªðOV7ösÔó?¾Mø�ßcþ÷Þò_¸M¯m¤3àÖ÷ëö)|¾ý.G¡æ%c |Îgô³=f.cÚßòÀnm¤¿å÷¾_×~ÜèÃ-þJì^õ_ɧ O¥ÿI±;Á (\Ç%Ãïù]80øÜá¿Àgªçï:íõÞ öx×yüè<ùÝu ýÆÙwÝ|ÓíôÊi®8Ms$ýxÇeºä¶gWí_c®X¿8 üS×aÅý|Õu^u»=õ3ð4À;Vô3ì¿êû3Äãv¶mÇ_ÃG7; ù9Ò~íj×#©3xø\Ë ß¶ë_ï¶v06%zón%þ|2ºlï¨uÛó¸Åq¼=ÏiRaÆÕ}fÄý÷²nOhØoï¾ ¬~ïP±uÁ¤ë7zðx;æ7ýÇ6¾ÛÊWX |5¦j¬¼ªq Ù6m]òóð¯Ô¼»Ø35nÙ»¡äÙº¯×x%fí¯Õ¼ Æ颢¿ÝR-¶|ÖÛ{·ïxôùi;£÷jùÝleúØÏ\ñýAæݶ¶UÖØÿËÏLN9~1üNãìr|ô_ç >«ªØ³ÖI¶ÀIO#Îä ëħ ¶B,óQÅ×ÝÞxÈýs´ Æ ßMt bæ[>ó rSKÊ|ÁG½ÝïáK©kÝÏ^ÿñ¹àÏä~å[ãÛþÅwðC_W¸�g¿6¸?Wí½Gñ9Îûú\û Ö Î:íëÿ¯á£Qd,}@,=bLößVqGòu®ÈN ïÝp°åÖ¾[ù4mb\sÓþn&cñÆ à4ÄñÔñ;
oÎÊ> ÈC¸;|Ì+ö#ÆþöN_ ±Y/¬U®íJ|"² >bsàF̹ÿÉñ9¡·û ê9ªøvgLëeÍ{ðû Á·þÃOÀ>%£ÊÞÌkÆ|'ô!°äå N¬c\_ÕèxÌ\_Ö/8(çÝþÌqÅÏÊ|Üïç÷\ö7óaÔí?b Lè
ÌñS 7éë|ZÙ3bIG[¬+ã !ë)ëøúB«Ç²Î!spChã9ÅÇön¿¯±Ï^ S->XCYÐ3gÿmO¾k6óy wnJñ¥]õAZ+ÕÇN ü4úó¸¢àÃigæË| 'ý_hÚ}4¡È~Æ)eTä?¼1d ó±4®Ì+d,qÔù´UÃc,ß>ßoeQüþ=/æ&y2®ùßñ^dºß>eî Kñ£ÌÙ£p%I½/°#ºLÆva|M(>:½\zoÍùÊwÈyu»¨ñôË}Åe(»ô+Å>'û¶{Ã&aYRìó bJá/²NSÖï] g ~òaÑï?{{áG¿L¥]KpAª¯\6zõÛ2oöiØÿ?¾/?Ïeϵëg1ÉÖ¡QEBo¦Î(Ü]¸×p4·wб)\_ÇnÁÖÆO<Ù£qÏ a±ýGƯÅ]|Öõë.1×î1VwøvH=wÎiÀSÌú]mpCSøêýËü¾¢aý{TÙÓ§¯¬£gXmæ ²>2 k¾O¸¬ðà±{ªã2ïà[Rô9
ç>ËZ{A٥ߦø5ëZë3»}8\¸ìð»þv<ñ¢¼}ÿ�»5wYcýUe_tl|µp®ÑÅO¶v-¹VrábV8)^gtÇQ·IçÈ]WÖÔ=_÷@;óØÙË~®êÞ�¦½SÙ§Êz oyH:Ìã9_»«Ì½uÅ'?÷ºù¿óþ£ÛÁm×üfþVä=z,²�Ûþ¡óEnß¿ý{aU¬
øç:·®cý6ýíAÀ\WàAÝþD¯G©¶º ð$À èst3Þ/h¯²ÖtÝ| ~?oyÑè£ËíXlýÎÜ©ë{?øËàÓWyáÁlå�Ãe®T(èä"s¹n=¥à-§[Þ¬µé{awÕs¬WØèÄÁçcíA~áÔ1ÞÞ¡îÈôìõe
o ÖÜ8ÐÒ[ÊŸ¿µ¼ úmÏ\¶8¼JÖ¢ëÊ^Vìõ^rÐm?ª¬e¬ëÌ#rÀì»>±C!¼·ò®´ôXÓjSÖNæN 1ôɲýg0GðaÀ£|Ä!RäZµçM\_7EÄNèò�=9×íÿý\_Ðó«ok^Á¥¢=Ñiô eß )EfþiùÎ(<10#ø-ÓíÀ×dÒíÜåd¥ÅúrÒmËͶäv§°«W\³ ü|wûY¸=§.\µ û\_õså/ß9ÌîY¶Ã ¬ë \Gdò>ëÑy
ïSiWྱîo}@?Ã+§ ¹7åtx\_À¢Ëü¢ÿöyªÏµ4ÁEÁ¿ðkÖ¼)·ÿ¾örÍ(xË EN°Ï!>vìã|Óm¿¾8ò²ëÍØ
· RöòþVYK{GZ¾'\_v6¾²9
óÊúÍ1©Ì¾8û ¡§1\_Quu>UÖg.{ùßÓ-Þ[¡|ù[®(1îO5¼håOûÀmÈþ2ÑƯñü\Ù³5¾Þ¹îçË¿\_cå²ë\¶zù¬¸Ø'åÒFÛl©2ÿ¢Ä³¿RøXÕww]æ Õñ¹Í¯¹Sòo\_ |gøT^8¯ë.ËÏß±ÄE/(ûk]q_ÿy£ÝõÁöÏ/¾Û¬²Ç÷Ê>5ìsÍé]÷];ÎlªÛG.ó9×ÿÏìEQiUíľ?¸._¹Ý«|_8Íý ýÏý§¾rY~ðÿêjõýÁéÒ×ÿ>¾Sdãi
ü«©6ø²GÃÏn»]Öu?3å¼¾Wö1zÐÚ±òZsÔ¥ìScò¸ÏØû»¬ðKÏ?ïw=Ø£î×÷öO]¯WþÉç/}ï¥ó¨ßWÜWðX«?kíûG÷ßOTí1é÷FÝ^5áD½t=>U¸%5W9:py>VöK¨yñûaÎçyeoí;øüù¬wu>£Ä»£g;øF¥À®2+{ Uywº~èkç} õ>lydi½æúeý ýü@±ÕÎ8]bÑn+ü@pÓ>Kì ~eüàqÄÀÃf=YÖ°~ V>>Æ×V\_)öü¬}#èmàHó-/xø¤±è?KÕ³tãí æÛõHÚ c/Ó¼sK|/ö8÷1ÌúcC±F£W¢l.:9¾klí®+qoga¾ïk
ág|þ£µöú7ùÀBèÜôÇQôêκù]¸æó-mÖõÃÝæÜfc®/¼nîe·ÿã7XhíaA±»zÙÎ)vÕ9´7ë)8ÑØèÃèo#nËwï¹?Þóõííóñ¾Ë &uVõ@·ÄÞ".¸nlCÆç9·¾W þ3±½S-)÷=ÅÎSîw¢åMð¿à#0ñMÐöøº[±8±2¥Cõ»æöÖÖÆõûBø¨~Ú¥áÝ N/ºÊ¶MÈ10k¸à¼Ôò£+)¶\4ìrl4dö¢²5ó¯úis4coB Ç^ òy |¡øùÌ=ðÝE§"õÿÆ8á¶
?øs7§p¯9=±û¿5² Ù îVíný¦c´)?cõtÌå]SÏíPæ4x¾Ùпîn÷Éh;(/mÍÍZt;oN\<¥^gÚ3ãh\ñ¯mUö¹Þê{È°=¾Æyº=ó¯íݽþ þ1ªÄÐÃåÀ v0¥ìs@»c£·ßØtøÝÁ!&}:²Ö3ö´ráwé²5¹o¶ÚªøCú¸BÖLûçìRæ!ã¦ûséÇJq¾gSýG]nìØê»w\3í4®|£d"ÇÛþyhïïj×wµô=º´ ²¹9ãkÁÖñÐÛÄÕû Î;[¡OPFæ y3Oà%÷½Æ<{^á
Õõoeýweßï;/øÃÐì_±[Á'ÿîc¿ÛáúÖý¨àìwp·ÖZyµ4§Úûøìöo: ÑpgôÇ8cÜqG»ýwãg^´>oó~¿üK, Ï¿µ9´]á°1'ÚoÒcÝáÞT»6ÓÞߣÈÕÙV7ÚÞ |%tiì.ðj008'È] öL£ígõ¸ÆÑ(¸08Ø&vóòh¿©×ÿß®pAàÿ/ߣ_'ÿ§PæϲfpmÓOÕýõüP0ãÞ¾ÇS×ÿÐú1ÍT¿ç~ÒÅVBç¨<:w¶gmh¿û³íYt[l¬÷%ròa[iïã§aÚn³¾«añð1È ÚuØÞï2qßuæ>7ìeBà»SF_}_±ó¾õó/]µ0b¾ ¿)yµ¢ÄiT>Í>ÃMéþJôù~Ì<ØíÀGØKqKØØàFpÜÇÄà×oö Ã^×7ðBñ³o >Bþoï|¡ð À4Ð_î)útaUWý,¶Ãi
«îÑ9×çLá¬ÛûàCÅO»s@Ùo; 79CMàwÅ÷D?»ã#DÇçÿ¢Õ~ï} c+]]Wö5!v¢ò?¶¶«4 {+¼\û«Ç¼ªø£I>¢|ö_Ã!¹öÿÉË{O±éö;/øUæ+þ½ßÏ|ätú§Oñwc§Ãs@÷BOcNÊÁc¯¡.Sáì ϼҰA=X'+ÝUìv8·ènøk¶+ëk/¸òÞégÞ¹¢àeuþÀé^Vü¼ÌE
#ÿãnSxmpàÕox4Äs®)±3];äô¬ /ªqE\0þ¾ËÊwȲ#í>¯aoi#óðyûVýtGñÒxUæȲ7õd<ѯ%ÅÿI:Øi
ózJÃã eE#^T8U¯«ÊþMSÙ÷¤BV£ÝáÆKÃoÅþC¬}¦ð^;^¶,¼¬p]ZßÃÿg>P6Æ cÌVxÌZó(|¸¡¬µSʼdo!tAô øÁÕ»ÞòÞ,|]æû¾V¸ ps(åÀÏ~$]8Høúº~WÏSxèmàÈgݾó.ó ûR©÷Dn!ߺÞQåØ¡á}îX¿á Ö35o=¤ìý¼¤p¯ëù/(ãê\_[Ï=híVÇe4t9ÿ¤ìé '£'ΣäÙ¤²ßbÓÌKd~¹þ¼½«ÃÊø.¿Ö8º¢eìyt#taúE¿Ï+óþKl?ÚÞéçUen ,)º[cøöÝÎ1Xk r2ã³î Á¬áçs,´4çÎ \eîïU8¹½C;ªìÏ®á@²®0Wxú±W$cvaü wçÏALk÷GPú] ~±
3+Èà«è¢èìøÈ:ùN\íÑy±ÈôhÖ9|E ßc½»5=vè8_çÈÅÁóæw;g9¤rhø»ò¿Àß?þ¨Þ/¬´æNe-XQ í¼=®èoÈzì¾Zã¿óøYñëá «YhiÀgÿG^Ìpî?¯c|rä¹ÙWÖÕ}0ÈqÎèêpnWy\Ã'Ñç'Ø<¾áùVÖ«n/ìÙËÇó®Üç@3F±ËÇÚ5ÖCðìéöµ8æçË7¥ÄЧW8�àV¹Vò}yX__9Q×>Tö>YwmSöÃ!îibÛwi±^uÒe»ª¬õàÛ7Ü ¼ [ì¶ëxÅÏñ=Ìn3w\V°ÖJæ?:>5æ÷ª²þO\ã
þb\öxÖ t ÆúñÖ.¬´-²ò¸²ÿí-Å6ëº.c¹¹ªá8Æ2óyñ E·aþÑ&Ó1ÆoÊ|kÏ0þ± (ë9òu^Ùk=[¶ÄWË\ à´ßñö¶ø\_gíàïFßFÇ[%&§Þû@Y(G¥óf¡Á5hiÛá&oü Ó ¦Ã E3ÖÐïY;ðÇ[¹Ñy×ÛïSí¼¬è
àÏûÊܼ¬·ÉZr¬Ýg½b,uóû]:æe ó©Ævëm»Úò£/+ë-ë|½õ²ÆvºëÊJKõ . éáG�"¶}>k^÷UÚ-òêø,ãõa·¿y=}pjìmäk=ÜÅS .=ÎA;÷ÿé÷æabRÆý
~Ý̵Ãï\e-§Ìû[öµ þrôxð¦Ú»à%ÇZZ#§ÏSø®± ;ï¿m¥¹CûèüäUey¤ÄÅ27ñwÿ%¾]ë 6H_¢s°öÒ.pòXSû·§¾0y|ÈXøbèÝø?ÜÃj£]JÁuÆ]Þz\xd ~)æ$Ø z(:Ä9#amÄo 7�n㯽¯Äݱ¶bó¶~" Tý>¡ÈgÆë ߣǿg~̹_²®Ç ÃÁSèëØ:ø&ÄkÊ^àóî¾w{QÔ{ì#2¥èì¥N¾ßíTuoÎÝê\ä
v È0æÜ6e~p¬\²|§}G¬Ø»K-êN9ºN²Û÷+Æu¢µã)
ë|ÆÇMÉÜÿÜ>¢è&uïBU¾Ùò'å©÷o\Uxì§\_m~bÓìe½ïK°cæY×gà|ð<ÍýÖÃe\ñý¶nÇå/´<Û}Ú¨ÇðpÈ1E~vï>q6¬µ[Þ%k´cA£àbìWÁ5Ök°?ö½ 6ûÙÈ3øµû(ÌYpO8FØÕp]ÖÍ´ß¼Ûÿ#ñÀ}ÁZ±ÁܪÏúþ-e.bcõ¾£EÇK1©¬Ø_ãÊ þ®s¬å F\F~t{¥¯Ø¯
m{@á°ß$x̪ËqYYïÀ5úþàÙÌ¡.WøÑÀjØÿ-¶Ø®cwuÿæx 꺤Ä3>áB ?5Pbuéïn¿÷¼±ãÐÅ°÷èêÜ^wù;¹¦à4øY5¬·ÒØrèt{vçÇ!gð@Ð1áì\_¢û2i\ÇCE/ÓCõuhpMlßJø|Kç"OèGd¼öÊësôµM}EÙcï¡Ë »k·9;ÖÒº±ñþ öl¬#N9·¤ì=¶Þ¾aù0øÆò¿9VúMþ[Þ´ã¿kÖ¿ÿÛãöØò6täJ]ÿèÍõ½çA}é÷S¾åþ¬ÿÅû;çòbóV{<ó>¼µNVÌé]ç[zÃN£½½ïÏäÇnûjoïn _>S0¥No ]ó³ã.ÛçùµóøÜõ<¡pêÞÏ §ãïW
ÓÕ\©±þ¡²×ÿK,+ż®±?ÕîcO1»>�ß¾ë6k1x6k96g»ûùPi(Xá¬W Íà /|Û¿s¹îmSö3Ä÷KºÓBf3FnTüRä §¸ pÝqß;¤Ìé?ï+:$ó,ûóËwª¥Í¾sø£X )ïA
;^ÇïÿNóèÿè^äp~Ðñè#Òv;Ò¬¯È1 û°¾W;8½¥ö \|ð¸ï;ïð·ÿpº ¾=²X½]Ñ9<IJÍ)q6JÇü.ú3v(:1Ö@tyt-y]¨y|Oᢰ½i-5ßKÞ½~óû¦Ëhyüv¿s~=~sû¼ðw]JFþ¨ègÈ:jÌL{æÿUþ;?\_²äVÙ
î{u½ë´(ßqy?ôµ/Ø{^2¶±à¦À}GO�¹¢a}}N±=°UèØ»ð6ð
Ï\ÜQ¸I&|qøç|C ùY÷y·2áçé-otË}·IõׯN¯ú©ì·Zwn\ßµ©¾øÅ÷\_øKÏlùg§ýÏÜ¿¹?¹~õNéÔ×ýÌ>W~ëç¿tiéM»|Õ÷QÿK9ç:Ôxð{Uþ·ÜϬ½O|ïÓ¨ó=Y3«~?:ßOÝU÷Zï¾iéýÁíþÄÏTÝj~~®|Óé ?OÞßùþ §QmûSûÿ«ÿÁ×òoêñ¹ßñ÷×è«}ýò}Ï6Þ|ÛöúÄõª½0¾w¹>igïÚïö«v}üàG·ù ÷÷]?[eþ¯÷G¾^:Ç ÷ÅÇ®Ã7~®òû£ûó©óþÎeþÜíRuGWyå:|êÿ\_)rz>vÏ=þÝWÏo&}ïß\_¸ Õçûà¯}é4o)߯a±Nå5êç?Qöƹíçëÿ×ójë¯úöÕ
#sOù¾MÞ!¦ ûl¸§~ýX\_ÐðÃóNõ?Ø2ul¼dN··À¬;vQÏmWpÜúOÜ +>Î ×!þ ܬ~Á¢¢£Ïwà¿O)8À>% Ü´îSM ¸~ÖóI%þk¦åA:ϾM×ñe-´<mÊs\ñ3ãçZðõÝLÔßïÐülÿÁ%'þ)ºº:öf÷wp0fIw´¥ß£çvýdç?¨7<8tö1ëb\àÝÛÚêDË1ÕË;¢øÀð'¶üûyLÃ1w5Fß÷µm qL¾#>Ã) ²Û)ø#Êx«Çg�^îNL !ýþ|¤ÄѱçàAeÜP°¡Î¡ÿã9ø5pæ}úúu!åÿemýÆÓ}ºõUù¾ò|Ë8ñúÍå´ÏÙ¼å× ³äÚmesÐÞØ®ÌïÎ :¢Ø\p5U̸ûu_u\qÙ µû@÷Õ×?ÿö{AÁ©'F¥IÜ>xí<®a]ݹfEº¬-pàQò\8úìóI+uWö«õûèÉÈK0bìKpðCü7£Ø²Õî¶{Ùß»ÄEï|{ærõùH¬x/ø>s±;ý®Ï»õú<ÏØô>!üäØÊÈå9e_U
cZÇ
ÃAÀóë{<ÅuN¹MñÉíRìa°^9¢ìKìá¬bûO´zs¾ewÕ¤Ð:V_AÙ8zì?æÕ6#Øír×±CÁ Àv:§zKÒøng½�ïSö@8¨pNÐ]¨÷Y[(ßEìPÖlÆ(¸:{ctïÙ0תïK7ůVï£ú{o} Ä»®)\w¯L«ìk.Oéðçýù¾Ó,Ûî¦2Ø ç+õ¬¿]wÜTdö-ûÎáOdïü¯>:(\û¾'Ûß&ÖQh7æÁmÅÆ;Ïî¹Ó¬ø¡±¡ _m×ð
¯){)s{1¨;ó=õßì³Ùì/W
~^á vü9Ò9OøfÈ5 ¿ñÕ¶7¸Ö}¸Ý¾
|]k}NGS¾ÇC]àMÃe×°ÿn¥Õ~ñU%æ·òA±ßÁ¢¯Ä¯wFÁĸ~Ä N+~"ÖzÖä|¬·¶ý
¼ïø2L~�[o÷N+ß°ÓÅÚ7¡èE¬aÄó¯ã+Ã^±·1s9QIèñQcí7~[üüÄoR¸¥C?#ãL¶ÁF÷màoMÊÔù¤¬£ÌÏMøsûb;¯+òf\ÁpiZçu¢7ÿÜ9ÞèSø[íWÆü5
>¯|ÛÙtUá&p9à/ôõ {;_ò]À·lÖ]·ûD)º#óû´²w%ýÚ¹IÌob3ûÙ.zÒ}½ªpyª sʺn»¤ð°°éËi·ÇNeýC_Ã~.õ»Æ'ëýò¦<¶<¿è&£zooK§Ûè§Ø[ù
Ï�ý ¼w²¥[zÁ
;£¬ ÄקG_ty4§øý_ø2Æ}²x ûX¶¾?ã YMIWÄàEMqÄnVÙkÜUø$¤Ñ×98ßK-_xøÑoÑ[ÑY)/~¹Jk»UÖ|ô y¡£ïWb|h[dø²¶71¯'ÜÄ>¶¾ó5 ïq4´KoÆ>BÏæ9öè½À®{çÊ~îÈNtIqƽÜ~Ã;hu½g½ÃÇ-ßÓêÇ
ÿ½ä6¯ëìá[Ga´×]¦ë- çè×PFÀØ{b¾µ+sq0çô/¶g&
½}á1Ùh¿±çÐ?á"c«QÖ8ìOë>r¼ù>÷±uÁî+k/\¡
ö.k>õ=¢ì± &Ç^ ì[v®n ¶þAå;9pÞæÛõ|úñ6ãú\û>¾>ÑÇÒ¡&>ó±vç¦Ûÿ 5¾Ü}ÚYÁxSæÜDkoò\nÿé[0Êjç y¢=Ûßc CC9ñùÓ×cÝw)ãSÕu]ñ1\_Ç1ÙÒoý°Ke~nîìhx®u½ßZºÔïhËsWû}HámPOxräÁ¸A1GÐ{ÁÕYà%½üÓ~/\ºGï³ ïyPý
ÿò2ÿ¶ô#mé±æð=n~çEÔû#>ôë!%Ö18§ÈCt äóþ7åPÆ3kzø ¸yF9)ÝÖúôÁbX§·:nÓÛõðZw¯¹,7x£:³=1 ðñÑsÊNÇÏw±À%ÀoúùêSÖ½-/Ö)É6ûýôýàÇbòÞ´ó¸ØÒ½¥ÄýÀ±¥ÞýMyó¢{(¶èwÕq¯38ßî-øå ç5ü½!¸¾ûÛ»Ï^@~uF×áϯ¶ßÄgjy°°)tKÆθËÓu\ôSÆgxöT{øÖ øÙå\_EÅß/�=³Úô"ûÄø9å«/uÉgx.àåC®qrWÙKLý�}ÞRû6>Pöaì ë¹ò×OüÛ}ðMJaô|¢?¸\ß)1'Ø.|tÁõgNt[wBÁ!°§ºß¾ÙÑú/úÚ¼2κ Üý¾ÒWÖ"ÖüÀu~W±][õ<üRä3òϨ2Ç\_+vëtË{TÁK^ÃcÙí¶s[ÝÐp,\_ç"Sßzq>ü¤õ:|½S² [÷±à1×ñ²Ö~?{A± ·+ñeÄea¨ôKî î , < 64¶ >Îyç¼ã}8 èrno ÃÙd08}?xØ\ÇFW| >Ø{îZk;°}J\:ó\_\f:ÆÛýE®ü¦è£c-ß#í7x[×Ï[ØðRá|{ì~Wþö{×}Óp½éæÚ¯ <ê¶øEÁnm´ßÕ[,¿xÙoùÿWøSð\_lc|ÀØûèªàHø&Ûl2'ê~çO¬�ß)ûPÃÝ/.üο+cÔ~Õ-oÚa XXt׳N9?t!pmì:tü.}¼2Áª%Ãû>ØZ[¹í1©ÄÅîUÆíR»Oºþ3·xòÝtæ}Nùº¢·1fØ«¡ãÓÛÀÃp¹ÑuÑûÆ[þè«ÝOPí|Zár¼ýó ~O^«Ê÷{{¶ü'Ú½Ê>îý8ÞòD.Û®öeOäë÷Jì åتìÕÞÓP8Ƭ/øõЩuÞæzv½{T3{ØÛ]ßñtk_æ�\¬í~ß(s_by:æ:a\SÆqɳOýÞH{¯Û»FÝ·9O°þCÎßðÖö^ó=Å×Üý×ït0OYoÀ°ßé#ä%ÜøøÃ{ !ë!: ó ,ïÃ+¿Oøkãã#Pe¯õæþªcéuv_ú÷÷¶ýPgàɵN¾rÙK·aïbòïm¼·åM[þ]Á¬Èʾ´K.û%e/ÿ·õ,EÖUð#æi÷ýUݺªè^Yy¢ýF¯; a ë´äU:d6z\pÀ�á½#±éÁà¹<[ïíT8=}/tàÝæ [kL)k1Ýg9ÑÒ'/p´yeÝøIÙ×þo)ô xNà£|qZÁû^µø
θî+6æªÏp.9¯;.ɾ ¤]ç+í=ö^XqÙÐ ×»8
cNþ¸×ú¨òd-BãÇ®«²=p9O+{7 ëð+²W ~9ì¬eÅͽOÙ3ºv�\_Þee½ú³gØQÅæ¬1x×Ç)¿{OÙWÿÿ]
k|ÌõºÜúnzÄ¢2áí"N¶÷î)v< d(û']PöÆD×/sºÕí¦bASk.3õÀ~#ÿLÙH³ß{:.»j|| ì· =¬û ½ïî¹Óß;=ëè3øÄØ¡¤Óm/ð¬ÝÊø!6®sSyßÖI÷ÿ9eþßQÖ8ø!×uå[éø±®¶ôÀº= ¾wLásîú,òeÁi®(¼öx½àwn)kCû}ehéò5üù¬3ôße
gB, |dôº÷@\_>¯pJZYÕV(òØðDnÁGU'~[ûòªûnDÙoü±'1¯ª.Ö]ßxYÁÎ)û ÒÿpÞÙ§êûÜe¸®È6úÝpVñ%·6Ç\_Î�øVá&ϲÛcZY/\ÃÛþ¿ |O6=ïºNµv�£©¶a ³ nÚé/(6{RWöGæÛô³¾vÙï3WÀñÉ\Uðp$ôH8%àÉcÊZÆ\¿¨Ì+ôö§'h}þPáÎL¸ýϸõÞenQðDø øÑsÍþÃOrÙÇj;W?\SbY7«G\_4ò9xYYç|î<º O�¹NuAïÄFìT0i85~ÀäXÃ+ó½:°©ñ±££±3~©;vÌ ×ñ?1&®@}ëÌXè¼ |îðسG[Tx¶ðÀ¥:FomÑípÞõÅoÌaoàÃ~ÚmyÆmµE« ¶ÅØÓ1«àðàkXñQ·eßTüËÊ·àu�ºßm¹£ØUpV°ó¾VtVY÷ñe+{ÍaG°nûvO+q¿·îËɸ@×ny ûàP_Ãn:ãrñ=s0lÕn[¸·ôþäI§{IÑ{¦(ãXtÚû_¾÷jxxÇÝæ³î0bô?æ6ôËËz°¨Ä¸Pì3äþa%fcÝi±Î#ÐÁEÁN:>·pÑ{úþïK ?¹Yu×Ê:M,ö"¼¥ bw^TÖj×Êú¸®ø!Sè8Èðn39,6.$ümÓpÔÐñy:¯X^µ¬èÙè�øñ(Ã-Å\Ñ° ß±ìnGÖ«Ë ¶Jû¢¿¢{W®(ë×ÐÑ·ÀU»þìB®S-YÃàïw. |Rög¹¤ÌIößé ð-8!àø·ÀØW¥êM > ãÊ\/FÔ.}4ßú¾>-l§Å|§ù ?µëìuKÚô9ß\_Bÿ@nOµkÈ^ìGÖwÆ[zðÀ|Ø x¸v »#OÐõñ7ÓÔ µ5<6óõõ¡ÃÇN\li §Úø×pàXó\{¥QGæ þWdÃ
³[óS w¸åÉ÷®ªMÁq\¯ßüÉß´\_Ò0æîweLÙÿYÁþ Ø(7Üð\_ohg8£èÙpÒàF°®[ÒÇè
fU£ÝØþõÌG V¸ ìÁöÖù´ØÈÄÌ \_¹,¸±áQÁ1XUxTèÌìAÖ¸®»2?hü$û[kAø+¼PîWúw\~üÈ3N9J;UÖ7ö¥äcç±Í×MÅ·¯vªþ=
?e3îoû÷º¦Àÿ>L;0\_h?æ\<Ú´~×ü wªóýéxl¬¹ÝVÃ^yêëã-ïÎßCÇá7¸56?±£ÊxKÿv8¦øiú:Ï.ÏÇ#¾fñ6ÀCAÇCñv½ ¿Íî]ãdAÿ6ë/¾+øN+ÈFæ5¶öùSEçp}ÊæQ°vê0ªàE{Äx?ïçvµö"³süá@:Öéw; Þk :ëNÇ;=ÀÀÔ».VåCÆ7Xm³¦p «-ßÕ0ªàèúÈÒ/
Ý^Ãç~b¡¯ô5üô%yrYH¿àS8Ø;NÄóÓí90Qüèøêè?bÒÀH 9þÏE ϸ¿½Þ{_ñ÷ÀåÅGÏû~²Ù1e?(ä3><ê¶ì2£1ÑÞ&1L3çÇZÞø¸&9ºSñq3þáî Cv\ã)~ümkqQùþò¿)>w°É=¼ÒxOEW8ÞàOs\ñëíl} f¿¯ïV¼Ágѽ¨ 1×Ø¡{ßæûÚ9Ä}G¿Ã}\_©4öùØ¡aο¹\_ò´Öbt¾ÒÃÐÀ^À5À¬jߪà ðÅE
GxÈϦè²{A±±½Ô±tUáf¼§ÈÖ§±Väï7 ¯ê§wÎm²Uñ=ÐÞ»|}DÙ®òüBYWüþ+µägÁøÁfá×¹¤ìkN~È¥Ãí~ß{öaM(:Æ6eM3D×Æ¿x18Tè\Þ¬9¿Sìcü^åÆÙ¼~c#'P×j<ÝÝôNµ}ñÅN·kèlý?¾TøÚønµç®é?§ÿÇnéi¯·ëËL§6=½t²=ÞBý°3ÖÿÁ»¯.{gS§Ûox[·[ݱýÀ6¥ &;¯øÐA(þ´ú¾K,6¾nüpªáI¯·÷èvývÿ¬ßçؼï®êÍ.Ïß.þ útºý·ÜÒA8¹)ýËËÞk¿)>æ3Ê^|ØÈ?°zê²ÜòWíÎYkïãKìïã«BçÅ^·mÈ>Q²Ì68éôùÎ:ûRoì¾õ°ÜúºÊ-grRáÜt»÷9~A×®?oQ1.iÊGN+¸ã1Å\_Þ±LÚ[¤·+m\_Á!Ýæ'òDûÏh¿)ÓME£ÈªïucÆÇ\f¾bïð»d<\p>¿zß) Ë 0Òç¹>FYWu to)¼5Ú»áBËýÞ9>ØùÖÈ#æ :>Á7i¿sÚ×k׿ڼs¯ü¯(kþ;¸+ènÈ6ÊAL墧b;\me;îwüõÖþäy~²õÙ§ ·ø&üùû«ã»±øß·ý§Ù¼sEÁÄ/(ë~f¸[mÏ ïð³\lõ=¥È#pwd>}Ê|¾¡ÌQú~^Ãs Ùò]ô5÷1cþ^pømúúÆ÷X+À ¨ëÛw@ÑíÁ °Cº-M|ú
OB{e/µ6¶'æüòtø£ ï~\Á\_ðtü{ªýÇþ|¢ðfð
ÔõÛ /çµbOLl¼»å¿(cL{»pÊ}~Wçô²þëAG/í±¾pØg¹ÛõîS ¯ÇøÒáéÀþSEOkø½¿brzpMë9bìp{GKulW¾}|Àÿñ/Røéàë/8
Ó|¿ªÄÏÖ/AÓ¿¬éÆe«Ë\_ß Q%æ\\_û÷% ï:�8áö[SÆ ø>¸÷²§åK×|¿¢ÜVX3YûßÕ&ØVà$ðqë÷^ÅOµßù£«QðíN·c àRØ:ð¿\ûð±fu'õÈÿgÝ¡M¿¹GÌô^
}·C]³·ï°ë:¢Ä# ¿í¸²÷ý»Ó}AàW£æµÃÖöým·x¢ÝʾÌqxõ{l<8
[L¶tÙKþv>Ø� "k¯)8'zcq°îc¾Ïï{yUßWì³ e/E%2L(¾ù=Ö¡Áü·|¦4£ðHÁãÀö)ó¨ýJ\±8ã>£ópo¢µ9qPàë¬9;,®Hµ×ºëK}hGüõ# ¯÷;ö7¡èYÈG°rôð0Ö@Ö¦ÊÚ\_¸Úl,ñ¢ÒÇð ÑצØ#æçnÅ7Õ÷ $ìx
ßølk'0 øÍØÊèùØó<íþ\k ~8Ð6Ä©rT¾·½þ�z1zÜUøSèd³-íÎÛÓ~û·ëæÎ 6vøT]¿``:K Øûèç7½;«ìQ~'Leìú[sNá©õÑ>èÅàÝØÖà-kN< {2Áÿ OÌÏ ·àÖóøC/µüê³×[,µ><ëÿ#ÊÅwy@áaãÓÄ·ÌÚȺ?X3ì\x袬ÑUo0ÊÔ9Ú·>]ýGNÔÜÛ®øW¦]Æ15}À@wC'GÁn@\Ðð>å6]«2ðý1
@]Ñ;ÐßXÏ'èÇ|ó Ù·KÁèm½Å Yð±±VCuÝÁ宵t±µÐÏá nz2£ÛòÎק6]ß«áïóà¬vØ¡èÄ\Oµ´à£ÒÎÇyP\_Ö¸ÍøðÅs CÍ´ç;' ß \µÇZö;Ew p#ȳ·é÷5m´6=Ç:´¤ØÆøûÞÕ·#ʺ]Ç£MÏÍ·|Ð÷·ôN\<¸þ|ñ#íYÚp¶½¬í>kÓa%®pRY¿×ý»ÛhðÐÀbÑyx¬¬Õ÷Þ¼ô}ʺ\ëÒ÷:gwú<¢¬ëp\_èàøÀ à0³ °ç6ÜMö¿ªô\_\¾ç &Þ9J¾ÿ²þlUøvðκÜ;Ú=ì.ã´9¸¾¾söU¥¢çéð D0ÖÅÞdå:~ îw§~5n)öæ=ükèØèr}5
Û}ɽÓí7(ðæ¾Aðñ%¾ú ²· eméÇãMé÷6@ê¾/t®®[¡½X/Û^ß;>7ó
ú#ßøYªOàIcOMj1ç[¦ÊxTÃ6ØãÖþ¼Kyµ ¹<ÝÞRðC¶/\g ÃZIösyBÑÄfÐÏðw´wá-À{aÍàþIÖ(d&kôñgÇ8fZ~ðZ&}PNÉát» > æ){×£íjé;Ác»¢p÷{Zºø;ù.·OÑèg¸>Síytì³Êú ÎÐ÷�oc,¢ÓîWlïåõ?Sëkx¸ 8,ûÀaTö<Ú¯ðO(ã |6ô8öª«L\�øëðDË ¹÷ õfAµ®¾ZS0ÓÊ÷©°óá7¢s°~ ãO¸'\50ûºM»à èÌðú×0ÁÜdÎW:Û,¾@ßc¼¡É0/Ö|h¯q[OÜc¿bwmWôÓ) ï´_á6¡+ÀQ>ªØSuÞÕú\|NÃ{ð\jåBï[R°cd%rcߦ, Lý¶sñ-àìò~ܪÏÎùGN7}Î(¼}ì±évôëc-=0ZxÒôë´¢Ëjù1¾¸.ÊØç×ëüÁ©ÖÏpáê̸¡=×è}Ê·èº_¦¿ ~ËXÜÛúÞ6\6æ2 %k¾$ìäÈXK¿c/ G=ßumêµÈz³Wá43W÷¶<Àp'[^èÚ´6÷-Ø$Ûq··=ÇÜbrýÈØ}ê¾$Ú1UròòÍÛmvixmíü4Æë.emwí1!³ï9¾O . oÑE¦[:ØuýÔûÛw ¿�UÎ6Çù»¨»ïëEn×m#píM¼¨¯%ýÚ ÆT OÊQLy&v¥&GÕÎö[¾gÚïÜñw¼ô÷¼êhx3iÐþwéø{áú{@ûÄ<èÀ<1P¢ÃA é8¯~J îp?ûñp3æÑH±åêáV}´�ÇjÊÅýF\_«÷«ù°:þmþ[¨o¶²¤ÿÕmë|Oú}h 9}¼@:ôgÎïÀÜ>p$ v¡¹fº¹èºñ¤ë¡¥Îi;0Î0ÆÞZÇYöýi(?Tî¢ßéÙf½$9¦XÈ¡·ñáU¹Úø·ò-þ]yî×Åt2ÁÚd´Ñ�«×Ò±A\ÖZP\jscÞÛË�ºÉé¤ôƸ2c^þ4Úí×Ll\_Üpp\_ÅÄî�Mð6éÿ1¿\_8êÌ{[øõM©#~m?Þ ¿Àqnj®Ï¯û¹Nù\_iï7òXÎoÔiUehî\_O/AZÙÎ1¿Ö²ÝR3my-¾[«þ»×õçJïç{¸£BþÒ»,ÔD_ô'Ð(¼Ì/лE{)f3Þ\?Ðn à 6é8Æu>4i ü¡B×i\¶üß ¡H2�èx1@:ÐÇBlP ô6ý+û~ÿÿ…T§}c@paìþ÷qNá–ù-ËýWò[£uùÿ[ÑRl¼oàåþ é¿·ïaýþ²fï¿ZN·ý-ømEÿ[rM¥w¯í_J7îþ°Å~ßdb§æó§ÛDæ¶Z£r &æXÚ-y©ý}ürèc¦V0Óõ¼ßoÕ¯äílö~fñwaã¥ï}äÀXvlhüuÒñ%üJ4³¨/R«ÓVµ\:S´ÍïgN ¤Í4V|ºàßd;§87OQø³¤u´L.¡ó²å8÷/û- ÌÜ@Yë &³¥ã<_gb®¾ëá9i,|¢4i.ξ;hßñýrÔh=}¸e3TSq6U¦Ðxåêsp®¼°P 0ûBO3¼Þ¸zýtÑv± m·+§hº8ÆKûÐËé3=ÕwhºZKCtzÊPªRQoe8|µ ÝÇ+WNÐ5æãÜ£Óewо ¾ß!ÊV¢©HqëÏ ðgÓXÈ<é¡ ÊÁÙ1Gï1~Üs1õTr¡ CPcÎøy¬ à!KÙ Þy¸2ªD-è{<Æu>öQøÕ~´= IEò2ë¦Ï3çð~åjÊk([I±òzJUÖ#½ªc ò3åùÔ2ï)cÏÅùs;ÐßpÌyuSi8Ʀ#ÔD6å²Xê)dPOqù&êþýÑvi |°1©æè÷@ëXsþO@MÆzj¤OÕbêzYÍoeóITÍ¡à̱æ)ert¤cê4èétwÐXÕÖ¢L-dò=Íãìq(~ѯ3^]¥BõÚ^¥ýê> zÏ!úá}רo¢lÖqíò ~Ó4q=EZH÷¼7 % s2÷6ë7îñÊdæó{çKç÷qýo½ E8Ǭ3ç$ePߧÍÍ1ïMÏ5Êð{ÌtÆ¢?·ð?Ø{ÿ(¯Ïì¾ï3?+¤Õj7Ûn¶»ÊVÖÊLØ1%<z<LÆ1ã1¡Ofdz³ åÉPL0ËR|(áPJ1¦ª"«²jÊTÕrLuUYÕ¨d½U·Ê®}îçû~qïwöí$=§ýã9ßÏû<Ï}î÷s?ÚË÷Cæ¼æáþò¾ËqGK+i±p¥îîÓo¹!õëåøüÐ{u]äE×jð9«ýÍÕlIϪ~ø¬3ð}î¾Ü YËY}ûF£õýÃ)¯·÷[\×r\_y¡\_õíF^è^µ:Øx¿åkðêGKz³Ñ^÷qW¬wð2/éHI»ýáýV'oÓõF\_ÕukUo7üßN?ï#oû·[>RÒ¾F{j8¾à¬Ê÷¶÷© ßTþoªÎ¥üEEÖ^Tú³½ÐºíÍFÕé¨Ê~Rå^O}·ÚçcçiµïQµãPIß\éÚ´[t<£¶ïV?Rý|Ì<ÒxîôªiæßÏéþ ç´ºª2Jj-c½õòÀ¿¿G´ñ>i¤º?:4ÖXóàymOéJeU/V¯¨^'÷Z|>9´èê÷¼ÝówZ>\Ò³ùÁR÷óËTéik\_¨î~¿¿ñ~ëFjéÖxvÄJÑïêïñ¸îkôÓÈçF뽺~Q8 'Ï sA>zçíÆØçßYݵgËFõ÷FÖ8ªÏ#êã9ûîø0Pß'åÁHý+ÁÿT ¾ó¢vçíÎXK{ukõ>8+·9V¥¯qNK4©»ÖÚl´¹34ÆýÝ-jÛæ4®6>ïÝ0¨¶ÿ=uþÿéO|íóùïküØ×E\_CúIéßt}ÿ4Éy´Ï+CE©ç¯Í¾Þø<Mí½C¸CT~Þ£´UeùÑ×!{òõTiºñmíÒ»ÎߤmªïKJ¾;ßcÛð]ÒÐû¤å9 óí\{§¾ÛÒ½ëïhÍ«¯=ùè²Ç)ÿ ã黥Ñ;$ç«{Ds=v)ïÑsêërñÃ\ï¾óòO\íRΧ÷N+ù»ü¾óçÏõîÕá ¹Î×Ê÷è¶KëÍÙ;\Ó=I¨.3ʺ?¡:<¡<mÐ{.¿WIu¸¨:?¢£×õ¼
Ü6®zîÔõÅF¹µ¦¥ÓKòD- ñ¨ò<Ê> ò]ñõx\_øSò¾Ùܺ÷Ö¡:NEIm×nÍxò8 m/VÝu:W ¶]¼u³íÅ[7¬«æÛ«êuÚXMWÕw>ÝÞýéÒÅ?UÎãïãÛwÚGãûʳ¿ÝyõÝ»åÝw|OæD9ÿ
rÜ÷ëÔñÃjÿμêî}á|aLcÊû¹ïòë×ô¾ËlÃÖà;ÞÎo4>].Ú®<]vئþؼ}^¸Ü0¥q1¯{ãz·Oï ö[u=¯ºL¨®Ó\ÏëÖ©ãê´Ce¯×XÐ÷ýª[¯Úï÷ª¬MªË¤î¿¨<½.ô² çõ¬Ú²\_ymMzõÞ%}?§ämwýçÊC~Ø¢¶ k,ÒùêÛ©²§,xáÛôýa]ßÌ^S¢Ïy{Z+½ÃçÈJÕyú×}kt=&:(otÉ~Õq2©ç#jû°êËs×máK®ÎªÛU·ÎFêó.å?(úÖý Êëê켶Oõ<ª27ªÝ
6-÷6ú¾eD÷7ªzg£úhòW}.«nÔ¡Ô?Çô~¯òa=^õîÑõdjÃúæIåï}»^yM¨ÆÙ¦T\_?¾Æ1îéOWTîÕÏizî%½; ú3^Fï!µÑËíê9sÏß¿ª¾êKånIôôµê±ÔÏ]zoòQ{õ¾ß?¢ºÒ§;ôÝh¶ÙByJ´& µÔ1)k¾¹x]õ =N]wµoñ}U×ÞjsJ|RtÇ8E·þ÷QÔy;6qî¨þ®äëßÕï¨zZ¾yëµ®ºõÏê8;Õ|ë[7˽Ö×u¼²Ý^\{ßÿ¸hÅ>ö\_×1~Î)ÖÏãg§bý£ÇàY²¯¿½ä5Vêû^×qz.W8ÒÎ:¦ÒsØJ·§ÐöfÛdyÿLµ¾N;ËyI§íByGÇ:6Ó+U/ù×q}.ÞºÿÇ÷y¶¸«úÈÙªºÜ®ÓHuªýÁêÔªÚ/®Úýó[÷/:\_Òûï:[õÜõù2®¼uÓö¹à1£ÊychSÕ{»O·WÓuÚPæ¿w¢ê]2W®IÃåþÁrÿt¹Ï» ß\_p½t¦^:PÍ6u~÷t5};õë«åxNG\_?XÞ¿Y¾-é®oUÓw=
·ìåòÎGJË«Þ
×¥¯NÙWË=ÒK¥~GÊ÷WË»øþí¹ë%o%ûvɯ£ä×÷þï/½Têz¢ÔùŪ÷î7ßÿ=bXy´n¬u¼ºWéöù¢ò¼¤%®]òµïJ¿Ù/¶üÝ\_ºº¤Þ8.>Tºë¿/mÞUÚy±Ü+´ñoxñ¶òüo>ø>ùùXz¿gKúËøRZ¼½/Äõ÷ºçÇ¥½·nÞ5ýÞãÝ=NãéFZz%ι^úvIï-óà8Ú ÜLǯDZô¹rïÞ¹~u¤ûïüÎÒ£¥¼§¼mü÷Ëqõû½mwMÄqñ¦RÖÙÆó
ï¶<[ j½uóµUï=¥î^ZÒ'ãxÏîÆñî=%{ïñWKz; çì]goMxZjDyK/ô¿Äñ=åm+é×Þ{üAË[ØÿNN4¦÷h¤ÖUÕ)Ï×^ï5oé¼ûË
^?WMßóÙòÞßîýÍjúÞ¿{ëæÿqÕûÁßzïûÌ>Tæó§þà¯g¾\ÍÞûʳÿ½¾ïëÕì}fýÙ~'½«÷>t³äõtyþGåÞ/g?^Ê\u¸ï+Uï½ÿM5ûRÒ\_.Ï>Põúù¨äQÎï-sìeüÀߨ¦?ðãÕì=?Z®µ¼óK
vU[ʽºþ¿Zèçmøyÿ/7ÚÀy>ÞûOKZÔho¾¯Ï«¤R§»¡ðÿÜó²TÞírõçßtrú|øÏãï4þ®¿Ç÷~~ïÿUÒßÕ½5ʯÛùÑF[<¿ºíô¶Ý÷¥ßlÅ·w¿ÛhÇ?Ô uÿªjþ®z«É¥pk¾?¬?ü¡jöCe.Üóùjþ¾Ï~(ëÁgLz°:Úv¬)©hU[yÛ.4Ò-+É£#·¹Òýïe8ÖHþ¿[ïäëáj¼µÈ°å8åç%Mû¿7Êq¢ýÊ·[Uý%,éÁî/icIwîmë¨Ö´¼R-o½R¾?V-ó#É÷²wVµ÷TÛ¯ÕqK?<¢^F«ûn_ûTóro«ó²Ò²¾Zܱ±.+²Pï²Ì£È,\/-|ù|[(G"_}/yóO-?. ¹xy)«¤
×·ãEî+×ûÞ{¸_\ôJ)ï÷^×ñ:ý¿(Êùôo/¼^(/×1'Tn_/븣
«DùùÉj¶í¥rÞòëBºúzÓÖò'ýR§¢Ú'n-iþÖ]åøõrüzûDu_ûó%éXÆþfUëøZ9þ;Öðݹí7Jú`IEgn}´¤û÷Zo£ûEÿ°¤·¤QÒÇKú¡4òhý %ý°å6åW¿GZÖÐïÜ×ò|z§äg¼O>²¢ëµ}²uºñ¬uMÃ&è~b÷Ö¾Ç5úÖý|õl´¡Gº_§'J>xUG/óÆ~÷Ö û¹ZG¿¾,iÑQÛÎÔÑÀø¹ãZ74|¡õû®5·ÞÔ³5üõ;å¼uyã÷Öïú~´©v÷[ôå¦Û«Cåè1Öwµ;ªñlgI%MèÜïmkwüdáM% ä6Ã%õê]¿?TïÁ©~ÚxÇÿÛ°®¤Í%éÚ¿Ù¢ã·õÞF;Ú«ú?}%]ª÷zWþ_¢¢7WÕ¸ÊY¾ÑÑë8[ÒK%ùåWKòvíÖóú¶R>]õ>æFÖÔÙ E÷ÕeCJ´³[´ð4 ä6vÕûÁ>ª½]E³yµÿEµéù×qHÇ ¢ßêÖêØ©ç]¢K®×5µ<Ò;-ßhëz\yÌ·üZÎêý'ÓÑéÚ§õæüí84öÜùî×? øgCWáÝ[ôïu÷ZüÝùÖs·.è½ þ+±ív¾Õ±)ç[ÊÚQoë§C% µokmmßÖ¶övy²^Õ^ÀÇTï÷ëäë¬z××m/ù>Á[¯.º¯ò|¯éûóÊÃ÷W¾Î¿®TÖÆñÖþ¬íêc%joüO©Ï÷d:
<ôgRùÓÖåýr¥Ñ/í5ú¥ý#wî¶Ô/Q¿üåù/ÜúNS¿|rAV×}ò:|×1òÕågþ´u©ªwöÖýðèèî4\Tý'%(iKÞmÔTÏ÷êkù ïw^kùKõ1ýü¾ñÔß/úçªû¿ÏXêËÛþ^õ#m3Õ'Ú¯6/:P}-ÅR_¿ úÏÞ!zÿw¥ÞþÍj{ûj®;y÷Ãí/µü|ë"Ëù=Pu´>Ù¨[[WÕ_׳ú¥R·Þ¶¯½7Î{ÛêëWs×qÞ[)uøxõ×KZZ¾o{ª?_EßoÄyoëfÚnæÅj¼õÍrÖ¤Oµ½[Úü»Õ_jÿP}üDÛ?©þíEK«l;]ÒêWUe}ÙSõýÞ©¾ÐÞYèäi¢îùÕR¯W_²ÿ¡ú«KF«¯øÿ|¶ú¢ÿª\ï,÷ÿÛªméÃUÿ§¡Nÿkõw°ú;w?Zµ}౪mÉCUÛ]/Ç%åØRÒhÕv÷ß(ïÿLõÿÃÁê³eýþ/ù~ :çÿtXroµgñoU_ZòåÞþòÍÊ÷}~ÉïT·±RÞgʽªv--õ\2W¾][ÞûZõyÿïÃTÿóáówýêów/õüüS¿÷?%.mgªïñ}©úHÛߪzÛßóîªO··ôÁêÓ~¾è׫X4]ÎoTî·ý^#µ¸útëÊnûýÒxÑÿ立˵§§«jýïªO·üwÿö' ¿ûkåç÷báy¯U±Ôµ£í«Õ_h¸Q®çY¿ëÿCø§Õí}ÕçÿXécÿ~þ!¼ç?¿W®ò»üaþÿA诶üïøų¼÷ëÕß®>gXm^ú£ÕÏاÊ÷ÿsõ%7ª¹gQõ
{V{÷V_\ò©ê v·Ò¯T_\üÇÕgìCå\iÉrmJ«Ê7%-ÆîþÛ%¯«/,]Q-ùvµµõóÖ^Î?Qʾ§ÿj³}¾óçíý þÑü_M~yÁ¿!~±ÐbThò
ãuòCüJM~ù}þ QTߣü÷öÉóåúÎÿ¦ølÞ¯OÞç¿-÷½ûÕ¶·ÞýêÒ¯Úýçe-¼ÙÝ~òÖÛ~ÿæö#·¾PtÉ/Õ÷üÜß©:J_vø¹'Ëe }ºôá§ëóÆï^¥ÛçÖÀ»Íý9ù\PòýàÕ½
qtÌ£ãtÏë[Çáña ºcIëÝ}0N éXÈ+zæØϽÊk^Çý<ÚJjíià[>ZÒ;ÚxHuÓ~ÃoíøÇçupï÷«Ëóz8Vý·~¨<÷}µ¿o¯òùêLß9nrZ4ØrcêTÇé¹c7±ÀhÇö}!¥þ-ÄÔf\_¼®útcÿc]®ï×ô=uųãD/êÙúÅ÷¯P6'äGß©zÏë¾cG\_QûÆUþ> ¬ýnÑÅÛÑßhKë°¾UǸë+ºÿÅ>Câï. \í!õÁ.Ñk·ÅÞÓ£{kéÿj«ÛÚrFçSö'Dgö=b «=©þñøÝ>nºÕ'ËíµÀéz~ÔU¸å1µ»Gùl]Ò;ìI;ì4Þªr÷;£|gyLè<ÇA½EÇ-°Ýkuîù®VÝ7ª\í?ªÛ:ªãÕyÊU=zwÅ~¤uÊkÌö}F½¢ÇFÑwLùè|BõÛ>c\_g\_ÕÑyÊg=òëÔ{]êðÐÝÊo<õ ¼b^ù¥V\_ô½&Cªÿ&Ñs³eKJ]{Àç÷[¶ýY_jãÞéS¹\üþNõáÞÙ«ò^è½A=Û¬ºZài6¤þݨ÷¦ÔÝßu0ê1¥gì
p¬¿«'Øu0;é¸Kuí»-öd°÷µ\_iC¢û¦tíSá3ìÃZ§º¨ ¯ûGô.uéV½t\+ºoNiØb\_I¿Êa÷YàögDÉTß>µu¹¾Öù:Áõ3vèÞÝó~½Õ£Kïv$z÷©ðnÕ£Ciuè]¾L×Ã:×?¬~}Pôaûfú-öØ°GÞ¶AçKîw[̯׼ãé»1µí~õǪÜaѤO4ÔÚr{ßBžÆz½Ç¾öIõ§òüýYÃÍêwÚº,þ²Ý^/k½kDíu=éÛÿ±[ÏÐY|ì¹Þ´Muö²^I4°Ø÷=¡úÏþÓÊþ÷¬úg®§¬í´íÙËéą̊¬Aýcûþ'õ2SWz¶ÛB'Úa±Fy~õÞv=È;ãikº'zÀk÷y¬úÅPæ²lwúí²Ð
^S\ÅG \eMçWEwô}ïE'l}ÃbãAÑÑiqBuFï=¬{,x{
6¾\_²\_tqýÜǺûÇÕþíª¿çå:£ý+¢ï>õÅ´Êa/=ÿk"Ê ÕaJõQÿ!çN[è,öinSÙ¬ÿ6Õue>z|>|\ço©Ózßëq\ßùxxTßPÙsûò¯ë9ûQWßz]|LïUû¼NÈDðúnµÞ|0õÃúñ¨Å8Ú¡÷ö[È0ì9¶»Õ×,ø0|̯ÙãºFçÌ-ïßA yÑÓE]OvÄðzôªÿ6[ÕÖKx,{b;õ16úõy]qê6ì½ÏZ¬Èßýê3þûåãïiµí1å9kCã¸úóuÕõÅzìå°·ýE½OüÃêë=ÖlACçgßðäúÀ§ éö;¯BTÿ \_Ão¶vØ ¦Uw¯··'ÕGÄõò±ô°5Ë%ðÇÝ¢7óo»Å?ÒZðòüTü¸»Æõ϶Füªº|lO6úÃyTm|Sï8Ý\_SzCÇ«:wÛ×#KÞªyÖk8Wõ¥¸ßõ÷¾&×ý'ýZÇýºOýåýr@mÿ¦î¿¨ö¾n1V½Ý>~OããhrQ}áýù¾=«¾Ïël5Ô£¯ÇµÓðå¿®ÓFùµó£êûW4jjïë¼_°à¹ã \ýíxTÄñgë5f(o/#:{,¬z¬n´X§EÑá9ãçèIdC½ò͹ì¼Veo±°í õʯ²fÝ|¹Ê\mͲõ ù-ÇQÙßPý°½øù)]{?Ó8¯û[C1þèlGü øÇ óy\uܤ¾÷ge½oyÀB^q/³Ðí´æ×cÁ×LGÄ·qõ··OÎ.+ùóªÛëóéqåóªÞ{Yqü]qæ6'S>Ïéè´òØDOëz¨Ñ:ƽoZÈuÓð°û?+:?¯DIÑÏëHLYÕÁåÄ×D'÷E{Ï÷òQþTßÃjÿ³jÃSjÃ5µÛóLÏÏs¹ñnÃÃíÇG×è±O÷o¦7TöU=¿¢¤xu}½E»Wù? þíõTÿ+ºO\Kż¬Çøµ^ѳW"ÿÄåá¹×ãMåqIôxQÏzDÇKzþm{N߸|ø²Þy,¥WÔÖWU¯SÓM[ü/ô¢¾}J×WÔÏ¥z2æÎ>6vª=§U×ÇuÜ-º;íÓwU?GÿB½«èhì5kêQ½÷¼hÁº¡ç¯$Z¾¤kÏëh¢×y1MøûèÞeÝ{Rm~^uB×¹ªzø·ÇTÏyÑð[±Ö7×ô½·×çÿIÑßÑi¹Kå¾"Z=ª²Ò»¨Ì%?î±ü{óªÞe\?ª<¯©¬×uô:n¶g%ÞR×÷¬ÊïÕwÌ¢å©.×tïhÍ8º®ö=«ûGU7wØbl½)úMõ|]õ>):^×w¯[ø2ýÇ$þ+ñV|¬×ókª£¿ã¼LñfëkÏwZíño|.½£2}ÌÐ=ú{}¢Û%å}N÷Cü/ämÕ¬·ø¬Ä-~oªM,øé å÷êp]e{Û^Ñóm:/XÄóúMª¬×ÔîSÿ ¾®rSçEûózï¢ò{Ñ"&ì3±è&U¿ëZ«Né}¯«ÏçãªÏhèÿÃz÷)øM«¾§uïq½ÝùøÅÜyX´ð|Ñw>¶à½Ðàzÿ¸Úú¸hÊØò|^Õ{Ò3ÆΨèzEù16½®ÄGg,<¦¾òë]ªºÍ!ÑCÎv³:ÿêþMXzo¯ûÛãó¬WUÖ¸úðHªç=§k§ |ðqõ«·áQ^G-dçÓ³Çtÿîù·kＤûÄô»d1>ÓùëzÆ|½":æxÊ'Ô§¯ª¯«mߪ÷ëüµyp^tc}OZÈg-b2¿/XÄ&ô÷¿´§÷mµÕÏqoõ´5ÏÃDÖÌ£:÷º±~-^Ò7'TúyèM ~|JßS]öèÁÿïç/rFÇcq_°à;ØÖ=×=|,úxu9ËåÎ~}6Ñ
>sðê¿ÉbÃFåc}«hpVïû»>Wö©ÍüóÚóAÿ=ïQ»·ë1BqÖ¹øÇtµé¤ò8ÚëXÌï½m·ãM×÷6ìt·c>{ÿ_×µ¿oÙíе=êªêt©Ñ×µîü};åó®¿m±¼å<w}¾¾¦ó«éú9}ûz£ÎuÞOëþ yôªî1gü2Ùkº~ÒBo_8ß¾^;}ÞÀ×±¯ûø'þ±JXøÅñSûпÃ-?1Bý8dáÅ6aá LÏñß Ço²ðí{]\ܾ ákæIE|ø©¹UuÃ6?j{yPm̱æ°+ÏYÄÔÆW9jÓ0Çô¹àãè6µß :>¾6lw;to&·Y°CîPÄ$þ!8üÃØÀuxÙĽ;f·ÀlLt²ðiã¤\_w¤öR¾'lZÓ¢á3êì[-ì8øl±m°§ønÀºXø%fR¿?·Çí#k,|PSx®Á\àNí?½oê1oáãë¶Ë N\_;ØpþþÃz9cf.ïW·[ØÙw©Ü-q±Ì¨îÇ-°PÐ|·ÅXé³kà¦-lIszÞm1þ÷ê>>ï=>#ü½ø{sã_j¶ÿ¥2·XªÀñácp®Ó÷غ§tÜfá·O[îF ëÅܯ72w[Ø»Ù;báßsº÷¤¾Ãö»Í~ ñºôZø³6Y3Nÿ&ßNª½»,÷3á)÷2 >-±#úq½Ê
y¾Ì! \Ø2ìÛô=sÞû¹[GÚáù¿vÄ7?¿4í×ìçk,Ú2üúæ
¹¹ÓoeZÐÏð'b¯îP¹¬mø®9 Nì¾ÄêCq=·à¬#^70é¼|È-xké¼Îë4Å ß°Ï^i ,£·L0<²\_ùð6ºíÖ×t)ì²ûhö\íPþ^ÖJ Xê©êoV=ü]ç/,ðúõúúóª·wÀbn°f0¶vXðy 9b
ÅÚ?ÝÊ#þñÅxVù^7lð-°BC¸ ½qYøs×>½»Þb¶Ó >°¿ã2Öôv³Ü,¾Ó.øµàë{?ßfÇe,Gá¸ÁbmS9³éþ^ û<|p«ú¿"þ?Ö0øä¬EÜsüøEæä¬?p
l5¥ã! jÎ÷ê3x8¥y ùi¯Å?À ¥r¼m¹]õÚeáS/x¿5¯ñûÔ¯»u³EÌÿszÿǹ[up9ï¤
ÿ%ß?b!ï©ÁÇðiîNçÞ^a5ncÀÁæ¶ÐXç{{ÕN¥C[hÿ0l'
L? úì·ÀXø>ù�òÉ^µUtʽiÑ9J8ÿ¦5;ˬõ¬³óë>kÃ^ ù0çÎ/í ÅØÅ/:ÃFA/ÓùAÝ©ßösiÎBQ?ÁñÝ#ÍXì?µÐöYÌ{üÚȵ©lÖÖMªkeÆÝ#7Zð®¬?DxþóÇBÖÿ´ ^³Ö¬G²?
,ǸħH}ÁÁ#?Ð;ðth¸ÙßÝÖç 8Iä>úri;0ÄçÝAõØ$äÖ®N Ù~AVêµÀ壷f] ÿ2ú7´@çëµËÀ@mNåíOyB0±ýj×Z ú ÇM(d'ä·åÜkFDþm¨Åí"O©?êý§,tZþW0§'zú÷úöXÈ Èpô:Ý^k§ÞÅöäõ=«<Î¥þGµðiOëÜiáöÕ{ÎX«À³½CKe¨/ýz¿úÇëÈúpH÷'E{vZà>n¨OºD7æ{-àgùß(-0Ýu©ý$¿)ì_g?çëªÃ5Ñ}ÌGï;w>nÎXàIÑ»#~¬'uD®ì³àÈÈW
ݧü©?{U}ôröõñÃ_UÛ^°ÀG{Û_Q½éû+¢ëýÑFÖe²R}}ná³xRôr¿ =8'E#öJè9zÚ]ïUy^/êþ^uÀSW
ÛQ»ù'§ó¢Óû5÷ªü]~²Ç-°q'ríVÑÈÏ{R૽¨w¯^ðÇS¢º¾¿ëóËÇäNѬ_Øø ¢ï ¿Gh?©¼^QB®ó÷æDëk¾ôk7:êÊ÷ÅЯs½úÞýo6>Îé{x6ûç'²ÎKÊó%ÕÛûÆçÝ»ýÍ:Ï3Øô60/êÏï¤êq Ñ¿uûN)?áõ5å9¯þ}o³QÑõuB}Ç!±ðílI4~B}èù¯Öý7T×ãÊÿÿ0VövWßPêçuÜl!\_À½]oZØYÈ|êßI}TeûïÁV ólû^¦õÎI9ÏZïͪ Gíö?ÀZ6¦wgôÌ×ô÷o[àOv49×í|Ôê}?y¬ÿÇüÉÆw·ñWDÓÃ|jl©ÿ¿n½^¢ÞþºêáuðyÉØ'ïÖùì¢ÆC{ÚoãÀ7òo¹_ÏýÞ«>lRþ×-öîyÿ9?x̧ï¿f¡wb/xÔëèãÐyøi} ;øðLþþ)å=¨qàß·°=R¯ªÌõí¼ê
üæãìe ½ùètMyÎ(Oa$ë~<©:íÁÙ9ý}óõñòØkáJmQ?RwnépØ;ß´ÏÐÿ¼ï}ΰ7Ûññ ÛúxìÐ=|TÓÑÃàéÑêt½±ØHü=|4
ï Ýu£îgÝ=¹yÅþ[ìCÔ±Ku°ðÓðµ¬{nÕ»ø3¶¦wñ9õYìaÆ}fìûÁÞ˺ÝkÍ~#|ô{Èü|¥ÅþEt]lÒ~ÜÒØktãS©¥YÙ«ïÇ5çW>nÁ°ÃÛ%K^ÿÏ»ú[b78?×}¯ìY;¨{>/Øã´ÏB~;n±ßò\g>}w?Ût|,û·§ò¥:]Ô;u$>Èè Ý¡[¥ýwTO|6ؤÁãÿ©|vÐo«
]o[ª¶k¯'ü�ÿ¶ÐìX©÷Ù_Í>oø8ã±3n1ÿ)¼@wÊ{^}ðÈ},ü²þ þxó®Ó«ÇÇ;Gì ëõÓ{
ÏqL÷GÓ9¶Nì¬ÃºÀBçÖ=|Ô,x,}À^zl <Øc²ÎߪçFåÿ]+²Êc7 ¦2°µ¯±ø44Ñc1æù¿]Æ1°Ó°ÇÓG-ð&ØYÃ�?°RiÌB?�cÄxX¾Ék!: ó
uÃëµb"¼ >öæUÿSã³×nǪÛÙeáÛaÿB=Ø'm_Nd<æ/ë=²ø~Væc§Þ_g¡3?ѺDoük-üÔ ¹9¾Ý´]mµ1i16§>·Oø YOY×°¯§v[¬sðfæØhþÜj!àÃéHGæË@Ê<À {.8¿@h¼ êµàýSêÐgÁÿÁÁk¥yeß$í
?=dÍc[.¾}þ+ÝêÚaá÷Ïe17)qÍu¶÷«ýÖ[Ìx)~ç9k-°äIì ôÌÑ]_½>?º¾ìr¯Ïù£ÿ!çùÛ
§óª÷AÕÁõÒç,ôsöq°ð½¡°ÖLZè7{,pýøEÖ tJ?_mÙßÐ7Yø4hó&Õ?û?Ùóyci[ë pSþ~b{áG¿O
®Ì%° Þ,ttï·y³ÞýàCàIÈ ø2ñó´Ã£éþ.1É5ëP
,Ü vZwÁ^ÑJß £S²½]? üd
-îI :DºFƯÅ]|úý#²ÄhzÆXaÞáÛ!OäÜQå^�L1ë·Óà¨Å'äß°àßYþî²é×ÖÑ{µ+ðjðhð4|P¬Qø>Áj²~7Ç îÉÓIyÞb»
<1jóAeÝkÅß\_fá×ô{>®Ï¬±ÃÁ8PÀ²ÐÞõÛãõÍãûûÀa ×±fð-kç¿Ã"® 26¾Z0×ÈÛ]¯9\_8n¡ÎZRöYàÚ8";v&#wÄbM]ká×H1=¥÷¼Ø°i¯²ÓÆ}ë1¸aæ!ù0GuïÅÜ·ðÉàÏ=bÍøß±×Èvðµ0¿¿Cü9^îþÊ
oQÞJ}ù}a-Ä?±°1&OÁaÓOÈè´ +ó {PÖ?ë±£8öZà$°ÐçÈ$;9^Ð:µ&ËFØWüüR\þ@Ji"õ;sÇﯵàñÃþ2ð4èÁ^\_p0ÛR=°á2W·Yì£ ?àÌMæ²u¿
½e6ÍZËØNeC½Ëßc½BDÞdx>Öø6O=éÚ/AnA\_²À[ckan ¤ü±-EeOìLtA¿õê½. Ý;¸JÖ¢#±¬õ½9dP´ï²XËX×GìËÁfå<ì+-xßͨ¾3)?Ö4\_'»-ÖNæ Ñ¡ü)¿/]3F#ø0ÀQ?ö!m°àkNÏcºÝù=!óäø\Öÿ;S¿ ç{ßú¼K=¡?rO¯EÜ ~ 9ù·1»É'Í|ËpJø:Àbé}°g>éy±¾l-àÓÙÑ9ÑÓ&Ewè^=£vì²ÀÁ?¡Ü5zlϬÎiB¼ñCzÏýÅã3Ùó}C{1]wxÄ-Ë j#<~Fõhæ{,xÏñg,lf¬;{RÐÏàÊ¡!Ïúïá P_°ÿfuìOÇÝ)Oì¢Ø¿&-ö¿fÍëýקkêµÉƽeÚOç;zÄV ùh¾¼86ä)µ± nÞoËû%µ±·1;máëBÏÆW6jye}DNAçè³798qÐÓ¯ð(?v©¯ëª»ûË8mQl÷å·´Øãþ5Çõäþ´ó¢!ñeÞ¶Æ÷ñ|É"fk¼sDï»ßÇʵÙuu÷Y²¿A¯ó ýì×,ðXÞw§T«Öòô6ßçcÜ¿íx÷a+À§rEeQ]ÞlüǺÞ=n\_ë úú[ ºÕíÁvÝb\_ðññÔÞëʾöòQzÙ7ÎZà·ÀÕ8 Þ¶Ñð¦h÷¦ê:¯wúUÖ+qÎ&:zYs¢ñk±u|LnÕ}ÿÕÜ¢Ëù{ôíEÌ%bqøsáúnÿÃ1µë®÷ëø¬]U~~P}Õiüt¢ïê¿×,ô §G¾ë½| ºªv?b;ä6ìHc©,pøä1¡ÅÛé²ñ o#¡6|¹Nc{nÙß>{êÍö£C±F#W",¬29¾ktí,+ñlµ
= ½t¹î¹Ý?«óÉD;ô/äoÊ,DÊØäÇ{-äÚ̺ð[°æc)oÖõ¡D·QѬ[íwu"òP]Öè¿Áx¢;8q ½+×m·
^ÕkÍ èÍzhzò0ò[héãîAõǺ¿"¥n¥åª/6©]{=-Ñ·ØľntCÆg·
Í¡'õ vöpdoo\§\_õ}ÐBÏSì7ʦLð\_à¿ø& =¾î©Tgt�Ò¸ÅXé·° ù¹Ïíý=Bt(y?¶æ=[k,ì$ø¢½nË,ø| 5Øì¼´ú#+t[èFÑÐËÑÑàÙ±¨ÞO ÷I3öz-x \ üyþB;ñó1'{Øw'?8 ü´üsZ´?ø�s7jy<¬üØ5a¡ÿã·7Á;±»9=°¢à£É¶?h1ÊzÚzqD¯ñ÷VZÌiìøf³Ýþõ|רOºR¢¾Ð14!:ßì³À)å6C+£ ÿÚCq®Ò3xØZÝã8ÞùHúvαtYì¡ËÿÛA¿EèÞÎÑéð»cèÓsÚnÈZÏXê
ß%óÖø¸hõ
?$+xMEüÕóqý¹ô£çÉ8_» ý]ª7z¬÷Ýv]ЩÇ%«-øaO: éôýêtuÊÞMÀýº×úÛ:~�h±M¬Y¹ü¸Ñy:2O(y.9Çfè±°gYàüþÇS]?n±úùr~4ñ+ÖXØ'Wê´ðípe¢ÃE<ðÊìÖ\ªÏgú]ç#6!ú�Ìá&{ïg,c7dÜA·{,lüÌ«Ô=)1ïWZóààô9ó:±V1VXØ¿½éüXwxÖîmJ߯µà«#©mÐÜ x%diô.ìÕØÀÀÀV[ØþáiÐ~ÅzìãèvaìxØ6Ñ#Ôo Ó®º^aÿ?añ¯9úµÏBÆßh§ëµ¼ËÁ½Ê /G{3°ÏS²¿y~ZïbsÀÆÞí©°K ûб±}T³ßÏA/Üï9nà
¸-ø¹Þ¾¹lÀ¦ürÚBv[Õ!½î0kÕGöÈë=6
]¢>/àCÅO;c@·�¡?°MàwÅ÷D?bwÇGÿrjøÞ3Öl[ÉüèE\öNxùàÇfD;ÏÓmon¯Ä®½ÛWx
?=É-þ?�½À×0F(g4]ã?T}O[èt\ ü×ù Î;õÎãÊ'Ïyú7z:8d/ä4pRƱ®¨Nn·$.<óºßu ÚÁ:éù~ÄBosì¿f
ÅúÇÚ]þî´¼ÓzÐ^æÇóÊ÷
ù0aÄÿ¢)¸6°AàÃü û9ç,öÎëbÜÁ§G,°AØ|\±/ßÿþ0/áeyºû5Û3wXØ=Áà÷ݨú³/ð]Ç,lkþç-9[NåöYØKöè[o8ù¾½! ¸£u{è¨'x5öFÑ^Æ×t .sëE¼È-ØIÆ×ÅÁO ×<òçáófï÷ÓI (} ÏÛ·MF;Oô+cÒÿÉ>d°Y ÌëvkØPf,öOXĽ],7eLeI¯ÿÝÁÆÞøC¬}ØLÁ½f{¶]xá ¬Ëxê{p0ØÿÔ±8c Å&DzÀ!0|Í°À eí·ÄBD®�ìôXmaG�7¿ÃN n¼.ó}}ªX°9ÔzàçÁ¿N¿c$_0Høòº|çïï¶Àe ·aGÞ%ú©ÎGÙìð-ø[;¼+9Îë7xAÇç-{E7XÄ~´À^ûû>^Î[«Ãûký½³nNZÄAC6ÕõÃ1ÁdxzTe8?ë³·èë4óÞ_±¡ÁoOY!ñÁ\Äþêãè ÅXFG6B¦/áYôûż?h±Þ¾9};,æ²À¤
ìù¾½ Ñ:1} ¯R'ìø¬;ðGlÖàów«OÆSc2'°Ê<_gÉ°æ½wÈ-3aWÀãÀ^ÀKGS>ô×Ãå¦)±PWpEø±¹^y\_%ó}5ø>Á@²®0Wøö+1 ]?ðÝQ="y>=ÙA=èdylXàØ[¸Éù 6dì«È¢ÈìøÈ2ùξ èq±ðähÖ9|EÔ ßc¹½59:d;_ÆÈcæ>ÌyÞwe
MþÅì¤>ì_MïTîÖKl´ç\µÆb|¡/á·[-ä7x=z_Þêã?ãøYñëá ÄV3ò�Ï þ²Ø²ÿ{ã²àsø$û-ög.ôÕaëÊ>ø8Gdu0·3ü¾O"ÏOlóøÇR]^賬yYÈÊð}|1^Þî±bAÏN笵ìa~[ìåë·ØC zÌM»õ¯9$.ë ãk¯ð{XÄ>YÄÃaßÓj Ý>ãL}³«¶©n,ÖzìÛG-ìVØÛÐÅN¨õÿÃÌ:zÇ [k%ó ó{Åúßg1^è/Æö¼5Y±¾5Ñ
õÚ+·ZÄ¿=n¡eY1 ËÜÜaÍû8ËÌGæ >DÆ'vTø(² ó [ócü¦Ì·Ñôã½:°Ã_Ç,bí"£+@K|µÌèÌÄî ý¶¦oÐÅG-ðQØ×Y;'ø»·1±²'Ç¿;o±>PϧǶ'o:¯§Sé>:<4c¶ø{.£Q]v,H{Ó9¾Ítü?¯ãköY9bã×z×ç MñåÍY³|ì_!?xøhÌ~cçB
fíÝoaÆÎv6õ|oKóëòX³Þ7v;üÃ?aئà E3ÖïYá;à{R½yçÓùöt² ±·z9Ë-æãÕy¸MÖ-é9ëc)ËéÜeÌÖ_dO5º[¦íT´>±Þ²Îûw©X/}|¡7 ±®Ì¤|Y¯À¢~ìSìC^Á}E÷>=[í³#Ö5ÖüÏôc§Fß±Ö]ÜnaBFæè=?Ê^¸g=)Ýò ýºkß ¾ÊZN;MÐEóZ N9ò8 pJýé[ì%[R^ÙFN1§ð]£fÜ ~[Ïs¥ÅÜGæ§,¯ËÃûbøȳÿß.ûý= Û }ÌÁÚ ]À䱦0öy!9
½ºÇb/õb¿ØXì^Èåýé/o<}æ#)!s"yýÏ)o°ªÛ,l^?×.ùÁ^ko;2¶Æý²ÃÑ~¾S4ë°Ð[¼Îk×YóÚíÌì+ÜîyyØøX£ñe�Ó ^
X z§Èà÷ZøjÐ[Sï6y|ðXðØÄ»Ð5ØC§Ñjý¬Ç |º?ã»N·
, n9,<<¿sÛ r(2Än k#~k°»uo¹Å¾;ÖVt®ÔOìGåçÓüqÁ:Ãÿè°cb¿g~ª¯Zðº¼G�>
¼®/©ÏbaXàcê;XþqDú-ä�b) wNÞbkdìVÆ"§blÃsË,æi/\_e;#ÖIôÝÉT>m§Y&Y£ç¾Çµ7Ñq»Öþ¹
ÿ¾½ÑB6ñûüz³Å?[¾añÔ3ÿ8dcßná æÓ r¼õ>c,±3ϲ<æ÷Á¤,ì7l=QÆÿoËz¶üñTæTzò¾þQ²Ûfï<ûlXkRÙÎ}t¤4n1G±¯{$ÖklĽo0ú¼wðkç(ÌYìЫ±Áe^·)óm¾£a÷ÅÖ>ÍÍé¼KÏ[ÌEt¬ÜWØhñÀRôY¬è\_= ê ùY{ÌT&6bø2ü#ë+y@çÈx-tÛ ¬ ñ&±ÇìP=X¬wØ5rü�ìÙÌ¡ÌWáøÑ° Z³ÿ]l"ÝGïÊ>ÿ
û5hë¤ÅþrÆ'Xä'l {ué﬿çJÙèqÈbè{ô moÏ«þì;9ga§ÁÏ2aÍr+4DC¦[gÃÎø8ø,ã2&ôKd\_Æ3ôñz³±é¡Àõ5ËÐØ5Ñ}YìKù´à'ô#¼\»õ
-¢O÷7é2öA{çTOð³ì¹ó¹vÜÏv§¼6¾¯÷Þcó÷Ø'È´=6þ!þPÿcùcÚ+]Êo)tlYÕ8¯×¬}ÿ²èq¹AVkÈ,ð¿ÿx¹¿®q¬ûëY}Ç>åãêO¿vÜßnÕ×éñ¤ðú:é{NO©\^UþîF¹uÌä ¢½Óæ%ÿnÀ\_¶°) (oþ5tXïö¨ngUæó\ãµsÚcäÏÞ´ÀtÕsÏÓít>W|¬?b±a#÷ëý¼¬óÚÇ~z>Å|Îò�xwð¬Û¬ÅسYËÑ9=±Ð?G
pį@C¸hìÃèæøíËþlE\9lø~É7Û´ààá[Û,üR ¦}Øu{ôlÅöñ³ÜBd^bDÿLåö§¼;?µúZÇ=ÝoáßÉ|ùÙ 9d<ú|EGúõ~ÐmÍ>Ïþ'¿· þÀúîy¬²Ø§7Þ½ìûÿðô¤k0 Y¿Þ^OÌ¿A¯Èö²Zì³Á¦´Eß"?£"a DGÖb?{|¶À¢°Z´ø|w~÷z9?¦:ßW°[ß#à-Ñçþë<òU ù ¾áÉǬó´'uíõw¾ãûç·9ßríÁÏè'籧÷YInR}ѽëÛ1/ÛèCSÀ¾#'9hÍòú¨
î® ÿ@NDß·/|Ä; 6Éó/þ¨áo£\ü3ó:®±Ïàs»í6n´åhýõMåçýäú¯;Ç,þkã}qCϯèýwZï·V_¼¥þxRíóo\¦>¢wÎëè}úÞVm¼òVý7üÚeÝj·^}çõ÷yq\ýÌÚû¨=ª<üxZGÖLoß«÷)õ·Ý×»R~/îêoÏÏg,þétYïSöËz~Ey8m_K×ßÔùºÿÎiÇ3úFÿ ¬ïÑWèþvÿ<Ýø¦þ·ôzBíòX¯¨^O¤#éåtþªò¾!o?xU4¿¢þ>¥w½Îoëe?®û.sL«/.ª /è=/ï õçc\ûeÕùÑÅÛ¬rMmxJ×ÏYðÚyAy^²SÀß=§1pÉI¯èü²êàôܯ2'ÕtïYåyÜÆÁÿkgÄÓñ²ºôþ±qNè}¿Þ¨vJýóòBÞ³°ù#sÚø6ù=Uè§ØÙ÷´Oïõù?<ßxÿcG²îbwõ-lÖÙí=ß[aaÇõköcŧ1!ØuØÿ
Ý[2ø }~«Úþ½ß°ÞbvSÆjöbÃÏÃzÞg±ÿkS}hÈ<ëÜÇ5Êز Ì ?3~®qÝ_êDÛñíñ ýÀy_ºÆ.Ùg?EBVGßÌþnl'$Æ,ùv¥üó9rnOVÝ¡ÝààÙ»-l] ü°ë]ñ¡ö¦2S¹¾¾�ìøºRùùØmÍ{î|.×½eLÆ1åvè¦ýYOÁçÀ|ì°OØêñ¯GöF&f!ýü¼1%öÑ+"æà Å8jaÊú1Î_ÃXÁμ~AÓ¾1¤^\¯´ÇS¿aãÀ}º~½ÃÿÊc©<öû91Ê¡OáÍ-×ÕnlÎ×NXcôFwe~gLÐF ¬¼yvßáË 3ª#±P³~ïýwÄÏ¿6[Øi'6!üäèÊðåQ¸0;,0¦öZ1hö|¿Ggl±Ø:ûES|r«-ôal½¼³Ñ"N.{G,tÿÞÔ-\×õ.Ø¡òC¶à·?À>zô?æÕ2 ÁÕÛÓJ {¶é¦èô¾[ö³^ïï¶0h9Av¡]1pØ¢ú,ö]°wÓëîóà<ò:æ{2 ._%ò-r+2+õÅ/çyÐù.u&}BYÈè{| -<|ÒbmÅÞļîU°÷©ËBÃw>gÍ1:·ËÍèGÈÙ¼GÖ ôºRfËUxîðNd>Ùã.{ ;éMm½g½ÃÇ.óÊé¨þ{R4÷ûÄðõä6Ú#ªÓ ]ÌÑN ¼F¯Å¾lÄKteN1Fÿ¾ôN¯
ؾà°Ù ¤sô9äO°Èè\;-Ö8ôöúsø(eó£ëbk8c±öOß²æÓÞ1±ÉK¸IèÑeGSÁ¦ ëZü'ÌÛX:"òþa\_¤}C3î¦sd8ä ää<6¤<ñw§g¼7®Ç¹N¬ÉàåÀîCKxã¹ÛbÎõ&zSæTº¦o±Q:÷YØ"§Ó»ù;Ö 04Ô?}Ýè¾ÚbÜsò¶Î[øÙ\_GêKù÷¤~Xm±ù¹°Ð£Á¹ú3äJÎR¾´os\su:ßÛ àä(qb ÷bWga/Éõ^p>a!{ä>ëµæÞ_øÿ©/ósÊ}¤Ø¡FR~¬9#Íϸÿ¾#µ~ݱ×18jÁ-àÏ©lêÕk1YË3°¿×¢löQö.K}NþØbX§RÙíõ°Õ×ÝêËQýF~$~={Àã#ç¸ÿÿbaÀLï{²îMe±NúRÞÄûÉñÀÇ¢òÝ°ÊØò=n±ï-íôüÎ,(;/²W
.º1ím\g3Øëõ ǬùC};Ó·Ï\@y~DÖáïïHçìÏÞÊfÀø2-oØtz¬¹>Y¦BÎØnÍöÞÝÞaëøl÷¯"ÇoLÌé4=lÁØ÷>ñ aÝWï>jçÏà\°{¸ÙÇÉ)X:Ød¨cÇmûïD÷ú$?{#ð^W½^¶Øs®Í
«ý̬ëöZØ!Ч²ß¾Yú/òÚÅ8Ë~,d8ì.È÷wÅZÄÚØXè.© |)ü~ §Ëb¿n¡·§²»,ìÎίÁÄbcY#ZtVGy/\_Æ"Ó^q<ühê7dxÿÆyºß£75c®ã;%V§ÞÝk¡®°Ø\_ƾ,t@°c)=±»= :4º >Î1 ¿ã{0 Èr¬o ÙÃH 009 8lYø�Ç=w8Ñ[áz}éÌo|Ù×Ñ[ð·,äÑîTîÆt½-ËgÌ-tLp©¾±{Ì~W®gõÝ¥-ÖÜnú¹öM rhqÃp¼A¿Øm[¾ã²oãÿZì?Åþn}Y;¾^ ÝlµÅðç>°°à;%5Ø}Ç2ï¼i1FåWm)thÁÖ-:ËYÛU²vmô:dü.Ø òxe c+òv:Ïqѵ²Kè}ûb×YÛÉtÄ{/¶xÅ×+OlÌG,ä6Æ ±²}:c9 Y¹¯'¼ýNçY l@¥,Ï
ý²vXü0Ó«+ß²ãÓÖT&|߶ÓÈ^×W,öP,bµç|{-0Ƭ/øõ©á~\¦vf¹»Ëbά³Ø{»Úï8èË�µBßàeîáKôº1/áQ[Ô&ôÃãØùÙSú®#}õ]£íËT&¶þ \ßðCé;¯ç¾æ쿲øOóõûú;}¿ÛÞxÞCÈz Ã<Ã'Ë÷àÁ¯àÞha×ÄV× ¿VyÞmt¹î[êË.ïZÇ>bÁÏ°'û:yMuwÙ[ ±Kغñ]Kɳåã6«s6óqi'U÷ý±}þ·X°±Guûó4ûþ¼íì:d!{í³àÓé¹nÀí�Y¦¥,±ÀTÁ³ã<QîN Ë»þÝ\ LO
¼ÆñÈÖØ5ú-Öötdeoʲ°£Y¬¯YÄ5¦?-
Î û(ÿb¶°çXµø
vªÎXè;t³´\_e´%Ë@Þ~<¾#öê\8o¡w°Oaò£?N§>ò2Y#±ñ£×yÝΪ³±°7Àëð+«¿zÖ
{½EÌxÚí�øòX¬µÕ~bm¶Ð9} RÚ®oO[ÄÆÿÊk¼Eí:úlrÄÅü· ؾ;m¡·?i¯ElLd}ð2³©mÇ,ôöçïµSsª3í@£üG,l ðFúxcÄt||·· }¶"ô+ä"þ»Í=mø=ÐYG'F%¬{aÏZc1~ر©|okú·Åü?i±Æñ²Xü+?Ö¡6 ¬OcßÛb7ç>Ø/äYø˸ò±À=ã ]t¯¾9naãamðz/·£)¿íº?uþ;3a/ xxô¼5Ç@^Þc)Ouan:Î[ð?öæOdß-6Gø·ãQ½.~yH}×aoü
=yåmÙ¯¼NYØÀ7°°=í¶7Hÿy'þ·÷êpÄ·Ñ¿È#¾¤ùDsüåÄÀ· 6r¦Daõqæð®Ç-þ'M÷¨ýØh6!ö¬¥V¾ØÇ-t>bRlµÌ¿éGtï¾g®sÄ'sÈ 9L öänµ¹¾Ïb^!×;IoêósØ^ѧêèßµ;-ìàð-"ç÷Zعé?ü$v¤£÷Ãa½Ì¬ÌUÒF \_4ü9xÀbÔ1ãèÎYààKÈT{-æ;{£±¬²°IÁðñM5l«Å<%V:5>vdqä2ÖaÆ/mGV;ø#Øäh¯ Ï|vsêãA -xìRÙFomBtØ£ö·çxÈB>ͺ<ãÖiáãå
m±¦cÄ�îL {ñq°W¼ËnË<>fáo²øØ°ì@öá߬-'-ô o vVl{t[Ñ.u\_FE¬9ôÖÍ.¸ÝÃû~[Øè[ìåd\ k§2à}(ñ¯¡7íT½ø96tÕ}¢-Ø[úròÝo!÷ [àD_ì5@fèÏøð½;=ÀámÍGÔ'ØÿÛØ GU_Ö =nÓú|ÈbÏƼòb/!bŦ°Me䱿 Üã¿OZà3Þfp¬ÓìåA_·mèØ ÷Y¬N×sëã¼
9
Ï:36 Lþ¶áTG0jÈáè<×Ç^i ¹jÊF :·Ð#g¬Y϶ìm¢#ëÕ ÛôE~Eöö:´X'¸¬¼
]5ËÏð.øúný¬aà÷3<)ñYö[ÌIïaÉ ì[B°kßæ@\o7{àðmµëàÅÈvE¢/°'¥þ/O ÝÉy1ÿéOÍò ±nÉ>çÿKÈðíþtÞþÈúξ±8$ìÎöüuرgWH Ù~¬¿þ ]èT¬¬ùØ#ðg¡óÃc³¾³°GOHy §Zcácd^cÍ´F7ø\_á k-0»>ßÙ²Ý7ÊäWNSì8^Öy ßü¶TÞ°Þ´fv°+Ýñ\_àÄO@G9\:=jÍ8søÀN 9LØÖtKúÛ%r¡çékUW¢;ü�ÝßßyÜV8nÛ[ÆÓ¢#³gù˸ú½8\dMæ ú k\]Ð?Éz ]kÜí®àBI<÷üOªþø7)ø(tÞl±¾g)ÞG77zYÇ,|KøjûSÚo!Ó?h¿Ã²Ð-ÙßöYku
Nÿ y:0\_ s 4õs\_øcÁNe¼?}57ëjè+é~O\;ã÷q8ÇnÎÏÞÏ.ñÿtØbá/&ô3mf?xj°<x<òÈkß [àPñàãØx³Ü
È¿}:3²·7çùßL¬¿ø®À;ͤzÀ×èfèçYȽj뼶vÚÐea/f±Gw°øczou¢{83Æ ¤ö:µ>¡'{bAÆcÝÉãxE 0lêYóú!OãtÚÌY` X3'ìÈúðò/
Þ\Ãç~b!¯ä=jøUèKÊä9¼~Á§0ÞÇv÷Ó{ØDñ£ã«£ÿØ|°ã/Çÿ9aÍs¬@
¿Ý¿[náïË3úø$x'>²-ñ àÏøðhÛêüÉ~Þ2-± ZÈ|ahÌïNeããêµ£«,|Ü°;ðUã)~üeÖ<רãÅÿ?fásÇ6¹ÊBF?îyÓ©ûk,öycEö¢-ì¹F]káÛ\¾>ðùê;úìöÏc½ÒJkÆ
N-øÆiïx±Ùt-_ãK¯Oéxzï°½·>ù½%ç=îO¥:m_ðúÒ¶ô>öÚ1o3^ »ìÉuNçà¶N¤¶£ûaS\/6Ù1 2õÃæ×È»ìEBÃ×L58éùô={²^¿3=ߥïù'6ßã»óöc&!ã³Ç·?>$¬ÿM¥|¶-Èÿ@ªãÔoýÞ[>æ±øÐáØêiËT _)º;cd.}/1¯ }ܺ!kö çÅØL Á6åÏÖ±wv£§ð¯©èk¯+ºÉ> È1ÑÕ±oÌñ½û5kìEÀÀ¿¨ÐúQY »ã y¶eB[tLWhƾ9 o³ÀQÀ/¦Ó5s°7S§còíOx{XØñÃ1¯è;ç1o�G
Ý#ϯ'ã·ßù6òç½sJ]¼ÎÛ-Ôó=n[Öè {S9È?àÎñÁ¥¾1g1ð ¼[gu¾vد6fs÷òZ¬9øïÀ® »ÁÛ¨{'-äTtC©NüÇý¤
¿�| i>Ñòð0Ï·¥>Û¨<Á±¿ >ãþ㿱øßÇ-Oóùæ
M|¯Åú
,÷v¤wàwøYö¥ön·àGØÝáùô)óù¨Å¥ïǬyn#·ø/úú±J0/¦U~¼¾ñ?Ö l´µ×boß
l�=$ëÒìO¿b'Þز'MöìùÃ~Ð?èà+º,p÷=öü&ÙþÝ®Ñ?µÀÍà ñû',p9¯[è½o[±M}½°_}|WÆô²þ ëAG.Í{}Á(g9ëþícÖ¼ãKw ¦ýr v1l ÷[ø¯cÛXS=vM°=Ì#lKVXüûð®ñ/m·À§c_¿j±OdÆÓ þ~Èbÿ¬IðÐþeM÷ßÇî=cÓú¥kÞi!°ØßÄÉÚÿv [a'ëçë,üT\Yu£
}ûòͶìRè:à¿-ð{±ùÔúÞdá¯Û°àgì~ÖB®B¿[i±wÌؾ!µµÃFÜaoÛjûþ]¥¾ ìWÙF͵Óá¡tM|)±hEÜLæ8¸úµ¶=ì±ìSÈv¾/±DÀ?¡çc�IÀJõXÄOcr/{ëø¿8{÷áÍÓé¼Ïw½»zy8 VXتá[S~ؾàÙØÛv¨ýø«ºÒ÷ØA±b·ÆçÙ\_Þ
íÞ"I »ñ´Å|¡ì£À~³%Ç5¼ã´E\\äËN2e»4̾,°ØÄ/ÔºaS0òÀzÏ#Ûáµ-ìÈ]möcëÞ¢çÌñËËÛ¸ÜB?ëµE1a±:ôZøæ×Z³ Ío(ÓoÁ6YàH±Ça;Zo1ÙÕi±/½8=:"óð¬7ÑÄ>(ìë¬9«,lqE^ój/íøë;,p|m½rü[9r ö0Ö@Ö¦£kxa§/2=¶al-äSú4òÚ°ÅÞ#æç ßTH-lì±Àç7Þè |3º2r>ú<ï¡;#?ûZ_ 46ö©¼Üãr3øäbä°«à§ÉFRÞ·¦ùúoÍ©;t0ô"ìS~ÿ
Îdªú>2áßXĨD¾M=Ëèôºæ¨&¶>è\½Ý{ËòÁ¾IÀ¿á_§ßwà[àý}ü¡ûSyþÏÙ#©O&SîÒuÅÅw9ÃƧoµu|{ÍÐsÁ! ²F{»± P§Ñ>a§ókøϽþaÕ±[õe cE0ÝÉcÐÇ9N9Çñ÷¼ü¿§Û@[;ßXÏá'ÈÇüó Þ·ÚF/ýì¶uk%>¤.µ»Üá/ºò9´®ïPgd[ÎÁq¿ÁýuÖüüNr {®ûS^àQ¡óV eÐ^Ö°ÍøðÅ¡6¥÷3& ß Xµ¼'µì£²õ�Anä×´®¼Ç:4i¡ãkα=I޷붧¼7ÊAÞJùm³ÀÁåïÀoLïBÃr]¥çàPÉìMÕÕûÞÇ¥Ûs< 5¯·û
\_m·° Óí·À¹=n± #2¯|Þ?¦ëc|ù2a5\_<ÈÈOZà½ñbOG÷üÏZèÀóÕ1 Æ,ö«°GÙéýßãu}ÛnȾ!çeÎ Yèû¬MCû û,Öcøïg ür¶Xäcö<\°X«O[àÀ¥¯·X}]êµæX§þî\;,Öu0/dpü Ø À0³ sì&ñ¯<ÿ«¾K6ñQêÓó,Ö,ðvàv©Þ+Ó3ô.ö?3Æ¡9v |}»ÙW-9ÿ@ÄÆ:c¡o²Ör?Pö»Ó>CÇ-0Äæ;ükÈèÈr9ʶ ýg³éLöæ¾Aìã3ûKi/vAbÛP×/e\X¦²Tö}!seÙ?ôb½H {!~SôðÜÌÚ=þÆxÀÏ}N}ªÏSÍ~Ì1 ]Æë¸Ùu° þ|Kùµ ¾<¾ï·°l°f½ þ}Ö°0¬ì3B?b.÷ZÈ¿ðCöfÐÏàwÒ·àÀ½°fpÿ$k<5zk\3ÛHRyàZú¨'ûÉÁd½ > æ)±ë¿V§|±;ÁÃÆvлËHùïäYn½
|D?õéOï#cï²X?±3äxØÛÈ´º÷T\vñÄúÆÚ½» vXaí³yÔi·¶Oø\À³!Ç«¬2û°\_±÷¦²à¿yYoÆ-öZ{_ÍYØåÇÕ©_°Zs ý©^ȽØú-lÇðJøÆú [6QäÛÄ·O0óúñ!ñ1ÿð)ìMØ>7YàöÑÇSÊ÷»S~ØhÁIӯòÜTãûØE»Ø¹°ëeüêg°p~dÜ@Oæ5r\_¯Å¿è²\_&ý±¸.õ¸m°lÌ x>JÖ\|Iè ðî?~Ç\¿emÚµÈz³ÎÓÌ\]ÊÀÛÊBÖNèX<· ¬°wëÒ{Ì-Æ)÷7ZìGdl°>e\_ôaL9ÜkñÏÛe«ymÍø4Æëjµ»kÞ2bá»f¯·Ø^@ß" §|Ð3ýþö%ïÜÖñ¿Àçà³ôMüÌot\ÚlÇ.³Ýÿ2¾ò¾ïT¢2¼>nü½ïLï/´¦{´o?)OÆçF,fÆ6 ø\_FV'OûG¹ßrÓb!±vö§{ð
ü^}¬ îú|÷&lËÓ½t>hÍ~Xdæô¡Ýz~ ²°?öYs³õÖLìi¬àÖYð}Æ!vJÏ{Yêß«]Ë-1´ùת¿ü98uö8xþSÞüwh
Ý2·:e7czÁ·&-ÖÊ{¬û+ÆtÄßÿdMÿýj½ÅÜB^bþo¶fÙªÓ/Yó~H|yÌó¼×>C ?½NüÃtÅ~@xóyÜiGîè·fÞ@»D l½)?ðþ<ïýV[¬Sø~©?[cÍø0én'Ãã2¾ðá¯ù~¯Õøì+ôzãX§{Gø5{zA×/éxµ¼c%µ¦ïôãD[Úß[lX¢þÝÛ©ìk:¾®{Ê£.ïƽº¼×uï\·|ø®ÎÛß{§ÑWõ½ôηuªóp<ºó¹S¦r\_Ò»o4ʪã$¿®²ßIu¼©÷/«}^î\ëÚõòxCô{Nõ¾¡g~ÿ]3ú¢ÎÿFs~õ5í¹¬ò®©®/èU'/ãE=÷}¯êøî]S~þÝ%}ûoÿ?í½ ¼Uåyç¿Ö¾.¢(HäÉ1pH)¥8x©RC(ÁA¥ß!ÌB(%HOd !Zµæ¸uðZc3'ãÛfRG±~Ú¤©µiæ¢ÿçÙçýú¼{MóÉ9ÛòòáwÞµÖ^k½÷ç}®ïÿvôQÛoOú<¥ ÷}ôp¸ïp¨ÛáPf¥m
gC mñLø¦üsÞoàOw¨Ûs!2þ(äõD¨÷þPOðBÈϾ§=¶wÚÚ¸>ò\<ûýp｡ÇÁð×=Ïr?äÍ7í e ö=µñ³áýÑá¯r¼îy"×CZvE÷ÇsrohÇcëé×W{Ïk¿ßõÍ Q»=Ú÷¨ßÏ#|=Ü÷²ø¸}UüûñO
ç»C;x,¤víðüË¡ÞqøÆü·B?|+àÅ>ÊdϼÞi4ý¶ðºdt Øò3z~$ø£ÕÑsø >k¬iðbö¾±Íßô ·ù.-Þ$nÿk×mÝ7;ÄúÐfë0Þ[>zÃPø[{Íý[Äùó»ÅíÞØ÷ûo°¥xÞò¾%Ôµ#´3öW|ïYhír@ÜþeqßÆÈUQ½ÑÝ+õú^ã3
òÞ.n¯Çÿ{+ãA©÷y¿C|ÏjlÃèû¬ë1ºÄíAÅ}Nö
úîýùø:ýL�úfl¼èÇwëêÑñ"ÿcÄç{$å¹1´;þúè¾wïq-"^×ñyèßý}hü7¶ûà{ÀXd'ôìôÁúè÷X×Nlq+Å}8°ÇaCá¼SÜ× ßíÏ 68vߺ}t5+Ä}4�~øõ/«Äååx®PflöÃi{~¬ ÷[²J\~bÿHúúÓ)î×ï¾0ø¶m!¦~Ä·}Wwø¾èÚ±=àAÌí=½<Üò9eZ!nCÇ (¶tvüZXcûí7ôQØðç@î ýmNïðx(ç·ÅõãøÉM}Ü.®ßÀ?=ñ¾çFùRV|áÛçËø¤Ü&>þÑË0§Ïñ_? áªðNËo£¿À/Ýúd»ÔËìÄõ²ègJýØ÷ñFþ±þ©k BÝ÷ó:OÚ¹ßlèmðQ¤-¾Ê79Ô 9è}1-ØÓv©×ÇǶ i«Ø_¿©Øg zÛuæØy+÷ccý k5úc¼Ñʨ¯ ý¾è¶¿ÑyL#ñU9WÜ/=c?ih3üªèþYëî¢è:¾ú üÇc;¾¬Ø У¾,×£µv~tÝ{öc'ý[lÇ^8[Ü>Ç^.øãà7�ÿ³]ÜååÑ9å½F\7º><ÎéÐöðÐ ìñ{ù5q_7K\/×!õôý¨lýÚ½Ý5R¿WÚ:©§#Ä_àLS¥^OÏÊ77à
#1ݲ÷-nßa§.£ý
f¯Áú.ñ=^b»E\ÿûÅ㸺ÄyÚÖÊr¸ÝڮÿÆqð(¤ð"¤øä3¿÷F¿±ÖOÇ ?~~fsÔÿ9~ËC9.¿¿|Á]!{˺~½¸ÌÍXc|ò=üÞð£.¾¯x,S_aº¸+ñ²Ï÷#Né{¡ M~ÊtD?§÷]±èÈbÄZ×e÷à
ìbca¦8MÝ%Nñ}ÁFË<íJ3Åm\øéÄ¿?k8(¾#³¥ëÄÇ?<ìη¡ÎR»jô|òük(| < ówj�²öÔÄ×v
yC{Ó>SÅãC £ÄùÜëC]ð'3;ÑÉÑ»°cã³L)×°eÓ.ü>GÜFxÿÿ ùùúàa¼":¤éÑuÎ2,ǯn8»\ÊòrÏpmqô~ô¬ûÁ,\_)¾>²> ]'Îû?'®#Gĸ\_³¯�tø»ìí¯{ÈÏþføòX]/îgÛòDWIÌ>úIä²ùáÚËûW½Êg<À
RÿÍ%/
ç4~õÃÊ.Úû¬À£0ï/çcæz '°÷ØX1{]E|¯:ûýqZëW
ò°oë2<&²ð´pî×ê?]Ü^$dM|i2{ ®í.=�rúHdJhæJñ=8©q<ô÷Vñý®à¯ó¨ãÅõûÍÞo <+~èb5׿/® AVcüÞ!¾ö¯9¡.èùw
¼ÙcXEì!ö£øÀ£×
¿BØ){/ó÷¢st@Å})á9¡ÓìE²^|Èk¤~ñkÄå1Ö±åáY\_ßÓ»cßãÀwòqn§x<\:0lQ¬IÛÄãµYóAĶNð-BÜ/.Ýú×hË4q[¼ sÏl¾]ãc8m> ¯Õ{lÃ{ضé ù£¯³1i2%ßç[)¾ÈqÛü=r+÷ãZòc}Û!>îÁÀÇëÐÞSÄ÷EÀ¯1OöyüFî0Øì&wlífå{0ô >pQ{ïÚ3[Åõ«1ðÓÅÎÀ=KpïÐ×3BÅ×ñ7%æ¹Ç|àÿËcÜÈ1vùª7¸4¼ßKÅ÷ÔdïöNÅMìØ-zÆ-:¹8=ùnñ8}pUt=àrqºþ¾áÀXßúÛæ[k8/¾/BE<¦iø·6vǾ
ý¿T|J¡aÈØnÐaYùÇ^"Whå±q} ´Ãfñ=�m|·'ä·#äÞ±r{xÇqÙÔʲ?6'VØÖ{CÆ3n × ï»KÜÁøÓ/ÌÖH['î¯^|1J¯¾ñozÿx=cüCOÙ/ ¾qnù#_RbXcÅÏ=øPYwgdà%�²ÍþÐGÕÐvWIý¾"ø²Cìºä[Ä÷Z_=¿'\ß/®Wæ^νo ÏÜÎï·
ߦæ¿ÈØ#Î#n Ïñ>báñw
¾ >Ò|í¦ù,ÆB{óÐtä2ä¼EñøwÅ1è±õC³±Á¡gkCï¼Áwá kÅe Öhë(íN¿·ëc° 1é?üø\@;X=ÿYÞ¿"´ëmú&ËÛtÊOsç6mìÚ³ùùXhÏ{Ãov¯Í=ümíæ?iãû ø¦èÞïAÞ ]!<=6¹Ø&°2úøyd ÅmaøaX=K;L÷5 Ö2|·ÛºñÉ þ¹~5soT^ÖftW
úãcÏ·É\GqPü;+Ø« GÐWä.{ß eC§GDÄO]Êi²Í]±ÉQõ½¡LF{m.¼,îÄÅ_>v¹ø7()ËøÄç¶7ø^|ÈV¿Y¾§[¤>û2|uã} û¯Îo�ó0¾¶ì÷!¯Ìå5Ú ?p¸ï±Mt$øÖUÄubøRÚ;_Õ5Åê´C\ÜaôÉt%è&¡ÏEõ~FüÛC°ÞãónèìÞÍRï7h¸~»v=ì6«ÄýY»»gGïF¿p£øwÑIaS¢Ïã¹�OÇ#X¤KÅùì¡}O÷ÅF¿^|ÿ2øat7¬£«¢sô@ø$³G9úÆNq²Sê÷:~C£nmd<ÄAqV~\ÜG.Åö
¸P\?²\vÁ^²´ÐÆÆëO¯]/n¸÷HyßI¿ ¯Æ|.lHY óF´6»Mê÷"cn®WÞòÆ_vø÷įÿ^qºÁ¿¹«S7jþ6G/x&ô5ñ2_×£ëeïtzÛÄ}5ºÄ÷ãbàY:Äýùð¹cn4_Êa©¸ úüY°Ì eÜ!®G±{á9¡K3s6b¿¯uáyôÆSþÉÖörÚú¥[\6°wcëÐúÐv÷r<ÚéöU¡í·»Äe7øk¯ëÅ÷ü¤]®·M÷'^\î·8I\?^ca8¶wl{£¸-Þ¼¡ÕÈ¥ÈHøtaK·ßX¬Løa'~ÃÏçÙÊ¿l¼ ã÷,Ü\]ç à Ã{o÷#vÃòÞúîÞÞwå&ÝogèsôèA¬ñqü±¸.ïà/m)±+ö§Ã»mÿíýà;ׯÃ/Ç�ãÓÇ|ç۬ȴÄIfóÛÖzËî }V ï:(.Z9,vï)q¯»füÛ>ñ½Ìæî·]±ö5~ÍÆ¿{£ \_ X~ÛÚî®?þQ> ¹ÃêõÕpmè×jhgó{<¼£+ä^ÙæðÖ?2Ïý¡û²gçög mzW¸ïpýÊðÞýá=ûÃó«CîïW5J<æStW|LC÷Kÿ¯×"³n¢«
¾ãKÆfN³ C k÷@ßdeA6Àý :ktmÈUwë¬ìúNñýjàñËCr©Ôëhð]
@FLþPÌ3|æ® eÃÿ¦Cü[/ðì± m¾�¾|¸nä°ø÷6Ð ÿ¬ ï·µÓÆ
w6[÷·6y0ô=:l]V§CO:?ÞgÏÜòØ-n@÷yMÈsU¨/:#{ÿð tÝîY ®º4a¿ø¾«·¶fÝîÿnÍmö=öc½¼J\¿m:?£è2nÞAL,1öÛÝséÅpl´í®î ÷°ÿ°·\_kz,Sß í°P<þÛÖÆúÌjõÑÞÐþ6uÞ®ÝÚÄî»H\æß!.X^ËC{[»Ãµ]á³6ä¹/ ;´ÏÞ@?mTÄýCf=BßÈïøÔ1¶ó ðôÈ,è]Xø 0/±ã>1¤÷ç³B\_¯¼#Ôs¨7¶ëC»¢´ßàöKýÞÒØåb¾yx¾¸\_@ì+o~m£¸¾Ú ?}¸nm¸D|%tø:xiöwªó.±\_6¾èèÑw¬÷s
eþ^">'á[Ðùáó ÿÎí¾ÿ8ô¬èÑàÓ ÿν4>÷ðKå }RMOm9vøwCðãÁÞ(?zÔNqýbø¸öÒ®ö® ñèÅ;áCi_l°ômA ºªq½eGÿîó�_bæI\½c¸Ý&¶1ÆX«WÇ=pÿUó9²±«KܯùñøÊU4wЬӱ]ócu86j´õ°¸¯SÜÏ~Àx·Åu Ð$æý,p¥¸}~µöûø\´t¸¾j¸n9ûY"®û´²Úxx4GyȼUÞà-kûW<òß.ÎwÏ×ïX]ÙhþO
:ÝÞÍ^ÆÇÞê÷bèÏï
ë[sþ¿mëËñïu ôÝæñ{ÄiÕiø>qÎ1ß.KmårìEÖØOVï LcùÎçÍßÀ·¿ÆëÇèZÜ?¬èùç Äã|JýjìG³4z'úüùRoû¾JÜ®ÃÚÈÜŦ?±oØñ¡ÈÆÈÒÌdÏ«ÄcµÐ]có
GÇFô¬cÈܬQèñAÿÿkuá;�بæ¦C<}4þkÌCtöÜ-64dæÿûlãÀƬͳ6N·zÿcq0k×¥ãÝ5>dð'ØXK±Ãû£û
¿ÄxôððÝð!+ÅeSÖyìüßVñµ?öaÝØ/¾F@ñ¿³Þ,Ñ¿Óßè^Ñýðg1 fcç?/Ç|¯Ãֱżã Õ"q[ö[bÇcðV±®5¯¿èCÌ]æ%6Eä¸XO}öfÓb¿oçµb^üxXY;c;-í¸ZÜLÛáßÇÏL¾ÄÊxcô;c :ÌÜCVçÚÎpØ¢gÅ¿-I¿îßãÛÚ¹]ÜÖ±^|M.ÑÖðDÐSô%öÛt©þ¾ø~ñø\üëKý¡ö^ôUËÅyh6%ÑuxcôgÅc×[È8ÐcüÄ°ð>tq¬ÿ6nñØ%[oúÆûaX=íØô3_zîeqÛ)>6±ík¹ø~ð?ÄcÄqøÙ.oÐJ¾Ç{ìüCR¯û¶gã8øä½Ø
}½�ë#üø2ñ=rXobzX¦ÝÀÆêoúÄXνC.mðÛÚÝx£¡ÞµöÃ>D&ãk·þøz8,]áYãÓæ<ºÅyhÍ\©ß÷¹zü /cmÏ$¶Wø)x\ÖËÅ}/,¯ö'ø\x¸ËÅyѵØ8ʺPÃöxeô¾§Vtðy´ËñùÊø¯é°Õ_ì·|¯Õúw·x:ý´åkávÏ3züÑí®p ÚQès+Ã-á|kèg#¯_ïóù;Åå laøà'M½�k"õömçWÑaÓÀ dYÖ|V÷ǶwóLFyAO ë»Ä¿ -ïçûÇöD¾_Ï üȨðYYÚ5ÚÉúÊzqIê?¸U\AÛ¶ üBñ;ï·»Óoè:ø¶k=|&t¸C|¯IÖ}äHüiÙsÙ¾Cço¹C²¾£?÷êúµþC&FÀzîÅêº@ܯþEïA~F÷5ôuÞ n'Ä7²á¯Æ¼Ø�¯/I,Ý(¿Ø÷½y¡Cö0Úlsû.q]ùcQ;cïÅǸuôð_øt·Í´s¾'¯´ñHì7íc8ï@3þh÷WÅéBìo5_ÜÏÌÞ ÿ ä-h%¼)|Û!qÿ 'ßçXï×ËðÈ Èè6ûc²®BÐc¡·BWbϯóè^°Åðíd|çÛ^/N3® }|¹¸}gøºok±M±g¸üïÿÞ ø¤wý·ç÷6Þ/N/áéãá9¾·Qê¿ÏÞ&ºuÆq+¾&ø¯eØ.þýG«ËݾVk£ßXçá·ë¡ ØT_Èlè4WDïØ#õqY«ùýáx{hotì±9ôÛÐ;ÅeOæÙNñ5öÀæñù¶Hü[ÊÄkáÇ mÛÙ#¾Ïãpç2qKù°ËÄ&ËÚ¿ ºnã¹/ ÷Þµ c?æ7ù"¼Z%\»#\ÃJñý05dL;5Þvø¾Ð½µßù¢í,EׯÝ×G}iå±¹Ëxî.wì÷hùÆ2ÏVyCOõÆ 1h6\bfûÑuï·<Ê¿K|Í/BX-¾?æZq?|;´ù�ÍýVÑS²>k ·\_$Îc"Ï¢OZ!Î+ag@fåzì#+Ä÷]'.ÿaÆfÄú˼¥<ÐvÚ^§]Ü?hìõ¼ þ>
}ø"ü. ýë4ÑIàO-]µå<ñýé¶3µ7¶ÅzFÑ cåþ!¿vñ¸P|Xó°·aSEö!î=s¡ø7fïÞ~1ËöÜפ~UèàÆ(Ý)þm½FÓNqÿ\_h:¿ÅúXìx»Äã´Èÿ|7óxß
¿\½5»3\<}£Ï¬¢¼XßX?ë}µw®Ï{U|Ýe>Ò¿Ô>®C\ÝBOÄ3ð¥ÄM®Ç ÀÞÕ%õô}UÇ\_³ðôÍØá Wzp\_,W3'¡ÛÌ)ò[½k¥¸/2íh}u =ð´ß«¶õÃæ¾ÃÐüákºÄý%I½Ýÿybô±ãÀ[fDýÁ<
¾à<8\_ܾ4~WJ=ËZÅ»Ñ ù=Q>èËAØdìúöëyr+²ëïÚ!®Ïî÷bÝ}ýÜC§_7OêíÙÈP¬Øµ;°ol·ý0~x/²4í²L\ï»B|¥¿iOJѱ1GðèÈ8W¯ÓøD+>ú0ÞVDýNÌü/6 ¶¹Q×uÑoðñ8Á¾¯k~º7Fí
nÑhÇ,q½#>»ËÅ×HìÓ35óXïHý÷×ÛfÛÐá2÷ÇëSö¸4Ö,߬qF¶Æ=;ª#ã ùÛ´½/u@çjúÐ×ÄýrÍÖô¦¦O7»ÌL7Ê÷ÌNa>¶>OÙî÷]5»ù7þÑðÞï¿8|b÷w<üðu5þßd¢o£ÃøblÕÄnÇ>6ÐIÆ)zidëÖ&츴-|ú@h1lèç¶BÏí è ¾G1ýf³w?ó{}¨7ú;Äc:C^ø,ç-}Lêy,xè2"¼¼5¯3¤>Ö½v!~ÃG�=:ÄÎè7xLøôýÿx~³cÄõ/1ß=KÜG¹É~³Ä}ðÑÿÐØ©3íØòhß YØ�ë;| ÑËÚ5x|¬ÿ ¾}ëò>~öØY» ·6·Vû[Y§Ç³u\_¾#}ýå\ö¯BßI,1þSÈLÐÑyRO#à ÇßÑ«°ÞCÃÚj§ø\¸&êSÚ"Ö
Ï· ÿøhY»¶ïo6C\´z"×ؽìûIb#²#4Y ?Fõ¶¾eÿ äFÚ~|°Ð·t¯}Ðe3Ý´ña¢{Q ¶¥Z¼kóÖP¾÷ÀX³v¨2W¹¬·½rëÏés=ö
¡X?õñ+àBñu>/¾¯'r4zÇ1´5i¦¸Ì\¬ M6£[$î÷�Mò!þdm¤Üè+¡>¥>F2MÜ^}}H§Û¤ñ[$¾sùÓ~ÐÆí<ª¥·ïµ±RÜwËÊ ñ¬µ¸Ö_»{Û¬d1
ßïí¿¼óð;¾÷ØÜëc^}öSRï×¹6´kg(|åñoF¾îÝ->oø ľFS·ö°uÞx=6Mt5ø¢w9,îÛbï3þÃøbj&¬²+|ÊÚì¥ÞòÕ¾OµC<ÞoóìÏnßÓ ~Êê=I\VkÍÇ|RxæUqõííq(/ö´>2ûËýùõ¿y)?ê76ôÉÓ¡ß7
¼¸ö8ooëH5ÔËÞýµpí«ÕPË ý»-ø9ÜÊlcÈøáúþð®ÃáüªPNÚÃÁ¿õÁÇAqÙzO(ó¡~/ß^ y}]üHw
s+ǺP®WC{ì mxGh»îPÇÅ㶲½Ú[!ßÃá·çsÄ:ìÞEÌÕ·B/Ô§o·xLÕ¸CáwMhý¡ÕplÏììÍ;7ºùÃÿz÷©õ/CßY»»÷·Ú=FëY£mþÚÄf oi°ù
ßÁ2¾QVfö1z±\Üÿ¿9ì6'aEwkãdSè[BÐ!cF§¸P\ç¯ ¾<¼
ß>ú ôQY¨åØ .×ÁË¿[.Îÿb§A.[Ú6Þs¦¸ü~Q¨Ëtqû#1øö.§
Ø\X»°/ßw}|ð²ÐGø<^)ï»RÜoÛyÃW_$¾>Ä>èðY!¾.¢Ç?áFqKh0¼k$¾²6~m¼tP[ûEv}lc_UlíàR©÷ëÇ×}j¸N\ýftÜÆåôð;6tè6Ûl¬Ùz3QÞàqr;F·Ïåû¤¸/UGè;ÆÈ£ßTÛ$nÓ¿$ÛøTôÍÅ÷x^(®ÅnjôÞö«B×ßÅ<ñѽëÀb=ü6þBøAÃmì$®¶gÑáÏ£ìÓeïm_KàñÄÿiøú±:ê[ÖvñýìC× ];õaP6ôø²£%.kÚõÙá|¸þÿ
{3ã/Ä ØàÑÍØ};Ä×yR¿w"ñz+CÝÐ{Á+Ì÷ .½&²vúkB:-zÇRO7/ ¿£«FMÝñÛDw61jW|\¬OñÛýuÐw³Ø,þMìKÅå}äèw«øx#¾6?Ý)îÛʺìü
¾±ÛÉcÛ8r ëÚtñõ #z>Æaì·ÎÞuØð#µg %øË1]G^6.ÿXçzO|Ñð CoÄÊóE×Ñûp>¹Ä½ÒöÄ5+õò¶¿·±'kï)\_:KÜWo}+ªÏ|ñ±½8Ücëé®÷þvÈçåоÏòØîy42Üc÷Úzb4õÞ·;[ƒºÃï«B;"GØZccÅÖsãðK@‡mëÖ‹qðŒØ}°«¡g±Yœ?Þ$¾mƒú}Æ$¶°iú.Ƽϥ¡¡Qк«Ä}@ðë„Ïžkå³u´U|ÞcW€?ÇNX7L95'ÐK\ʃ¾’±‹¿z?ʉœ�ßÇž+õq{V¿I߆O×QµŠ¯ivÿ‚è=è¥ðEE¾Ÿ,®e@F²wØž(¾7:º7êÀuæõšÝ‹¾åeº™)zúY$VVÓ0æ
ûÏ %®W¡=ï}½#3h·¶Àß155´%4yETwlmóÅy9kÈtñï¿Ä~ð¹Á>FÑnøÃÓèêñ·éßurÛþòÏqþ_fè%c¹ ?Çýèi_ѱØ{ÇGý¶RÜ ïÅ< ηËèôð%eí§ÑEÍ÷E²<Æû®\/î³ï"ôõ= ¶ldéQyðñ?}ú:X½LWÁ:ï3kà ñx|6fËèéXìäÈ)øÀbwB®C@ ýOÜ4|{ºÓÅuq»¢÷1Æ7ójȦÈóÄÞ,÷¥Ý(Në°C\_.>à1XæóZ±,°,þ´ãÅ¿÷¯<=¹\_\ßÏØ/ÌÝêÙ)Nwº¥ÞÏ Ù9
þj¸ÇúáÐf;Ä¿qó x|ñ}¼ÚC>ØEñÿ¾\¼çñ/ú|ãùM\_f2mR¯1+ªõ7´jZègøócÄã±Ð/ÑU\(>>ÛÃûf÷¡×nO¬¯Ð¥KÅé=¼´óÒ(Óå½ nKy�|CøßË·cL¾úz(÷«¡ín }E\ÉtF÷
ç÷v'¾ÆÊr8<Ãwí ÷[¿ÜÊr(ÜswøýÑ(¯»CÝæ{|. åZ$õ{áNÿV²~,ÈÚè#à?§ñam¦¿á§Û>Ú¼!õzëÛÉûB¿°\)>ßð3gNÇty¸ïÝ;F\¶AÞkÔûÞØx=A\díÆ~¯fwÝ">î°ËPbÁÐÑ [0ñ¯ocm²côõÐce l¬MÐ2d#Ç(3¶D}Þacxß?ÿ®Ü®wkÐìNè/ÐXÙvGõÇ^wyô«B\_ïuØ"õ²þ èéÐ@gX#w2G¡IÝá^ìÐÞ«B9ñi´2Mów'NúÌZ\o>[ÜïÃøý]Çèf¬OðG0<(. ûJ/ ï_÷iBÀc~Ã;Ey¡#ìRÖôÐq x¤õ!ݵ%Àv·\ÜsY8F.ÄOÛã¼pÞ>fgòWBÙì|ÔËö¬ñ4¢kørÁ'cB®3îJ<®±85äóGð.ÄmKñïðèhÑ ó5\_ëϼà{ès'5<Ǻ²"µc³³ß(îo9MO´ùs·øû¤Ì\_?élÈDð0ȼ BùXçix=;gýFÞþOnkãiºü¼nñI~Ìd9¾©ðè3ºÄ}mo鯽½co ÏáÛmt{ú%ìûtûų,ßÿbgàÏWfà·»3äÓ%®?{ø¿:Íî»-¼ÙvBwfubÿ©´q+ü°½c»x©åqP\^DϹL<ÖÈêp·¸Úê2E<®k¥xì±q¬¡k8ù\qÿ|ì³;Å÷ÀuÔ}¦¸^ÚMü-:?b=¯ÿnÑoø¥ó~¹{Bsìú¨}Wï5½^\×±,´éúÐw´AÔïaaקäkí\_�Ï1vïCìuÀçxX+
kÆT¢öCA?ñÄ]\Ml¾ÔÇʢߴß÷ËĨí çû» ØíÚñïÜÞ¿O\>ãÒè=øé.oø÷
í>Õà=nßC{8MEÎa~ì ù\Ú}e§8¯¸,ü¶2ôÿñ¸ZËç.ñø'x'æ
ýü¶K|b«Ð¡£ÓVX¹·ûÐ÷»TÖòÛ ñ}ò·û¢'¸^<.çðÌñýn÷qûn¸;öx7dÛy¡Oã
6Ïkcm¿x%ã}2°õ òN
fmÍ^Þ;[®¡Cܾ¹A|ß@dNæãã(ø<ö-/ˬðø/2:ÅåÝ+RϽ"ª´ 'ÖsËo¹øÞ-ЩÙ>6ì9üÎ=12ò ôm¡ø¸Ý$î·]?tƬiV¦»Äå=«û-ö»7Ë�Ø2c[.mD¿Ç±={Äubð&\_{Ï q[!ú\_x'ü:C^ÛCþøã OÀÚL\<±Y±~²N¼wÅvqûò¢PoÞµC|Õû<ÄüÚ2qþ(¶ äÞ969|CÐk£ß£ èb±;À#ZÝn÷»\ê÷äèY ƾEçè¼E¢6ëè/Ú ÃBïz£¸N�Yt^HïçWgGÇØpE×°ËOý¾qíÄÄ÷è {Ø¡Ûçû4àWN)Îøsü8úhdíØ·ú§¹V\WBsÛÏ°'Ú{ÑQ^(®fݼRÜÉ~BøÄ>'æÚ!þ-sttGeÁ®
ú_/îB|22q»ÔëÀÐ¥·L7Ì«ÅõWq°1·ûÖàsE,}kª8ïý?¢wÛõ}¾y1OÅþYèÇr~gx½[°1b#f¼[{ ÀWs²'û°Äk!åf/,|¯ß_ Ç%Ñ{¶GùûÉ^]¬[�YÿkxSxr;?}bðÍÀ'$¦Å´}»Nü[¨ØÉà3ãuÿö>Y&õ1>ÐØYñ«ÄÍþxÔïWò¢ø>cèÁmÏ tý0k/ö üæyxÝ?GÜ ºû!·Ã¢oz\tjÌMôϬÁðp±^».z×v©ß3y^%¶=1÷±/×ÓØ}ر9ñº+Ë^ JäÖ7ôÎJ½?³½ùÏ Èþ&Vï={xÐèc#2qGôìºpPxux,Ö!ttèó:0'JçDçÿûÚ£ã
G¸ÿÞ5á]è�çD)ÇèøðøV^]G¼!üÏ$ëáLñu]5þnsÀFwHþñþñµÑ1yR|pÑÆï}÷á±Ð#Ni¸÷p >h¾ÔAì7|Lî±1a±9FÇ̧1Iñ³ñeñðÛÅ÷DÄ'ymcëIñ9ËX³=GG^"¾·2sùOr#ºë<ôýò"å\/Îò÷�võQ®©CÜðRq[¶?;ÞÊt(z?º¡qþ2öäº^\w®êvq9\öCoˬ#ÄmWÖ¦ÝQßÒ&ðÆøÉÁWÀç§!Ïïaëa\DZøÐÈÃè'Ç)/d-'ÆñzͲg°ï¢+£O }ðÄØMñEWZ n#'ïvñ8/t7ñ±ûÓÈϱ¯ þÔÇ®aSÀO&¶Zävú¾i~$Ôï\q>õB©ÛdDW aÆ4{þ~ØÔàUì>=æÛcq/oöÔb$ì>£ æ×cqoasØ|m×l<Ån>î1ï
c»¾#Ücë§ÍûÃïöÞ¯ëVBÛCþOkæ;ïàFqÕ¾4¬ËãÅi-ºnä$d-Óa{
ÄgÓúËæ)ô9ÔÇuYþ¶^ïoøá¯\_î¹/¹\_êyÌíqìØ<±ã@·óðèWØ3ÏÎÙãoøÁØ$çëÐ
ãwå6ñyýô9«+üZL÷ñ½µ¾çbþÑnøZÍ÷fn3°Æ±ovLÜö5R³ÇaÐ\_¥~¿£8¦ SÄÏ ýo6dlpÌGxÚ;X»øØC>e¿$æù2qúÍ8Æ?=Ø)Ô9?C»øÚ-ø B£w¤mì7³Mþ×æ§ÛËÅuîÄ[Æï ÇöÑ!ì=16î± Ø³ûCùìÛÅ÷máåè(çëwñéÁÄèí ß±g n :¿6ô1+»¢>Çïùªè|ø>uè}O ë"º"æ ëkôq5zfüXcù¬U\¿Å·\ôsÓê¯1góÅ}¶¡ðùKÅý áS/è¤(+s«=º?øG¬i²r/ù ù~[Ügög }\,OÓû¡ÃÇÇ~»9övÆýߤPVôGwrÞ-®gD/ÇX[þ]ÔðtëÅãÓb#Ú{}¼Rü[÷èå¡16N«á|Æ�/÷VÀ\_>ý2¾ ¬ è¿ðñ-ã,±§2WzþP\oÀ¾ÚÄí/Ì5ô߬õÐtãAl>cWêÁcor|$·7,_ìÌidRäö6,Êg4ÇæñB#j8ïk<Óäü¾òhöú¿¼+ó¾F³ëgëtl£ßÑpÞ×èìç÷7ÖMÖ6÷5 5UÃy\_c}Cþ]ý\¿Î&£cÅk×îÎû¶ï°¯á¼¯±úWìÿ_øÖ5 ìAö5÷5V6Ôwôo{ÍÞá¶ó¾±0½ôïú¾X|'¾E¼ê×·À¾ 4÷5öúY}I?mÐÕpÞ×°1¯Ok¥×?£Ù1=Ý.ýK¯ÝÍÆ^©§7û¥é~\wÃy_Ãú0æ·÷Jÿòów5´ï~ýüþFlzþwô/m: FøËó¾ÆsÍE>¬ùy¦úõaý¿ýeè×úMnrÒ\zF<Ø%ýËO,iÈ¿«ë×ÙÏõùuã¶79?á»"j8ïkìhèÏÍ¿dÿÿ²x;ôKÍÌÏxÞX^Ú,ýÙ÷O·e]úAêê÷sýv~iCÃùª~Î/ñK}¿þí°»á¼¯aúÏö5÷5¿Ô·fÁhèkMÄ£MÆMFª\_ªß/[¿fæ×ìõ¶Qþ[+ý\ÿÕhZ3íqwHséõMî¿fýµÀFé\_ûX£=n[?çgõy+á¾Fcûmö8kÓØÿt³ô¯«ù DzîÒ¿²´í³ýtgÎûùõ7~Ýê×ü£¹~ÍÖ/üûìþõGiÔ/þå\_\_jvÿ5Æóö§>éHú¥.Iö¸¾ÛÁ/5qþål+Iï~&}
Wßú÷>Ïïà×®~/ý?Ö¿æõ{[ø¥Û"li8ïklhÈ¿«ë×Èõ÷úw´ÛãÞnÿ¥{¥ù¥ÆøÍþFú\_ªhwÔhxû¯Ioý{Óÿ\_ÀOüÊõIõ«G³×÷ÀS4Gsý¿ÔÇ°öקMÒ¿ë߯~iw=Ò¿ñNüÒé\_~iǯØÿ\_ªÇÖæ¶g#¿Ïêg~1éÁkR~ÿoç×̽ýÔ¯Ùú ¾Ó,ÿ¥Æ÷÷w~뤩òíQ¯\_2"ö÷Ù,ýëO´¢á¼¿ýùé-Ò¿k|cûmþKØøå¿ôôî¬?ïs\ú ¬k8ïkX|q,OtJÿÊ+õë<ÊêwcëwUCþëú¹~W5äßßãoè¥ ç}fë'l}õ?]Ò¿ú%ÛG7^ÖJÿ®¦?Kú¥¾ñ ·DØÓpÞ×0ýRì¿ß)ý°ñWìÿ\_Õö=Úø¥þÞ¿ª&s~+3 ç})R¿lürûÍþ²hÿÞ{¯îG¼õ¿u¸²á¼¯1EúW\_ÝËoy ç})ûÓ7õ»¤ë×ìõv±ÔÏMÒ¿óÏößá鿽w &ß6^7»ÿnor~¶Çòüé\_}élæxiì¿£\_2%w7KÿÊÓFOú3^¥/ör¯7÷5Ìæq8þó¾ÆOßæúÝÝÏõ3ÄûSß-ý»ÿµ½?¯ýíÏp´×ïíö\_êoýßj¿ÇÐß8ÚõK¦óiæ|°ý¹býÏé\_ýR£ÿRë~üþlÜï»QÛ?9ÞÏîYéßýòl>ÜagÃy\_Ãdê»"ìi8ïkíõÛö6Ô/^/:¤×#³¿ÅúÒ¿ú ÓÄëï\éÿ5¾ôÌÚ´üÒ²&ç×h;ÚôKÍÍ··Ó·Gú\_÷3Ø%}»?B#=®oñ}iî÷Pì{Y7FXÞpÞ×°ïT\_aeÃy\_ãh¯ÙWbþ¢¿¿ùvôßõV7÷5¬~Ë",m8ïkX}IÏlüÇô¦KúY~ñ~Ê[¤öiß4óKG»=ÎÆK3í+G»=®£ßßfûízÿµÒ¿þ/ÝROo6IÿÒ3ûÿög¼Õ/¤çû&·§Õ/ÿ[¤éå×þ^ºÎãÅûõëoíú¥µÒ\ÿº&Ç3ÿZèbH3öh¶~©ôìh×/5¶gã{ÒC h8ïkM'Þ¯}»ôÝÞïo¶|3óÛÕäü]¿gßWÍê·©óközÛèÝßú%˯üÙý\·\_j\o±þ5\_jôWêo~©Ùüucÿ=ÞÏý·µßß«cloi<ïküPêãcþ¿ýÞQ¿û¿ of¯·æÕüJ#49?«ßÑÌ/5ÛOØ-ý«7ûQ3¿×Ól~é×!>®íg½{.½Æó>ÆÏéöJÿÊÓþ×{¥ýSýúöé;¤ç4lk8ïk4»~o~©üË ç[û9¿fóKGû~Ö\_±½s·ô¯=µq¿¥þÖ/5ÛþÞؾ\_úÕñÌÝMÄ£!Ïfa_ó;Úë×ìüÌï´^úg_"ßõõwKÿÚõs}ÞîýmÛÐäü¿n&½NþKGÿR£ÙöÔMίÙüé ðxÃy\_#Ú¾3ù&EA1Xj{fÖÎ'+¦+ö+LÎ~MÓ1³£¢ófôÛwÖr¥ùBÅåáÊ£åûzó, ùl y¾êç
!á~~$Fõ"×vËï ß~yV¡ó.×u¡ u+÷Äߺ8(õߤ0»#:NhÛö£Þ¶©o ߬±vQZ«ü·}Ý0ÅÄÐv>^±2ZoùlZ¤p¼1ÜÃ^mkÃXXÎã=6F[x9ß_ã÷½Ñu[#Ýê¾>[þç÷#óþíß_Z+î·¤õÙöØb_ÍQÙbÙ =?1¡Ïýà½ø÷ï åml§Ft
1°I,nï½Î÷ªßÈý9~ÙóÆrnúýHåd½Ú=w¤ròük<ðï=qý×6ôûÊiý½Qüè'ûþæ¹ïÓÒ»ÊÓÏ ¯4Çãö¿r¤öäeoÕïáù7úýÇÿ¼ôç¼÷¶¨]ºÞ"ÿx|Þ+o®¯ íõy7~OóçξÕøÆùsó´'Ißj|nåÇ'ßÚ°ßø¦A§ø7t¾.ºoÄíI|ôþHådRëÒXN®³Ç5{3[üíÊè=Ñç-úýzÙ£÷oÍ^³oÖø ÓÞoVNÖ�ì¡/KïÊÔÖøÜdãxO¼=G8olOÖÉ7ëw³öÜʲñMî³ú°§{õܳsË;}¶êå$}³öû=ÞëÍúu=�.ýÙxN =ç×sbÁã÷vÉÏÇ¿Y¿Ó¿h|ÃH[óæú¶(_ÒÆñ·'x³rÆý¾GÞ|¾ó<±[Ä}¦Ùtü|{þ¢~?R9¹ïñè>|ñÙûUýöñVý¾[|ýýü (¶×7;ß®aSäüHåÜo^Nl"GjÏ#õ;zä¯þáîè|yC9Õ~?ºÙ+f½ÿæýb>ú °?Ë
zv¿5Êï:2Z>Ùc®SéòÝwEøN øÇÀïÿ</®Ç±ïû÷cÈ¡^ ýt ëβãîu _©8ØÞ¡Ðwøý,;ñÒ,;IÏG. Ð2ü©,}o{1forGBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBÿåȳì+_ËWg³¿Êg
lX6-û@ k=f@VÎò¹ó,?%gòK=YVú³ln¾¹gøèÊÜCùæ¹ç?ñ¤w|óý³þc'¾zýõë;×笿x}aýÇF=õ¿ôúG×éU7èÿx½þ¹öºG_|ÝÕ×]Ýç¯+e×v^{ÓµÝ×¼6¿öºÎß;ù¶Ñ
3¬\úwbyÅ+²µfs
ìÌül-ÊÙÙ"E1?;Y=aÄ;îËÏÊgÌ=GÖ¬Õ?¿»Rÿ,ÿþùðFw®Ø±ÉÏ(ùá|ìó¯Ø¼úäQ'þ§óF¿QQ¸ïõç[&ô;´ræ¡ñ=Ç_9§íwhQ®hyOöME!¤çǶWLËÆh:FÓ©Ös²m-ïÊv)×{f3õÉwµ}©erv§ÔôMP´´¼»eTµÐ:÷PËèêà¡ZF·È2}ÏÔáÕbë¸C-'VO©×gµ¤ù¶¶l9©Zj]Ü6@Ïól«þý\íq-'õqfE8©g̸Þtø V½qf6[QÐÏòôÅ
ÖKÛFµ ׳1-§´Í· iÚ2LÓ)-ïliÕjMl9µeRvl6¯¥¨.ZåïWª´ o)_ËJZå¿ËYkùCúãþ¤üæ0îPù¥GW~óòKZÒqå׫'ªÐU1º¢¯ÜÕsÜ öÿÒ3ìʼÃåÿ¢mwcY'hùò-Ú´ \ß]¡¸VQÌnÕ¿ùëß(¦gèq:P«úÀ'íoùîò͵üR-m§{fU\µtÜËcgUó°LwÚhÔvrùOmÛßòå=åÿª ÷åíåOkà \ߪWÿ¨|{ùµÁ¾PÞWþbvìë?Þ3¹µRn\þ¸>ú/µ¿Ëk²+ å³zÊøÑm'?ý¶ZÅzÅ6E9;ü»ÙìòªìÅõø³²¶ïÒA#\ë(¯Ô ×õËzÎ:ÇÊ~YUG÷CåÿX^TkÀEå
½ xuuÈqzýêòUúÖòÅåKtÜúPùìN
ßk{ÆO²7\Û3üDKUÏ8«r\_ùÃåô |@´QzUuÌizuaùýïÈCµ¾Òöîòuåë³!åÊ«³¡eSüq 7hÏÛßCzVÈ>¦×èÙ6Mw{Ê: ¯Óy6ÀuÙjO Õ£±VÅl
]ùì@y
¾cnù7ªZƶ÷?X¾´ü´æ£¼@{¡¥üA-e©<\_3|Póúö%E9{Bÿþ
^ý®¦E}Û߸çÅz¼TÓõÞYóåß)_Sþöçå+Ë¿Ó}tù úWdÅEI§Ã\Íq^ù\ZçfQµÎ«ê8¿¯üÞòD7Úïì9e\E[«µgÜøÊ˧k×M)ZëÉåI½7U7I:UÏkÃqROe¶uĤê¸NÓËã³³²ñå3ßH§kz¨<]Ûmº§ µì.k^}DQ(O-Q¦ísJyly¦ò{ʳµ>3Êggj}§jé^ÉþSéÙï+þ¶< ûaÙµzv¾b·õ{õê¿OÐ)>®ô/='¿£R>\ú¡>}~é_j#cDÏÓ+ÚÞS>>¥(d-ͶGèÑ{Jÿ 8Tz¨vüñ:ÿFè�¨óøìòÒ÷jcõØÖÔ楴%¤eMÐ{ï+ýcïõÒ÷J« öѶåaµü4û ¢PVú[=o-4µç Úý¯÷gÚL§Õ~ßZû»[ÿÞ¡(þ©ôjé³Á¥¿)½Xú©ß,ýMv¢PúYéµÒëÙ±¥-ý¨ôoÖx¥'³{Kÿ3+¼þ|éVOd¤BÞ1& Vi{gé¯Kß6r]úvéÏkéÿ.=UKÿ²ô§µôî±Ò éWKÕZí=^Kÿ[É(Vké¿ëïVújéO«Ò:°í¥¿ÊòÒ_iÑ«ß}¥öë_þGí-ÿCïÖÁUúóðÔ¥kO;¤Îö¶!¥õýáÞý}!=TºG×»ÛÓó¼ÔS: Ég3V±@Q,ýYé1ëÃJz&V)µ_úBv Å·ßUüDÑôïbEáõGJ_è>²2¬mDi_¶HqVE){Dÿ~CñE±´·ôùlæõùOªCÆv¶\º-Û¡ø¼ËOZô=zu6Õ©¥ÿ}BñMEñõovõ R¹BÝ¥wiyve¯(JÙ Òîl´B^él®b©ÅMré³¥cª?¡mbiG6A±LQÔîÐûwdg+¿§¸Iq³b¯¢[1@+ÓÝ©(dJÑ»¹4¡úαÚÆþPóüCmØ?Ìf+nUÜ©h©»ú¢¤W¶êú+J¿¯ïØVR3öþÀæYi{ÏIcÇj×}Zïü´Þùi}öÓÙÅ6E¶ò§z¯dmCK2jXÚÍS,V|FñE¹ôÇ¥Û«§½¡mxév½çæÚß³K×»>Q|FqPQÖ o¨.øÊC¥ ¥ ÙHmð ¥U§]Ö6¬ô1½õcZÎOèßÏÔ>SÚ¨±±Ö¶¨|>öÒÚc´SÆm\ú¨>öQÍó£ÚãÍWulÕR®Õ_Öjÿ(«õÿBº^ÓS4ýO!½1¤ÿ_i]õ±ótðÓ¯«eÖ以ëõï ý;ZѪ(꼡gÀ±kÛ~/[¯(d:´Í:²PüDQÒÜ¡/êÐzth_QúÙµêëtTÛ®Õ±p-+ý®×ßÕ£oêßïÔ®(-×'ëõåúüÕ¥ÃPZÝ_²
ëÒ'³ëW褧)nV<¬xNQÖø>s«þ½Sa´å¡'WÎm;½tµöÐR-ôÕÚRK¿£Y]U¹Z+qµ>rµÀRéJÄ:®Ìö~[ûð·µðWjá¯ÔV¹2è×ÆÑe=Wn}¸tft½Ë´)^=}JEIã)ÚÙ´
Çj:VÓqNÓt¼¦NÒôNÔt¦§i:XÓÉZÞjñ'Tû|¨4AÁ"½ðHéÅ@½bY ÒÔ²¬éÒ¡þ¦Ã4}¦ÇijY ×Ô²:^SËêX#Æz@OsÍï ãë ©©¹¯øoÅéÚv]ñ³¡ÅUü(«ÇÓjø¡_?Òú#íÃ?Rd\ñÇY^ü²Åé¯#²Aú{í(ÚJ1Gÿ^¬¸Zq½VÅ %EÅCúûÅ{²5BöIýûDíèsůë_,þÑàßÿw-}!ÿ>S¼Û(~ñ!}2¤÷ÿ¬þi8ÿóWjé}½ç¯?_¼»:üøÊCÅ»õE-µ ߩάؤ&ëÁ_¿Ó3ôxmÿî9£ÝÒ'zÞ1¡²¬mñ%-íKY¡øxñ1+
>óXuô)µ¿RmªßÐ+ÓÅ¢øíPÒ¿ÒÔJðtH\þI ÕD r¸øåbwÕ¾å
VÏ;m|ÛE
E¶°.(\TKôwÊø¹m Ò¿Qè(Ô ª|·mpa¾¾aQaqúy¶¦½þ¼êÈQV°B[uV¼ÐVmk©^[4¹öËÜêc\49oòøC
9=³TIÒZ9éÁû²¯( zûûª'¬=÷¾ªÎ ï)ÌÒ©ÒZ¨fézy桬³\\Ë?ÜsÊ)½©Ö´T9ó¡l©ByÓü¥êã\òz¾\lÛ6 ÿ:yþ½Úþþ¬VñC
û{© }2ÅÜÁªÖø¾×ɧö:¥2í¸|jvyÅëR6Nÿv+^Qõo>÷´|îëùÒ×ö¾ö×í×ÊgþléÏnþÙ#?+e?=ó§KzóOK?m?}ü îÈF+ö+J
KzÎ:¾µmxá£Oú÷Ú ¿¡ç ÈÖ(îT ï·[µÞß3txå¶ ï7v¤°PÿZ»ýýû]E¡ð[
ómü.Ö´Tëó£S8»Ö3 gkkÒ~=[ t¶æ|¶æ|¶ætvV.[x¯òe?y ð^m¥
³ª§¶n;£pæñHíïÙú÷ÅÅMnE9Û¾£øB¸þ§X¦¸¡vå'
éúü2ý»FqPQÌæ
¾\úrYUûòPñÃ-äø-ä[Ê #£´FFe4]¤éRMjz¦7h:0ûAþÍçóùY¿¿P>öóä/èÉäw+ºãp¾«6ô¯võ®¤ùè0ÌF}µphî{t(Ì}uiW?[hÍþ&ûÕaÃ+_Ú_j½éù÷[oÚïûB¹õ v¸7ß«I¶Øþ¥ûoØ_j;§ðZáǵú¦:Y ?ÕÔ&ÜOBúã+µôµ?Ö&ï{çØýÅÙÚù{4Õß¾[SíլΠéÙÅs´J
¶ï(©Ý9º8¦öãÇÕÈÄpMíú° ×SrQh;¥Ð]ør,_.Ü]9wþ¤vþ'
;ké]Úõ;BúÇ!½½pgæµ[Ø SS©«X¤h)lééµfms Ïæ( Ù°²3KE#§dPìWõ¯é¯NпóËPòïåÿ`$§ø[Å÷×j¶HS«ÁÅ!½(¤ôZM3/(\8ß_ÝYj=ªvYr¸º© ÉCÕÍ
Ûd¯Ï[Ý|ÖX#3³«c&TzP;xwµýüÚÁ,ÎéѹîÕ§äFùéùd-ÍÜCùä³frsrõ Jî&Ï©õñ¯ZÐ~F1÷ßù®Êv[½þHÏÎe©ÔÒË~»7}ÿ-ýo;Û~³²³k Ý3÷®çTº>·~ú³åÖ=+·Î½uÌØÊÜÏé[õÊçÿE±KqõÙ3¦Uæ~ö3õϸ úGërqW~ñçr]ÙþkqOnÕÔ:ewqOmÀ)~¶ØUëÎÏhj¿üaHw»¬»¼æÈ?^ÑÚêñJu¼Êà ß×Ic?(|ÁÞPø¦v¾?¤_ÔTLa_H÷ôóáþÛ _°«oüBuV¥ÒvJñ¬»jÓoº¦V¦35µ²L é!ª© ÅÖ¾³ø.«Í}¯¿¢ÇÙò?²8ªvçIÅQ½«ñ¨O©Ú©µhjw´´®ÇÔiaSÏæÚ¹
Ûú{ýá²l¢[Q,.>¤£µxuorIÏC¦¤ÈN<ÝÎü/z¨úpþÙbÅwÅü©$¥è§¶½£0I'Õ$fjSïÔÚ1Aû µUg®ÃãLm¬OU|BQÌ¿]_Ó]åÝ3ppePÛ°üi[ò¯gË
ìÛùÿÔ
"Ë_Ëfecóïæ¯ý¦ó¿ÏnVôTgWÛoç×l~ñ¼ZeÏéV=þÄÊ}ùÎÜTè½7kQo°¿ùÎ7X>ùÍåÖóåÕ3Çkò{½ÉjKÌoPwüëÏç«{N8±=¯Î)ûÏo¨go^HK¡ÿÚÄ¿¦6ñ?Ô£_çàÒÓ[+W· ÉÖVý/S¹þv WV7רËoå3¿/o¶ÌØñü=Õ ÕêðjÛyáúYµ÷VßÿppÞoöô¼sºåØ^=é¤Ú
¶jå=áJk8ur8PVÊæTçÌ Ùá@GïÁÔiáÜp-i=Væ.Tµ6òÖùÄÍ-?P<LñÕc1ª_4QfiÛ´¾ìÅM{ÝGßP®_Ò群ëÀ²ÿ¨x]Ñ¢¿ì×w+~ÑÞ«¿Qù/ê 0ýwk'Íí¬µî9kVåf¶fdùhíªÑÊÖq7ZÛüýk3ºúSÃýÇTiüÝáN!óãõèxåï×gÏöº(QB}|¶H±TqCÃ]Çhï̾¬xXQÌ.Ö¿W+®Wtv(^W´è[FöLf]5²zÖ{kåX]´(5OIöÀ-[µ ÍÔêaÇåÇèßóý;6/k˪JóÜ 6[òB±õóÿ£sÑØ/ëé :Á^ÿJþg[,Q¯üÝæ¼õÏ5ûÐÜÃs.>üÐÀÖêlß6°u«bÛæcj+MsæÕV´U-ý¸ÃZzÞ|KçÞðñÓϨ||c©u£à&ÅVlPÌíüÀ+úOiö[t<|bs©uÑÍ:¨nÚ5bä9#FÌ1üìCg|ÖÓG´9¢8mDvÆÓ&9}òÐw¶yWëÐ :qè)c;thÛ±ùóZi3¢õï|kþûÙ©µ)òû='ªÌm{^Xª¸Iq³¢[QÎ/ϯÈäó¬ð°¶ý¡ëKåÇéõiù0í«aÚWôu鸵ûsûmÈÁBñÇ#óWõWòÒËßÏÿñOÌ=~Jo 2¥Ö&ÓJS¦V;nðàc 0pÐàcdp±T¬tpðõ§æã&|cBaîEðüW&íÓ&èxZqrkbÎÐ|hñ
ÑùcGsò±#tìðÒ Ç.w¿ »q{÷ñ¹¦hïÑzÁ¡¸KºÏj½ []±ä<ÿôezµ»ð©CJ
»K:TÐdøy\_±äP>Ê~Þº»´ý²Ë.ëuÁ¢%vçecº] ·Þ4æ²î³ìàæ1eú¯£Ãþ¬±¿þ¯£µvµÄþÝsúiçw¿óüßé~×ùKçÅ7çõÏú¿5ñ4£5½×5;½²fíZ=Y[»ª§kßä-µ×¼Q }¬·ä>NÍô¦Kñy=N|ò>]R¬4ÚNãôÒðD~^uܸJkëeuåê°X:{;ÇôÌeÓsÚÞtÄÉ[ïÓ¼VÏѽ÷ ?±òLíÚðón;8¥8¡ÆlÒÉÅIµ5î´N ×'ôÔé¸-N¸'Úà2j<¬ø®iÓ+Ãiª5®¥ZMK«2 b·¾?Ûb<ϯñyKTñí F?:}ZÅø£Ñk©zÈëÁ°ãj/Xku´ûÞÕ{ß)cÃ}ÇWZëÚ±V$]ºªgL«ô\_ mÔU>¢òFÁõÇ=5{ô;ÕZvOuÄH{ãè¶aÙgܯ8Xã)íïW߬Ó;uÍÒ··ekÖ¼É@ ÿ8ªk8~ûyK\.,Öäk-Æû«§¯çýÕÖ3zzTþäzÇ-&Ô^0ºm@Ö¢RG^{²Ô.è}²ÆØÁ £zÛÎ'®ó<¬mgM sgîìPkù;iù;åíàê£+½c|tí¼çäZÛßÙóÎwõ¦½}qgè ï kWTì«-¡v0qR¥6¾©2¡
ÞÛÎé°2N¾§#G+ÇÐ;¬ZçT§õvÙ9Æ÷7òçFÍ56Ö0çÝ;j-@K¬ÑiûÆ\[Ó;ûìRÞÛNa×èÐä$&J½Ä#ëhÍ£µKõEÉ;ÖÜW>stream x]ͪÛ0F÷~ -[º°-Kr.Ù´²èMZîÖÇÁÐÈFqyû\óµs¡Ð±GÌhFõÇçC7SÏK<òf¦9oË=G6g¾Ì©jç¸ý] ãuX«º·¯4-Õ~oêåãmËóîtzýм¯êoyä<§K1ÎþüUÌñ¾®¿ùÊi3MEdFÊV\_õëpeSKà<=V6VÖ-2ËÈ·utájßöËC§ñ¿Ï]¨óôö{GJÛ¨RÚ ªmIiÊÒNP]Ťì΢¬#¥k¡)mOJç Ð!/Ié^ FRºAT×ÒPÞBÉÆ ÇöÎ2 UHªCTÝÁCIvPÈ[½ÎËM¤ QRÀ¼ô è·¤ è>=ºíq'á){Ôè¥:°G^ª{ÔèRö= ö;ÊÓ÷ÏçÕÑq÷ËMû%óþô9±^ÁuYQ¦¼ÕLùë endstream endobj 32 0 obj <> endobj 33 0 obj <> endobj 34 0 obj <>stream xsqpqp . ,® pIX ��àªQ endstream endobj 35 0 obj <>stream x+ä��î�| endstream endobj 36 0 obj <>stream x+ä��î�| endstream endobj 37 0 obj <>stream xßoÚ0ÇßøühOãq"U(°i°neÝêC iB«ý÷;ŧJ¨vÎ;ß}m_Ü[×åC>¯ÑÅEÜ«ë|þT,Ð,®^îé"¾Î˯ËUßlïk;õ¹ÈÅúò] úèjÚíÄ9MºüqdeB!£4
ÏóéÆ ÇM·ÃÐc3J÷£OÝÎ ßÈàáO$JðW9îMH¤ñÜ¡éng¬³.%Í=Ìpä454kYJ4ön~(Ã?H$ðp<§ÈASi(bͧó|¸%cÏ0kW¸æÄÍlÛQ6Z.v=;9é,©Êuu¼»ßÝÎ÷æWKEQ]¹¢)ZÝί¨j%oùð¿iÐÐ;ü¢á¸P|m:îµ'N
'YBl¥4UgÀïxüwI²Fbá²$"=R@¢2\CÚÃúªs{I<BÕAÅIÝf4 \NæEMPq8óPtTÄ£X Çe{$«%MDñhVs8ÚAzãÑjƨÔAlUª©2Ana¡pÖTUÚ£\ק «GºJhÕHnÊ)W )Îhv¨ÑmYH«e^Ù|áz
±×kkµû çp:Z¦ÒõXsHÍí¢±å-^7CíÙ>¤ÍMеóýçpí0'«Êe-!«:lcBf4åmÛQe|Ímå.óýPÙ Qÿ©´/v´ÞlögøÍöeýd{¿½ÁÜdx»Ûí¨ª ¢ñry¨uSoóåaLï:£74ÍÌ@M1(ÀÝkSÿÕ ´+ÿ²¼°Îh�ÍV¹Éïm¯W6Þ˺,rãêfà
ˤåȯWÏ-)nN nC:Ú\@¯u.¬¶vÎ]¶ó¿V¥ endstream endobj 38 0 obj <>stream xSä\är áÒÈ4P04PIã2T0�BCSs @. C CCC(C0C¸fHkW �¨ò X endstream endobj 39 0 obj <>stream xuAKÃ@
ä¶7=x,sµPÝ4¤PzhÓD±µnµh/"Xªøÿñ5XifvvßÌì7ÔFmÄúhZ,¶Jjy^I05}SýRJ7ÒÖq¥+5,U0[?½K·è^OúÙ ¦ãB1L\_5?¤Çò¹Tg8Á#ä¸+\àîáSÆï9ïw´C|{cýJ4O9Õ}#ÌØ×@cפR qF«Y¾ñïå|8Þá´¶í'¿¨;ôÉþeMͲ?åÿ6¼ÙCéáçháæÁbñÅ°]4+úôÕÆ9ëÉ£8òÃô¦SßWbRÍ endstream endobj 44 0 obj <> endobj 45 0 obj [250 0 0 0 0 0 778 0 333 0 0 0 250 0 250 0 500 500 500 500 500 0 500 0 0 500 333 0 0 0 0 0 0 611 611 667 722 611 0 0 0 333 444 667 0 833 667 0 611 722 611 500 556 0 611 0 0 0 0 0 0 0 0 0 0 500 500 444 500 444 278 500 500 278 278 444 278 722 500 500 500 500 389 389 278 500 444 667 444 444] endobj 46 0 obj <>stream xXÛrã6 }ÏLþR'¦ER×íÎxsÙ¦mÒm×M}-%ÖÔ\]îß$ÇÍØLXA8�8uöÏköáÃxR×ñ|&ìq<-ÖßÆÓïëtü%~Éò¸Î|üµÕøè§4NÒòãGöéê}oMÏÏsàG°@rGº,p=/Vçg{ÁÏçgÖ= ¬-] ZæuuÙÝE²7@M~bõ ÁP}p)Ù/ 6Ôw5Ôë8 Ïúlÿ=ÑÜç˼B¤aMMM¾ÅB|ÍÍ 3~ H^sñ¦KÞ·×wÀK9§ ü?uµ8îÖìî\0 ¡CnÄh¹L
ä:vÆ;A}¼¶º®¦¡¾®aÖPÛP¡!ØvòêfÍ©©jã´ÔÍ3;=«1Q7»ñY9ÀÐ{ùÛoúXjv!qèP ½uË!æ°=ØpßÉı¿Þ·qh Äáæ"ápùHk[.²+]ß|±Û¯a0rû¹ø×º¡ùõÅvL0pÙlç~ Fèä~µ+UÙüÍrö¸ûaÝíþvg3x/÷#pá×û!wG~Ë»ÎfX¼
»/Ãݨb8áu &¸û.\
áz ?À5¡ÃJÝÆÝõüwÜ\_FmÄp ë1ðÌÞéï#ÊÝ]w@1îZîFðàòîóty.¸ü6¦Gqù-\'Ëoæ׿K²ý+ùô/ãº\_Ê/¿é!\~×Éòßòëq û]\_nç¶ìÆÅQÜ®ahÀaìFݧîFrjà>Î]H´Í¸¼(·ÄÓuåÎx^£+wùÍÛñ^§þJÌE¼2?ö3k XKó¤¤æáVéjËO÷ðþy R2¶Fbm$ê¶ùE^BWz¼Òã{§WPB#r'R Sêúx(uåj[öp{¤´\_TúüÔ&ó{éyVòÙTèL
TØ r ¤4Wv#ª÷Ô{ëSê÷ M&oSJÓ^¡ÚSíN©Þ+xJ¼%)%{
,O7+%k¯àð8¼Ç^¡½êHÕªsUB}USUkK×ÓíÏÎUr·çÇºí±¹# ¥£èN=ò=Hq ÏB\AjBèä,¤¤j¤z$ïxà rÇp¯ß£´ño§¿hg8ñÃÝÅyÕ¥rëö 1|öal?¬ô\Qê;÷)õÕÃý;zòîaÊ=æfsR R=R3HαéxDLçÏGî@jF:$°¦³éôiLéa.ësbDù×þ£Ljc©êÐôp@á;¾Yá% O#'ê/OÔ?;QÏD}2h zxHá.Y[ª?Zª¯.Õ§êñiV"z£psø'+<]vé¿wé¿qé¿réãÒ·¸ô×¹ø}ñ¸õ4ZáZÎa'\Ü'kúúéý¾T»G'e OTxçðõQC¹hNÁפÝTôJáî@)f¡îÀ8Ì»»1û¡;ð+Çiø .w'õ;JcàLøõ7ÃùW0ü"æó0ßGàÅüÉîÀ:Þÿ ¼ÿ×xý[Vóþå¾=0A©ÿÍð}»ºÓqÔǺÓWᨿ&éʨt§÷cí¯ºÓ7cöwúbÌÚ»½\À
Ý4G© æ$ÊûÎ&^Ê%©q<>y1æãn®èNçwó0¦ÛY2ò4xH2£Û£L2xGÄ"tñy$áõÄäênÏ:|tÔÛïø6pOü Ý»?óÃîC×Opuu;Î¥ÁÛëø³çãŤ LëvMª±áLz1G¹ûRèuIçxÚ£´îõ+.õ@ã1ÏLÇ£^¼îv¬K?ÍÅ ·á§asmúhGUàc¬7Ø,p09QìYê(êAÐsÈä¢dã3õ:ÒpDGåÆ'iQÁ 9]µ\Õ¨¦º^5J§ÊP9U ªxU´:JmTGªuêµZ-©5Uut0Ü'ûù+¤hÉÈ3Ià\PÊFÊ9ÛÄ@PSÜ;fVI+§AgT%©¼¡¬s¿2¨ Oéé¯ìT×Ü4£ àþZ¼ê¤÷¢zà 4P^µ1Gµ'@ÖÆmq<\_³q[m-TvM\§<"Ð;=e6bYYb+m\[þ\_XÃ0÷ÿü±ù¯ýØ:wTNÑùTBmg./j+;Çñxøm¡Må'h3ÏjgÕ´¥b ¯ÕåµW»7mÆn$À3Þ¸y7¥[Ò ÍÔ]QÞåvuz&ðNh>Ï+æ =+ ÀgÕð »ÑD¤<+&ònhC3\û0ò0(纼^ìîå]ºFx±CwÒ|èçfwHZUÆñB2ÀÏ}Rú ÷¡jìãÿÿùSöÿÐzf}pëlþV¢ÁS1©¡sëÊù¶Î¶F§³ëÖ\_WøgÏçù¬9xæwÞê)wvÍý\_góæYò.2»]³å9åݳäYYåµ=ûZÇTþb¬ÍWÇÓú\_ÖÊ6µ¯ò¿4Wòæ}|¬J>V%k¼O«rJTÖÌèR²Z k¼j#p?4ĹjË,ÆæÑÊæå²ÝwR xliýµ:OY§7efò&ܼ)¿wn²Ý5Êw 7±Úä)#~b«XP~õgÙ²eËq¶b
ùò6¥r9îZ×ÔÊαØÀï'D¬"¤x%R#áú>M \_Oha¸ñ?U{zø\_ï {H¹9äyr±|:;5ä!2#GH$Y¯¢:=èc@Äp ò%VÉ£ä=r3YJ>!}=W! SA1j, ¼Ü>½"Èò 9 a\ÉòxªðöðYb%)á?
ßÅ«ßO )ÜEÆcéSbB/½<áôBòÇðÿY¦F²ÖÀè]5B¾°%¼"ÇÈ[P¥Idø®æú'À gÃgñ4OZOîE»ÉYÉÆÄI|ä:2ÌÖ;É{&ÃeáG±v?ùúéKL
røÉRO¶ÇQo~t´P>Î!L¯Ã¿ÄwQ¶J²¬&m(ù>¼÷099Cè!Ra¹ÛÚÉ^¿J¨
³ðÛ+fJÑáðgá0I#3P=ä9ãdc¹Ùr!QX.æ®ÃÞJvóäuãCÔûÈwé#zm O²¨Ñ{I®'3IYIn'¿ÅU}¼@¾©{^Wân}¤ e¯ÆÞSñÙ[qºIÓÛ8K8q#a2LyÐ; ïÁ{T¢.<,¿dìUöP(áb|
Gôh%ÓÉ|\»PÛ|É+>ÈÀ½÷_¦£h9¦'è9ú!ÛÈÚ
+=¡¾Ð?B?·ZÙ8Ôà òjáßARa!,¿£äÛéQÉÌà X)»Õ²{ÙCìeöga©pHx\_ Ωf
^W7(r%tOF ýÌEkZò5cZJÖud ¹íåAÒo!¯·ÈßÈ?q¸Pæ8úmhuá~Lax^Wà#¸ÌucJ¡
´¡cé<ºÓCô<}~ÎÙlÃÛ0íf½ì=ÄiA¹Æ[ÅýÒ«ªÕxU£úµ+iµH(6tShGè¹ÐgáiáU(¿dLtJù(Úà^LO¡%öÈkäEÖ¯oZC:®Z CgcLë1Ýi:ÌÄ4 a>¦Vhõp7lmð°vÜöAèÅtNbz .À§ð%|MÑ)CköÒdEp¦cè8ZM§G05Ó¥t%®Ð~ÚCOзyng±ö({=ÏÞdß TH²0M'Ü-^Þ~b
8_Ü->/ÅIùÒÒBi§tDú\º¢T5è°®Q½© «½VÀyûÅ·YYÒ9X&F wÐ ¸/l¬YÜ7¢Æ$z[ÌîgçEæ÷a [À
céw¬ ¦Ñ3àf±Í%÷0¢ÑKô3!n \_@ð�§Ml Ætøßb»ÅÏÑ×}Óµp¾ÈîfwGÅÝpAÜM\_'N¡ÉÜÕè#xÓéºÌòÅÉÔûAñÔ÷hz/¤±7
Ýäæ¡ß|µQãO0QH¢·Ð"8;d�ZH3åÒ|5§èÊý.r^!Ó ò÷ã(oM8®W1½lÖb¹@Ö!eK~,|g7^¸4\Ê %%ÆOs²¡Eq|ü0¶<ÌW\_kV1Î%×À|ß iLVV©°$³´4 汴تªJÿJvi¯Î(忱wÙi\v³ëxÐþ²ý[ÓÃßõx¼ùJÁðç=(2 \_°·!ì[¨tVßÓ£ßУb±wG3S:ª×G|nØX«)6á÷ÖÛ Lãr-\ñçåºÁËCÓë«Ü-uÃ!ÎRÉ×ÌÖuíÔ©\h¡#3ýEæ¢PãKAFzql!ó@Ò»½¤¸8çÆÙ¡¿BÊêt¹xTNòý¡÷ÐDÈøðËÒl7Ãýè&9¤dêÑÒ>ÝAãAï¾ôÙ'¤^Ý ã ooúlÝjFi%ËÑó#|�uÄ6º0nlDÔØ Ôö L ¹c1¯cǦ¥Q¾8hGþ¡pdM&3åL¤Êú2ÕHÇ1##íe7Ì´ùÑÜ&¡ ëZ0#%%Æ�¦Ác�³ã� b8mÊô¯5¾m³JnKó½Ù>¯Wöô¥{^£Ïå³Zl»
I:o7/§0&N"?F¥bÉlI nÑ�±Z,Åk°T�%>d#2Èu)È¢ Q àYôq $Q\8öÁÓ®Ö:¨<_]?Ù"_£B«[¶Z¬¼\þkÉ>ȵü²]¥tø©Yl¾TË>¸r㦬¬ÞP]³ul^µÅãÈvçæ8ÙÁqÛ&=µ|æêêÍã²=îì̤ìl¸è:ñ¾yvúá º¦¼ãìÖ #£ò'}vRÞàçu¹Ñg¦ý}ϸ¼ºOWæ¡T?D[ÈAow®TöURc%H8Óç´JãÒRòÊVÇIB.TòÌÑ|ËéQù©r^rvåäFUC|CjCîës_÷¾S§ÓûÍÒèÜÒÔx¦IRªd÷èøèѣ㥼ø4)Ý?Ú\ªÑäì$t'ÚW]täXÓi«¬5;g±µ93 »Tø¤ßÁãd4üÁ&¾t<¾ÄQiµÆêp·l¶;|à[×P�§Fu59Ô 9®ýü¨¡£C15ðñþ¦Oqfbøsv\O$ì¥vAêÅéõ±A0ÈQX}Tö5ø:|}>ÁgÕõ¤ Ýòv sEb÷DTíðjL6Öµ\40¼mðì$O; #} ¨¤òÆÇ%Ä%Æ1Éì3xµ>Ýj§K05ÛÑX²è°ÎFS#2Æ4{²«GjBiiëÌùQ S4E 'ûF l8.&D'Þ·|fî5ÝûFãóën{¡¢¨¥pybfvRQjqyÁø|ºûs¨RºçÅÐzþä¹oCw=ÿ0ÑeÜ¢ #-o¬·LøÃã?nZ:2z½4!g5ýà¡4g¢Ûa:ÎñÎ1æËëU6mÕ]¾MFfçÌh±¤ªª ªIvÞ$ÌTßdi[¤^nZµKûÈGMµ#__±¾l{ÏúÏù½ð½5&»c·Ø 6ƪµiòíãìíNÍN©5Ö®³Kzf§¢d³rwÕ,à=_Öhäh]I4A'ëbl»ø¡Lí'z%°¨.1Û0&>®6×ÌfÁl濱K²³ÍÉNê´ïqéA£ëim¥íô =G/ÐS5µ;Ný?ÛsÈ¢ë&á¶2ò50X×NwKÄKéx»ÎhÎi(©k©õ÷sSVÏtjêrt}Ûk#Ú"\_àÁÉÒ:\±¡�
¹ :½Ua\_]RQ+·°p;T¥fs÷[÷ø¼ösíý[öÄ}ßÆÅÓÇÆúÑ e°óàº}+ZN¼ôæöyó~{,tq¤1¿à»|®g.T á¾n]»a]Q©¦"b¬¶Ò-Ó@jêÈT9¿!ÿ\~\_þ·\¥VÏȩ̂N$Ì|%óç÷¯\_º¿ðê&¨Sp\_OJiÏùlȲücL4ZÀ=ÇdV~BÆôõ©)§>&úwY[k@·+k+ÙÓ©]¶c}F[ÝÑA3°þX½ªç¤Èr>täͧOÁèã²ùíyp>¿º@Êê ð Y?F£=þ¥%uuøwLvVï¿®á±ùÄÍ/¬ùÐîÞ;;×(ONHW%v÷êÊ )¥·l¨×ºc¸wë \ÝJßS,z0 ½l²Òt£BÀ¦%tZ=tzAÒéqWÅËQ\u´J¥V3A%éÔÄ¡ý)ØE$¢
=²^I£$µ(èt)ûE seFc°aá[Ù%Êö2@U$Y{ä5{¨% ¬P�7?5òw%EYCqqpiÀTdR6 l°[l0Ñ¢7Ò²b<&ú$°½{§+ì %Á¥ûC¿¹mlýûèãõ¿ÑÞWUÄò'ªM\ضJ ÷ ÛT´Àu#»Ñ9ݵ(~¥¸~Ý»%þ v@Óáéó?m±XÕÑxò2®Ó
G®àtÅÆÅ3M±vOÓé2D$±1³:
ýØåB¯ì$&q0îXªÛ1üíز§ÁC=¸A¾ï5Ò¸øCdS6v©Ñî> ÃÆúëæu\;i÷_"FÔç(³IéQ]_ ¬_ KéRçzXO×;%D43cn!k MQ·&6Í b]-:YJà,I×øXÃƶ lÕäÐüZÐ<¶qúëZÝéMΪ´¢k÷ÖÛN V=Õ¼ûÞà¢Þ¶äSsãýFW~Wëog¨¨[ç \.´NI!Wä´·G®×¼çý+I Ö²ÕjËF«P§H"óØSìsÖ£+ØÑëÄ°Ìg@çl[Ü9é1è+ó5£´±$MN£rZCZGZ_fÒ;6³Ñì4geóvsYe¶§þì¢\A³ØGQ µñÎ~ÖåQ'QE
éñ^MTB|b<L^½Ï«ñ Bã+KI¾Fr6·¹ö§ CÄD2ÕO¸Î}S~TRaðí'#ø³ö?±(iû[_·æµ³} ß-|-jÜؼ Ó7ß»Ö7]ïÕWÿög÷u>ußS7÷@B/Í,ß4µá£²¬'wúÁ» ÜÏö.ÐçN!Ü×c-¿»D¦)#²¾Aß¡ÿ#¼Bß
wiU Z zYϨ( Gù+9ÑhƨÀô¢<®@ü$̤Í<övhAk×'éçÑÏd,Ô(¦Ý°Þy4Õ¯Àõ%~úþ!ÿtSäÚW³\.m7H°á ¹õ8SnùÏôP mmɾ!/A¬òýð¬ðb\f¿Ç\ö¶íÍN|$VË'k1´Ësä¥%7åv·iÛtm±mqë½m¾-ym{c÷{{tGcûN%¿ñ¢ö½EE"@ÒÓXM²Eoõê½pÜßyD"ÅPIaBJ=Ü|sÞB²Ðy¾
Éóóî5É+Ó×äµ íbªM½Þ´>ª=ºÝ²SØ¡~È´#ê1Ë>ßÓÉOç
^õÚ/u_D~üEnªJ¯I.&E02W,W]l² 0£UñÅ%1gf}B©q]Ï)ËFÄb#) ¨\ÐPÐQÐW xNcÃ={ "Û[·[Õþ5 ,ÊË Tú/ yèÜàG]hä¹þ¬D·É"¨c¼.ÑңÓIfnÈDîû-$Ë1dêöÎÏG6-|Õ|?lu(öQ^J^ûjþé%
ò®kóãu¯|òåÅ:ªÞïznñ´Us¼rîܶ©5Ûn[¼Þ7ÚÐ1mÃî¥U»Ý;ynKû««f-ÙõöµÕ n_Y??+ôÙؽ ë[=}|ÑBÄ ëq'@°dÐÉyw&¿'¾ã~/Y/¬×ªWkn×Ý¡_e¾Ý¹U}·9B£nO¥£Ôb²ÍlY¢W ñ$Ì&6&×àÉÈ$k²¼M^ôI"_H1ê¾£V+ÑÛ8Åá82F9£XTæ ¥Ê©m©LNmHíHíKRc »Ég"h=åþÌÀC38ú%ÃàdT^n0ªhh½ÒÔ&Ïè÷y|½«$xؤÆS± [ã½øB)g¿Å1ü#è|VH¦Åëû^OýMkûksï|iÿí~øÒãÏÒ¼¨²Ujï©-ϼ+ÞKW@Ò9;Þ½õàC?~Zµn!=±~ò¬îèØýÆíÓÒyÔQóvÖxd%e]ÌÎí,A?¯p»½?¨tè9éüí114æ4xñÜø !ô¸¤øÞÃobüpM8m¾6´v ¿ÅOÏ-ã9ë±3KÍeC¥2~2Ý:±uÊã!³å8÷9Ó¼ /¸©Ng1jt½6+:È&ËDÙq¾AãÀà¾0ÎXlp9\m.æz9ÎÄC}å¥4Jä=RfûuR Z~!0û¿¼ÛØaáoù?_r°.yHvùþ÷ë ÉäpðúnßÁÀJlU@ÔÿCÉ·À9QÝûsf&3I6É{I2$d7»É°I²,ì "Ú]uy,²"V,»+"H){ëc
ÚBµ¢à-Òë¶".hmõzm¯Eí½Ðþ¶ÔVa³÷,oûìîsæäñÙüßßã¦ÕQÈç³ô¿Rö°Ø0iÓná¼Y8T½·è å éÄ\´!l|.8Boaÿ,¼ØÌlçVpë¹å,àLmæ'·Cð62¹^a×Í!ì ®¦é¦Ã´Ð²ÉVXÖ[eOS§{08rÍí_Y«/,ßsïç^<¸¤seß¾ç漸iò<=½hÁ×·$úå_Ñu9ùô-K¿#ò|ûW/¿eÿÌ åáÕ÷toçàÒÑßp"¶ÕI̬Ñ<$µÆ¼ºEÜ}X|ؽ_|Úm¢ø¿ëµ¾Õþ¯3ýßf¶gkãdQä¦áòäJ8Èçö!µ?¶Ë8îsewKPf¦ìÛäxÈÃLÞÌ{äÈÍÒÐ.¨ºÚ]È2±�ZÛb2eUF2y¦¾ä³=ý´fôqNûÏì;ÝsúxûÉNcCÕW({c>ÅRÇë¡=å×-µ Ôùð ¹&h 8�8ǹjÓß×= Jtäóºk¥t6#¨;IÜ&áÜxö U|ü;o¯[uòÛºF]Z9õLå÷ÃöÝ»©ÁxCvîJágû7T~yt¸ò§Í}»¼ûv}zðì«°ëKüÅ ^2½$É<úqlÇÝvŹSºOú[%òJx¶ú^Q^üRdÛ2¼î¢Q ÅyUqhT3N§3~?mnX ( ·éæÜãÿµÐÐ=3Atqr{ÉLÀXö&HIhªªJnÒ8R'am´ÐMKlZ¶Ç×x@tq8öìÏ~Lr^åÊcEE¤{SQ1<|x¸Ô9Pñ礼°Æôô>«1ÖíxÆTØVb½Hæ$ýa¢ '½0ôBåæwÖÏ9+ÿ~jþMúxí&fùúX£¾±òÜ/ï=÷Ë/átAxqÈ:ìSØb¶¥ë·4gc%aN°×Ò˯Ö[,ü&aÕT×U%«%DÐNÕª1í+4g®åÐÈDne%sM8©LÄò×Uôd^ÃôýK4£W~uC§O$Õ0GÉ&Øðd&? £æx}IµÚvç²³/gßb~}}ßv=c³ör½õÇÜeæ±ÀÛ¬ ×êêaÊt Q·¦zN±8©ïªJJKd36¡°Õü&HAi=N!@Hg3C z£Þ¬ïgë7[,\;xøµ¢Rê¸T¯Ì¹N ºe]»Y^]WÞ\Ø!üº{ïcoS=fCÁt($Y ¢XÙ²¨eµÐ+÷W7l7wË»ö¦ü\û [r¡ÎÞ»Ù°ÇZØ:æNû¡æóEþsÜM¸(Rÿ³ú¸#ÀYæÔqÛ²·dg&n
Y5ZG(¥DZ£q«Ó½ÙDwÛ¹Þ=êfÝ,áÐw¢týÍ´ÛÚ:SPL©)Ð)S"oÈã©¥%k¼ëY>KYE¹ö.MG.Icªf%±,/ z&]nH3: DDÍ5ÆTÉÅgmMÀÀsNÖ´¥ ÚugS-AG9ml .,Ku@È0cbW=Àæ#Y$ ãêÖ°«gßÇlïZó\ed°oËfÝ=Er%r?ñÞtlCå׶ÎYºç¾W/]³bÇ£0ØÅuí¸æ×ð+îKéð®¥íZ\UÔo¬,ÜzöGÝûȯ+×ûÌyí¾5u¼¥>=Ã$DúðèÇûGôðèYÓME\ûEÊ¢¢?ÁômSÞÅ©¾ÄGOTaôñÐ G|5+¯¨Ã£\_íøjÃ1¦}H&sP.lC =O#Ë×q@ùÑGty½%óf3²Ù×;cÇÃ3Í=3¶ÍN¼DXìZ<®a-^T¦µ8Rð¨¢%°ãhÒ%U¤%0ɤ2yÒ$»ÝrMMá°"¸=qdÆáÑ8Åxo|GüpüXÜF13$ÍX4ãÐ &6ΦÇKÅEETÜ>}ñ¯åìåÒé~ÒD-õõSc@Û©kQþ1TZÍë¼nS1Àrð÷ʯýssP{ ÜVa;5 t15ÞØ4ÆȳÆU©àÈFúиgj&?¦a"b ÷+xû²ªaHS½ï¼ß®\s͸á§QÀi59[^0WhÔ kTt43S j]KZUAZ\V·ª ÔVUqi · jA""5AHI%/ ƽpLFh"Y( À§ T
áÑ¿=E^3BÀX¯6 ÓCëÔiÌ!í°S>9AÍ4Vú¾þ¦ºÊ2êÿ®c@«ÿt¨¤úgì)Yý]S\3úÆ
)�粒áBÏB/Zè ÜQ÷¸xHçÜ24tSG!¡J¨%_Kþ Èð^Ôé
Þaƶ/qX#ááÑOéçÆÓOz
©ãV«!&á!á {^8bª¡>0½L~J¿~£þcI}Û«û.äÞí¡ ¤§£ÇNìéko«VõÇ°£Rlu¡ºðDh·)öàDýXµs¤¦ÏsróEZu´FÝ×(婬üüò Ö+¤I%OéÉ¥)G®ßúÜ×{ÚÆÕ«[.êb¶£iÛ«#¦SÑnsxó9'Zàzp3Zë\e¬)ÝÚò¼í C¸@7;-E°ÍF×¢´Áܶ{O9Nýǯn;dȸæ¯Áæíîp¾Ñ,Øi0äêTkÔÑ�t·¶[;¬wo?E«=h7 ÌÌÎiÑNs?ÚoÛ}Ñëà×à0ü%zù|OÁ¿ØNÕýÉ!û þb±Ù(vÁà[-Í÷µ°ãѶè´}Àg §´ì²EêSJzbó1òHÜGiS̲ÅÁ+eµx^U2Z¼mÊ$¥cY
©×UU%%&[¸ÓáÅÁÈ�HbªË(z £ £8
f)E¶Õù?wö:w"ggyÞïÉm'f2éIõõ©¡´X,\qBÛ½¬Ó0òì�{9È £ féèt ÜíaôÙ)¢©/©ãØÙi¬9{ªÓÑíO{¶ÑÔBp,µ00¨ê'íZ41ÿîØi¥ó¿ÕL¡<¦Õg{´IR ϵöÐç³aµµÆáèë:Ï´æ MSò5Me{º{²¤ïÖ"ûí¶·Ü<Å´ÚnøcEÛn6ÃfÈ[8$1&ºHFquÒ9]N¢a²À^'_Á )w@˹)Zn" ù<¾k0mRò·®V©,Ë®²,¹Ë@ËáÑÃûåá-W[7éÕ ±Ó¨^~V+ @ðDpÁÃ>rZÑ;JÆô{³®ð#ø.ÈÏMúÃzg~ä17áù {óÙUëÔ]/ÆúUó3´~æÞÝxîg¾ýÜè{£àì2/³6º¡{Èì(mBÐA0<<«= £hñxÜ-×3¯ /ákívAî¸×íc ý®)¦ Íj
H n+CùQç¾ÊåIdJÍÙS.Ìi fMVIÛëi6¹½dÖÃùb«cõ¨Þã%oáÓ4#Åq4@Ñ? Á$³Ñ¸ YüÝ1¢sy¼×ÇiKa×'OÖúpØ\¦,æI+?è!æ9cuÝõ°ÝàR÷B0ß½|Ñ}«ûAø|îs¿ ?
î? H"¯nc뾩ä»UÐ讽Qw;")slÍqyb?3\&Ë=µI¡ÓþcT²GöØËømW§¿íóé³ùçRÏDª@
ªøTø{)£)ö2ÄÀ#Dg¿ª¤:°A8ibd"wÙYqÊ ìÅgtNpÖè±DcùÕôd¦4ÇÝïÞÅ?f{LbokøAxÏNÀø2«ÜF¾ FbÈQáßP{)1#(j#ߪ-3õd/I+^.õe?®æÇN4C
¥<)g« (Pn^¯ü^I6G "<¸_°¾&p!±²ÿ{42§å\ôÛ%¥Sè$àm[+VNTnûõóÝÿ¥ ߸qïóløzWT~Yyµ² ~¶Á©¯=9spWåÙÊS{ï p \ðø]$·KOY³4Õ@Ô{[KùÜÍòJeeøËÞÜ}a~ütòæåðÛIK0-å2©²^NO̹ùéëÓ½¹ýe�Cáúð¬ð¯ï(Ü® üiòÀÛÉ·ÒG2&-a3ÉNbJãPUx- OKH¬±!iOt$P"Áû2~¿ ¼à!)dÌPo ÍÌÕòÁ ÍÜîz(w(w8Çä!uºBH$NªmµõÎíM¹axË^¤¨>£júÖs9©Ó¤ªu©ÓTV´\CÚËîª%¹d} ,ëT} UÉ0ÒÁÔDá\ñÌ.D±ùILdãÑØDÌB@ÙjCY?ì'êýV\ÆÎH¦ýçë,<|$º¼8ò öÏ^ûgøÇý?ßüοëRº2²ìþKnï\t¢µTì'¨+åd5kÏvΰپ30ïþY¢Ü¬>#fý48¿2'WKá\x'Ü ¶ÿK|XYÑAfÀÜÏ£Ãf^ðg$DàÒÁ+ÁÁdãmª'ïAç;U9ÏÌÌeÌ ÊÚ$GÌDê@é¤AïZXÖ£¶vrÝ´¦bz8e¯³Õ!¬'zYT\_¼ F¬!¬<"R.|«y£MøS)ux² A§¿ &Üx =ïcmï äM#(4L¦k'ñ®y<§jÕ7)æÎ¥'ïßXy¹ò»¥»n!,ð¬{·î\_q÷7¾´ïÙ/-ÿHÜýh]»vïµSCåhÀ{\7V^ÿ¤rûÁW®ì®<½gÃï¶??:°f¬º² k2÷$e"¶:ÞÁ8tßz)ñR33ù½&$«ÜÒ$c
V=¥Ï�óà ´"¹®E7©7ÅVÅWëáì¦ÇáãúÓ©gF>Kìvxwòöô¶äNøz4ùDÓóMG?469ÜÀCÈÁZ6®5×j,M^·5(>Uµ8Ð3 Àq¸Gà${0Q£LÆô"C(øú´À ÿ./ñü"ÙLÛÃ2.Ãobs& #ÑéÄa¤à¦ óª Ó:J@{BC!mÔÍÞÃ-LKQ -P:T³
¸ßG5ÛG7}T³}ÛKÐðè3g©ä²ôëòUÎ×´ºÈjõç³$J ¤ctCr-ÊÒs-Õ£Öá¦qѪ7%ò8.\¼±�I#Ö\L²pÜß_ÑQÿªÓ)8:µÇ[Î8ÉK&^Ú' IÄ.V=c79'«iªüÿÉ$ð$nÍçú«ynYeK¥T9¢R8uY Þáï¼¾éáÇ¡¼hã³½ÞnÜòZ»Ö¡=ÀݯìUx#Q?éãVb=¼�DAFLfp¶»Eüvà rÌ�eÌ¥vÎJMÚ9k<àW³Qbä \EFwDÙèAþÑL;±~jÿüøÝ÷ÆôT ¨§{Ábk/7ØcÀÙw¥¶ZApö¾Æ:N¡IÃCéêq}ÐSˤªe· ìMÄð û1e÷¨×u=y~äÏ^)^ʧ$î²Ê]ÉÖñgNAq¶ÎéY¾�N&TµãÄTÍÁÛ� ù¢A»Út6»üábÆÒj¹Ì²FdõnN4§§%¦¥w¦ùút9:öµ¶ôóé¿¥,mÎjªXU o cI&phýÒ3kÑþø¡^§] ÖHM²Z³®,¤Evy½Ä÷P?d¡¹bôUîô?½¸½$°×Øaì6¬¡Æ(3c1ÊÌXÜí^ï+<ÐC}ÇIóDÉcþôùø¯g,oCÎðÐ0Ûs>CÏóÔhκbÍ㬺)-cs³È"êi=é5ɪ«ov&éM c×IT«àöþÀ¬ ¨,ü»ìt:
}ÍgÒýTÿjy+tf}W|í¿±i¤Å§Ø F.Û´ì_=IWOUûFÞ~í7ßÙöÕî¿ ÷ºÏéz)Ù?òdÇký®ÜwéëcXÜ8û!Ñ.ä~Ê&ZTT¶=åQÉ1Ìüöi§ü¼ RûjF>ó¤y²Î-iÐ/ØËù!Erµ-²Pª¶E6æélÞKÿì>£Òò3¡ÝÚ'<÷Xp(ô,·ßrç¾Ï}Ïòÿ}ß÷üÜüfq³{³Æ]ï[XÉ®± hÜ|ÿÜ@§våz»ï®¶}ÞÙíãLt1s¹«,\L+²|ÓÁL'§[êùñeüF¡-Òk-±N-fóü ~ÆÏ;ÈGTØóê$9íöiä¥^"©Új´éTèÑ)à'«¨¢.?o <Ñ£Îb!\òÈñhUÄ0 Þb=ß~Ó¿ÙÊÏúO>Ó×éÛí;åãb¾E¾^ßõ £÷Ç´-é¦ÄÆ£'xºçÝ ×2ôñxé7PVäÿ¡G6ýÄä[m²»,î2K²RY
ÇäI@bÒ%ò¥!Ø;@X;ÁW~x^úoƾ>ª^xÎÌîÌìììîì{ö=ûÞÍd_Ín²¤ÝI&%MÛ@6¡F¨<l¤Ø(ZðÒ(Öx¥6Àõµ<|D{½ÊU¿[©~\D1×êE~(MúÿMüÝï÷ûÒî3gÎÌîù¿_]N' iÉwÉ:ºéj"оÁ(héï~ÌðéË]¸14åk>xî?ûGr¡ Dó¿ænÇXc:ì7¡j6å²c覦bÀvÐf¬·V'x}&Ö¼¡ÈÖ8UvÖDü � ò¦r�bqÄöî´
~Ó¨aÄô bU¾Ù3Î?hʤKlÍ_.ö±=Ü&óÆÀ;ð£8)®cwsׯõ_øPË^Ã^v/·W¸Ù|xÿæÀíÁ û ï~²ðÉÁÒg¸Í÷9ïô>ðÙÌý
Ïå7=n~Üÿhà±àã¡oOsþÙKÅ¿ò5 ýU鿶ðµ¥&C5p}xOäÃ9øð×M¦ÈÆ̦a4°£piäùËÍ£,f=
¦6RjæwNrt´¦ Ál×ïlÀÁsfdæki=ûNø/¼p!( 5AÞdXî yÅàô»ÎL!È8D|t8H×JÕ@mîüÄS³ Ìߣ¹<§fs,güÁØ$Ä!Äó1°Ô %ãàH°X»%§#É%¢ÍÀó©ãËìÃ%üÌidJFS¹b¹X.Í¥]¥ñÒÙ9S:[Koð¿5]f<í7?K+ýM3k xJdÄo´wÌÑ|JG4HôI¯ÉÒÒ[DIQ^¿ 4¬»+ó.vøÛWáÿWo9ÉÚÉãÔI²þ©ã ¨+ñXêaØ(E¼Èsj¸jÆF;Ö@ÇFêì ¤3 j3qÔ\lài¼Ý^Yv©Ëwe°¼X¾!'º6t ·åJµQ°çôùYZJTË9d@tsÈZ18UßùîsÌUç¾d¸úcÞT2,Æ[èS;פϡl˾¥ýæGÞ o%XmÃúô«ëLs£Ø À8=˶·eÜRÊ¥ò¯J§hÏòTÿ}3Ò¹ .ÝbG)Õn¦þ!jtD Ãj«$WMêÒTéÀ8÷R7ÛoM¨·æ=="=},öXüÜc
çÏ%K=[=^Yz!ðòrm¾ëg)5í : ) 5/ÖJEGQ鶦j
Õô.¥«ØuªËðb}$÷Ñ~õÁУQÆ÷Å=ëê]üÝiÖáÊ£DþÑ£ç ü]ZÖÊÓv7p+~ÖÏ䪩Gõ
Ü´)¯P.ª¤]9êÊ9ê¬ÄX¿¯Îe0ÖÛ5#2>ð¥bpÕB[ IR å\å0åt/Rp)J!µS²A±Zµ ±h~eüµuTW)ÈÏQP½£ tÍ£3è,Ðý7ÍÖ£lSv+²Ñ±9ú»Çµ®Vë1¬ a¶:Ȥ Á$k¥þ¿ÐgõÖÿ 5k KÑÇðdsü&»GP6VXÇ]$ì뢻Õ=
ñ.+EX÷!Xf»:}]n¯i\²ü Í)Ö²¹ÇøFO-UôÀøüqO-ñ�g~~Øì£HEÊ�^H;¢gdØáû{¥Ó\_®ÚªR±ßvá å 2©LI´
Ú¶» WrãΦ¸ ç];¸içcÔcÃç©9ô¼0gqÙ$ }FaíFmb&Á
åÄ £¤@¦µð9n·SÓËA^Øí0kýgݹÕÉ8ík I÷ò>óèÆ·c3ê±Rùú)°´ÄQ/uJ¯7 éJtÇ¢>²Ê �?KI"q S;ÿκÓ(¤0Ñ Æt³ÀCçöÓÍÓ+QíÝ£ÌÕË[®;4¼ËN|]~0ix¥0z+ÚF¡ó\î58w_=CÑP²1T«F×ú6ʪôN½ä þ%ÑJÔàx÷´!¾Ü;6óKÎ/2'¨5ÔZæFÞR'±vu ðÃàòIÞlK:&)±E§z¸ÅSðþ¯I PÈ ÓBæ¶Ô8Òr9vRLø| 6deQ3áZ([;>$Îÿ©I¢hØ'#Êd,%Ã\g³,Ö±Ä6樸Í
ÕOÕTÀ;ÏÏVÕÒO �/ í1ïn¡ÛZCÔ¦ë CuÜNÝÞruù rxÚ¥útÝØ ÊØ@»V?+§P±KÐ&áó%MmÝí¬Ýa¾Æ¼ßt§p§Ù6ä¹ÃCGê»êô8ßB;óÙ\ù9,dY©&fÌ5ølÚ+icÇEF!Í^Ñ vÊàÎk[å]ò)ÈûdZþ&V°b§ÖIãeO@ù£\ß·9¦W³ÌùùÊ'©(ËøÆÃOnyÞdC%á5IN'g-y6IO'QRIÉçèn£ÜXÔÜsè-(ÔJf)Ü 7Í1Îrt¯ëþ°n"R¡6-A3d¬DG½=¡ ªJõE¦Rí5£ªcDé×b£%±öU:q£³ÚZ¥Y/ð4)1k e9ÃiX(ï0ÖT/+¨R6;RYcxÓÎv)¥�³iÃ.ÓÔIY>b:&§(1êU a ¯4DãÖZb^ôR¹ÔeÅ\óO ÝoÆúµ¹ÖVÀ[nMüò+£xÉz9ÀîuÇëö®ÎÆõzÜ$r+ܺ?©uA÷ýc¢uí®ÛÙüaǶz2ERÉÑ÷né:¯MÝWÚÑç·öl¯ÜyÝ÷vznÞ8xu,ÖÜ_SÎmÉFÖ«û¿wG³tVè¹uúÇkwaÌ?ÿîù×g÷R~¢cþa#°¸ltL+d |è" à¹ó-0\_e/e MNÐ ì.ͧ¹ÜT i2GGiXÓë§UÝNðô´:/½+ 3rÁÀçÁ9pnØhL%)Èce ~Î;³°;ÿuD1´z ï[ШÚ-R }=Î>Íý>b0¦º-cJê&f¯á.æáæ<×Ç¡vÞ¶t9îÙ+R¢è/)E3zÙ'ñMÑCQrB%Ë e2c1LãÍQ CYÀ)[ÄÝyË) gÁØ¢³bO~gS# "
õ¥±)ÝÏ0U·{k¤NAOaÌ\Ja ò ròÉf1Èã½!ª 9¤Bl@iíhX3?þq ð$#jt]¬%aKwx§-v»ç¢³EûúÇ|õo>¼Ý¦ÈÁ&+ræZn¨íüÒvWúígþô¯oÝ?ÝÞÎ<ýÅ~)>±Yú5-/?ô[æá½ú1÷¢¿ã h
Ðþ÷¤Àz67ÒR$áÿѦø³N=;ß?%Tb0Çä[«\$²�yÞO:H&ÛM¹2§çµì0ÒAçaq;ÄFÜ5ƽÆij::ÕSÊê?¦6Ô¶ËÃÁ]ñqy<¸W Þí¸×9cA\_§?
¾^^òý-ø{å-$³t¿cãÈ=Êtül³+èÏ¡üA
( ÀE ãÑé(ME¥¨B'¢3«ÑÎF-Ñ«C¯Úí%OÒÄ
@pÕ Ñª^¤9úöDZ,H$rq f¨£ÔN0Á�M=~£ÿ?=èGýÈ?DÍqE+±º¼bd»cÝÏÐÖX965¹49öÚ$+U/.NÒý£b¶ÐU¡CÌ}!Õ1nT«UT%å^ X1°"ÝíìqgÍ(I+Á§ÊªàSÆÅInºR¦ZÖ¬ðiÓ$ Ó6¦?ùó;¾øB³þW©¹#l7Çãëv¯½ô+¯ÜÒVFï{ú»}õçÈzhsªrïû¯üÊ×ßíÎßWßsþ5S¨£75+U Ù�YV&@Åë�FRBB° É#À¨!+ §f²ÒÃNÄ®¹!P®þ qËó8!]¤qo/sà³(ÅÌ0pjp"´ÐxEÌø2f¡a,ü ]Ð\ oi8ê|Ì!cY¶'TmAÕ$©¨@.N-Ô!Ã8ÏP
óóOõõ "ëÕ|y¼ðQÃGw¦ Oæ V.ÐTÁÓäVÃüú�ÇmäRhúíç ßh:Ràæ gUZQ(%ú,v3æ:ÊÊÕõÊÊaê°ò8÷ ÷b9Å;Ób#ììqÒ®8ÔÁ§ Ínr×"ͨ¹9#9 8ÜãiÏ&ñÐ7³,u3ù2´'ú\lw¾{\_óyqi Ê¡ÀÄÝNá%cò(úHIɤ?¥øt2ÅgJ5àMK\¨ÉجP cPcUpOp0'ÙÌ+)£Î½ÆxÅNÔ Ó/uO÷?pæïÞ²SH¿jAö-ê äÌËgólçU
;^¿óÞµï¾ðêÛüØ¡|÷ôWúöøä÷ÐÏ{&j[¯}ùûÿ!j®mcR.\ÄÜÞè ïÁüNDxÊJFn »¨Qhàçç fJ2vrJi
³9B]ñ<Î0wþrî|ÿ¡d6�4ÉLwÙkÌ óqÈ nRÌJ\5ùú7ê¹° �aS¸£Cqã34p1|ÕpÌÀÀWqxi)�g+Æë.^-{X-n°î¬ÖHø½,]8\|ìäغF¯q
Á¸ »ä1ß85îz
1ú Ó5¬EHáîþ2! )ámMùrõFWxvy/wú9ÄXÎÄF÷%ìAúSìñnièkô7å§?¥aû¥ôýßÓ1ÎóxuMßæ^¶å0§ã,wÒ ðÅxÒßjê¥ûL[#CôéJz>è<è{ÐùuÓ×
9þiÓQá%ú·ôñ-ÁÅÅèIháÞAÏQån7¸¨¢Ç ?Õé¨9v¹÷¹»_uÜîÀ¿A
ó§0¯Ï1ÝÍ£mtÔà¿/àp?ä=@ÍæA{<û<<ç-kfxºÈ_å×x¼þ(gùÇnuàiÖE+Ta(«dU¬ÌY+²/1á{iíw7$¬l^±e&±OJnMH©Svü°¬½Çem¨iD6MrÝTµ Y Ý#³,
hzr(Äv6E¬þ6s¼&j¹àÇ2hı¾Ð5ö}OÐ÷LdO³jnÉWó)öE!³H}>::êd½ çÓxÜÉhJø%Ú½ûÀåûs÷÷?ÿð:þÐKУFÉwUë¶;è~ä#WÝì:øk~ñ&~ðxûH¢ª}ËC[)¹Õø)J¥ùv's\_å4;9¢WT$YYÄ[³'q+äP9�AúzÀ%û¬É¨ |"öR-kCcêã,ÎKóõ
EiQgJó N^'IæY¡lä ª
²l_Ï" DD®&?ãç#Çû¿$òµÕk^aA§a¿~aAÍ hëîQt?bzq£o?³\_4>d@
ܾ(¼Êù0ØôeéËö£9Äb:µ«iJyëlÿL ͹9×"ñðáðóa:lO$½HÄÊo±)ë°³<'HÀçÐeO ïýö1Ô¤Î!I³d²Èa³K±ÙPõ©ññ2iÛÛõ¶^×ÛD´'-ÏXø.ëuÞzÊÊZ}ÍÏ2,Ã5t ܼöª;qóúØkSÄ×Ù¹4ÕY\_m¡¿êH¦]TÒJz2A\íJÑßE®a!i ÿñJ ¼§£Q ^ÛÖüÜ-nôH0¹nÛÒélf½ïر§'¯i/½-ýH\¯ÿÀ ,=2kN$2=WÒoì<ø/7õäªáJô§³tÍ+ë7BµË½ÌÿÆ2yu 5Ê|NûÃ3ø¹Ô vÒ{ön£©&6Ï^vb¨·mݹ§í¦ÔÄN¨Jy÷NùPåîuwl8´é®÷{ïÜ:gxÆ8ë¿WþÞ¦ù§vÙyvgÀ¯¸[¤«5²Óø ¾¿µ wìtRaìFÝsLI}ýBÁ}Ó>Ú÷-ú'k3Õ ,ç»]ÚÜlÛü/Ló»0ÞÖ¨ÍLQHE´§x¨x¸Èeà¯EP¢X©åé!4k³lÅïÏJ.Òù±Å éé "¡»¯¿|(¶f&2óSCÆ 33+6hÜù/Í#s²³¸SÛyßsãN85hË;èE½ÄÓ[R<Èæðüû¹óÖì$IÁÀC~£gþæ|°ê¥"3ÈÐ p|[é I¯ÊÀ× °FæºËw>nÆzðäAp¬UL-΢:õ¤Nê9ª^gRz$Á/J ¦°ô:°º¥)¡þÄóñdÌ%f}5Jc>1õÖ"8ÀHòÕ$Z±7 æ$b»uÓö J0ä1\Sj)K ÛÚÊ'RÛCAì©MÍ µÕj±¤sÔeêzäÞ Nï¢í;Bí<=ÐA ú´©¿ÒªÑÝPw¡3¶. RÛ²ÔowP¯cº£×ؼ÷M¤À) ?IÉ$¬MòÑä�[ÓÙ'H½<À z:7t(Ý[Dþ]¨¬çí÷º´¢
7IQTvõÞ¯ ]¾päñï¨V526õª'îékDÁÛóÁ/¾ûíýÌö ·«¬Ö»wOypàÊ -Ëïí»¿5ûÍòC¿F[²÷~ò¤fdM^¿d7NLw¥j.»£É2qÙäUÙ±¦UëMWEJøô½·~yÇú©[\_¾þÜÇ[FÅĺ}Ë3}ÊÓcm®>Ôࡪ+ v0BANÀ¾Lñd°ò�NÈ#l SÀ-#0+éDkDs2#îÅ·É·c¸óÍF2¹^a-¬KÀ¬Ö?IüÉàOCèJ
رTÚj6«P�]T�#¿®¡U:ù餪,ñä\Ýp¤ì�¬-þÆt\_.iO ý å - Kjؾäjá(á(^ÍYBmpçϳp�wÎc¹\µÁµ Ónô@è«Ðc¤ø(XÉ U©"TDZÜlKÚRÓÕªáhu¾zªÊ¨,¬W'H«"
³aûcÓì±\6î Ù°Ôfé9ƪåãt¾«®ô %ÝJUb±Ên0Íè¨lpXø±H%sT4äsã¹a:7£æÔÏÊrãmìÓ FÒ ®.°äµÆ{.ÌÙåy6H¾ x?öÜ°Ã0TF;½UÏ#odnê¼Äë±³D5ìZ½ÖMCcD÷|¢kËDÀiÚò:·¶F"=ÅÒûݵÞåöµqløÝ+rï]ºòÖ Ûߧ=¾üÏ;9ÒÔóÀ
òÖåàùH"áªÛµºöN¼á0¾©ÝðÌ+åcñc_Ó_Õ¯À¬wQS1ç
jN^úÁ ºYxuYä pÝÚzáE0®³MMM󺻻óö3Vä2Þ ::£KÊödç0zÿýÍÏ èóW-÷¢CÆCj³èdÎ3N¢9#ðüÞM: ð<³?×:|ê°-À°kïËäÜù9wþSLpJb³iW1£1/õ~$Q
Å°ÚI¯,4ÀRUW�S}ë.Ç¿èG¬©p§ëmz ?MTgÔG¨¬w¦UF#§TÆÏgÒJW:éñÁØa§ßÔä (YóÌ!«f(Jäð7Û±×hMúcÖúL^õzýøùêPKL°R¼MD"3 ²)Þ>sVa
XçÎÿk<Ö¤þk9I¹kxTL^(÷~úÉóª×u<;Ê.f ¼-NB¶Æ4jã
@zÅØj %¶H
rÐþBÞ4÷÷�³Ê
åiù;¸É¨éì)Eýûû£rÞszÃMjç²rîê7_[¯±p;;>Mêsj@¢ìe1ÝkcoÀê'ìßG¶z0¸]ÏPÔÃE @¶X6xÀt4UZÓùj¤ÒK%Cðÿ¼+¿"'äWäÍãÎßÏçkÕr k@YÀ èómX²!Úîp:%_Äö屨 ´),¤·¦ ]pkU Ìî$KA´Á²aÑ.h(Eq"C¶ $ måCÐdñf³ÀbÈ÷ùdÁ\§Eú¬NgDzB×{XF²?RC5,ÚPBÊKù£$ÝÖ8GÓù<¯ÖæÐÍOEù0AíÉ)È]ÀÒåi jmlrU¥'xÅK§ J©³áYWªXõ÷½ÔäXLt�ä8Rdj>S0VjÒ³Ø%6.¬¯D2Xó/c#¡tÇÌ534öMoLzOÚkhU(ìjWI°Ç9NÓ²Rñ¼lMÛEi ÉÖßõ|4
î½ì®7ß¼2VLøÖ-w§åßúòó½q·ÙfUüî&;÷üiC]!ZQè|Ç/ÿý¶hÁ$Èíô¶ kOVeHØÍÞè¥ÌúÃ}{(ÍZ,aÙ0¥q£O¯ÈW^,^ùÊ%²kØçÍ@f Äì/ü÷DÃWD(- ;ÿ«§wÜø-Lx¨wM9¡ó_r1N«k.ïtä$XðVi i'\ÄYnq;ÝfGü(]èuE:ºÐ#^Ï{ÿ:ñM91ã÷õ2^b,ë-C«µ×:ÊÈ{̲»uÐ4ï wÜ;áñÁ91æúc(fÓñG9þI+P(aÑÃ]å h\gÄ#YÑ(ó¬[tñ½ÞyQPÁ3±9å½²É dÜæ+÷-×ëy¿5"û3vd7Þûn×öjÈ!ö
>]z&\-2OP;kpï(Ñ6G Ök'Ö>
]ýï|ßwþçÿiq=X1ÄÞL1èëì
ã½{©^|Ì?|åbõêèðÈ051¼oÃÒ05ÁÔ^ÏðغÑ)j=³vÊSpãýÄ$ôÁ~ù)³ ÉñôW'ÿÉ�ÖÆÃ9½í³w$ì¢C'öX:ŧ>ßgß×A :ÆQcý¯8î±DÕÖÀy=2wØ:Ï£¿À-ÃîÍå+îõÝôU+¶Äü¡º°9àY L0uEåÖÕåë\_ÖìZ]³±±üjå²Òµª¹ Þ;7%Bo:½QLf7n¸{Õªµý÷6ï¼Bó#? Å]#ðÁñ¢QYnË5W¯J¢c]F8ßÛô¯àµð/äÛö°�ú?&+Ss¬B4Y'±»LNôDZJ(Wñp"ÃÔb6$úóðZkµ«ý/ÁÏ&Ó#ÐI|ºÉÅar£0¹E8C¢kb8gÚrÆ4ÑHåÃÖJP¤Û|E¨D'V$|ñ̺º8 ¦íì0ãmÝV5o¢ÄJ%ÅjÆóôdÙΫkJ~'ñï.R' è2ï/&82zrDSpDkp~¿ðC~òû+= LÎ aòf|QÐh« V&øL¦Òó·ÛmÚ_1²®ûge¤2V¯ì«°¤>^¬XVNV¨8¯ÐaÎfà-$Vvpse<ÄÍÀ[W;ØéZñî2ùÆx\BÀ°îãàAÜ87ɽÁ1¼3åp"ÍdÆ0kñDf_æ)C2&uøÌX|ËýíÁ7·¬ÐFWè@²UÛÝØ$Ëo�;½ý7L?ïày# WýÓgWݦù¶®¡æQÁá»î´9qGô.ë£í~8óUW ÜÛÜ~eT!17q ¼kÇO6à õ´6Ë¿¶\% ¤´OÑGQ?A²·zZ&¸s¦O'a0´]epßÁoáÁrÐ9¤sd4!lfÔá<¸ÇïãóT|qËÊxÄyí±à$b¾1ÄÀUØí&H EX¸ÐXÚÓ°KÝ>ø¸ÿ°ÿeø=~:üoqÿRËù¥þ+}÷Ãø=;AkÔè®01
¯ø¾§RF®àÚqÔ 9dÿ¯A¢ÈÀx?1ãÌ>æ caNã$\uÃ>\9\�ÆãÀlnÕÁôe«\²þ{dÅ3QfÅ¥ë×½ÐA[ôÜq<.^÷M ÒÝ^ºû}éýà¼ht=ÏùT
a·îLRz()è¤Kôj U úyT¨æqH Òhç³4 °h×òÛ d I\¼Îpm£¶YîîqÞã¾Û¿MÞ£ô|HrÕhóá9QCf4fzºj�ÏÖzÝ üÄw¾±ó{nÚñÚe[&?yí'6]Døòî?;ñµO=ý?Ý5Xÿò½¯6öØwÏ<4±·j®¤\_@²5ª£%koß-dq§ðGñìÑÜ^Ãsm{è]m
«ÑéqZÔÌD; EÝYµXI|D ¤iXd¹Í
{,÷¸ô R¬¥ ÐmGA÷¹³ÏcAì°L ,èG#rë!:Ò£c�^ýÖcMCg¥-Î5Æ[@0º©áúçd þÃRý aÖ´BºJÚãbvåá|}ÁªüUù[\·äo綻¶çïã¾f}ûïè\°®<Ús[c,%NgÜdV)»:<ȸJÅA\¶&K(w.M3E© qK(+n";»»¢>&-üF£H/¨i#¶:îiB<ÙØX?ôg,1±¼Xl.K;q3AE©bupzOÒìÔ+Ön hWæ«ì²µ¿HPAæÒzÙ7LÈaªmÀýóæ$XSaÅ<-Cjò¢½k¼zËãÿ²²îÔV55¥7åñIñ¬ÃÞùwm\tÉÕƺÎR®m}{ûµ·Ý÷æÌ£;}b¡ùþ5å¦ê´um¤¯í;ÿ²9Þ¿îþpËŲÏS,i®d�å0ÈÁ7[²¬&ªLú%°¹Ò/íÄ> Af¶øÄÄAbǾ;~a_0]gC²-bÄ×eKfÔm³:M¹A"Sï<'k Íñ«ÐËa0ePÕÈ ÄäÖäÔH2
N?V:ÕÎX=Û[#ª[]['¨£ØúìÜfé:õºØæì½ÒugdKlgî~õÓ¹/W¿ù|lö˹'ü\_W =;êÿjÁOr§så²Závýöô^Ï<\_ð/X/óÀÎ<èTËÊb$JÇÕ Ä\_+®e«Õ A4êÄbWQ¸Rcp4ä¿ÿdäñQßö½áûö$oq¾Ä,6³¹ÆV3-@8Ø3õY,î6˲H{@RiÚéþ¸S^ ¡lÇ~1xËÖ¾çλ\-ô$ öw/íj§ÄñúV¹¼²Ùíé {å«Xqÿ¿AïwkcÉþÊ?¤6ÖÇûêí ®¦|tãºî®K¶2}o[ó¼uM %fKðh¼þÖKG9cLAÏ·±Y¢#-Ñ+EÓ
-W~¾çm۵ѶEÅÚ $¢Ä16J<^r"hٯ஠HìkRS;St\mí4RV'°;üÛÿdY.ééÞ8¾]]»ßÉS<ºlA-%ÒEúÀë¯/ΧªK½k+V§ñðêúÇ°äeã¹SÔ[H^º¼¤ÊD^ʶÎ(HæJ!+
bPåRv|<ÛêGÄi7aî²r)1Ƹs,ÜÎÛXÈê%aÖªÜ×GD×T8¦«ê¶út£l QÑÀÐl,"Èî;ñæ éMs$î¬?.²T¶ËjÞFq¯báìÇYÕ³Ö%¸1rGènÄ-ü¡biÅr·Ê9rã"w·FÌi³Æ¸ÙÞ¤ééF]&«®Z+a2|^ÉSnwÑ°Õòi[MöÚ×'N°UH ±òxy¢lËSP3v#uùÇÓiýGñ·ïäßcÞ¿x?os×óüÇ ;ò{á^j/=áÃ'B{ {Ìl&мÝò¯v|/Î
h¿×òL0ÿÿð¨ö¹øç6wÎί̯)o(ß¹;¿ËùDü@ùWô{!{ë©In{¼XªáÊÊåÅDªPR5ôäðÊ~üfÛ;l"ÿ ¥l�ø¡ªP/àðúKøÁR¯¹!tc(Òï0Òö¶q§~\Ü'Ò¬JJUQrùÉ#~i¼J½�5Ð µgVµ;æ#ÎÑ,FÁÁÆhìÊgÏATŨSgHº^B4>X Y¥òÓ×á°µiÆFM±ÆÖ ÆPµµ&¸¨ñ3Ò¨&¹,Ö¨+ 8m °¦Ùl+vì{á5õY?>t}f£p+![g(p¤'m\_tìóíS÷÷
éøB|²' ï·!ë [)^J|ÿhÑ<ÛÅF³+)5>Ô !Ô(´M@®Jæñ
Zʯ٥»îÔð/ ÖH¡Ô&¬9nvLmã©å[Ü/ϺÍ{nônôîZ^sãk>0D&ÖhÉ>Çoðáv Ïq sÐ&»ÈöKô.üæ=L6Ô80ç rÛ¦©ùÄiÔ¾Xò®«]¡E7|ö/n»ü¶/àÅB_¾né×6V(<úñêpÙ%¹íôæ«»ee¡/)^týWv<TxÑC¹¤¶ô}ýµ+·ìNé0ï¹ßSÌK gÛ°áF:,L¦Ðmv±û<õª d6ZÊÓW÷àgaeÙ¸¼è÷2: d³'Of¦[cØOÛ«ðÎë'%ÉÞ7¯Ä³ÄjWlÏéq´AèÛä
+¼|D}¶-Y°$Qõá# i)ÿ<í>'L!ë�ê³'ãÒ iºÑÆ4 5x8Píµ pEÕøQ¾íû¶Jùb Ã=\\c\_ãØßàøwµÈ>9%Ó~¬¨4Ä;oð1Hû:[¥;) ZìÜhÿ¾wu7ø°áy¿¼Ïb)|0L
Ã&¼#8á¤õ ç¸ç¤ççg,ôÔ¶kÐJ#Ð8ÓÀÀÈ3x¥ìì)s&½u ºpÆWl¹HF{Iû.bSõÅÄØáɹòí·ËéØ"W>±¤¸.ûßzo/2ÌKÍÿ¹lö£2éë®/o¸º9æß´åöD¹.ª-Èn³Vb-étáù]úz1æ¦Íßî¼wéð|¨w9çþnUm=ýðì[d5Ôw&F v¡>ûèÄHu5e½¸ÒñÑ$£õRCªÛBr»Ð¦¡¢ÿï3�(fN®ÐP:Së>Èñ|G ]góà¯×cqÿÏå¶Päêß©hø>'rçÿPãÒOOHdUÁ»/ÖÉW)´B(Ù+xºÖWñ^5Îw1æNÈ¢©ý|Mèwãt ýêJn¿DX/UV¨¸/qðÿ]ýbp²ãIð÷5þ+ÊWÔ'ßç å#Ê ê±àñ·ä
åÔ$;Æl¬¹.³dÌò¢Ì22ËxÜ,].Rê;î[áVj½Wû{ö~×Þ¾ëzäZðËñØUëy·B÷ºËGöF< ¨E[pEP/ØeäyUÑdEé4¹3ªà9Tã¬a8dyÜÈlU±ÉS O( aR8,¼)°>
X2,¥Ç¸£Üë¨÷îàm&FÐ�Ú+º{øº+¸8b¯�þ8r¦à·Kp¢Ã|è,\==1¬X)ÃY°°¾Pgå÷0ñ|FÁåVyfvEÁÚu÷ßÀ¼gÒymi[DôMƽçÍï¨#åõ«#¨ä6 Ëû9²R K<5NCf ÚZèh®.js¨x<&¹OÜ$÷!<|ËRß[o8[GÌõxã¡æ±Ló¨?uuÓëI-ÞÙ´P¾°mºÎ¸"ËÎþf«%ç°o|îûê-yúD«·$cÊã ðIcÒzÔ"Z°×ë¥ÉÈÕ¦ãjõhô\BÐ!R½L ¹<ÒäæÛó0¶éP·mKôͼ{>\_Å
V\Ð5ê J>Ìã§|ÆMh@Bõ?
LÒøñûúûé?êLÛñ Ïz\ä¼WÑX±ÙX +J4 'óÉZrmàñÀãòãIΦ÷&zSkÀj8l]Á]XNg°NH®Oë¤ÈL¿(=OÖIGõ£éo\_Õ\_M¿£¿>Y±Z|L×)>mÉT¥Å®öRëò¥=¶½ÒòeOüýäD1°ßؤü(¼KºËÅ >~M] õ )àHZ<Ñ@&¢àQ%Aný®C88unaÈzBã¬oMdÒÞL&¤AOur¼ãxd(¾ {A'²e%+ÈYGýO@¿Ã1xu¢<}( E~%'²MÐ((IÈ×�
BG§ N\·�pðë6Pc´M;+Þ êç2q¼jÆgK# |L/\o(ï"÷ÙD uïàMÔ¡~ôÖjý@øP·BiCÉ ~ß\q/ nÎ!sJÐ@N¤?À2ÑØ.M?f%Õ À¹2¥123Ç3'3ÖÌXaÎjÁ3Ê:3{ 9=[Z}RÑô¶|JE¦ÞÚ©ÊU
M¬6)Y1ý¬9þMÚêäofÄ&áõÁ12Hò8=Ä a°c,y°ã|0s
<¨é¸ðWÏøÎó£E F¨�MµÑV$×0NzÄ'Ð0<ýròÀçG¼Üɼ©]i¾ùEóßõæO}H0P4?û{øôîó¤¸×7ûøQUóD(]wl:ûjÅìZQvNO�è\_" ÓÇ0-Ñä$S�èV%ÌXðHT8BÄe\RÉäýmõ}ûo·{©�÷:ö:÷ºv'w÷¼m{;ðÔOʼXL º-aß\l³½×m õÅõU¦XgëRÝÕ¬§k=ý+lk¤5®eÉÕéU=FÿÊúHÿ6ëNÛNi§k§gàÒ¤ëqùX2dEItù¨uEó!(õ RÿZ~}u¤¿EL voï}øÜY¥b²GPÄß!R kÅbÐJ¥!ÖhÇÍ=þND}3à÷§zz\Ín/Ë8Áì©ô+º{¯¿ä® 2Kýöðe$#%}s|gïø¢µráLª<ö ¬°¬UW¬ÖDE÷V\ºÝJuíÞrÙ~y·Ê)]±õ²@Û{¬VyZ+(~Rÿ h(w¹ðø\,hø]¦�
H$,ØÉùüf?ôõ)è<¤)PÁ·°KC9¨ü\ù@að<:+Ǩ(+¼éÙJ1
ôÃ!PåcÔK ú©áC±{LúÍ3ÅÚÈmAþÙ¹öðWðKÄ�Å9Ü5ê¸ÿ ݵ%ù´tªú)òèq®µ:"Ò½§QÍÊIÎÝNiÇô4.¦¹i+\8tÔ̱FèOÚFêeÆ,þñ_ สÿ géña·¹ê#(´Cè
LxÎ:g9º³kýØ8Ae&-»}pX¬éM�Ì hÅ][4S·9ÐrÇ+Þ\è _óú³áY·Y¸L#"è¨Iè¸Ð0Ü5Ik.´å Nõß,ÜxpôáhÈÇW«r¾ZºÓ[Ë ÍÅù1ÝÌ\_Ë.´ùjÝxCÀ6÷<æþûË ¼à ¢Ú¡3q\Ë¢±zæ0ÈÒIYç1ööbË'dbqpÕò$¬v%ºÖî8uùòZs¤ x][R(4ßJëcå% ª än©ãæ¯W}a¤¨ä7§¶wÑ×4¦§¯rÉ)\½á»Î½cdËè3HWuÏͦ"{5¥ÁÝ) #D~½XU¹H¦HÕnRíj»¹Üiô¯^:Ñh+±îð9öº¨{ºa7p#
¿èõè)ÏA?mL#Oh 3hPZuùºAðÜrî "Õ/H-PØS<^#èÌýcòôý«ÿÀÞo¡xus §ò9¯äîÌõÁª»¼È}3³°I¹Q½>xsþnn»°]¹K½#xw~°GÙöó_P?;NöüGVJ.ÏfHlwüùîÁä4EU;³Ïå©Ë¢K²Ï\ ²=¸xËè'4NÔÚT)^ =ªû!¸Wï àéÓqáw-ìÀ©-7ð4¿ÃK#¹·E¼ÄaR£´½ò°¯ç©¼RîyÉ0¬±uøTcË©Ù3 Ì0Û ÏʵRܶó0póÆrÌè#Püÿ ×p V ¹ÿÊ8'Ö¹eÞòìÛöB×ò ±wO¸¬\Gfõ´Ì+ÍOU\²uog¬"%GÄ"ÊHSáÔÒf7üs&íæmd¾Ë1gýìÇîI>[öF'©CÑbÜ.ÙÍ ômHz}ðI£äæd&Î')Æ:À6GWu¬G|t 8=5Ì¥»ÌIÑÚÊ4¤~Z¤¬} ?Îv¥ØN»eïáq³H�ê³HIÝù¼á5äì%iÐÁ®ÑÍ²Ï ãÅÍxi¡m#B»3àÀ°ítØ-ÒÄcÔ"à µÈÈÓ°8¾VqÄ;cÜA;ÔR X ö¢(H)þÀ?ÈÅg¶ Á¬8\_ÑÆ)éÎIÖÚã]»ÕÀÈÛ½cZne©oDõ9䶽ïÏ®0¿f,{¨ÅëäÛåÛm
{»äû )>^x¢h\_\XQUèZq¼T¥»åªÚ§Y9¦¤TMÈI5Wä´é¡àäëX8=4+ñØôN\¹é¡$y«sz¨J Lõ¢QïpM9>ÝÁMÙ?÷\ÈÊNUË+j¾ä¡ûõÔ~=¬ÅÊþjµoo¯Ý\·Ûó º\_ô,ôPa<Ú î\Taÿ-EX|ZQܨ§/Èç´%HÁå²"¦& V^ØùSTÖpsô~Ê¿?PTîæ²·ëå�ÁøC±u÷û)³SÈIA^2f3D©Í¥EcÅð,±Ã2@<&ùoÛ좳'ç´.1Cyç»Ç|¶ÉÉAà ÄÍÔ#æLÒ ÕqÊI'åk\_DU\0nFú.ÀÜWÊÔýÚëW¥ÆWo;¡ ò0Ч»+[d<ÍZó5äÎ'¯÷¥Ü®t»ú{Ëξxy ÚÜ¥ÔúÅߦtèÍ/kæ×fDú³ðôìǨ=ÍÈîÙ¥ð«Ó>=~ñQöc5CD èà[~ 0g§ï£¢øC±\_üsµ5dg¦ÆqUi¾Ó±ýXóq8Ñ\5fÚÊÖ Iñ ¤ªËS«vc§º±Ó- úÚ8Éÿ§øûEco³ò].\_¾ÞÏ~ØhNcx¦ÔÀÐïÜüEr±yuÊR¨× Ãh¾V¬ùQgkF¡u¬UVÉçÌjæGÈô ýÜÝð.2ù#xÆn³NÑ©#mðÎ $W!cað9Ö2ô®aÃTdÖ ×i ém NÑCG¨ý¬~ÐI§�GýÒ \I¨w�6 Xà# x®¡ÑXàcðÊÛqË
C»}HY4a¡-/PþÞP(Ø#¢U²vZéÍÖÖ½VÚzµmÝÏL's¨ÌÌ6È&$®g@iFUfgUõ,ĪB2q¶Üp¦UÁ´à[tX>ùbÌÊ޺ή¤ô:ÓóÑkW/¯Üؼ«yGó~vGøûì¨ÆÏó§k ê£×õô&pµ¾ÜCßÍÜGÝÇ"E4ü@ À§YÆ˲ C3,zx )04û4²t±�Õ u îD=úg$¹ôºéQ'mS Q ·mÐÀTX§ï4BÝG[ËÙ[ÿüKú_ÏÖ�<÷[ø2ã¢H¾Ã&g¯�ʶ {q±ô!ãa¯ öW ëfàËÿ¦ë8÷ü/N/¯DÏf1}gn~P�Dæp2r@FÓË6:\í0Ð.¡ æXLÌ5Äÿchø®è:'´ãþ^oi Þq({¶àÑ4Ýl£e¼Õ±ö;33ÏÔh¡ }s¬åyýEÿëá²u«ÞáÈ»3 Wö.¾í~êª ÍnËû3 ûnÙÅÞ)n\w8Å
ùÎ¥w¬Ýøt2Ùõ¢Ó)-Èu-ߺvÓÓxôk=Hi$5\_è;ã´H¨,ºÖ=Géà¯Ñ³RÁF uBÒÀAÀH,ç5HJ8¡-Ì>Ïöè0 bàp¡b|øüs>3cÚ®À¤¨ÁêçéÄ$ð7êA»MÜAWzQ4W[r˺ì¹ETEE+?P([/¿óZ·õ%ÔÖ?¶®1¼äx/8ì0¼¤l¿&ð/=$ýV¶G%bj·~$ÜFO¥zGØj\«©¿½IWmv§Í&fûß:º¾¡´°¬DÍ&v[Haó©sߧboF;8ez}3u7õýGÊBOQ×¢ þ&RKúçgá{Ì T?êHÏÌ·[ÊRO¥goþóuìñ·WЯ¸ý 0à¹6 >±øñ|®Xô Q1 eä8 M0>d¶Pd �;uî=#Af¦ Ë �9#@VÈäu £-ÇèÿYCTù AléúÐà YçñÿUGµv¼4ÞC,ÕKV,=Y}lèùê+ÕSUáæÞ±¡ñ¡\_WÝû§êG½Ö!¨qb&"¤:ôç"Ú®6áSñÀsè®xF¯öè.±Ú·Mì¢zøF צ t)¤3i¤¸£/tXIL0ìä¨ãCo QCF ܬïÕ)ý³)eÙW=õÕ6×9^ÓNMr²scaë|ffËÄín]3-ÅÛ ÿ@=\_T\_X§,¹ä@ÞÐ@=»@#+û²]ÁaÈî-·Ìó¹°ã$®7Ù²YÖ\þV- u¸îå+fßu!¯Àîy×ÎòZ¶iO,[ôÌ37¼²ã¢µÊoð¨ôÃÙþÍuÈEn¿hxD×ìÿ¸M]¡Dbø>êòUGNÜYMuã|¥åÏáñ%d¬d, þc_åFu¦[·ª$vÖJ©TUÚ¥Ò®º[Rïív»»1MÛØ÷¥íl1Ø,1ÄI ;$gC&cÈ2ô&Ïv6N973lw&/ç%Vñî-©í¶2쪺µ´êþÿ÷¯w+Oü±Ò^ô$³ N´ ¸YÚt
¨.G#Kx\À[Äçïöø?¼ùT¼ù|PÌpõ\ÊWñÓæÎÚkàÁÛ\ÞÌKx3!qþÞº¢®AÛÕn\_E£~~¯üoøãèº ëê07àß é+N®'ñíu'CÊq5ö&Ähöý|æ+òiùMùײöå&ÏÇ,Ç<$ãõG1@Zy\abN%¢;eo¹¿6Öt0}gH«f÷ (Ư5«+èºÓE¸þ¶z³¹å)oè¶,þu/ÿºÒ/ÙÙ¿TwM«æCÎ߸£a0 n·;Ñ·ª2øãÛUÉìöÐÐü÷SÕV?4®ÕÌX³û1ÞÑ+XÌUOfGoéguYÌ BKôNuD³Kݺ.·NCQ Ð^CéOwT3¾¬GÇò±Íÿ÷ò±Íç ]0¹¤ µuF8ã!¨³¨xóù~æYæ?"ÄÌ0xn631$ÓÉ
éÎätgrQ¼¾Å1Ï+jpr®a³ÎéÔQÏf ªÃ½hu¸óC¦ÿ0áhÌ3nzÞÓíÆ,vºa4ÌwéÌ,hb4|Èd,·ú.
xöOM½©í&GðGa|îÅ®oýµõeü¸\_w¯õ^qÈz³ãNìtÁë=ønÝv+ñüQëSÐáEú.ýÍÁw¤3Í£õ¤e3a> ö¿Í¯¹¾0øAäaêµÿ«ï"X¡|aFJÁ]ZuT§-È=SñpúÄB9KZòôM+ÎÉ÷1Óׯi]×?´4ªÁ¯ÉMàu5:K7Gá :½×dþ=ÿo?íºT0êº8¿ÉËZDÙåÎ[uC-õ°»6ô-\mbØÍùäó0<®cÀ±ÀÇx\¢´AãÃÆ KóÏ/£Ù¤²¾ÁpÌn§qÌëÅ(«èõaî®G£ñD°ÏíÛ� ÑGQdÚÝ;º{Ow¿t]wYùòýå×uîKËÒeé÷·²ÅböhóûB®ÑRÉÃÏ× ¹\_¶m:ß}áÞö3Ùb>/\±ßE hû{t÷'Ññ(ÜdaIy»PÈý Àcðýµ[á¼Kíqxô©l¶º7):xð+ôØ?³Å <HBóÝ= 6Õ4û¹fÓaF Pûm3h F´¢èrñÐV LS¨}ÔIÉÐÔ#ݾ£é.$±M[LcÕï|Ò
<»ë¿\¾3¨ù©ÛÎ+ õ;zSA5øI×wt±¥pÁ¬§®J=¦ËªufXöÉ!/tjçÇ÷1ÄóÁæÔÚxSoòm¬ñïòãßeÁw={¿ßØÁ}&pØSàãà&h¶ +4
¸·ÓtÄNì4ñÖ8ÛØkUçc¨j]öy´î¸¼T¬ã°ÑÒÇuXÎàà?·|ñzÏÞ'æ·>~ ¾ðçÖì[íÜ7.¤Ví&Þ9ñêíûȵ3'¾µïöotèºÊîOoä¦7|zÍúOÌWa½¯ÝCüc±Ü>«Ó´§Á¦!äÌ:q§°ìibñ5£¯ø¾sûÝ7lï¾Oa5÷ _ ù[(û _L(á{iëBÅiÆjÄæÈXµýÍjïEÔ_Æ·äj^D¶l!µSØæ&Ï@#§hüI¥¡õñet:oÈEa×ý QÀx½¦ÀHIHÐóÈlmçÏ´òö¹<¢mAV{u1/EMÊ-Ô©ãKu¢R'!Qk ]zë\ø±;3áõí|ýõ;¤T7ñ½3òì¡dPçWniÌìã«·¼r×Ér ?oÙºALN4+¾ÔT¯°fH÷Ô
Pyv{fðð=ßAB[Wö%X?æþàgÊq/QÙD|1f¬ç%^gÐhtÎÌ:#0@þöÈã(\Ä?·>øò�<³ö_Ïiæq-Ê�A> )!NA4°Á¦ ??¿ø§É§ ¿Áh4¯B�ÌD¼÷Ãh¡QP)~¶GÊÒù,ÊÛº@ð"º¨8¥$v/\õèhÁ:DüÄü/ɧ!H6PCo·¾Ë-yL±ï¡òUÊÄS$ ßÖa j]ÛïxK½Ã@wÌ@ìîeæü3Τ ¾ªÖÇ¥§àW©O¸ÿÊ!xýeõºG½¾^? ½Y-ƨåKèÚó ¸U½îû ߯L½Îª×¯×RË~µÜk'1¤EÐ[Þ®9Ma;°µm«Áê Û'Nà AÈ%xsÔz ËD¢¶+Ü Ö}ýµÈö3´9/¢Þ¨öpÞ01=' cÛ¦§ v þD£}&/ïc¯6Ú ´òOß°µÎ 'ÞA¾¡vª( h¡dÆr£:GP]êpeÍ_RBy¤:©Ê«Nh²óTTE«%oM_Ó]^+äg¾@¶ìÓ<ý´'3o·SýZÙazS ìØï¾ÛÄ÷¬¹uu~ýhÜQ]¹QyÑ óbÊ_ªÃàs8e6YCL²_´\F#x¤pj¼{jEû3c«D$£+ð]íÏôMÄdlà»VÈ×íÝ>qG+!2Ð þ»' {2ÓóÕûÖóÅdÔ¥Þho×<9s{²9|óèÇwÝÛ¶6S£ñ¸Ü_bÅ£'MÀ¤ÑDyQAV£ò¨{Ç7~ýøUh=±qíEf¯}¨JA½YÄëãPí0j==ÑÌMMmpÌzãñÙz}xrÄ.³ÅXÐaÃÙa0;<;¼ýàÆÛ7#»¶o\¿²Þ;
XÜÐ?iH×_ÃàR©2ØöÙì.»¨Å:Ìm,8êç+X$éÈ!µmáLa1ÓË('#½ÞZ:ÐÏTk%ñÌù7}íƪ= Øí!7y$]Ý}-þôÚysûéÄd5¼$:+îùÖm½beÀ/¤õÞ!±>]~éEeTó[H×°a7g(lR¤Îdh½¹¾sÌoܲô˳þì~@ï§÷ó0ö»ED§±ìÕ=Ú=ç¶]k=9t¨»]ǤÆt $üó>=òhUVäEªý B'Ú.pÀV¥.#AùU\áÃt'dEáÏ1\¦gþ"}wûiÐ:ÓS+»¥MÇ óS/=Bù
+U%1ñê¿Ñ¨·"îJ½Cï#?1 êÍ3Í0$ú¼LÌBm¤®§
ÿNUHP5 g) §À÷(p e ÄãÅÅIRõbQWÔá:xÕ謦©ÁE Ðhb¨ä±,ÅÐDN²t+Ak©äÔ/@I©WV?ù¸ õd7Ø[4<ð4DÈ
óhgô9йL£ë¹lùÿ �8b -ÅH4Ê«+"ÑüjtV@T°öC°öil]3j±Ð6[ øädH((Y0 ¦ÓKÇÆé¤$FÉ@ã:ÉÅNU/VøÌRÏæÕêº^¬jeyU/É÷ðÄPt&FËÊbÕ¨EÜ®<¨õ÷¾§Òi¹fáÙ´K¹úKÊ#É>ɦq¡OÝé!hcCYÐöÏ.ö z2 �£0O ¡E¶É�0"TÛ´³ùýv-vUÔ\_ôØ% ;TBGÚåsjKösj°¥ {ÛrQ¦;Å,Ë¥:áW<é\õ5'v|bcZj¬¾v}L¬6µ<ËÊcíëî}ûÑ«&þá±ò¾±)»Û¬Å%ü«¥mÙ{ÿzvõÝ;VÈní �9ÒCêø¾ÜÿËV®>ñÖßylÓ¥7jq«Ñ«í>PÚ%éÅ6{µn·Ð;"¤úóÑTIìåÄ&Y¹ 'sÓÜIîqNÛè|ÓpTÊëµ®±ðèqF9·¼±Nî�ùd07,"<¼»$ íî"¢Ôáù¹?Þu%2\øeHËõÁÿ.ÕCS¸?¯ØRUºP£ªx+ NÇ5zÀ/ËÍw½ÌøÁ-´ÿWi$N¢Öß[Áv|PñÉq\BPwf$ó+D(Ü,Bª°X¬ib.Q5ͦù\|Å9ìbþÙËwMø+ßTXTjrO@O²Õx£TeI} G]z õÉÙ³PûµX=áB¿íÿàçøføÛ2h\Á(¦ÓQ,ÍÉð·\_ôùçlÈÜäeõ×ϵÏÙÞEßsjµH^{.s>K<¾¹Xõ¤Ñ訮<Õ¼t4c·R4«w1\Ï7JàÇÄ \nPTàúÙìX1ùÝg,~¹Æ%H-Uw@jGàÃXxlx«½Ôô:}¶§æìÑ{j5èé¬: 3Ò´WoÔ¬×Ézõ¬Õj4ÆGe1ç{X1Îö:0$ç¹Ü0¬lÓർÈÑá.¤ ÃÖU^ìk¤ 0bß9®Q}´À !A²ZìzBèø:¤p+5im·ûÄËÖ¾ÆñV§ìFÉÅ9 ÛgTd´ÔïÙx\°Ä%uÅõýÃ!-Cè1ñII\sþÜ
úü ñ+9,·?ááz]lnJyOääÞ�>Ä÷ˬúvaÈÿ?À·ËcÓ_Ç{¸ZKXé ÓÔ^aÅ\$É 0¶ ¸àÿ+ À>áÛ00ÍbhÎ -4F8ÈbI[OFìV£}iCÀïÏ1^/ãð
¶} oålçÕ¼H«ïèßü¹Vk ¨tlgZhb´êú1_¡t�¿W¢¬Î½ÆË}ÆßߣôåJ^TççI[´UWzY áë«o·Sø[??kÿkt(ïE¾²"¯9ÿáoµØL úÖVýý~BúE±Ï7·èÙÇ0p+vïÅ@m§E9¦{�l ²�À²|Öæ^«Pvå¼ô±ím^AÇéù=ö#v|½ÛAÕìÓ«{kQ%Qúú·à!rú¼±u¦Õ>§¦àÅsÇ»!O«Å´Qx . áR±®ü³æp#ö
ÊЮÃ%å¾jä³ñ¸§ ;º T2çBèÚ?Dùh~sc)|pëä
"«]VÛ}ñ}M¹Ëàã1PÐx¯°c,¾Ö1ïÀ×Ù÷Øñ9óN3~n_Gì!p²á4&Úĺ\Ô×E(ÜðìKX@²F Õ/sH[-Ç
TÇùîÆ7rÅıÓ×ïåØÄÊc§¯ËØ¿kZ~ ¥á=«VíhBã ïÜ2òêA{þÔ}SÓ|sÿÞ×N̸rWÝüåõÎkvï[ßÛ»~ßü¬3¼y×|b¼µkw'B&nHa·òeL«!ØäËr ê²Jb@RB²LEù ¢$,s²Û @®ÆYr¶A7hCi@ĵëJWûB¼^öÌnÝyu±ãRckV/ôµÛáºÌÃ50G#IgrE!íKcNm\4¶ïà¬FåR\_Ü
Æ5|"3wJ4PZ¾wu®qMÞ¥Ó!þFódòÇú±ß4'Çj¼V¤Àä$¾#zSRA%Öp,�Êî97^¶9z'}&J4¨¯5ï665¸V îá+çD̯dïòý«ÌøÁdý@ëwû#~$ݾ[ÜOøð²Ïeq[6}(Kd³0zÖùMxÖ�¾h�û G ' ¡"AÃÿÞKqÉOyCkÏTì ´t¼n¾½ûA9¦+aG£Méb.k4Fà2¡gWÜpt6ãÖLÞ{úº^¹gÅqÍsåÍëæRç¿×ÜGTÀ¹¢#KSÎã·ÞîºjǾôþ×î[uÕ¯î¹åF]úÄ¿ÿáøQ�É©|{¸é¼¿î¯$TQb:éL§ " Éý¹\ú£ip( 6§Á@¤óýýμYW¬ÕY(I±$Hj0 J>ίót3¬fîsÛoÕFª¹u±²º"l{µxQ;µñ@«íø2kÝqt Éinw¹¼¬Eð+²8hKæJ¼Ðà¾QOV]µZØT¹,[�ð$Ù@{ÐÈ\Òo WÉþ² ¼"!åwÕz@#ÎÔpÊ#)@ZlCof´(2RyÝäZüÇ5Y¨xS NÍ:¬~Ý[ñØß½DÜ+!TÝhnCkÕV ñb±>¢Q:7[ÿöß}Ã7aû0<+T{;éî¡Ájz+}&ô04òÐ]10| 1[ Ūò\hUAoue¯²:ihrr|AXØH$Ä¢w)i©ãNUüÒöå¥Ô0 9«Õï³jÏe D"K|¨#þÔÆxfI§D3ÄàÕøâ!6$ù0- l¨ÊëFä¡ÕÙi÷ÒFR¬ùZoAÒ#ÅP°ººß2Fv ÇêÙíaüäúB2î2Þh³VãÙæô¦ÞÔÖ«+6»Íh¥ur¨ÕÇ[G¤7îëõþ|k,a¡- òCºöIõ ¶©Ùä¾�G'x=0À9Üxûñ;q<ภá8n¥DÛ2ࡤòm/¼Ó δϼèv6oëî/øx\\ħ-Ëô?i Õd%]èå¨G 1ÍÄÛ¿© ¦vÔH{EÑnHøîd#þøö´·Á·¯b[¥Ã)°65OÐà~8fw@O/i³yËòÙ<ÏÛ{}!±³ èáª>øÞßAcÌ êT£h¢kû;û al×QêÄËcNhZ:®òmµfPk÷ð¶oýèûOü²ÜQ¸5Ô»~(<9ÖpÙÝF¨IkYè³¹¡r¤÷p¼:³¼üøJÇ=p¥FrÊsep¤ÙT-Wðd¨u«>ÖT={£hY7a U¡ h)JdN2°XMSÆGù YÆçdà´a}Æ�Ï8µÀ@s®G¹n@z ÌÔ?jh´«Éõ$+Ô3jP¡(¢N±àp ¥éRM,¥úEËd,Få,�[X.å%å³.IDn�>gÔÓ>
ÓþX1zAàz¯)·Äh¡%QåûàçZè^buìûÍ»o1D"äpp¢¦1S\ϬªàPÞòÊ2NÁ½p£h ÀP®=OOñÇàH¬("Ø]8\¸¿@ôÀ¡ %£&t\_ yqSÈêib^ì©×{ ÁtËÆþtÆ\Ù;äºÐþ¬NÈ´I^^j|¶«®¶úYvؽC½³²¾Tê¡ÜhYHtÈ U4Aä\¢èãtÀm6~ñ/!NpÔÈ26ÊlѾûÄÿgï[ÀÛ\®ç^=·mÙmY¾lù%Ëå§üòCü@í8$!²-ÇJlÉH²B)4$áÑà<(YÁäµ:@Y~hJù[¶À,¥ýÙ.R ¦üiKìÿÌÜ+YÒîþÿv¿Ïø33gÎ9çÌÌ{¯$¸ôMÌT),çæ´ù×Ló?;FkÊÍKcód©Ij£þÒsÔó Àh¤%©)Ò,AqéÔ%.Û HÍLL FAbaGÃçèñK÷ò k³EØ:Ø% àJÜp~¾µ¾¾L£AV
ÃÚ^ª/³¶3ú2Ä¥/Ó·È
Lá0H|eeßãyÀö ª Jú2µ^_¦×ÃJÒ¶Únf$;ÔõVìºI 5"¡¬$§Ýªno·Òm GWGG{»£¡Áê ²:ÉZ ϯ´»ôéJ¡Ñk¿Ì~ÉÆ>ÑànFÆl×Ü®8{D6ÖãëÕJ¥2NäÙñ!s¦rÑex ¹³6÷TÖØåÊJñæÕ&}2\_6z{Ìdú|>:ÿLçZV£ã½ùùp¶½Ó2ÿ1ï@RÉUõ )rÁoø~6ªEa¸z-§ìå«Êi{ɪzbÁ@Yã ±¡tOé¡RAi©¦>×£aTiéSQ®2Ý 1·÷ÚÊ\&þûî \_«ýåYî1{ëÝ)¯ð(|±²S²/¤¡¿?Ò(h«¬ß¶íúóë+éîÝä¡F 9ÿ¿$Æ"«&ó?Ø"6r®ø°QäÙƼÕ6ío,,°Ï7ÿLä7rx]Ô ª¨Ó TåÓ¼ß#جÊÜlîÛÙo³ @á.(OîxnrZT÷Ü÷W\Di¤<µ§æ4ðhËD[ÐDXm¨¬¨6+V4PMñòÉýXÁmÀ} 7Þ>ÝÁòæfS(Cû>9Î¥P»ó-:¤¤i^rã>OFviÆTaJ5åkÛ××èZK÷t/O"§ªîËðîôò«ª4c\ÓÖlͯqèÔÅþñ>¾r¬xÐÀ¸êÐÚ|]]úvAööª:AU §ZYT¤TÕÕQu:¤R"H (Ó3Ô$¡O¤À>�9Ç=çM´# ï¦ì ÿÜ#óä¸áäE#Z<%÷Í_¿6>:6Û/Î,ë®Ím«2nª°gò:"+òT9j¹ÐaÍI-rôòKÜJ®(C¥HLª²å¦äÛs;;Õ55V±umk¡LjRSrWèV4å'|_]0~ R ¶áÅ\çQÛ2_Ý©T$HQ µ¢ñwKá¸×<ðÈ×÷Ù÷�;kSo[&¯§ÀÖTN°ÿ2¼iþ$õ"þ¨$¼ =·àÛçðïÒR6ÖAS¸¾8{?o£¸DoÀÑƦñ5ÍÚÌùªºÜüÖkkÛ tÕ}¡Ý}ó»y¥¡º@]dÔȲr«´yõh½"MWÑ]^¾¡°¤'ØPºÙ7Ðl¿]Unìï(O4Õ§rX3Èn0J°¾VCÅñÑH.MHxàûï!¢ýM^ĶE_ÄƦÉlTWÒóç7|ö¾,+'?µ [pç¥4úÛ¼¤\ì8Y 6ËCUh5ºª!WI·æ«ËÁáÕÖVcG9%ÔSÆj¤/¨ó%ÍHOéqDzhà F>=Cn@|Äý¶2~¸Ì~4+yñ¦Ùrã§{e/¸KN÷¸çÍô2sFvmÍæ©7mU©øI±©©EÙÚ&2§þÞ¬Ô¼BeEKU-ÕëUrSy»8£¼oL4Øs +Wdê,v=Sa6È2mvCaM§/mSºµü¼cb¾E+¬.Îmb¦ÅéyeÆ´ ³®Î&ÏÊ+Í)ìJu éí±nCteà ÊMBHGé°IX¨baÅÞ+±}d#KMeüÃd.¾(îr1öA [Óø�Uë°j¸ùùíÔµTƪ¸ ¢H-Õ5m
VªÕÞꡦEé5®rTºB v\Íé:ú·<b5BzP÷(,Å=¾îzÔºHèK ×%!^Eê:¾PW R5¤®3Zgru2hç"u]±ñ6Øk e ýÔvG[Ê£-³¡®´¼Ö2)¦«j+IKw´%-Ó'´ì!uÒ9BzÒô2Éd+æC\ïìä}:þUïï5¬ïìYi¿:Ǻ¡Øjïw8õÒ!pHè| êèp&«=NµÚÙcårLwê©ãTÍ8URWÃ¥JÎ+4Å{÷?ààÿXûRáD¿oIXð±Ä/ Ëd7ÊþUÿH±^¢LQ $þ.i6Ù«^bHÍÒÔjÎi·¦¦ÿSÆï2Å{õ÷fõ3r ÿÝøZ®Ú´ÿBKák±ôiá§æ-ªÖÙÒÛÛ¦ÊçªÓìöÚÝú/4N5w$µ¬líoÿ¾ó½Õ®sóWyÜßëyk¥^~Ë&è«So]¯¿wn1õ¹¿f:ñ
ôÏ}ÿÚ?Ú}ÿÏ×b©M·¬Ý˦«¿Þ¼úÍuU$ù¾,|íT¸ÓÿÅôárúGLÿ!YNËé0Ù7x6ÓrZNËi9-§å´ÓrZNËi9}tÍt\úùrúJÿc9-§å´6vlÜãEÞo³×åõxïñî÷>ì=æó>ë=ë}Ùûº÷ï{Þ½¼þdèÌðQ\_ßȺMÔ¦C£%£Çü
þû76?±ù÷[Ú·üi¬}ìýñÐøGÁqð_K_{˵ ?±Gv.¦É+¤Ò%ÉK_HÓ_;íþH÷ÿ?I¦lSnÍÚºqëÓÛ²¶¹·ýzº}ºÚ7Þ>½gú»Ó'¦ÿÛôO§ß¾î§×ß´Órú?Oä[F=ðZêf$¤Lrvã÷Ï~¥@;µ sHÔ�sxr'õüEö
u�k�F®%ô:ΤHð9üýÔÀS}ah'åµ 6ë hë@6i � Ìòrhû-Õ ³ÚîXKàº
¨Zý dV"ðWB[Lg.ì¨ÎJ³)át,\ °Àv] OtºÐ}^Cèµ^°ôUjÒW5J$t&ôX }aCJ 5ÈÇÐúW#7)ï%ô×AÈ´þÐÖúc:¬aæN21tÁØí 3¦{ ½Àu ¹X²èY 23&:$×L sH¹¼P 21tAïµ Ó=ÇCxÖ8�åNämèIt-êOT-¼ 0Úö¢UPÛ µ.4�µ×L¸Ðk ½Ðë}= ÚbØDý áIQ[;ýï
áWÔÓÉ[-<Ízã7×ô ÍCkè=Íã -ý>G #BS11²ò¬n4p´\n¼ý®jJô!GSHü'¦HÝÁÑÈðqcÖ-@}Ä¢áب+ ¯ØÛ¥³²%w£äùO!ò·ázwñ}±¥W@ÛÀÄZ Üibv4Fı8?Al7µH¬-IÛ¨5[à ɱ»8öhÍ[ÃÐËè'ÚJú"qt¥~Ù¼Ä×$¶W<6lW\?ÁEC¬2Ìõåç$ q²ØÑãd¾0ò ±æ4Yü0ïã}þez¾ ûë[)>¢¢~Íb1tåѳ½Q¯8à°cþ¢kLDá4±þ%�yÞ/)ki﫲3(ÈAvT,çò7£±¶S±ÈeåN¼n|µ¢+ZtÝÄÚ=çÅÒ9h&6özóèçøåó6¬uXc;ÓJH
ô±
ÌdñÊðX7G´®yÍeëFÑÄ m'HoìÄ=ªÍß²2ÍÑ]&ÃÁdÆr3±úßGv1n]Ó¯ZÝ£ñõå+|Ô{ݱyÛCÙøb#ÆÇõ·Df-f2î·ú²×x
ð°¾Fe´àöi¶ HeWÛ@,Z¼hqÊüôGÌJ^2vl;?ÙßY+Ir ºÆï~²²I|r:~¹É~°dc2¹(®>NCÿ9ñµåUÚOÚE¹¯¼V/[«¢¶¿¼õ¹zò\_6î¨^× 3g26Ç£>4Õ;Hzå}qW!ÖCafí¬ÖD3ã\º°>,á<&3e,¦Ctn/¥¯oÕÅ¢£ß7Æô¢%¶;ÿ~®íø)ÀYfé~DìuÓ¢]6ÇPÜNù5]ÇÉ¢ûý «¹¤ÉÊså+OöÊ(ºo,Ú(º7-Ú)~]YÚ\LÖ Ö\_Üد¼z¿Ä«¡ÜõVÌá1¢®ß£ÿÞ(ßëÚáê st¡VÈõovÇ%N(c5uCMäPê<àèáêóÇúÉÔ|½d¿ce¸vB~¬u!y[ ü ·mAkH- pºì(unáøpf(é
<¦ÛÈjÈö× Øki'·?²z p©VNÒcT³È¹A~;WÛ²DÖ÷ßJèΦÄFX2ÙÌ]ºIi/ànàë!ý71³Úv1´B=;¢îÙåÃöéãj°°~.H£j$6h'Ú,Ú¯p7hå·Aì]ÐÒAFÚC¬×Ù ÖEr£b=ÕLFmà�ºþÚb¶sÈê¶Ôvý¤~_#åºHõF3Éy¯pó¥ãò^ûI$¶®F2X´èeµF'ÛGW&lØ·ñºD£ù9JÖ÷rþ¢]°ÕM°^=±¿L²å1¦Ôº¢ñúð1ÍÁÐD0äø Ó86ƸýF#aÆí ûBS¾a #·ûC¾L×/àÁm\Þéàd nò1CÁénÃñVc¨Ò̸½c£L»70Ú¥«£¦}r8{òúÃÌX¼iòù¼c ×#ð¡S& ù�D¶zC>f20ì 1<§qù|°¯ û|o|Ð7<ìfÆØRfØ ù'ð�Iþ×?¶xüã¾0Ó ½¸ãÞ�îËËDBÞa߸7´
|¹¢
öËe¸}&Ǽ!&¿Ã? bu ú|¡0îºRUMØ0w÷txbÒY!ïVÓ52º3ÅLOÄóM!?XÍÌôù"074ì DÕ¶Òà$3îf&a¨lÔ ÔxÃÌ/4îDÀÓÄT-½®Fb'"?ÀlõƵì Mb#FÌ°?<b¼ahå!àî- í<fòý¬õãe¢ÜWTusÈtª¸î¡yLV Ñ ß½D|ãØ!?ô:Ü zã;¥½¬ª$0Þ tp22Á6ìÆQßØÄe#KäìõàØXø"33è BÁ@,ê¢ñ?LØKJ|ËVÿÿoØïµCJp®8¯á³ÀÌx'&Æü1Ð;så u¥ðÇá¯a[nxü¾)ßLbÓ¥SÛkɤÃÃëÆ>{a|ÐnSÈ 63#!BúC£ÞÐ&56e;0ÁA:l/öóoVÉü^ÃÁ¡Éq0¼þ1°M>¸d¼L7ï\_+ û@ õÄùþÈ(.\3UXûhõíË ±+ô0=GhfÆÃþ}Ä 0 ð¨O =80.äFXû) ØÛ®¨i»dçgi¢ÄÖÑàøWGûd(�Êpó4ë#Ñe³o( ±ÅHöùe¹w08å[Àa1shgÓÄb¬pUáQ/kзdzã aÝøÁI0KÙýU&g]{ ÓÓÕêéot·0ΦÛÝÕçt´8¼ÆÈç~§§½«×Ã�»±Ó3Àtµ2Ìjg§ÃÌ´¬év·ôô0]nÆÙÑír¶@³³ÙÕëpv¶1MЮ³ v 'ÌGêébp(gKÖÑnnlcÓåô V§§Ël¡Lw£Ûãl5ÔÍt÷º»»zZ {ítv¶º¡Nz
2¦¥2LO{£ËEºjìíÝD¿æ®î·³Ýôw¹-PØÔ56¹ZØ®PÍ®Fgq4v4¶µV] ÅMØ8íúÛ[Hô×ÿ=ήN<æ®N²f¥ÛkÚïìi13ng6H«» ÄcsB."Úu¶°R°©%ïíiYÔÅÑÒèY=¸q<³.BäPrxDÓ!ÿ[r¼ÖGo�³7vyûyOðþ÷øû>ïÞñ8Y^r¬æÿ'í[Òo4"¯ç¯à¯æ·ñë�V·kø!{Ä¥§þ #_#ðÈmu,#ú<¡
l¿ò?Onµ°)"ÔAÿ¦F¼ûj\gØKè¿øV.Ì{::ÝV+B·±Ï¨\_ùg¦Ë@ÚZ v"¾Þxô~z?Ðè@¤ý�ý] ¤?úSú"м$^ñy@·ñVíÝôM¼Í»wè?ò>ú?(~A<þ$èëø×}=ÿn ïáß ô}üûþÿ;@ï%(Ô Vè@°Eò
.aÐ~ ××�= ¼èuÐI §[Þ&ü¢
· oúá@ïFèÑ#':": ôFDÄOüøcD/¾�ô@r@VÄKØ&M@T"#T!ͺ@jºLú(гÒÇ~Bú<Ð/HÏý#éO~YzÑÒW¤ïý[éPþô<ÐþèϤýoÒú¢ôO@ÿY Þ!Ù ýPö"Ð?} ôyÙ-» W"J®kO&ïºO¾è Rx^D+UÅuo ¾FÅS@?8åÏ+~x¿_)~ô¿(_Fòò#ò}åoü@ùпS¿W~ô§$1>/4Ê&d}ÇzóØÐ ÖóÁ/5b°xxÐ^ñÀñÀ)ñ4ÀëÄ7@íMí�oß%ßè[Å·}»øN w¿ ôð öÈyÎþ4X¾h³´ìfZmAsé錄#v; ðG2¯ìE°!¶X ÀTyØJ#×�Å6äÆ#AòGoÈ;¡éÐÚ¶)äÛv
:8æÐ,Ò!~+¬¿(¿Ã5À ²XQC¯ÛÏÌARßü"´©¸ß"æ!ÌRGQÊäèôĪ8Ï'¨QV\
¤(NSÑüëRD»Dz{5YpÒ05(Ëñiñ¯)P2¢¡ðzÀ÷üÀRô\_(@ÉÌ!°@cÞJà÷¸ÀøØB=DàI!ðE\_%ðW¾GàE é\]nß2¾
x·¸Àû{ðíç=ÿ@Ãkbµú®Õøá1e¿ððë¿ÿðüaÕáÃöîÃO¾ãðAv<¨ÉyÄNhé#³<ûÈ«|ðÈüõó#}GGnaÇø¨ðQÝ£ev\Í£ï7ûöìg©3?ÖúØÆÇ®{lïc§ûÙc5u>zÃÑû<úÒÑ\_ýüöXÙ1÷±±c·{èØsÇÞ9vñ¸ú¸õxçñÑã·?tü¹ãï?Brp~s"pbÇÃ';ñöÏØñd=A=>ÃáS~èÇ?cñçÙzáÔ§>˾WÅÍçts¥Î½<÷îÜ
ÓÓºÓVv¬§ï9}äô3§_aå>Y϶}³ÄSºñ©û:Îjó´CÀ
.eãûi;[9ìáð '8|#ïàð=>Èá#~ÃÏrø'~ÿáðϳøû2§s8Ãnàp'×qx3·qøVsú}ÿOrÓëû/røg~Ãïqøc_dñ349=Iå0ÃáBq¸ÃÝÞÀúá1?{Å?àVÀ|Î3ï°øù¿ðøÏróþGbsò^äÖsõ/)8|X8|¹(ýéZ¿| QFY#E½H}NÓô}'æµòñü×vÁaÁIÁY^¼"ThÕÞ+ü(ÉÙ ¼Ò[·DzHE¯.2ȧs±I¬½ÿF|1¡5a"áIÑ« HR%:É ÒuÒééY¾Ì"Ƚ$;/×ÉKåò ÍÊgÅ°ü!Å=J¾rJù¾êÕ!Õs ®M¼.ñý$ZþPþnIæ'?W¿¢MY
kSÖ¥lx!õ¤F%H3¥Ù©yRsAªe´Úaíí³ÚµÓÒÓÒ¦ÒæÒ^IGétUº%½;}Cú é¯d3ÖfÜñîÝO2µPKÑp§7ؤû 2µ8çHæ ü=ù:¿Ö#}«~§þÎéèç ÍgɲNe½õ³9Ï^½+K}F?ÏìÈÞ¯tY§ !æaÖðvöì37ûU(ÿ®]ðgð'ðçð§jæ¨OvS¿¿,ì¦)øKXø-Y£ sgæ{À¢º®}÷9À9cÐ@#ÆRJ± J1ÖJ %ÖzyÖX¡ÖRJµê³kÄgZ£"3 á Ã0Ì ÃÌk"ZªBþÌk X_>>b)×ky¿µç 4·½y¯ïãû½ö>{ïµöÚk¯½öÑ=Ù¨#ð;ÑüNÝ ûtîBÐMºA· dï£ê×cãwÔÅÞu7EmQ6Ü�~àÝGÀ-àÀ(êüó¦ß£¯èÝ¢õæª @9ð&P¼T@ÀHÉÿï?ݶ »tÓîYÐ- ý¿Q»ÄÙ [ dKl)-²åB¶\È Ùr![dKlI-'!@·4èF,Z�'ràM x ¨�qvwÐz6¿ÍAw9è&ÝãÈô!׫+r\Ç ×1Èõäzr½ ¹^
\Ç ×1Èu r\Ç ×1Èu r½Êïxø=ºAw@èÝÿ Ût÷n~нºõAw>èÆÝ÷ Ût×nzdzñdª÷z=êý@p8@y9ð&P¼Tÿwu ¨Ìpà 8òM@3ð¯À 8 ú7àî@Þ(ßhg|4Hé Ý0¡û%t»îÐͺWB·JèNI&l«�¶w(Þ\·JÀȸv p 'tïnкB÷Mè¶ Ý5¡&tÏndb6 ÀéP¼ T�o àî hÅð±<ª%
ßBñqI÷Oèö Ý=¡'tïn<»{N"¿B·SBøíºB7SÒxi:ÞÓíº{òð§²N$ÉÛ'Es¼´Ó}~
n³ð», §?ÐÄ'½Ib2|s¼PÌÀÊ{°¥¯J¤µ?õüð§²úûÿbÕLÇüa~Ã;,]øÓøxtIÆÏÑ^úãñ3êññ3R0sü _ðùð÷zø{½úco
zÜ[!
�3½¾Zx·9SßNÌ}í B¯IxÒkCiÞ1lü7¢íÝ)bí±Þè5IÓ�µ7Ö"i@?Dã a1è% ½<",ò& _ðnxçI ÿ4n
´ ã±C¨fÀF"°cÍ"h7K|wGþqÌ6ÚAnîiA4p~D@:ùGFzÓpúî³ê|×½1Êè¨7ôàA;Ák ´ºkOcTÆöYAæø=Xï"¯ Þ2xË°<ðÁK/+ô&ãÁïßYÌ® ~dÅ2øɣ̦£ôr½\A#èa¶~5¯ 8äGúwMCß#,è\i´÷÷èwýöEY<�+û³õä\_G´HÜ®ì¼æ�îWP3ó
ó"Åz¤EÀ%-Cú5ÄØÚöû!Vë-dFo'³y;¥xXoº0þZ¤KK¼¿Eti9ʾü³Þ¥,{Þjê±ï Iñ@èEÀ2%òO_ö^¾^S¼CLVáù|ms¤Ç¼fië->4e#Mð¤'.áý.öJHËPôÀÀJ´_ô+Þ"i Òg}i°xÖûs) ÆX-àzÒ7£c´ §Ó¾òV£·!ôrãîdÓÑ"òFA;/C®(´údBË ´Ì@\ÔüøD±¹iºÑk±¾¤#ÿ°Î[Èá7·ðn³wHü¶W'¾�:éHóQ÷%×9ÓÁ´9Ó9õàFZNé/Ë7W\ÞVymjôòUÎyGÁy]àü¾ø ½g ÞóÞSbòßÃûÐó£àãÅmàhGÆiÃmà: ®£àjW¸Ú8Ç�6
Óæ�.?=çòÞýøçDÖóÅz1ë
àôÌò¸ýD± JXP'æÀFêá%5ÐíÃ@ÕBÇ ´Í E .kQþu¤Ï#¿t&¤Ëýw«¸ô÷@ç ý>¶yÉK ÿÒÝÀHý2ìço]qîϱ(ñ[H_@ú±ua]ümdúĵ©Ø IÜ ¾~�Ý sÉ¿ I^m}¼Ëÿ{lw6ôóÌÒd¨Fï4;4ÿ6ÿüÿÝ32gBîç¼½°°jôø8Tà@+zE¯ÜnɳøÞ¼ÀyñRØ¿}è1~½b¥X({Îû¸é7;|åÏ{_.¤Ì ùµxìY~¶<ȯ-Ç»§§ñn .!ÝûÐÃëØÛ4^Ëd '?60Vßçpv)#®EJ¤s):!÷{ Ø&Æðc´¯È.H:!Mb99ÐUtEöÕÉ(zb_å³1^/ð±eÎÀ¬òE ß·Þ¡¡!pz\_þY~" «PUxAá\ȵë[}C×na¾Õ§ç¶°ôVnqz¬>òy9P¼ÈmC¡wò¨X¤CJ²ÊBHVñÿãÿ1[ÌÌÉÉ© }øîÆ\_dK°Èö£¸7^xø[Ü7DÁ !]øßá²Æßab6èï�[ï¢þV¤ßC¤ßrAÿûtHÄí w�;=ÀËØd¿øKÓ'l$ÏVÌ!$% hçìT,´³ÕÙºÐkÑu®DùÓ(OÁz W<+Ôq§óqçr_G;MYu6Ècã+,Ï].·ï _B¯þÜìk$Ï«ÿð]|d
×�åa
b¬6hÏ 4xÓ¬ Å9Êÿü.y$$ÎàÞÈç ù|¯caT _ðʯ7"gS<Ïøªô[Ô7¼©Ü÷lýW¸ßcX Cð(Q:x3FA}Ëæ£èĺæ{ëxeqÊ^iøgHi1gy4 ÿ"Êò¹?oF<4Ðz
úc´Ös~ß9÷¤þEìÁQäGq^¤ñofTw¾ïJ>ph%\Á.ýme¡ÔËËñ<Ág! ÐÛìйî%ï»lµzQë=ïsãz}uò¸Î·ÿùc;ò¼ y\_+ír3ÚÃcµfaü䥳¨,çÔ×4Hí.Pm°
NÇWû}¥&oä|Ô~ïÿ>ê¡Q÷rÎjè2ºÌÐQ6¯ÅçÞ@ÌSöøÄg@3±ç{ÞäV@3S=13ÛÙQ)VîÛ·A?òþ÷®ôa Ðíg[°QlÚzhÜÆu(¬Ðd>/ßÌò³ý{Ày÷ÿCh[Þê¬skÍ_6Æ&FfD¿!· ¹eg'ÆÙ©ì06ÚaØ,B_;pÿ¿2dþ
¯ê[À¯ÿP7ò/ $K dI,)Æ¿pY|ÿPG߯à_füW¿]ò_¬ ü.è¾ rä²Ùø¿nüã¿kÎò/ '¤|¯þ3§|ñïCo}s z;Þ=Ð÷êS¾ìÊö(z{ôz uÅÀºb]1ôÕQ½Ò4@ä]£|§£/¯Òo"\_9ü/F¦þØ»P=î](
�3½ ÙC÷}±°Ûø¾ÚêÑVÏÐû/Ðû/Ðã/xOüzù³/}Où¾ç×YJ\Æú7}£^Äïoþ^Dô^o"zID/zô^ôè
äND/zô¢gªOø]ðÐO/bòkÒɹ¹û¹i¼ÿüg°{Óéô#ìØѪB¿÷}Ý©Ýà;/³\_¢¬
íéD48ÿ}Qüÿb !Ä@C輺1Y?h1 ÷!B 4ø±qÐ !Ä !èÌÊϹA DgÆÄ@C\_.N9u\DÏ?ñÔñ8¬¯ÖWë+¤ØÇq¾1øc¹®¿ËuMÄrKÐûd<×5ÏùÆéh,qÝs×} ýøb;çÄ·¥a.UÒxïü¼F½úzƸ>8×ó³Û#ü 㳩çaÌ-îÒ1wÃü«P¬÷»8[Ôa\ßåãYÔÿuÈw¶g 4Í¥Cséô]ê¾+
| ø.4~ø]È?·ÊÜZxm_D9¨Ì徯ÁR<´´%IO#ý³^9뼦D¦o+gý3ÏkP£3%tö!½ò(õ3'sèr(«±ð{µ|\r¤8~ÎrÊösà10,Ëçî;£Ç$̳sÚò¨ý§èå|eø¬Þ?«2ª¨eE-ëÑlå'¸ô¥SæG>t¾9á ·?J.AeSìåY¢ú¿×³ð'°XF¿ªö8ýëÿíè$
°eøS±/ï!þQCÙJü
ѯ ²éì¿á/=Ï6cüô{l û.ÁÞ\_3±zì8&CçÍø{aí,Ã\_4ëÀßö¿ñ7WøW!Xfóµ f1FаùBÅæsØ!Fac±ìsBPÆ
&¡=.ΰáׯÙ;;l¡pE¸z¶HxOx}Aøð;¶Xx\_x->>?°$áßgK
? b_>>fËE& ìKbÕl¥¨5ìq¶øKçsYª#Æ°µb¬Ë Y¸Hüûº¸DLdÏKÅeì~SÜne{Å1½"æ¹l'ng
nq;$ýT,ØauºýL]¨.dGÔ¯ª_e¯©ªþ);ªþúgìú5õk¬X}L]Ì^W¨KØÏÕ'Ô¬TmW;^ýú2«P¨¯²Jõ ú÷ìmõzÕ¨G0³úcõÇÌ¢þõ8³J¢$2,
°Fi49¤éÒt&KjIÃR45KHQì4GËNKó¡¿b¥Ï±3Òç±2ÏJ X¿JKÙo¥å°ÓKÒ i5{Gzë¡OJÖ2·&¥±÷èîEx2ý¿xaìu\_úàÞô\0ú Ò[JJ¸@ßcìýY½+¡6RY´ùJ§!c1~7mò Èõ¥î2¤ËñÎt¤¡¼éz£ÒÒLÁ]¥¼Ëäã!¦dôɹ�/ɸ eÀ|. ¤vÛ}ô»»@7#-øäzSòþ6ì}I,ëßÝÏVvôsì?ÇqÇ-v»ÅwãÝ¡{Ý»==a==a½îå>ômõÜìËótÇõçt/ìÏïNDº¼?¿o¶ç:¡{Uÿîþ=}ÛQo7êû/r¬B½Ô/è/ã8ÙÐ{Êcêmóغ÷÷sXQp4AMS"ägq&,MHw/çØ à�ÆE(Rsáp¹K�ÝD¾ùäo&¸þüh ¹OýE¨Üm=áî5îµ@òȯ»¯qõ\_ì)wßèÙãvsT!o@¾Ï3л z'äzò8¶¹opì÷ìæ8)ÐyJ ݧ + ·ÃãêíòØz»=-½nO{ß\ÌA?¤cHïúçsRôb°ÿ>·»; s¶s¶i&ÒUýû0ýsÙ·�ýìEß0çGС ²J@:@ ú«8Nö÷qìï7pXû¯røëw¡nW@{¾ ßÌq²ðnæ°óN°40i+ï¤l'c6èÙTß=c{Gº;EAÇ&÷z,ØW··[AòÜÛÝ[8î¥ ¸ ìWpH/¯GB¥¿¶»+8&mø$?4aÒþü]ØðÝ�Û
mn]fØ&Áì= OäA7ä[a/°Éú£¨g"ïÏáýAØ6á°½,- £Ø{ÏUØ?Ð<Ð3<Ð<µ1лʳÐó!ÚßB{ÅOõÜFþòëð~.ÞÏí]îY #¿�ù S{WëQ7ÔÖ¾ ëmÖÁ/�]ðùíÛÜ£:µxÐ[õI8©@ç©$àÝUB¯e@ %ô^óï½u<ì¹Ô; Üñ\XË~,R°TÁ «¤\HW@> hõEZõÚ ¾ xGШúòFðôÒÀ¾ä¾¤×ì¬cÒ7ÊwÆ@¾® ¾®~jo︧§/Ø3Ð+Aϳvc?h4A×ßÌq m ý'ûosTÀºáOwçBçÕð«ïó¾}q;#ݽÿLê¢Ðõ,ÆZJÆØssùÀÊ)kË¿ÖvLYkf÷xv9¿u4Ú }Íuà×_ïFÐ�ÖM<ÖM<Ö@:°¡÷öB ö~Ï#£:O5Á¯^+ìèvC@¯<Ð} yê¾ón5Æ\ø%}Àø/aüܺ�½Åò 9üõ{PB¨CH
þî-ØcrûwoCº«ÿp¶Z [½¹x¾ùaä)? [¹½GèÛ;&d)¸ Û !½; W"_®ÈDðç¯zãà{zfTºK}[bÁ/¼®üóäyYyYÚèÙ¬è3A^ä
¼@ ùäÈ}íÈG~Ò¾ïìS"üY¼1áá {ÖpT6¸§Îc¿Ê³#ÜSDèÛ;°{÷]Äj yŽ{Áïsú#2t÷ß7gü¦y(¿c>-$)d)B<Åfð[á³TëTÿÄæ¨6ª¾Åã÷Á?ËïeÇñ[Õ\_àw¥ñ{ÐOóÐk4ËìM¿æ÷ìY~Çyçôó@PLÐ&}>h1S%%³Aÿ3hÍ YÈBVªb¯©V©^Seª¾/ü\õÕ·T/ªò
JÕÕN¡jzÃtp2\ß&è&¶M·¯Å ÍÛb>baO³ëH3¡g:R@"MÖ+)ac�Mg�ÄÞó;w ÆïĹ¥Ê ìWÒC 0övDw Íü&èð¥8 uèð.ip0¢ç¢+ +m(=Å!tÊ»S|<$ÓT>9Û�ð:A2v¡m(3 ñ%åt׬ Úÿ£c¡¯¬£upëprû)yÖ y~½åbýúûÿ%û={[WÝÎ%E�K[¿«è¬T\gLÒIo0Qº,]aÁRÔËT[òøî¦Ño (-w+-¿LR-u² ½~4ÊCÖ<ËBUªe,Lµ\ÁR¿ÌfpÇ,©\zÍEþEr~Ñß\Îoæ´¦Í×Ñ´³ o!( :i'×÷·aï´ð\_´[Ï6ó Ùð#¬U2#³³ÖÁ.³vÝbw
`A"
a¥\¬²\a°W8$³ ]¹®XW¥+Ó´w¨×
éu3@éVjïêA}¨½ ÕÞuMðzÚíuí.ÔEÝ,í n>¨6mÖ Ê¥cÚK¼?³¶¹mX ê¤v¸íºö(½¶Zkk» ªXÛ§5´¹Aµ%Ú T¥¶][ªmU®Ý§=¬=J§µj÷k JµyÚÝÚí¼m¹6\_[\ ·åZ5i7i«Y0dîÓÞÒ^Õ^Ôj/kcQ2¨-Ci9J«´·Ë¡vv6AKcÕ®×Îצ± íQí¶ÅÚ"¾le·ÊîµÅkEP7´ÚæiãA ÝlU6ê²6º-L; TGYwÙ5-ôRÖ¢UõhÙÿÓýhÿÆ?
~ÍäÙ4Õ>Õ«LÍ[ä3üAá¿ý%¹¤eÑü·<`9Úe°^ö\_]ú�ì/¿¾ò+øß\_ÝRRí�þüm|y[(�ÜFùhó4N|pùD·%Ð~`/(ÇúoÃÞÖ@¦6ìkm÷f\XïKI^Î{\°ßµa¿;Nr`O9ûÑý°à¯×ÁÞóµÒÒ4`ý¼Ò'¶fn)Í-ÝVº¾t9»Òýø;Tz%¥ºÒÒ¥FZKeù[ú;Rzª4®ôÈÝ'ö8p¢¨4ñôÀ黧ÇN¾yúzkBëÖäÖëZ3Z7·f·æ´æ·îhÝ¿}¿Ãååÿ9°´QÜøoLÿVÌNÅî!nuá°º/Áö°½a{ßdªÎåø¨j³j3ØüéfØáç`÷XüôqXã°Æo³'aí, vØž¨¹¤y}IsEó.{ 6éf«4«lµææ}Xé ¬ô«ÜJ
Ñÿ`)I¾§¸|OsùVsùÖpùèVv¥µÀ¶Ðo´ï�ö�ûÀÖÚÅ@RN(ª÷\_JhiEÞ<"J¿n\)+)/)}ì¿ìøeWKXËM%åúë-YúK-[¹ßø£øGèé¶x»êLTmPm`Aðϳ`Õ·á%B¦×O¯gªéÿ1ý?ØCêLxPé,¼D÷Ó5¿Õü
k.ÃW¨5=>&i<{Xó;ÍïXæºæ6G3.'!qÎc"ÆfnBL.´°|Æ ØAu ¥�v¢¢¼® )¢�¢�]¥êµO@°JL|]åÃGêïwñ§Î2"'-ErdÍ:.m
eL^Q£cõC¹Õ°Ürc\ÅGêl WÖ\µFÈk '-w¡8q;jRhK)4à£S,zо4K¥c·ò\¯ÐÕxn´Ðgà3Óbò ÔÓ<ÖÑ(jgX7Ó¬Ù4"kÎäè¬ù]7ÙOílÙÚ¹^Àgy#×?ßÚxëÇÃëÇXí"¢krzi¦v5iV¼Ûs6´´Ï±¡6ÕzPfµé¤ÕÚ Ü2¹}r«¨ÝMÚ ×HµY¤ÕÚγsj·[åüÚÝÖ2yGí^ÒCíÒCmi ú\_©][Êi=ý\k9¸Dsz ·ä¾Fr9}ÃLÜù\ÄqzÑämªäÚJkÊ7Z íµÕV³c ,Ôj?ÎjMVÕñdÁ° jmÜ¢|+ÛʱRÈS½'÷Ôº¬Í½µ-ÖVxx-yùª;µíÖsÇIÿòA\_Mò`òaòUwoZ.®=oÙ+Õ^ëÏEmÑäÙÐ|\^;Àõô\_{ÓzQ®ª±^ |lò»Ú±�úî¤þÉVÉäyd³Yû&£ÊzdÒíÆpë Ü\wÞÖÓ[S0Ѧ0NÏàôìÀV{äðuÖ[ð´ÖÛöp´½AGØ°ë�Ù°«HYéÜ;ù¬×4×ì:jZ`)u\¾È·¢wñ9åz6Åûõ\u´çÒw¹\ÉǺªÍ-Fr>:ð²MÞÞeRüjÌWñy.VyδÉy¯WMö\_N}ÖGPÇïYïÉççÛ ºQN0F6ÊóJZåËÆØ %ófÉ· Ñòbãùr1¹!N¾ZÚ°Ðe\Ù(¢ærÞjj®iH?,ifkÒä[kÃzù¶q]ÃFù1£!Ó)77lqÖ\lÈu\7f7lsJÇï5ì:Î9 Çù û³;9£{8çW¦3θ¯¡DNV$?Ø s.4n¨p&N:£¾ ~̹ª>ÒÜÖTåÛe §)Æò6g±ÊzÙ¹¾L´UFCCl Ú¹ÑhnÀ> ºÑ³½ÁíÌ467\snñí°¾½ÌØÚpë"Z¥Ws\×]eßã«c¤º¸F1³î8wá9'YWRÎ-XN6öÙPÎ×ñªm|Ë8hí,PÚæ:÷oÙ8oÛ± PÇTxïÙ9Ô¶¥ÎºPÛ ç¶ÚvÛjôCq=åä:ÉêU7ËîØ]mÛuoÛtßs}Ý|[SWgÛê¬àÏËáÉ}²ïY·Ðç4÷Ù¶Ãnm»Öºå¶½N¹níóT]-ÞÙVf+3ÅvÔÙQ·ÞVêìxêÝumNw]¦Úy Oó/çº-6sXyæÚ\ÎQ
ÞfkqÎòÍÆÕ¾¢í¼óNÝ.Û%çx]çxrÝ~ÛÀqVwÈvôÛMDY/ÐÁu%¶ãóðãÏ»4 Ì5ÃE×éU®Ù>=×U4»æ¥4F¸ÔltÅûRhÒØ8ϵÈØÜ Úß= ®¥urãb× ÐÉ®Õu§WºRëÚ׸Òë:׺6Ôu5®smªënÌpeÕ¹7»¶Ö]kÌvåÕÝhÌqm§=BÎç{Í´´QDM¢ñó¦æ\Ì]Géìàját»i5ÅB¦T¥çw½½×nwñ¸Èu(Wiè¢åbÓ&Ð×ÑvÜu[ï) ûÎX %×f5tä¶6vì®k,U(1Ö)ÖMà7p hbJùöÆ2\_9vU\WÝÎO =±i7ùÓ^î@æ²ÓED7EPЩìqE¦HÓÑF³KÏËçQyS,§8½ØTÚhw5éÕ¦JNWM§¤¦dÉß´Òdã'ÃS¤ñö"²ç¦5D7%ÚÕÃéu>;7t5¶:ò GÏ9z ¢K6-%?cr¡häí½4epºÓM-)2i¼ÈoS6YxS©½±ÏQi:ßxË N\_"ê7eS}ÔYJç5SOã NFðWMùdùo/%óºiG ã{½É·×OF5ÇbþA7í1 4~è¨ü·0G8V¢ÓV3]aèTØ´Î\_o/2Ä5ÞÆÞl¼uä£GìbÓAx6NRÌ``ÉC6¦õÕTÌé2¢1n 妻öPìôÇ|¨göYrssGsws9Úx¯¹]ßo¯p®7î³ë/ÛÍ ëó]»ëFw¸öÖ÷ÙÍõWíróò5öSÍ«êímÍ)GGì®j%\_kïjN#Í7¯çÞ{cýÿaïk ¢J®oÕënºi@aBÖaÐaãdYbød±uÃmG{ Í8ݯ"Ð4®aK !c<,ñø9Ã|5®!c\c\5×°Æuüüü!Æq'ßzïu¿×4jÌ$Ùs6§Î}UïÞ[U÷Þªºõӯ߻>tnd£´Ã÷¶Ò®V»cÍVv©ïÜÐîUå¯ÞºáßáMøÜ·` ~ BÉBh%;É. {ÉOáIø¥Î¦û<¼¯ëÕ} ~«;ªût§u?!&Ý5Ý{ä Ý^ OéÓõóÉs£d¾á¸á{¤Òð}Ã÷ÕpÒðc²ÁðÏ1òÙSÌÓä+1ÏĤÞgc|¤Ïä3¨ÞÔfÚCãM\_5uѧMß0¢5}Ût¾húi®4ýÌ4E?mz?6ÖMfm2'gÑfs¢ùi0\_2ÿîÛ÷ º7îN<¡?OO¡?FÏÅ¿ÿý×øEñèÅY ³èÏ ulü¿S»Ê`DH&ÚdlJhJjJi×´ iQSVSnÓ²¦&KSiSEÓƦꦺ¦-M M®&S iò²7+óã ã FÑìûêü¹k ¹4]J¥/Ñ@ ùtûç2-]MWC ]K×®£V0Ñ×éëO«éÀÇM?O?OP'uboQ<ÉÊ~É~ A.À%®ûí¼uPííôöx{½½ÞaïQï¨÷¤wÜ{Î;á½ì½ØÞÛÞ»Þ|:\_¬o¶oo®/Íáó-öeûz·øû }«¼ ¾2\_¥¯ÊWã³y]>¯Ñ'úvø^¿¯Ý×áëò|û¼uª°E rp
\_ ¾>ï.ß!ïß yßß1ß ß)ßßEßß5ï¦ïÖtÏë÷¿A3æïÑs4½}M.ØwÛØóóyoÿöòXýü;P½ü§ði¸¡ÛhMÌs1ó¡,æùçamÌ1/BEÌÇbúÌLXÖ¥1KaCL^LlYS¯ÅlÙ¯ÇTÅTá¨!Ð }ÜÊi pBá1Ès÷»Ü#îãî1÷i÷Y÷÷%÷¤ûºû{Ê}ß="RÑ(&Iî11E'.pa:KÌ¢E,+ÄbµX'nDè.qØ)ö½Aq@£Iñªxq'1ç¸xN@ãF' v¬ÆZocÈaø+ø9ûÿ ×0äÆÆ'bÖƬ
¥1515ðI qwãøÛB @óQQ DÇtB¯°¤y¸uWóÑÖ=Xz´µ³ùdkO3oím>×z0DhÑ>¥]¡]n¥þjëÑæ£ÍÒ n·ä ¤ï¶h (²(| XùJùÞ ËÄêUî0ú£"Z®GÅFL§C¡+ò#®E'ÉÊb±½jÐäWéqËll´/é Ôñ³6Rß3{òpY1OË)ɦ#Û¶enë9u\uñX¥%uÇCuEļ¬?˲ta6XÜzuZ>]D½Ù7Z¶ÞnYÞzWSÑ!¤K4Y¾¢Me¯BÕ=³ÓMy{¨îC}RÝe=\˪ÖZÊ:M»³øhtý£É¤¹WÆÇ<n «ó¶Tc[ú-Uêv}XØÿht5ß´¶zåÝGÚ9Òyû©ïQïãáp¬.'p@²ÓÃÊ¥Ö#ZÇZKUpvKMpOËqÈ/Ëc°ÅqÓøk f¨ýu\ܲ#Ím6î-ÁàÒöàruÿké¶tWµì
üìZ5|üªü ¯ïHÐÆó :Býéx"ØÈÛíx[IË© ÈÓcmåÓmVÖ\_gÛ6.´Õ.µÙmÛØ=÷ó?0>Q¢µedÛ\Ǻd?¸®#DjÛ¸ßælûflÄØ~¿Ö²Zi[s«±m§"7³mkBÛnµB²ÎàÕXæc;lo%æk?LÏøZ®ÿliqs4bEÎÿÁûÀ6mnO×#r¾}dy¾§Ìóê ëÈöTÆV¨>ÖÿÐÞ½#½Ç§m1nMjÛØÝØÛVÔÒÖÝ:¯m¿fÍÈÊeÀleµ.h;ÃÌnêõ±2þu,O뢶~6O´fµ
Æ=Ã綰ñ§Îߺ¬íxH¾È²±ÜÖ¶1×\_jÿ¤ø¢ÐÚÉ\ÚvZ¡·V´Uü{ëƶ !»É2·V·]R¯;¶6´ÝÒ´1ëÊÈòÕµM¶ni»ÎÒl'oüñoÌç\_m»a¾Ä>ÿÇ=iÑ ð[~¢ò:?Qùá¸áû¤¥tò³ýü,å,?Kù7~òs/6æó üä\_ø É¿òã'$ï±!
ìDxìDø8;!Ø·Jzá`øaë(ÚzmëÍw¶ÞsÃàÛr$:©tÇBGæÖ3z"¬2HW s C 4bø(x1Ì?gà4TÔý'ð$$À³üÉ`oXWWåÊså»\%®rÕµÉUë²cØæÚîò"¥q;]»]{]ÝÞï:¸~×kÄuÜ5ØÓ®³® ®ü¹®KXʤë:^o¹¦\÷ÝÔmt'¸ö»Ü)îyîîE3Ï
µ°R±P¨/ÀrxØ»5×½Ì]ÀKQ~%¸s]Ínëû+ Ëðº7º \_ïoÐwDiIV{w;×t]r0ïï¼Ü$\Öã-nS©\_)m}²Þá»æ»é.uÏóÝa%dÙ¦ÛUã^£7ÁäÚëMqyçy`Xä¾áÍ+ÎñÜ÷.så{ ¶æz-Xã$Æ¥Þ ïFoµ·ñ¼ ®[ü¬6àÝ
a;CvSåÙ}K:êÝjñtí&O¸Þ«ü<ù6rÞÅ:®æðɲwoîöqFa'ÌÒù²{t¾ì+ñÆz[¾ë¸xBìOgÄóèpÅ+5ñ¦xcQ¼§ØË'Îȵ·¢Ú¤Oª2<é
LO·+kõmbO³g'ë;ݽ¼)iônÏ~ÏO¿gÈ3ä÷÷q=þÑsÚsJÏñíyú¥~ç¹äô\÷Üò¹Ð.px¦¾Ç¼ï;÷DZy¢?%/ò§¢êö§ûú3qLìuõç >Sè¹7FtÁçÏ÷¡\_IpMùK¼ãþr¿ÕÖÃkßîßØí~¯¿Ù¿Ó¿Û×á½êßëïöï÷ð÷»Nûü#þãþ±úeþÓÜÿåI}×uÉÖÛåö\_¼%Úi7úÒÿ%ÿ$ ?óÞÿë(Ô¶üô|^¡.BRÝ s1¤aÈÀ°C6¥c(Ä° Cʺöº\ 5l1v`Ö±¯ÃQãëÆMüÛ¯ÀJ´k1| ׫qu`WÑzf´ókð$¸qS\"þ«ýÆ6'%¶\û¸F¶!8ä4FQEÛ¡¢)|5rÚ¦¢UioGèå4.ô>ME.®JnWÕY!OLTèHYj"äT@¡«ñ6UluUCÕ Ð%ówÉm ªì®£C¦«ïUyjäݼMW=7±¾°Ý>m~ÛìÛB¥f9Þ°a¯}úܤÍÝp\_¦àö«ì\Ï¥ ôG´é¬ÛJçH`}︤/ÓqóªÌÓZ{m>pá$u[S÷±ã\_N^Íø8oÌËhùE2¾¡Áú°I\GPyµüìrÙ ¶iy#ò-·6ZEëkÐÚní°vY÷Yû0ZïAÜ1NXOYÏXÏ[/Z!þõõ¦õµÐZøª]Y,§î±ë´aÙî²Ý¬°CË&Ë&\_MÚ¬ qìùÏéÏ�óe 3V+á)£Ë(£ÏøEø±ÉØsÆ6x?ý;?ýû1ó æáãæLs&dß3¿÷ý¸77¹ñ³ãOÄ?ÿ4ÇÏ·>Bô$í¼°~�`Íb Ö#E ãfõ'ÆïeùþjýÆÃËáÅõë-¡tÍPºB\_¿QI«8¢lþ7Ëëðg¼õüïXþw¼±Ñø6$ww í´ýNãßÀ<ó"ó\_Bù?Ì×aAÜ~�ñsçÀ ñÉñÉ°ðW.ÙzØ>²èÏðG zº¢¥Y4 ÓÙt=ø¶%ÌM°ÀÇ°efcËäÿÉ%ý¯tüê@þù5ä7tÄÆ¿ðQx¨Îzrÿ ?Ã(¬iµ\_ u¸Úg+ügqUÿmxÝáy\Õ \Õ\_¿+raÃ'p
ÿ XÊ¿ôðIxC®ö§°¤»ðXïcXÿáeþ|þÄ@ P@į31ÃJþe"þebòy,äIò$|#sì�Ç ÎÑû�öÌIøEûJI �¯DUËSUZ-Ïʨòjy¢J¬å)"3ETÎ)éeQQ%é§ó|JËÒOçYÁ³? ÏêÞ(<%Zé5a.Ræb`\jéH®R^BãC¸Öp®·UƹÞ~׫kÖódÞ9«<ªÍ#¹ÖFX¢1\WE×[Q¹ÖEp½«RË
z°¯ñÌ ñI-´>ôÓ¹¬Q¤Îµ!ôÓ¹6F~:×kQ¤gã`ÿRy?é]»ÓzÅt¾×£öé|¢öé|Ú7ù[ty\óUGm÷é|oDmùé|5QÛ~:ßg£¶~rÈ|µQ[v:ß碶ít¾º¨;¯>|:ΧpJýÀE¾h|£ÈÏE¾h|&,\o¼\0ËYæ¬tV9k6v§ÃÙè;Á5'X>Úåé·è·p¯ø}1ߦßJé ô0= ::FÇ@ÏÎÁ`þù8Ä ×I:rÅñ'Hᥠ\`÷$^PÃ6Üi¸ÒpÞYRºÜçgrÃ5gÞæcÎm ÷ùÎZgªß|¬ábé\§ÝYÄxÛE ¾´¬á"Çì Ç°´#oö9»YýR©Iêq&:ór�iEÎíÎ~ÌWô:îa9ÍΡWR+ÙúJª³Ü¹Ó9îK箼çLuZ»ÇéθAçNäÞäÜë,/í[ÝÇ$-Ïä9` æ`²¬>$Õ«åcqH>9VêWÊ)n·¬3Ù yk±¢½>Ì^²Ð>,\_Ø>vyD{(räëSÊQôQìT~ 9óc«Yî,¯Ð!%ÝçÌd-élfå3\_kdù°¦8gN©øêMgú«7Ë3±ÃLýIíúéèW°ï~LæQó 5ÿÚükGZ ÝG¿´Qz Ré»ôð¼á-Ã[ÀÏÒa ³ÞuVñïÈIk¸ >>·°'ý5e³Rcùhûýi½i=S© ¨é Ó l¦- 3½iz¹iþ%Jò+ó¯À,yËøÊ�`«cOz/m\ú»FÆ}xR0oÑÁ¯{ð:û¯ù$å7¿ £ãyX®æ9WqZÚeu± Sky??Á©Ù\ò½ÕªåVnOÉÇrJøª8[ñR Ùl2zÉtRe¼$»î ·?}%ì@
ÆBlõ¦FÐñHã-ðܬÿ;ëÃË¥è£sh/óMkCÆÞ¡±ýèB®ÆÛ@É]Z¤Á]LR ÙìI,Cwù.£dTÝ:2¡÷j|ôlg¡th([°Ý¤÷jl%ã"U ürÄWc($5ø ,ßa1BLÈB5îÌaîáu¶vã{p,Ö`G±'á$Ld þ Ù hð{@#d:®i(ì<¼/k(U`àÿªðãØõc>5õ·èÑàX+Å ±ïoKyPì§ío½ÁW¸ÆôÏ«²gùõ]©fý/ìàµnq ýba:¡C²P©ÆÓ{È 0¢Á:C°ÑëBЫÁ½P&ѳB@Ø¥¡ ¢G´Ð,;öj%¡A,;AH íàè¥B¡PµlqnL >ô¸:Iè}!YCAýébzEÐi°¨?í¤Éô4½©Æ{XÐ!z^C¹cå\¥ 4ØSXN)9CKél ~ËÉ¥¹ä] ¥GQM¥eT£/ñc P]Jk(µ8Vn`°t^DýÕ¢Áfc#d)¡É\û1¤!ÜÐP(R$>eÓ\ úd æ9%$
vbH)³x ~¼WSûq lRG5ø½X×\è&sQr»"b]ìôm^sH¶ K«
«$
äiðE8Çq&Ç<éÊbÍÌd3@bC4Þ
@Æ(³R,u`Ú`z çã×MX£©ÆTS©bLõ¦z0¶¶É´Íô55Èã£Ubé7C³Ú38«ô½tg¸O ðçóyîe>ÏåóyÎòØ ºGyWö+añ÷üúu º }îpßô4%9GAur, P]OwM]oÝpQWÝp}cîhîh]Å©áSõº«\ê®®<ÇSünÅdñ/Ê´áy,|ç¯Fð+wϯäà|P>NSjX1Y/®%¸±ò\Áp½ør©mûëÕu/[º^¶Ô8\_©\`&µtºIwáú 4 k¯ë©ëzÚÖSs®8=£F»{t+÷ØòÙ]}FAOý±aÉÖ+ÏÕ÷<%I¸ÓÚE}g3ìµíÔÞEêrä¬<÷J!ÏÇ$+gË·5Ë´¼h²Ò\_гҿÒÏÛåCêÑzÁíþþýÛÖhK.°ØJêzê\H[lKE[¦b>WÝ@}¡-½¨ËVìs1Zýi½uL#[~½hÛ&i¤í/Ö÷Ù6uïÚFj[QAO³8/ÝôÔMh7^.]q©^D x õõí}J¾W ÑÒ]}cq¾V©·ÇpA¦QNÛaÛBoq}lݨ-˳Jú±;ÔHÑv-u·£
Ñìû8ôëèáûNb²¢§èÇ?¼}· }·fqßýÉ>õIîG¾÷v$æ)ó¬þzµ|g i&ùcíopwZqÞq tår2/ó´zh1p×W5¸S¸`³,®^IªÃ~\_¸@4¸vO`ÐÖ²Ïûj ;ìó\ÆA £,5ölË&اÁÍÁýÓy%ØRVãw¶ÝãÍÝìjþg~½ÀWlÅÅV%\'Ç-vGÁ
{0Vù ¿aaÕ½¢Áï·>§³Þv[ÃI½k!FVy·\Û\<
Cpï&Ô%à¯ÛÆ4% cØ@`@S Øÿü:5%ì 8÷ÍØÿÕó-2à�ÞUhpI¸c;,÷»õ´)Ñbظç="÷Y¼GÌÆñ<Á{D&ïç¶Íu}ÖuX{GvÔÞ!𶶩ôNK {?;Ø¡ë!Ç 6Åý& 5g#hã7>L± ¹e8p=nN¶3Í ió ÄU ì á©!ÖûÌA cü] ¿þ¶}w)Ú^gØkØ«Ðcï3$ ¨rúãú1 ú;XhÈVå©6Tëûô(~iÕp.ýNÃ2ýn¤CJ¡þU®Ez!]ß4Ü«ê'C½!Ѩ¯0ÄêquªÇ»þT8~J¿TS^@¿(Þõó¡ÜúTýY=ú}R'ô{ÐcÁÏÆÔ-ýTXBÝ5}·Ç«}
þþR8n\wñi¡?¦?¡ÒÊ¡ÒáÑ]Ã<½ú\ù\ôº.úPÝ8Òv2ÝC¹ê¼:ÜaéYZ¿Eß ÊªOÕUë꺩©ú2}¥JÆÝ]zÞ¡Ùú¥áºI]î¢n1RqůOÖ§-¢;!Nw\Ôå¨Ñ}=Uå<(ÜÔíî - ë8¨»¤SµnnpF×,à\¢Ó!}3¶0 ×&±Ô}ºa¾SèAüiÄu½a¿.[ e)]@E©Õ¥Ø
.¤T趩ÊZ%àÞRðcíF]®Lç¦%ä" ç;].7¬pC²VV±§Ñt©ª\Gè}- d²´pC§²Ð-tÓËgl! ©ÝájXéìto )^aPWYCPì
ô"×}HU¢E°ÐvvLGæºT¹ÓmgoÚÇÒBµàUåKiÅÞOw"5Q(ªÃ9i.½AÑP;¦ï²pN½æÒ3}-Åã4í(?+Á1@q Ð[AEÛG&i'¹´d,u=G¯«tßAwðSÃôtXrn"Ò £ÝtPUæ\²ôEäá§a¯¤§44Ð Bê¤ÖP\g"© ±ìì þ4ÒaiH.AÙyµ¨$I#ç ú?RÁõ sDz3KÌAÉ«ïÇýAÿ'½ðºùS=löÅÈg/RëÿèÿHΪòTjèC|Ʀxw$ vbÝÌ ÿ#
¤[k8H:Ρì ?ì[$D¨ ±À´FÿGê¹` êb9)LUá||
ÁÞ0Î×|&H¦L}¼õîö×i´çÊ÷¼E}Òò.;s'86ßf½¢JG5½43¶2o]åØuÁuCë³S+3×õsT.déÌØ%UëVs, .iÌê]×±òòº%U9«²S×÷¤!¦oI0§cÉì¬ÞÊ¡¬á¬ËYÃKªVeõætTfætdήY²tIÙÙKªrë\s\½ °ò<¿c]ó©`FÙúÖÁd É#ËM^Ëò1,«É´¾'kWeNVÏ»ÏÎ[Rs ñåyë\©Ü[Ù¼Þ¨Ë;鿱U~cÓóßØbM ¦0óöãíýø¿çìoÐÞMxå§HlÏHôTîÛº×ßÛô' ã æÞ¨ FÃÞýõÓ°F ü?µxÿ Óexÿ½h½ÿõhØßüKTì òMmj]4ìW4ì/¦»Qå\ ;µ/öWÓÛíw?¥½G¢Ú¯ô¿h/øÓbeÞfû¯ü>ýÎ¥Y8fãplHZ�TìD�ãÒù¥iéÒ,Å9Z¼gøvÆ®ñËfAe±¿¸)ÓG$Ŷ ò#GÓo,î)(îµd¤ÌO¿T|Ù`¥YXhñ K¥9]C»TÎß<Èiɶò´ùFKe°x¼x Ãeid!iÅó;;ç ×ù)¬<õUªWºJõJW©|)ïôë£hÊuHt´3éÈ´ãz¥Iz©5z°.R-RRR.I6I»Î§Y¦c×Æù)ÅãÌ®¬f[,mãßi ZöYÚ14bÓ4ja-ð;®å´ýë1 \`÷$¨UþæÎ=¾ªêû³oçBBnr3BH19'U@îr "RT T®R^¥)ZD¤RD\"´)b@JK 1`HQ)9ï¬ïÞlÕVìÏç|ÎoV~³fÍë={ÎNphËäZ§'×µK,Hv Æ´ä1U-«b·N/(\Hò£ÉUÓE±5W°'8´`RAÄ%ÁÚ`æ»ó/;éT)ZÖµTp àÈ-«²/3ýÉÁFÁ¥Á\ù´~18+Ó®k뢶)s[#¹bïjtóõr'\_]ûnÚÅÿ¤¤-kÚNj=+㨯-£.«õ·\WèËææèÚôRá[º6£t¼.iãÜàòàxm'98YêUìHݶ.J®MmozpnëÖ9íº&¿ü¯%rsüïô¯/n8Få¸yØÿ庶§b\_\gTQ^ǼүúĦ~5ϧãÕzf7¾\_iíkmÍÛñ˺¹»mqåLaõë3j
¼©ÛW}zù~=s¹VÛÚÔwæ3æ3º\_m6Ëm¾c¾£|æ~ó]íÃ{æ{Zó°yXEè¼ÎÈ]1Ø(e' Aîm¢:ò5ô·(¯Ldë·ø¾WñôÿñÝñ ö¾Îæ·ñíZýúç´ÿ翱û5§?¾m\_ئv\¹ ©¿oQÏôÄ=Ç&éÞÓÔtôÄ1Ißn~¬çXýIMõ')ºyRÒ ùðaR´û¹ÚZyÅ®X½VõZÓk}¯5b!~üþæIî·çùöÚ¤c×ôÚ¢®íÌ¡R½ßXQÚÕ|#»´ë~»ì~{õº\kîÿ\_S"t¿ïq$7²ÇÕ=D´ÉMíq0²G¥0úsB¸f «õß¹Í"zTÆM×|¥|r#s#%¦YDÜt÷#ñÿ°Öã«mѽb£çáÇôçXkX7½ÅÅf»ziqQ¬ÄMïy¶ç¹zÊi÷tÀwSêõR É9µ9¯èomC%rC%ß1\_Ø¿ö:%ϧt^
ª¡RÊkþF6Sä~åS.rBÑáIÓµîK#ÒoQ|×øõ9ÃrfįO¸KË5ñs³wÅçÇäÆ7oÔãpüÜ9%Z£$~[sãÇmÏ×£|²weïÉÉÍ)Ûî~®¶¿4§¦Ç©/lmå ®Ï×v»j[g{tILÐvt]zË©päÌSW4|ùÇÁ·({G\rܬؽöKÎ^Ñ´VeóÙdïÝs¶ÙòìqÛ²Kã¶EîÒ:Û$®N\½A>YYZï¸mÁ\_e-{Æ[±{ÅX¹b#®8.¹ùFm{Gö¼¸õÍó²7¸¶{DäºVWÿwJëKä[=¨YYÖêcÅoV&a÷êØ®±]³Å-Ëå$g}Û5k^Öþî«|/ç´ÏUÓ½6{°|ºW»:¾Æý\m-ëÐ[qú¯¬OÅÊYËéû³Øꬪ¸nÙbÎÅÝÓ)qk¥{í5öMiy àtúVߢîºï\_Öý¡fë»O߯ñB`Óóݧ]wGLfë3\_Y³µû´Ì12wÄlÙÚôHL¯ +åÓô|ÓóMú5é×}æø\m-fõ?l5é×l½Ø+WltÔ}§N9 û
ñVÆ&%¬é°2+7[eò¬¬»Æ§ßázeåñVƪýÔ[¯®<'¸ýê/ç5odÓóW>M;yaÌ5¯h¾Ñ;ºò[ïÑ;¿c^.WvÁÍ ÿoQÙéG§ijT#¥W6ÝdöÎeDÉði~öÕ^/ß«¥W^±%kå̽2ëϱ&³c/Æ^l~3ÖÊ7ó=yö;íÀ»¨~0ïÿöÇÛÿ¶åT¶÷N4('ìN
ÓEÐ8Ô»¥Î -§Èî¼,ïÜÏ;íäwW"«:gr|òV
81å+T¾gxæ »}cÁe =>y#ÅEÿl0¾ YÎæñqþÈÇ7å¼sÓØT¿uø1+û±<òÉiíO¤\ºrµ;
¬Äsk²ON7ò½-²øiìtuàçú
Ü$ñÐê(hn÷çi¬4O6KÎi
. îtùÐJt´M;]ä@Rh2|Ëü°Oó«%Ötu¤f-õpJÚHY /±æAgÔ³Ö¬ÓÖªEÇ·ÌëÖäIÏ\ùýóêæ>ma¶qR㳬ãöoH¾ÈSÀ p»ðæI±à1ó,Á{
±¢yÒü¤^[ûXFÎÜSN0Öº$¿×3/p^m¤äÇuÏk¹ø+òç"ä(ýð¼yô+[ß\_ÉÆFùÕ!'Ìë¬röÛÜm&Ië¢kç PN!Õ o,²¶4Óü!ù²ÙXJmÈñÆb³¥Qd)©ÕE¹þLZéÉñf
[Am¡«¥KaZz dÍÙcéöµÞ²j\+Ö&ÎÝ:óxOé£NÝ·ÊÉH»¯£G¢OÙW8z9®ëèYß÷ºÈÎP'\_ë'Ù÷i¹½ÝW˶äò¤ÓOË+ÑY,²3ËÑ~úû9ZÇèèºuV9´ÎLѱ~aþVcëvw:÷kå8wYÃ4f[z5ß¹Zã\_jùnçAmaüU3 (õxKêäÖ4γ©f3k©æ\_°¤¯>c=)yYk¼ÍZ.gz%3Ú»ÛÚ¬ù6ÅÖë÷7Xí´
ÏÀZA« ~~Ïgé^du4¦[¿ÆC±ù¤Ô³ÝÐZ¬uZ¯jÁÌKÑ2ìö´î?ÎTgD«µt¤EÆغ·Ø;ä$´}JNÚêú¯±½s'í2¹SJÇi[bͶèOÐÖ^uiü,ÇÚÄ®Ö÷ßFXIÚÓÓͦ$[Sä$ÌÀcÍÑ8ÑéI+<,¼%ççYU[Çà\_²;¿Ó§[SÑJ+Èìt§×si?ÈoØíÓ´Å«UN[lÑø¾õßð^³áeàéÎIÈÏ3¹±Müß\_ %¼:ÎRôïF.CÎdþ/b4qÁ¥zkî9ÓLÃæÛ2ZO¢ÖWk=òSÎsÖ¯Áå=ÖúPPNuºþ×SòTÙZÍ9ó`7´ºö
U;{þØ.ç!gu§ø>Óø?MÊîkéçÈ.úÃË|õÄZ÷ïÆNV£.uO0ÛȬe¶¡=Ô±f9³Ùoa. ÝTrÞÞT<ûÚÉÕʻ޽Äõ«È¯¹} ßIYt=2u|îIy» \åz(Ì^ë±rb\Ë-~Û$©gmÎ+S3 ]ÙÕqOõÚ ðzºä/üzl9\àÿ169§ípFÁ7ù1\_@;Ë©.óоwÁÖà8®ûõ] ÃßáÚíÃo ¿ÕX$v¨W÷I ¦Æ ½w`èy5ü)± £e×ë¢ðz×xéôO½fà=!Îã{ÌéÕ¶ôö#º¯îöë9Ð$h§ æ~m
N tRÍ
ß<þYôí÷`'öîXF^üSbß1a:`ÓfþÁ²sü¤r˲þ|", CîL¬ 3f+òbp 9¶ÿÌEô#À<øð§ÁûaF!ï??©a;<¨ ¾wuK]ý4ø¾ðóÁ `½?Y"A{Õ uüèTÁ<ócpi©[û�å]H¾nîào\_Ó ¦XAÚÑà<}çà0èØÈáÒß»¤¿)|³by¬ÉÍb{¨Gº)#ÚÙ-«hg }\hn´ ú]|´Y{>P÷ízu ý¹¾]BoL!m.©v O Õ³XC¶ßs-£ÿ8:Àt'häÈ?%ö ¦ýðÇaàórAvn¿Tny7ÀïS
aȵaÃlE^ ®!Çvð¿¹~?þ4x?Ì(äàÇàg `'ãuCÓ÷.±n©Ë°ß~>8¤¥¬÷ÃàGnÛIZ ¹%Ì'%Ì6%Ì<KD3¶«7@ä�ìOÄN)L[K¢Añ?ócp¹Óöêg!~ºÞv¿~L+>`iG#ÿÞAéí0¤rþ>¦íÊ[Y9ðûN³XÐÙ-ëg }\hn´ ú]|´Y½>ÐI!6~òøgÑG¶ßyØO`ºc!y%òO}ÆéM?üqø3kÈÎ ð7Ê-Ëøsð=`°0 ¹3±6Ì`ÈÁ5äØþW0Ñ�óà'ï
¼üü ¤í òdü¡6|húÞ%Ö-uöÓàûÏg´õ>rüf`íUo`Öñ'¢S SóÌÁ9¤¥níw!ùº¹w¿~L+>`iGó@ô?OÀ c»ò@A÷´s5ZýÁ£¬E"\_bý¹ FÛ»`ÖÁ°º°eMaøRÐ\_KÚya½6³I¢WóËc·z:¬R¬¤íÜ àr3ÍÆ\b#Ý}vfÆbß
6ÒdÝbõgÍÜ 5¹QÜ\|Øá7üƱcÖ{Yd\_.|)øCôû6LX¶ÿ3¶ Ú'B3Åsð ù¸;)Ôs:Ö:á³#ë1cèCIu¯¨7½;62º×1·VÑ!í�¼\äb-^¹m×?HµùÄÆÎa®'mÖRüÕ½÷!v~.¤5ï¦ÔoÏót,óQ)ùÚéX8L÷áóIvèç&ùZ3å·ö6¿÷6S%.ûZʾz[+kKäà[ÜÝ\@îÏÊsXåHø¹ÿ ^¬à®m2úñPôÞ"`�æEö¬^$ßÃàbó} ÞþN,[&ÞïvÇïv¬íB>&öÍDê3¤WÙÒv» ïWò6Ok[ÐÏC±y ObÀÛÑ,¡¶@÷îò{øã'í`ì/7êTh90ãÀ�\_ÃóGi¯hwwAÝçÕ¤Fǽ\3DSҽįü¯#IÅ f¼ÍÑXëæ]¹D> ]o±Ó
~ÈLn>ÓØ{ÑÏ#÷4ryØ //jù>ðNð7¤]Ftv`aø<£>WÑO¦]°LíYÀWc J½ßzïñfë¹@9£¦\L.ïa^á\_OÏ´/Ïí0YØÀÍãõí&ãÎw7¿é\ÌÅôè Ãñ\ o'û0}ã^ä² aÓúJ>ï¶;^ÑßÌü\(;æãÔõfùAw'yúmiÁÝ踣ò5ÄÎ÷ÚZ|¸ØMhÞI©½Ð,C§+rx/úí«©Ö-º§I:?ÝñV Uëëì¬;Yë;«Ù+à÷rG¯tîîÓÙ%`O@4U'sÇ-÷úÎz
ìÛTzE&ò%túÈøxs:X n#ä p 54a#Çàø'H[Ésç¢Ìüàyîz¦l6Fn ¦ÀHbX(g¹O1äIe"ÐÉD>|Áÿý7JA®´bÀ¥øvØ O3 ó0]ýѦRö[4Þ%#Gs6£ðö.J±YùÚh|Öõ &Ø Ù}²R3qöͱÈe1±#È·£ë~~?#W !ÿ y7^ý 9'ö'M4§SÞL®È϶PÞnaE§6æÔÃjæøo/Àó®¾ïk.»£æßïeþ§t¯aú;aI¬ÙØ×¼äirÞÜIUC^)´
bÿ >DÝð0YÈS\_Ȧ)å\êê×ä[K)vÒsÞ¦çD°ï×Tà³ìÞ oý0[Ùºn±xV0½WblJ--h t-ø%w:)çòÖz}È]Øy ~�ãÝTÎÿ6?Åò`tÆÈ1ó:zB
XM|ðÆÓàà@v&kÑo\&RÃ6:XÊY\36±©çl½f3×6¹ûånyIþ þ1JÑÁ+K¡Æã¤UøyÒó¶'h$¯©g#]ú±MûMy{ã´:ØåJ-ûÖZǯcÇ9ûµ|±Éfjò,5ö ù¾@ÏtÇ/ýü�=Áôä�²ô¢Fßw\ÐӪܹ¾½Áëè±?gæ¡çá?¤]v»3ì^CÀÆàßÈýFéÕz¼Hª-ôÞ8ÙôŠþÁH䱦GeñË'%î¸g²Æ,°ûñH»ý<ñÑyÝÏØdìøk#ë29úñÊb¼ÿñø·ãüEÐw@jÒ×Q/èßsRd'Jоñõ¬xh5Ãf\òÈ·7°@kËÑ:ÕYA=¤Þ²e,«Ûðä0I;¾>Íåî¨a÷ÉÚ OÆŵÀ¥þkeGÝL~b<Æ<ßÛ~+=ªÐ'ºx£@ôÓSèÿÀþk´æyç>-¿C^ ;è«É}ôXI{+©6µð.ÙW·kÝüÜ ® :-Àú9.Lt}!\¹ªá!@>Aìp® ¿ò$bw¹6eWßòSÛ9"Û>,$ÁWAÕ× ÑoKì<Úh±ýÁyàZAs;¸OPÏóÑÈÎEt.K:ò&äir¥p69à%Þ´?;lär ¾RÏ}NíàAsý eÖãð¿~$h39ðä'0Sy¨ì ÚcÔØnJÔØÒýÇnI^ÍØSmüò8ü¼|?ñÅj¦'± ±¹þ9Ôa.õYLïDÓæ J÷7ÆÏìÎÎaÙ}¢Æ¡ÓÛþ\Ç'Õ£hÎk5´SdgØæ ètÞÌ°\_c¯ e-=\_};ÃUwÚh$ëU)»ó¦c #{}¥Núë\W4{£
¾Öñe¢yRòÒúÉLkìñÄsª0¾!Ê×Xdç¨Ø7¶ËÅÜ3Ë¡Îeµ[½íOdɵ)Ï}ûDÇɦªy~ýï"}Àÿ'iÁ!÷GòüÔ¼$etÎa?MÒú!ë ÝVèeµ·Võ9(õóô çSvkt¾Î»`® ¹QßýÖ@ñM{(òÃvoA]4oÍ aô:í&Ù?}0ÖEð±æDX§uìß©·d©]3R?wv#v>7òÄ|µuÆÓn¬øi®¦Ôö8l'±¯¤²ÆKé4A].K3»>ó¬¹Rº¢?KìÛÝäjk½ÉþÆdx²ù&P`6õÿSziõvʾNó¦Ø´ç;KjÃ)
BÉËI§MÓéÛ}ÜÖè±KéçoÐÏïG~Gd»Þ¾¹"æ ,ÜàêÐÏ ³¾%;ÉÍ`®ÃÚ9rÍø}TÐ,`qu»;rNI÷ YkõÁ|UoÄú (æØDFt©` Y:ÜõE3[§M ÔÉ)s}èÕ0 å ²Îw £CVS»Ç¸¦TÈÚÕyÙøgÔü£ÎëÓ×IÕ«ÈYûmd[aKYÞ6uæ¹s»X³ úÌ(zE)=ê$=ía[÷çvAº\_]dí7þ Wó ü;kÈð§Þ'h!³4»^ÓÁÚðii/äEàçt¸ÌÝÙ·°o64];ß43Å �¾´5`¼ÈçswßÍ8Î\Ì ñrýBÏXÎd¿t§»ÓNȽwO`ùYçÖQÏ7Áðý¸ë\_Ä>Õt¬¢yØ'ðáëÜã[¸ë7èT¾:3ýD·tXÎÄ«ð7 "ð³ò äØ ùgh@ÓÀÎ@¼ºØläh÷ò¬£òйùy,ì#¯ 9`@ß=¡HìlFg L1±xK^eàÓà ÛgÚÖaJmaÓ¸µÐY æÁ$í�Ô×ñgø7·Ñlnvs!Õqp'üçR£ÙÝsNA³ }æ0ü6vßä}à7%ÕýGñßìqÈ×á2±n{E~:y|½©U` í»Áë!"WÁk=MIµÁëcH%Ès;» Íñ¼yònOÌHY NÆ~>v>)ür,ÇyLYÊ5tȦ·?,: 8)qTø¼Ìóv-#�¹0.`λãy'Ťm ÓFìþƦ¹ù oÄ:¯S¢³þ\a|¿¦uþÞNïm&|`iic.©rñy¹Û«Áñ´ûüq÷¨w;ÀX4/×3§fÞ[OIS Ù¥ÝÞôBÊØ;C){$}õ ¹ÀJ° <¦`á¤} ||é¥æXÛ?LÞ5i£ïM·ûi|'ª¯±[øg ®BuѯcÕæä¾'Ð Êy¤=þ NÌî�»JrÚs>v'±©ÌmðMcx 'ûj4áoo`X\_ûÔKòÌKçzS©ËÀ2vSy0CÁuà6ð<' qöf(Þ¶ ,aÎ37²º(W=}w-¼}Vãë ®ùÜÏÙªý²GªË¢j!W7Åh{0Ú}û¥£9SNoÏ× EFk8mõ¹-+[ÛåÕõ`oéQrÓgËy§¾÷åªnõK Ï�·úÁþð=³³ÐYÎBäÅÈr2LÓ üDз´aòÁ4©{¿¤uþîÅÎêj)òp ?iߧH
W²Ô±·êr=a? þ\Ðù!ø¢ XÐ ³ùåNÍXçüÍuh®ã^{>äí«}ò\_0Où ?eOCË)nõ q«ów]º>GPN¬±\+v´¼\N\_PꤣßdTÀýþãì¤H+ky ¸GPÞÝ«ù÷À3¤úÇÚãÁ"4%ßû"µÑNdßÊ.ïy:ç5»æ[½Mbeiû¿¥HUç"§ßëH¼�9Þ¹WÐ~¹§Ì$¬´ÝQY}NJÓ»ê|0óÁàë´øÛô(=úY¶ÙÞ&hFÉå»GÉ^qÄʪÞØÎr¦"OE¾ù~ä§F~ùQìhûæ-ä|÷:0K@ÞhnÇÎ:±oErà<Ï!-§2À±x¼öDë8¹a¹'^öy5oSFë�gî@æ?"ØÈ(¥~ø¿ZýA^=è5kZPøßzÞ{h)Áózõ±àµ¦¬%¡Y&¨WÏ~Ê+ÖÁl#/® uä ÖÞ+À¹øãCÞèùÆÚFP§OÿO!¨{Î|ÆÅ|ìÌyM½íÇÏF0ÔùLjVsò½ÿÈáÐBô¢gÏB¬ÑR!N[]æTÆeÎ\æÍeNÁ]¦}/Óõ8¤MͶRÝî{¨=òBjC×0µèó!Z0äýoKFws~é Tñr¡FÞWZ¢\î.=AU=¢Ô°KOdD«Deç÷줺õíóÃ$Õÿö~yIjÒÀþµoá°¼ÇNÕIÝ j°JWw«6ÄÔW>ÕTcgÕUõR·¨!\CUß÷ü\FEªT=Cߤ ÕP©Æ©¶^\@Ū(¦º©<}½¼KuWãU;/®S2² nV}Ô0¥&è&cÈõ¦¥WT¶ê«ú«ÿá=¡%\_ÄÖWIzÆë¨Û&GõS·«b5R×ÄÄ/-ÕLÏ
Ãå7µ}TÊþ½´wnÚVº.®SßS=Ôjú!ÿ{{q\Y5W)ª±ÊÕý½H Tw¨ÑÚÚ=^lkÝ'¯WMTOU nSÔjÎs²2²ËÔæp ¸¬�÷^2Í< OgÁs#GNl^´L0À`Wp8 1ªdÜÝÖ\pø¸ \ n·/;À=à¾QîhU` X7fܤáÖ§à%AÛ`1¥ÃGÚÑ` °=ØÌ7iÜ4;ìöÁ¡ã¦ÞSbÇÁéà,íÚp{.¸\®·LúÑDûpø¸¬,¹gd}<VµàyðÄÑ£Æ9 ô`c0N)uZmÀ`\Øí8Y`Ø�,8 N§³Ju¹àp)¸\®:qädg=X¾ î÷NæU` X7Mrù ú|`PNëÊÿlW ßB2ôØÿzÿCÐcùÛHV®51z,uÒ³Jª!ºèY§«Cºéù ]Ï/z®ÈÒ£?GÒ\=Þz})åWI¦ÑÿQØFæ®/BC]÷ìØßõ¿-=[´TIßB2TÔ7bÄ7 g»ïé+údo¥òµÈÙAÅÊF±&Ó× A·öcÁÈoĤo@SÏÈÿÐPm¿~#¦PÓÍôÜßü[Hò¿Ã¾M}½h÷oÃAªL½¨^Qj¯ªTGõJ³ÖH6R®Fgô1QF1ÍXj¬4Ö-ÆËÆNã-c¿qØ8aÔç¦iFÍ3Ùìhv13Í^f¡Ùß¼Ãkn0ËÍmæs¹Ïº5õíÔSi\-&SZ~Úдi¤¥¹ëh£Ë ×zÕ^èÕm#nx[·F× ÷ï½Ðó\/Ç kÝ0¬®p{HþzÒÇåoÊß\_\_[(hQÐ ¨ ¤`vÁ²=G Î#m9Áà´à²àºàËÁ½n®·ltݲ¿£o©.T
1
ó .\\S¸poá©K½£{wìÝ«wqïɽêýºÏ"7ußDþ®ßwXß}õ]ßw{ßý}«ûúE÷ëØ/W¯¬u=Ü:Àí·Ê[èt[׺i:ya//ä
^ýÜ6ÖÕ¿mþ¶Mn½Þ¶Ç-ÿmU^Xã
u^ºOÝ°´¾/|Ñ ouÕ;êÛcó&y¡^;÷ÕôóÆóÊôgú3y~c{á·gòGìGèÐý-P(VM3³TLàÁÀøIà'\10?°P5, ¬VIÿ S××ïPÿ:V?µ~ªêZÿLý3êÆÈ×#w©n\=ªQTS«ä]ãõõ lvµòíb=sfêëu¡^·ß¡×íãU©ÉÿiX¦Vê{Þ ªüK³a:«Î«KmDú\_Tÿ9ÿo ËüÛ7û\_",÷¿¬Ãç´ô;çüÛ Ëü¿'Üì
°Üÿª®çü;ô\_eZû5çü; Ëü¯nöï",÷¿¡µËüú¯ÍZ{7ásþ=eþ7 7ûß",÷ÿAkoö¿ÿ\×Ú$|ο°Ìÿáfÿ>rÿ´vù?ÕÈX}Ç0CÍþjd?%ßä׫^ͼçÕL¥W3u>ü¼úyß«Ã^½üÙ«#^üÅ«£^|àÕÈ1¯FS#'¼9éÕHW#õjäW#§©j¯F>ôj¤Æ«3^õjä£S#ÿ½3»©jÿÿµö:kßçÜgïµÍIdL¦$$!¤$ó¯ç÷Ï×~Ýë,×{ïµ÷¹öº>kØûì=&Ðô¿µùÉ÷ÈUß#×|üì{äºïðÈMß#¿ø5æïÛ¾gîøù5æ7ß?¿ûþùÃ÷˾\_îúù+æ0Å<1GÒx$y$¬b Û1b c Gþl¦tN°G.ÓuúMX"9óH8óHØy$ìÆ<Ö1=ãpªG©c §y$6æpºGéGb ?óH8c¬Æy&)æðC¦Æ3ÇüÎûçaß?Y}¿ªr¶!«zêIýãñ=}ßÓø¾ïéd|O§{:ßÓ{{r.s®R¸<Û~ðs#+È=ǶýÜ?;¢³ñ#:?¢óñ#ú.~DGt1~DßÇèRü~ÑÕø]ÑÏñ#ØEAîôg²ÌÝk9x\p[m½éO^çaf>î²\¯ëR{®Ñi.kÜ÷¼¦MLCuÛEzÚ·T¥6,ÜÛsëq®Ï{=aí\°¼\_».ÖVxJ\Fgæ67±ÝLKb\_°ëئ¶NØÆìÁºi·yìmǺnÖ´îÄölödÝ2GgÝ@)5ÍX?ß1ª§UIö
y"µ´?²ûYItjd²L63ÞÒÇÉò~ö̱XÖëÇØ6Þ\_ÞÝTmBûÍÑéÑé$¼?½?Ñ 2sÉB`ÿ\à}Ån¶L°IqU V§iÍl;ßÜÁ!ƶýóëþ¾ Öè%
ííeö óÊÄ2k s °B ̼«±
ebz¢eä?+P¦-2Ó½ë~\Ç® s¥þDbü?³¯nM,oX&-&sWÅø@x1£±þ2ûc1w¼t iÆiµeÖÇû¸+Êä®;µåÿ Y ûÄ¿OÆØ9zl¼uÝúÕ\¥àúP²ÝÏþOü½ÿÔÜÃ%t7ä§"ÇeÆüRÃqÍ)n3[Lù¯ìI¿«òEèvV;»óÉ´[^ÙdY@
e1ÙGàýd9P¦È¡<-?äd9]Îså|/Ëåj¹Nn[å¹[îåqyR¸¬+ò'\_Wy¼?Ô3ªzVWÏ©JêyõzI½ª^WuTÕDµPmTÕYuSïª÷UoõúP}¤¨Õ`5D S#Ô(õ«Æ©ÏÔçj¦f©yjZ¦V¨ÕjÚ 6«mj§Ú«ö«ê:¡N«óê{uE]S7ÕõMvÈÛÊNc§³3Úìíö#ö£v^;¿ýý¸ý]Ä~Ò~Ú.i±µËÛõìFv3»stqtit¹c9¶ìh'ÁÉä<ìäpr;y|N§SÔ)îrÊ:ÊNUçe§SË©ë4t:-ÖN[ý¾þ@¤?Öõ0=BÒcõgz¢§éz§éeú+ýµÞ£è#úoª7Óë-ôz+½µÞo³·ÍÛáíäsµK¥é¼eYYw%óËü|6QH>! %O-{ËÞ$?PX~(?¤üH~DÉr@Q9XfÕ"+Gp=ÐrE+ÇR\ùüRËIr¥Óä4J+gÉYNÎs(½'çQ¹@. ä"¹2Êer=(WÉUI~!¿ äòKÊ,·È-EnÛéa¹KrÜGÙä!y²ËcòåßÊo)§<#Ï°Æ'¿£Gä%yrËäô¨üQþHyäUyòÊåÏOåQy(¿\ PUJ¢ÇTiU ª²ª,=®Ê©rTHUPè UQU¤ª²ªLETU
ªjª=©ª«êT̼åRµUm\®ê«úô´j¬S Õ\5§ª5 K©öª==£:©NTZuU]©ê®ºSYÕSõ¤gU/ÕÊ©>ªW}U\_ª ú©~ôê¯úSE5P ¤JjDUJ¡çÕP5ª¨áj8½ FªTUV£éE5F¡jêSõ)½¤Æ«ñô² &Ð+j¢HÕÕT5^U3ÕLª¡æª¹ôZ¨RMµT-¥×Õrµjq ¬¡7Ôzµê¨MjÕU\_©¯¨úZ}MõÕµ¨}j5TÔjÄr«oÔ7ÔDR§¨©:§ÎQ3uQ]¤æêõµPWÕUj©n¨ÔJÝV·©µú]ýNmÔ\_ê/jkK[R;;ÉN¢övÔRÛ³=êh§¶SÓvZ;-u²° ÎööÔÅÎbg¡·ìvêjç²sQ7;·ÛÜsgç¡wì|v>ên° лvA» õ° Ù
è=»°]zÚEí¢ô¾]Ü.N½ìv êm¶KS»¬]>°ËÙ娯]×®KÚ íÔÏnj7¥ìNv'ê]]D¢K¢Kh`tEt}ìGÐ G9;'B)ë¸4ÄIí¤¦¡Nz'= st¤áN' p²;Ùi¤óóru¥ÑN^'/}äwòÓçqçqëqЧÎSÎS4Î)é¤ñN§ }æwÊÓ§S>w^p^ ÎKÎK4ÉyÕy&;¯;¯Ó§S¦: 4Íi4¡éN §ÍpZ9h¦ÓÆiC³tOÝfë>ºÍÑýt?«ê4OÒh¾ªÒ=\§
z¤Iô=ëñz<-ÑõDZª'ëÉ´LOÕSi¹®§Ó =SϤz®K«ôB½Vë¥z)Ñ[õVZ«wèô
ÞwÓ:½\_ï§õú°>Lôq}¾ô¦xSh£7ÃA¼9ÞÚì-ðÐo·¶z+¼ô·Æ[CÛ¼õÞzÚîmò6Ñï+ï+úÚÛîm§Þ×Þ×´Û"sÈ|²,\oÊAr¸#ÇËrª)ÊrÜ 7Ëmr§Ü+Ê£ò<-ÏËï¹e¹¢òÊ\¯Ê/?V/ªWÔkê UO5RÍT+ÕN½©ÞRï¨÷Ô5CÍQ ÔU\¿Z§6ªjÚ-òçau\TgÕuYý¤®«[ê7u׶lÛN¶µü^½hg9ìÌv[»ýÌn7°Ø-¢+v'ÎÉèdv²9¹Na§S)ís\:UjNu§¦SÛ©ï4v;íuoý¡ èOô8ý9Ò9z^¢WèízÞ§écú[oº7Ûï-ö{«½uÞFo+kþ`¨=AítÞÎKè|z® ä64< G áÉÐð(´ÚV»Ðj ö Õ© Õ©¡Õi Õi¡Õé Õé¡Õ Õ@«3B«VgV?Î Î~\ ü87øQ(p(p^(p>(p~(p(ðcÐÆÐÆÇ¡
O@ C@B6>m,m|ÚXÚXÚX Úø ´±4´± ´±,´ñYhc9hcyhchãsÐÆÐÆJÐÆÊÐÆç¡U /@«B\_6V6¾m|û\_Yé¨\uèÛ«Ð·Ð´× i5¡i¯CÁjAÁÞÕÕÕ
ÕÕ555^5^5^5
^5^5^µ^µ^µ^µ^µ^µ
^µ^µ^u^uF½ êê êuz ÔÔ ô6Tè¨Pw¨Ð»P¡P¡÷ B=¡BïC
zA
zC
ú@
> õ
}êú\Ô\4�\4\ô1ThTh0kN."³Ë¼òqYDÞËaò9N~.§Èr\!×Èõ[ä×r< Èoä)yN^4µ5çkN>ÖªêeUCÕRuUCÕTµTmUGÕE½z¨Éjºæ«Å\Wª|ê õ¥Ú¢¶«]ò�RÇÔ·êúN]R?ªÕ/êWõ§-leGlW^TUíô¬4ÙmìbªçêÛíæêLt3¨ã9i¬NNç1ç çIçiççYç9çyçEçç5ç §ÓÈiæ´Ó½t\_Ý\_§èÑúS=él=\_/ÖËõ6½SïÕõQ}æÍòæy¼eÞ\ï ïKo «NÊÿªÎÿªNLuЪíyÚSÚSÚSÚSÚSÚÓ�ÚÓÚÓÚÓÚÓÚÓÚÓ ÚÓÚÓÚÓÚÓ ÚÓÚÓÚÓÚÓÚÓÚÓÚÓÚó&´§´§3´§ ´ç-hOWhO7hÏÛÐw =Ý¡=ïB{z@{Þöôö¼íéíé íéíù�ÚÓÚó!´§´ç#hOhÏ�hÏ@hÏÇÐAÐÁÐóNáLlæb3IÞ ï¦U^ö~Å\Ì&8ÖÌÌÄÌRkþYµËò2%á¬
íVv+sN&M9¨ m¥½tÎÓOtGX|2l"(JÉ2óy¨ ¥¸;¬ªü
Ëï#osÚWòy|ÿ;§)v²Ô3öÛ±»sú¬ÝÓòz%YÍÖòòý¼mîXæT³¹°v\^#\_7ä¯iïó9f{ìÛ[±ß¸³wM æúPtouC¦° b+Ý(¶vìS^ÆVóx+3%¢|TgØÅï= ÅÏYl¶¨"óRÌ2I\Û³¹üÔ±YOF¦a5¯$\_ \¢VÅUIJe?g¿@éìjö«ô]Ó~²ÛoØu(gtft!åþýwkºõèIý¥ÞBÏèú$óvyû©¼wÌ»Hϧ©Õq%á8ËqZª¹n]×?$ÿx3ûµ,vÜp#=iüHOÁáÛÿä·ð¨&·¹^Üÿºr¾õáÜ@ÊùÑþdlMóγ²TóÕ¹²Ôs¾ÿ-+ãÛ¬Azß©¼vÿÛFw|ôfü;ÿýt Ò³ÿa/¤Ã÷ïJ=©/ÿ ä¼¹ÚÕ&ÐTíç²Õ<Õ{ïtþù¯B/ó\_MÎ?VñKåz°µïçÿé¯'ÿ¿¼Ït[êDo³?ÞfO ÆÓäÿÍdÙmWYþ3õ¥>5
îÿ¯+óªà[ CºÔÿ÷Ïà/ÌG:%A.øÞûÏúEàNý\tï¾ÈTþ÷y3ÖLêy>+È°5ªúVsQAÙ-ÑIÑÉDÑ©Ñ©ÜÒ\Ä\úý¹p·ØûÕûÅ»åÝöîx¿yܲÐÊZÜ y$÷Bj}-kSìÚXxÖì§)=®3ë,úaUgÓÙuSçÒèÜúQGçÕùt~]@?¦ êÇu!ý.¬è¢úI]L?¥ë§u ]RÒÏèÒº.«Õåty]A?§+êJº²~^WÁÕ¨Ö|Ðýþh\Sv÷OmiO§ÕétzA? ÔÝßÝ?Ü»î\_´ÐR´Ò¶NÒaÑÉ:ªíjSéÔ:Τµ¿ü1vàÏ6çµ~~÷{Ð=äq¿qO¸'ÝSîy÷û{ÅýŽåîv¹«Ý5î:w½»ÁÝìnu·¸\_¹ÛÜíî÷kw§»ÇÝëîw»GÝcîq÷[÷´{ÑýÞ½ä^vtr¯»wÜ\_ÝßÜkîU÷g÷¦{Îý½S{YæÞß\ô ks-ÿ!ÑaÜ6Í¥l]ÍKî}¤¼tt§¹'v£ö"/%é/¥è^¡[¼¦;ô+¡ßyyþä¥÷Ù,\/B"DwDè9ª(¢"J
+\z^x£\"µHM/´"-UéEzzQ< jAñ ½$ÑË"ÈB¯¬"+UÙEvzUä9©xD` @ÄV¡&6MÆããõOD«õ·x5Ñ«±øÖæ]Ü!QPáSYiɶÒ[ù)Ùd ¢ì¹Éÿê7í?-'ñ[³ßkê×u-ý®ë躺í,$xƺÈjjîG6O¼6¿½oÖ¥8õÍVj Tó-¥¸Ôþ°åa[°;ÒDßfÊ\̨5ØÅüõ Eð6Ñd¼M4·:x¨·j¼M4-Þ&oM½½Mxµ5þ{o#ô4#MïÕZ+ÙC¿½ÝÍÿ0²¦ór
÷a~y'Á"µ
§ñßé¸=duååµùo['÷ërÜfYñVÃüw8aÍÕügæmÈ\°nfË<µ%ç.ÇÞjå¯mÁýhk¿uÔ:y}qÅogg²ÚÚ`Þçߢ6ÿ=ÌöüÝj'l±Ú2ÏZ±ô·RL>¾E/xaÁ[[¶5m±`\_K«UEÜM°ÆýTûÚ
û¬hÞ¬(㮬ûÖBV\Ü/e»¢P=£¸çHqA\ïº+Ìï\_«0ÙÉ\x4a¬æn\óîøùãþÓLÜõe~%-Á>{hqOÀS¾&¾Ü£dÎÞµûçúQ?ÑVq¥yÍÑÒVT
yÆyòϺ8Q<º¬.}cw»¥îmUÛ²ÌÜ.ß7{«îo[Q'§´÷Y,¸î[ã¬qäF
FÆûf½pÅpEJÅ1ÑR£îBݱSØ»ì]¦"xÓlQÄ¢=ÒþÕ·±ZÖÞó½HL¶&Å·ÍÛ¦Ü\3ÏR!ë<SÒîbw¡2(,J{¥CiUJ$¬iHÇ`|}ù HWÅÚ(gCÌ
î©bÕe"×4ï¼Ì{b39Ç¡èFóölä6Ås3OÎI±B a< mcðÞV÷óý|¦FñãsÓ
ºs¸ ¢á4Æ«1÷ô¯%ó{ùÝ<®9Σ ¸Cù×Þ÷jRq¯
Ï=÷B ¢¢(Ëõ§\×¢Z¢¾h\Z¢«è!úþ"EäñÄD/̵óñ¶R8M±p:Ų·êZìá·Fp:½bY#Yq-k9SÖhÖ]Õw ÎàXã>5kÅ5Ȳßd
+²ªZ\jsZ;RÓ:ºÖÔã´Ô¶~¤§æÝÆV¤¡©EF¬¬æmÑM ÅM9miÆi³HsNGZpÚ"iá;Ò\ÒÓÖ6¶´å´m¤§íXÁHûHNÍHG>ûkz'NÍ[HgóNêHsäHWN»r·"ݸX\÷áô®õVô6×zëþUNMÝ·<ÌtqM¼ÿkØ/Øä¿-2ѽyn'z½:ȤÖ%ÔÕ
$ô|Ö4¬ ECÑÏð|îíX¬BV«©ÕÏÒbk§u),\«É²/˽òr(z8TÙJÎ\,ùåäÖÉ}'$¯LÞ|%jG³EKDkDÛGûG'G×FG¯9ÉN.§´SËéä r¦;ãÎMW»yÜrn]·+fóØä¤{GdxØPw×#ÙóaÒÎÔðð ÔóKH¿�2dH 24@ ÃA #@FÈH #dÈ(Q2d4Èè�ùäOd È12d,ÈØ�2d\L21@&L A&L) S@¦ÈT© SdÈ4i2d:Èô�2dFÌ 23@fÌ ³AfÌ9 s@æÈ\¹ sdÈ<y2d>Èü�Y�²�dA,Y²0@,Y A,% K@ÈR¥ KdÈ2e²d9Èò�Y²dE¬Y ²2@V¬Y «AV¬5 k@ÖÈZµ kä /@¾ A6È&M d3ÈfͲd È�Ù ²dk|òÈW² dȶ�Ù²d{ì�Ù²#@vìÙ »Avì= {@öÈ^½ {dÈ>}²d?Èþ�9�r�ä@9r0@9 A# G@ÈQ£ GäÈ1crä8Èñ�ùäoäÈ ò-È· ßÈI 'äÈ)Srä4Èé�9räL9 r6@Î9 çAÎï@¾ù.@.\�¹ A.\L$!´§!´§¡@{jz°áA&µ.!½OºtéHÐßdRëÒ81}àð ZÞ'µAjÔ: u@êH]º u¤H=zR¤>Hý�i�Ò�¤A4iÒ0@4i A4& M@HS¦ M¤H3fÒ¤9Hó�iÒ¤E´i Ò2@Z´i AZ´6 m@ÚH[¶ m¤H;vÒ¤=Hû�é�Ò¤CtéÒ1@ÞyäÍ�éÒ ¤Sté Ò9@ºté ¢$3bÒ¤k ~"H ~xeF6CÞ'hÑ£hÑ£===hÑ£hÑ£hÑ£===hÑ£hÑ£hÑ£=ºdȺ�Y²d}l�Ù�²!@nÜ�¹ ¿üòKܹ r;QixIKHï@~�ù!@®\¹ ?üòcüòÈOrä\ÈÕ�¹r äE©ÇÄ/Òó)ù¢'1J~£äW0ÛScåW1V®Îùéqsßø¹'¯Ñû<.L.e£|«þé#uý¿pä7K"ÀçÿGh3-{"Á¥hÉô¹ç à³hK¿ûv&Ç×ÞóÄE¼ÁÏÏ/À/ÈÏ8;ãÁø$¡ü¤?ÝóþWvdq¢;Ð÷]O½ ô÷»ßqöÚI¨
cÜ=?k¾½ûî?þR¾GµÓux ±§äy,6ÐÊÑ~À®Û�·ú�úúîàóàþyëÿr§UDüùɶ?ÐØöºrÔb°QïÀ¾Æ¾Å¾Ç~ÄNÁ8Tg`2,Ë° LÁº°!lomá°¼ öà]0 >ÇÀ p gÃyðE¸ ¾WÃõð¸~wÁ½ð:?>ã 7nN'ÄÉãqÈL/Å�ÆÆ<?«'�LLúÑÃØiég3³�³s�sϼ�X�XXX x°°ðàeÀ+eåW+�¯VÞ¼ X Xxð.`-=à=À&Àû�>| ø° Ø øðàsÀ/Û;�;»�»{�\_¾|Ã!ôw0¼³æ¢ = àYsÔÐ ÐÐeÖÜutô�ôôôþ óèÿc®ó1��Àr¾ @?ê?qz�z�z�z�zp° ðÆÏ\ú8lÎ~!ÀÀ·CÃ#þü½éþw9à¦!4àåHãÈñ¹lK,ÍÊÅb0TÇ͵cŸ Ŀð´°\b,#!..19ä@tÔ©rèß"3«2ph¢,\_Kvòëpz¼çü1óÙci¬y,
»°à i0ÅÔQãVî³9=zµ ̱Ã
Ú Ì7¸IØVT"ÈIðð¼¼ý<t?ª�Bädòx6¸»ùÒýܨ8'OgêíÊ Ð=Òút?áÌôFÏA¤8å0IôG¹µ·¯;Åéìë/m¡¿KÕ "UJC4íФBI"%ÿËæ"N935·Ø@],NJúé{û{¹3¤×XHX-ÓÓ6Ч¨#«i-\F]ÈM"ñ\_~"+wF·«;ÈÎna ³0üDz0¨ð£IÖ5Úõ½Û´MÌ/å¿ñß^¥úú¶ê[z^ë¾{x]eG+î>ãwíóþ"ãUc£Næ÷¹
zu+l7w¥Äo¬²'Ù9oß]õ°_þB"qHtôcàh@BR§ÒíãDݹ^U8çBS{Hñ|¯K!V2úDg õËõu1{zÜÛS¶6\Wºs°©a+ÞØtÓ±ðÛX»,öƽÁ4óyÕï¤V¸êYrzû¥@JÛ=Q?®ýxS÷ÜDËüñ '¥ù´AãX¥§XGgY>´E¸ ´I%82N8¤£o Ï ka¸ ÈðdÊ-Z/aCr8Q.,Gûøt¡?~Pw"h¢dqÑm~ÄsÎY¬M3J3Ô÷b2ý©ªº2|T|¿÷+ÝWÕ»7'WÕAw te¨Ît#§A'¢£R=±åæE'&3AÖ!ÆßÓ6rÅ· ÿêîR3!qì]à¿W óþ4!aÎ(QÞ\Ydé¨Xîk²w¿nÿÚ3^5N?ñÄí~Ú¶§~ÑÕòÒ7ãO»úê<áù£=+1¸ëj/IÜñi´½ ¥Wy¢<þsBçÃ&íÉ q&q²Ø¢À&ï«j÷Øg)ÉîYóÕ§÷TÛD~[©rÆÔ6}îçG Ëãtó>ÏFu7½Êý]¾Hmú UÄî&H·Ö¸¬Q´ÖNÍɨÊÉ÷m©S¯{¦ó\ävGvuWóîl|fîø7×E²Q{ÊØã2Á/ -e{Çà±ÐÿXÈ!2Ó^lv¹»´·§Zë!C4U¦FÓà&²t&DìûoÙ·Ãá?9ü\_ Ó±IºfE'\®ØbØá|5Ëp¼RåÑûoï&+XV>käçÌôzÃÑÔúP©Ekô½tç¬RxÎà\æxMh¯¸Õ}jôÑGgtáÁkø\NÌ®ÆAÎççWït9róÆù [z+!~-± ¬}ï.óR#_ÇñËã'HËM¾ xil¶Bø|Ü¥G,;Äxö#NçLíNëW¶É]½ÁÏ^/púéeÆþó=§rtZÎãEs?e%×WBµEÆ-öSÚìC®«úXsYèÄüÝ~û½.Hj+Ngg}&'´E~5QáYjuh OíW¸÷Ú÷îÑ£å Q¢ÎZ4ÄS,Cõ"ÓYósÿ¨#TN¤¬NÕXª±¡!ZKÕ5 UÓÃBsÕD(..¥nj.TW7Dc©6íØ Ø[ÿ°l©ÓRQ'ËMRðRÍ´�#¨¦¡iðo :ÑbGD¢N¥¨¡ 0@»Yh "8KWþ5 üº¿Ò»ì\«£íË1SÎ[¸í8 ~lì\çl[¾Ák»;ÌÉ#{bG Ö³ò3'²Ï9~Htª «®ÚuÒ{Puóݽõ.ØsÅÙVB-'6 kTÎIÓOÇgQȧn2yÚÆZ³ÖjZ nKÈFS^7 ºV
ªg ³j^êÎçW?¹kõ¹©©°ýäºn1®< #ըϡҹ&¶ fà®rRÜÞ n17X$FxpÏÊÐA®.ïÐèÝ4Ü9@¾-z»dù}çý]×J8°¸®¡z9wxgu1àQ9G x¾c ß/qÃÌ'ÃhÃSç#"ÈäéY< ò?taÚQÞÞ®ÎLwéÕL/:Ã Ä A´Õ©jjÔ¥êj¨©}Kªq§Îþ+»ÈØì0q«HqÖK²òY¹
ÞØð®ûT"Y £}sØÕ4µ·__ÞÒ3{Ú46áÕH¯ñÊ3]Uº~ÇI#g_¶JÁªwçùuï ïkÐN- vÈInµYq3Ï~ô]Á¶¢-bðsÁ|Åq!0úÏ,õú¥pÄJCG$"ÄÍ÷-4ÁqQ÷w&Ë©åK3ÕìáãINËs©ôì×Z)Èü±¸9xÈ DÃ}hõ´ÇrÒµYøj!iR¹ou|sO&b1ek#Ä M?mu䪿®e3Å ths$¨õ,3FPQ¥bÚÿ#Ç0úÓµþ£~a1Ðæ¥+÷\ÐuÕ.m ªúå®p 4YNaëç¦ú(Ô³r»,e¶äé¨\Íȵ9Õí\_QVú1ôÒZÆøÊþÕ{ë;çx7d¦|&XTÛÜ£t¯{xÍÿMîÜ 8˦£ìðúMïëz7:<Ô)E[Qf2b%w@9%ßÀ#ñ¾ËìcTz}/)ë¨Yí±ãÊ;|O}±jñlýê q/#ªrÑÅPW5÷>õµµyqk°FÕqìYÁcßdæqRÏ÷sK®×. ºÇ$·}Èø,¤Àç®ðn§ÔºæNÞ!'Dê4È/%ÖÆP®çÓÖ Ì¶¼Ð°iJºË7teîK$éìR2>Åhõ©¿ùÖÿì¦cv'D§-0íÆïõÄ3³4)ª"µ¯ZBcô¬O.F«Ý%\_¼t£7>~$ÒZ+}4±¤]êpj~´H7¿çSç¹½\_ñpªTÑúÅ«ôWFùnI\_-È\»ª¸dvv\_½Èûv|ìñºèÖ2sNdDzí³R´8ÿö;y|ùÁûÛr¨ª)Ï»wè°¡=.FÍMëÊD?Ñ7Ïêbu·}õ>y¢K W TË·å¶æAõ{ø»~½h@¿ÅÿýF´PßUl ud)G¿© ©p\_¸ÿ¯ÔûLºOq{ñ1å]ÛUæwVvu×$[ÊYä7½5çjÎi6Ég"Ò§Ú-k£äê<\OKü&äÕ¡n-ëyu©¨Þµ4Äëñj¼¢¬ %i'Åû(«ªËëfµ°\mx½Lˤy]ÊÐÚq;ýawwík¿þÄ&ªÒWb½C«éôÖϪ©«T6·×Î~¡²pèÆå1n¿üë¯M~|QÍq×¾_>:gö7Ô¨?R]:ã ~?GEEÄÓd̳§üƯòßpwe·Y§Ì¡Ú ÖiÊá)ä ¹B ¾äö4d
BþhÊÍwFßyA¡é áò:YCýé g¯PÕ mIc[þ^`lÜ]ø%%ôm·\ij²&åË£Z1gtN¾Ý´EÿYµèZßxne�O³vBÙê«¢%¶WïjVXhxk1nÚ» ßðëý~~~~~~~~~~~~~~~~ |
8640 | https://xaktly.com/OrbitalHybridization.html | MATH CHEMISTRY PHYSICS BIOLOGY EDUCATION
xaktly | Chemistry | Molecules
Orbital hybridization
From atomic orbitals to molecular orbitals
By now you've spent some time learning about how electrons are arranged in atoms into atomic orbitals. Now it shouldn't be surprising that when atoms come close enough to share electrons in covalent bonds, those orbitals change their shapes. After all, it is the process of being bound to a nucleus that creates the atomic orbitals that are the bound electrons. The close proximity of another nucleus from a neighboring atom must also have a profound effect.
The mathematics of quantum mechanics tell us that the atomic orbitals of individual atoms "mix" (in very specific, mathematically defined ways) to form new orbitals we'll call molecular orbitals.
In this section we'll mainly look at the carbon atom and the compounds it can form with other carbon and hydrogen atoms (hydrocarbons). This will give us a pretty complete view of these rearrangements of atomic orbitals into "hybrid" molecular orbitals.
We'll look here only at what we call "open-shell" interactions, bonds that form between two or more atoms with incomplete valence shells, such as the four single bonds formed between a carbon atom and four hydrogen atoms to form methane (CH4).
The atomic orbitals of the n=2 shell of the carbon (or any) atom might be represented like this. A spherical s-orbital and three orthogonal (mutually perpendicular) p orbitals, px, py and pz. Each orbital can contain two spin-paired electrons. The ground state of carbon has two electrons in the s-orbital and one in each of two p orbitals, for a total of four valence electrons.
Pro tip:
Always remember, that even though we often speak of electrons "filling" orbitals, the orbitals actually are the electrons. A 3-D orbital does not exist until it is formed by an electron bound to a nucleus, or in the case of molecules, a group of nuclei.
Formation of molecular orbitals
We won't be able to dive into the complete mathematics here – you'll need to learn a little more math for that (don't worry, you can do it) – but we can illustrate the results.
Let's first think about two hydrogen atoms, each consisting of a proton and one electron – the simplest neutral atoms with an open valence shell. We'll think about how these atoms form a bond. First we ought to think about why they form a bond. The answer has three parts:
The valence shell of each atom is incomplete, so according to the Pauli exclusion principle, there is room for one more electron in each, and we know that a full shell is more energetically stable,
At short distances, the electron of each atom is attracted to the nucleus of the other atom, and
At very short distances the electron clouds begin to overlap and repel.
Any chemical bond arises from a combination of sufficient attraction and strong overlap repulsion.
Now when we bring two H atoms together to form H2, we're bringing two atomic orbitals together. The laws of quantum mechanics tell us that when we "mix" two atomic orbitals (two s-orbitals in this case) we have to produce two molecular orbitals.
In terms of energy, these look like this:
Each of the 1s orbitals of the H atoms has mixed into two molecular orbitals. One is of lower energy than the individual 1s energy levels and is called the bonding molecular orbital (BMO). The other is of higher energy. If either of the electrons in a bonding MO are excited into it (the anti-bonding MO, ABMO), the molecule flies apart into two atoms once again. The energy spacings here are just schematic, but the idea is the same for any bond. You can look at it like a conservation of orbitals. N atomic orbitals will always combine into N molecular orbitals, some bonding and some anti-bonding.
X
Pauli exclusion principle
The Pauli principle states that no two electrons bound in an atom or molecule may have exactly the same set of quantum numbers, n, L, mL and ms. It is the reason that each orbital can hold no more than two electrons, and that each of those has opposite spin.
Formation of a bond from two atomic orbitals means "mixing" those orbitals according to the rules of quantum mechanics, which results in two molecular orbitals (MOs), a bonding MO (BMO) and an anti-bonding MO (ABMO).
When carbon bonds, it forms hybrid orbitals
A neutral carbon atom in its ground state has four electrons in its outer (n = 2) shell, two electrons in the 2s orbital and two in the 2p orbitals. Four valence electrons means (among other scenarios we'll get to) four possible single bonds to a single carbon. For the moment, we'll think about methane, CH4, for simplicity.
Now those four electrons are in different kinds of orbitals (s and p), but we know from experiments that the four C–H bonds of CH4 are equivalent. It turns out that nature creates hybrid orbitals, consisting of three 2p orbitals and one 2s orbital, called sp3 orbitals (one part s, three parts p)
The diagram below shows how it works. After hybridization, all four valence electrons of the carbon atom occupy equivalent sp3 hybrid orbitals, ready to bond to four hydrogen atoms.
The diagram below shows the approximate difference between the atomic and hybrid orbitals of a carbon atom.
Valence shell electron-pair repulsion (VSEPR) theory predicts that the four main lobes of each of the four hybrid orbitals will move as far from one another as possible. That means they'll assume the tetrahedral arrangement shown above. Ultimately, this leads to the tetrahedral structure of CH4, and to the fact that most carbon "centers" in organic (carbon containing) molecules are tetrahedral.
The bonding and structure of methane, CH4
Now the molecular orbitals of methane form from the four 1s orbitals of each hydrogen atom and each of the four hybrid orbitals. Here's a schematic diagram of the approach of the hydrogens.
As each hydrogen bonds, an sp3 orbital of the carbon mixes with a hydrogen 1s orbital to create a bonding molecular orbital (BMO) and an anti-bonding orbital (ABMO). So long as the two electrons forming the bond occupy the BMO – the lowest energy state, a bond exists.
The fully bonded structure might look something like this. VSEPR theory explains the tetrahedral arrangement of the H atoms around the central carbon: The electron-dense orbitals move under the force of like-charge repulsion to create the largest-possible separation.
Except where double or triple bonds exist in molecules, every carbon atom is a tetrahedral center, with just such a structure, because of sp3 orbital hybridization. The figure below depicts one of the structures of glucose. The five carbon atoms in red are tetrahedral centers, with sp3 orbital hybridization. The carbon atom with the double bond is not sp3 hybridized, and does not form a tetrahedral center. This will be the subject of the next section.
Chirality
One of the most interesting implications of this embedded tetrahedral structure in carbon-containing molecules is that of chirality, or handedness. We can, in fact, have right-handed and left-handed molecules. Take a look at the two chiral centers below. They are mirror images of each other. From the point of view of the ligand labeled "1" in each, counting around ligands 2, 3 and 4 produces a clockwise circle on the left and a counterclockwise circle on the right. This actually has a huge effect on the chemistry of carbon, including all of biochemistry – but it's a subject for another section.
Molecules that have this mirror-image symmetry are called chiral. Note that the ligands 1, 2, 3 and 4 must be distinct.
X
ligand
A ligand is anything that bonds to an atom or molecule of interest. Which atom or molecule is in the context of the statement. The four hydrogen atoms of CH4 are ligands of the carbon atom. An oxygen molecule, O2, is a ligand for the heme group in hemoglobin, and so on.
C=C double bonds and sp2 hybridization
Now we turn to a second hybrid orbital, this one composed of one part s-orbital and two parts p orbital. The figure below shows the energy-level diagrams that lead to such hybrid orbitals.
Notice that in this case, hybridization first requires excitation of one electron (up and down arrows in the figure) from the s-orbital to a p-orbital first.
These sp2 hybrid orbitals will give rise to a new kind of bonding. You know it as double bonding, but these bonds are also called π-bonds, while single bonds are called σ-bonds.
To form an sp2 hybrid orbital in a carbon atom, one 2s electron is excited to the 2p orbital, and three new sp2 orbitals are formed, leaving one 2p orbital intact. This orbital will give us our double or π bond.
The resulting sp2 orbitals arrange themselves around the 2p orbital as shown below, with 120˚ bond angles between each, all lying in a plane perpendicular to the unhybridized p-orbital.
Let's consider the formation of a double bond (π-bond) between two sp2 hybridized carbon atoms. We begin with the two atoms separated like this.
Overlap of two of the sp2 orbitals, one from each atom, forms part of the bond. The other part comes from an interaction of the electrons in the unhybridized p-orbitals. As these are drawn nearer to each other, the molecular orbitals that form above and below the central sp2-sp2 bond create the rest of the π-bond.
A more realistic impression of the orbitals of the full π-bond is shown below. This bonding arrangement, involving four total electrons, causes the molecule to be planar and rigid, with the C=C π-bond shorter than a C–C single (or sigma, σ) bond.
If we add four hydrogen atoms to each of the non-bonded sp2 orbitals still unoccupied in the drawing above, we get ethylene. Here is a simplified drawing of ethylene, C2H4, which has a C=C double bond with four hydrogen atoms bound to the other unpaired electrons (in sp2 hybrid orbitals) of the carbon atoms. Ethylene is a planar molecule.
The C=C bond of ethylene, because of the way the molecular orbitals that form it are arranged, is stiff and does not rotate about the C=C axis. Thus ethylene is a fairly rigid planar molecule. Contrast that to ethane, C2H6, below, which has only sp3 hybridization and σ bonds.
A typical C–C single bond is about 154 pm. The length of a double bond is about 20 pm (13%) shorter. Next we'll construct a triple bond, which will be even shorter, typically 120 pm for a C-C triple bond.
Triple bonding and sp hybridization
Well, we've seen hybrid orbitals of carbon formed from 1 part s and 3 parts p orbitals, and from 1 part s and two parts p orbitals. Why not 1:1 ?
This schematic diagram shows how sp (sp1, but we leave the 1 off) orbitals are formed.
The resulting orbital arrangement looks like this. Two sp hybrid orbitals (yellow and blue in the figure) are orthogonal (perpendicular in both y & z directions, as shown) to two non-hybridized p-orbitals, each containing one electron.
Now when two of these hybridized carbon atoms approach, two of the sp orbitals can form a molecular orbital and thus a bond,
Now in close proximity, the unhybridized p-orbitals of each atom can overlap and form their own molecular orbitals, a triple bond. Notice that in this drawing one unhybridized p-orbital of each carbon is seen end-on (gray circle).
The complete picture of the bonding orbitals might look something like this. A central core of the merged sp orbitals is surrounded on two sides by pairs of molecular orbitals formed from the unhybridized p-orbitals. This is the nature of the triple bond. It occurs in carbon compounds and in the nitrogen molecule, N2, among others.
If we add two hydrogen atoms to bond with the remaining valence electron of each carbon atom, our completed molecular orbital diagram looks (very schematically) like this:
Finally, the ethyne molecule (or acetylene), C2H2, is a rigid, linear molecule,
Examples of hybrid orbital systems
Nitrogen (N2)
The energy level diagram for the formation of the sp hybrid orbitals of the nitrogen atoms in N2 is shown below. There are five electrons in four orbitals before hybridization, and the same afterward. Notice that one of the sp hybrid orbitals is full; it contains two spin-paired electrons (up and down arrows).
The hybridized and unhybridized orbitals look schematically like this. The yellow orbital is full, and won't form a bond. The other three orbitals are available for bonding, and we know from the Lewis structure that N2 forms a triple bond.
The figure below shows two hybridized nitrogen atoms lined up for bond formation. As in the last section, two of the unhybridized orbitals are meant to be poking into and out of the screen (gray circles in the middle).
Finally, the complete bonding picture will look something like this.
The stick drawing of N2 looks like this. I like to leave myself a reminder that there are lone pairs of electrons on either end, These will have an effect on the properties of the molecule.
The N≡N triple bond is one of the strongest bonds in nature, and leads to some interesting consequences in biology.
PCl5—sp3d hybridization
We can also have further levels of hybridization of orbitals, such as that found in the phosphorus atom of PCl5. In this molecule, the five ligands (Cl) all bond to equivalent sp3d orbitals, composed of one part s, three parts p and 1 part d orbital in the third (n = 3) shell.
For this to occur, we first consider an excited state in which one 3s orbital is excited into a 3d orbital:
The hybridization of the five occupied orbitals looks like this. The energy levels aren't to scale; this is just a schematic picture.
As predicted by VSEPR theory, those orbitals arrange themselves in a trigonal bipyramid, with three orbitals around the "equator" and two forming the "poles" of this nearly spherically-symmetric hybridized atom.
The molecule PCl5 looks like this. A trigonal bipyramid consists of two back-to-back three-sided pyramids.
Octahedral symmetry — sp3d2 hybridization.
A further orbital hybridization, sp3d2, consisting of one part s-orbital, three parts p-orbital and two parts d-orbital is possible, and leads to octahedral symmetry by forming six identical orbitals.
Examples of molecules with octahedral symmetry include SF6 and the ion [Co(NH3)6]3+
xaktly.com by Dr. Jeff Cruzan is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. © 2012, Jeff Cruzan. All text and images on this website not specifically attributed to another source were created by me and I reserve all rights as to their use. Any opinions expressed on this website are entirely mine, and do not necessarily reflect the views of any of my employers. Please feel free to send any questions or comments to jeff.cruzan@verizon.net. |
8641 | https://ximera.osu.edu/csccmathematics/precalculus2/precalculus2/complexExponentials/rootsOfUnity | Warning
You are about to erase your work on this activity. Are you sure you want to do this?
Updated Version Available
There is an updated version of this activity. If you update to the most recent version of this activity, then your current progress on this activity will be erased. Regardless, your record of completion will remain. How would you like to proceed?
Mathematical Expression Editor
CSCC PreCalculus 2
Creative Commons
Ximera tutorial
How to use Ximera
How is my work scored?
Exponential Functions
Theory
Shifted Theory
Basic Forms
Algebra
Logarithmic Functions
Theory
Basic Forms
Polynomial Functions
Theory
Factoring
FTA
Rational Functions
Theory
Analysis
Function Forms
Categories
Operations
Pieces
Composition
Composition
Steps
Formulas
Behavior
Rates of Rates
Analysis
More Analysis
One More
Function Type
the Derivative
Refining
Algebra
Critical Numbers+
Analysis
Analyzing Functions
Analyzing
Analyzing
Analyzing
Analyzing More Functions
Analyzing
Analyzing
Quadratic Functions
Theory
iRoC
Missing Numbers
2D Numbers
Real Numbers
Vectors
Numbers
Complex Numbers
Notation
Complex Arithmetic
Operations
Complex Multiplication
Complex Fractions
Fractions
Algebraic Geometry
Polar Form
Direction Vectors
The Unit Circle
The Key
Right Triangles
Similarity
Side Lengths
Angles
3 Views
General Triangles
Law of Sines
Law of Cosine
Trig Functions
Sine & Cosine
Analysis
Analysis
Tangent
Around the Circle
Easy Angles
As Input
Cartesian Graphs
Transform
Polar Graphs
Graphing
Gallery
Rate Of Change
Infinitesimal
Sine and Cosine
More Trig Functions
Reciprocals
Cotangent
Secant
Cosecant
Algebra
Inverse Trig Functions
Inverse Functions
Exponential
ArcSine & ArcCosine
Arctangent
Complex Bridge
Euler’s Formula
Bridge
Trig Identities
Function Analysis
Analyzing
Analyzing
More Zeros
Quadratic
Pairs
Reals
Rationals
FTA
More Function Analysis
Analyzing
Implicit Functions
Implicit Functions
Rate of Change
Modeling
Length
Angles
Volume
Periodic
Model Analysis
C&M Trucking
Sequences
Sequences
Visualize
Sequences as functions
Limits of Sequences
Dominant Terms
Existence
Sums of sequences
What is a series
Special Series
Geometric Series
The Series
Convergence Intervals
Closed Form
Rational Functions
Complex Exponentials
Roots of Unity
Roots
Powers
Complex Logarithm
Inverse
Complex Powers
We have seen that every Complex number can be written as (r \cdot (\cos (\theta ) + i \, \sin (\theta ))). This is a scalar times a
Complex number on the unit circle.
We have also seen that if (z = r \cdot (\cos (\theta ) + i \, \sin (\theta ))), then (z^n = r^n \cdot (\cos (n\theta ) + i \, \sin (n\theta ))). Raising complex numbers to powers is
accomplished by raising the modulus to the power and then multiplying the
angle.
(\blacktriangleright ) Roots of Unity
Roots or unity refer to roots or zeros of the polynomial (x^n - 1).
Any such root of unity would be a solution to the equation (z^n = 1), which is why they are
called roots of unity.
For instance, the square roots of unity are the solutions to (z^2 = 1). The two solutions are (1)
and (-1).
For instance, the (4^{th}) roots of unity are the solutions to (z^4 = 1). The four solutions are (1), (-1), (i), and
(-i).
(1) is always a root of unity for any power. Then, there are (n-1) other (n^{th}) roots of
unity.
If (z = r \cdot (\cos (\theta ) + i \, \sin (\theta ))) is going to be a root of unity, then (z^n = r^n \cdot (\cos (n\theta ) + i \, \sin (n\theta )) = 1), which means (r=1).
(n^{th}) roots of unity all look like (\cos (\theta ) + i \, \sin (\theta )). They all lie on the unit circle.
In addition, if (\cos (n\theta ) + i \, \sin (n\theta ) = 1), then (n \theta = 2 k \pi ) (a mulitple of (2 \pi )), because (1) is on the positive real axis.
Therefore, (\theta = \frac {2 k \pi }{n})
We need multiples of (\frac {2 \pi }{4} = \frac {\pi }{2}).
The fourth roots of unity are:
The roots of unity are spread out equidistant along the unit circle.
We need multiples of (\frac {2 \pi }{3}).
The fourth roots of unity are:
(\cos \left ( \frac {\pi }{6} \right ) + i \, \sin \left ( \frac {\pi }{6} \right )) is which root of unity?
Using the quadratic formula, we know that (\frac {-3 + \sqrt {11} \, i}{2}) is a root of (x^2 + 3x + 5).
We know that (\frac {-3 + \sqrt {11} \, i}{2}) can be written in the form (r (\cos (\theta ) + i \, \sin (\theta ))).
(r) is the modulus of (\frac {-3 + \sqrt {11} \, i}{2}).
(r = \answer {\sqrt {5}})
The angle for (\frac {-3 + \sqrt {11} \, i}{2}) is in which quadrant?
(\arctan \left ( -\frac {\sqrt {11}}{3} \right )) gives the angle, but in quadrant IV. What must be added to (\arctan \left ( -\frac {\sqrt {11}}{3} \right )) to get (\theta )?
Add (\answer {\pi })
As long as we are here...
In the example above, (x^2 + 3x + 5) is a polynomial with real coefficients. Therefore, if (\frac {-3 + \sqrt {11} \, i}{2}) is a root,
then so is (\frac {-3 - \sqrt {11} \, i}{2}).
Therefore, (x^2 + 3x + 5) factors as
The constant terms have to be equal, which tells us that
The constant term of the polynomial is the product of the roots, which are
conjugates. The constant term is the square of the modulus of either root.
ooooo-=-=-=-ooOoo-=-=-=-ooooo
more examples can be found by following this link
More Examples of Complex Exponentials
Courses
About
Social
Built at The Ohio State UniversityOSU with support from NSF Grant DUE-1245433, the Shuttleworth Foundation, the Department of Mathematics, and the Affordable Learning ExchangeALX.
© 2013–2025, The Ohio State University — Ximera team
100 Math Tower, 231 West 18th Avenue, Columbus OH, 43210–1174
Phone: (773) 809–5659 | Contact
If you have trouble accessing this page and need to request an alternate format, contact ximera@math.osu.edu. |
8642 | https://discrete.openmathbooks.org/dmoi3.html | Discrete Mathematics
An Open Introduction, 3rd edition
Read online
Download PDF
Buy paperback
Now Available: 4th edition!
The links here are for the previously released 3rd edition, which some might still find useful. Everyone else, please see the 4th edition page for the much improved book.
Runestone Academy
The book is also available on Runestone Academy. Instructors using this free service can assign and auto-grade the embedded interactive exercises. It even connects to Canvas, Blackboard, and other Learning Management Systems. Check it out!
Discrete Mathematics: An Open Introduction is a free, open source textbook appropriate for a first or second year undergraduate course for math and computer science majors. The book is especially well-suited for courses that incorporate inquiry-based learning. Since Spring 2013, the book has been used as the primary textbook or a supplemental resource at more than 200 colleges and universities around the world (see the partial adoptions list). The text is endorsed by the American Institute of Mathematics' Open Textbook Initiative and is well reviewed on the Open Textbook Library.
This 3rd edition brings many improvements, including nearly 100 new exercises (including many interactive exercises providing instant feedback), a new section on trees in the graph theory chapter, and improved exposition throughout. Previous editions will continue to be available indefinitely. A few times a year, the text is updated with a new "printing" to correct errors. See the errata list for more information.
Online homework sets are available through Runestone Academy (free), Edfinity (inexpensive) or as WeBWorK sets from the author (the WeBWorK exercises are available in the Contrib folder of the OPL as well).
Please contact the author with feedback and suggestions, or if you are decide to use the book in a course you are teaching. You can also easily submit feedback about an error or typo by creating a GitHub issue.
Get the book
The entire book is available for free as an interactive online ebook. This should work well on all screen sizes, including smart phones, and work will with screen readers for visually impaired students. Hints and solutions to examples and exercises are hidden but easily revealed by clicking on their links. Some exercises also allow you to enter and check your work, so you can try multiple times without spoiling the answer.
For offline use, a free pdf version, suitable for reading on a tablet or computer, is available for download. This should be searchable and easy to navigate using embedded links. Hints and solutions (when available) can be accessed by clicking on the exercise number, and clicking on the number of the hint or solution will bring you back to the exercise.
If you prefer a physical copy, an inexpensive print version of the text is available on Amazon. This should be cheaper than printing the entire book and binding it yourself. Page numbers match the pdf version. There is also a cheap Kindle version available; it works great on Android tablets; unfortunately it seems the math does not render well on Amazon's e-ink readers.
PreTeXt (and LaTeX) source
The source files for this book are available on GitHub.
Instructor and student resources
There are a number of videos instructors have made for classes they taught using the book. Check out the playlists below. If you make videos or know of others, please share them.
Mathematical Visual Proofs. Beautifully animated videos illustrating ideas from the book.
Dr. Trevor Bazett's Course. A complete course that uses the textbook as a basis for the videos.
If you are using the book in a class you are teaching, instructor resources are available by request. Just contact the author. You can also request WeBWorK homework sets if you have access to a WeBWorK server.
About the book
The text began as a set of lecture notes for the discrete mathematics course at the University of Northern Colorado. This course serves both as an introduction to topics in discrete math and as the "introduction to proofs" course for math majors. The course is usually taught with a large amount of student inquiry, and this text is written to help facilitate this. Originally designed to support future math teachers, the text has a friendly and informal tone, and puts an emphasis on understanding the included concepts, rather than simply memorizing procedures. The book has also been successfully used in courses catering to computer science students, who also benefit from the deeper understanding it promotes.
Four main topics are covered: counting, sequences, logic, and graph theory. Along the way, proofs are introduced, including proofs by contradiction, proofs by induction, and combinatorial proofs. An introductory chapter covering mathematical statements, sets, and functions helps students gain familiarity with the language of mathematics, and two additional topics (generating functions and number theory) are also included.
While the book began as a set of lecture notes, it now contains a number of features that should support its use as a primary textbook:
473 exercises, including 275 with solutions and another 109 with hints. Exercises range from easy to quite involved, with many problems suitable for homework.
Investigate! activities throughout the text to support active, inquiry-based learning.
A full index and list of symbols.
Consistent and helpful page layout and formatting (i.e., examples are easy to identify, important definitions and theorems in boxes, etc.).
About the author
Oscar Levin is a professor at the University of Northern Colorado. He has taught mathematics and computer science at the college level for over 15 years and received multiple teaching awards. He received his Ph.D. in mathematical logic from the University of Connecticut in 2009.
License
Discrete Mathematics: An Open Introduction by Oscar Levin is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License. You are free to download, use, print, and even sell this work as you wish to. You can also modify the text as much as you like (create a custom edition for your students, for example), as long as you attribute the parts of the text you use to the author.
If you are interested in using parts of the book combined with another text with a similar but different license (GFDL, for example), please reach out to get permission to modify the license. |
8643 | https://www.vedantu.com/maths/representing-real-numbers-on-the-number-line | Representing Real Numbers on the Number Line Explained
Sign In
Courses
Explore all Vedantu courses by class or target exam, starting at ₹9000
Find courses by class
Find courses by target
Long Term Courses Guaranteed improvement in marks or get your fees back One-to-one LIVE classes Learn one-to-one with a teacher for a personalised experience Topic specific courses Master any topic at just ₹1
Courses for Kids
Courses for Kids
Confidence-building & personalised learning courses for Class LKG-8 students
English Superstar Age 4 - 8 Level based holistic English program Summer Camp For Lkg - Grade 10 Limited-time summer learning experience Spoken English Class 3 - 5 See your child speak fluently Learn Maths Class 1 - 5 Turn your child into a Math wizard Coding Classes Class 1 - 8 Learn to build apps and games, be future ready
Free study material
Get class-wise, author-wise, & board-wise free study material for exam preparation
NCERT SolutionsCBSEJEE MainJEE AdvancedNEETQuestion and AnswersPopular Book Solutions
Subject wise Concepts
ICSE & State Boards
Kids Concept
Online TuitionCompetative Exams and Others
Offline Centres
Online Tuition
Get class-wise, subject-wise, & location-wise online tuition for exam preparation
Online Tuition By Class
Online Tuition By Subject
Online Tuition By Location
More
Know about our results, initiatives, resources, events, and much more
Our results A celebration of all our success stories Child safety Creating a safe learning environment for every child Help India Learn Helps in learning for Children affected by the Pandemic WAVE Highly-interactive classroom that makes learning fun Vedantu Improvement Promise (VIP) We guarantee improvement in school and competitive exams Master talks Heartfelt and insightful conversations with super achievers
Our initiatives
Resources
About us Know more about our passion to revolutionise online education Careers Check out the roles we're currently hiring for Our Culture Dive into Vedantu's Essence - Living by Values, Guided by Principles Become a teacher Apply now to join the team of passionate teachers Contact us Got questions? Please get in touch with us
Vedantu Store
Maths
How to Represent Real Numbers on the Number Line
How to Represent Real Numbers on the Number Line
Reviewed by:
Rama Sharma
Download PDF
Study Materials
NCERT Solutions for Class 9
Important Questions for Class 9
Revision Notes for Class 9
NCERT Books
Worksheets
Maths Formula For Class 9
Sample Papers
Sample Paper for Class 9 Maths
Sample Paper for Class 9 Science
Sample Paper for Class 9 English
Sample Paper for Class 9 Hindi
Sample Paper for Class 9 Social Science
Syllabus
Maths Syllabus
Science Syllabus
English Syllabus
Hindi Syllabus
Social Science Syllabus
Textbook Solutions
RD Sharma Solutions
RS Aggarwal Solutions
Lakhmir Singh Solutions
NCERT Examplar
Step-by-Step Guide: Plotting Real Numbers with Examples
A number line is a horizontal line used to represent integers in Mathematics. Positive and negative integers are both represented on the number line, which is endlessly long in both directions. Real numbers can be represented on a number line for the purposes of comparison and ordering.
Real numbers can be represented visually by assigning them to specific points on a line using a real number line, also known as a number line. This article will go into great detail on how to express the real number on a number line.
Real Numbers Definition
Rational numbersQ and irrational numbersQ′ are the components of real numbers. R represents an array of real numbers. Real numbers are a category that includes all types of numbers, including whole numbers, natural numbers, integers, decimals, and rational and irrational numbers among others. As a result, R=Q+Q′ can be written.
Real numbers are shown in the following graph:
Real Numbers
Real Number Line
Every real number can be represented by a different point on the number line since we know that real numbers can be either rational or irrational. A real number line, also known as a number line, represents real numbers on the line with distinct points assigned to each of the numbers.
A coordinate is a point that is related to real numbers.
The origin of the number line is referred to as the number 0.
The left side of the origin represents all the negative numbers or integers, while the right side represents all the positive numbers or integers.
Representation of Real Numbers on a Number Line
How to Represent Real Numbers on the Number Line?
The steps that are taken to represent real numbers on the number line are as follows:
Step 1: Mark a point anywhere along a horizontal straight line with arrows at both ends. The origin is referred to as zero.
Step 2: Find real numbers on both sides of the origin spaced equally apart, and give them scale labels.
Step 3: Numbers that are positive are on the right side of the origin, and whole numbers that are negative are on the left.
Step 4: By finding the specific point on the number line, you may quickly mark whole numbers, natural numbers, and integers. To represent large numbers efficiently, the number line's scale can be increased. Suppose a specified point is a big number, like 120. In order to designate 120, we can choose 1 unit as 20, which requires us to move 6 steps to the right of the origin.
Step 5: Mark both rational and irrational values on the number line after converting them to decimal counterparts. Sometimes it is impossible to mark a number exactly where it should be. In this situation, we can apply the technique of successive magnification, using the magnifying glass to see the numbers on the number line.
Real Numbers on a Number Line
Integers on a Number Line
Integers are represented on a number line as a combination of both positive and negative values. On the number line, integers rise from left to right. Between positive and negative integers, 0 resides in the middle. Negative numbers are −1 , −2 , −3 , etc., while positive numbers are 1 , 2 , 3 , etc.
The number line below displays the integers.
Integers on Number Line
Rational Numbers on the Number Line
Numbers in the form of p q where p and q are numerators and denominators, respectively, having integer values are considered rational numbers on a number line. To represent the rational numbers, we divide the two adjacent integers into n parts.
Rational Numbers on Number Line
Decimals on Number Line
To represent decimal numbers, we divide pairs of successive integers into tenths (10), hundredths (100), and so forth.
Decimals on number line
Number Line with Irrational Numbers
Real numbers that cannot be expressed as simple fractions are referred to as irrational numbers. They cannot be formulated as a ratio.
Irrational numbers on number line
Ordered Real Numbers on a Number Line
A number line can also be used for comparing any given real numbers. Always, the larger numbers are to the right of the origin, while the smaller numbers are to the left. Although we are aware that 19 is bigger than 10, the sequence is different for negative real numbers. For instance, since negative integers start from the left of the origin, −2 is greater than −6.
It becomes easier to compare numbers after we graph each one on a number line. Real numbers are compared using symbols like less than (<) , greater than (>) , and equal to (=) .
The example of comparing two real numbers with the formula shown above is depicted in the image below.
Ordering Real Numbers on a Number Line
A Number Line with Opposing Real Numbers
On a number line, a negative real number is the opposite of a positive real number, and a positive real number is the opposite of a negative real number, i.e., the opposite of −8 is 8, which is the opposite of −8. According to their signs, the numbers are plotted at the same distance from the original but in the opposite direction. As an illustration, 8 is the reciprocal of −8. As a guide, take a look at the illustration below:
Number Line with Opposite Rreal Numbers
Real Numbers' Absolute Values on a Number Line
The distance between the origin and the real number's graph on the number line is known as the absolute value of a real number, represented by the symbol |x|. It is always positive because it is a distance. As an illustration, |−3|=3 and |3|=3 . View the illustration below as a guide:
Absolute real values on a number line
Real Number Line Examples
Q.1.Express 2 on a number line.
Solution. Consider the irrational number 2.
Mark the origin and other integers on a number line.
Use a compass to determine the distance between 0 and 1.
Draw a parallel to 1 that has the same length as the distance between 0 and 1
Connect the opposite end of the new line—the perpendicular end—to the origin. You will receive the indicated figure.
The number 1 on the number line, the perpendicular end, and the name origin are all A.
2 is the distance between A and C.
Calculate the distance between A and C using a compass.
Set the compass at A and indicate on the number line the same distance between A and C.
You also reach the 2 point at D.
Irrational Number 2 on a Number Line
Q.2. On a number line, show that x<5.
Solution. The procedures listed below will help you represent x<5 on a number line.
Draw a number line, mark 0, and then draw equal distances to the right and left.
The value of x is less than 5 in the inequality x<5. We must then move to the left of 5.
Create a black empty circle on number five and shade the line from number five to the left until it reaches the arrow as shown in the illustration. In order to demonstrate that point 5 is not included, the circle at that location will be left unfilled. This proves that x<5.
On the number line, this mark will reflect the necessary inequality.
Practice Questions
Add two numbers 5 and 8 using the number line.
Ans:13
Express the subtraction of 7 from 14 on a number line.
Ans: 7
Summary
In this topic, we studied what real numbers are as well as their classifications. We discussed what a number line is and how to use it to represent real numbers. We have also discussed how to express real numbers on a number line using examples. There are also examples with solutions to help you understand the topic better.
Competitive Exams after 12th Science
CBSE JEE Main JEE Advanced NEET Olympiad CUET
Best Seller - Grade 11 - JEE
View More>
### Vedantu JEE 2025 - 26 QR Revision Cards – Physics, Chemistry, Mathematics | Flash Cards for JEE Main & Advanced | Quick Concept Recap & Practice Booklet ₹1999.00 Sale ₹1499.00 + ADD TO CART
### Vedantu JEE Tatva Book Set – Physics, Chemistry, Mathematics | Set Of 11 Volumes For Class 11 | Chapterwise PYQs, Concept Videos, Theory & Graded Exercises | Latest Edition ₹3999.00 Sale Sold Out ₹2999.00 Out of Stock
### Vedantu JEE Advanced Rank Accelerator 2025 Books Set Of 3 | Physics, Chemistry, Mathematics | Chapterwise Practice, PYQs, Mock Tests For JEE Advanced Aspirants ₹1999.00 Sale ₹1499.00 + ADD TO CART
### Vedantu JEE Main 2025 Crash Course Book Set Of 3 – Physics, Chemistry, Mathematics | Latest Syllabus | Includes Free Recorded Course ₹1999.00 Sale ₹999.00 + ADD TO CART
### Vedantu's Instasolve - 1 Month - 24 hours Unlimited Instant Doubt Solving ₹2998.00 Sale ₹1999.00 + ADD TO CART
### Vedantu's Instasolve - 3 Months - 24 hours Unlimited Instant Doubt Solving ₹9998.00 Sale ₹5499.00 + ADD TO CART
### Vedantu's Instasolve - 12 Months - 24 hours Unlimited Instant Doubt Solving ₹17998.00 Sale ₹12000.00 + ADD TO CART
### Dream Hustle Achieve - Men's Round Neck T-Shirt ₹998.00 Sale ₹499.00 VIEW DETAILS
### Dream Hustle Achieve - Men's T Shirt ₹998.00 Sale ₹499.00 VIEW DETAILS
### Dream Hustle Achieve - Women's Round Neck T-Shirt ₹998.00 Sale ₹499.00 VIEW DETAILS
### Dream Hustle Achieve - Men's Hooded Sweatshirt ₹1598.00 Sale ₹799.00 VIEW DETAILS
### Forever Vedan - Women's Round Neck T-Shirt ₹998.00 Sale ₹499.00 VIEW DETAILS
### Dream Hustle Achieve - Women's Hooded Sweatshirt ₹1598.00 Sale ₹799.00 VIEW DETAILS
FAQs on How to Represent Real Numbers on the Number Line
What is the basic principle for representing real numbers on a number line?
The basic principle is that every real number corresponds to a unique point on the number line, and conversely, every point on the number line represents a unique real number. To set it up, you draw a horizontal line, mark a point as the origin (0), place positive numbers to the right and negative numbers to the left at equally spaced intervals.
How are the two main types of real numbers, rational and irrational, located on a number line?
Both rational and irrational numbers are located on the number line.
Rational numbers, like fractions (e.g., 1/2) or terminating decimals (e.g., 2.5), can be precisely marked by dividing the segments between integers into equal parts.
Irrational numbers, like √2 or π, have non-terminating, non-repeating decimal expansions. They are located using geometric constructions (like the Pythagorean theorem for square roots) or by approximating their decimal value.
What is the method to represent a fraction like 7/4 on the number line?
To represent the fraction 7/4, you first convert it to a mixed fraction, which is 1 ¾. This tells you the number lies between the integers 1 and 2. Then, you divide the segment between 1 and 2 into four equal parts. The third mark after 1 will represent the point 7/4 on the number line.
How is an irrational number like √5 represented on the number line using a geometric method?
You can represent √5 using the Pythagoras theorem. Follow these steps:
On the number line, mark a point A at 2 units from the origin 0.
Draw a perpendicular line segment AB of 1 unit at point A.
Join the origin O to point B. The length of OB will be √(2² + 1²) = √5.
With O as the centre and OB as the radius, draw an arc that intersects the number line. The point of intersection is √5.
What is the process of successive magnification and when is it used?
The process of successive magnification is a technique used to visualise and locate real numbers with non-terminating decimal expansions on the number line with increasing accuracy. It involves 'zooming in' on the segment of the number line where the number lies. For example, to locate 4.26, you first locate the interval between 4 and 5, then magnify it to find the interval between 4.2 and 4.3, then magnify that to find 4.26. This process is particularly useful for representing non-terminating recurring decimals.
What is the main difference in the representation of a terminating decimal (e.g., 2.5) versus a non-terminating recurring decimal (e.g., 2.555...)?
The main difference lies in the precision of placement. A terminating decimal like 2.5 can be marked exactly at the midpoint between 2 and 3. A non-terminating recurring decimal like 2.555..., however, can only be visualised approximately. We use the process of successive magnification to show its position more and more accurately within progressively smaller intervals (e.g., between 2.5 and 2.6, then 2.55 and 2.56), but we can never mark its exact final point as the decimal continues infinitely.
Why is it said that the number line is 'complete' with the inclusion of both rational and irrational numbers?
The number line is considered 'complete' because it represents the entire set of real numbers without any gaps. If you only consider rational numbers, there are infinite 'holes' on the line where irrational numbers like √2 and π should be. By including both rational and irrational numbers, we establish a one-to-one correspondence where every single point on the line corresponds to a real number, and every real number has a specific point on the line, thus making it a continuous and complete line.
Recently Updated Pages
Knot Theory in Maths: Concepts, Applications & Examples Mobius Strip Explained: Definition, Properties & Uses Fermat’s Theorem: Meaning, Proofs, Examples & Applications Infinitesimal in Maths: Concepts, Examples & Applications Fuzzy Logic in Maths: Concepts, Uses & Examples Explained Point Estimation in Statistics: Methods & Formulas Explained
Knot Theory in Maths: Concepts, Applications & Examples Mobius Strip Explained: Definition, Properties & Uses Fermat’s Theorem: Meaning, Proofs, Examples & Applications
Infinitesimal in Maths: Concepts, Examples & Applications Fuzzy Logic in Maths: Concepts, Uses & Examples Explained Point Estimation in Statistics: Methods & Formulas Explained
1
2
Trending topics
1 Billion in Rupees: Explained with Examples & Conversion Maths Quiz Questions with Answers: Practice for Class 6 to 10 Exams Logarithm Values From 1 to 10: Complete Table Maths Made Easy: Key Concepts, Formulas & Smart Practice Maths Project Ideas for Classes 6 to 12 Geometry Symbols: Complete Guide for Students
1 Billion in Rupees: Explained with Examples & Conversion Maths Quiz Questions with Answers: Practice for Class 6 to 10 Exams Logarithm Values From 1 to 10: Complete Table
Maths Made Easy: Key Concepts, Formulas & Smart Practice Maths Project Ideas for Classes 6 to 12 Geometry Symbols: Complete Guide for Students
1
2
Other Pages
Teachers Day Speech 2025 in English for School Celebrations Important Days and Dates in August Independence Day Speech in English for Students 2025 Important Days in September 2025: Festivals and Events You Should Know What Is Teachers Day and Why Do We Celebrate It? Essay on Ganesh Chaturthi 2025 : Sample Essays in English
Teachers Day Speech 2025 in English for School Celebrations Important Days and Dates in August Independence Day Speech in English for Students 2025
Important Days in September 2025: Festivals and Events You Should Know What Is Teachers Day and Why Do We Celebrate It? Essay on Ganesh Chaturthi 2025 : Sample Essays in English
1
2
Study material
NCERT
NCERT
NCERT Solutions
NCERT Solutions for Class 12
NCERT Solutions for Class 12 Maths
NCERT Solutions for Class 12 Physics
NCERT Solutions for Class 12 Chemistry
NCERT Solutions for Class 12 Biology
NCERT Solutions for Class 12 Business Studies
NCERT Solutions for Class 12 Economics
NCERT Solutions for Class 12 Accountancy
NCERT Solutions for Class 12 English
NCERT Solutions for Class 12 Hindi
NCERT Solutions for Class 11
NCERT Solutions for Class 11 Maths
NCERT Solutions for Class 11 Physics
NCERT Solutions for Class 11 Chemistry
NCERT Solutions for Class 11 Biology
NCERT Solutions for Class 11 Business Studies
NCERT Solutions for Class 11 Economics
NCERT Solutions for Class 11 Accountancy
NCERT Solutions for Class 11 English
NCERT Solutions for Class 11 Hindi
NCERT Solutions for Class 10
NCERT Solutions for Class 10 Maths
NCERT Solutions for Class 10 Science
NCERT Solutions for Class 10 English
NCERT Solutions for Class 10 Social Science
NCERT Solutions for Class 10 Hindi
NCERT Solutions for Class 9
NCERT Solutions for Class 9 Maths
NCERT Solutions for Class 9 Science
NCERT Solutions for Class 9 English
NCERT Solutions for Class 9 Social Science
NCERT Solutions for Class 9 Hindi
NCERT Solutions for Class 8
NCERT Solutions for Class 8 Maths
NCERT Solutions for Class 8 Science
NCERT Solutions for Class 8 English
NCERT Solutions for Class 8 Social Science
NCERT Solutions for Class 8 Hindi
NCERT Books
NCERT Books Class 12
NCERT Books Class 11
NCERT Books Class 10
NCERT Books Class 9
NCERT Books Class 8
Reference book solutions
Reference Book Solutions
HC Verma Solutions
RD Sharma Solutions
RS Aggarwal Solutions
NCERT Exemplar Solutions
Lakhmir Singh Solutions
DK Goel Solutions
TS Grewal Solutions
Sandeep Garg
Competitive Exams
Competitive Exams
JEE Main
JEE Advanced
NEET
Olympiad Preparation
NDA
KVPY
NTSE
CBSE
CBSE
CBSE Syllabus
CBSE Sample Paper
CBSE Worksheets
CBSE Important Questions
CBSE Previous Year Question Papers Class 12
CBSE Previous Year Question Papers Class 10
CBSE Important Formulas
ICSE
ICSE
ICSE Solutions
ICSE Class 10 Solutions
ICSE Class 9 Solutions
ICSE Class 8 Solutions
State boards
State Boards
AP Board
Bihar Board
Gujarat Board
Karnataka Board
Kerala Board
Maharashtra Board
MP Board
Rajasthan Board
Telangana Board
TN Board
UP Board
WB Board
Free Study Material
Free Study Material
Previous Year Question Papers
Sample Papers
JEE Main Study Materials
JEE Advanced Study Materials
NEET Study Materials
Olympiad Study Materials
Kids Learning
Ask Questions
Important Subjects
Physics
Biology
Chemistry
Maths
English
Commerce
Geography
Civics
Revision Notes
Revision Notes
CBSE Class 12 Notes
CBSE Class 11 Notes
CBSE Class 10 Notes
CBSE Class 9 Notes
CBSE Class 8 Notes
JEE MAIN
JEE Main
JEE Main Important Questions
JEE Main Important Chapters
JEE Main Notes
JEE Main Formulas
JEE Main Difference between
JEE Main Syllabus
JEE Main Physics Syllabus
JEE Main Mathematics Syllabus
JEE Main Chemistry Syllabus
JEE Main Previous Year Question Paper
JEE ADVANCED
JEE Advanced
JEE Advanced Important Questions
JEE Advanced Important Chapters
JEE Advanced Notes
JEE Advanced Formulas
JEE Advanced Difference between
JEE Advanced Syllabus
JEE Advanced Physics Syllabus
JEE Advanced Mathematics Syllabus
JEE Advanced Chemistry Syllabus
JEE Advanced Previous Year Question Paper
NEET
NEET
NEET Important Questions
NEET Important Chapters
NEET Notes
NEET Diagrams
NEET Difference between
NEET Syllabus
NEET Physics Syllabus
NEET Chemistry Syllabus
NEET Biology Syllabus
NEET Previous Year Question Paper
OFFLINE CENTRES
Muzaffarpur
Chennai
Bangalore
Patiala
Delhi
Hyderabad
Vijayawada
Visakhapatnam
Tiruchirapalli
Madurai
Coimbatore
Pune
Nagpur
Guntur
Aurangabad
Raipur
Bilaspur
Srinagar
Jammu
Bhubaneswar
Jhunjhunu
© 2025.Vedantu.com. All rights reserved
Privacy policyTerms and conditions |
8644 | https://documat.unirioja.es/descarga/articulo/866181.pdf | KUIL, vol. 26, 2003, 429-474 THE HISTORY OF CHINESE MATHEMATICS: THE PAST 25 YEARS ANDREA EBERHARD-BRÉARD REHSEIS (CNRS), France • Ph. D. Program in History • The Graduate Center, CUNY JOSEPH W. DAUBEN Ph. D. Program in History • The Graduate Center, CUNY XU YIBAO Ph.D. Program in History • The Graduate Center, CUNY RESUMEN El presente artículo presenta los más importantes logros de la investigacián sobre historia de las matemáticas chinas en el ŭltimo cuarto del siglo XX. Comienza con una breve panordmica sobre el estado de conocimientos y las principales figuras que realizaron las prirneras contribuciones fundamentales antes de 1975 para después examinar más detenidamente las aporta-ciones Ilevadas a cabo durante el ŭltimo cuarto de siglo: autores europeos, publica-ciones en inglés y, finalmente, la extraor-dinaria produccián en chino, en su mayor parte tras la Revolución Cultural.
ABSTRACT This paper presents the major accom-plishments of research over the past quar-ter century in the field of Chinese mathe-matics and its history. We begin with a brief overview of the progress of our knowledge of that history, and the major figures who contributed the fundamental early works prior to 1975, and then exam-ine more carefully the achievements of the past quarter century, beginning with a general overview of the subject before surveying in more detail the contributions made in the past twenty-five years, first by Europeans, then by scholars publishing in English, after which this survey turns to examine the extraordinary production of scholarship written in Chinese, most of it since the end of the Cultural Revolution.
Palabras Clave: Historiografia, Matemáticas, China, Siglo XX.
Recibido el 27 de febrero de 2003 I.S.S.N. 0210-8615 430 A. EBERHARD-BRÉARD, J.W. DAUBEN, XU YIBAO LLULL, 26 The history of mathematics has experienced an unprecedented increase in levels of interest and exponential growth in the number of publications in the past twenty-five years, but nowhere has this been more dramatic and unex-pected than in the amount of new material, including the discovery and editing of primary texts and a burgeoning secondary literature, all related to the histo-ry of Chinese mathematics. Whether one considers the collation and analysis of ancient texts, or the proliferating secondary literature covering virtually all periods of Chinese history, the number of publications and scholars working in China and the West on the history of Chinese mathematics is truly remark-able. It is thus particularly appropriate that in celebrating the past twenty-five years of Llull and its own contributions to the history of science, we reflect in what follows on the major accomplishments of research over the past quarter century in the field of Chinese mathematics and its history. We begin with a brief synopsis of the progress of our knowledge of that history, and the major figures who contributed the fundarnental early works prior to 1975, and then examine more carefully the achievements of the past quarter century, beginning with a general overview of the subject before surveying in more detail the con-tributions made in the past twenty-five years, first by Europeans, then by scholars publishing in English, after which this survey turns to examine the extraordinary production of scholarship written in Chinese, most of it since the end of the Cultural Revolution.
History of Mathematics: East and West It will come as no surprise that interest in the history of mathematics in China is of very long duration, and reaches back to antiquity itself, when schol-ars of the Han dynasty (206 BCE-220 CE) were already engaged in the retrieval, collation and study of even earlier works written in pre-Qin times (before 221 BCE), and went so far as to ascribe the beginnings of mathematics to mythological origins in Chinese prehistory. Later, systematic interest in his-tory of mathematics was stimulated by the arrival of the Jesuits in China.
Following the first translations of Western works into Chinese, beginning with the first six books of Euclid's Elements translated by Matteo Ricci and XU Guangqi in 1607, some scholars argued that this offered nothing new and they in turn began the search for early works of Chinese mathematics to prove their point'. MEI Wending (1633-1721) and the philological Qian-Jia School of scholars became the first serious historians of Chinese mathematics, followed in the last century by two scholars in particular, LI Yan (1892-1963) and QIAN LLULL, 26 THE HISTORY OF CHINESE MATHEMATICS: THE PAST 25 YEARS 431 Baocong (1892-1974), to whom the origins of all modern work on the history of Chinese mathematics may be traced2.
The history of Chinese mathematics as written by Western authors began in the nineteenth century, when French sinologists like Edouard Biot and mis-sionaries like the British writer Alexander Wylie began to take an active interest in making the history of Chinese science and mathematics better known in the West. Wylie's accounts in «Jottings on the Science of Chinese Arithmetic» appeared as a series of articles in the North China Herald in 1852, and were included as part of Wylie's book, Chinese Researches, in 1897 (reprinted as recerit-ly as 1966) 3. Wylie was the first in English to give a reliable account, for the most part, of Chinese mathematics. Early in the twentieth century, MIKAMI Yoshio wrote the first book devoted to The Development of Mathematics in China and Japan (Leipzig: Teubner, 1913; repr. New York: Chelsea, 1974) ), but more substantial treatments awaited the monumental work of Joseph Needham and the still on-going project he inspired following World War II to produce a series of volumes devoted to a thorough study of Chinese science and technology. This is discussed below in the section devoted to works on history of Chinese mathematics written in English, but we begin this survey of works devoted to the subject by European authors, before turning to those writing in English, including Chinese authors. The final section of this survey is devoted to the substantial literature on the subject produced in Chinese.
Continental European Contributions to History of Chinese Mathematics Communist China, before opening up to the West in the 1980s, had closer contacts in fields of science and technology with the former Soviet Union than with any other part of the world. Due to the political climate, Russian scholars could easily access sources and arrange for scholarly exchanges with historians of mathematics inside China in the 1950s. The Marxist approach, comparable to that of Joseph Needham in England, assumes that modern science is essentially international and based on contributions from different civilizations and epochs—a view that links the development of mathematics in China closely to socio-political factors.
Consequently, the first non-Asian scholarly group interested in the his-tory of mathematics in China was formed in Moscow around Professors ÉPvi-ra Iranovna Berezkina and Adol'f Pavlovich Yushkevich (1906-1993) at the Russian Academy of Sciences. In the 1960s Berezkina discussed and translated 432 A. EBERHARD-BREARD, J.W. DAUBEN, XU YIBAO LLULL, 26 seven out of the Ten Mathematical Classics (Suan jing shi shu, compiled during the seventh century CE) [4274-4283], and published the first general survey of the history of Chinese mathematics in Russian in 1980 . Yushkevich pub-lished a more limited overview of medieval mathematics in China in his Geschichte der Mathematik im Mittelalter , the major drawback of which was that he relied solely on secondary sources. Thus his treatment of the math-ematical content in the Chinese classics did not include the important commen-taries written by LIU Hui in 263 and by LI Chunfeng (602-670), both of which accompany all now extant editions. Only later, mainly through publications by Donald Wagner in the 1970s and Yushkevich's doctoral student Alexeï Volkov, currently the most active scholar of Russian origin, did Yushkevich learn more about the theoretical aspects of these commentaries written in early and medieval China on the mathematical classics. For a detailed evaluation of Yushkevich's contributions to the field of history of mathematics in China, see Karine Chemla, «Les travaux de A.P. Youschkevitch sur l'histoire des mathe-matiques en Chine,» in Studies in History of Matherrzatics Dedicated to A.P Y ouschkevitch, Eberhard Knobloch, et al., eds., Proceedings of the XXth International Congress of History of Science, Lige (Belgium), 20-26 July 1997, vol. 13 (Turnhout: Brepols, 2002), pp. 25-31. For Volkov's many publications in Russian on history of Chinese mathematics, see the CD-ROM version of The History of Mathematics from Antiquity to the Present: A Selective Annotated Bibliography, as described in note 3 above, items [4256-4259], [4300-4301], , , [4365-4366], , [4431-4432], , and .
Yushkevitch also maintained a very active correspondence with the German historian of mathematics Kurt Vogel (1888-1985), who published a German translation of the Nine Chapters on Mathematical Procedures (fiu zhang suan shu) in 1968: Chiu chang suan shu: Neun Biicher arithmetischer Technik. Ein chinesisches Rechenbuch fiir den praktischen Gebrauch aus der friihen Hanzeit (Braunschweig: Ostwalds Klassiker, Friedrich Vieweg Verlag, 1968) .4 0ne of their common interests was the transmission of problems between China and the Arabic world. Vogel's article on the circulation of a sur-veying problem («Ein Vermessungsproblem reist von China nach Paris,» Historia Mathematica, 10 (1983), pp. 360-367 ) constituted the point of departure for a conference on the transmission of mathematical ideas recently held in Italy in May of 2000. Four contributions to that meeting concerned China, and sought to shed new light on possible routes of transmission, although a problem-oriented approach seems to be less fruitful than an analy-sis based on the comparison of algorithms. Volkov's paper on the «Hundred LLULL, 26 THE HISTORY OF CHINESE MATHEMATICS: THE PAST 25 YEARS 433 Fowls Problem» and the solution of simultaneous indeterminate equations in different cultures reflects his recent investigations into hitherto unknown Vietnamese sources, i.e. an unpublished manuscript currently preserved in Hanoi and written around 1468. Volkov has recently suggested, based upon an examination paper in a Vietnamese treatise (the Chi minh lap thanh toan phap—Chinese: Zhiming licheng suanfa, preface dated 1822), that the suggest-ed solution procedures cannot be interpreted merely as mathematical demon-strations aimed at «proving» certain statements and results of the mathemati-cal problem to be solved. He argues that they must have also served as models for the state examination essays. Volkov presented a preliminary account of his work on this material in a lecture, «Demonstration in Chinese Mathematics: Argumentation for State Examinations,» at the Workshop on History and Historiography of Mathematical Proof in Ancient Traditions, 17-19 May 2002, a Joint Project with REHSEIS CNRS—University of Paris VII, the Maison des Sciences de l'Homme, and the Columbia University Institute for Scholars at Reid Hall, Paris.
In French, Volkov has also published a number of papers, including «Recherches sur les structures des textes chinois anciens en URSS,» in Modéles et structures des textes chinois anciens, Karine Chemla, Vera Dorofeeva-Lichtman, and Alexei Volkov, eds., Extréme-Orient, Extréme-Occident, 13 (Saint-Denis: Presses Universitaires de Vincennes, 1991), pp. 11-30; «Structure d'un traite mathematique: l'exemple du Hai dao suan jing,» in Modéles et struc-tures des textes chinois anciens, Karine Chemla, Vera Dorofeeva-Lichtman, and Alexei Volkov, eds., Extréme-Orient, Extréme-Occident, 13 (Saint-Denis: Presses Universitaires de Vincennes, 1991), pp. 93-99; and «Nombres et nom-bres,» in Sous les nombres, le monde: Matériaux pour l'histoire culturelle du nombre en Chine ancienne, Alexe'i Volkov, ed., Extréme-Orient, Extréme-Occident, 16 (Saint-Denis: Presses Universitaires de Vincennes, 1994), pp. 5-11.
In Germany, where Kurt Vogel was one of the first to institutionalize the history of science in the 1930s, there was no circle of historians of Chinese math-ematics. But Vogel's special project, to revise for yet a fourth time the Geschichte der Elementarmathematik by Johannes Tropfke (1866-1939), which Vogel planned as early as the 1960s and pursued until 1987 at the Lehrstuhl der Geschichte der Naturwissenschaften (Institute for History of Science) in Munich at the Ludwig-Maximilians-Universitát, did attract the interest of a sinologist, Hermann Kogelschatz. Kogelschatz worked on the sections of Tropfke's History concerning geometry in China and submitted contributions on LIU Hui's treat-434 A. EBERHARD-BREARD, J.W. DAUBEN, XU YIBAO LLULL, 26 ment of circle measurements, stereometric proof methods in ancient China, and geometric definitions found in the Mohist canon. Publication of the volume on geometry was unfortunately abandoned after Vogel's death in 1985. Kogelschatz, mainly interested in critical textual studies, nevertheless published two important papers in German on editorial problems in the edition by DAI Zhen (1723-1777) of the Nine Chapters [4241 and 4242], and compiled bibliographic data for the Ten Books of Mathematical Classics .
It was only at the end of the 1990s that another young German mathe-matician and sinologist, Andrea Breard, published her Ph.D. thesis in the field of Chinese science: Re-Kreation eines mathematischen Konzeptes im chinesis-chen Diskurs, which appeared as volume 42 of Boethius (Stuttgart: Franz Steiner Verlag, 1999) . Breard's analysis of the history of arithmetical series takes a diachronic approach to studying the development of this mathe-matical concept from the first to the nineteenth century. She discusses the treat-ment of series within the conceptual framework of piles of discrete obj ects that mathematically correspond to finite summations. Although written in German, some results of her book are also available to a larger public in English (see below), as well as in French: «La recomposition des mathematiques chez Zhu Shijie: la constitution d'un domaine autour du nombre quatre'», in Oriens—Occidens, 3 (2000), pp. 259-277. In German she has also written three articles, «Abakus,» «Chinesische Mathematik,» and «Japanische Mathematik,» as well as ten biographies of Chinese mathematicians ranging in time from antiquity to the twentieth century, for the Lexikon der Mathematik, Guido Walz, ed. (Heidelberg: Spektrum Akademischer Verlag, 2003, in 6 volumes or 1 CD). Subsequent to her Ph.D., she has investigated aspects of the introduc-tion of Western symbolical algebra into Chinese discourse, and she is now the only scholar working on the history of probability and statistics in China.
Breard owes much to one of her Ph.D. advisors, Karine Chemla in Paris, who can be considered one of the most active and innovative scholars studying the history of mathematical philosophy of ancient China. Her Ph.D. thesis, on LI Ye's Ce yuan hai jing (Sea Mirror of Circle Measurements) of 1248 , analyzes the relationship between the description of method (fa) and the details of method (cao) in the context of polynomial algebra used to solve geometrical problems. Her subsequent studies have centered around translation of the Nine Chapters with their above-mentioned commentaries into French. She has also examined modes of expression of theoretical content in ancient Chinese mathe-matical writings, such as the specific conduct of proof, the meaning of LLULL, 26 THE HISTORY OF CHINESE MATHEMATICS: THE PAST 25 YEARS 435 problem-structured texts, and the significance of diagrams found in Song-Yuan printings, as in «Les problemes comme champ d'interpretation des algorithmes dans Les neuf chapitres sur les procédures mathématiques et leurs commentaires.
De la resolution des systemes d'equations lineares,» Oriens—Occidens, 3 (2000), pp. 189-234; and «Variete des modes d'utilisation des tu dans les textes mathe-matiques des Song et des Yuan,» From Image to Action: The Function of Tu-Representations in East Asian Intellectual Culture (Paris: September 3-5, 2001, in press). Her conclusions on interactions with the Indian and Arabic worlds are mainly based on internal textual characteristics, for example, as she explained in «Interactions mathematiques entre la Chine, l'Inde et le Monde Arabe: quelles perspectives?» at a conference, Legons de l'histoire des sciences Arabes: Interaction scientifique des cultures, Book of Abstracts (Tripoli-Beyrouth, October 30-November 1, 2002), pp. 15-19.
In collaboration with Francesca Bray, FU Daiwie, HUANG Yilong, and Georges Metailie, Chemla serves as co-editor for the sections on China in Storia della scienza, Sandro Petruccioli, ed. (Roma: Enciclopedia Italiana, 2001, in eight volumes). Section I of volume II is devoted to «La scienza in Cina» (Science in China), pp. 1-608. A second section on the history of science in China after the seventeenth century is in preparation. A partial translation into Chinese of the volume already published has appeared as an issue of the jour-nal of the Ecole Frangaise d'Extreme-Orient: Faguo hanxue (French Sinology), 6 (2002), pp. 1-397.
Also working in Paris, at present the main geographic center for research on the history of Chinese mathematics and astronomy outside of China, are Catherine Jami and Jean-Claude Martzloff, among others. The lat-ter is best-known for his synoptic A History of Chinese Mathematics, the most authoritative introduction in a Western language for a general readership (French 1988, English 1997). Martzloff's Ph.D. thesis was devoted to the great-est Chinese mathematician and astronomer of seventeenth-century China, MEI Wending (1633-1721). The last part of the published version ana-lyzes the synthesis that MEI Wending forged between Euclidean geometry and ancient Chinese geometry, a topic that Martzloff elaborated in several articles [4484-4487] and extended into the modern period (see his article with LI Wenlin, «Apergu sur les echanges mathematiques entre la Chine et la France (1880-1949),» Archive for History of Exact Sciences, 53(3/4)(1998), pp. 181-200. Recently he has turned more towards the history of mathematical astron-orny in China, a field hitherto little-dealt with by non-Chinese authors, and 436 A. EBERHARD-BREARD, J.W. DAUBEN, XU YIBAO LLULL, 26 has recently published «Le calendrier chinois: cadre historique général, struc-ture, typologie et calcul,» in Les Calendriers. Leurs enjeux dans l'espace et dans le temps, Jacques Le Goff, Jean Lefort, and Perrine Mane, eds. (Paris: Somogy Editions d'Art, 2002), 3.XIV.
Catherine Jami is the main Western authority on the history of mathe-matics in China during the late Ming and early Qing dynasties (sixteenth to eighteenth centuries). Since her Ph.D. on MING Antu's development of trigonometric functions into power series: «Les methodes rapides pour la trigonometrie et le rapport precis du cercle (1774). Tradition chinoise et apport occidental en mathematiques,» Mémoires de l'Institut des Hautes Études Chinoises, 32 (Paris: College de France, 1990) , she has combined her skills as an historian of China with her mathematical knowledge to study the interaction of Western mathematics with Chinese tradition from an externalist point of view. Many of her articles are related to Jesuit activities at the court of the Emperor Kangxi (ruled 1662-1722), and to socio-political factors that played a crucial role in the circulation of scientific knowledge in China.
Jami recently co-edited a collection of articles on XU Guangqi (1562-1633), who in cooperation with Matteo Ricci translated the first six books of Euclid's Elements into Chinese (1607). Jami, in concert with Keizo HASHIMOTO, contributed a jointly-written article on XU's shaping of math-ematics and astronomy in the early seventeenth century. One of the co-editors, Peter Engelfriet from the Netherlands, has also worked extensively on the Chinese translation of Euclid's Elements. His Ph.D. thesis, published in English, places this work into the proper intellectual and cultural context, and offers translations of all statements of the propositions of the first six books with the newly forged geometric terminology and their counterparts in modern Chinese.
To conclude this section on European works on the history of Chinese mathematics, two additional authors should be mentioned who have also made important contributions but are no longer actively working on the subject. The Belgian scholar, Ulrich Libbrecht, analyzed six mathematical manuscripts found by the sinologist Paul Pelliot and the archaeologist Sir Aurel Stein at the beginning of the twentieth century in the caves of Dunhuang (Gansu) and Turfan (Xinjiang). Libbrecht shows . how the manuscripts provide important material related to metrology and decimal values, and to mathematical termi-nology in general . His major publication, Chinese Mathematics in the Thirteenth Century , is a commentary and annotated translation of the Mathematical Treatise in Nine Sections by QIN Jiushao (1257 AD). One of the LLULL, 26 THE HISTORY OF CHINESE MATHEMATICS: THE PAST 25 YEARS 437 most important advances made in this Song Dynasty work is a generalization of the method for extracting roots, applicable to quadratic equations with pos-itive, negative, integral, or decimal coefficients. Jock HOE received his Ph.D.
in Paris for his internalist study of the treatment of polynomial equa-tions in ZHU Shijie's Jade Mirror of Four Unknowns (1303 AD).
Unfortunately, the published version of HOE's thesis (under the name John HOE) does not include the extensive appendices, which include ZHU's Chinese text and translation of its 284 problems into modern symbolic terminology with a «semi-symbolic» language.
Contributions in English to the History of Chinese Mathematics As already noted, it is easy to date the revolution in the general level of awareness of and appreciation for ancient Chinese mathematics in the English-speaking world: the appearance in 1959 of volume three of Joseph Needham's monumental series of books devoted to Science and Civilisation in China. The third volume, written by Needham and WANG Ling, is devoted specifically to Mathematics and the Sciences of the Heavens and the Earth (Cambridge: Cambridge University Press, 1959) . Earlier, WANG had written his doctoral thesis (unpublished) on The «Chiu Chang Suan Shu» and the History of Chinese Mathematics During the Han Dynasty (Trinity College, Cambridge, 1956) .
Bibliographic guides One measure of the growing interest in history of Chinese mathematics was the appearance in 1984 of «A Brief Chronological and Bibliographic Guide to the History of Chinese Mathematics,» published by ANG Tian-Se and Frank J. Swetz in Historia Mathematica, 11 (1984), pp. 39-56 . A.P. Yushkevich added another 23 titles to this list in a subsequent issue of the journal, Historia Mathematica, 13 (1986), pp. 36-38 . Meanwhile, LAM Lay Yong con-tributed the entries for history of Chinese mathematics as part of a comprehen-sive bibliography (a special project of the International Commission on History of Mathematics) devoted to the history of mathematics, The History of Mathematics from Antiquity to the Present, Joseph W. Dauben, ed. (New York: Garland Press, 1985), pp. 423-425. Also useful are the sections written by Christopher Cullen and devoted to mathematical texts (the Zhou bi suan jing and the Jiu zhang suan shu) in Michael Loewe's Early Chinese Texts: 438 A. EBERHARD-BREARD, J.W. DAUBEN, XU YIBAO LLULL, 26 Bibliographical Guide (Berkeley, CA: The Society for the Study of Early China and The Institute of East Asian Studies, University of California, 1993) .
Special Encyclopedias, General Reference Works, and Collected Papers: The classic encyclopedic reference work for historians of science is the Dictionary of Scientific Biography, published under the general editorship of Charles C. Gillispie and comprised of sixteen volumes issued between 1970 and 1980, to which two supplemental volumes were added in 1990. These include scholarly articles devoted to a number of major figures in the history of Chinese mathematics, as do articles in the more recent Encyclopedia of the History of Science, Technology and Medicine in Non-Western Cultures, edited by Helaine Selin (Dordrecht: Kluwer, 1997) . For a single very general overview of the history of Chinese mathematics, see the recent article by Jean-Claude Martzloff, «Chinese Mathematics,» in the Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences, edited by Ivor Grattan-Guinness (London: Routledge, 1994, vol. 1, pp. 93-103) .
There have also been several works published recently that offer collec-tions of articles by various expert scholars on subjects dealing with the history of Chinese mathematics. These include two volumes published in the series of Boston Studies in the Philosophy of Science (under the editorship of Robert Cohen at Boston University), the first of which offers selected papers devoted to Philosophy and Conceptual History of Science in Taiwan, edited by LIN Cheng-Hung and FU Daiwie (Dordrecht: Kluwer Academic Publishers, 1993) [4504 and 4524]. Several years later, a similar volume appeared devoted to lead-ing contributors to history of science in mainland China, edited by FAN Dainian and Robert S. Cohen, Chinese Studies in the History and Philosophy of Science, trans. Kathleen Dugan and JIANG Mingshan (Dordrecht: Kluwer Academic Publishers, 1996) [4243, 4337, 4391, 4529, 4533, and 4534]. There is also the somewhat earlier volume of essays written by many of the leading fig-ures of history of science in mainland China, Ancient China's Technology and Science (Beijing: Foreign Languages Press, 1983) .
Comprehensive general histories The earliest comprehensive yet reliable history of Chinese mathematics in English was written by MIKAMI Yoshio and published as The Development of Mathematics in China and Japan (Leipzig: Teubner, 1913; LLULL, 26 THE HISTORY OF CHINESE MATHEMATICS: THE PAST 25 YEARS 439 repr. New York: Chelsea, 1974) . But it was Needham and WANG's third volume in the series devoted to Science and Civilisation in China (Cambridge: Cambridge University Press, 1959) , devoted in part to mathematics, that set the stage for the sudden proliferation of studies devoted to the subject over the past quarter century. Of the more recent books, LI Yan and DU Shiran's Chinese Mathematics, A Concise History, was translated into English by John N. Crossley and Anthony W.-C. LUN in 1987 (Oxford: Oxford University Press) . A decade later, Jean-Claude Martzloff's mas-terful survey, A History of Chinese Mathematics, was transiated by Stephen S.
Wilson (Berlin: Springer-Verlag, 1997) . This book is especially useful for the beginner, because it seeks to place the history of Chinese mathematics into its proper social and historical context. It is also a substantially revised version of the original published in French in 1987.
Continental European Contributions in English There is no general survey in English specifically devoted to history of ancient Chinese mathematics, but B.L. van der Waerden devoted two sections of his Geometry and Algebra in Ancient Civilizations (Berlin: Springer-Verlag, 1983) exclusively to Chinese mathematics, including Liu Hui's measurement of the circle and his methods of finding the volumes of a pyramid and sphere.
Originally trained in Spain at the University of Zaragoza but now teach-ing in Mexico, Jose Antonio Cervera devoted his dissertation to Spanish mis-sionaries in China, and focused in particular on mathematics and astronomy: Los Misioneros Españoles como Vía para los Intercambios Cientificos y Culturales entre el Extremo Oriente y Europa en los Siglos XVI y XVII (Zaragoza: University of Zaragoza, Faculty of Sciences, 1999). This has now been published as Ciencia Misionera en Oriente. Los misioneros españoles como vía para los intercambios científicos y culturales entre el Extremo Oriente y Europa en los sig-los XVI y XVII (Zaragoza: University of Zaragoza, «Cuadernos de Historia de la Ciencia,» no. 12, 2001). Among his publications in English are: «Spanish Friars in the Far East: Fray Juan Cobo and His Book Shi Lu,» Historia Scientiarum, 7(3)(1998), pp. 181-198; «Dominican Contributions to Science in the 16th and 17th Centuries: the Example of Fray Juan Cobo in East Asia,» in The Spread of the Scientific Revolution in the European Periphery, Latin America and Asia, A.
Lertora, E. Nicolaidis, and J. Vandersmissen, eds. (Turnhout, Belgium: Brepols Publishers, 2000), pp. 183-192; and «Two Spanish Cosmographers in the Philippines: Andres de Urdaneta and Martin de Rada,» in Historical Perspectives 440 A. EBERHARD-BREARD, J.W. DAUBEN, XU YIBAO LLULL, 26 on East Asian Science, Technology, and Medicine, Alan CHAN, Gregory Clancey, and Loy Hui CHIEH, eds. (Singapore: Singapore University Press and World Scientific Publishing, 2002), pp. 317-326.
Andrea Breard, already mentioned above, currently resides in New York City and is a Visiting Professor in the Ph.D. Program in History at the Graduate Center of the City University of New York. In addition to her pub-lications cited earlier in French and German, she has written in English as well about ZHU Shijie, SHEN Gua, and problems of mathematical terminology: «Shen Gua's Cuts,» Taiwanese Journal for Philosophy and History of Science, 10 (1998), pp. 141-162; «The Reading of Zhu Shijie,» in Current Perspectives in the History of Science in East Asia, selected papers from the 8th International Conference on the History of Science in East Asia held in Seoul, 26-31 August 1996, Yung Sik KIM and Francesca Bray, eds. (Seoul: Seoul National University Press, 1999), pp. 291-306 (Part V is devoted to Mathematical Sciences); «On Mathematical Terminology—Culture Crossing in 19th-Century China,» in New Terms for New Ideas: Western Knowledge & Lexical Change in Late Imperial China, Michael Lackner, Iwo Amelung, and Joachim Kurtz, eds. (Sinica Leidensia, vol. 52) (Leiden: Brill, 2001), pp. 305-326; and «Problems of Pursuit: Recreational Mathematics or Astronomy?» in From China to Paris: 2000 Y ears of Mathematical Transmission, Proceedings of a Conference Held at the Rockefeller Foundation Research and Conference Center, Bellagio, Italy, May, 2000, Yvonne Dold-Samplonius, Joseph W.
Dauben, Menso Folkerts, and Benno van Dalen, eds., a special issue of Boethius, vol. 46 (Stuttgart: Steiner Verlag, 2002), pp. 57-86. Also soon to appear by Andrea Breard are «Mediating Tradition and Innovation: The intro-duction of Probability Theory into Late Imperial Chinese Mathematical Culture,» Historia Mathematica (in press), and «Institutionalising Statistics in Early Modern Bureaucratic China: Sir Robert Hart and the Statistical Department of the Inspectorate General of Customs,» Journal of Modern Asian Studies (in press).
Karine Chemla, in addition to her many works in French, is also one of the most prolific authors writing in English on ancient Chinese mathematics.
She has published extensively on aspects of proof and demonstration in the Nine Chapters on Mathematical Procedures, and on the mathematician LIU Hui in particular. See for example her articles: «Different Concepts of Equations in The Nine Chapters on Mathematical Procedures and in the Commentary on it by Liu Hui (3rd Century),» Historia Scientiarum, 4(1994), pp. 113-137 ; LLULL, 26 THE HISTORY OF CHINESE MATHEMATICS: THE PAST 25 YEARS 441 «Relations between Procedure and Demonstration. Measuring the Circle in the Nine Chapters on Mathematical Procedures and the commentary by Liu Hui (3rd century),» in H.N. Jahnke, N. Knoche, and M. Otte, eds., History of Mathematics and Education: Ideas and Experiences (G&tingen: Vandenhoeck and Ruprecht, 1996), pp. 69-112 ; and «What is at Stake in Mathematical Proofs from Third Century China?» Science in Context,10 (2)(1997), pp. 227-251 . She has written as well about algorithmic methods in Chinese math-ematics, and notably influential examples are the articles «Should they read FORTRAN as if it were English?» Bulletin of Chinese Studies, 1 (1987), pp.
301-316 ; and «Theoretical Aspects of the Chinese Algorithmic Tradition (first to third century),» Historia Scientiarum, 42 (1991), pp. 75-98 .
Chemla has also treated root extraction methods and the problem of irra-tional numbers in such studies as «Fractions and Irrationals Between Algorithm And Proof In Ancient China,» Studies in History of Medicine and Science (New Series), 15(1-2)(1997-98), pp. 31-54 ; «Similarities between Chinese and Arabic Mathematical Documents (I): Root Extraction,» Arabic Sciences and Philosophy 2(4)(1994), pp. 207-266 ; and with Agathe Keller, «The Sanskrit karanis and the Chinese mian,» in From China to Paris: 2000 Y ears of Mathematical Transmission, Proceedings of a Conference Held at the Rockefeller Foundation Research and Conference Center, Bellagio, Italy, May, 2000, Yvonne Dold-Samplonius, Joseph W. Dauben, Menso Folkerts, and Benno van Dalen, eds., a special issue of Boethius, vol. 46 (Stuttgart: Steiner Verlag, 2002), pp. 87-132. Her most recent publication was issued in conjunction with the International Congress of Mathematicians held in Beijing in August of 2002: «China and World Mathematics,» Beijing Intelligencer (Beijing: Higher Education Press, and Heidelberg: Springer Verlag, 2002), pp. 42-48.
Other subjects on which Chemla has written include «Reflections on the World-Wide History of the Rule of False Double Position, or: How a Loop was Closed,» Centaurus, 39 (1997), pp. 97-120 ; and «History of Mathematics in China: A Factor in World History and a Source for New Questions,» (Proceedings of the International Congress of •Mathematicians, Berlin, August 17-27, 1998), Documenta Mathematica, Journal der Deutschen Mathematiker-Vereinigung, Extra-Volume ICM 1998, Vol.III (Invited Lectures), pp. 789-798 . Finally, on historiographic matters Chemla has written about Joseph Needham in particular: «The Rivers and the Sea: Analyzing Needham's Metaphor for the World History of Science,» in Situating the History of Science: Dialogues with Joseph Needham, S. Irfan 442 A. EBERHARD-BRÉARD, J.W. DAUBEN, XU YIBAO LLULL, 26 Habib and Dhruv Raina, eds. (New Delhi: Oxford University Press, 1999), pp.
220-244 .
Benno van Dalen is a member of the Institute for History of Science at the Johann Wolfgang Goethe University in Frankfurt, Germany. He has writ-ten primarily on the history of Chinese astronomy and astronomical tables, which are directly related to questions about the transmission of mathematical ideas between China and the Arabic and Persian speaking worlds. His most recent work on these matters is: «Islamic and Chinese Astronomy under the Mongols: a Little-Known Case of Transmission,» in From China to Paris: 2000 Y ears of Mathematical Transmission, Proceedings of a Conference Held at the Rockefeller Foundation Research and Conference Center, Bellagio, Italy, May, 2000, Yvonne Dold-Samplonius, Joseph W. Dauben, Menso Folkerts, and Benno van Dalen, eds., a special issue of Boethius, vol. 46 (Stuttgart: Steiner Verlag, 2002), pp. 327-356.
Peter M. Engelfriet received his Ph.D. in Chinese studies from the University of Leiden, and is the author of Euclid in China: the Genesis of the First Chinese Translation of Euclid's Elements, Books I-VI (Jihe yuanben, Beijing, 1607) and its Reception up to 1723 (Leiden: Brill, 1998). Earlier he published a related article describing the historical background of medieval and Renaissance Eucliciean texts: «The Chinese Euclid and its European Context,» in L'Europe en Chine. Interactions scientifiques, religieuses et cul-turelles aux XVIIe et XVIlle sQcles, C. Jami, and H. Delahaye, eds., Mémoires de l'Institut des Hautes Études Chinoises, vol. 34 (Paris: College de France, 1993), pp. 111-135 .
Another of the Parisian circle of historians of Chinese mathematics is Catherine Jami, who concentrates as noted previously on the Ming and early Qing dynasties. Among others, she has written «Western Influence and Chinese Tradition in an Eighteenth-Century Chinese Mathematical Work,» Historia Mathematica, 15 (1988), pp. 311-331 ; «Scholars and Mathematical Knowledge in the Late Ming and Early Qing,» Historia Scientiarum, 42 (1991), pp. 99-109 ; «Learning the Mathematical Sciences in the Late Ming and Early Ch'ing,» in Education and Society in Late Imperial China, B. Elman and A. Woodside, eds. (Berkeley: University of California Press, 1994), pp. 223-256 ; «From Louis XIV's Court to Kangxi's Court: A Institutional Analysis of the French Jesuit Mission to China (1662-1722),» in East Asian Science: Tradition and Beyond, Keizo HASHIMOTO, et al., eds.
(Osaka: Kansai University Press, 1995), pp. 493-499 ; «Western Devices LLULL, 26 THE HISTORY OF CHINESE MATHEMATICS: THE PAST 25 YEARS 443 for Measuring Time and Space: Clocks and Euclidean Geometry in Late Ming and Ch'ing China,» in Time and Space in Chinese Culture, Chun-Chieh HUANG and Erik archer, eds. (Leiden: Brill, 1995), pp. 169-200 ; and «Western Mathematics in China, Seventeenth Century and Nineteenth Century,» in Science and Empires, P. Petitjean, C. Jami, and A.-M. Moulin, eds.
(Dordrecht: Kluwer Academic Publishers, 1992), pp. 79-88 .
Jami has also focused some of her writing on specific individuals, including: «The French Mission and Verbiest's Scientific Legacy,» in Ferdinand Verbiest. Jesuit Missionary, Scientist, Engineer and Diplomat, John Witek S.J., ed. (Nettetal: Steyler Verlag, 1995), pp. 531-542 ; «Aleni's Contribution to Geometry in China. A study of the Jihe yaofa,» in Scholar from the West.
Giulio Aleni S.J. (1582-1649) and the Dialogue between Christianity and China, T. Lipiello and R.Malek, eds. (Nettetal: Steyler Verlag, 1997), pp. 555-572 ; «Mathematical Knowledge in the Chongzhen lishu,» in Western Learning and Christianity in China. The Contribution and Impact of Johann Adam Schall von Bell S.J. (1592-1666), Roman Malek, ed. (Nettetal: Steyler Verlag, 1998), pp. 661-674 ; and «History of Mathematics in Mei Wending's (1633-1721) Work,» Historia Scientiarum, 4 (1994), pp. 159-174 . On historiographical topics, Jami has written: «In Memoriam. Joseph Needham (December 9, 1900—March 24,1995),» Historia Mathematica, 24 (1996), pp. 1-5 ; and «Joseph Needham and the Historiography of Chinese Mathematics,» in Situating the History of Science: Dialogues with Joseph Needham, S. Irfan Habib and Dhruv Raina, eds. (New Delhi: Oxford University Press, 1999), pp. 260-278.
Also working in Paris, and leader of a team studying the famous Dun Huang manuscripts, is Jean-Claude Martzloff. In addition to his book on the history of Chinese mathematics already noted, he has written several articles in English, including a «Note on the Recent Chinese and Mongolian Translations of Euclid's Elements,» Historia Mathematica, 24 (1997), pp. 200-202 ; and «Matteo Ricci's Mathematical Works and their Influence,» in Proceedings of the International Symposium on Chinese-Western Cultural Interchange in Commemoration of the 400th Anniversary of the Arrival of Matteo Ricci S.J. in China (Taipei, September 11-16 1983) (Taipei: Furen Daxue Chubanshe,1983), pp. 889-895 . He also contributed an article on «Chinese Mathematical Astronomy» to the volume Mathematics Across Cultures. The History of Non-Western Mathematics, Helaine Selin and Ubiratan D'Ambrosio, eds.
(Dordrecht: Kluwer Academic Publishers, 2000), pp. 373-407.
444 A. EBERHARD-BREARD, J.W. DAUBEN, XU YIBAO LLULL, 26 The Belgian historian Ulrich Libbrecht has written several works in English, most notably his translation and commentary of the Song dynasty work by QIN Jiushao, Mathematical Treatise in Nine Sections (1257 AD), that includes among other things, an important generalization for the method of extracting roots: Chinese Mathematics in the Thirteenth Century. The Shu-shu Chiu-chang of Ch'in Chiu-shao (Cambridge, MA: MIT Press, 1973) .
Libbrecht has also written about the Dunhuang manuscripts; see for example «Mathematical Manuscripts from the Tunhuang Caves,» in Explorations in The History of Science and Technology in China, LI Guohao, et al., trans.
(Shanghai: Chinese Classics Publishing House, 1982), pp. 203-229 . On historiographic matters, he too has written about Joseph Needham: «Joseph Needham's Work in the Area of Chinese Mathematics,» Past and Present, 87 (1980), pp. 30-39 . Libbrecht's «New Studies on Chinese Mathematics. A Review Essay,» Chinese Science, 4 (1980), pp. 65-68 , surveys recent studies of the period from the Southern Song to the Yuan dynasties, including LAM Lay Yong's Critical Study of the Y ang Hui suan fa [item 4449], and Jock HOE's doctoral thesis at the University of Paris [items 4442 and 4538].
Donald B. Wagner wrote his dissertation on Proof in Ancient Chinese Mathematics: Liu Hui on the Volumes of Rectilinear Solids, a thesis he present-ed at the University of Copenhagen in 1975 . Wagner has written a num-ber of articles related to LIU Hui, especially on matters concerning the approx-imation of pi and derivations of the volume of the sphere. He has also questioned the authenticity of LIU Hui's commentary on the famous Nine Chapters: «Doubts Concerning the Attribution of Liu Hui's Commentary on the Chiu-chang suan-shu,» Acta Orientalia, 39 (1978), pp. 199-212 .
Among his other works: «Liu Hui and Tsu Keng-chih on the Volume of a Sphere,» Chinese Science, 3 (1978), pp. 59-79 (which draws parallels with Cavalieri's method and procedures used by LIU Hui and ZU Gengzhi); «An Early Derivation of the Volume of a Pyramid: Liu Hui, Third Century AD,» Historia Mathematica, 6 (1979), pp. 164-188 ; and «Proof of the Pythagorean Theorem by Liu Hui (Third Century A.D.),» Historia Mathematica, 12 (1985), pp. 71-73 .
MA Li, who now teaches in Sweden, wrote her dissertation with Joran Friberg at the Chalmers University of Technology on The Rule of False: Early Applications and Conjectured Transmissions (Department of Mathematics, Chalmers University of Technology, The University of GOteborg, 1993) .
LLULL, 26 THE HISTORY OF CHINESE MATHEMATICS: THE PAST 25 YEARS 445 British Contributions Of historians of science in England, Christopher Cullen and G.E.R.
Lloyd are noteworthy, the former for careful textual studies, including his recent translation with commentary of the early astronomical and mathemati-cal classic, the Zhou bi suan jing , and the latter for his comparative stud-ies of ancient Greek and Chinese mathematics. Cullen, who is Director of the Needham Research Institute in Cambridge, England, for study of the history of East Asian science, technology, and medicine, has written on such subjects as Chinese tangent tables , and recently published a general survey of the Chinese gou-gu (Pythagorean) theorem in which he considers the question of whether ancient Chinese mathematics included the notion of rigorous proof: «Learning from Liu Hui? A Different Way to Do Mathematics,» Notices of the AMS, 49(7)(August, 2002), pp. 783-790.
Cullen also considers historiographic issues in «How Can We Do the Comparative History of Mathematics? Proof in Liu Hui and the Zhou Bi,» Philosophy and History of Science. A Taiwanese Journal, 4 (1995), pp. 59-94.
Here Cullen is critical of the use of the word «proof» by J.W. Crossley and A.
W.-C. LUN in an article they devoted to «proof» in Chinese mathematics . Cullen argues that the word cannot be applied in the same sense to LIU Hui as to Euclid, and considers how best to understand LIU Hui's commen-taries on the Jiu zhang suan shu (Nine Chapters on Mathematical Procedures) and the even earlier Han dynasty text, the Zhou bi suan jing (Mathematical Classic of the Zhou Gnomon).
Geoffrey Lloyd has written about Chinese mathematics in comparative terms as well, contrasting the content and methods of ancient Greek and Chinese mathematics in his Adversaries and Authorities: Investigations into Ancient Greek and Chinese Science (Cambridge: Cambridge University Press, 1996) . Chapter 7 in particular is devoted to the different conceptions of the infinite that arise in Greek as opposed to Chinese mathematics. He has also written about history of mathematics more generally, and in «Learning by Numbers,» Extréme-Orient, Extréme-Occident, 16 (1994), pp. 153-167 , he emphasizes that historians should be primarily interested in the «problems investigated, in the styles of reasoning cultivated, in the relationships between `mathematics (however interpreted) and other types of inquiry, [and ] in the roles and value of mathematics' in society,» (p.153). Lloyd also offers a com-parative analysis in «Finite and Infinite in Greece and China,» Chinese Science, 13 (1996), pp. 11-34 .
446 A. EBERHARD-BRÉARD, J.W. DAUBEN, XU YIBAO LLULL, 26 Studies of Chinese Mathematics in the United States and Canada In the United States, after a four-month exchange under the auspices of the U.S. National Academy of Sciences and the Chinese Academia Sinica in 1988, Joseph W. Dauben undertook serious study of both the Chinese language and the history of Chinese mathematics. He has also worked with two graduate students in New York, HORNG Wann-Sheng and XU Yibao, in the Ph.D.
Program in History at the Graduate Center of the City University of New York. Among other subjects, Dauben has written about aspects of argumenta-tion and proof in ancient Chinese mathematics, «The Pythagorean Theorem' and Chinese Mathematics: Liu Hui's Commentary on the Gou-Gu Theorem in Chapter Nine of the Jiu Zhang Suan Shu,» in Amphora: Festschrift in Honor of Hans Wussing (Leipzig: B.G.Teubner, 1992), pp. 133-155 ; and «Ancient Chinese Mathematics: The Jiu Zhang Suan Shu versus Euclid's Elements.
Aspects of Proof and the Linguistic Limits of Knowledge,» Chandrasekhar Memorial Issue, International Journal of Engineering Science, 36 (1998), pp.
1339-1359 . He is currently working with colleagues in China, Singapore, and Taiwan to produce a critical edition in English with commentary of selections from the Ten Classics of ancient Chinese mathematics.
Dauben is also interested in the introduction of modern mathematics to China, and has written «Internationalizing Mathematics East and West: Individuals and Institutions in the Emergence of a Modern Mathematical Community in China,» in Mathematics Unbound: The Evolution of an International Mathematical Research Community, 1800-1945, Karen H.
Parshall and Adrian C. Rice, eds. (Providence, R.I., and London, England: The American Mathematical Society and the London Mathematical Society, 2002), pp, 253-285. His contribution to a symposium at the University of Zaragoza, Spain, is due to appear shortly as an article in Llull: «Mathematics and Ideology: The Politics of Infinitesimals/Marx, Mao and Mathematics: Nonstandard Analysis and the Cultural Revolution,» to appear as well in Proceedings of the III Simposio Internacional Galdeano (Zaragoza: University of Zaragoza, Spain,.September, 1996; in press) .
Roger Hart is a member of the Departments of History and of Asian Studies at the University of Texas at Austin. His work has concentrated large-ly on Ming and early Qing dynasty mathematics. Among his publications are «Translating the Untranslatable: From Copula to Incommensurable Worlds,» in Tokens of Exchange: The Problem of Translation in Global Circulations, Lydia H. LIU, ed. (Durham, N.C.: Duke University Press, 2000), pp. 45-73, LLULL, 26 THE HISTORY OF CHINESE MATHEMATICS: THE PAST 25 YEARS 447 which takes up, among other topics, the problem of translating Euclid's Elements from Latin into Chinese. He is currently working on a collection of essays, Cultural Studies of Chinese Science, Technology and Medicine, which he is editing and will include his own study of «Quantifying Ritual: Political Cosmology, Courtly Music, and Precision Mathematics in Seventeenth-Century China.» Frank Swetz is Professor of Mathematics and Education at Pennsylvania State University, and has a special interest in the history of Chinese mathemat-ics. He has published translations of several Chinese classical texts, including a translation with commentary of the Hai dao suan jing (The Sea Island Mathematical Classic) and Chapter 9 from the Jiu zhang suan shu (The Nine Chapters on Mathematical Procedures) on right triangles and the Chinese ver-sion of the Pythagorean theorem. The first he published with ANG Tianse, «A Chinese Mathematical Classic of the Third Century: The Sea Island Manual of Liu Hui,» Historia Mathematica, 13 (1986), pp. 99-117 , which led to Swetz's The Sea Island Mathematical Manual: Surveying and Mathematics in Ancient China (University Park, PA: Pennsylvania State University Press, 1992) ; the latter he published with T.I. KAO, Was Pythagoras Chinese?
An Examination of Right Triangle Theory in Ancient China (University Park, PA: The Pennsylvania State University Press,1977) . Swetz followed this some years later with a further account, «Right Triangle Concepts in Ancient China: From Application to Theory,» History of Science, 31 (1993), pp. 421-439 , and in collaboration with ANG Tianse, «A New Suggestion on Tzu Chu'ng-Chih's Method of Finding the Value of rt and its Significance in the History of Mathematics,» Papers on Far Eastern History, 16 (1977), pp.
161-165 .
Swetz has also published on topics of interest to classroom teaching, for example: «The Amazing Chiu Chang Suan Shu» and «The Piling Up of Squares in Ancient China,» both in Mathematics Teacher, 65 (1972), pp. 425-430 , and 70 (1977), pp. 72-79 , respectively; and «The Evolution of Mathematics in Ancient China,» Mathematics Magazine, 52 (1979), pp. 10-19 . He has also studied the progress of mathematics since the late Qing, in «The Introduction of Mathematics in Higher Education in China, 1865-1887,» Historia Mathematica, 1 (1974), pp. 167-179 ; and Mathematics Education in China, its Growth and Development (Cambridge, MA: The MIT Press, 1974) .
448 A. EBERHARD-BREARD, J.W. DAUBEN, XU YIBAO LLULL, 26 XU Yibao is writing his doctoral dissertation on the subject of the infinite in Chinese mathematics at the Ph.D. Program in History at the Graduate Center of the City University of New York. Previously, in 1991 he completed his Master's degree in history of Chinese mathematics under LI Di at Inner Mongolia Normal University in Huhehot, China. In addition to his publications in Chinese, he has written a number of works in English, including «Chinese-U.S. Mathematical Relations: 1859-1949,» in Mathematics Unbound: The Evolution of an International Mathematical Community, 1800-1945, Karen H.
Parshall and Adrian C. Rice, eds. (Providence, Rhode Island: American Mathematical Society & London: The London Mathematical Society, 2002,) pp.
287-309. «Bertrand Russell and the Introduction of Mathematical Logic in China,» History and Philosophy of Logic 24, no. 3 (2003), 181-196; and «The First Chinese Translation of the Last Nine Books of Euclid's Elements and Its Source,» Historia Mathematica (in press). Also in press is an historiographic study of Joseph Needham: «Joseph Needham and the Historiography of the History of Chinese Mathematics,» in Proceedings of the Symposium on the History of Science in Commemoration of the Centennial of the Birth of Joseph Needham, December 8-9, 2000, in Kaohsiung, Taiwan (to appear).
Alexeï Volkov, who as noted above studied with Adol'f Pavlovich Yushkevich at the Institute for History of Science in Moscow, has worked in Paris and Hong Kong, but currently teaches in Montreal, Canada. Many of his most recent publications are in English, and range from studies of the Chinese classic mathematical texts to works he has recently discovered in Vietnam. His dissertation, Matematika v drevnern Kitae II I-VII vv (Mathematics in Ancient China during the 3rd-7th Centuries A.D.) was completed in 1988 , a short abstract of which appeared in English in Historia Mathematica, 8 (1991), pp. 185-187. Subsequently, Volkov has published extensively in Russian, French, and English. Among his works in English are a number of studies devoted to calculation of pi: «Supplementary Data on the Values of n in the History of Chinese Mathematics,» Philosophy and the History of Science. A Taiwanese Journal, 3 (1994), pp. 95-110 ; «Calculation of 7c in Ancient China: from Liu Hui to Zu Chongzhi,» Historia Scientiarum, 4 (1994), pp.
139-157 ; «Zhao Youqin and his calculation of 7c,» Historia Mathematica, 24 (1997), pp. 301-331 [4395/4457]; and «The Mathematical Work of Zhao Youqin: Remote Surveying and the Computation of n,» Taiwanese Journal for Philosophy and History of Science, 8 (1997), pp. 129-189 . This was a spe-cial issue on science in fourteenth-century China, including a case study of the Daoist polymath ZHAO Youqin (1271-1335?), of which Volkov was the guest LLULL, 26 THE HISTORY OF CHINESE MATHEMATICS: THE PAST 25 YEARS 449 editor; see A. Volkov, «Science and Daoism: An Introduction,» Taiwanese Journal for Philosophy and History of Science, 8 (1997), pp. 1-58.
Volkov has also written about the Chinese means of computation using rods on a counting board: «Counting Rods,» in Instruments of Science: An Historical Encyclopedia, R. Bud and D.J. Warner, eds. (New York: Garland, 1998), pp. 155-156 . Recently, he has turned his attention to mathemati-cal works in Vietnam: «On the Origins of the Toan phap dai thanh (Great Compendium of Mathematical Methods),» in From China to Paris: 2000 Years of Mathematical Transmission, Proceedings of a Conference Held at the Rockefeller Foundation Research and Conference Center, Bellagio, Italy, May, 2000, Yvonne Dold-Samplonius, Joseph W. Dauben, Menso Folkerts, and Benno van Dalen, eds., a special issue of Boethius, vol. 46 (Stuttgart: Steiner Verlag, 2002), pp. 369-410.
The Canadian-American linguist Brendan Gillon contributed «Introduction, Translation, and Discussion of Chao Chun-Ch'ing's Notes to the Diagrams of Short Legs and Long Legs and of Squares and Circles',» Historia Mathematica, 4 (1977), pp. 253-293 .
Australia and New Zealand Jock Hoe, who teaches at Massey University in New Zealand, obtained his doctorate at the University of Paris. Among his writings on history of Chinese mathematics in English is: «A Problem in the Siyuan yujian: The Jade Mirror of the Four Unknowns,» in Proceedings of the First New Zealand International Conference on Chinese Studies, Waikato University China Papers, vol. 2 (Hamilton, New Zealand: University of Waikato, 1972) .
In Australia, John N. Crossley and Anthony W.-C. LUN have collabo-rated in translating two important works into English, LI Yan and DU Shiran's Chinese Mathematics, A Concise History (Oxford: Clarendon Press, 1987) ; and with SHEN Kangshen , a translation and edition with substantial commentary: The Nine Chapters on the Mathematical Art—Companion and Commentary (Oxford: Oxford University Press, 1999). Crossley and Lun have also published a joint work on the concept of proof in early Chinese mathematics, «The Logic of Liu Hui and Euclid as Exemplified in Their Proofs of the Volume of a Pyramid,» Philosophy and the History of Science. A Taiwanese Journal, 3 (1994), pp. 11-27 . For a critique of and reply to this work, see the article by Christopher Cullen noted above.
450 A. EBERHARD-BREARD, J.W. DAUBEN, XU YIBAO LLULL, 26 Chinese Authors in Australia, Hong Kong, Malaysia, and Singapore ANG Tian-Se wrote his doctoral dissertation, Study of the Mathematical Manual of Chang Ch'iu-chien (unpublished), at the University of Malaysia in 1969 . He has published several articles since on aspects of computational astronomy and right triangles: «The Use of Interpolation Techniques in Chinese Calendar,» Oriens Extremus, 23 (1976), pp. 135-151 ; and «Chinese Interest in Right-Angled Triangles,» Historia Mathematica, 5 (1978), pp. 264-265 . His most ambitious work is the book he wrote with LAM Lay Yong, Fleeting Footsteps. Tracing the Conception of Arithmetic and Algebra in Ancient China (Singapore: World Scientific, 1992) , in which the authors argue in terms of a careful trans-lation and study of the Sunzi suan jing (Master Sun's Mathematical Classic) that the decimal place-valued number system actually originated in China and was transmitted therefrom to India and then via Islamic mathematics to the West. LAM and ANG have also jointly authored several articles together, including «The Conceptual Origin of our Numeral System and the Symbolic Form of Algebra,» Archive for History of Exact Sciences, 36 (1986), pp. 183-195 ; and «The Earliest Negative Numbers: How They Emerged from a Solution of Simultaneous Linear Equations,» Archives internationales d'his-toire des sciences, 37 (1987), pp. 222-262 . ANG also collaborated with Frank Swetz on an article devoted to the double difference method, «A Chinese Mathematical Classic of the Third Century: The Sea Island Manual of Li Hui,» Historia Mathematica, 13 (1986), pp. 99-117 .
HO Peng-Yoke, who now resides in Australia, is the author of a num-ber of biographies of Chinese mathematicians included in the Dictionary of Scientific Biography, «Ch'in Chiu-shao» [Qin Jiushao] , «Chu Shih-chieh» [Zhu Shijie] , «Li Chih» [Li Ye] , «Liu Hui» , and «Yang Hui» . He also wrote «The Lost Problem of the Chang Ch'iu-chien Suan Ching, a Fifth-Century Chinese Mathematical Manual,» Oriens Extremus, 12 (1965), pp. 37-53 . This is of particular interest because the extant versions of the Zhang Qiujian suan jing are incomplete; in this article the author discusses the missing portions that occur between the end of the second chapter and the beginning of the third chapter. HO Peng Yoke's book, Li, Qi, and Shit • An Introduction to Science and Civilization in China (Hong Kong: Hong Kong University Press, 1985; revised, New York: Dover, 2000), devotes the second of its four parts to mathematics (pp. 53-112).
LLULL, 26 THE HISTORY OF CH1NESE MATHEMATICS: THE PAST 25 YEARS 451 Of all Chinese authors writing in English, none has been as prolific or as influential as LAM Lay Yong, Professor Emerita from the National University of Singapore. In addition to writing two books, Critical Study of the Y ang Hui Suan Fa, a 13th-Century Mathematical Treatise (Singapore: Singapore University Press, 1977) ; and (with ANG Tian-Se) Fleeting Footsteps. Tracing the Conception of Arithmetic and Algebra in Ancient China (Singapore: World Scientific, 1992) , she has also written extensively on the most famous of the Ten Classics, the Nine Chapters: «Yang Hui's com-mentary on the ying nu chapter of the Chiu Chang Suan Shu,» Historia Mathematica, 1 (1974), pp. 54-55 ; and «Jiu Zhang Suanshu (Nine Chapters on the Mathematical Art): An Overview,» Archive for History of Exact Sciences, 47 (1994), pp. 1-51 . Among the other Ten Classics of ancient Chinese mathematics, in addition to the Sunzi suan jing (translated with commentary in Fleeting Footsteps), she has written on the «Zhang Qiujian Suanjing (The Mathematical Classic of Zhang Qiujian): An Overview,» Archive for History of Exact Sciences, 50 (1997), pp. 201-240 ; and (with SHEN Kangshen), «Right-angled Triangles in Ancient China,» Archive for History of Exact Sciences, 30 (1984), pp. 87-112 .
LAM Lay Yong is also the author of numerous articles, many dealing with the Y ang Hui suan fa and other thirteenth-century works, including «On the Existing Fragments of Yang Hui's Hsiang Chieh Suan Fa [Xiang jie suan fa],» Archive for History of Exact Sciences, 6 (1969), pp. 82-88 ; «The Jih Y ung Suan Fa [Ri yong suan fa]: an Elementary Textbook of the Thirteenth Century,» Isis, 63 (1972), pp. 370-383 ; and «Chu shih-chieh's Suan Hs'ueh Ch'i-meng (Introduction to Mathematical Studies),» Archive for History of Exact Sciences, 21 (1979), pp. 1-31 . This work gives a gener-al introduction to Zhu Shijie's Suan xue qi meng of 1299, including aspects of numeration, metrology and special terminology found in this book, as well as the influence of the Jiu zhang suan shu (The Nine Chapters) on Zhu Shijie's book. LAM Lay Yong has also written about another thirteenth-century work, in collaboration with ANG Tian-Se: «Li Ye and his Y i Gu Y an Duan (Old Mathematics in Expanded Sections),» Archive for History of Exact Sciences, 29 (1984), pp. 237-266 , a study devoted to a work by Li Ye written in 1259 (printed in 1282).
She has also written extensively about the Chinese origins of the deci-mal place-value system: «Linkages: Exploring the Similarities Between the Chinese Rod Numeral System and our Numeral System,» Archive for History 452 A. EBERHARD-BREARD, J.W. DAUBEN, XU YIBAO LLULL, 26 of Exact Sciences, 37 (1979), pp. 365-392 ; «The Conceptual Origin of our Numeral System and the Symbolic Form of Algebra,» Archive for History of Exact Sciences, 36 (1986), pp. 183-195 ; «A Chinese Genesis: Rewriting the History of Our Numeral System,» Archive for History of Exact Sciences, 38 (1988), pp. 101-108 ; and «The Development of Hindu-Arabic and Traditional Chinese Arithmetic,» Chinese Science, 13 (1996), pp. 35-54 .
LAM Lay Yong has also studied the different computational methods of the ancient Chinese: «The Geometrical Basis of the Ancient Chinese Square Root Method,» Isis, 61 (1969), pp. 96-102 ; «The Chinese Connexion Between the Pascal Triangle and the Solution of Numerical Equations of any Degree,» Historia Mathematica, 7 (1980), pp. 407-424 , and «Chinese Polynomial Equations in the Thirteenth Century,» in Explorations in The History of Science and Technology in China, LI Guohao, et al., eds. (Shanghai: Chinese Classics Publishing House, 1982), pp. 231-272 .
In Hong Kong, SIU Man-Keung teaches at the University of Hong Kong, and has written on matters related to education and history of mathe-matics, for example: «Mathematics Education in Ancient China: What Lesson Do We Learn From It?» Historia Scientiarum, 4 (1995), pp. 223-232 .
Among his studies of ancient Chinese mathematics: «Pyramid, Pile, and Sum of Squares,» Historia Mathematica, 8 (1981), pp. 61-66; and «Proof and Pedagogy in Ancient China: Examples from Liu Hui's Commentary on the Jiu Zhang Suan Shu,» Educational Studies in Mathematics, 24 (1993), pp. 345-357 .
English Works by Chinese Authors in the PRC and Taiwan Both mainland China and Taiwan have active groups of historians of mathematics, most of whom work on one aspect or another of the history of Chinese mathematics. Of authors in Taiwan, the best-known is HORNG Wann-Sheng, who has written extensively on ancient and modern mathemat-ics. His doctoral dissertation was devoted to Li Shanlan, The Impact of Western Mathematics in China During the late Nineteenth Century (New York: City University of New York, 1991) . His articles on ancient Chinese mathematics include: «How Did Liu Hui Perceive the Concept of Infinity: A Revisit,» Historia Scientiarum, 4 (1995), pp. 207-222 , where the author considers questions of circle measurement and infinite divisibility.
Concerning mathematics in the modern period, he has also written: «Hua Hengfang (1833-1902) and His Notebook on Learning Mathematics Xue Suan LLULL, 26 THE HISTORY OF CHINESE MATHEMATICS: THE PAST 25 YEARS 453 Bi Tan',» Philosophy and the History of Science. A Taiwanese Journal, 2(2)(1993), pp. 27-76 ; and «Chinese Mathematics at the Turn of the 19th Century: Jiao Xun, Wang Lai and Li Rui,» in Philosophy and Conceptual History of Science in Taiwan, LIN Cheng-Hung and FU Daiwie, eds., Boston Studies in the Philosophy of Science, vol. 141 (Dordrecht: Kluwer Academic Publishers: 1993), pp. 167-208 . This work analyzes the advances made by Chinese mathematicians working in the nineteenth century in the tradition of the Qian-Jia school, which revitalized interest in ancient Chinese mathe-matics to show that the results and methods of Western mathematics were already known in traditional Chinese mathematics.
Professor of mathematics at the Academia Sinica in Nangang, Taiwan, LIH Ko-Wei (LI Guo-wei) has written on both ancient and modern Chinese mathematics alike: «From One Gnomon to Two Gnomons: A Methodological Study of the Method of Double Differences,» in Philosophy and Conceptual History of Science in Taiwan, LIN Cheng-Hung and FU Daiwie, eds., Boston Studies in the Philosophy of Science, vol. 141 (Dordrecht: Kluwer Academic Publishers: 1993), pp. 149-165 ; «Bao Qi-Shou (Bao Qishou) and His Polyhedral Hun Y uan Tu,» in Science and Technology in Chinese Civilization, CHEN Cheng-Yih, Roger Cliff, and CHEN Kuei-Mei, eds. (Singapore: World Scientific Publishing Co., 1987), pp. 93-107 ; and «Bao Qi-shou (Bao Qishou) and His Polyhedral Hun Y uan Tu,» in Philosophy and Conceptual History of Science in Taiwan, LIN Cheng-Hung and FU Daiwie, eds., Boston Studies in the Philosophy of Science, vol. 141 (Dordrecht: Kluwer Academic Publishers: 1993), 209-220 .
FU Daiwie, a graduate of Columbia University (New York), teaches at Tsing Hua University in Hsinchu (Taiwan), and is the author of «Why Did Liu Hui Fail to Derive the Volume of a Sphere?» Historia Mathematica, 18 (1991), pp. 212-238 .
The People's Republic of China No mathematician in China has done more to promote history of math-ematics than the topologist and proof theorist WU Wen-tsun (Wu Wenjun/Wu Wenchun). Among his contributions are «Recent Studies of the History of Chinese Mathematics,» Proceedings of the International Congress of Mathematicians (Berkeley, California, 1986,) pp. 1657-1667 . WU sets out two principles for studying ancient Chinese mathematics, i.e. «All conclusions 454 A. EBERHARD-BREARD, J.W. DAUBEN, XU YIBAO LLULL, 26 drawn should be based on original texts fortunately preserved up to the present time,» and «All conclusions drawn should be based on reasoning in the manner of our ancestors in making use of knowledge and in utilizing auxiliary tools and methods available only at that ancient time,» (p.1657). Among his earliest con-tributions to analysis of the methods of ancient Chinese mathematics is «The Out-In Complementary Principle,» in Ancient China's Technology and Science (Beijing: Foreign Languages Press, 1983), pp. 66-89 . His most recent pub-lication was a contribution to the Beijing Intelligencer published in conjunction with the International Congress of Mathematicians held in Beijing in August of 2002: «A Tentative Comparative Study of Mathematics. Developments in Ancient China and Ancient Greece,» Beijing Intelligencer (Beijing: Higher Education Press, and Heideiberg: Springer Verlag, 2002), pp. 23-41.
A leading member of the Institute for History of Natural Science of the Chinese Academy of Sciences in Beijing is Professor GUO Shuchun, an expert on Liu Hui and the Nine Chapters on Mathematical Procedures. He has writ-ten extensively on this subject in Chinese, and edited numerous texts related to the history of Chinese mathematics from antiquity to the present. Among his publications in English is «The Numerical Solution of Higher Equations and the Tianyuan Method,» in Ancient China's Technology and Science (Beijing: Foreign Languages Press, 1983), pp. 111-123 . This work studies the «kai fang shu,» a Chinese method for solving algebraic equations developed in the Song Dynasty (960-1279), as well as related methods for extracting roots by successive additions and multiplications.
LI Di, Professor of the History of Mathematics at Inner Mongolia Normal University, has trained several generations of historians of mathemat-ics in China, and is editor of the Journal of the Cultural History of Mathematics, which is published in English, but not on an annual basis. LI Di has written on all periods of the history of Chinese mathematics, and in English has published: «The Literary Problems in the Mathematical Works of Ming Dynasty in China,» Journal of the Cultural History of Mathematics, 1 (1991), pp. 19-27 ; and «400 Years of the History of Mathematics in China. An Introduction to the Major Historians of Mathematics Since 1592,» Historia Scientiarum, 4 (1994), pp. 103-111 . In 1978, with his colleague BAI Shangshu, LI Di discovered ten calculators in a storeroom of the Palace Museum in Beijing, and together with Michael R. Williams they have pub-lished: «Chinese Calculators Made During the Kangxi Reign in the Qing Dynasty,» Annals of the History of Computing, 14 (4)(1992), pp. 63-67 .
LLULL, 26 THE HISTORY OF CHINESE MATHEMATICS: THE PAST 25 YEARS 455 LI Wenlin is a member of the Institute of Mathematics of the Chinese Academy of Sciences in Beijing, and has devoted much of his research to the history of modern mathematics. In English he has written: «The Chinese Indigenous Tradition of Mathematics and the Conceptual Foundation to Adopt Modern Mathematics in the 19 11< Century,» in Zhongguo shuxueshi lun-wenji (Collected Papers on the History of Chinese Mathematics), BAI Shangshu, LI Di, and SHEN Kangshen, eds. (Jinan: Shandong Educational Press, 1996), pp. 146-156 , and with YUAN Xiangdong, «The Chinese Remainder Theorem,» in Ancient China's Technology and Science (Beijing: Foreign Languages Press, 1983), pp. 99-110 .
LIU Dun is Director of the Institute for History of Natural Science of the Chinese Academy of Sciences in Beijing, and an authority on ancient and modern Chinese mathematics. Among other studies, in English he has pub-lished: «A Comparison of Archimedes and Liu Hui's Studies of Circles,» in Chinese Studies in the History and Philosophy of Science and Technology, FAN Dainian and Robert S. Cohen, eds., Kathleen Dugan and JIANG Mingshan, trans. (Dordrecht: Kluwer Academic Publishers, 1996), pp. 279-289 ; and «A Homecoming Stranger: Transmission of the Method of Double False Position and the Story of Hiero's Crown,» in From China to Paris: 2000 Y ears of Mathematical Transmission, Proceedings of a Conference Held at the Rockefeller Foundation Research and Conference Center, Bellagio, Italy, May, 2000, Yvonne Dold-Samplonius, Joseph W. Dauben, Menso Folkerts, and Benno van Dalen, eds., a special issue of Boethius, vol. 46 (Stuttgart: Steiner Verlag, 2002), pp. 157-166. He has also written, in collaboration with Joseph W. Dauben, «Historians of Mathematics in China,» in Writing the History of Mathematics: Its Historical Development, Joseph Dauben and Christoph J.
Scriba, eds. (Basel: Birkháuser, 2000), pp. 297-306 . LIU Dun also con-tributed the biographies of LI Yan and QIAN Baocong to this same volume.
MEI Rongzhao, also a member of the Institute for History of Natural Science of the Chinese Academy of Sciences in Beijing, has written on aspects of proof and argumentation in Chinese mathematics, and on the origins of arithmetic in ancient China: «The Decimal Place-Value Numeration and the Rod and Bead Arithmetics,» in Ancient China's Technology and Science (Beijing: Foreign Language Press, 1983), pp. 57-65 ; «Mathematical Formalism in Ancient China,» in Science and Technology in Chinese Civilization, CHEN Cheng-Yih, Roger Cliff, and CHEN Kuei-Mei, eds.
456 A. EBERHARD-BRÉARD, J.W. DAUBEN, XU YIBAO LLULL, 26 (Singapore: World Scientific Publishing Co., 1987, pp. 53-75 ; and «Liu Hui's Theories of Mathematics,» in Chinese Studies in the History and Philosophy of Science and Technology, FAN Dainian and Robert S.Cohen, eds., Kathleen Dugan and JIANG Mingshan, trans. (Dordrecht: Kluwer Academic Publishers, 1996), pp. 243-254 .
QU Anjing teaches in the Department of Mathematics at Northwest University in Xi'an, China, and has written in English on various aspects of ancient Chinese mathematics, including: «Bian Gang: A Mathematician of the 9th Century,» Historia Scientiarum, 6 (1996), pp. 17-30 , in which the author discusses calendrical research and the last calendar of the Tang Dynasty, the Chong Xuan Calendar (892 A.D.); and «On Hypotenuse Diagrams in Ancient China,» Centaurus, 39 (1997), pp. 193-210 .
SHEN Kangshen, of Hangzhou University (since 1998, merged to become part of Zhejiang University), is an expert on the Nine Chapters, among other topics related to history of ancient Chinese mathematics. His most ambi-tious work, a translation with commentary of the Nine Chapters, was recently published in English with the collaboration of John N. Crossley and Anthony W.-C. LUN, The Nine Chapters on the Mathematical Art. Companion and Commentary (Oxford: Oxford University Press, 1999). In addition to numerous works in Chinese, in English SHEN has published: «Historical Development of the Chinese Remainder Theorem,» Archive for History of Exact Sciences, 38 (1988), pp. 285-305 . He has also collaborated on several occasions with LAM Lay Yong, and together they have written: «The Chinese Concept of Cavalieri's Principle and its Applications,» Historia Mathematica, 12 (1985), pp.
219-228 , which presents a method developed by LIU Hui to calculate the volume of a sphere that is similar to Cavalieri's method; «Mathematical Problems on Surveying in Ancient China,» Archive for History of Exact Sciences, 36 (1986), pp. 1-20 ; and «Methods of Solving Linear Equations in Traditional China,» Historia Mathematica, 16 (1989), pp. 107-122 .
WANG Yusheng is a member of the Institute for History of Natural Science of the Chinese Academy of Sciences and Director of the Beijing Science and Technology Museum. He has written in English about the development of modern mathematics in China, including: «Hua Hengfang: Forerunner and Disseminator of Modern Science in China,» in Chinese Studies in the History and Philosophy of Science and Technology, FAN Dainian and Robert S. Cohen, eds., Kathleen Dugan and JIANG Mingshan, trans. (Dordrecht: Kluwer Academic Publishers, 1996), pp. 369-394 ; and «Li Shanlan: Forerunner LLULL, 26 THE HISTORY OF CHINESE MATHEMATICS: THE PAST 25 YEARS 457 of Modern Science in China,» also in Chinese Studies in the History and Philosophy of Science and Technology, pp. 345-368 .
ZHANG Dianzhou teaches in the Department of Mathematics at East-China Normal University in Shanghai, and has written primarily about the histo-ry of modern mathematics in the West and in China. In English, he collaborated with Joseph W. Dauben to write: «Mathematical Exchanges Between the United States and China,» in History of Modern Mathematics, Eberhard Knobloch and David E. Rowe, eds. (Orlando: Academic Press, 1994), pp. 263-297 .
Journals In closing, it is worth noting that there are a number of journals, includ-ing Llull, that publish material related to history of Chinese mathematics. Of those publishing articles in English, the most important are The Archive for History of Exact Sciences, Historia Mathematica, Philosophy and the History of Science. A Taiwanese Journal, and Historia Scientiarum. Other journals that from time to time include works devoted to history of Chinese mathematics are Centaurus; East Asian Science, Technology and Medicine (formerly Chinese Science); Extthne-Orient, Extrème-Occident; Science in Context; and I sis.
There is also the journal in English edited by LI Di that appears occasionally, as mentioned above:Journal of the Cultural History of Mathernatics (Huhehot, Inner Mongolia Normal University, PRC).
Contributions by Historians of Chinese Mathematics Writing in Chinese The earliest examples of documents devoted to the history of mathe-matics in China can be traced back to the sixteenth, or the eleventh, or even much earlier to the eighth century, depending upon how the word «historical» is interpreted [LIU and Dauben 2002, p. 253; LI Di 1998, p. 64; LI Di 2001, p.
412; Martzloff 1997, p. 10]. Many early historical works now seem to be lost and unfortunately no longer exist, but among extant documents, history of mathematics in any sense approaching our contemporary understanding of the term did not occur until the seventeenth century, when Chinese mathemati-cians, including the most influential of all, MEI Wending, launched a campaign to «prove» that Western mathematics as introduced by the Jesuits actually originated in ancient China. However lacking in evidence such studies may have been, they nevertheless prompted systematic investigation of the most 458 A. EBERHARD-BRÉARD, J.W. DAUBEN, XU YIBAO LLULL, 26 basic issues on and methods of ancient Chinese mathematics. One example is that the true meaning of the Tian yuan shu, a method which was developed in the twelfth century (possibly even earlier) for using unknown elements in Chinese algebra, had been forgotten. By the sixteenth and seventeenth cen-turies, no one understood the method, but it was subsequently rediscovered as mathematicians tried to discover the Chinese «origin» of algebra itself.
Historical studies of mathematics were also pursued under the Emperors Qianlong (reigned 1735-1796) ancl Jiaqing (reigned 1796-1820), when textual criticism of the Chinese classics, including ancient mathematical texts, was popular. One major result was that classics of mathematics such as the Suan jing shi shu, the ten books of mathematical classics, were brought under scrutiny, and better versions of these books were subsequently pub-lished. Another result of historical research at this time was the appearance of a book entitled Chouren zhuan, a compilation of biographies and excerpts from the writings of the most important mathematicians and astronomers first printed in 1799. Three supplements were issued in the following century, bringing the total of individuals covered to nearly seven hundred, and ranging from legendary figures in prehistory to such eminent nineteenth-century mathematicians as LI Shanlan. Nearly two hundred Japanese and Western scholars were also included [FU 1990, pp. 219-260; LIU and Dauben 2002, pp.
255-259].
By the twentieth century, not only the example of MIKAMI Yoshio's The Development of Mathematics in China and Japan (Leipzig: Teubner, 1913; repr.
New York: Chelsea, 1974) , but the exposure to Western methods of Chinese students studying abroad, stimulated interest in more technical, scientif-ic historical research. In the decades prior to World War II, a new generation of historians of mathematics emerged, including LI Yan, QIAN Baocong, XU Chunfang, YAN Dunjie, and ZHANG Yong [Martzloff 1997, pp. 11-12; LI Di 1998, p. 69; and Dauben and Scriba 2002, pp. 259-260]. One consequence of this approach was an excessive use of symbols and formulas. The influence of Chouren zhuan, however, was still evident because the new generation still focused more on presenting historical evidence than on interpretation. But the research done by a handful of scholars due largely to their own interests built a solid foundation for future studies. In particular, they provided a concrete body of new historical works to which Joseph Needham and his collaborator, WANG Ling, could turn for support as they wrote their account of Chinese mathematics in the third volume of Science and Civilisation in China (1959) .
LLULL, 26 THE HISTORY OF CHINESE MATHEMATICS: THE PAST 25 YEARS 459 When the People's Republic of China was founded in 1949, historical studies on mathematics, like other studies, were encouraged. LI Yan, QIAN Baocong, and YAN Dunjie were called to Beijing to form a research group devoted to history of mathematics within the newly-founded Chinese Academy of Sciences. From this small group, a more ambitious research pro-gram for the history of science generally was established in 1957, and with it, the Institute for History of Natural Science. To help stimulate the study of his-tory of mathematics, the Science Press in Beijing reprinted articles and books published by LI Yan in five volumes: Zhongsuanshi luncong (Series on History of Chinese Mathematics, 1954-1955). In the following decade, a number of high-quality studies were produced, including LI Yan's studies on Chinese methods of interpolation (1957), QIAN Baocong's collation and commentary on the Suan jing shi shu (Ten Books of Mathematical Classics), and his influential Zhongguo shuxueshi (A History of Chinese Mathematics, 1964).
Unfortunately, this promising beginning was soon brought to a halt by the Cultural Revolution, and only after the Revolution came to an end did a new phase in the history of science in China begin, a phase that would also see the unprecedented development of history of mathematics. The last twenty-five years has indeed been extraordinary, not only for the status of history of Chinese mathematics in China, but abroad as well, as outlined in the preced-ing sections of this article. What follows is a brief survey of studies on the his-tory of Chinese mathematics in the last quarter of the twentieth century.
Surveying publications in Chinese by scholars in both mainland China and Taiwan, it does not pretend to cover all of the work done in this period, but is selective, partial, and incomplete. Readers interested in more information are encouraged to consult bibliographies in such recent works as HORNG 1991 , Martzloff 1997 , and Breard 1999 .
Changing Academic Environments The Cultural Revolution officially ended in October of 1976. The sweeping political changes that followed resulted in especially favorable poli-cies on science, technology, and education, and the situation for scholars and intellectuals generally improved dramatically. Rather than the ideology of class struggle, the government regarded science and technology as keys to social progress, and correspondingly, the value of historical studies of science and technology was increasingly appreciated. The program for the history of sci-ence mentioned above was upgraded to the status of an Institute in 1978, at the 460 A. EBERHARD-BRÉARD, J.W. DAUBEN, XU YIBAO LLULL, 26 same level as the Institute of Mathematics in the Chinese Academy of Sciences.
That same year, the Institute again started to recruit graduate students in his-tory of Chinese mathematics, first for the Masters degree, and soon thereafter for the Ph.D. (prior to 1978, there were only two graduate students in this field in all of China). Outside the Institute, a number of universities also began to train graduate students, first at Inner Mongolia Normal University, then at Beijing Normal University, followed by Hangzhou University (now part of Zhejiang University), Northwest University, and Tianjin Normal University.
Higher education in the field of the history of Chinese mathematics provided much needed new blood. In the decade following the Cultural Revolution, the number of professional historians and researchers in this field has increased steadily, and so has the number of publications devoted to the subject [LI Di 1998, p. 83; LIU and Dauben 2002, p. 261].
In addition to more favorable national policies, impetus also came from Western scholars, especially from Joseph Needham and the great stimulus his series of volumes on Chinese science and technology exerted, even in China. The heated debate that ensued over the so-called Needham question (Why did mod-ern science not occur in China?) generated considerable interest both in China and abroad. And the account Needham and WANG Ling gave of Chinese math-ematics in volume three of the series directly inspired research around the globe [jami 1999; XU, in press]. Studies on the history of Chinese mathematics in China also benefited from increasingly active international exchanges through international conferences exclusively devoted (or closely related) to the subject, and from increasingly frequent visits by foreign scholars to China, and by Chinese scholars and graduate students going abroad, primarily to Europe and North America [LI Di 1998, pp. 80-83; LI Di 2001, p. 417].
The flourishing of studies on the history of Chinese mathematics in the past twenty-five years has also been due to the strong support prestigious mathematicians have given the subject, most notably WU Wenjun, a topologist and expert on proofs by computer. WU studied ancient mathematical texts in his spare time during the Cultural Revolution, and found that one of the major characteristics of traditional Chinese mathematics was what he termed the «mechanization of algorithms.» Subsequently WU has not only published intensively himself in this field, but has also provided leadership and help in many ways [LI Di 2001, pp. 419-421]. In 2001, WU received a prestigious award from the Chinese government, a part of which he donated to establish LLULL, 26 THE HISTORY OF CHINESE MATHEMATICS: THE PAST 25 YEARS 461 the «Silk Road Project,» a program which supports research and study of the transmission of mathematics between China and other cultures.
Studies on the history of Chinese mathematics have become increasing-ly important in Taiwan as well, specifically at such institutions as Taiwan Normal University, Tsinghua University, and the Academia Sinica. This has been due primarily to the return to Taiwan of several students trained in the United States in the history and philosophy of science.
Primary Sources One of the most basic tasks for historians is the discovery, study, colla-tion, editing, and publication of primary sources. Historians of Chinese math-ematics, along with scholars in other fields, have made considerable efforts to publish traditional Chinese mathematical texts. In 1993, a group of historians, mainly from the Institute for the History of Natural Science (IHNS) in Beijing, under the leadership of GUO Shuchun, a research professor at IHNS, selected and published in facsimile nearly ninety Chinese mathematical texts from the earliest, the Zhou bi suan jing, to works published at the end of the Qing dynasty late in the nineteenth century. The best editions they could find from libraries in all parts of China were used to produce this five-volume col-lection. All extant Chinese mathematical texts prior to 1368 are included, and their major contents, the different versions or editions, as well as the basic scholarship related to each text are clearly described in the introductions to each work. The Chinese translations of Euclid's Elements (the first six books by Matteo Ricci and XU Guangqi, as well as the last nine books by Alexander Wylie and LI Shanlan) are also included as an appendix [GUO Shuchun 1993].
A year later, JING Yushu, a rare book collector whose primary interest is in the Chinese abacus, published another set in four volumes of Chinese mathematical texts, also in facsimile. Though much of the material in this edi-tion overlaps with the collection published by GUO Shuchun, JING's edition offers different versions of some texts, and is especially valuable for the texts it includes on the abacus [JING 1994].
Inspired by the popular source books for mathematics edited by David E. Smith and Dirk J. Struik, LI Di, Professor of the History of Chinese Mathematics at the Institute for the History of Science at Inner Mongolia Normal University, together with some of his students and SHEN Kangshen, 462 A. EBERHARD-BRÉARD, J.W. DAUBEN, XU YIBAO LLULL, 26 published a series of source books on Chinese mathematics [SHEN 1997, KONG 1997, GUO Xihan 1997, LI Di 1999, and GUO Shirong 2000]. Unlike the collections by GUO and JING, these works focus on only the most impor-tant and interesting parts of the major mathematical texts included, from the earliest times to the modern period, with some texts on calendar-making as well. The original texts were reset and punctuated, along with scholarly com-mentaries and detailed explanations. The main purpose of this series is to help readers to study traditional Chinese mathematics through carefully edited primary sources, rather than from the original documents.
Among traditional Chinese mathematical texts, the Jiu zhang suan shu (Nine Chapters on Mathematical Procedures) is one of the most important, both for its style and for the commentaries by LIU Hui, which had a decisive influence upon the subsequent development of virtually all mathematics in China. Collating the extant different versions of the text, amending errors and corruptions, and explaining obscure terminologies and methods have long occu-pied Chinese scholars from the time of DAI Zhen, an erudite scholar from the reign of Qianlong, down to QIAN Baocong [LIU and Dauben 2002, p 256].
Prominent scholars in the last quarter century have been especially active in producing new editions of the Nine Chapters, including the late Professors BAI Shangshu of Beijing Normal University and LI Jimin of Northwest University, as well as GUO Shuchun [BAI 1983, GUO Shuchun 1990, LI Jimin 1993].
The most sensational discovery related to primary sources in the last twen-ty-five years was the excavation of a book on bamboo slips, the Suan shu shu (Book on Numbers and Computation), unearthed from a second-century BCE tomb between December of 1983 and January of 1984. The Suan shu shu, com-prised of some 190 bamboo strips, was not published in toto until 2000. With only a few words and several problems from isolated strips to go on, much speculation about the contents and nature of the Suan shu shu was produced until a prelimi-nary transcription was published [Jiangling 2000], followed a year later by a more careful edition by PENG Hao [PENG 2001] and a large volume providing photocopies of the entire set of bamboo strips [Zhangjiashan 2001].
Journals The most prestigious journal in China is published by the Institute for History of Natural Science [IHNS] and the Chinese Society for the History of Science and Technology [CSHST]: Ziran kexueshi yanjiu (Studies on the LLULL, 26 THE HISTORY OF CHINESE MATHEMATICS: THE PAST 25 YEARS 463 History of Natural Sciences). The journal, a quarterly (founded in 1980; first issue dated 1982), publishes research on all facets of the history of science and technology, including the history of medicine. In the past two decades, how-ever, the history of Chinese mathematics has been one of the journal's priori-ties, and from 1982 to 1996, more than eighty articles on history of Chinese mathematics have been published [Editorial Staff 1997, pp. 80-82].
The IHNS and the CSHST are also responsible for issuing another quarterly, Zhongguo keji shiliao (Chinese Historical Materials on Science and Technology), also created in 1980. This journal emphasizes documentation and covers largely the modern period. Thus far it has published nearly one hundred articles on history of mathematics, including topics ranging from interviews with contemporary mathematicians, to obituaries and histories of important institutions and organizations.
Due to the growing interest in history of mathematics, these two jour-nals alone could not accommodate the increasing number of historians of mathematics and their graduate students in China, and in order to meet this need, LI Di created a new journal in 1990: Shuxueshi yanjiu wenji (Collected Papers of Research on the History of Mathematics), published jointly by Inner Mongolia University Press and the Nine Chapters Publishing House in Taipei.
This journal is irregular, and published six issues in the 1990s. Beginning in 1999, the Chinese Society for History of Mathematics, founded in 1981, adopt-ed this series as its official journal. Since then, Wenji has been dropped from its title, and the latest issue of Research on History of Mathematics (December, 2001) includes twenty-three articles, fifteen of which are on history of Chinese mathematics.
Studies on the history of Chinese mathematics may also be found in numerous other journals published by universities and colleges throughout China. Among such journals at institutions where there are a substantial num-ber of historians of mathematics, and thus a correspondingly large number of articles related to the subject, are Neimenggu shida xuebao (Journal of Inner Mongolia Normal University, natural science series); Beijing shifan daxue xue-bao (Journal of Beijing Normal University, natural science series), Xibei daxue xuebao (Journal of Northwest University, natural science series), Hangzhou daxue xuebao (Journal of Hangzhou University, natural science series, until 1998), and Zhejiang daxue xuebao (Journal of Zhejiang University, natural sci-ence series, since 1998). In addition to these regular publications, some univer-sity journals produce special or supplementary issues exclusively devoted to 464 A. EBERHARD-BREARD, J.W. DAUBEN, XU YIBAO LLULL, 26 history of mathematics. Among these, for example, are the 1986 issue of Liaoning shida xuebao (Journal of Liaoning Normal University, natural science series), the third issue for 1991 of the Journal of Beijing Normal University, and the third issue for 1993 of the Journal of Inner Mongolia Normal University.
Because history of science enjoys a close relationship to philosophy of science, journals devoted to philosophy from time to time also welcome his-torical contributions. In the last twenty-five years, studies on Chinese mathe-matics emphasizing philosophical issues may be found in two journals in par-ticular: Ziran bianzhengfa tongxun (Journal of Dialectics of Nature), a bimonthly journal founded in 1978 by the Institute on Science Policy and Scientific Management of the Chinese Academy of Sciences, and Kexue jishu yu bianzhengfa (Science, Technology and Dialectics), also a bimonthly journal, founded in 1984 and issued by Lanzhou University.
In Taiwan, three journals have given particular emphasis to history of science, including history of mathematics: Qinghua xuebao (Tsinghua Journal of Chinese Studies, new series), published by Tsinghua University in Hsinchu); Hanxue yanjiu (Chinese Studies), published by the Central Library in Taipei; and Kexueshi tongxun (Newsletter for the History of Science), issued by the Taiwan History of Science Society.
Collected Papers The predecessor of the journal Ziran kexueshi yanjiu (Studies on the History of Natural Sciences), was Kejishi jikan (Collected Papers on the History of Science), an irregular publication that first appeared in 1958. Kejishi jikan ceased publication during the Cultural Revolution, and did not resume until 1982. Volume eleven, the last volume of Kejishi jikan, appeared in 1984 and is a special issue on Chinese mathematics.
Meanwhile, beginning in the 1980s, the Institute for History of Natural Science in Beijing has issued a series of volumes, Kejishi wenji (Collected Essays on History of Science and Technology), one of which (vol-ume 8 for 1982) is exclusively devoted to mathematics. Following the tradi-tional example of publishing collected papers on mathematics begun during the Song and Yuan periods (960-1363 AD) [QIAN 1966], the IHNS has pro-duced a recent volume of collected essays on mathematics during the Ming and Qing dynasties (1368-1911) under the editorship of MEI Rongzhao, a research professor of the IHNS [MEI 1990].
LLULL, 26 THE HISTORY OF CHINESE MATHEMATICS: THE PAST 25 YEARS 465 Other collections of papers have been edited by WU Wenjun: Zhongguo shuxueshi lunwenji (Collected Papers on the History of Chinese Mathematics), the first volume of which appeared in 1985; to date, six volumes have been published in this series. Articles on specific ancient texts or mathematicians may also be found in such collections of essays as those on the Jiu zhang suan shu (The Nine Chapters) and LIU Hui [WU 1982 and 1993], on the Shu shu jiu zhang (Mathematical Treatise in Nine Sections) and its author QIN Jiushao [WU 1987], and on three mathematicians of the nineteenth century, WANG Lai, LI Rui and JIA0 Xun, and their works [HORNG 1993].
Collected papers of prominent mathematicians and historians of math-ematics have also been published. Those of greatest interest to historians of mathematics are the collected papers of WU Wenjun [WU 1986], LI Di [LI Di 1991], LI Zhaohua [LI Zhaohua 2000], HORNG Wann-Sheng [HORNG 1999], and YAN Dunjie [YAN 2000]. The works of the two leading historians of Chinese mathematics of the last century, LI Yan and QIAN Baocong, have also been reproduced in a set of ten volumes edited by members of the Institute for History of Natural Science [DU, GUO, and LIU 1998].
Several volumes in honor of late or retired eminent historians or math-ematicians contain valuable contributions to Chinese mathematics, including those in honor of YAN Dunjie [BO 1992], DU Shiran [LIU and HAN 1997], and WU Wenjun [LIN, LI, and LU 2001]. Moreover, conference proceedings and articles published in celebration of institutional anniversaries also contain papers on Chinese mathematics, for example: Proceedings of the Third International Symposium on the History of Chinese Science and Technology [DU 1990], Proceedings of the International Symposium on the History of Chinese Science and Technology [CHEN 1992], and Exploration of the Roads to Modern Science (commemorating the tenth anniversary of Inner Mongolia Normal University) [LI Di 1993].
Books Numerous works covering the history of Chinese mathematics, from ancient to modern times, have been written by Chinese authors, of which some of the most representative are those by LI Di, A Brief History of Chinese Mathematics [LI Di 1984]; SHEN Kangshen, An Introduction to Chinese Mathematics [SHEN 1986]; LIU Dun, Great to Talk About Mathematics [LIU 1993]; and LI Zhaohua, A History of Chinese Mathematics [LI Zhaohua 1995].
466 A. EBERHARD-BRÉARD, J.W. DAUBEN, XU YIBAO LLULL, 26 LI Di is revising and significantly enlarging his studies of the history of math-ematics into a four-volume set. The first two volumes, which cover the periods from ancient times to 589 CE, and from then up to 960 CE, respectively, have already been published [LI Di 1997 and 1999].
Many other books are devoted to specific topics, such as the fiu zhang suan shu and LIU Hui [LI Jimin 1990, GUO Shuchun 1992]; the three most prominent mathematicians of the Qing dynasty, MEI Wending, MING Antu, and WANG Lai [LI and GUO 1988, LUO 1998, and LI Zhaohua 1998]; math-ematical methods involved in calendar making [QU, JI, and WANG 1994; CHEN Meidong 1995]; the Chinese abacus [HUA 1987]; and surveying [FENG 1995].
At the end of the last century and in anticipation of the new millennium, historians of mathematics in China decided it was timely and appropriate to synthesize the work done in the field in the past century. Accordingly, two sim-ilar projects were organized, each planning to produce ten-volume sets of his-torical studies in chronological order. The project under the leadership of WU Wenjun, LI Di, and SHEN Kangshen thus far has produced nine volumes yvu 1998-1. The other, organized by LIU Dun and WANG Yusheng, has produced five volumes to date [WANG and LIU 2000, JI 2000, KONG 2000, LAO 2000, ZHANG 2000, and ZOU 2002]. Although their contents overlap to a great extent, these two series in some ways complement each other. For instance, a bibliography of traditional Chinese mathematical texts is included only in the first project [WU and LI Di 2000], whereas an account of the development of modern mathematics in China is available only in the latter [ZHANG 2000].
Special Topics One of the major reasons for studying mathematics in ancient China was for the determination of calendars. Consequently, research on extant cal-endars and the documents related to their construction in the official records and histories from the Han to Qing dynasties has provided considerable mate-rial for the history of mathematics. Although this has been a subject of study for centuries, in the 1990s research on calendars became an especially active area of study among younger Chinese historians of mathematics.
Similarly, studies of the Jiu zhang suan shu and especially the commen-tary by LIU Hui have focused primarily on two aspects: collation of texts and LLULL, 26 THE HISTORY OF CHINESE MATHEMATICS: THE PAST 25 YEARS 467 study of the theoretical aspects of LIU Hui's commentaries. Apart from the practical nature of applied mathematics in ancient China, LIU Hui's commen-taries raise important issues concerning the nature of proof and questions about whether ancient Chinese mathematics included such concepts as irra-tional numbers, about which there has been considerable debate over the past twenty-five years.
Traditional Chinese mathematics reached its climax during the Song and Yuan periods. As a result, the most important mathematicians of this period, YANG Hui, QIN Jiushao, LI Ye and ZHU Shijie have all been subjects of his-torical research. Much attention has been paid to biographical studies of QIN Jiushao and LI Ye in particular, including the details of the mathematical pro-cedures and derivations they used that were omitted from their published texts, and possible links between the methods they used and those of their predecessors, all of which may be traced back to thefiu zhang suan shu.
As for the results and works published by Chinese mathematicians of the Qing dynasty, these were of much less significance than what was being produced in Europe during the same period. Nevertheless, it was in this peri-od that Western mathematics was first introduced to China, and exposure to the new mathematics served to revive interest among Chinese scholars in tra-ditional Chinese mathematics. Throughout the Qing dynasty a large number of mathematical texts was produced, providing especially rich material for historical study. Scholars in the past quarter-century have focused to a large extent on the following topics: 1) the mathematics introduced by the Jesuits in the seventeenth and early eighteenth centuries, and later by Protestant and other missionaries in the second half of the nineteenth century; 2) the influ-ence of Western mathematics on Chinese mathematicians; 3) major works by prominent Chinese mathematicians of the Qing dynasty, such as MEI Wending, MING Antu, WANG Lai, LI Rui, DONG Youcheng, XIANG Mingda, LUO Shilin, DAI Xu, XU Youren, LI Shanlan, and HUA Hengfang, among many others.
Conclusion The history of Chinese mathematics is arguably the most active area among studies of the history of science and technology in China today. Thanks to LI Yan, QIAN Baocong, YAN Dunjie, and other pioneering historians working in the last century prior to the Cultural Revolution, the subject was 468 A. EBERHARD-BREARD, J.W. DAUBEN, XU YIBAO LLULL, 26 given a solid foundation. During the past twenty-five years, history of mathe-matics has developed significantly in terms of the growing number of profes-sionals and the substantial output of literature due mainly to scholars affiliated with the Institute for the History of Natural Science in Beijing, with the Chinese Academy of Sciences, and with universities and research institutes throughout the country. Also, favorable government policies towards science and technology, and the proliferation of higher education throughout China, enhanced by the increasing frequency of international exchanges, have all served to support the history of science in general and the history of mathematics in particular.
Although much research in China still relies on the tradition of collating texts, or using contemporary mathematics to interpret earlier results, there are strong indications that in the first quarter of this century the history of Chinese mathematics as a field of Chinese scholarship will continue to prosper, and w ill encourage the younger generation to develop diverse research methodologies using more sophisticated interpretations of historical material. Such optimism is based on the following indicators: 1) In the aftermath of the Cultural Revolution, Chinese intellectuals quickly set about the task of rebuilding China's academic infrastructure, and in colleges, universities, and research institutes throughout the country, Chinese scholars have enjoyed a generation of relatively peaceful and stable political and social environments in which to work; 2) National and local policies have favored studies on the history of science and technology, resulting for example in the creation of new centers for research and study, including the Department of History and Philosophy of Science at Shanghai Jiaotong University in 1999, the Department of History of Science, Technology, and Scientific Archeology at the China University of Science and Technology in Hefei in 1999, and the Department of History of Science and Scientific and Technological Administration (formerly the Institute for History of Science at Inner Mongolia Normal University, Huhehot) in 2001. These new institutions, together with the many already existing ones, will educate more students and promote the study of history of Chinese science, including mathematics, at much higher levels. 3) Strong support from prominent mathematicians. WU Wenjun, for example, when he was awarded China's highest Award for Science and Technology (approximately US$600,000) in 2001, used a portion of the award to establish a foundation to support research on the transmission of mathematics and astronomy along the Silk Road, including mathematical exchanges with Japan, Korea, Vietnam, and other neighboring countries. 4) Last but not least, younger scholars and students of the current generation have more opportunities LLULL, 26 THE HISTORY OF CHINESE MATHEMATICS: THE PAST 25 YEARS 469 to master foreign languages, which are prerequisites for Chinese scholars to con-sult primary documents in foreign languages, and to enjoy the benefits of free exchanges with their foreign colleagues not only in the field of Chinese mathe-matics, but with respect to the history of mathematics as a whole.
In closing, in helping to celebrate the twenty-fifth anniversary of the journal Llull, it is gratifying to note how international the study of history of mathematics has become, and how much the history of Chinese mathematics in particular has become a matter of world-wide interest. For historians of Chinese mathematics both in and out of China, the celebration of Llull's twen-ty-fifth anniversary is a strong indicator of the success the history of science has enjoyed in the past quarter-century, and we look forward to Llull as a pub-lication of continuing international importance in the years and decades ahead.
NOTES 1.We follow throughout this article the convention of using all capital letters to denote Asian surnames. Thus XU is the surname, Guangqi is the given name of XU Guangqi.
2. For a brief account of the history of history of mathematics in China, see Joseph W.
Dauben and LIU Dun, «China,» Chapter 17 in Writing the History of Mathematics: Its Historical Development, Joseph W. Dauben and Christoph J. Scriba, eds. (Basel: Birkháuser, 2002), pp. 297-306.
3.Numbers in square brackets are keyed to entries in the section on «China» compiled by Karine Chemla, Joseph W. Dauben, Alexeï Volkov, and XU Yibao for the revised edi-tion on CD-ROM of The History of Mathematics from Antiquity to the Present: A Selective Annotated Bibliography, Joseph W. Dauben, ed. (New York: Garland Press, 1985); revised edition by Albert C. Lewis, ed. (Providence, RI: The American Mathematical Society, 2000), available on CD-ROM from The American Mathematical Society The bibliography for «China» compris-es pp. 737-851, and updates the 1985 bibliography with a total of 358 entries (-) covering the literature in Western languages primarily between 1985 and 1999.
4. Yushkevitch's correspondence with Vogel has been published by Menso Folkerts, M.M. Rozanskaja, and I. Luther, eds., Mathematikgeschichte ohne Grenzen. Die Korrespondenz zwischen K. Vogel und A.P. Juschkewitsch, Algorismus, 22 (Miinchen: Institut fiir Geschichte der Naturwissenschaften, 1997).
470 A. EBERHARD-BREARD, J.W. DAUBEN, XU YIBAO LLULL, 26 BIBLIOGRAPHY OF WORKS IN CHINESE BAI Shangshu 5511w, Jiu zhang suan shu zhushi (t,iitym) avy (Commentaries and Explanations for the Nine Chapters on Mathematical Procedures), Beijing: Kexue chubanshe, 1983.
BO Shuren ed. Zhonsguo kefishi tan sheng mstst.ant (A New Exploration of Chinese Science and Technology), Beijing: Kexue chubanshe, 1992.
CHEN Meidong Guli xintan (An Exploration of Andent Chinese Cakndars), Shenyang: Liaoning jiaoyu chubanshe, 1995. , ed., Zhonggua kexue jishu shi guoji .>cueshu taolunhui lunwenji 14:1 1/11,14ItiffligamsmIntigrat3t (Proceedings of the International Symposium on the Histog of Chinese Science and Technology), Beijing: Zhongguo kexue jishu chubanshe, 1992.
DU Shiran 4±01, ed. Di san jie guoji Zhongguo kijishi taolunhui lunweiji ggEszuglizwattorermsc (Proceeclings of the Third International Symposium on the Histog of Chinese Sdence and Technology), Beijing: Kexue chubanshe, 1990.
DU Shiran 1±Z GUO Shuchun and LIU Dun gijit, eds. Li Y an Qian Baocong kexueshi quaryi 01WW118151.±111 (Complete Works of Li Y an and Qian Baocong), 10 vols. Shenyang: Liaoning jiaoyu chubanshe, 1998.
FENG Lisheng )19 311-1-, Zhongguo gudai celiang xueshi l'il ffift11111559. (The Histog of Andent Chinese Survging), Huhehot: Neimenggu daxue chubanshe, 1995.
GUO Shirong Suanfa tong zong dao du 1 1 ht nt (A Guide to the Complete Compendium ofArithmetic Methods), in LI Di, ed. Zhonghua chuantong shuxue wenxian jingxuan daodu (A Guide to Selected Readings on Traditional Chinese Mathematics), Wuhan• Hubei jiaoyu chubanshe, 2000.
GUO Shuchun 9514, Huijiao Jiu zhang suan shu tan </tiityion (A Critical Textual Sturb, of the Nine Chapters on Mathematical Procedures), Shenyang: Liamaing jiaoyu chubanshe, 1990. , Gudai shifie shuxue taidou Liu Hui titty104, -ntipme (Liu Hui: The Greatest Mathematician in the Ancient Worla), Jinan: Shandong kexue jishu chubanshe, 1992. , ed. Zhongguo kexue jishu clianji tonghui: Shuxue juan 1:11 Mi14515111913013112: (A Compendium of Chinese Classics of Science and Technology: Mathematics Section), 5 vols., Zhenzhow Henan jiaoyu chubanshe, 1993.
GUO Shuchun 91/ and LIU Dun 9.91, Jiaodian Suan jing shi shu <WEI-10 (Collations of the Ten Mathematical Classics), Shenyang: Liaoning jiaoyu chubanshe, 1998.
GUO Xihan 911111%, Y ang Hui suan fa daodu <MILX2) 241(11 (A Guide to Y ang Hui's Mathematical Methods), in LI Di, ed. Zhonghua chuantong shuxue wenxian jingxuan daodu M4110,1119511d1911.114511 (A Guide to Selected Readings on Traditional Chinese Mathematics), Wuhan- Hubei jiaoyu chubanshe, 1997.
HORNG Wannsheng 1111, Kongzi yu shuxue 515ZMNI (Confucius and Mathematics), Taipei: Mingwen, 1991; enlarged edition, 1999. , ed. Tantian sanyou (Three ChattingFriends), Taipei: Mingwen shuju, 1993.
LLULL, 26 THE HISTORY OF CHINESE MATHEMATICS: THE PAST 25 YEARS 471 HUA Yinthun fflEP191, Zhonggito zbusuan shigao P1:9111,11.111 (A Dmft History of the Chinese Abaeus), Beijing: Zhongguo caizheng jingji chubanshe, 1987.
JI nigang IlJ, Nanbei elmo Sui Tang shuxue Mdb#1014M115 (Mathematics During the Southem ond Northern, Sui, and TangDynasties), in WANG Yusheng it± and LIU Dun eds. Zhongguo shroareshi daxi 14:1 (Comprebensive Series on the History of Cbinese Mathematies), Shiliazhuang: Heibei kexue jishu chubanshe, 2000.
JIANGLING ZIVA1-45021tElf lfill, Jiangling Zhangliashan banjian Suan shu shu shiwell5MR1.1.121115 RB3Z (Texts of the Han Dynasy Book on Calarlating with Numbers, on Bamboo Strips, Excavated from Zhangliashan, jiangling County), lVen 117u tlo (Cultural Relics), 9 (2000), pp. 78-84.
JING Yushu iŠ40, Zhongguo lidai suan-vie jieheng Pln al 1-01§0u1 (Colleeted Cbinese Mathematical Texts from all Dynasties), 4 vols. Jinan: Shandong jiaoyu chubanshe, 1994.
KONG Guoping L F 211 , Li l'e Zbu Sbijie yu jin Y uan sbrwre ;1 2 .15K-#0151Zkit ygzs (Li Y e, Zhu Skie and inathematies during the Jin and Y rian &nasties), in WANG Yusheng It± and LIU Dun eds. Zhongsuo shuwieshi aaxi esst.)çg (Comprehensive Series on the History of ainese Mathernatics), Shiliazhuang: Heibei kexue lishu chubanshe, 2000. , Ceyllan haijing daodu lAdnitnt) 3/451 (A guide to the Sea Mirror of Cirele Measurements), in LI Di, ed., Zhonghua elmantong shuxue wenxian jingxuan daodu PriM14111195Mitilikeitit (A Guide to Seketed Readings on Traditional ainese Mathematies), Wuhan: Hubei jiaoyu chubanshe, 1997.
LAO Hansheng Zhiisiianjii shiyong suanshu (The Aberrus and Applied Arithmetic) in WANG Yusheng Eat and LIU Dun 9111, eds. Zhongguo shuxueshi daxi 11114Milicg (Comprehensive Series on the History of Chinese Mathenuaies), Shiliazhuang: Heibei kexue jishu chubanshe, 2000.
LUO Jianlin Gryllan milii jiefa ykhu (Wdal112) Wit (A Modem ainese Translation with Commentaries on IMing Antrr's] Quick methods for FindingAceurate Ratios ly Cratinga Circle), Huhehot: Neimenggu jiaoyu chubanshe, 1998.
LI Di Zbongguo shu.xweshi jianbian 11:1F, 'MVŠIMMII (A Brief History of ainese Mathernaties), Shenyang: Liaoning rerunin chubanshe 1984. , Zhongguo kexue jishusbi lunwenji 11:1 [411515t011it3t111 (Colleeted Papers on the History of Chinese Sdenee and Technology), Huhehot: Neimenggu jiaoyu chubanshe, 199 • . , ed. Tong.xijangxiandai kexue thilu de tansuoArn ig,1-11111t 1111911k71 (Explorations of the Roads to Modern Seienee), Huhehot: Neimenggu daxue chubanshe, 1993. , Zhongguo shuameshi 112 (A General History of Chinese Mathematies), 4 vols.
Nanjing: Jiangsu jiaoyu chubanshe, 1997-. , ed. Zbonghua ehuantong shraire wenxian jingxwan daodu 1:13 M111111MakeiliMit (A Guide to Selected Readings on Traditional ainese illathernatics), Wuhan: Hubei jiaoyu chubanshe, 1999.
LI Di and GUO Shirong Ig100, Qingthi hurning tianwen shuvurjia Mei Vending 211-14z Sztk§/tlYzffil (Mei Vending: A Prominent Astronomer and Mathernatirian), Shanghai: Shanghai kexue lishu chubanshe, 1988.
472 A. EBERHARD-BRÉARD, J.W. DAUBEN, XU YIBAO LLULL, 26 LI Jimin /10101, Dongfang shuxue diaiji Jin Oang suan sbu jigi Liu Hui .z•hu yalyiu ((hX11/ RIU1011.1M (Studies on the Otiental Mathematical Classic: Nine Chapters on Mathematical Procedures and its Cotnmentaries by Liu Hui), Xi'an: Shanxi renmin jiaoyu chubanshe, 1990. , Jiu zhang suan shu jianheng 01,10019 ria (A Collation of the Nine Chapters on Mathematical Procedures), Xi'an: Shanxi kexue jishu chubanshe, 1993.
LI Zhaohua 9/5, Zbongguo shuxue sbi IkSit (A Histog of Chinese Matbematics), Taipei: Wenjing chubanshe, 1995. ,Hengzhai suanxue jiao0eng 11E (A Collation of Matbematical Tods by Wang Lai falso HenMail), Xi'an: Shanxi kexue jishu chubanshe, 1998. , Gusuan jinlun -131"4-it (Contemporag Views of Traditional Mathematics), Tianjin: Tianjin kexue jishu chubanshe, 2000.
LIN Dongdai 19M 4 1, LI Wenlin 3t491, and LU Yanlin 7, eds. Shuxue yu sbuxue jixiehua 11-14#94 1 4figt (Mathematies and Mechanization of Mathematics), Jinan: Shandong jiaoyu chubanshe, 2001 LIU Dun «131, Da aijan shtt icUTY1/ (Great to Talk About Mathematics), Shenyang: Liaoning jiaoyu chubanshe, 1993.
LIU Dun «11$1 and HAN Qi eds. Keshi xinchuan 11111114 (Passing tbe Torch of tbe Histog of Science), Shenyang: Liaoning jiaoyu chubanshe, 1997.
MEI Rongzhao 119111, ed. Ming Qing shuxueshi lunweliji 11101.51:WOr (Collected Papers on the Histog of Matbematics During the Ming and ,Qing Dynasties), Nanjing: Jiangsu jiaoyu chubanshe, 1990.
PENG Hao Zhangijashan hanjian Suan shu shu zbushiŠR11JAM att111) 51114 (Commentaries on the Book on Calculating with Numbers, on Bamboo S trips, unearthed from Zhangijasban), Beijing: Kexue chubanshe, 2001.
Qian Baocong ed. Song Y uan shuwieshi lunwenji (Collected Papers on the Histog of Mathematics During the Song and Y uan Dynasties), Beijing: Kexue chubanshe, 1966.
QU Anqing JI Zhigang 112,1411, and WANG Rongbin Zhongguo gudai shuli tianwenvie tanxi 11:i gitt4103151P41 (An E\plotution of Ancient Chinese Mathematical Astronong), Xi'an: Xibei daxue chubanshe, 1994.
SHEN Kangshen 5t1114, Zhongsuan daolun 11:9111th (An Introduction to Chinese Mathematics), Shanghai: Shanghai jiaoyu chubanshe, 1986. , Jiu zhang suan shu daodu 0.1.,M1141) (A Guide to the Nine Chapters on Mathematical Procedures), in LI Di, ed. Zbonghtta chuantong shumie wenxian jingxuan daodu 14:1410111t5Y11111W1312 (A Guide to Selected Readings on Traditional Chinese Mathematics), Wuhan: Hubei jiaoyu chubanshe, 1997.
WANG Yusheng LIU Dun eds. Zhongguo shmscueshi daxi 1115M5.51)C1 (Comptrhensive Series on the Histog of Chinese Mathematics), Shijiazhuang: Heibei kexue jishu chubanshe, 2000- .
WU Wenjun ed. Jiu .zhang suan shu yu Liu Hui 1 t.Y1111) 141111 (The Nine Chapters on illathematical Procedures and Litt Hut), Beijing: Beijing shifan daxue chubanshe, 1982.
LLULL, 26 THE HISTORY OF CHINESE MATHEMATICS: THE PAST 25 YEARS 473 , 117u Venjun wenji Sdk3Z#1 (Collected Papers of rVu Venjitn), Jinan: Shandong jiaoyu chubanshe, 1986.
, ed. Zhongsuo shuxueshi ltanvenji 11:1 ilit51103K (Collected Papers on the Histog of Chinese Mathemalics), 6 issues, Jinan: Shandong jiaoyu chubanshe, 1985-1996. (The sixth issue was edited by Bai Shangshu tái 9 2, Li Di and Shen Kangshen , ed. Qin Jiushao yu Shu shu jiu .zhang 4/1.114 (ellit211 (Qin Jiushao and the Mathematical Treatise in Nine Sections), Beijing: Beiling shifan daxue chubanshe, 1987.
, ed. Lill Hui yanjiu $(1.1010M (Studies on Lin Hut), Xi'an: Shanxi renrnin jiaoyu chubanshe, 1993.
, ed. Zbongguo shumreshi daxi Elk011.)cg (Comprehensive Series on the History of Chinese Mathematics), 10 vols. Beijing: Beijing shifan daxue chubanshe, 1998-. (Vol. I, Shansgu dao Xi Han_ht 1.112 From Pre-historical Times to the Instern Han, ed., LI Di, 1998; Vol. II, Zhongguo gudai shitxue mingzhu Jiu .zhang suan shu ff11 -11-11119 (iutiNit) The Ancient Chinese Mathematical Chssic: The Nine Chapiers on Mathematical Procedures, ed., SHEN Kangshen, 1998; Dong Han San Guo 511E. Eastern Han and Three Kingdom ed., BAI Shangshu, 1998; Xi Jin hi 11711 Dai nstmEt From Western Jin to the Five Dynasties, ed., SHEN Kangshen, 1999; Liang Song W512 The Two Song Dynasties, ed., SHEN Kangshen, 2000; Xi Xia Jin Y uan Ming nit5-GBA IVestern Xia, Jin, Y uan and MingDynasties, ed., LI Di, 1999; Mingmo dao Qing thongqi 11.151111:1 A9 From the End of the Ming Dynasg to the Mid- Qing Period, ed., LI Di, 2000; Qing fchongqi dao Qing mo 511 Šijx From the Mid-Qing Period to the End of the ,QingDynasty, ed., LI Zhaohua, 2000; Zhongguo suan,me shuinu huibian 1:12 r:),K§ai IEI 112151 Bibliognoly of Tmditional Chinese Matheinatical Texis, ed., LI Di, 2000).
YAN Dunjie ZII Chongzhi kewie zbuzuo jiaoshi talt,t4elvtlnyip (Collations and Cornmentaties on the Vorks of Chongzin), Shenyang: Liaoning jiaoyu chubanshe, 2000.
ZHANGJIASHAN ŠR111=1:51-tlitAW'rffig l@dNIIII, Zbangiiashan Han mu zhigian 111111Eltfit (Bamboo Strips Excavated from Han Tornbs in Zhangiiashan), Beijing: Wenwu chubanshe, 2001, pp. 83-96; 249-272.
ZHANG Dianzhou 11101, Zhongguo jinviandai shuxue de fnhan Plu IRDÉ11.113tWORIR (The Development of Modern and Contempomg Mathematics in China (since the 1840s)), in WANG Yusheng iik± and LIU Dun 11#1, eds. Zhonsguo shumieshi daxi 142 F.) (Comprebensive Series on the Histog of Chinese Mathematics), Shijiazhuang: Heibei kexue jishu chubanshe, 2000.
ZOU Dahai Zhongguo sbuxue de xingqi yu xianqin shivale 11:1 1: ilit§110olkozstorks (The Rise of Chinese Mathematics to Pre-Qin Mathematics), in WANG Yusheng and LIU Dun 1#1, eds. Zhongguo shuxueshi daxi (Comprehensive Series on the Hisiory of Chinese Mathematics), Shijiazhuang: Heibei kexue jishu chubanshe, 2002.
474 A. EBERHARD-BREARD, J.W. DAUBEN, XU YIBAO LLULL, 26 REFERENCES Breard, Andrea, Re-Kreation eines mathematischen Koneptes i,n chinesiscben Diskurs "Reihen" vom 1. bis zum 19. Jahrhundert. Boethius, 42 (Stuttgart: Franz Steiner Vedag, 1999).
Editorial Staff of the Studies in the Histog of Natural Science, "A Complete Classified Table of Contents for Studies in Me History of Natural Science (1982-1996)," Zimn kemieshi yalyiu 1 9 1,1f4VIIIM (Studies in tbe Histog of Natuml Sciences) 16 (1) (1997): 78-100.
FU Zuohua fellFM, Chouren Ottan yanjiu <Igual) TffR (A Sludy of the Biograpies of Mathematicians and Astronomers)," in MEI Rongzhao 1111, ed. Ming Qing shuNweshi lumvenfi IMEMst.lkSdll (Collected Papers on the Histog of Mathematics tbuing the Ming and „QingDynasties) (Nanjing: Jiangsu jiaoyu chubanshe, 1990), pp. 219-260.
HORNG Wannsheng 101±, Li Shanlan: The impact of 117estem Mathematics in China During the Late Nineteenth Centug. Dissertation, die Graduate Center of the City University of New York, 1991.
Jami, Cadierine, "Joseph Needham and the Historiography of Chinese Mathematics," in Situating the .Histog of Science: Dialogues with Joseph Needham, S. Irfan Habib and Dlisuy Raina, eds. (Oxford: Oxford University Press, 1999), pp. 260-278.
LI Di «Zhongguo shuxueshi yanjiu de lishi yu xianzhuang 11:1 OÉSIŠiffR119 114Vitk (History and Current State of Studies on die History of Chinese Mathematics)," in WU Wenjun ed. Zhongguo shuxuesbi da.vi 11:1 11151ic..1 (Comprehensive Series on the Histool of Chinese Alatbernatics), vol. 1.
(Beijing: Beijing shifan daxue chubanshe, 1998), pp. 63-83. "Zhongguo shuxueshi yanjiu de huigu yu zhanwang 1tgailffiRiih1111ffl 1;12 (Retrospect and Prospect of Studies on die History of Chinese Madiematics)," in Shuxue yu shuxue j&iehua wzmageoptruct (Mathematics and Mechanization of Mathematics), LIN Dongdai 41M1:1/, LI Wenfin Sz.491, and LU Yanlin 11X7 1 491, eds.
Shandong jiaoyu chubanshe, 2001), pp. 407-426.
LIU Dun 9.11, and Joseph W. Dauben, «China,» in iVriting the Histog of Mathematics: Its Historical Development, Joseph W. Dauben and Christoph J. Scriba, eds. (Basel: Birkháuser Verlag, 2002), pp. 53-261.
Martzloff, Jean-Claude, A Histog of Cbinese Mathematics, Stephen S. Wilson, trans. (New York: Springer-Verlag,, 1997).
XU Yibao "Joseph Needham and die Historiography of the History of Chinese Mathematics," in Proceedings of the Symposium on the Histog of Science in Commemomtion of the Centennial of the Birth of Josoh Needham, Decenriber 8-9, 2000, in Kaohsiung, Taiwan, in press. |
8645 | https://zhuanlan.zhihu.com/p/609863478 | 研究方法的类型有哪些？(实例与技巧) - 知乎
首页
知乎直答 焕新
知乎知学堂
等你来答
切换模式
登录/注册
研究方法的类型有哪些？(实例与技巧)
首发于英文论文写作干货分享
切换模式
登录/注册
研究方法的类型有哪些？(实例与技巧)
Wordvice 英文润色 中英翻译
已认证机构号
收录于 · 英文论文写作干货分享
2 人赞同了该文章
什么是研究方法？
研究方法是用于收集和分析数据来回答研究问题和检验研究假说的技巧和流程。研究方法包括几种不同的类型。每种方法都有其优点和缺点。
研究方法的常见类型
学术写作中采用的研究方法有几种主要类型，应采用怎样的研究方法，则主要取决于要收集分析的数据性质，可利用的研究资源是否有限制或约束等。我们建议大家在选择一种或多种研究方法之前，首先调查目标期刊的已出版文章，仔细查看这些学术文章中常用的研究方法。
调查（survey）调查是一种通过问卷或访谈从大量人群中收集数据的研究方法。调查通常用于收集关于人们的态度、信念和行为的信息。
实验（experiment）实验是一种操控一个或多个变量以观察其对另一个变量的影响的研究方法。实验通常用于检验因果关系。
案例研究（case study）案例研究是一种对个人、群体或事件进行深入查证的研究方法。案例研究通常用于收集关于某个特定现象的详细信息。
观察（observation）观察是一种查看和记录个人或群体行为的研究方法。观察经常被用于收集关于自然行为的信息。
内容分析（content analysis）内容分析是一种分析和解释书面文字或口语内容的研究方法。内容分析通常用于分析大量数据，例如新闻报道或社交媒体帖子。
历史研究（historical research）历史研究是一种通过一次文献和二次文献（如文件、文物和照片）研究过去的事件的研究方法。
需要注意的是，不同的研究方法可以结合起来，以便更加全面地理解科研问题或假说。例如，在开展实验之后可以进行调查，以便收集关于参与受试者的态度和行为的更多信息。
总的来说，选择研究方法取决于所研究的问题、所需的数据类型和研究者可以利用的资源。
数据收集方法
数据是为了回答研究问题而收集的信息。收集的数据类型取决于研究问题的性质和研究目的。研究人员收集的数据主要包括以下几种。
定量数据和定性数据
定量数据和定性数据是研究中经常使用的两种数据类型。它们在特征、收集方式和分析方法等方面均有所不同。
定量数据是通过调查、投票和实验等方法收集的数值性数据。它经常被用来测量和描述一大群人或物体的特征。可以使用统计方法对定量数据进行分析，从而发现分布规律和趋势。
定性数据是非数值性的，是通过访谈、观察和焦点小组（focus group）等方法收集的。定性数据通常被用来理解个人或小团体的经历、态度和看法。定性数据的分析方法包括内容分析、主题分析和话语分析，用以确定数据的规律和主题。
总的来说，定量数据能够对现象提供更为客观和一般化的理解，而定性数据能够提供更为主观和深入的理解。这两种类型的数据都很重要，可以结合使用，以便获得对某个问题的较为全面的理解。
优点 缺点
定性数据 研究方法可以随研究进展而调整，以便回答不同的问题。可以通过较小规模的研究或样本量进行先导性数据采集。无法针对更加广泛的人群或现象进行统计学分析。由于较难将调查指标标准化，研究结果可能会出现较高的偏差。
定量数据 产生的数据非常系统和具体。生成的知识经得起测试并且可重复。分析数据时需要了解统计学知识。需要较大的样本量才能获得相关数据。
科研人员应当注意在研究工作中善于同时使用定性和定量的研究方法。
一次数据（primary data）和二次数据（secondary data）
一次研究和二次研究是学术界使用的两种不同类型的研究方法，分别对应于一次文献和二次文献。在大多数研究中，一次文献和二次文献均会被用到。
一次研究是由个人或组织亲自进行的研究。它包括通过调查、访谈或实验等方法收集原始数据。通过一次研究收集到的数据是专门针对某个研究问题和研究目标的，因此通常无法通过其他渠道获得。
二次研究是使用别人已经收集到的数据进行研究。二次研究的数据包括政府报告、学术期刊或行业出版物中的数据。二次研究的优点是通常比一次研究更加省时和便宜，因为数据已经被别人收集好了。然而，二次数据可能与研究问题和研究目标不那么直接相关。
在选择使用一次研究和二次研究的方法或数据时，需要取决于研究问题、研究预算和项目的时间限制，而且还需要考虑想要投稿的目标期刊的要求。
优点 缺点
一次数据 能够更直接地回答研究问题。研究人员对约束条件和数据的控制更为有效。收集数据需要耗费大量时间和资源。需要深入了解如何收集数据。
二次数据 比一次数据在获取上更加方便和快速。数据能够来自不同的时段和地点。无法调整或控制数据的创建方式。处理和验证相关数据需要耗费较多时间。
实验性数据和描述性数据的收集
实验性数据是通过控制实验来收集的。在控制实验中，研究者操控一个或多个变量，观察其对另一个变量的影响。实验性数据的目的是确定因果关系。例如，在某项关于一种新药治疗某种疾病的有效性研究中，研究人员随机将参与受试者分配到接受药物治疗的一个组和接受安慰剂的另一个组，然后比较两组之间的结果。这样收集到的数据就是实验性数据。
描述性数据是通过观察或调查收集的数据，用于描述人群或现象的特征。描述性数据的目标是抓取特定人群或现象在当前状态下的一个快照或截图（snapshot），而不是为了确定因果关系。例如，在某项关于特定人群饮食习惯的研究中，研究人员调查收集参与受试者通常吃什么类型的食物，以及吃食物的频率。这种数据就是描述性数据。
综上所述，实验性数据是通过受控实验来收集的，用以确定因果关系。描述性数据是通过观察或调查来收集的，用以描述人群或现象的特征。
描述性数据的示例：
某项询问人们最喜欢的音乐类型的调查；
统计某一地区人口数量的人口普查；
某项询问人们政治立场的民意调查。
实验性数据的示例：
比较两种不同的药物在治疗某种疾病时的疗效研究；
测定不同浓度的某种化学物质对植物生长影响的实验；
对某种疾病的新疗法与标准疗法的副作用进行比较的临床试验。
数据收集方法示例
研究方法 一次数据二次数据 定性数据定量数据 使用场合
实验 一次数据 定量数据 检验因果关系。
案例研究 二者均可 二者均可 深入分析某个特定案例，通常用于在没有资源进行大样本量研究时。
观察 一次数据 二者均可 分析某种现象在自然状态下如何起作用。
文献综述 二次数据 二者均可 将自己的工作在研究领域中定位，或在研究主题中发现某种趋势。
内容分析 二次数据 二者均可 在给定的定性数据（通常是文本数据）中确定某些词、主题或概念的存在。
使用专业学术英文润色服务准备你的论文！
为了确保研究方法在你的论文中被准确地表达出来，而且也要为了确保论文没有语言错误，建议使用Wordvice的 学术论文英文润色服务 帮助您完成 期刊论文润色 等其它类型的学术性文章润色。
如果你需要对论文进行实时的 免费的英语语法检查，不妨尝试使用Wordvice.ai，看看它与人工智能英语校对领域的其他著名品牌相比效果如何。
如果您正在寻找专业的英文论文润色修改服务，可知乎私信我们，也可在我们知乎主页确认官网地址，访问了解哦！期待和您相识，竭诚为您服务。:)
发布于 2023-02-27 08:51・中国香港
学术研究
研究方法
正规论文润色
赞同 2添加评论
分享
喜欢收藏申请转载
写下你的评论...
还没有评论，发表第一个评论吧
关于作者
Wordvice 英文润色 中英翻译
专业英文润色 & 中英翻译 (学术论文/留学文书/商业文件)
已认证机构号
回答 575文章 441关注者 984
关注发私信
想来知乎工作？请发送邮件到 jobs@zhihu.com
打开知乎App
在「我的页」右上角打开扫一扫
其他扫码方式：微信
下载知乎App
无障碍模式
验证码登录
密码登录
开通机构号
中国 +86
获取短信验证码
获取语音验证码
登录/注册
其他方式登录
未注册手机验证后自动登录，注册即代表同意《知乎协议》《隐私保护指引》
扫码下载知乎 App
关闭二维码
打开知乎App
在「我的页」右上角打开扫一扫
其他扫码方式：微信
下载知乎App
无障碍模式
验证码登录
密码登录
开通机构号
中国 +86
获取短信验证码
获取语音验证码
登录/注册
其他方式登录
未注册手机验证后自动登录，注册即代表同意《知乎协议》《隐私保护指引》
扫码下载知乎 App
关闭二维码 |
8646 | https://www.collegesimply.com/colleges/minnesota/for-profit-colleges/ | For Profit Colleges in Minnesota
C CollegeSimply
Search colleges, majors, states, cities, GPA, SAT or ACT score...
Enter a college, major, state, city, GPA or SAT/ACT score
Search colleges, majors, states, cities, GPA, SAT or ACT score...
Enter a college, major, state, city, GPA or SAT/ACT score
Search
Search All Colleges
Colleges by SAT/ACT scores
Colleges by GPA
Colleges by State
Colleges by City
Application Deadlines
College Cost
Best Colleges
2025 Best Colleges Ranking
2025 Best Public Universities
2025 Best Private Colleges
2025 Best Value Colleges
2025 Best Small Colleges
2025 Best Large Colleges
2025 Best Test Optional Colleges
Compare
Majors
My List 0
Admission Calculator Ad
24 Results
Filter
24 Results
Done
Home
Colleges
Minnesota
For Profit
Minnesota For Profit Colleges
List of For Profit Colleges and Universities in Minnesota
There are 24 for profit colleges and universities in Minnesota enrolling 48,364 students. Explore the list below to find the for profit college match for you. For-profit colleges differ from traditional colleges as they are run by private or public corporations as a business, and ultimately seek to earn a profit as on outcome of the educational services they provide.
Discover more
CollegeSimply
educational
Education
College Simply
School supplies
College Rankings Subscription
Resume building workshops
Study abroad programs
Career counseling
ACT Prep Course
Map Colleges
Filter List
Years
[x] 4 year
[x] 2 year
Type
[x] Public
[x] Private
[x] For-Profit
Admission
Your GPA
-
Your SAT Score
-
Your ACT Score
-
[x] Test Optional
Admission Chance for your GPA/SAT/ACT
All Matches
Within Academic Range
Safety Colleges
Target Colleges
Reach Colleges
Major
Field of Study
All
Specific Major(clear);)
All
Location
Within Radius
300 miles
of Zip Code
State/Region
Minnesota
Size
Students
All
Cost
Tuition + Fees
Any
Avg. Net Price
Any
Discover more
School supplies
College Simply
Education
CollegeSimply
educational
Purchase college visit booking service
Best college lists
SAT practice tests
College ranking reports
Hire college planning consultant
Minnesota;)For-Profit4 year2 year
24 results
Sort
Popular Matches
C
Minneapolis, MN
;)
Capella University
Private 4 Year
0 reviews
43,915 enrollment
$16,975 net price
100%acceptance rate
-SAT range
-ACT range
NA Avg GPA
CompareCalculate my chances
A
Minneapolis, MN
;)
Academy College
Private 4 Year
0 reviews
118 enrollment
$29,280 net price
100%acceptance rate
-SAT range
-ACT range
NA Avg GPA
CompareCalculate my chances
D
Edina, Minnesota
;)
DeVry University Minnesota
Private 4 Year
0 reviews
567 enrollment
-net price
73%acceptance rate
-SAT range
-ACT range
NA Avg GPA
CompareCalculate my chances
G
Minneapolis, MN
;)
Globe University Minneapolis
Private 4 Year
0 reviews
206 enrollment
$17,395 net price
100%acceptance rate
-SAT range
-ACT range
NA Avg GPA
CompareCalculate my chances
G
Woodbury, MN
;)
Globe University Woodbury
Private 4 Year
0 reviews
547 enrollment
$19,582 net price
100%acceptance rate
-SAT range
-ACT range
NA Avg GPA
CompareCalculate my chances
M
Blaine, MN
;)
Minnesota School of Business Blaine
Private 4 Year
0 reviews
181 enrollment
$17,017 net price
100%acceptance rate
-SAT range
-ACT range
NA Avg GPA
CompareCalculate my chances
Discover more
School supplies
CollegeSimply
College Simply
educational
Education
Buy college tuition payment plan
Pay for college counseling near me
Hire college planning consultant
education
Student loan options
M
Roseville, MN
;)
Minneapolis Business College
Private 2 Year
0 reviews
135 enrollment
-net price
100%acceptance rate
-SAT range
-ACT range
NA Avg GPA
CompareCalculate my chances
N
Eagan, Minnesota
;)
Northwest Technical Institute
Private 2 Year
0 reviews
77 enrollment
$25,130 net price
100%acceptance rate
-SAT range
-ACT range
NA Avg GPA
CompareCalculate my chances
M
Moorhead, MN
;)
Minnesota School of Business Moorhead
Private 4 Year
0 reviews
97 enrollment
$21,604 net price
100%acceptance rate
-SAT range
-ACT range
NA Avg GPA
CompareCalculate my chances
M
Lakeville, MN
;)
Minnesota School of Business Lakeville
Private 4 Year
0 reviews
140 enrollment
$18,889 net price
100%acceptance rate
-SAT range
-ACT range
NA Avg GPA
CompareCalculate my chances
M
Richfield, MN
;)
Minnesota School of Business Richfield
Private 4 Year
0 reviews
581 enrollment
$19,167 net price
100%acceptance rate
-SAT range
-ACT range
NA Avg GPA
CompareCalculate my chances
M
Elk River, MN
;)
Minnesota School of Business Elk River
Private 4 Year
0 reviews
135 enrollment
$17,095 net price
100%acceptance rate
-SAT range
-ACT range
NA Avg GPA
CompareCalculate my chances
N
Roseville, MN
;)
National American University Roseville
Private 4 Year
0 reviews
135 enrollment
-net price
100%acceptance rate
-SAT range
-ACT range
NA Avg GPA
CompareCalculate my chances
D
Dultuh, MN
;)
Duluth Business University
Private 4 Year
0 reviews
47 enrollment
$25,119 net price
100%acceptance rate
-SAT range
-ACT range
NA Avg GPA
CompareCalculate my chances
M
Waite Park, MN
;)
Minnesota School of Business Waite Park
Private 4 Year
0 reviews
141 enrollment
$16,701 net price
100%acceptance rate
-SAT range
-ACT range
NA Avg GPA
CompareCalculate my chances
M
Brooklyn Center, MN
;)
Minnesota School of Business Brooklyn Center
Private 4 Year
0 reviews
80 enrollment
$19,159 net price
100%acceptance rate
-SAT range
-ACT range
NA Avg GPA
CompareCalculate my chances
M
Shakopee, Minnesota
;)
Minnesota School of Business Shakopee
Private 4 Year
0 reviews
304 enrollment
$20,966 net price
100%acceptance rate
-SAT range
-ACT range
NA Avg GPA
CompareCalculate my chances
I
Minneapolis, MN
;)
Institute of Production and Recording
Private 2 Year
0 reviews
128 enrollment
$22,763 net price
100%acceptance rate
-SAT range
-ACT range
NA Avg GPA
CompareCalculate my chances
Top 50 Results
Close Map
1
2
>
Best Public Colleges in Minnesota
Best Private Colleges in Minnesota
Private Colleges in Minnesota
Public Colleges in Minnesota
Public Four Year Colleges in Minnesota
Four Year Colleges in Minnesota
Community Colleges in Minnesota
Junior Colleges in Minnesota
Religious Colleges in Minnesota
Catholic Colleges in Minnesota
Christian Colleges in Minnesota
Colleges in Minnesota that are Easy to Get Into
Colleges with Low Acceptance Rates in Minnesota
Best Minnesota Colleges with High Acceptance Rates
Popular Minnesota Colleges with High Acceptance Rates
The Best Value Colleges in Minnesota
The Best Public Colleges in Minnesota
The Best Private Colleges in Minnesota
Colleges with Dorms in Minnesota
Colleges with Daycare in Minnesota
Night and Weekend Colleges in Minnesota
For-Profit Colleges in Minnesota
CollegeSimply
College Search Simplified.
About
Data Sources
Ranking Methodology
Contact Us
Privacy Policy and Cookies
Terms and Conditions
Applying to College
2025 Deadlines
Colleges that are Easy to Get Into
The Best HBCUs Ranking
Degrees Offered
Search Colleges by Majors
Local College Search
Find Colleges Near Me
College Rankings
2025 Best Colleges Rankings
Rank Index
Highest GPA Rankings
Highest ACT Rankings
Highest SAT Rankings
Popular States
Colleges in California
Colleges in Colorado
Colleges in Florida
Colleges in Georgia
Colleges in Illinois
Colleges in Michigan
Colleges in New York
Colleges in North Carolina
Colleges in Ohio
Colleges in Pennsylvania
Colleges in Texas
Search SAT Scores
Colleges for a 1600 SAT
Colleges for a 1550 SAT
Colleges for a 1500 SAT
Colleges for a 1450 SAT
Colleges for a 1400 SAT
Colleges for a 1350 SAT
Colleges for a 1300 SAT
Colleges for a 1250 SAT
Colleges for a 1200 SAT
Colleges for a 1150 SAT
Colleges for a 1100 SAT
Colleges for a 1050 SAT
Colleges for a 1000 SAT
Colleges for a 950 SAT
Colleges for a 900 SAT
Colleges that accept low SAT scores
Search ACT Scores
Colleges for a 36 ACT
Colleges for a 35 ACT
Colleges for a 34 ACT
Colleges for a 33 ACT
Colleges for a 32 ACT
Colleges for a 31 ACT
Colleges for a 30 ACT
Colleges for a 29 ACT
Colleges for a 28 ACT
Colleges for a 27 ACT
Colleges for a 26 ACT
Colleges for a 25 ACT
Colleges for a 24 ACT
Colleges for a 23 ACT
Colleges for a 22 ACT
Colleges for a 21 ACT
Colleges for a 20 ACT
Colleges for a 19 ACT
Colleges for a 18 ACT
Colleges that accept low ACT scores
Your GPA
Find Colleges by GPA
Understanding the GPA Scale
About Letter Grades
GPA Tools
GPA Calculators
College GPA Calculator
High School GPA Calculator
Raise my GPA Calculator
Raise your GPA to a 3.0
Percentage Grade Calculator
Public domain college data is sourced from the U.S. Department of Education National Center for Education Statistics survey for academic years 2023-2024. This website is not affiliated with or endorsed by the U.S. Department of Education.
Some Directory Data is copyrighted material which is reproduced in this publication by permission of Peterson's Data. Copyright by Peterson's Data. All rights reserved.
SAT® is a trademark registered and/or owned by the College Board, which was not involved in the production of, and does not endorse, this site. ACT® is a registered trademark of ACT, Inc. and is not affiliated with this website.
CollegeSimply.com is not affiliated with or endorsed by any of the academic institutions presented on this website.
© 2025 CollegeSimply.com. All rights reserved.
You have 0 colleges saved. View My List |
8647 | https://ocw.mit.edu/courses/res-6-012-introduction-to-probability-spring-2018/resources/variance/ | Browse Course Material
Course Info
Instructors
Prof. John Tsitsiklis
Prof. Patrick Jaillet
Departments
Electrical Engineering and Computer Science
As Taught In
Spring 2018
Level
Undergraduate
Topics
Engineering
Systems Engineering
Mathematics
Probability and Statistics
Learning Resource Types
theaters Lecture Videos
notes Lecture Notes
Download Course
search GIVE NOW about ocw help & faqs contact us
RES.6-012 | Spring 2018 | Undergraduate
Introduction to Probability
Part I: The Fundamentals
Variance
Instructor: John Tsitsiklis
Download video
Download transcript
Course Info
Instructors
Prof. John Tsitsiklis
Prof. Patrick Jaillet
Departments
Electrical Engineering and Computer Science
As Taught In
Spring 2018
Level
Undergraduate
Topics
Engineering
Systems Engineering
Mathematics
Probability and Statistics
Learning Resource Types
theaters Lecture Videos
notes Lecture Notes
Download Course
Over 2,500 courses & materials
Freely sharing knowledge with learners and educators around the world. Learn more
© 2001–2025 Massachusetts Institute of Technology
Creative Commons License
Terms and Conditions
Proud member of:
© 2001–2025 Massachusetts Institute of Technology
You are leaving MIT OpenCourseWare
Please be advised that external sites may have terms and conditions, including license rights, that differ from ours. MIT OCW is not responsible for any content on third party sites, nor does a link suggest an endorsement of those sites and/or their content.
Continue |
8648 | https://www.gamedev.net/forums/topic/186838-dot-product-and-its-derivation/ | dot product and it's derivation - Math and Physics - GameDev.net
GameDev.net uses cookies to ensure you have the best experience on our platform. Questions?
I agree. Let's Make Games
Log In
Join
Login
Username / Email
Password
[x] Remember me
Forgot password?
Login
or
Don't have a GameDev.net account? Sign up
Forgot your password?
Email Address
Reset Password
Please contact us if you have any trouble resetting your password.
Home
Blogs
Forums
News
Portfolios
Projects
Tutorials
12 users logged in
New? Learn about game development.
Before posting, review our community guidelines.
Support GameDev.net with a monthly GDNet+ subscription!
Follow Us
Chat in the GameDev.net Discord!
Back to Math and Physics
Advertisement
dot product and it's derivation
Math and PhysicsProgramming
Started by boomjiOctober 21, 2003 01:57 PM
14 comments, last by boomji 21 years, 10 months ago
1 (current)
2
Next
boomji
Author
1
October 21, 2003 01:57 PM
hi, googled around first but did not find what i was looking for. any way,could any one tell me why is the dot product ||A||||B||cos theta. Every where i read up this was a given(cos theta). now i know i may not yet be abel to digest the proof even if it was explained but i''m hoping some one could post a visual explanation or point me to one...please. thanks for your time. b
remember what you see to see what you rememberb
Cancel Save
Sneftel
1
October 21, 2003 02:15 PM
The definition of the dot product for two vectors and is AxBx + AyBy. Also, Aθ is arctan(Ay/Ax) and Bθ is arctan(By/Bx). SO:
dot(A,B) = AxBx + AyBy
= ||A||cos(Aθ ) ||B||cos(Bθ ) + ||A||sin(Aθ ) ||B||sin(Bθ )
= ||A|| ||B|| cos(Aθ ) cos(Bθ ) + ||A|| ||B|| sin(Aθ ) sin(Bθ )
= (||A|| ||B||)(cos(Aθ ) cos(Bθ ) + sin(Aθ ) sin(Bθ ))
and, by some product-to-sum trig identities:
= (||A|| ||B||)((1/2 cos(Aθ - Bθ ) + cos(Aθ + Bθ ))+(1/2 cos(Aθ - Bθ ) - cos(Aθ + Bθ )))
= (||A|| ||B||) 1/2 (cos(Aθ - Bθ ) + cos(Aθ + Bθ )+cos(Aθ - Bθ ) - cos(Aθ + Bθ ))
= (||A|| ||B||) 1/2 (cos(Aθ - Bθ )+cos(Aθ - Bθ ))
= (||A|| ||B||) cos(Aθ - Bθ )
TADA!
How appropriate. You fight like a cow.
Cancel Save
SpaceDude
1
October 21, 2003 03:16 PM
what about 3D?
Cancel Save
Tron3k
1
October 21, 2003 11:09 PM
Let''s do it backwards.
Let''s say you have 2 vectors in 3D, and you want to find |A||B|cos theta. Does this reduce to Ax Bx + Ay By + Az Bz? Let''s see.
|B|cos theta is the length of the projection of vector B onto vector A. Let''s find the point on vector A of this projection.
Let''s call the coordinates of this point (Cx, Cy, Cz).
Now, let''s create a plane that has its normal as vector A - and let''s set this plane to intersect with the endpoint of vector B. The intersection of this plane with vector A will yield (Cx, Cy, Cz)!
This plane is:
Ax x + Ay y + Az z = K, where K is some constant
Now, this plane must intersect with (Bx, By, Bz) as stated above.
Ax Bx + Ay By + Az Bz = K
So the equation of the plane is:
Ax x + Ay y + Az z = Ax Bx + Ay By + Az Bz
Let''s find out how far this plane is along vector A.
Let''s represent vector A as a parametric line:
x = Ax t / |A|
y = Ay t / |A|
z = Az t / |A|
We must divide by |A| in order to make t the actual distance along vector A. (i.e. an increment of 1 in t travels 1 unit)
Sub that in to the plane:
Ax Ax t / |A| + Ay Ay t / |A| + Az Az t / |A| = Ax Bx + Ay By + Az Bz
Ax^2 t / |A| + Ay^2 t / |A| + Az^2 t / |A| = Ax Bx + Ay By + Az Bz
Ax^2 t + Ay^2 t + Az^2 t = |A| (Ax Bx + Ay By + Az Bz)
Since Ax^2 + Ay^2 + Az^2 = |A|^2:
|A|^2 t = |A| (Ax Bx + Ay By + Az Bz)
t = (Ax Bx + Ay By + Az Bz)/|A|
Since t is the distance along vector A of the projection of vector B, t = |B| cos theta
Therefore:
|A| |B| cos theta
= |A| t
= |A| (Ax Bx + Ay By + Az Bz)/|A|
= Ax Bx + Ay By + Az Bz
Wow ... that actually worked!
“[The clergy] believe that any portion of power confided to me, will be exerted in opposition to their schemes. And they believe rightly: for I have sworn upon the altar of God, eternal hostility against every form of tyranny over the mind of man” - Thomas Jefferson
Cancel Save
Advertisement
Tron3k
1
October 21, 2003 11:15 PM
oh lol... you wanted a visual explanation.
Well, have fun with that crazy stuff I wrote.
“[The clergy] believe that any portion of power confided to me, will be exerted in opposition to their schemes. And they believe rightly: for I have sworn upon the altar of God, eternal hostility against every form of tyranny over the mind of man” - Thomas Jefferson
Cancel Save
Atheist
1
October 22, 2003 01:52 AM
Take A = |A|a, B = |B|b so that the vectors a=(ax,ay) and b=(bx,by) are unit vectors.
1) I think the factors |A| and |B| are clear then (put it into the equations if you don´t believe it, also the scalar product is bilinear by definition).
2) ab is the projection of vector a on vector b (which happenes to be the same as the projection of b on a). Let both a and b lie in the xy-plane and let b be a unit vector in the x-direction (you might want to think about why/if that assumption is ok in a general case after you understood the following).
Draw both vectors in your coordinate system. Draw a triangle with points (0,0), (ax, ay), (ax, 0) which is a recangular triangle (vector (ax,0) is the projection of a on b as you will see).
By the geometrical definition of cos you´ll see that cos(theta) = |(ax,0)|/|(ax,ay)| = |projection(a -> b)|/1 = ab.
EDIT: The |(ax,0)| above is not completely correct since it can also be negative when a is in the 2nd or 3rd quadrant but you´ll see from your picture what I meant.
[edited by - Atheist on October 22, 2003 2:56:34 AM]
Cancel Save
FReY
1
October 22, 2003 03:35 AM
Actually guys, the dot product between 2 vectors A, B is DEFINED as |A||B|cos theta. Most people (mistakingly) get it the wrong way round, A.xB.x + A.yB.y + A.zB.z is NOT the definition of the dot product, it is merely a reduction of the definition into component form. The reduction is just an easier way of computing the dot product, and it follows from several different methods of reduction: one of which is Tron3k''s reduction.
Here is another reduction which I found interesting and in my opinion more visually and mathematically intuitive.
(1) A.B = |A||B|cos(theta) (by defn of dot product)
If you think of the length of the 3 vectors |A|,|B| and |B-A| as the lengths of the sides of a triangle, you can apply the law of cosines here too (boomji, to visualize this, draw the 2 vectors A and B onto a graph, now the vector from A to B will be given by B-A. The triangle formed by these 3 vectors is applied to the law of cosines for a triangle :D)
c^2 = a^2 + b^2 - 2.a.b.cos(C)
In this case, we substitute: |B-A| for c, |A| for a, |B| for b
and we obtain:
(2) |B-A|^2 = |A|^2 + |B|^2 - 2 |A|.|B|.cos(Theta) (by law of cosines)
Remember now, that Theta is the angle between the 2 vectors A, B.
Notice the common term |A||B|cos(Theta) in both equations. We now equate equation (1) and (2), and obtain
A.B = (-(|B-A|^2) + |A|^2 + |B|^2) / 2
and hence
A.B = (-((B.x^2 - 2A.xB.x + A.x^2) + (B.y^2 - 2A.yB.y + A.y^2) + (B.z^2 - 2A.zB.z + A.z^2)) + A.x^2 + A.y^2 + A.z^2 + B.x^2 + B.y^2 + B.z^2) / 2
(by pythagorean length of a vector) and thus
A.B = A.xB.x + A.yB.y + A.zB.z
This one I think is more visually explainable. Unfortunately, I don''t have the time to draw a diagram, sorry!
Hope this helped!
do unto others... and then run like hell.
Cancel Save
Atheist
1
October 22, 2003 04:49 AM
Where did you get that way of definition (and what makes you think your source is THE definition)? I always thought the dot product is a special scalar product which is (in the form I know) given as a multilinear positive definite symmetric form over a vector space.
Also if the two forms are equal (which is only true in the case of a real vector space because you can´t define an angle in general) where is the point in marking one form as THE definition?
Cancel Save
boomji
Author
1
October 22, 2003 12:50 PM
hi there,
thanks so much folks...and pretty detailed as well.
the cosine rule and it's involvement here has made most of the cogs in my block head turn so i will definately look into that area more(did not do math beyound 10th grade...came right back at me and bit me in the moon there didn't it ).
thanks a ton guys...going to save this page and thoroughly go through it.
b
[edited by - boomji on October 22, 2003 1:51:18 PM]
remember what you see to see what you rememberb
Cancel Save
Advertisement
FReY
1
October 23, 2003 02:19 AM
Hey Athiest, don''t be mistaken - the idea that that is the definition of dot-product is not just simply an idea that popped into my head. It was an idea strictly enforced by the mathematics academics whilst I was doing my degree. More generally, though the definition pertains to Euclidean space rather than Inner-product spaces.Sorry to have made such a broad statement - I guess that''s just asking for trouble (even if I am right :D) ... but by all means, prove me wrong
The point of proper definitions: learning and common language, although in this case they may clear up confusion of certain concepts. Such as the origin of the projection of a vector onto another vector, which has it''s basis in geometric and trigonometric methods and which is more easily explained by the definition |A||B|cosC of the dot product.
do unto others... and then run like hell.
Cancel Save
1 (current)
2
Next
This topic is closed to new replies.
Advertisement
Popular Topics
Using a Variable Inertia Law (NKTg) to Interpolate the Masses of 8 Planets for Advanced Motion Simulation in GamesMath and Physics
OpenGL Crash on DrawGraphics and GPU Programming
I’m Luuu — a composer passionate about bringing games to life through musicMusic and Sound FX
Advertisement
Recommended Tutorials
The Game Industry and the Economics of FailureBusiness and Law
Advanced Python Part One: List ComprehensionsGeneral and Gameplay Programming
How to Pitch Angry Birds, If It Were an Unknown Indie GameBusiness and Law
Cel-ShadingGraphics and GPU Programming
fmx/07 Coverage- Day One/Day TwoEvent Coverage
Advertisement
Reticulating splines
About GameDev.net
Leaderboard
Community Guidelines
Terms of Service
Privacy Policy
Contact Us
Copyright (c) 1999-2025 GameDev.net, LLC
Back to Top |
8649 | https://en.wikipedia.org/wiki/List_of_quantum_logic_gates | Published Time: 2022-09-10T14:57:23Z
List of quantum logic gates - Wikipedia
Jump to content
[x] Main menu
Main menu
move to sidebar hide
Navigation
Main page
Contents
Current events
Random article
About Wikipedia
Contact us
Contribute
Help
Learn to edit
Community portal
Recent changes
Upload file
Special pages
Search
Search
[x] Appearance
Appearance
move to sidebar hide
Text
Small Standard Large
This page always uses small font size
Width
Standard Wide
The content is as wide as possible for your browser window.
Color (beta)
Automatic Light Dark
This page is always in light mode.
Donate
Create account
Log in
[x] Personal tools
Donate
Create account
Log in
Pages for logged out editors learn more
Contributions
Talk
[x] Toggle the table of contents
Contents
move to sidebar hide
(Top)
1 Identity gate and global phase
2 Clifford qubit gates
3 Non-Clifford qubit gatesToggle Non-Clifford qubit gates subsection
3.1 Relative phase gates
3.2 Rotation operator gates
3.3 Two-qubit interaction gates
3.4 Non-Clifford swap gates
3.5 Other named gates
4 Notes
5 References
List of quantum logic gates
[x] 3 languages
Català
Deutsch
Italiano
Edit links
Article
Talk
[x] English
Read
Edit
View history
[x] Tools
Tools
move to sidebar hide
Actions
Read
Edit
View history
General
What links here
Related changes
Upload file
Permanent link
Page information
Cite this page
Get shortened URL
Download QR code
Edit interlanguage links
Print/export
Download as PDF
Printable version
In other projects
Wikidata item
From Wikipedia, the free encyclopedia
This article contains dynamic lists that may never be able to satisfy particular standards for completeness. You can help by adding missing items with reliable sources.
In gate-based quantum computing, various sets of quantum logic gates are commonly used to express quantum operations. The following tables list several unitary quantum logic gates, together with their common name, how they are represented, and some of their properties. Controlled or conjugate transpose (adjoint) versions of some of these gates may not be listed.
Identity gate and global phase
[edit]
| Name | # qubits | Operator symbol | Matrix | Circuit diagram | Properties | Refs |
--- --- ---
| Identity, no-op | 1 (any) | I,I{\textstyle I,\;\mathbb {I} }, 𝟙 | [1 0 0 1]{\displaystyle {\begin{bmatrix}1&0\0&1\end{bmatrix}}} | or | Hermitian Identity element | |
| Global phase | 1 (any) | P h{\textstyle \mathrm {Ph} }, P h a s e{\textstyle \mathrm {Phase} } or e i δ I{\textstyle \mathrm {e} ^{i\delta }I} | e i δ[1 0 0 1]{\displaystyle \mathrm {e} ^{i\delta }{\begin{bmatrix}1&0\0&1\end{bmatrix}}} | | Continuous parameters: δ{\displaystyle \delta } (period 2 π{\displaystyle 2\pi }) Exponential form: exp(i δ I){\displaystyle \exp(i\delta I)} | |
The identity gate is the identity operationI|ψ⟩=|ψ⟩{\displaystyle I|\psi \rangle =|\psi \rangle }, most of the times this gate is not indicated in circuit diagrams, but it is useful when describing mathematical results. It has been described as being a "wait cycle", and a NOP.
The global phase gate introduces a global phase e i φ{\displaystyle e^{i\varphi }} to the whole qubit quantum state. A quantum state is uniquely defined up to a phase. Because of the Born rule, a phase factor has no effect on a measurement outcome: |e i φ|=1{\displaystyle |e^{i\varphi }|=1} for any φ{\displaystyle \varphi }. Because e i δ|ψ⟩⊗|ϕ⟩=e i δ(|ψ⟩⊗|ϕ⟩),{\displaystyle e^{i\delta }|\psi \rangle \otimes |\phi \rangle =e^{i\delta }(|\psi \rangle \otimes |\phi \rangle ),} when the global phase gate is applied to a single qubit in a quantum register, the entire register's global phase is changed. Also, P h(0)=I.{\displaystyle \mathrm {Ph} (0)=I.}
These gates can be extended to any number of qubits or qudits.
Clifford qubit gates
[edit]
This table includes commonly used Clifford gates for qubits.
| Names | # qubits | Operator symbol | Matrix | Circuit diagram | Some properties | Refs |
--- --- ---
| Pauli X, NOT, bit flip | 1 | X,N O T,σ x{\textstyle X,\;\mathrm {NOT} ,\;\sigma {x}} | [0 1 1 0]{\displaystyle {\begin{bmatrix}0&1\1&0\end{bmatrix}}} | or | Hermitian Pauli group Traceless Involutory | |
| Pauli _Y | 1 | Y,σ y{\textstyle Y,\;\sigma {y}} | [0−i i 0]{\displaystyle {\begin{bmatrix}0&-i\i&0\end{bmatrix}}} | | Hermitian Pauli group Traceless Involutory | |
| Pauli _Z, phase flip | 1 | Z,σ z{\textstyle Z,\;\sigma {z}} | [1 0 0−1]{\displaystyle {\begin{bmatrix}1&0\0&-1\end{bmatrix}}} | | Hermitian Pauli group Traceless Involutory | |
| Phase gate _S, square root of Z | 1 | S,P,Z{\textstyle S,\;P,\;{\sqrt {Z}}} | [1 0 0 i]{\displaystyle {\begin{bmatrix}1&0\0&i\end{bmatrix}}} | | | |
| Square root of X, square root of NOT | 1 | X{\textstyle {\sqrt {X}}}, V{\textstyle V}, N O T,S X{\textstyle {\sqrt {\mathrm {NOT} }},\;\mathrm {SX} } | 1 2[1+i 1−i 1−i 1+i]{\displaystyle {\frac {1}{2}}{\begin{bmatrix}1+i&1-i\1-i&1+i\end{bmatrix}}} | | | |
| Hadamard, Walsh-Hadamard | 1 | H{\textstyle H} | 1 2[1 1 1−1]{\displaystyle {\frac {1}{\sqrt {2}}}{\begin{bmatrix}1&1\1&-1\end{bmatrix}}} | | Hermitian Traceless Involutory | |
| Controlled NOT, controlled-X, controlled-bit flip, reversible exclusive OR, Feynman | 2 | C N O T{\textstyle \mathrm {CNOT} }, X O R,C X{\textstyle \mathrm {XOR} ,\;\mathrm {CX} } | [1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0]{\displaystyle {\begin{bmatrix}1&0&0&0\0&1&0&0\0&0&0&1\0&0&1&0\end{bmatrix}}} [1 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0]{\displaystyle {\begin{bmatrix}1&0&0&0\0&0&0&1\0&0&1&0\0&1&0&0\end{bmatrix}}} | | Hermitian Involutory Implementation: Cirac–Zoller gate | |
| Anticontrolled-NOT, anticontrolled-X, open-controlled-CNOT, zero control, control-on-0-NOT, reversible exclusive NOR | 2 | C¯X{\textstyle {\overline {\mathrm {C} }}\mathrm {X} }, controlled-NOT{\textstyle {\text{controlled-NOT}}}, X N O R{\textstyle \mathrm {XNOR} } | [0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 1]{\displaystyle {\begin{bmatrix}0&1&0&0\1&0&0&0\0&0&1&0\0&0&0&1\end{bmatrix}}} | | Hermitian Involutory | |
| Controlled-Z, controlled sign flip, controlled phase flip | 2 | C Z{\textstyle \mathrm {CZ} }, C P F{\textstyle \mathrm {CPF} }, C S I G N{\textstyle \mathrm {CSIGN} }, C P H A S E{\textstyle \mathrm {CPHASE} } | [1 0 0 0 0 1 0 0 0 0 1 0 0 0 0−1]{\displaystyle {\begin{bmatrix}1&0&0&0\0&1&0&0\0&0&1&0\0&0&0&-1\end{bmatrix}}} | | Hermitian Involutory Symmetrical Implementation: Duan-Kimble gate | |
| Double-controlled NOT | 2 | D C N O T{\textstyle \mathrm {DCNOT} } | [1 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0]{\displaystyle {\begin{bmatrix}1&0&0&0\0&0&1&0\0&0&0&1\0&1&0&0\end{bmatrix}}} | | Special unitary | |
| Swap | 2 | S W A P{\textstyle \mathrm {SWAP} } | [1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1]{\displaystyle {\begin{bmatrix}1&0&0&0\0&0&1&0\0&1&0&0\0&0&0&1\end{bmatrix}}} | or | Hermitian Involutory Symmetrical | |
| Imaginary swap | 2 | iSWAP{\displaystyle {\mbox{iSWAP}}} | [1 0 0 0 0 0 i 0 0 i 0 0 0 0 0 1]{\displaystyle {\begin{bmatrix}1&0&0&0\0&0&i&0\0&i&0&0\0&0&0&1\end{bmatrix}}} | or | Special unitary Symmetrical | |
Other Clifford gates, including higher dimensional ones are not included here but by definition can be generated using H,S{\textstyle H,S} and C N O T{\textstyle \mathrm {CNOT} }.
Note that if a Clifford gate A is not in the Pauli group, A{\displaystyle {\sqrt {A}}} or controlled-A are not in the Clifford gates.[citation needed]
The Clifford set is not a universal quantum gate set.
Non-Clifford qubit gates
[edit]
Relative phase gates
[edit]
| Names | # qubits | Operator symbol | Matrix | Circuit diagram | Properties | Refs |
--- --- ---
| Phase shift | 1 | P(φ),R(φ),u 1(φ){\textstyle P(\varphi ),\;R(\varphi ),\;u_{1}(\varphi )} | [1 0 0 e i φ]{\displaystyle {\begin{bmatrix}1&0\0&\mathrm {e} ^{i\varphi }\end{bmatrix}}} | | Continuous parameters: φ{\displaystyle \varphi } (period 2 π{\displaystyle 2\pi }) | |
| Phase gate T, π/8 gate, fourth root of Z | 1 | T,P(π/4){\textstyle T,P(\pi /4)} or Z 4{\textstyle {\sqrt[{4}]{Z}}} | [1 0 0 e i π/4]{\displaystyle {\begin{bmatrix}1&0\0&\mathrm {e} ^{i\pi /4}\end{bmatrix}}} | | | |
| Controlled phase | 2 | C P h a s e(φ),C R(φ){\textstyle \mathrm {CPhase} (\varphi ),\mathrm {CR} (\varphi )} | [1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 e i φ]{\displaystyle {\begin{bmatrix}1&0&0&0\0&1&0&0\0&0&1&0\0&0&0&e^{i\varphi }\end{bmatrix}}} | | Continuous parameters: φ{\displaystyle \varphi } (period 2 π{\displaystyle 2\pi }) Symmetrical Implementation: Jaksch Gate | |
| Controlled phase S | 2 | C S,controlled-S{\displaystyle \mathrm {CS} ,{\text{controlled-}}S} | [1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 i]{\displaystyle {\begin{bmatrix}1&0&0&0\0&1&0&0\0&0&1&0\0&0&0&i\end{bmatrix}}} | | Symmetrical | |
The phase shift is a family of single-qubit gates that map the basis states P(φ)|0⟩=|0⟩{\displaystyle P(\varphi )|0\rangle =|0\rangle } and P(φ)|1⟩=e i φ|1⟩{\displaystyle P(\varphi )|1\rangle =e^{i\varphi }|1\rangle }. The probability of measuring a |0⟩{\displaystyle |0\rangle } or |1⟩{\displaystyle |1\rangle } is unchanged after applying this gate, however it modifies the phase of the quantum state. This is equivalent to tracing a horizontal circle (a line of latitude), or a rotation along the z-axis on the Bloch sphere by φ{\displaystyle \varphi } radians. A common example is the T gate where φ=π 4{\textstyle \varphi ={\frac {\pi }{4}}} (historically known as the π/8{\displaystyle \pi /8} gate), the phase gate. Note that some Clifford gates are special cases of the phase shift gate: P(0)=I,P(π)=Z;P(π/2)=S.{\displaystyle P(0)=I,\;P(\pi )=Z;P(\pi /2)=S.}
The argument to the phase shift gate is in U(1), and the gate performs a phase rotation in U(1) along the specified basis state (e.g. P(φ){\displaystyle P(\varphi )} rotates the phase about |1⟩{\displaystyle |1\rangle }). Extending P(φ){\displaystyle P(\varphi )} to a rotation about a generic phase of both basis states of a 2-level quantum system (a qubit) can be done with a series circuit: P(β)⋅X⋅P(α)⋅X=[e i α 0 0 e i β]{\displaystyle P(\beta )\cdot X\cdot P(\alpha )\cdot X={\begin{bmatrix}e^{i\alpha }&0\0&e^{i\beta }\end{bmatrix}}}. When α=−β{\displaystyle \alpha =-\beta } this gate is the rotation operatorR z(2 β){\displaystyle R_{z}(2\beta )} gate and if α=β{\displaystyle \alpha =\beta } it is a global phase.[a][b]
The T gate's historic name of π/8{\displaystyle \pi /8} gate comes from the identity R z(π/4)Ph(π 8)=P(π/4){\displaystyle R_{z}(\pi /4)\operatorname {Ph} \left({\frac {\pi }{8}}\right)=P(\pi /4)}, where R z(π/4)=[e−i π/8 0 0 e i π/8]{\displaystyle R_{z}(\pi /4)={\begin{bmatrix}e^{-i\pi /8}&0\0&e^{i\pi /8}\end{bmatrix}}}.
Arbitrary single-qubit phase shift gates P(φ){\displaystyle P(\varphi )} are natively available for transmon quantum processors through timing of microwave control pulses. It can be explained in terms of change of frame.
As with any single qubit gate one can build a controlled version of the phase shift gate. With respect to the computational basis, the 2-qubit controlled phase shift gate is: shifts the phase with φ{\displaystyle \varphi } only if it acts on the state |11⟩{\displaystyle |11\rangle }:
|a,b⟩↦{e i φ|a,b⟩for a=b=1|a,b⟩otherwise.{\displaystyle |a,b\rangle \mapsto {\begin{cases}e^{i\varphi }|a,b\rangle &{\mbox{for }}a=b=1\|a,b\rangle &{\mbox{otherwise.}}\end{cases}}}
The controlled-Z (or CZ) gate is the special case where φ=π{\displaystyle \varphi =\pi }.
The controlled-S gate is the case of the controlled-P(φ){\displaystyle P(\varphi )} when φ=π/2{\displaystyle \varphi =\pi /2} and is a commonly used gate.
Rotation operator gates
[edit]
Further information: Bloch sphere §Rotations, and Rotation operator (quantum mechanics)
| Names | # qubits | Operator symbol | Exponential form | Matrix | Circuit diagram | Properties | Refs |
--- --- --- --- |
| Rotation about x-axis | 1 | R x(θ){\textstyle R_{x}(\theta )} | exp(−i X θ/2){\displaystyle \exp(-iX\theta /2)} | [cos(θ/2)−i sin(θ/2)−i sin(θ/2)cos(θ/2)]{\displaystyle {\begin{bmatrix}\cos(\theta /2)&-i\sin(\theta /2)\-i\sin(\theta /2)&\cos(\theta /2)\end{bmatrix}}} | | Special unitary Continuous parameters: θ{\displaystyle \theta } (period 4 π{\displaystyle 4\pi }) | |
| Rotation about y-axis | 1 | R y(θ){\textstyle R_{y}(\theta )} | exp(−i Y θ/2){\displaystyle \exp(-iY\theta /2)} | [cos(θ/2)−sin(θ/2)sin(θ/2)cos(θ/2)]{\displaystyle {\begin{bmatrix}\cos(\theta /2)&-\sin(\theta /2)\\sin(\theta /2)&\cos(\theta /2)\end{bmatrix}}} | | Special unitary Continuous parameters: θ{\displaystyle \theta } (period 4 π{\displaystyle 4\pi }) | |
| Rotation about z-axis | 1 | R z(θ){\textstyle R_{z}(\theta )} | exp(−i Z θ/2){\displaystyle \exp(-iZ\theta /2)} | [exp(−i θ/2)0 0 exp(i θ/2)]{\displaystyle {\begin{bmatrix}\exp(-i\theta /2)&0\0&\exp(i\theta /2)\end{bmatrix}}} | | Special unitary Continuous parameters: θ{\displaystyle \theta } (period 4 π{\displaystyle 4\pi }) | |
The rotation operator gates R x(θ),R y(θ){\displaystyle R_{x}(\theta ),R_{y}(\theta )} and R z(θ){\displaystyle R_{z}(\theta )} are the analog rotation matrices in three Cartesian axes of SO(3),[c] along the x, y or z-axes of the Bloch sphere projection.
As Pauli matrices are related to the generator of rotations, these rotation operators can be written as matrix exponentials with Pauli matrices in the argument. Any 2×2{\displaystyle 2\times 2}unitary matrix in SU(2) can be written as a product (i.e. series circuit) of three rotation gates or less. Note that for two-level systems such as qubits and spinors, these rotations have a period of 4π. A rotation of 2π (360 degrees) returns the same statevector with a different phase.
We also have R b(−θ)=R b(θ)†{\displaystyle R_{b}(-\theta )=R_{b}(\theta )^{\dagger }} and R b(0)=I{\displaystyle R_{b}(0)=I} for all b∈{x,y,z}.{\displaystyle b\in {x,y,z}.}
The rotation matrices are related to the Pauli matrices in the following way: R x(π)=−i X,R y(π)=−i Y,R z(π)=−i Z.{\displaystyle R_{x}(\pi )=-iX,R_{y}(\pi )=-iY,R_{z}(\pi )=-iZ.}
It is possible to work out the adjoint action of rotations on the Pauli vector, namely rotation effectively by double the angle a to apply Rodrigues' rotation formula:
R n(−a)σ→R n(a)=e i a 2(n^⋅σ→)σ→e−i a 2(n^⋅σ→)=σ→cos(a)+n^×σ→sin(a)+n^n^⋅σ→(1−cos(a)).{\displaystyle R_{n}(-a){\vec {\sigma }}R_{n}(a)=e^{i{\frac {a}{2}}\left({\hat {n}}\cdot {\vec {\sigma }}\right)}~{\vec {\sigma }}~e^{-i{\frac {a}{2}}\left({\hat {n}}\cdot {\vec {\sigma }}\right)}={\vec {\sigma }}\cos(a)+{\hat {n}}\times {\vec {\sigma }}~\sin(a)+{\hat {n}}~{\hat {n}}\cdot {\vec {\sigma }}~(1-\cos(a))~.}
Taking the dot product of any unit vector with the above formula generates the expression of any single qubit gate when sandwiched within adjoint rotation gates. For example, it can be shown that R y(−π/2)X R y(π/2)=x^⋅(y^×σ→)=Z{\displaystyle R_{y}(-\pi /2)XR_{y}(\pi /2)={\hat {x}}\cdot ({\hat {y}}\times {\vec {\sigma }})=Z}. Also, using the anticommuting relation we have R y(−π/2)X R y(π/2)=X R y(+π/2)R y(π/2)=X(−i Y)=Z{\displaystyle R_{y}(-\pi /2)XR_{y}(\pi /2)=XR_{y}(+\pi /2)R_{y}(\pi /2)=X(-iY)=Z}.
Rotation operators have interesting identities. For example, R y(π/2)Z=H{\displaystyle R_{y}(\pi /2)Z=H} and X R y(π/2)=H.{\displaystyle XR_{y}(\pi /2)=H.} Also, using the anticommuting relations we have Z R y(−π/2)=H{\displaystyle ZR_{y}(-\pi /2)=H} and R y(−π/2)X=H.{\displaystyle R_{y}(-\pi /2)X=H.}
Global phase and phase shift can be transformed into each other's with the Z-rotation operator: R z(γ)Ph(γ 2)=P(γ){\displaystyle R_{z}(\gamma )\operatorname {Ph} \left({\frac {\gamma }{2}}\right)=P(\gamma )}.: 11 : 77–83
The X{\displaystyle {\sqrt {X}}} gate represents a rotation of π/2 about the x axis at the Bloch sphere X=e i π/4 R x(π/2){\displaystyle {\sqrt {X}}=e^{i\pi /4}R_{x}(\pi /2)}.
Similar rotation operator gates exist for SU(3) using Gell-Mann matrices. They are the rotation operators used with qutrits.
Two-qubit interaction gates
[edit]
| Names | # qubits | Operator symbol | Exponential form | Matrix | Circuit diagram | Properties | Refs |
--- --- --- --- |
| XX interaction | 2 | R x x(ϕ){\displaystyle R_{xx}(\phi )}, XX(ϕ){\displaystyle {\text{XX}}(\phi )} | exp(−i ϕ 2 X⊗X){\displaystyle \exp \left(-i{\frac {\phi }{2}}X\otimes X\right)} | [cos(ϕ 2)0 0−i sin(ϕ 2)0 cos(ϕ 2)−i sin(ϕ 2)0 0−i sin(ϕ 2)cos(ϕ 2)0−i sin(ϕ 2)0 0 cos(ϕ 2)]{\displaystyle {\begin{bmatrix}\cos \left({\frac {\phi }{2}}\right)&0&0&-i\sin \left({\frac {\phi }{2}}\right)\0&\cos \left({\frac {\phi }{2}}\right)&-i\sin \left({\frac {\phi }{2}}\right)&0\0&-i\sin \left({\frac {\phi }{2}}\right)&\cos \left({\frac {\phi }{2}}\right)&0\-i\sin \left({\frac {\phi }{2}}\right)&0&0&\cos \left({\frac {\phi }{2}}\right)\\end{bmatrix}}} | | Special unitary Continuous parameters: θ{\displaystyle \theta } (period 4 π{\displaystyle 4\pi }) Implementation: Mølmer–Sørensen gate | [citation needed] |
| YY interaction | 2 | R y y(ϕ){\displaystyle R_{yy}(\phi )}, YY(ϕ){\displaystyle {\text{YY}}(\phi )} | exp(−i ϕ 2 Y⊗Y){\displaystyle \exp \left(-i{\frac {\phi }{2}}Y\otimes Y\right)} | [cos(ϕ 2)0 0 i sin(ϕ 2)0 cos(ϕ 2)−i sin(ϕ 2)0 0−i sin(ϕ 2)cos(ϕ 2)0 i sin(ϕ 2)0 0 cos(ϕ 2)]{\displaystyle {\begin{bmatrix}\cos \left({\frac {\phi }{2}}\right)&0&0&i\sin \left({\frac {\phi }{2}}\right)\0&\cos \left({\frac {\phi }{2}}\right)&-i\sin \left({\frac {\phi }{2}}\right)&0\0&-i\sin \left({\frac {\phi }{2}}\right)&\cos \left({\frac {\phi }{2}}\right)&0\i\sin \left({\frac {\phi }{2}}\right)&0&0&\cos \left({\frac {\phi }{2}}\right)\\end{bmatrix}}} | | Special unitary Continuous parameters: θ{\displaystyle \theta } (period 4 π{\displaystyle 4\pi }) Implementation: Mølmer–Sørensen gate | [citation needed] |
| ZZ interaction | 2 | R z z(ϕ){\textstyle {\displaystyle R_{zz}(\phi )}}, ZZ(ϕ){\displaystyle {\text{ZZ}}(\phi )} | exp(−i ϕ 2 Z⊗Z){\displaystyle {\displaystyle \exp \left(-i{\frac {\phi }{2}}Z\otimes Z\right)}} | [e−i ϕ/2 0 0 0 0 e i ϕ/2 0 0 0 0 e i ϕ/2 0 0 0 0 e−i ϕ/2]{\displaystyle {\begin{bmatrix}e^{-i\phi /2}&0&0&0\0&e^{i\phi /2}&0&0\0&0&e^{i\phi /2}&0\0&0&0&e^{-i\phi /2}\\end{bmatrix}}} | | Special unitary Continuous parameters: θ{\displaystyle \theta } (period 4 π{\displaystyle 4\pi }) | [citation needed] |
| XY, XX plus YY | 2 | R x y(ϕ){\textstyle {\displaystyle R_{xy}(\phi )}}, XY(ϕ){\displaystyle {\text{XY}}(\phi )} | exp[−i ϕ 4(X⊗X+Y⊗Y)]{\displaystyle {\displaystyle \exp \left[-i{\frac {\phi }{4}}(X\otimes X+Y\otimes Y)\right]}} | [1 0 0 0 0 cos(ϕ/2)−i sin(ϕ/2)0 0−i sin(ϕ/2)cos(ϕ/2)0 0 0 0 1]{\displaystyle {\begin{bmatrix}1&0&0&0\0&\cos(\phi /2)&-i\sin(\phi /2)&0\0&-i\sin(\phi /2)&\cos(\phi /2)&0\0&0&0&1\\end{bmatrix}}} | | Special unitary Continuous parameters: θ{\displaystyle \theta } (period 4 π{\displaystyle 4\pi }) | [citation needed] |
The qubit-qubit Ising coupling or Heisenberg interaction gates R xx, R yy and R zz are 2-qubit gates that are implemented natively in some trapped-ion quantum computers, using for example the Mølmer–Sørensen gate procedure.
Note that these gates can be expressed in sinusoidal form also, for example R x x(ϕ)=exp(−i ϕ 2 X⊗X)=cos(ϕ 2)I⊗I−i sin(ϕ 2)X⊗X{\displaystyle R_{xx}(\phi )=\exp \left(-i{\frac {\phi }{2}}X\otimes X\right)=\cos \left({\frac {\phi }{2}}\right)I\otimes I-i\sin \left({\frac {\phi }{2}}\right)X\otimes X}.
The CNOT gate can be further decomposed as products of rotation operator gates and exactly a single two-qubit interaction gate, for example
CNOT=e−i π 4 R y 1(−π/2)R x 1(−π/2)R x 2(−π/2)R x x(π/2)R y 1(π/2).{\displaystyle {\mbox{CNOT}}=e^{-i{\frac {\pi }{4}}}R_{y_{1}}(-\pi /2)R_{x_{1}}(-\pi /2)R_{x_{2}}(-\pi /2)R_{xx}(\pi /2)R_{y_{1}}(\pi /2).}
The SWAP gate can be constructed from other gates, for example using the two-qubit interaction gates: SWAP=e i π 4 R x x(π/2)R y y(π/2)R z z(π/2){\displaystyle {\text{SWAP}}=e^{i{\frac {\pi }{4}}}R_{xx}(\pi /2)R_{yy}(\pi /2)R_{zz}(\pi /2)}.
In superconducting circuits, the family of gates resulting from Heisenberg interactions is sometimes called the fSim gate set. They can be realized using flux-tunable qubits with flux-tunable coupling, or using microwave drives in fixed-frequency qubits with fixed coupling.
Non-Clifford swap gates
[edit]
| Names | # qubits | Operator symbol | Matrix | Circuit diagram | Properties | Refs |
--- --- ---
| Square root swap | 2 | S W A P{\displaystyle {\sqrt {\mathrm {SWAP} }}} | [1 0 0 0 0 1 2(1+i)1 2(1−i)0 0 1 2(1−i)1 2(1+i)0 0 0 0 1]{\displaystyle {\begin{bmatrix}1&0&0&0\0&{\frac {1}{2}}(1+i)&{\frac {1}{2}}(1-i)&0\0&{\frac {1}{2}}(1-i)&{\frac {1}{2}}(1+i)&0\0&0&0&1\\end{bmatrix}}} | | | |
| Square root imaginary swap | 2 | iSWAP{\displaystyle {\sqrt {\mbox{iSWAP}}}} | [1 0 0 0 0 1 2 i 2 0 0 i 2 1 2 0 0 0 0 1]{\displaystyle {\begin{bmatrix}1&0&0&0\0&{\frac {1}{\sqrt {2}}}&{\frac {i}{\sqrt {2}}}&0\0&{\frac {i}{\sqrt {2}}}&{\frac {1}{\sqrt {2}}}&0\0&0&0&1\end{bmatrix}}} | | Special unitary | |
| Swap (raised to a power) | 2 | SWAP α{\displaystyle {\mbox{SWAP}}^{\alpha }} | [1 0 0 0 0 1+e i π α 2 1−e i π α 2 0 0 1−e i π α 2 1+e i π α 2 0 0 0 0 1]{\displaystyle {\begin{bmatrix}1&0&0&0\0&{\frac {1+e^{i\pi \alpha }}{2}}&{\frac {1-e^{i\pi \alpha }}{2}}&0\0&{\frac {1-e^{i\pi \alpha }}{2}}&{\frac {1+e^{i\pi \alpha }}{2}}&0\0&0&0&1\end{bmatrix}}} | | Continuous parameters: α{\displaystyle \alpha } (period 2{\displaystyle 2}) | |
| Fredkin, controlled swap | 3 | C S W A P{\displaystyle \mathrm {CSWAP} }, F R E D K I N{\displaystyle \mathrm {FREDKIN} } | [1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1]{\displaystyle {\begin{bmatrix}1&0&0&0&0&0&0&0\0&1&0&0&0&0&0&0\0&0&1&0&0&0&0&0\0&0&0&1&0&0&0&0\0&0&0&0&1&0&0&0\0&0&0&0&0&0&1&0\0&0&0&0&0&1&0&0\0&0&0&0&0&0&0&1\\end{bmatrix}}} | or | Hermitian Involutory Functionally complete reversible gate for Boolean algebra | |
The √SWAP gate performs half-way of a two-qubit swap (see Clifford gates). It is universal such that any many-qubit gate can be constructed from only √SWAP and single qubit gates. More than one application of the √SWAP is required to produce a Bell state from product states. The √SWAP gate arises naturally in systems that exploit exchange interaction.
For systems with Ising like interactions, it is sometimes more natural to introduce the imaginary swap or iSWAP. Note that i SWAP=R x x(−π/2)R y y(−π/2){\displaystyle i{\mbox{SWAP}}=R_{xx}(-\pi /2)R_{yy}(-\pi /2)} and i SWAP=R x x(−π/4)R y y(−π/4){\displaystyle {\sqrt {i{\mbox{SWAP}}}}=R_{xx}(-\pi /4)R_{yy}(-\pi /4)}, or more generally i SWAP n=R x x(−π/2 n)R y y(−π/2 n){\displaystyle {\sqrt[{n}]{i{\mbox{SWAP}}}}=R_{xx}(-\pi /2n)R_{yy}(-\pi /2n)} for all real n except 0.
SWAP α arises naturally in spintronic quantum computers.
The Fredkin gate (also CSWAP or CS gate), named after Edward Fredkin, is a 3-bit gate that performs a controlledswap. It is universal for classical computation. It has the useful property that the numbers of 0s and 1s are conserved throughout, which in the billiard ball model means the same number of balls are output as input.
Other named gates
[edit]
| Names | # qubits | Operator symbol | Matrix | Circuit diagram | Properties | Named after | Refs |
--- --- --- --- |
| General single qubit rotation | 1 | U(θ,ϕ,λ){\displaystyle U(\theta ,\phi ,\lambda )} | [cos(θ/2)−e i λ sin(θ/2)e i ϕ sin(θ/2)e i(λ+ϕ)cos(θ/2)]{\displaystyle {\begin{bmatrix}\cos(\theta /2)&-e^{i\lambda }\sin(\theta /2)\e^{i\phi }\sin(\theta /2)&e^{i(\lambda +\phi )}\cos(\theta /2)\end{bmatrix}}} | | Implements an arbitrary single-qubit rotation Continuous parameters: θ,ϕ,λ{\displaystyle \theta ,\phi ,\lambda } (period 2 π{\displaystyle 2\pi }) | OpenQASM U gate[d] | |
| Barenco | 2 | B A R E N C O(α,ϕ,θ){\displaystyle \mathrm {BARENCO} (\alpha ,\phi ,\theta )} | [1 0 0 0 0 1 0 0 0 0 e i α cosθ−i e i(α−ϕ)sinθ 0 0−i e i(α+ϕ)sinθ e i α cosθ]{\displaystyle {\begin{bmatrix}1&0&0&0\0&1&0&0\0&0&e^{i\alpha }\cos \theta &-\mathrm {i} e^{\mathrm {i} (\alpha -\phi )}\sin \theta \0&0&-\mathrm {i} e^{\mathrm {i} (\alpha +\phi )}\sin \theta &e^{i\alpha }\cos \theta \end{bmatrix}}} | | Implements a controlled arbitrary qubit rotation Universal quantum gate Continuous parameters: α,ϕ,θ{\displaystyle \alpha ,\phi ,\theta } (period 2 π{\displaystyle 2\pi }) | Adriano Barenco | |
| Berkeley B | 2 | B{\displaystyle B} | [cos(π/8)0 0 i sin(π/8)0 cos(3 π/8)i sin(3 π/8)0 0 i sin(3 π/8)cos(3 π/8)0 i sin(π/8)0 0 cos(π/8)]{\displaystyle {\begin{bmatrix}\cos(\pi /8)&0&0&i\sin(\pi /8)\0&\cos(3\pi /8)&i\sin(3\pi /8)&0\0&i\sin(3\pi /8)&\cos(3\pi /8)&0\i\sin(\pi /8)&0&0&\cos(\pi /8)\\end{bmatrix}}} | | Special unitary Exponential form: exp[i π 8(2 X⊗X+Y⊗Y)]{\displaystyle \exp \left[i{\frac {\pi }{8}}(2X\otimes X+Y\otimes Y)\right]} | University of California Berkeley | |
| Controlled-V, controlled square root NOT | 2 | C S X,controlled-X,{\displaystyle \mathrm {CSX} ,{\text{controlled-}}{\sqrt {X}},}controlled-V{\displaystyle {\text{controlled-}}V} | [1 0 0 0 0 1 0 0 0 0 e i π/4 e−i π/4 0 0 e−i π/4 e i π/4]{\displaystyle {\begin{bmatrix}1&0&0&0\0&1&0&0\0&0&e^{i\pi /4}&e^{-i\pi /4}\0&0&e^{-i\pi /4}&e^{i\pi /4}\end{bmatrix}}} | | | | |
| Core entangling, canonical decomposition | 2 | N(a,b,c){\displaystyle N(a,b,c)}, c a n(a,b,c){\displaystyle \mathrm {can} (a,b,c)} | [e i c cos(a−b)0 0 i e i c sin(a−b)0 e−i c cos(a+b)i e−i c sin(a+b)0 0 i e−i c sin(a+b)e−i c cos(a+b)0 i e i c sin(a−b)0 0 e i c cos(a−b)]{\displaystyle {\begin{bmatrix}e^{ic}\cos(a-b)&0&0&ie^{ic}\sin(a-b)\0&e^{-ic}\cos(a+b)&ie^{-ic}\sin(a+b)&0\0&ie^{-ic}\sin(a+b)&e^{-ic}\cos(a+b)&0\ie^{ic}\sin(a-b)&0&0&e^{ic}\cos(a-b)\\end{bmatrix}}} | | Special unitary Universal quantum gate Exponential form exp[i(a X⊗X+b Y⊗Y+c Z⊗Z)]{\displaystyle \exp \left[i(aX\otimes X+bY\otimes Y+cZ\otimes Z)\right]} Continuous parameters: a,b,c{\displaystyle a,b,c} (period 2 π{\displaystyle 2\pi }) | | |
| Dagwood Bumstead | 2 | DB{\displaystyle {\text{DB}}} | [1 0 0 0 0 cos(3 π/8)−i sin(3 π/8)0 0−i sin(3 π/8)cos(3 π/8)0 0 0 0 1]{\displaystyle {\begin{bmatrix}1&0&0&0\0&\cos(3\pi /8)&-i\sin(3\pi /8)&0\0&-i\sin(3\pi /8)&\cos(3\pi /8)&0\0&0&0&1\\end{bmatrix}}} | | Special unitary Exponential form: exp[−i 3 π 16(X⊗X+Y⊗Y)]{\displaystyle \exp \left[-i{\frac {3\pi }{16}}(X\otimes X+Y\otimes Y)\right]} | Comicbook Dagwood Bumstead | |
| Echoed cross resonance | 2 | ECR{\displaystyle {\text{ECR}}} | 1 2[0 0 1 i 0 0 i 1 1−i 0 0−i 1 0 0]{\displaystyle {\frac {1}{\sqrt {2}}}{\begin{bmatrix}0&0&1&i\0&0&i&1\1&-i&0&0\-i&1&0&0\\end{bmatrix}}} | | Special unitary | | |
| Fermionic simulation | 2 | U fSim(θ,ϕ){\displaystyle U_{\text{fSim}}(\theta ,\phi )}, fSim(θ,ϕ){\displaystyle {\text{fSim}}(\theta ,\phi )} | [1 0 0 0 0 cos(θ)−i sin(θ)0 0−i sin(θ)cos(θ)0 0 0 0 e i ϕ]{\displaystyle {\begin{bmatrix}1&0&0&0\0&\cos(\theta )&-i\sin(\theta )&0\0&-i\sin(\theta )&\cos(\theta )&0\0&0&0&e^{i\phi }\\end{bmatrix}}} | | Special unitary Continuous parameters: θ,ϕ{\displaystyle \theta ,\phi } (period 2 π{\displaystyle 2\pi }) | | |
| Givens | 2 | G(θ){\displaystyle G(\theta )}, Givens(θ){\displaystyle {\text{Givens}}(\theta )} | [1 0 0 0 0 cos(θ)−sin(θ)0 0 sin(θ)cos(θ)0 0 0 0 1]{\displaystyle {\begin{bmatrix}1&0&0&0\0&\cos(\theta )&-\sin(\theta )&0\0&\sin(\theta )&\cos(\theta )&0\0&0&0&1\\end{bmatrix}}} | | Special unitary Exponential form: exp[−i θ 2(Y⊗X−X⊗Y)]{\displaystyle \exp \left[-i{\frac {\theta }{2}}(Y\otimes X-X\otimes Y)\right]} Continuous parameters: θ,ϕ{\displaystyle \theta ,\phi } (period 2 π{\displaystyle 2\pi }) | Givens rotations | |
| Magic | 2 | M{\displaystyle {\mathcal {M}}} | 1 2[1 i 0 0 0 0 i 1 0 0 i−1 1−i 0 0]{\displaystyle {\frac {1}{\sqrt {2}}}{\begin{bmatrix}1&i&0&0\0&0&i&1\0&0&i&-1\1&-i&0&0\\end{bmatrix}}} | | | | |
| Sycamore | 2 | syc{\displaystyle {\text{syc}}}, fSim(π/2,π/6){\displaystyle {\text{fSim}}(\pi /2,\pi /6)} | [1 0 0 0 0 0−i 0 0−i 0 0 0 0 0 e−i π/6]{\displaystyle {\begin{bmatrix}1&0&0&0\0&0&-i&0\0&-i&0&0\0&0&0&\mathrm {e} ^{-i\pi /6}\end{bmatrix}}} | | | Google's Sycamore processor | |
| CZ-SWAP | 2 | CZS(θ,ϕ,γ){\displaystyle {\text{CZS}}(\theta ,\phi ,\gamma )}, | [1 0 0 0 0−e i γ sin 2(θ/2)+cos 2(θ/2)1 2(1+e i γ)e−i ϕ sin(θ)0 0 1 2(1+e i γ)e i ϕ sin(θ)−e i γ cos 2(θ/2)+sin 2(θ/2)0 0 0 0−e i γ]{\displaystyle {\begin{bmatrix}1&0&0&0\0&-e^{i\gamma }\sin ^{2}(\theta /2)+\cos ^{2}(\theta /2)&{\frac {1}{2}}(1+e^{i\gamma })e^{-i\phi }\sin(\theta )&0\0&{\frac {1}{2}}(1+e^{i\gamma })e^{i\phi }\sin(\theta )&-e^{i\gamma }\cos ^{2}(\theta /2)+\sin ^{2}(\theta /2)&0\0&0&0&\mathrm {-} e^{i\gamma }\end{bmatrix}}} | | Continuous parameters: θ,ϕ,γ{\displaystyle \theta ,\phi ,\gamma } Submatrix of a controlled-CZS (CCZS) | | |
| Deutsch | 3 | D θ{\displaystyle D_{\theta }}, D(θ){\displaystyle D(\theta )} | [1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 i cosθ sinθ 0 0 0 0 0 0 sinθ i cosθ]{\displaystyle {\begin{bmatrix}1&0&0&0&0&0&0&0\0&1&0&0&0&0&0&0\0&0&1&0&0&0&0&0\0&0&0&1&0&0&0&0\0&0&0&0&1&0&0&0\0&0&0&0&0&1&0&0\0&0&0&0&0&0&i\cos \theta &\sin \theta \0&0&0&0&0&0&\sin \theta &i\cos \theta \\end{bmatrix}}} | | Continuous parameters: θ,ϕ{\displaystyle \theta ,\phi } (period 2 π{\displaystyle 2\pi }) Universal quantum gate | David Deutsch | |
| Margolus, simplified Toffoli | 3 | M{\displaystyle M}, RCCX{\displaystyle {\text{RCCX}}} | [1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0−1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0]{\displaystyle {\begin{bmatrix}1&0&0&0&0&0&0&0\0&1&0&0&0&0&0&0\0&0&1&0&0&0&0&0\0&0&0&1&0&0&0&0\0&0&0&0&1&0&0&0\0&0&0&0&0&-1&0&0\0&0&0&0&0&0&0&1\0&0&0&0&0&0&1&0\\end{bmatrix}}} | | Hermitian Involutory Special unitary Functionally complete reversible gate for Boolean algebra | Norman Margolus | |
| Peres | 3 | P G{\displaystyle \mathrm {PG} },P e r e s{\displaystyle \mathrm {Peres} } | [1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0]{\displaystyle {\begin{bmatrix}1&0&0&0&0&0&0&0\0&1&0&0&0&0&0&0\0&0&1&0&0&0&0&0\0&0&0&1&0&0&0&0\0&0&0&0&0&0&0&1\0&0&0&0&0&0&1&0\0&0&0&0&1&0&0&0\0&0&0&0&0&1&0&0\\end{bmatrix}}} | | Functionally complete reversible gate for Boolean algebra | Asher Peres | |
| Toffoli, controlled-controlled NOT | 3 | C C N O T,C C X,T o f f{\displaystyle \mathrm {CCNOT} ,\mathrm {CCX} ,\mathrm {Toff} } | [1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0]{\displaystyle {\begin{bmatrix}1&0&0&0&0&0&0&0\0&1&0&0&0&0&0&0\0&0&1&0&0&0&0&0\0&0&0&1&0&0&0&0\0&0&0&0&1&0&0&0\0&0&0&0&0&1&0&0\0&0&0&0&0&0&0&1\0&0&0&0&0&0&1&0\\end{bmatrix}}} | | Hermitian Involutory Functionally complete reversible gate for Boolean algebra | Tommaso Toffoli | |
| Fermionic-Fredkin, Controlled-fermionic SWAP | 3 | f F r e d k i n{\displaystyle \mathrm {fFredkin} }, C C Z S(π/2,0,0){\displaystyle \mathrm {CCZS} (\pi /2,0,0)},C f S W A P{\displaystyle \mathrm {CfSWAP} } | [1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0−1]{\displaystyle {\begin{bmatrix}1&0&0&0&0&0&0&0\0&1&0&0&0&0&0&0\0&0&1&0&0&0&0&0\0&0&0&1&0&0&0&0\0&0&0&0&1&0&0&0\0&0&0&0&0&0&1&0\0&0&0&0&0&1&0&0\0&0&0&0&0&0&0&-1\\end{bmatrix}}} | | | | |
Notes
[edit]
^α=−β{\displaystyle \alpha =-\beta } when P(α)=P(β)†{\displaystyle P(\alpha )=P(\beta )^{\dagger }}, where †{\displaystyle \dagger } is the conjugate transpose (or Hermitian adjoint).
^Also: (P(δ)⋅Y)2=e i δ I{\displaystyle \left(P(\delta )\cdot Y\right)^{2}=e^{i\delta }I}
^a SU(2)double cover. See also Hopf fibration.
^The matrix shown here is from openQASM 3.0, which differs from U(θ,ϕ,λ){\displaystyle U(\theta ,\phi ,\lambda )} from a global phase (OpenQASM 2.0 U gate is in SU(2) ) .
References
[edit]
^ Jump up to: abcdefghijklmnopqrstuvwxyzaaabacadaeWilliams, Colin P. (2011). Explorations in Quantum Computing. Springer. ISBN978-1-84628-887-6.
^"IGate". qiskit.org.Qiskit online documentation.
^"I operation". docs.microsoft.com. 28 July 2023.Q# online documentation.
^Feynman, Richard P. (1986). "Quantum mechanical computers". Foundations of Physics. 16 (6). Springer Science and Business Media LLC: 507–531. Bibcode:1986FoPh...16..507F. doi:10.1007/bf01886518. ISSN0015-9018. S2CID122076550.
^ Jump up to: abBarenco, Adriano; Bennett, Charles H.; Cleve, Richard; DiVincenzo, David P.; Margolus, Norman; Shor, Peter; Sleator, Tycho; Smolin, John A.; Weinfurter, Harald (1995-11-01). "Elementary gates for quantum computation". Physical Review A. 52 (5). American Physical Society (APS): 3457–3467. arXiv:quant-ph/9503016. Bibcode:1995PhRvA..52.3457B. doi:10.1103/physreva.52.3457. ISSN1050-2947. PMID9912645. S2CID8764584.
^ Jump up to: abcdefghijklmnopNielsen, Michael A. (2010). Quantum computation and quantum information. Isaac L. Chuang (10th anniversary ed.). Cambridge: Cambridge University Press. ISBN978-1-107-00217-3. OCLC665137861.
^Hung, W. N. N.; Song, Xiaoyu; Yang, Guowu; Yang, Jin; Perkowski, M. (September 2006). "Optimal synthesis of multiple output Boolean functions using a set of quantum gates by symbolic reachability analysis". IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems. 25 (9): 1652–1663. doi:10.1109/tcad.2005.858352. ISSN0278-0070. S2CID14123321.
^Collins, Daniel; Linden, Noah; Popescu, Sandu (2001-08-07). "Nonlocal content of quantum operations". Physical Review A. 64 (3): 032302. arXiv:quant-ph/0005102. Bibcode:2001PhRvA..64c2302C. doi:10.1103/PhysRevA.64.032302. ISSN1050-2947. S2CID29769034.
^ Jump up to: abPathak, Anirban (2013-06-20). Elements of Quantum Computation and Quantum Communication. Taylor & Francis. ISBN978-1-4665-1792-9.
^Yanofsky, Noson S.; Mannucci, Mirco A. (2008-08-11). Quantum Computing for Computer Scientists. Cambridge University Press. ISBN978-1-139-64390-0.
^ Jump up to: abcdStancil, Daniel D.; Byrd, Gregory T. (2022-04-19). Principles of Superconducting Quantum Computers. John Wiley & Sons. ISBN978-1-119-75074-1.
^Jaksch, D.; Cirac, J. I.; Zoller, P.; Rolston, S. L.; Côté, R.; and Lukin, M. D. (2000). "Fast Quantum Gates for Neutral Atoms". Phys. Rev. Lett. 85 (10): 2208–2211. arXiv:quant-ph/0004038. Bibcode:2000PhRvL..85.2208J. doi:10.1103/PhysRevLett.85.2208. PMID10970499.
^Chatterjee, Dibyendu; Roy, Arijit (2015). "A transmon-based quantum half-adder scheme". Progress of Theoretical and Experimental Physics. 2015 (9): 7–8. Bibcode:2015PTEP.2015i3A02C. doi:10.1093/ptep/ptv122.
^McKay, David C.; Wood, Christopher J.; Sheldon, Sarah; Chow, Jerry M.; Gambetta, Jay M. (31 August 2017). "Efficient Z gates for quantum computing". Physical Review A. 96 (2): 022330. arXiv:1612.00858. Bibcode:2015PTEP.2015i3A02C. doi:10.1093/ptep/ptv122.
^"qiskit.circuit.library.PhaseGate". IBM (qiskit documentation).
^Griffiths, D. J. (2008). Introduction to Elementary Particles (2nd ed.). John Wiley & Sons. pp.127–128. ISBN978-3-527-40601-2.
^"Monroe Conference"(PDF). online.kitp.ucsb.edu.
^"Demonstration of a small programmable quantum computer with atomic qubits"(PDF). Retrieved 2019-02-10.
^ Jump up to: abFoxen, B.; Neill, C.; Dunsworth, A.; Roushan, P.; Chiaro, B.; Megrant, A.; Kelly, J.; Chen, Zijun; Satzinger, K.; Barends, R.; Arute, F.; Arya, K.; Babbush, R.; Bacon, D.; Bardin, J. C.; Boixo, S.; Buell, D.; Burkett, B.; Chen, Yu; Collins, R.; Farhi, E.; Fowler, A.; Gidney, C.; Giustina, M.; Graff, R.; Harrigan, M.; Huang, T.; Isakov, S. V.; Jeffrey, E.; Jiang, Z.; Kafri, D.; Kechedzhi, K.; Klimov, P.; Korotkov, A.; Kostritsa, F.; Landhuis, D.; Lucero, E.; McClean, J.; McEwen, M.; Mi, X.; Mohseni, M.; Mutus, J. Y.; Naaman, O.; Neeley, M.; Niu, M.; Petukhov, A.; Quintana, C.; Rubin, N.; Sank, D.; Smelyanskiy, V.; Vainsencher, A.; White, T. C.; Yao, Z.; Yeh, P.; Zalcman, A.; Neven, H.; Martinis, J. M. (2020-09-15). "Demonstrating a Continuous Set of Two-qubit Gates for Near-term Quantum Algorithms". Physical Review Letters. 125 (12): 120504. arXiv:2001.08343. Bibcode:2020PhRvL.125l0504F. doi:10.1103/PhysRevLett.125.120504. ISSN0031-9007. PMID33016760.
^ Jump up to: abNguyen, L.B.; Kim, Y.; Hashim, A.; Goss, N.; Marinelli, B.; Bhandari, B.; Das, D.; Naik, R.K.; Kreikebaum, J.M.; Jordan, A.; Santiago, D.I.; Siddiqi, I. (16 January 2024). "Programmable Heisenberg interactions between Floquet qubits". Nature Physics. 20 (1): 240–246. arXiv:2211.10383. Bibcode:2024NatPh..20..240N. doi:10.1038/s41567-023-02326-7.
^Nemirovsky, Jonathan; Sagi, Yoav (2021), "Fast universal two-qubit gate for neutral fermionic atoms in optical tweezers", Physical Review Research, 3 (1): 013113, arXiv:2008.09819, Bibcode:2021PhRvR...3a3113N, doi:10.1103/PhysRevResearch.3.013113
^Rasmussen, S. E.; Zinner, N. T. (2020-07-17). "Simple implementation of high fidelity controlled- i swap gates and quantum circuit exponentiation of non-Hermitian gates". Physical Review Research. 2 (3): 033097. arXiv:2002.11728. Bibcode:2020PhRvR...2c3097R. doi:10.1103/PhysRevResearch.2.033097. ISSN2643-1564.
^Schuch, Norbert; Siewert, Jens (2003-03-10). "Natural two-qubit gate for quantum computation using the XY interaction". Physical Review A. 67 (3): 032301. arXiv:quant-ph/0209035. Bibcode:2003PhRvA..67c2301S. doi:10.1103/PhysRevA.67.032301. ISSN1050-2947. S2CID50823541.
^Dallaire-Demers, Pierre-Luc; Wilhelm, Frank K. (2016-12-05). "Quantum gates and architecture for the quantum simulation of the Fermi-Hubbard model". Physical Review A. 94 (6): 062304. arXiv:1606.00208. Bibcode:2016PhRvA..94f2304D. doi:10.1103/PhysRevA.94.062304. ISSN2469-9926. S2CID118408193.
^Cross, Andrew; Javadi-Abhari, Ali; Alexander, Thomas; De Beaudrap, Niel; Bishop, Lev S.; Heidel, Steven; Ryan, Colm A.; Sivarajah, Prasahnt; Smolin, John; Gambetta, Jay M.; Johnson, Blake R. (2022). "OpenQASM 3: A Broader and Deeper Quantum Assembly Language". ACM Transactions on Quantum Computing. 3 (3): 1–50. arXiv:2104.14722. doi:10.1145/3505636. ISSN2643-6809. S2CID233476587.
^Zhang, Jun; Vala, Jiri; Sastry, Shankar; Whaley, K. Birgitta (2004-07-07). "Minimum Construction of Two-Qubit Quantum Operations". Physical Review Letters. 93 (2): 020502. arXiv:quant-ph/0312193. Bibcode:2004PhRvL..93b0502Z. doi:10.1103/PhysRevLett.93.020502. ISSN0031-9007. PMID15323888. S2CID9632700.
^ Jump up to: abAbuGhanem, M. (2021-01-01). "Two-qubit Entangling Gate for Superconducting Quantum Computers". Rochester, NY. doi:10.2139/ssrn.4188257. S2CID252264545. SSRN4188257.{{cite journal}}: Cite journal requires |journal= (help)
^Peterson, Eric C.; Crooks, Gavin E.; Smith, Robert S. (2020-03-26). "Fixed-Depth Two-Qubit Circuits and the Monodromy Polytope". Quantum. 4: 247. arXiv:1904.10541. doi:10.22331/q-2020-03-26-247. S2CID214690323.
^Córcoles, A. D.; Magesan, Easwar; Srinivasan, Srikanth J.; Cross, Andrew W.; Steffen, M.; Gambetta, Jay M.; Chow, Jerry M. (2015-04-29). "Demonstration of a quantum error detection code using a square lattice of four superconducting qubits". Nature Communications. 6 (1): 6979. arXiv:1410.6419. Bibcode:2015NatCo...6.6979C. doi:10.1038/ncomms7979. ISSN2041-1723. PMC4421819. PMID25923200.
^Kyriienko, Oleksandr; Elfving, Vincent E. (2021-11-15). "Generalized quantum circuit differentiation rules". Physical Review A. 104 (5): 052417. arXiv:2108.01218. Bibcode:2021PhRvA.104e2417K. doi:10.1103/PhysRevA.104.052417. hdl:10871/127818. ISSN2469-9926. S2CID236881494.
^Arrazola, Juan Miguel; Matteo, Olivia Di; Quesada, Nicolás; Jahangiri, Soran; Delgado, Alain; Killoran, Nathan (2022-06-20). "Universal quantum circuits for quantum chemistry". Quantum. 6: 742. arXiv:2106.13839. Bibcode:2022Quant...6..742A. doi:10.22331/q-2022-06-20-742. S2CID235658488.
^Arute, Frank; Arya, Kunal; Babbush, Ryan; Bacon, Dave; Bardin, Joseph C.; Barends, Rami; Biswas, Rupak; Boixo, Sergio; Brandao, Fernando G. S. L.; Buell, David A.; Burkett, Brian; Chen, Yu; Chen, Zijun; Chiaro, Ben; Collins, Roberto (2019). "Quantum supremacy using a programmable superconducting processor". Nature. 574 (7779): 505–510. arXiv:1910.11333. Bibcode:2019Natur.574..505A. doi:10.1038/s41586-019-1666-5. ISSN1476-4687. PMID31645734. S2CID204836822.
^ Jump up to: abGu, Xiu; Fernández-Pendás, Jorge; Vikstål, Pontus; Abad, Tahereh; Warren, Christopher; Bengtsson, Andreas; Tancredi, Giovanna; Shumeiko, Vitaly; Bylander, Jonas; Johansson, Göran; Frisk Kockum, Anton (2021). "Fast Multiqubit Gates through Simultaneous Two-Qubit Gates". PRX Quantum. 2 (4): 040348. arXiv:2108.11358. Bibcode:2021PRXQ....2d0348G. doi:10.1103/PRXQuantum.2.040348. ISSN2691-3399.
^Maslov, Dmitri (2016-02-10). "Advantages of using relative-phase Toffoli gates with an application to multiple control Toffoli optimization". Physical Review A. 93 (2): 022311. arXiv:1508.03273. Bibcode:2016PhRvA..93b2311M. doi:10.1103/PhysRevA.93.022311. ISSN2469-9926. S2CID5226873.
^Song, Guang; Klappenecker, Andreas (2003-12-31). "The simplified Toffoli gate implementation by Margolus is optimal". arXiv:quant-ph/0312225. Bibcode:2003quant.ph.12225S.{{cite journal}}: Cite journal requires |journal= (help)
^Thapliyal, Himanshu; Ranganathan, Nagarajan (2009). "Design of Efficient Reversible Binary Subtractors Based on a New Reversible Gate". 2009 IEEE Computer Society Annual Symposium on VLSI. pp.229–234. doi:10.1109/ISVLSI.2009.49. ISBN978-1-4244-4408-3. S2CID16182781.
^Warren, Christopher; Fernández-Pendás, Jorge; Ahmed, Shahnawaz; Abad, Tahereh; Bengtsson, Andreas; Biznárová, Janka; Debnath, Kamanasish; Gu, Xiu; Križan, Christian; Osman, Amr; Fadavi Roudsari, Anita; Delsing, Per; Johansson, Göran; Frisk Kockum, Anton; Tancredi, Giovanna; Bylander, Jonas (2023). "Extensive characterization and implementation of a family of three-qubit gates at the coherence limit". npj Quantum Information. 9 (1): 44. arXiv:2207.02938. Bibcode:2023npjQI...9...44W. doi:10.1038/s41534-023-00711-x. ISSN2056-6387.
Retrieved from "
Categories:
Quantum computing
Quantum gates
Hidden categories:
CS1 errors: missing periodical
Articles with short description
Short description is different from Wikidata
Dynamic lists
All articles with unsourced statements
Articles with unsourced statements from September 2022
This page was last edited on 12 May 2025, at 08:09(UTC).
Text is available under the Creative Commons Attribution-ShareAlike 4.0 License; additional terms may apply. By using this site, you agree to the Terms of Use and Privacy Policy. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization.
Privacy policy
About Wikipedia
Disclaimers
Contact Wikipedia
Code of Conduct
Developers
Statistics
Cookie statement
Mobile view
Edit preview settings
Search
Search
[x] Toggle the table of contents
List of quantum logic gates
3 languagesAdd topic |
8650 | https://www.etymonline.com/word/pulchritudinous | Want to remove ads? Log in to see fewer ads, and become a Premium Member to remove all ads.
Origin and history of pulchritudinous
pulchritudinous(adj.)
"beautiful, fine or graceful in any way," from pulchritude (from Latin pulchritudo "beauty," genitive pulchritudinis) + -ous.
By 1840 in a humor piece in a Boston newspaper, composed almost entirely of p- words; in general use as a jocular word in newspapers in England, Canada, and U.S. by 1858.
Earlier English had the now-obsolete pulcrious "beautiful, fair" (c. 1500).
also from 1877
Entries linking to pulchritudinous
"beauty, fairness," c. 1400, pulcritude, from Latin pulchritudo "beauty; excellence, attractiveness," from pulcher "beautiful," a word of unknown origin. De Vaan writes that Latin the -ch- spelling is by Greek influence and posits a stem polkro- or pelkro-, but that's as far as he confidently takes it.
word-forming element making adjectives from nouns, meaning "having, full of, having to do with, doing, inclined to," from Old French -ous, -eux, from Latin -osus (compare -ose (1)). In chemistry, "having a lower valence than forms expressed in -ic."
Want to remove ads? Log in to see fewer ads, and become a Premium Member to remove all ads.
Trends of pulchritudinous
More to explore
Share pulchritudinous
Want to remove ads? Log in to see fewer ads, and become a Premium Member to remove all ads.
Want to remove ads? Log in to see fewer ads, and become a Premium Member to remove all ads.
Want to remove ads? Log in to see fewer ads, and become a Premium Member to remove all ads.
Quick and reliable accounts of the origin and history of English words. Scholarly, yet simple.
About
Support
Apps |
8651 | https://arxiv.org/html/2406.17504v1 | Back to arXiv
HTML conversions sometimes display errors due to content that did not convert correctly from the source. This paper uses the following packages that are not yet supported by the HTML conversion tool. Feedback on these issues are not necessary; they are known and are being worked on.
failed: complexity
Authors: achieve the best HTML results from your LaTeX submissions by following these best practices.
License: CC BY-NC-SA 4.0
arXiv:2406.17504v1 [cs.CG] 25 Jun 2024
\ArticleNo
The Complexity of Intersection Graphs of Lines in Space and Circle Orders
Jean Cardinal
Université libre de Bruxelles (ULB)
jean.cardinal@ulb.be
Abstract
We consider the complexity of the recognition problem for two families of combinatorial structures.
A graph is said to be an intersection graph of lines in space if every can be mapped to a straight line in so that is
an edge in if and only if and intersect.
A partially ordered set is said to be a circle order, or a 2-space-time order, if every can be mapped to a closed circular disk so that if and only if is contained in .
We prove that the recognition problems for intersection graphs of lines and circle orders are both -complete, hence polynomial-time equivalent to deciding whether a system of polynomial equalities and inequalities has a solution over the reals. The second result addresses an open problem posed by Brightwell and Luczak.
1 Introduction
The complexity class is the set of decision problems that are polynomial-time reducible to deciding the validity of a sentence in the existential theory of the reals.
This class was popularized by Schaefer and Stefankovic [32, 35], and since then many problems have been classified as -complete .
The class contains \NP, hence -complete problems are \NP-hard.
It is also known that .
Among known -complete problems, we find many computational geometry problems in which we are asked to decide whether some combinatorial structure has a geometric realization, hence recognition problems for combinatorial families defined by geometric constructions.
This includes, among others, realizable oriented matroids and order types [26, 27], linkages , simultaneously embeddable planar graphs [9, 34], and several classes of geometric intersection graphs, including segment intersection graphs [22, 23] and disk intersection graphs [19, 24]. More recently, other fundamental problems in computational geometry have been shown to be -complete, such as the art gallery problem , and several families of geometric packing problems .
In this paper, we prove the -completeness of the recognition problem for two families of combinatorial objects: (i) intersection graphs of lines in 3-space, and (ii) circle orders.
1.1 Lines in space
Arrangements of lines in 3-space are a classical topic in discrete geometry [14, 11, 29].
We consider the problem of deciding, given a graph , whether we can we map each of its vertices to a line in so that two vertices are adjacent if and only if the two corresponding lines intersect. Such graphs will be referred to as intersection graphs of lines in space.
Pach, Tardos, and Tóth studied complements of intersection graphs of lines in space, and proved that graphs of maximum degree three and line graphs (adjacency graphs of edges of a graph) are intersection graphs of lines in space . They also considered the computational complexity of various combinatorial optimization problems on these graphs.
Davies showed that intersection graphs of lines in space are not -bounded: There exist such graphs that have arbitrary large girth and chromatic number.
On the other hand, Cardinal, Payne, and Solomon showed that they satisfy a nice Erdős-Hajnal property: They either have a clique or an independent set of size .
Regarding the recognition problem, two closely related results to ours have been proved.
First, Matous̆ek and Kratochvíl showed that intersection graphs of line segments in the plane are -complete .
This also holds for the more restricted classes of outersegment, grounded segments, and ray intersection graphs .
Evans, Rzazewski, Saeedi, Shin, and Wolff proved that intersection graphs of line segments in are -complete as well.
We prove that we only need lines for this to hold.
Theorem 1.1.
Deciding whether a graph is an intersection graph of lines in space is -complete.
1.2 Geometric containement orders
Geometric containment orders are posets of the form , where is a countable set of (usually convex) subsets of , for some constant , and is the usual containment relation.
Geometric containment orders are well-studied .
The classical Dushnik-Miller dimension of a poset can be defined in terms of geometric containment orders, as the smallest such that the poset is isomorphic to a containment order of translates of the negative orthant in .
Circle orders are containment orders of closed disks in the plane, and more generally, -sphere orders are containment orders of closed balls in [39, 17, 5].
The corresponding notion of dimension is the Minkowski dimension .
The definition of -sphere orders is motivated by causal set theory in physics and the notion of space-time order [3, 31, 13, 40].
In 1992, Scheinerman proved that circle orders are also one-to-one with parabola orders and Loewner orders for Hermitian matrices, and gave a general equivalence statement for orders, an algebraically defined class of posets.
In a survey on the combinatorics of the causal set approach to quantum gravity , Brightwell and Luczak pose the problem of recognizing circle orders in those terms: “It is remarkable that this concrete question is apparently not easy to answer computationally; to the best of our knowledge, the complexity of determining whether a given finite partial order is (for instance) a circle order is unknown.”
We prove that the problem is -complete.
We recall that a poset is bipartite if it can be partitioned into two antichains, or subsets of pairwise incomparable elements.
Theorem 1.2.
Deciding whether a poset is a circle order is -complete, even when the input is restricted to bipartite posets.
Outline.
Theorem 1.1 is proved in Section 2, and Theorem 1.2 is proved in Section 3.
Our proofs build on the classical complexity-theoretic interpretation of Mnëv’s Universality Theorem [26, 27], and on recent hardness results from Kang and Müller , and Felsner and Scheucher .
Note that in both cases, containment in is easy to prove, and we concentrate on the hardness part.
Acknowledgments.
The proof of Theorem 1.1 is based on an idea due to Udo Hoffmann.
The author wishes to thank Till Miltzow and the EuroCG referees for their comments on a preliminary version.
2 -Completeness of intersection graphs of lines
Our proof is by reduction from the affine realizability problem for matroids of rank 3.
In short, we are given a ground set together with a collection of subsets of
defining the maximal independent sets of the matroid. In rank 3, the maximal independent sets have size three, hence is a collection of triples.
We wish to decide whether there exists a map such that for every triple , the points form a nondegenerate triangle if and only if .
It is somehow a folklore result that this problem is -complete, and a detailed proof was given recently by Kim, de Mesmay, and Miltzow .
Given a graph , a line realization of is a map from to the set of lines in such that .
For a subset , we use the notation .
The following observation will be useful.
Observation 1.
In a line realization of a complete graph on the vertex set , the lines of intersect in one point or are coplanar.
We also use the following statement, which allows us to force a collection of lines in a line realization to lie in a plane.
Lemma 2.1.
Let be the complete graph on vertices minus a perfect matching.
For all lines of a line realization of are coplanar.
Proof 2.2.
Let be the bipartition of the vertices of into two cliques of size , and consider a line realization of .
We show that the lines of are coplanar.
For contradiction, suppose otherwise.
Then from Observation 1, all lines of must intersect in one point .
Furthermore, there are three vertices such that the lines are not coplanar.
There is also a vertex adjacent to and in , but not adjacent to a fourth vertex .
Hence the line must intersect and but avoid .
The only possibility to intersect all three lines is in , but then also intersects , a contradiction.
By symmetry, all lines in must be coplanar as well, and lying in the same plane as those in .
Proof 2.3 (Proof of Theorem 1.1).
We reduce from realizability of a rank-3 matroid , as defined above.
It is convenient to let , and consider the set system on defined as the set of rank-2 flats of .
In a realization of , these flats correspond to all inclusionwise maximal subsets of collinear points.
Note that .
We start by considering a line realization of the graph defined in Lemma 2.1.
We consider the bipartition of the vertices of into the two cliques and , where is adjacent to if .
Then for each set , we add a new vertex and an edge for each .
We let be the resulting graph, defined on the vertex set .
We claim that is a intersection graph of lines if and only if the input matroid is realizable.
To construct a realization of from a line realization of we use planar point-line duality on the lines in .
From Lemma 2.1, the lines of all lie in a plane .
For every set , the vertices form a maximal clique in .
From Observation 1, this clique must be realized by an intersection point common to all lines , since otherwise lies in and intersects some lines in .
Point-line duality in maps lines with a common intersection point to collinear points.
We denote by the point that is dual to the line in .
Therefore, the dual points are collinear in .
From the maximality of the clique , the points form a maximal collinear set, and is a realization of in .
On the other hand, a realization of in the plane can be mapped to a dual arrangement of lines.
Let , and be the line representing element , for every .
Now each line can be represented by a line parallel to and intersecting all the other lines and for .
We embed this realization into a plane in and add a line for each that pierces in the common intersection point of the lines (see Figure 1).
This yields a line realization of .
3 -Completeness of circle orders
A pseudoline arrangement is a collection of -monotone curves that pairwise intersect exactly once.
A pseudoline arrangement is simple if no triple of lines intersect in the same point, and stretchable if it is isomorphic to an arrangement of straight lines, see Figure 2.
A well-known consequence of Mnëv’s Theorem – see Shor , and Richter-Gebert – is that deciding whether a given pseudoline arrangement is stretchable is -complete.
In order to deal with circle orders, we make use of recent analogous results on arrangement of circles due to Kang and Müller , and Felsner and Scheucher .
An arrangement of Jordan curves is said to be a pseudocircle arrangement if every pair of curves is either disjoint or intersects in exactly two points.
A triple of pseudocircles is said to form a Krupp if they pairwise intersect in distinct points and their interiors have a common point.
A pseudocircle arrangement is said to be a great-pseudocircle arrangement if every triple of pseudocircles forms a Krupp. See Figure 3 for illustrations.
Similarly to pseudoline arrangements, we are only concerned with the topological structure of the arrangement.
Great-pseudocircle arrangements constitute a combinatorial abstraction of arrangements of great circles on a sphere, hence of line arrangements in the projective plane.
In fact, great-pseudocircle arrangements and simple pseudoline arrangements are essentially the same combinatorial family, see Felsner and Scheucher .
Lemma 3.1.
Great-pseudocircle arrangements are one-to-one with simple pseudoline arrangements.
Every great-pseudocircle arrangement realizes two copies of a pseudoline arrangement, see Figure 3.
When stretchable, they correspond to realizations of pseudolines as great circles on the 2-sphere, or equivalently as lines in the projective plane.
We need a technical lemma on the structure of circle arrangements.
A collection of circles in the plane partitions the plane into connected subsets that we refer to as cells.
To each cell, one can assign a binary string of length indicating, for each circle, whether the cell lies in the interior or the exterior of the circle.
We refer to this string as the label of the cell.
Lemma 3.2.
The maximum number of dictinct cell labels of an arrangement of circles is equal to the maximum number of cells of a circle arrangement, and is .
This bound is attained by great-pseudocircle arrangements.
Proof 3.3.
One can proceed by induction on . We let be the maximum number of labels, with .
When adding the th circle to the arrangement, the number of additional labels is at most the number of newly created cells.
This number is in turn bounded by the number of new intersection points:
One can charge every new cell to the intersection point on its left in a clockwise sweep around the new circle.
Therefore , solving to the announced bound.
From this reasoning, tight examples must be such that no two cells have the same label, and one can check that this bound is attained by great-pseudocircle arrangements.
The circularizability problem is the problem of deciding whether a pseudocircle arrangement is isomorphic to a circle arrangement.
It is the analogue of the stretchability problem for pseudoline arrangements, and is in fact polynomial-time equivalent to it.
Theorem 3.4 (Kang and Müller , Felsner and Scheucher ).
Deciding whether a pseudocircle arrangement is circularizable is -complete, even when the input is restricted to great-pseudocircle arrangements.
With these tools at hand, we now prove our second main result.
Proof 3.5 (Proof of Theorem 1.2).
We reduce from the circularizability problem.
Given a great-pseudocircle arrangement , we construct a bipartite poset that is a circle order if and only if is circularizable.
We define the ground set of as , where is the set of great-pseudocircles of , and is the set of cell labels of .
Both and induce empty posets in , and for a pair , we let if and only if the cell corresponding to lies in the disk bounded by .
We show that is a circle order if and only if is circularizable.
Indeed, if is circularizable, then we can consider a circle realization of and add a small circle within each cell of the arrangement to realize .
For the other direction, suppose has a realization as a circle order and consider the arrangement induced by in this realization.
Since every has an associated circle as well, every cell label of is realized in . From Lemma 3.2, the number of cells is as large as can be for an arrangement of circles, hence no other cell can appear in . Therefore must be a circularization of .
Note that a similar proof was given by Tanenbaum, Goodrich, and Scheinerman to prove hardness of point-halfspace incidence orders.
Also note that from a theorem of Scheinerman , the vertex-edge incidence poset of a graph is a circle order if and only if the graph is planar.
Hence for vertex-edge incidence posets, the problem reduces to planarity testing and can therefore be solved in linear time.
References
Mikkel Abrahamsen, Anna Adamaszek, and Tillmann Miltzow.
The art gallery problem is -complete.
J. ACM, 69(1):4:1–4:70, 2022.
Mikkel Abrahamsen, Tillmann Miltzow, and Nadja Seiferth.
Framework for ER-completeness of two-dimensional packing problems.
In Sandy Irani, editor, 61st IEEE Annual Symposium on
Foundations of Computer Science, FOCS 2020, Durham, NC, USA, November
16-19, 2020, pages 1014–1021. IEEE, 2020.
Graham Brightwell and Ruth Gregory.
Structure of random discrete spacetime.
Phys. Rev. Lett., 66:260–263, Jan 1991.
Graham Brightwell and Malwina Luczak.
The mathematics of causal sets.
In Recent Trends in Combinatorics, volume 159 of The IMA
Volumes in Mathematics and its Applications. Springer, 2016.
Graham Brightwell and Peter Winkler.
Sphere orders.
Order, 6:235–240, 1989.
John Canny.
Some algebraic and geometric computations in PSPACE.
In Symposium on Theory of Computing (STOC), pages 460–467.
ACM, 1988.
Jean Cardinal.
Computational geometry column 62.
SIGACT News, 46(4):69–78, 2015.
Jean Cardinal, Stefan Felsner, Tillmann Miltzow, Casey Tompkins, and Birgit
Vogtenhuber.
Intersection graphs of rays and grounded segments.
J. Graph Algorithms Appl., 22(2):273–295, 2018.
Jean Cardinal and Vincent Kusters.
The complexity of simultaneous geometric graph embedding.
J. Graph Algorithms Appl., 19(1):259–272, 2015.
Jean Cardinal, Michael S. Payne, and Noam Solomon.
Ramsey-type theorems for lines in 3-space.
Discret. Math. Theor. Comput. Sci., 18(3), 2016.
Bernard Chazelle, Herbert Edelsbrunner, Leonidas J. Guibas, Micha Sharir, and
Jorge Stolfi.
Lines in space: Combinatorics and algorithms.
Algorithmica, 15(5):428–447, 1996.
James Davies.
Box and segment intersection graphs with large girth and chromatic
number.
Advances in combinatorics, 2021.
Fay Dowker.
Introduction to causal sets and their phenomenology.
General Relativity and Gravitation, 45(9):1651–1667, 2013.
Herbert Edelsbrunner.
Lines in space-a collection of results.
In Jacob E. Goodman, Richard Pollack, and William Steiger, editors,
Discrete and Computational Geometry: Papers from the DIMACS Special
Year, volume 6 of DIMACS Series in Discrete Mathematics and
Theoretical Computer Science, pages 77–94. DIMACS/AMS, 1990.
William S. Evans, Pawel Rzazewski, Noushin Saeedi, Chan-Su Shin, and
Alexander Wolff.
Representing graphs and hypergraphs by touching polygons in 3D.
In Daniel Archambault and Csaba D. Tóth, editors, Graph
Drawing and Network Visualization - 27th International Symposium, GD 2019,
Prague, Czech Republic, September 17-20, 2019, Proceedings, volume 11904 of
Lecture Notes in Computer Science, pages 18–32. Springer, 2019.
Stefan Felsner and Manfred Scheucher.
Arrangements of pseudocircles: On circularizability.
In Therese C. Biedl and Andreas Kerren, editors, Graph Drawing
and Network Visualization - 26th International Symposium, GD 2018,
Barcelona, Spain, September 26-28, 2018, Proceedings, volume 11282 of Lecture Notes in Computer Science, pages 555–568. Springer, 2018.
Peter C. Fishburn.
Interval orders and circle orders.
Order, 5:225–234, 1988.
Peter C. Fishburn and William T. Trotter.
Geometric containment orders: A survey.
Order, 15(2):167–182, 1998.
Ross J. Kang and Tobias Müller.
Sphere and dot product representations of graphs.
Discret. Comput. Geom., 47(3):548–568, 2012.
Ross J. Kang and Tobias Müller.
Arrangements of pseudocircles and circles.
Discret. Comput. Geom., 51(4):896–925, 2014.
Eunjung Kim, Arnaud de Mesmay, and Tillmann Miltzow.
Representing matroids over the reals is -complete.
ArXiv e-prints, 2023.
Jan Kratochvíl and Jirí Matousek.
Intersection graphs of segments.
J. Comb. Theory, Ser. B, 62(2):289–315, 1994.
Jiří Matoušek.
Intersection graphs of segments and .
ArXiv e-prints, 2014.
Colin McDiarmid and Tobias Müller.
Integer realizations of disk and segment graphs.
J. Comb. Theory, Ser. B, 103(1):114–143, 2013.
David A. Meyer.
Spherical containment and the Minkowski dimension of partial
orders.
Order, 10:227–237, 1993.
Nicolai E. Mnëv.
Varieties of combinatorial types of projective configurations and
convex polyhedra.
Dokl. Akad. Nauk SSSR, 283(6):1312–1314, 1985.
Nicolai E. Mnëv.
The universality theorems on the classification problem of
configuration varieties and convex polytopes varieties.
In Topology and geometry – Rohlin seminar, pages 527–543.
Springer, 1988.
János Pach, Gábor Tardos, and Géza Tóth.
Disjointness graphs of segments in the space.
Comb. Probab. Comput., 30(4):498–512, 2021.
Orit E. Raz.
Configurations of lines in space and combinatorial rigidity.
Discret. Comput. Geom., 58(4):986–1009, 2017.
Jürgen Richter-Gebert.
Mnëv’s universality theorem revisited.
In Proceedings of the Séminaire Lotharingien de
Combinatoire, pages 211–225, 1995.
David P. Rideout and Rafael D. Sorkin.
Classical sequential growth dynamics for causal sets.
Phys. Rev. D, 61:024002, Dec 1999.
Marcus Schaefer.
Complexity of some geometric and topological problems.
In David Eppstein and Emden R. Gansner, editors, Graph Drawing,
17th International Symposium, GD 2009, Chicago, IL, USA, September 22-25,
2009. Revised Papers, volume 5849 of Lecture Notes in Computer
Science, pages 334–344. Springer, 2009.
Marcus Schaefer.
Realizability of graphs and linkages.
In Thirty Essays on Geometric Graph Theory. Springer, 2012.
Marcus Schaefer.
On the complexity of some geometric problems with fixed parameters.
J. Graph Algorithms Appl., 25(1):195–218, 2021.
Marcus Schaefer and Daniel Stefankovic.
Fixed points, Nash equilibria, and the existential theory of the
reals.
Theory Comput. Syst., 60(2):172–193, 2017.
Edward R. Scheinerman.
A note on planar graphs and circle orders.
SIAM J. Discret. Math., 4(3):448–451, 1991.
Edward R. Scheinerman.
The many faces of circle orders.
Order, 9(4):343–348, 1992.
Peter W. Shor.
Stretchability of pseudolines is NP-hard.
Applied Geometry and Discrete Mathematics–The Victor Klee
Festschrift, 4:531–554, 1991.
Jeffrey B. Sidney, Stuart J. Sidney, and Jorge Urrutia.
Circle orders, N-gon orders and the crossing number.
Order, 5:1–10, 1988.
Sumati Surya.
The causal set approach to quantum gravity.
Living Reviews in Relativity, 22(1), 2019.
Paul J. Tanenbaum, Michael T. Goodrich, and Edward R. Scheinerman.
Characterization and recognition of point-halfspace and related
orders.
In Roberto Tamassia and Ioannis G. Tollis, editors, Graph
Drawing, DIMACS International Workshop, GD’94, Princeton, New Jersey,
USA, October 10-12, 1994, Proceedings, volume 894 of Lecture Notes in
Computer Science, pages 234–245. Springer, 1994.
William T. Trotter.
Combinatorics and Partially Ordered Sets: Dimension Theory.
Johns Hopkins Studies in Nineteenth C Architecture Series. Johns
Hopkins University Press, 1992. |
8652 | https://artofproblemsolving.com/wiki/index.php/Power_of_a_Point_Theorem?srsltid=AfmBOoqhMqVYHS92WiL61LynZLzo8ljw6cl5yXvk4FeVN1PvUWpZmlZU | Art of Problem Solving
Power of a Point Theorem - AoPS Wiki
Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical CampusVisit the Virtual Campus
Sign In
Register
online school
Class ScheduleRecommendationsOlympiad CoursesFree Sessions
books tore
AoPS CurriculumBeast AcademyOnline BooksRecommendationsOther Books & GearAll ProductsGift Certificates
community
ForumsContestsSearchHelp
resources
math training & toolsAlcumusVideosFor the Win!MATHCOUNTS TrainerAoPS Practice ContestsAoPS WikiLaTeX TeXeRMIT PRIMES/CrowdMathKeep LearningAll Ten
contests on aopsPractice Math ContestsUSABO
newsAoPS BlogWebinars
view all 0
Sign In
Register
AoPS Wiki
ResourcesAops Wiki Power of a Point Theorem
Page
ArticleDiscussionView sourceHistory
Toolbox
Recent changesRandom pageHelpWhat links hereSpecial pages
Search
Power of a Point Theorem
The Power of a Point Theorem is a relationship that holds between the lengths of the line segments formed when two linesintersect a circle and each other.
Contents
[hide]
1 Statement
1.1 Case 1 (Inside the Circle):
1.2 Case 2 (Outside the Circle):
1.2.1 Classic Configuration
1.2.2 Tangent Line
1.3 Case 3 (On the Border/Useless Case):
1.4 Alternate Formulation
1.5 Hint for Proof
2 Notes
3 Proof
3.1 Case 1 (Inside the Circle)
3.2 Case 2 (Outside the Circle)
3.3 Case 3 (On the Circle Border)
4 Problems
4.1 Introductory
4.2 Intermediate
4.3 Olympiad
5 See Also
5.1 External Links
Statement
There are three unique cases for this theorem. Each case expresses the relationship between the length of line segments that pass through a common point and touch a circle in at least one point. Can be useful with cyclic quadrilaterals as well however with a slightly different application.
Case 1 (Inside the Circle):
If two chords and intersect at a point within a circle, then
Case 2 (Outside the Circle):
Classic Configuration
Given lines and originate from two unique points on the circumference of a circle ( and ), intersect each other at point , outside the circle, and re-intersect the circle at points and respectively, then
Tangent Line
Given Lines and with tangent to the related circle at , lies outside the circle, and Line intersects the circle between and at ,
Case 3 (On the Border/Useless Case):
If two chords, and , have on the border of the circle, then the same property such that if two lines that intersect and touch a circle, then the product of each of the lines segments is the same. However since the intersection points lies on the border of the circle, one segment of each line is so no matter what, the constant product is .
Alternate Formulation
This alternate formulation is much more compact, convenient, and general.
Consider a circle and a point in the plane where is not on the circle. Now draw a line through that intersects the circle in two places. The power of a point theorem says that the product of the length from to the first point of intersection and the length from to the second point of intersection is constant for any choice of a line through that intersects the circle. This constant is called the power of point . For example, in the figure below
Hint for Proof
Draw extra lines to create similar triangles (Draw on all three figures. Draw another line as well.)
Notice how this definition still works if and coincide (as is the case with ). Consider also when is inside the circle. The definition still holds in this case.
Notes
One important result of this theorem is that both tangents from any point outside of a circle to that circle are equal in length.
The theorem generalizes to higher dimensions, as follows.
Let be a point, and let be an -sphere. Let two arbitrary lines passing through intersect at , respectively. Then
Proof. We have already proven the theorem for a -sphere (a circle), so it only remains to prove the theorem for more dimensions. Consider the plane containing both of the lines passing through . The intersection of and must be a circle. If we consider the lines and with respect simply to that circle, then we have reduced our claim to the case of two dimensions, in which we know the theorem holds.
Proof
Case 1 (Inside the Circle)
Join and .
In
(Angles subtended by the same segment are equal)
(Vertically opposite angles)
(Corresponding sides of similar triangles are in the same ratio)
Case 2 (Outside the Circle)
Join and
(Why?)
Now, In
(shown above)
(common angle)
(Corresponding sides of similar triangles are in the same ratio)
Case 3 (On the Circle Border)
Length of a point is zero so no proof needed:)
Problems
Introductory
Find the value of in the following diagram:
Solution
Find the value of in the following diagram:
Solution
(ARML) In a circle, chords and intersect at . If and , find the ratio .
Solution
(ARML) Chords and of a given circle are perpendicular to each other and intersect at a right angle at point . Given that , , and , find .
Solution
Let be a diameter in a circle of radius Let be a chord in the circle that intersects at a point such that and What is (Source)
Intermediate
Two tangents from an external point are drawn to a circle and intersect it at and . A third tangent meets the circle at , and the tangents and at points and , respectively (this means that T is on the minor arc ). If , find the perimeter of . (Source)
Square of side length has a circle inscribed in it. Let be the midpoint of . Find the length of that portion of the segment that lies outside of the circle. (Source)
is a chord of a circle such that and Let be the center of the circle. Join and extend to cut the circle at Given find the radius of the circle. (Source)
Triangle has The incircle of the triangle evenly trisects the median If the area of the triangle is where and are integers and is not divisible by the square of a prime, find (Source)
Let be a triangle inscribed in circle . Let the tangents to at and intersect at point , and let intersect at . If , , and , can be written as the form , where and are relatively prime integers. Find . (Source)
Olympiad
Given circles and intersecting at points and , let be a line through the center of intersecting at points and and let be a line through the center of intersecting at points and . Prove that if and lie on a circle then the center of this circle lies on line .
(Source)
Let be a point interior to triangle (with ). The lines , and meet again its circumcircle at , , respectively . The tangent line at to meets the line at . Show that from follows .
(Source)
See Also
Geometry
Planar figures
External Links
Handout on AoPS Forums
This article is a stub. Help us out by expanding it.
Retrieved from "
Categories:
Geometry
Theorems
Stubs
Art of Problem Solving is an
ACS WASC Accredited School
aops programs
AoPS Online
Beast Academy
AoPS Academy
About
About AoPS
Our Team
Our History
Jobs
AoPS Blog
Site Info
Terms
Privacy
Contact Us
follow us
Subscribe for news and updates
© 2025 AoPS Incorporated
© 2025 Art of Problem Solving
About Us•Contact Us•Terms•Privacy
Copyright © 2025 Art of Problem Solving
Something appears to not have loaded correctly.
Click to refresh. |
8653 | https://math.stackexchange.com/questions/3696711/obtaining-condition-for-existence-of-two-chords-through-a-single-point-with-thei | geometry - Obtaining condition for existence of two Chords through a single point with their Mid-points on the $x$-Axis - Mathematics Stack Exchange
Join Mathematics
By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy.
Sign up with Google
OR
Email
Password
Sign up
Already have an account? Log in
Skip to main content
Stack Exchange Network
Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
Visit Stack Exchange
Loading…
Tour Start here for a quick overview of the site
Help Center Detailed answers to any questions you might have
Meta Discuss the workings and policies of this site
About Us Learn more about Stack Overflow the company, and our products
current community
Mathematics helpchat
Mathematics Meta
your communities
Sign up or log in to customize your list.
more stack exchange communities
company blog
Log in
Sign up
Home
Questions
Unanswered
AI Assist Labs
Tags
Chat
Users
Teams
Ask questions, find answers and collaborate at work with Stack Overflow for Teams.
Try Teams for freeExplore Teams
3. Teams
4. Ask questions, find answers and collaborate at work with Stack Overflow for Teams. Explore Teams
Teams
Q&A for work
Connect and share knowledge within a single location that is structured and easy to search.
Learn more about Teams
Hang on, you can't upvote just yet.
You'll need to complete a few actions and gain 15 reputation points before being able to upvote. Upvoting indicates when questions and answers are useful. What's reputation and how do I get it?
Instead, you can save this post to reference later.
Save this post for later Not now
Thanks for your vote!
You now have 5 free votes weekly.
Free votes
count toward the total vote score
does not give reputation to the author
Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation.
Got it!Go to help center to learn more
Obtaining condition for existence of two Chords through a single point with their Mid-points on the x x-Axis
Ask Question
Asked 5 years, 4 months ago
Modified5 years, 4 months ago
Viewed 67 times
This question shows research effort; it is useful and clear
0
Save this question.
Show activity on this post.
Given a circle x 2+y 2=α x+β y x 2+y 2=α x+β y, obtain the condition on α α and β β, if two distinct chords can be drawn through the point (α,β)(α,β) such that their mid-points lie on the x x-axis.
For starters, it is evident that A(α,β)A(α,β) will lie on the circle. I have introduced a parameter θ θ to represent all lines passing through A A. Also, we need to make sure that every such line intersects the x x-axis inside the circle, as it is the mid-point that must be the intersection point, and therefore we must have tan−1 β α<θ<π 2 tan−1β α<θ<π 2. The equation of any such line is
x=y cot θ+α−β cot θ x=y cotθ+α−β cotθ
Putting this value of x x in the circle’s equation, I can get the y y-coordinate of the other end of the chord:
y=(β cot θ−α)sin θ cos θ y=(β cotθ−α)sinθ cosθ
And so the ordinate of the midpoint can be found and equated to zero.
(β cot θ−α)sin θ cos θ+β 2=0(β cotθ−α)sinθ cosθ+β 2=0
This can be transformed into a quadratic in cos 2 θ cos 2θ.
(α 2+β 2)cos 4 θ+(2 β 2−α 2)cos 2 θ+β 2=0(α 2+β 2)cos 4θ+(2 β 2−α 2)cos 2θ+β 2=0
According to the question, it is required that this equation have two distinct roots, and so D>0 D>0, i.e.
(2 β 2−α 2)2−4 β 2(α 2+β 2)>0⟹|α|>2 2–√|β|(2 β 2−α 2)2−4 β 2(α 2+β 2)>0⟹|α|>2 2|β|
I would like to ask if my solution is correct, and is there a faster and more efficient way? How would you tackle this problem?
geometry
circles
solution-verification
coordinate-systems
Share
Share a link to this question
Copy linkCC BY-SA 4.0
Cite
Follow
Follow this question to receive notifications
asked May 29, 2020 at 10:19
VishuVishu
14.6k 2 2 gold badges 17 17 silver badges 46 46 bronze badges
Add a comment|
1 Answer 1
Sorted by: Reset to default
This answer is useful
0
Save this answer.
Show activity on this post.
If the midpoint of a chord with one endpoint at (α,β)(α,β) lies on the x x-axis, then the y y-coordinate of the other endpoint must be −β−β. We therefore seek α α and β β such that x 2+β 2=α x−β 2 x 2+β 2=α x−β 2 has two distinct real solutions. The discriminant of this quadratic equation is α 2−8 β 2 α 2−8 β 2, from which we obtain the condition α 2>8 β 2 α 2>8 β 2. This agrees with your result.
Share
Share a link to this answer
Copy linkCC BY-SA 4.0
Cite
Follow
Follow this answer to receive notifications
answered May 29, 2020 at 23:53
amdamd
55.2k 3 3 gold badges 40 40 silver badges 100 100 bronze badges
Add a comment|
You must log in to answer this question.
Start asking to get answers
Find the answer to your question by asking.
Ask question
Explore related questions
geometry
circles
solution-verification
coordinate-systems
See similar questions with these tags.
Featured on Meta
Introducing a new proactive anti-spam measure
Spevacus has joined us as a Community Manager
stackoverflow.ai - rebuilt for attribution
Community Asks Sprint Announcement - September 2025
Report this ad
Related
0Find the x-coordinate as the chord of two points on a parabola touches the x-axis
0Number of chords of a circle having natural length
0equation of the circle which passes through two points and the chord joining them making an angle θ θ in the major segment of the circle
0Point Circle passing through two points?
1Locus of midpoint of a chord which along with a line makes equal angles with X axis
1Probability of chords of the unit circle in R 2 R 2 from the book A Modern Approach to Probability Theory, B. Fristedt, L. Gray
2Variable pairs of chords at right angles are drawn through a point P P (with eccentric angle π/4 π/4) on the ellipse.
1The line through a point P perpendicular to polar of P with respect to given parabola passes through fixed point. Find the polar of P.
2What is the locus of midpoints of chords which pass through point P P?
Hot Network Questions
Any knowledge on biodegradable lubes, greases and degreasers and how they perform long term?
What were "milk bars" in 1920s Japan?
"Unexpected"-type comic story. Aboard a space ark/colony ship. Everyone's a vampire/werewolf
How to home-make rubber feet stoppers for table legs?
Is encrypting the login keyring necessary if you have full disk encryption?
The geologic realities of a massive well out at Sea
How different is Roman Latin?
Can peaty/boggy/wet/soggy/marshy ground be solid enough to support several tonnes of foot traffic per minute but NOT support a road?
On being a Maître de conférence (France): Importance of Postdoc
What's the expectation around asking to be invited to invitation-only workshops?
I have a lot of PTO to take, which will make the deadline impossible
Passengers on a flight vote on the destination, "It's democracy!"
How to rsync a large file by comparing earlier versions on the sending end?
How do you create a no-attack area?
An odd question
Stress in "agentic"
Who is the target audience of Netanyahu's speech at the United Nations?
In Dwarf Fortress, why can't I farm any crops?
Direct train from Rotterdam to Lille Europe
How to locate a leak in an irrigation system?
Checking model assumptions at cluster level vs global level?
What is a "non-reversible filter"?
Exchange a file in a zip file quickly
Numbers Interpreted in Smallest Valid Base
more hot questions
Question feed
Subscribe to RSS
Question feed
To subscribe to this RSS feed, copy and paste this URL into your RSS reader.
Why are you flagging this comment?
It contains harassment, bigotry or abuse.
This comment attacks a person or group. Learn more in our Code of Conduct.
It's unfriendly or unkind.
This comment is rude or condescending. Learn more in our Code of Conduct.
Not needed.
This comment is not relevant to the post.
Enter at least 6 characters
Something else.
A problem not listed above. Try to be as specific as possible.
Enter at least 6 characters
Flag comment Cancel
You have 0 flags left today
Mathematics
Tour
Help
Chat
Contact
Feedback
Company
Stack Overflow
Teams
Advertising
Talent
About
Press
Legal
Privacy Policy
Terms of Service
Your Privacy Choices
Cookie Policy
Stack Exchange Network
Technology
Culture & recreation
Life & arts
Science
Professional
Business
API
Data
Blog
Facebook
Twitter
LinkedIn
Instagram
Site design / logo © 2025 Stack Exchange Inc; user contributions licensed under CC BY-SA. rev 2025.9.26.34547
By clicking “Accept all cookies”, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy.
Accept all cookies Necessary cookies only
Customize settings
Cookie Consent Preference Center
When you visit any of our websites, it may store or retrieve information on your browser, mostly in the form of cookies. This information might be about you, your preferences, or your device and is mostly used to make the site work as you expect it to. The information does not usually directly identify you, but it can give you a more personalized experience. Because we respect your right to privacy, you can choose not to allow some types of cookies. Click on the different category headings to find out more and manage your preferences. Please note, blocking some types of cookies may impact your experience of the site and the services we are able to offer.
Cookie Policy
Accept all cookies
Manage Consent Preferences
Strictly Necessary Cookies
Always Active
These cookies are necessary for the website to function and cannot be switched off in our systems. They are usually only set in response to actions made by you which amount to a request for services, such as setting your privacy preferences, logging in or filling in forms. You can set your browser to block or alert you about these cookies, but some parts of the site will not then work. These cookies do not store any personally identifiable information.
Cookies Details
Performance Cookies
[x] Performance Cookies
These cookies allow us to count visits and traffic sources so we can measure and improve the performance of our site. They help us to know which pages are the most and least popular and see how visitors move around the site. All information these cookies collect is aggregated and therefore anonymous. If you do not allow these cookies we will not know when you have visited our site, and will not be able to monitor its performance.
Cookies Details
Functional Cookies
[x] Functional Cookies
These cookies enable the website to provide enhanced functionality and personalisation. They may be set by us or by third party providers whose services we have added to our pages. If you do not allow these cookies then some or all of these services may not function properly.
Cookies Details
Targeting Cookies
[x] Targeting Cookies
These cookies are used to make advertising messages more relevant to you and may be set through our site by us or by our advertising partners. They may be used to build a profile of your interests and show you relevant advertising on our site or on other sites. They do not store directly personal information, but are based on uniquely identifying your browser and internet device.
Cookies Details
Cookie List
Clear
[x] checkbox label label
Apply Cancel
Consent Leg.Interest
[x] checkbox label label
[x] checkbox label label
[x] checkbox label label
Necessary cookies only Confirm my choices |
8654 | https://journals.plos.org/plospathogens/article?id=10.1371/journal.ppat.1012354 | The nuclear localization signal of CPSF6 governs post-nuclear import steps of HIV-1 infection | PLOS Pathogens
Skip to main content
Advertisement
plos.org
Create account
Sign in
Browse
Publish
About
Searchadvanced search
5 Save
Total Mendeley and Citeulike bookmarks.
3 Citation
Paper's citation count computed by Dimensions.
1,053 View
PLOS views and downloads.
0 Share
Sum of Facebook, Twitter, Reddit and Wikipedia activity.
Open Access
Peer-reviewed
Research Article
The nuclear localization signal of CPSF6 governs post-nuclear import steps of HIV-1 infection
Nicholas Rohlfes,Roles Formal analysis, Investigation, Methodology, Validation, Visualization, Writing – original draft, Writing – review & editing
Affiliation Integrative Cell Biology Graduate Program, Loyola University Chicago, Maywood, Illinois, United States of America
⨯
Rajalingam Radhakrishnan,Roles Formal analysis, Investigation, Methodology, Writing – review & editing
Affiliation Department of Cancer Immunology and Virology, Dana-Farber Cancer Institute, Boston, Massachusetts, United States of America
⨯
Parmit K. Singh,Roles Formal analysis, Investigation, Methodology, Writing – review & editing
Affiliations Department of Cancer Immunology and Virology, Dana-Farber Cancer Institute, Boston, Massachusetts, United States of America, Department of Medicine, Harvard Medical School, Boston, Massachusetts, United States of America
⨯
Gregory J. Bedwell,Roles Formal analysis, Investigation, Methodology, Writing – review & editing
Affiliations Department of Cancer Immunology and Virology, Dana-Farber Cancer Institute, Boston, Massachusetts, United States of America, Department of Medicine, Harvard Medical School, Boston, Massachusetts, United States of America
⨯
Alan N. Engelman,Roles Funding acquisition, Writing – review & editing
Affiliations Department of Cancer Immunology and Virology, Dana-Farber Cancer Institute, Boston, Massachusetts, United States of America, Department of Medicine, Harvard Medical School, Boston, Massachusetts, United States of America
⨯
Adarsh Dharan ,Roles Conceptualization, Funding acquisition, Investigation, Methodology, Supervision, Writing – review & editing
E-mail:adharan@iu.edu (AD); ecampbell@luc.edu (EMC)
Affiliation Department of Biology, Indiana University, Bloomington, Indiana, United States of America
⨯
Edward M. CampbellRoles Conceptualization, Funding acquisition, Supervision, Writing – review & editing
E-mail:adharan@iu.edu (AD); ecampbell@luc.edu (EMC)
Affiliations Integrative Cell Biology Graduate Program, Loyola University Chicago, Maywood, Illinois, United States of America, Department of Microbiology and Immunology, Loyola University Chicago, Maywood, Illinois, United States of America
⨯
The nuclear localization signal of CPSF6 governs post-nuclear import steps of HIV-1 infection
Nicholas Rohlfes,
Rajalingam Radhakrishnan,
Parmit K. Singh,
Gregory J. Bedwell,
Alan N. Engelman,
Adarsh Dharan,
Edward M. Campbell
x
Published: January 17, 2025
Article
Authors
Metrics
Comments
Media Coverage
Abstract
Author summary
Introduction
Results
Discussion
Materials and methods
Supporting information
References
Reader Comments
Figures
Accessible Data
See the data
This article includes the Accessible Data icon, an experimental feature to encourage data sharing and reuse.Find out how research articles qualify for this feature.
Abstract
The early stages of HIV-1 infection include the trafficking of the viral core into the nucleus of infected cells. However, much remains to be understood about how HIV-1 accomplishes nuclear import and the consequences of the import pathways utilized on nuclear events. The host factor cleavage and polyadenylation specificity factor 6 (CPSF6) assists HIV-1 nuclear localization and post-entry integration targeting. Here, we used a CPSF6 truncation mutant lacking a functional nuclear localization signal (NLS), CPSF6-358, and appended heterologous NLSs to rescue nuclear localization. We show that some, but not all, NLSs drive CPSF6-358 into the nucleus. Interestingly, we found that some nuclear localized CPSF6-NLS chimeras supported inefficient HIV-1 infection. We found that HIV-1 still enters the nucleus in these cell lines but fails to traffic to speckle-associated domains (SPADs). Additionally, we show that HIV-1 fails to efficiently integrate in these cell lines. Collectively, our results demonstrate that the NLS of CPSF6 facilitates steps of HIV-1 infection subsequent to nuclear import and additionally identify the ability of canonical NLS sequences to influence cargo localization in the nucleus following nuclear import.
Author summary
During HIV-1 infection, the viral capsid, which encloses the viral genome and essential proteins required for reverse transcription and integration, traffics towards the nucleus and enters through the nuclear pore complex (NPC). Following entry into the nucleus, reverse transcription is completed and viral capsid disassembles for integration of the viral DNA into a host chromosome. In this study, we investigated the role of the capsid-binding host factor CPSF6 in these events, specifically focusing on its nuclear localization signal (NLS) located at the protein’s C-terminus. Swapping the NLS for heterologous NLSs from other proteins significantly impacted HIV-1’s ability to infect cells. For NLSs that conveyed CPSF6 nuclear import, we further showed this defect in infection occurred at the level of viral DNA integration. This study highlights the importance of the NLS in CPSF6 to dictate nuclear import pathways that are congruent with functional HIV-1 infection. By extension, our study emphasizes that different NLSs can have comparatively dramatic post-nuclear effects on host cargo function.
Figures
Citation:Rohlfes N, Radhakrishnan R, Singh PK, Bedwell GJ, Engelman AN, Dharan A, et al. (2025) The nuclear localization signal of CPSF6 governs post-nuclear import steps of HIV-1 infection. PLoS Pathog 21(1): e1012354.
Editor:Li Wu, University of Iowa, UNITED STATES OF AMERICA
Received:June 18, 2024; Accepted:January 5, 2025; Published: January 17, 2025
Copyright: © 2025 Rohlfes et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability:Illumina Sequencing results can be found with Accession no: PRJNA1123763. All other relevant data are within the paper and its Supporting Information files.
Funding:This work was funded by the National Institute of Health (NIH, grant R01AI162694 to EMC and R01AI052014 and U54AI170791 awarded to ANE. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing interests: The authors have declared that no competing interests exist.
Introduction
A defining feature of lentiviruses, including human immunodeficiency virus-1 (HIV-1), is the ability to infect non-dividing cells, which requires import of the viral ribonucleoprotein complex (vRNPC) into the target cell nucleus. This requires that the virus traverse nuclear pore complexes (NPCs) that exist within the nuclear membranes of non-dividing cells. NPCs are large, macromolecular complexes composed of multiple copies of about 30 nucleoporins (Nups) [1–3]. Critically, NPCs are responsible for maintaining the nuclear permeability barrier, which restricts entry of many proteins while simultaneously allowing directed trafficking of hundreds to thousands of cargoes per second [4,5]. As a result, nucleocytoplasmic trafficking is a complex and tightly regulated process. To ensure the correct cargo is being transported into the nucleus, short amino acid sequences called nuclear localization signals (NLSs) are recognized by nuclear transport receptors to facilitate nuclear import [6,7].
The HIV-1 capsid core is comprised of approximately 250 hexamers and exactly 12 pentamers of capsid protein (CA) forming a conical structure that houses the viral genome and essential enzymes [8–12]. CA is the primary viral determinant driving nuclear import , and CA accordingly interacts with several Nups including Nup358 [14–16], Nup153 [15,17–22], Nup58 [15,21], Nup98 [15,21], Nup88 , and Nup62 [15,20]. Capsid additionally interacts with cleavage and polyadenylation specificity factor 6 (CPSF6) [15,17,19,21,24–26], which is a component of the cleavage factor I mammalian (CFIm) complex that helps to dictate pre-mRNA polyadenylation site selection . Following trafficking through the cytoplasm, the core docks at the cytoplasmic face of the NPC via interactions with Nup358 [16,28]. This promotes passage through the NPC [16,28,29], leading to docking interactions with Nup153 on the nucleoplasmic side of the NPC , where interactions with CPSF6 are thought to release the core into the nucleus [30,31]. CPSF6 traffics with the core further into the nucleus to subnuclear bodies called nuclear speckles (NSs) [30,32–34]. NSs are found near gene-dense regions of chromatin and prior work has shown that HIV-1 preferentially targets these regions for viral DNA integration [33,35]. Importantly, HIV-1 capsid mutants defective for CPSF6 binding display aberrant integration site selection, highlighting the impact of CPSF6 for integration [16,36].
CPSF6 contains a functional NLS at its C-terminus, and as a result is predominantly found in the nucleus of cells [26,37,38]. CPSF6 is expressed as two splice isoforms. Isoform 1, the major form, is composed of 551 residues while isoform 2, containing 588 amino acids, carries internal residues encoded by exon 6 that are removed from isoform 1 by alternative splicing . A truncation of CPSF6 isoform 2 containing amino acids 1–358 (CPSF6-358) was identified by the KewalRamani group, where they noted the protein was mislocalized to the cytoplasm and acted as an artificial restriction factor . This finding led to identification of the N74D capsid mutant, which is defective in its ability to bind to CPSF6 . Moreover, N74D CA mutant and wildtype (WT) CA viruses showed variable sensitivities to Nup knockdowns, highlighting that the mutant virus utilized a different nuclear import pathway than the WT . Investigation of the P90A capsid mutant, which disrupts binding of cyclophilin A (CypA) to CA, also found that it utilizes a nuclear import pathway distinct from WT virus [16,39]. Our previous studies using a nuclear pore blockade have also supported the idea of multiple nuclear import pathways . We observed that infection mediated by WT HIV-1 was potently inhibited by Nup62 blockade, but infection mediated by N74D and P90A CA mutant viruses was comparatively less sensitive or insensitive to the Nup62-mediated NPC blockade, suggesting the utilization of distinct nuclear import pathways, likely utilizing different NPCs . Recently, several groups reported that NLS swapping can alter sensitivity to the restriction factor MX2 [23,41]. Kane et al. reported that appending some NLSs to a GFP-LacZ fusion resulted in MX2-mediated restriction of nuclear import, while other NLSs resulted in GFP-LacZ nuclear import that was insensitive to MX2 restriction . Chai et al. modified MX2’s NLS by appending several heterologous NLSs to the N-terminus of MX2. They found that only some of these NLS-MX2 chimeras retained MX2’s ability to restrict nuclear import of HIV-1 . These findings showed that some, but not all, import pathways retained MX2 restriction activity and that the NLS determined which pathway was used.
To better understand the role of CPSF6’s NLS, we attached heterologous NLSs to CPSF6-358 to determine the impact of divergent NLSs on nuclear import and infection by HIV-1 and related lentiviruses. We observed that the addition of NLS sequences that restored the nuclear localization of CPSF6 also promoted the nuclear import of HIV-1 during infection. However, despite effective nuclear import, we noted that some NLSs promoted a dysfunctional form of nuclear import that did not facilitate or altered subsequent intranuclear steps of infection, including integration. These results demonstrate that the NLS promoting the nuclear import of HIV-1 can impact the nuclear stages of infection. Previous reports have shown that CPSF6’s C-terminal intrinsically disordered region (IDR), which is the protein’s NLS , confers NS colocalization and condensation to heterologous fusion partners such as green fluorescent protein (GFP) [37,42,43]. Our results support these findings, as we found only the full-length CPSF6, which contains an intact IDR, formed condensates following HIV-1 infection. However, our findings show re-localization to NSs can occur independently of the IDR, as CPSF6-NLS constructs containing the SV40 or C-MYC NLS effectively trafficked the HIV-1 RNPC to NSs despite lacking the IDR. These data show that specific NLSs license subsequent trafficking and integration by lentiviruses and further refines our understanding of the nuclear import pathways utilized by HIV-1. These data also provide evidence that NLSs possess the ability to influence post-nuclear import activities of their cargoes.
Results
Heterologous NLSs can confer CPSF6-358 nuclear localization
To understand more about CPSF6’s NLS and how it influences post-nuclear HIV-1 biology, we used the well described truncation mutant containing amino acids 1–358 (CPSF6-358), which lacks the C-terminal IDR, also called an RS-like domain (RSLD) . Because CPSF6-358 was initially derived from exon 6-containing isoform 2 , our isoform 1-based version harbors 321 total residues. The RSLD contains the functional NLS for CPSF6, and as a result, CPSF6-358 displays a significant cytoplasmic localization, in contrast to full-length CPSF6, which is predominantly nuclear [26,38]. We then introduced various NLS’s at the C-terminus of CPSF6-358 to rescue nuclear localization through several distinct nuclear import pathways (Fig 1A and Table 1). We chose representative NLSs, which were either cellular or viral in origin, and that spanned across the different NLS classifications. To understand the impact of these NLSs on infection, we first knocked out endogenous CPSF6 in HeLa cells and generated stable cell lines expressing the indicated CPSF6-NLS chimeras (Fig 1B and 1C). We observed that several of the NLSs (SV40, C-MYC, NP, Rac3, HNRNP K, HNRNP A1, and MX2) effectively rescued nuclear localization of our CPSF6-358 construct (Fig 1D and Table 1). Notably, some NLSs (DDX21, RB, HTLV-1 Rex, and UL79) were unable to confer effective nuclear localization, and the corresponding CPSF6-358 chimeric proteins retained cytoplasmic distribution (Fig 1D and 1E and Table 1). These results are in line with previous studies of MX2 localization, which showed variable degrees of nuclear localization of heterologous MX2 constructs across the tested NLSs [23,41]. Overall, some, but not all, of our chimeric CPSF6-NLS constructs functionally localized to HeLa cell nuclei.
Download:
PPT PowerPoint slide
PNG larger image
TIFF original image
Fig 1. Heterologous NLSs can confer CPSF6-358 nuclear localization.
(A) Schematic design of chimeric CPSF6-NLS constructs showing NLSs being fused to the C-terminus of CPSF6-358 derived from human isoform 1. (B) Western blot analysis of HeLa cells using anti-CPSF6 antibody to confirm CRISPR/Cas9-mediated knockout of endogenous CPSF6. (C) Western blot analysis of stably transduced HeLa cells using anti-CPSF6 antibody to detect CPSF6-NLS construct expression following 48 h doxycycline induction. Anti-GAPDH antibody used as loading control. (D) Immunofluorescent microscopic images stained using anti-HA antibodies (green) and Hoechst dye (blue) to visualize CPSF6-NLS construct localization. Representative of 3 independent experiments with at least 10 images per experiment. (E) Percent nuclear signal was determined using ImageJ. Statistical analysis in E was determined using one-way ANOVA. Significant differences are indicated: P<0.0001. RRM–RNA Recognition Motif. PRD–Proline Rich Domain. RSLD–Arginine Serine-Like Domain. HA–Hemagglutinin tag. NLS–Nuclear Localization Signal. CPSF6-FL–Full Length CPSF6.
Download:
PPT PowerPoint slide
PNG larger image
TIFF original image
Table 1. NLSs fused to CPSF6-358 and their effect on construct localization and WT HIV-1 gene expression.
Of note, our CPSF6-358, as well as the CPSF6-NLS chimeras with significant cytoplasmic distributions, exhibited partial nuclear accumulation, which was previously described for original isoform 2 CPSF6-358, as well as a similar isoform 1 truncation known as CPSF6-375 [26,44]. These C-terminal deletion mutants retain the N-terminal RRM domain that mediates binding of CPSF6 to its CFIm partner protein CPSF5 (Fig 1A) . CFIm in turn functions to regulate polyadenylation in the context of a much larger cleavage and polyadenylation (CPA) complex . We speculate that partial nuclear accumulation of CPSF6-358 and non-functional NLS derivatives is due to CFIm-mediated associated with CPA as a consequence of cell line construction, which necessitated multiple rounds of cell division and nuclear reorganization.
Nuclear localization of CPSF6-358 does not uniformly rescue HIV-1 infection
CPSF6 is an HIV-1 host factor that binds directly to the HIV-1 capsid to aid nuclear entry of the viral capsid and integration site targeting [17,24–26,30,31,35,54]. To further understand if changing CPSF6’s NLS impacts HIV-1 infection, we utilized single-round HIV-1 luciferase reporter virus to infect our chimeric CPSF6 cell lines, and infectivity was determined as a measurement of viral promoter activity driving reporter gene expression. To highlight phenotypes associated with NLS functionality, the predominance of experiments reported here were conducted with growth-arrested cells. Consistent with previous studies that depleted cellular CPSF6 via RNA interference , CPSF6-KO did not perturb HIV-1 infectivity in HeLa cells, and expression of CPSF6-358 in CPSF6-KO cells potently inhibited infection (Fig 2A). As might be expected, a similar inhibition was observed in cells expressing chimeric NLSs that failed to confer CPSF6-358 nuclear localization (DDX21, RB, HTLV-1 Rex, and UL79) (Fig 2A, striped bars). We, moreover, observed three distinct phenotypes amongst cells expressing CPSF6 chimeras with restored nuclear localization. We found that the C-MYC and Rac3 NLS not only fully rescued infection, but resulted in a small but significant increase in infectivity compared to the infection observed in CPSF6-KO and CPSF6-FL cells (Fig 2A). In contrast, the SV40, HNRNP K, and HNRNP A1 NLSs rescued infection to levels that were indistinguishable from CPSF6-FL (Fig 2A). Finally, HIV-1 infection was significantly reduced in cells expressing the CPSF6 chimeras with NP and MX2 NLSs (Fig 2A), despite these NLSs conferring effective CPSF6-358 nuclear localization. These phenotypes persisted across several viral titrations, showing that the differences in infectivity were concentration independent (Fig 2B). We observed similar infectivity findings using a fluorescent reporter virus, revealing that the results were independent of the viral reporter gene utilized (Fig 2C). Additionally, these results were independent of cell cycle arrest, as we observed similar phenotypes when cells were not growth arrested with aphidicolin (Fig 2D). These results show that simply conferring constitutive CPSF6-358 nuclear localization is insufficient to restore HIV-1 infectivity and is impacted by the NLS used to mediate CPSF6-358 nuclear localization.
Download:
PPT PowerPoint slide
PNG larger image
TIFF original image
Fig 2. Nuclear localized CPSF6-358 does not uniformly rescue HIV-1 infection.
(A) WT HIV-1 firefly luciferase reporter virus infectivity was measured 48 h post-infection in growth arrested HeLa cells. Infectivity was normalized to cells expressing full-length CPSF6. Striped bars represent constructs with cytoplasmic CPSF6-358-NLS fractions. (B) Serial dilutions show that infectivity phenotypes hold across several viral concentrations in growth arrested cells. (C) WT HIV-1 mCherry reporter virus was used to infect growth arrested HeLa cell lines and infectivity was determined using flow cytometry. (D) Dividing HeLa cells infected with WT HIV-1 firefly luciferase reporter virus display comparable phenotypes to non-dividing cells. Results (mean ± SEM) are representative from at least 3 independent experiments with at least technical duplicate samples per experiment. Statistical analysis was determined using one-way ANOVA. Significant differences are indicated: P<0.05, P<0.01, P<0.0001. CPSF6-FL–Full Length CPSF6.
NLSs are categorized into classes based on the structure of the NLS sequence . We accordingly wondered if NLS classification could explain the phenotypic differences observed amongst the CPSF6-NLS chimeras. However, the ability to influence infection positively or negatively was not dependent on the type of NLS. RB, Rac3, and NP are three examples of bipartite NLSs that conferred three distinct phenotypes under our assay conditions (Table 1). Rac3 and NP conferred CPSF6-358 nuclear localization, whereas the RB NLS chimera remained partially cytoplasmic. Further, Rac3 fully rescued infectivity, while NP resulted in an infectivity defect. Additionally, we tested two PY-NLSs, HNRNP A1 and UL79, and found that HNRNP A1 conferred CPSF6-358 nuclear localization and supported HIV-1 infection similarly as CPSF6-FL, while UL79 failed to rescue both CPSF6-358 nuclear localization and HIV-1 infection (Table 1). These findings suggest that the ability to influence HIV-1 infection cannot be explained by NLS classification, and that NLSs belonging to the same classification, such as bipartite or PY-NLS, functioned differently from others within the same NLS class. Together, these results display the importance of the nuclear import pathway utilized by CPSF6 for productive HIV-1 infection.
CPSF6-NLS chimeras show virus-specific influences on infection
To ensure that the observed changes in viral gene expression were driven through CA-CPSF6 interactions and not a deleterious effect of chimeric protein expression on cell function, we infected our cells with the CPSF6 binding deficient HIV-1 CA mutant N74D . The Asn>Asp substitution in CA occurs within the CPSF6 binding pocket, rendering CA unable to bind to CPSF6. Importantly, this CA mutant retains the ability to productively infect HeLa cells . When we infected our chimeric cell lines with N74D virus, we observed no differences in viral gene expression, demonstrating that the differences in WT HIV-1 infectivity observed in cells expressing CPSF6 chimeras are dependent on CA-CPSF6 binding (Fig 3A).
Download:
PPT PowerPoint slide
PNG larger image
TIFF original image
Fig 3. CPSF6-NLS chimeras confer virus-specific infectivity defects.
(A) Infection with HIV-1 CA mutant N74D luciferase-reporter virus in growth arrested HeLa cells reveals a dependence on CA-CPSF6 binding for the observed changes in WT HIV-1 infectivity. (B) Infection with P90A CA mutant in growth arrested HeLa cells results in less-severe infectivity defect with CPSF6-NP NLS chimera cells, while CPSF6-HNRNP K NLS cells trend to be more sensitive to infection relative to WT virus and C-MYC NLS cells only promote increased infection with WT HIV-1 and not P90A. Relative infectivity in CsA-treated growth arrested cells to untreated WT HIV-1-infected (C) and P90A infected (D) growth arrested cells displays a decrease to no change in infectivity for all CPSF6-NLS cell lines except for CPSF6-NP NLS cells during WT HIV-1 infection, which display an almost twofold increased infection. (E) Infection with non-HIV-1 primate lentiviral luciferase reporter viruses, HIV-2 (Red) and SIVmac (Purple), in growth arrested cells. Relative to HIV-1 infection (Green), there are decreases in HIV-2 infectivity in SV40 and HNRNP K cells, whereas SIVmac appears fairly similar to HIV-1. Results (mean ± SEM) are representative from 3 independent experiments with at least technical duplicate samples per experiment. Statistical analysis was determined using two-way ANOVA. Significant differences are indicated: P<0.05, P<0.01, P<0.001, P<0.0001. CsA–Cyclosporine A. CPSF6-FL–Full Length CPSF6.
The P90A HIV-1 CA mutant is defective in its ability to bind CypA [56–58], which is another host factor implicated in HIV-1 nuclear import . Importantly, P90A retains the ability to bind to CPSF6, but has been shown to utilize a nuclear import pathway distinct from WT HIV-1 . To further explore the role of the CA-CPSF6 interaction in HIV-1 infection, we infected our cell lines with the P90A mutant virus. Broadly, CPSF6 chimeras that were able to rescue infection by WT HIV-1 were also able to rescue P90A infection to a similar degree, relative to the CPSF6-FL complemented cell line (Fig 3B). However, the C-MYC NLS supported less efficient P90A infection relative to WT HIV-1 infection, while the opposite pattern was observed in cells expressing the CPSF6-HNRNP K chimera (Fig 3B). Additionally, we observed that cells expressing the CPSF6-NP chimera were more readily infectable with P90A virus, despite inhibiting WT HIV-1 infection (Fig 3B). To corroborate this observed difference, we infected our cell lines with WT HIV-1 in the presence of cyclosporin A (CsA). CsA blocks CypA binding to CA and thus displays a phenotype similar to the P90A mutation [56–58]. We found that CsA treatment of WT HIV-1 infected CPSF6-NP chimeric cells resulted in significantly increased infectivity relative to untreated cells, while other chimeric CPSF6-NLS cell lines displayed no change or decreased infectivity following CsA treatment (Fig 3C). Notably, CsA treatment of P90A infected cells resulted in little to no effect on infection in all cell lines tested, confirming that the observed changes were specific to the CA-cyclophilin interaction and not off-target CsA effects (Fig 3D). This demonstrates that disruption of CA-CypA binding can modulate the ability of some CPSF6-NLS chimeras to productively support HIV-1 infection.
Binding to CPSF6 is a shared characteristic amongst primate lentiviruses . We therefore wondered how CPSF6-NLS chimera expression would impact infection with HIV-2 and simian immunodeficiency virus from macaque (SIVmac). In general, HIV-2 and SIVmac were similarly influenced by the expression of CPSF6-NLS chimeras as WT HIV-1 (Fig 3E). However, some differences were noted. HIV-2 infection was relatively less efficient in CPSF6-SV40 NLS and CPSF6-HNRNP K NLS expressing cells, compared to HIV-1 and SIVmac (Fig 3E). Cells expressing the CPSF6-Rac3 NLS were less supportive of both HIV-2 and SIVmac expression, and SIVmac infection did not exhibit the same increase in infectivity as observed with HIV-1 and HIV-2 following infection of CPSF6-C-Myc NLS expressing cells (Fig 3E). These results show that the ability of CPSF6-NLS constructs to facilitate infection was generally conserved among primate lentiviruses that utilize CPSF6 as a host cofactor.
CPSF6-NLS chimeras display cell-type specific phenotypes
To test the CPSF6-NLS chimera constructs in more physiologically relevant settings, we depleted endogenous CPSF6 from T cell (SupT1) and monocytic (THP-1) cell lines and generated stable cells expressing four representative CPSF6-NLS chimeras that were exclusively nuclear in HeLa cells (S1A and S1B Fig). To determine if there were cell-type differences in construct localization, the SupT1 cells and differentiated THP-1 cells were analyzed by confocal microscopy, which revealed chimera CPSF6-NLS construct nuclear localizations similar to those observed in HeLa cells (S1C and S1D Fig). In general, the ability of these CPSF6-NLS chimeras to facilitate HIV-1 infection in SupT1 and THP-1 cells was similar to what was observed in HeLa cells (Fig 4). CPSF6-NP NLS and MX2 NLS constructs in T cells and macrophages behaved largely similarly as they did in HeLa cells (Fig 4). However, similar to the case of capsid mutations and other primate lentiviruses, there were some differences in the ability of some chimeras to promote infection in each cell line, most notably in T cells. In T cells, the CPSF6-SV40 NLS chimera rescued ~50% of infection (Fig 4), whereas it fully rescued infection of differentiated THP-1 cells (Fig 4). Additionally, CPSF6-C-MYC NLS expression increased THP-1 infectivity to about 200% relative to full length CPSF6 (Fig 4), but rescued T cell infectivity to about 100% (Fig 4). Together, these results reveal the target cell can impact the ability of HIV-1 to utilize chimeric CPSF6-NLS constructs during infection.
Download:
PPT PowerPoint slide
PNG larger image
TIFF original image
Fig 4. CPSF6-NLS chimeras display cell-type specific phenotypes.
CPSF6-KO SupT1 T-cells (yellow) and THP-1 monocytes (cyan) were stably transduced with dox-inducible CPSF6-NLS constructs. T cells and HeLa cells were growth arrested prior to infection and THP-1’s were terminally differentiated into macrophages. WT HIV-1 infection was determined by luciferase reporter gene expression. Notably, T cells expressing CPSF6-SV40 NLS were significantly less infectable than corresponding HeLa cells, while CPSF6-SV40 NLS-expressing THP-1 cells supported 100% infection. While CPSF6-C-MYC NLS strongly increased infectivity in HeLa and THP-1 cells relative to CPSF6-FL, this construct conveyed the CPSF6-FL level of infection in T cells. NP and MX2 cells resulted in defective infection in all cell types. Results (mean ± SEM) are representative from 3 independent experiments with at least technical duplicate samples per experiment. Statistical analysis was determined using one-way ANOVA. Significant differences are indicated: P<0.01, P<0.001, P<0.0001.
Heterologous NLSs significantly impact intranuclear stages of HIV-1 infection
Due to the CPSF6-NLS chimera constructs restoring nuclear localization of CPSF6-358 (Fig 1C), we next asked if defective HIV-1 gene expression was a result of defective nuclear import in these cells. HIV-1 nuclear import is commonly quantified by measuring the accumulation of 2-long terminal repeat (LTR)-containing circles, which form via host non-homologous end-joining machinery acting on unintegrated HIV-1 genomes . We infected the chimeric NLS HeLa cell lines with WT HIV-1 and measured reverse transcription (RT; minus-strand after initial strand transfer) and 2-LTR circles by qPCR. All tested chimeric NLS cell lines supported similar levels of HIV-1 DNA synthesis (Fig 5A). CPSF6-358 expression, relative to full-length CPSF6, significantly reduced 2-LTR circle formation (Fig 5B), while CPSF6-KO cells supported nuclear import of viral DNA, as measured by 2-LTR circle formation, consistent with previous studies [26,38]. Interestingly, all four CPSF6-NLS chimeric cell lines examined supported similar levels of 2-LTR circle accumulation (Fig 5B), despite the reduced levels of gene expression observed in CPSF6-358 NP NLS and MX2 NLS-expressing cells (Fig 2A). As integration can impact the amount of viral genomes available to be circularized by nonhomologous end-joining machinery, we repeated infections in the presence of the integrase inhibitor raltegravir (RAL) to block viral DNA integration. When comparing 2-LTR circle formation of RAL treated cells relative to untreated cells, we found no significant differences in nuclear import as measured by 2-LTR circles (S2 Fig). To further examine the ability of these constructs to mediate nuclear import of the vRNPC, we utilized fluorescently tagged Gag-Integrase Ruby (GIR) virus to track the movement of individual vRNPCs (Fig 5C). Using IMARIS software, we determined the percentage of our labeled particles within the nucleus by creating surface masks around the Hoecsht signal to define the nuclear boundary. Consistent with our 2-LTR circle measurements, the CPSF6-NLS chimeric cell lines supported similar levels of nuclear localization of GIR-labelled vRNPCs (Fig 5C and 5D). These results indicate that HIV-1 efficiently entered into the nuclei of our chimeric CPSF6-NLS cell lines and suggests changes in post-nuclear entry steps of infection underlie the differences observed between different CPSF6-NLS chimeras.
Download:
PPT PowerPoint slide
PNG larger image
TIFF original image
Fig 5. NLSs can significantly impact intranuclear stages of HIV-1 infection.
(A) RT was measured via amplification of firefly luciferase sequences within the viral genome. (B) 2-LTR circles were measured and normalized to levels obtained in CPSF6-FL-expressing cells. qPCR analysis was performed on genomic DNA isolated from WT HIV-1 infected growth arrested HeLa cells. Gag-Integrase Ruby (GIR) labeled HIV-1 viral particles were used to infect growth-arrested HeLa cell lines. (C) Representative images of GIR-infected HeLa cells 8 h post infection, with Hoechst (blue), GIR (yellow), and SC35 (red). Nuclear virus was determined using IMARIS software and creating a mask around the nuclear Hoechst stain, then determining percent of total GIR particles within the mask. Bottom three rows display inset from top images. (D) Percentage of nuclear GIR signals across the indicated cell lines, which highlighted lack of significant differences. (E) Nuclear penetration was determined via IMARIS software, measuring the distance of nuclear viral particles to the nearest Hoechst edge. GIR in CPSF6-KO cells remained nuclear envelope proximal, whereas other CPSF6 chimeras facilitated nuclear invasion. (F) NS colocalization was determined via GIR infection and subsequent SC35 staining. Percentage of nuclear GIR colocalizing with SC35 was determined using IMARIS software. HIV-1 in CPSF6-NP NLS, CPSF6-MX2 NLS, and CPSF6-KO cells displayed decreased NS colocalization. (G) Representative images showing CPSF6-FL distribution in uninfected versus HIV-1 infected cells, with Hoechst (blue), CPSF6 (green), and SC35 (red). (H) CPSF6 condensation was determined via HA-tag staining at 24 h post-infection. Masks were made around SC35 stain using IMARIS and fluorescent intensity of CPSF6 construct stain within SC35 masks was determined. CPSF6-FL supported much brighter spots consistent with condensation, whereas CPSF6 distribution remained comparable to uninfected cells across other cell lines. All imaging experiments were performed on growth arrested cells. Results in A,B,D and F (mean ± SEM) are representative from 3 independent experiments with at least technical duplicates. Results in E show median+95% CI. Results in H show box and whiskers from min to max. At least 10 images were taken for each condition across 3 independent experiments. Statistical analysis in A,B,D and F was determined using one-way ANOVA and in E and H using Kruskal-Wallis test. Significant differences are indicated: P<0.01, P<0.001, P<0.0001. GIR–Gag-Integrase Ruby. CPSF6-FL–Full-length CPSF6.
Following HIV-1 nuclear import, HIV-1 traffics away from the nuclear envelope and further into the nucleus [30,61]. This inward trafficking step is licensed by CPSF6, as HIV-1 does not efficiently traffic distal from the nuclear envelope in CPSF6-depleted cells [30,61,62]. We quantified viral nuclear penetration by measuring GIR distance to the nearest Hoechst edge at 8 h post-infection. As expected, HIV-1 in CPSF6-KO cells remained near the nuclear periphery (Fig 5E). Viruses in CPSF6-NLS chimera expressing cells, however, penetrated deeper into the nucleus, comparably to cells expressing full-length CPSF6 (Fig 5E).
The CA-CPSF6 interaction facilitates intranuclear trafficking of vRNPCs to NSs [32,33,35,63,64]. To determine if chimeric CPSF6-NLS constructs impair HIV-1 intranuclear trafficking to NSs, we utilized GIR virus in combination with an SC35 stain to visualize colocalization with NSs at 8 h post-infection (Fig 5C and 5F). CPSF6-KO cells displayed a reduction in GIR colocalization with NSs, as described previously using KO cells and/or CA mutant viruses [32–34,63]. However, the SV40 and C-MYC NLS constructs effectively facilitated localization to NSs (Fig 5C and 5F). Alternatively, the NP and MX2 NLS constructs conferred significant decreases in HIV-1 colocalization with NSs (Fig 5F). During infection, CA induces the formation of CPSF6 biomolecular condensates, which colocalize with NSs [63–65]. Due to the SV40-NLS and C-MYC NLS chimeras facilitating viral localization to NSs, we wondered if these mutants could induce the formation of biomolecular condensates, as previously described for full-length CPSF6 [63–65]. By measuring CPSF6 signal intensity colocalizing with SC35, we found that only full-length CPSF6 condensed with NSs, whereas the CPSF6-NLS chimeras retained diffuse nuclear localization similar to uninfected cells (Fig 5G and 5H). Additionally, we determined that there were no differences in the number of nuclear speckles per cell or the total volume of nuclear speckles per cell relative to uninfected cells (S3A and S3B Fig). These results were consistent with previously published findings . Taken together, our results show that a subset of our CPSF6-NLS constructs that supported maximal or partial HIV-1 infection nevertheless supported equivalent levels of HIV-1 nuclear entry. Moreover, following nuclear entry, the SV40 NLS and C-MYC NLS constructs facilitated HIV-1 nuclear trafficking and NS localization despite a lack of biomolecular condensation. Conversely, the NP and MX2 constructs, despite facilitating nuclear penetration, were unable to facilitate NS localization or biomolecular condensation of CPSF6.
CPSF6-NP and MX2 NLS chimeras reduce HIV-1 integration and influence integration site selection
The substrate for HIV-1 integration is the linear viral DNA made by RT. HIV-1 integration favors active chromatin including DNA in close proximity to NSs, called speckle-associated domains (SPADs) [33,35]. We asked if the observed defects in NS colocalization might correlate with changes in overall levels of HIV-1 integration as well as sites of integration within the human genome. To detect total levels of HIV-1 integration, we utilized Alu-Gag qPCR, which amplifies sequences that lay between the HIV-1 gag gene and genomic Alu-repeats; a second, nested qPCR was used to quantify the extent of first round reaction products [66,67]. We observed a significant reduction in integration in CPSF6-358 expressing cells, as expected, due to viral restriction in the cytoplasm (Fig 6A). Interestingly, we also observed a reduction in Alu-Gag products in cells expressing CPSF6-NP NLS and MX2 NLS (Fig 6A). CPSF6-SV40 NLS and C-MYC NLS by contrast supported Alu-Gag integration levels similarly as full-length CPSF6 (Fig 6A). These data are consistent with the notion that overall integration levels largely dictate levels of HIV-1 infection as assessed by viral reporter gene expression (compare Fig 2A and 2C with Fig 6A). To further interrogate the relationship between integration level and viral gene expression, we utilized a dual fluorescent reporter HIV, called HIVGKO . This virus contains a constitutive reporter gene (mKO2) driven by an internal promoter and a second GFP reporter under the control of the viral LTR promoter. We observed indistinguishable levels of expression of both reporter genes across our different cell types, further linking the relationship between overall levels of HIV-1 integration and viral gene expression (S4 Fig).
Download:
PPT PowerPoint slide
PNG larger image
TIFF original image
Fig 6. CPSF6-NP and MX2 NLS chimeras significantly impact HIV-1 integration.
(A) Alu-Gag qPCR detection of HIV-1 integration levels in growth arrested cells. CPSF6-NP NLS and CPSF6-MX2 NLS cells supported significantly reduced levels of HIV-1 integration. (B-E) HIV-1 integration in growth arrested HeLa cells into defined genomic annotations including genes (B), SPADs (C), LADs (D) and gene-dense regions (E). Integration into genes in CPSF6-C-MYC NLS, CPSF6-NP NLS, and CPSF6-MX2 NLS-expressing cells was reduced compared to cells expressing CPSF6-FL and CPSF6-SV40 NLS. Fractional LAD targeting in CPSF6-C-MYC, NLS CPSF6-NP NLS, and CPSF6-MX2 NLS-expressing cells was increased significantly compared to cells expressing CPSF6-FL and CPSF6-SV40 NLS. Integration in gene dense regions in CPSF6-NP NLS cells was reduced significantly compared to CPSF6-FL expressing cells. Integration in NLS CPSF6-NP NLS and CPSF6-MX2 NLS cells was interestingly statistically indistinguishable from the RIC value. Statistical analysis was determined using one-way ANOVA in A and chi-squared test followed by students t-test in B-E. P<0.05, P<0.001, P<0.0001 (black asterisks, vs CPSF6-FL cells; grey asterisks, vs corresponding RIC).
To further probe the granularity of the integration phenotypes, we mapped sites of HIV-1 integration within the human genome essentially as described previously [69,70]. In short, asymmetric linkers were ligated onto sheared genomic DNA fragments, and LTR-linker sequences amplified via semi-nested PCR were analyzed by Illumina sequencing. Unique sites of HIV-1 integration were then mapped with respect to several genomic features including genes, SPADs, lamina associated domains (LADs), alphoid repeat elements, LINE1 repeat elements, and gene dense regions [30,33,71–75]. Computationally-generated random integration control (RIC) values were also included for each genomic annotation.
The number of unique integration sites mapped from a total of 2 to 4 independent infection experiments across cell lines varied from 925 to >10,000 (S1 Table). The comparatively low number of sites recovered from MX2 NLS and NP NLS-expressing cells– 925 and 1,125, respectively–was, on the one hand, consistent with the comparatively low level of overall integration events detected by Alu-Gag PCR (Fig 6A). On the other hand, due to these comparatively low numbers, concern was raised as to whether the number of sites recovered could support meaningful conclusions regarding HIV-1 integration site targeting. Importantly, prior studies that analyzed much fewer sites of HIV-1 integration in the human genome, including 524 sites , 84 to 222 sites , or 110 to 1,485 sites , were able to reach meaningful conclusions. The ability to reach meaningful conclusions regarding integration targeting additionally relies on the degree of overlap between mapped integration sites and the genomic annotations of interest. The observed degree of overlap is itself dependent in part on the completeness of a given annotation-set for a given reference genome. To address these issues head-on, we conducted a power analysis for a chi-squared test assuming a power of (1-β) = 0.8 and a two-sided significance threshold of α = 0.05 to assess the relevance of MX2 NLS and NL NLS datasets for downstream analyses. The power analysis revealed that for gene, gene dense regions, and SPAD overlaps, there was sufficient depth across datasets to make conclusions with respect to CPSF6-FL and RIC at the stated power. For LAD overlaps, all comparisons except the MX2-RIC comparison met the stated power threshold. The MX2-RIC comparison had a calculated power of 0.75, which was reasonably close to the posited 0.8 value. Integration site overlap comparisons with LINE1 repeat elements showed sufficient power with respect to RIC, but were slightly underpowered with respect to CPSF6-FL (calculated power values of 0.76 for NP NLS and 0.71 for MX2 NLS), so we will restrict interpretations of LINE1 datasets to the RIC comparator. HIV-1 integration avoids alphoid repeats and alphoid repeat coordinates are relatively poorly annotated for human genome build hg19. Accordingly, quantified integration site overlaps with alphoid repeats were insufficient to draw any meaningful conclusions, and thus integration targeting of alphoid repeats will not be commented on further (S3 Table).
As expected, integration into genes in CPSF6-FL-expressing cells was significantly enriched compared to the RIC value (Fig 6B, grey asterisks). While genic integration in CPSF6-358 SV40 NLS-expressing cells was statistically indistinguishable from CPSF6-FL cells, ~2–7% reductions in integration into genes was observed in C-MYC, NP, and MX2 NLS-expressing cells (Fig 6B and S1 Table). Overall similar trends were observed for integration into SPADs, LADs, and gene dense regions. Thus, while SPAD-proximal integration was marginally reduced by ~1.7% in SV40 NLS-expressing cells (P = 0.03), SPAD targeting was reduced by ~5.4% to 8.7% in C-MYC, NP, and MX2 NLS-expressing cells (P<0.0001) (Fig 6C and S2 Table). Reciprocal upticks in integration into LADs in C-MYC, NP, and MX2 NLS-expressing cells differed significantly from LAD targeting values observed in CPSF6-FL and SV40-expressing cells (P<0.0001) (Fig 6D). In terms of gene dense regions, the 3.75 genes/Mb reduction in NP NLS-expressing cells was statistically significant versus the 15.4 genes/Mb value observed in CPSF6-FL cells. Interestingly, integration into gene dense regions in NP and MX2 NLS-expressing cells, compared to CPSF6-FL containing cells, was no longer enriched compared to the RIC 9.2 genes/Mb value (Fig 6E and S2 Table).
The SV40 NLS chimera behaved most similar to CPSF6-FL with respect to integration into genes, SPADs, and LADs, showing no or comparatively marginal differences across these annotations. The C-MYC, NP, and MX2 NLSs generally showed increasingly significant differences with respect to CPSF6-FL overlap with genes, SPADs, and LADs. Notably, CPSF6-FL and SV40 NLS showed indistinguishable overlap with LINE1 repeat elements relative to RIC. C-MYC, NP, and MX2 NLSs, conversely, showed slight increases in LINE1 integration relative to RIC (S2 Table). These apparent differences, however, were comparatively small with marginal statistical significance (P = 0.008 to 0.03). Altogether, our results revealed that the NP and MX2 constructs, which were defective in supporting overall levels of HIV-1 integration (Fig 6A), supported marginally less, though statistically significant, integration into genomic annotations associated with active chromatin, including genes and SPADs (Fig 6B and 6C). Collectively, our results establish the importance of CPSF6’s NLS in facilitating the intranuclear events of HIV-1 infection.
Discussion
In this study, we determined that the NLS utilized by CPSF6 can impact various stages of nuclear trafficking and integration. By attaching heterologous NLSs to a truncated CPSF6, CPSF6-358, we observed a range of outcomes that were dependent on the NLS utilized. In line with previous experiments that utilized MX2 as a model system, some NLSs failed to complement the nuclear localization defect of CPSF6-358, and as a result these chimeric proteins restricted HIV-1 infection in the cytoplasm [23,41]. This is generally consistent with the ability of different nuclear import pathways, perhaps through heterogenous NPCs, to accommodate different nuclear transport cargoes [23,40]. We hypothesize that multiple NPC subsets can be found within a cell, and these NPC subsets have alternative nuclear transport mechanisms .
Following infection with HIV-1, we observed differences in reporter gene expression amongst the nuclear-rescued CPSF6-358 constructs. Some of the NLSs (C-MYC, Rac3) displayed a slight, but significant increase in reporter gene expression relative to full-length CPSF6. Additionally, the C-MYC NLS construct resulted in marginally increased levels of integration as measured by Alu-Gag qPCR assays. Although omitted from downstream analyses, the Rac3 NLS protein behaved similar to the C-MYC protein in HIV-1 infection assays. Several additional constructs, including SV40, HNRNP K, and HNRNP A1, supported WT HIV-1 infection levels similar to CPSF6-FL.
Our results reveal that infection of CPSF6-NP and MX2 NLS cells resulted in functional nuclear import, as measured by both 2-LTR circles and localization of Gag-Integrase labeled viruses inside the nucleus of these infected cells. However, we found that infection in these cell lines resulted in decreased levels of viral integration as measured by Alu-Gag qPCR. The observation that 2-LTR circle formation was similar in cells with nuclear localized CPSF6 fusions suggests that the fusion constructs do not negatively impact core uncoating during infection, as 2-LTR circle formation requires the non-homologous end joining machinery. Additionally, it is known that abolishing the CA/CPSF6 interaction alters the spatiotemporal staging of infection, such that the viral core remains associated with the nuclear envelope [31,61,62]. However, we observed nuclear penetration of vRNPCs in cells expressing CPSF6-NLS fusions, regardless of their ability to facilitate gene expression (Fig 5E). Although we did not specifically interrogate the levels of CA associated with vRNPCs in these studies, the degree to which nuclear penetration is known to be mediated by CPSF6/CA interactions suggests that CA remains associated with the vRNPC in cells expressing CPSF6-NLS fusions.
Our integration site analyses revealed that HIV-1 in CPSF6-NP and MX2 NLS-expressing cells generally disfavored regions associated with active chromatin, such as genes and SPADs, which could potentially further reduce the levels of gene expression supported by these integrated proviruses. We speculate that the overall reduced levels of integration observed in CPSF6-NP and MX2 NLS cells combined with the propensity for these proviruses to disfavor active chromatin account for the comparatively poor expression profiles of these proviruses. Of note, our work fails to address why overall integration is significantly diminished in CPSF6-NP and MX2 NLS-expressing cells. Possibly, HIV-1 encounters a novel nuclear restriction factor(s) via the altered paths of nuclear import instilled by these NLSs or regions that have tightly packed chromatin compared to the regions that have loose or relaxed structures associated with active genes. Further work will need to be done to explore these hypotheses.
Prior studies have investigated the differences in the nuclear import pathways of WT HIV-1 and capsid mutant viruses P90A [16,39] and N74D , showing that binding to CypA/Nup153 and CPSF6, respectively, significantly impacted nuclear import pathway utilization. In our studies, the N74D CA mutant, which is defective for CPSF6 binding, predictably infected all CPSF6 chimeric construct expressing cells equally. However, infection with the P90A mutant resulted in differential infectivity relative to the WT virus. Specifically, P90A was less sensitive than WT HIV-1 to restriction in CPSF6-NP NLS cells, which was reproduceable following CsA treatment. Similarly, P90A displayed greater relative infectivity than WT HIV-1 in CPSF6-HNRNP K NLS cells, highlighting a potential role of cyclophilin binding influencing sensitivity in these cell lines. We speculate this may be due to P90A virus entering the nucleus via an alternative nuclear import pathway, as we and others have shown [16,39,40]. In this case, perhaps the activity of the CPSF6-NLS effects in the nucleus is modulated by the nuclear import pathway utilized.
We also observed differential sensitivities to the CPSF6-NLS chimeras amongst related lentiviruses, as infection with HIV-2 and SIVmac highlighted further virus-specific phenotypes. Additionally, CPSF6-NLS chimeras displayed differential phenotypes, specifically in the SV40 and C-MYC NLS chimeras, in T cell and macrophage cell lines, where we observed SV40-NLS only partially rescued infection in T cells but fully rescued infection in macrophages and C-MYC-NLS rescued infection to WT levels in T cells but doubled infectivity in macrophages.
Previous reports have shown that the CPSF6 C-terminal RSLD is an IDR that can phase separate as a GFP-RSLD fusion protein in vitro . Moreover, purified mCherry-CPSF6 displays liquid-liquid phase separation (LLPS) activity in vitro . These observations raise the possibility that the RSLD plays a role in CPSF6 biomolecular condensate formation during HIV-1 infection [43,63,65], which we have recently confirmed . The results reported here are fully consistent with this notion, as we found that only full-length CPSF6, which contains an intact RSLD, formed CPSF6 condensates following HIV-1 infection, while the CPSF6-358 constructs lacking an RSLD remained pan-nuclear in distribution. Interestingly, infection in cell lines expressing CPSF6-SV40 NLS and CPSF6-C-MYC NLS constructs retained HIV-1 NS colocalization. This was a surprising finding, suggesting that HIV-1 NS colocalization can occur independent of CPSF6 LLPS activity, as the constructs containing the SV40 and C-MYC NLS effectively trafficked HIV-1 to NSs despite lacking the RSLD.
Collectively, our results reveal that NLSs have roles beyond targeting cargo for nuclear import, such as impacting viral subnuclear localization following nuclear entry. To our knowledge, there is little precedent for post-nuclear entry roles for NLSs. As such, our results utilizing HIV-1 infection as a readout for the nuclear function of NLSs provide a template to understand the nuclear trafficking or other post-nuclear activities that may be imparted by specific NLSs.
Materials and methods
Cell lines
THP-1 and SupT1 cells were obtained from ATCC. 293T and HeLa cells were cultured in DMEM and THP-1 and SupT1 cells were cultured in RPMI. Both mediums were supplemented with 10% tetracycline-depleted fetal calf serum (R&D Systems), 1000 U/ml of penicillin, and 1000 U/ml of streptomycin. THP-1 cells were differentiated into macrophages with 100 ng/ml phorbol 12-myristate 13-acetate (PMA) (Sigma-Aldrich) for 48 h, followed by 48 h rest with normal media without PMA before experiments.
Constructs
Truncated CPSF6 containing amino acids 1–358 (CPSF6-358) from human isoform 1 was cloned into the pCW57.1 backbone (Addgene 71782). A hemagglutinin (HA) tag as well as heterologous NLSs (Table 1) were appended onto the C-terminus of CPSF6-358. These constructs were under a doxycycline inducible promoter, and expression was induced with 5 μg/ml doxycycline for 48 h before viral infection.
Virus and vector production
To generate viral vectors containing CPSF6-NLS constructs, 293T cells were seeded in 10 cm dishes and co-transfected with 4 μg pCW57-CPSF6-NLS, 4 μg psPAX packaging plasmid, and 2 μg pCMV-VSVg with polyethylenimine overnight. Media was changed the next morning and pseudotyped vector was collected 48 h post transfection. To generate pseudotyped HIV-1, HIV-2, SIVmac, N74D, and P90A viruses, 293T cells were seeded in 15 cm plates and transfected with 20 μg WT/N74D/P90A NL4.3 containing LTR driven luciferase and mCherry reporter genes (separated by a T2A ribosomal skip sequence), HIVGKO, HIV-2-Luciferase, or SIVmac-Luciferase constructs with 5 μg pCMV-VSVg using polyethylenimine. The HIV-2-Luciferase construct was made by swapping the GFP gene of HIV-2-GFP for the firefly luciferase gene, as previously described for analogous equine infectious anemia constructs . Media was changed the following morning and virus was collected at 48 and 72 h post transfection. NL4.3-based plasmid pNLXLuc.R-U3-tag was used to make virus for integration site distribution experiments. To generate Gag-integrase-Ruby viral particles, 293T cells were seeded in 10 cm plates and transfected with 4.5 μg R7ΔEnv, 3.5 μg Gag-integrase-Ruby, and 2 μg pCMV-VSVg using polyethylenimine. Viral particles were collected 48 h post transfection. All viruses and vectors were filtered through 0.45-μm filters (Merck Millipore). Virus was concentrated by spinning at 5000 g overnight at 4°C. Reverse transcriptase units of viral preps were determined using SG-PERT assay as previously described . Viral infections were synchronized by spinoculation at 1200 g for 2 h at 13°C.
Stable cell line generation
CPSF6-knockout cell lines were generated using pLentiCRISPRv2 with previously described guides under blasticidin selection. For HeLa cells, a concentration of 5 μg/ml blasticidin was used and for THP-1 and SupT1 cells, a concentration of 2 μg/ml was used. Following HeLa cell selection, cells were sorted by flow cytometry and single-cell clonal populations were verified by western blot. CPSF6 knockout cells were transduced with viral vectors containing CPSF6-NLS constructs under puromycin selection. 5 μg/ml puromycin was used for selection of HeLa and 2 μg/ml puromycin was used for THP-1 and SupT1 cells. Construct expression was verified by western blot after 48 h doxycycline induction.
Antibodies and chemicals
HIV-1 capsid, p24, was stained using mouse monoclonal anti-HIV-1 p24 from Santa Cruz Biotechnology (catalogue no. sc-69728). SC35 was stained using mouse monoclonal anti p-SC35 from Santa Cruz Biotechnology (catalogue no. sc-53518). GAPDH was stained with mouse monoclonal anti GAPDH from Santa Cruz Biotechnology (catalogue no. sc-47724). CPSF6 was stained using either CPSF6 polyclonal antibody from Invitrogen (catalogue no. PA5-41830) or from Abcam (catalogue no. ab175237). HA-tag was stained using either rabbit anti-HA-tag from Sigma-Aldrich (catalogue no. H6908) or mouse anti-HA-Tag (6E2) from Cell Signaling Technology (catalogue no. 2376). Secondary antibodies with conjugated fluorophores used in immunofluorescence experiments were purchased from Jackson Immunoresearch Laboratories. Aphidicolin (Cayman Chemicals, catalogue no. 14007) was used to growth arrest cells. Doxycycline hyclate (Sigma-Aldrich, catalogue no. D9891) was used to induce construct expression. CsA was purchased from Sigma-Aldrich. RAL was purchased from MedChemExpress (Catalogue no. HY-10353). Hoechst was purchased from ImmunoChemistry Technologies (Catalogue no. 639).
Western blot
Cell lysates were prepared using NP-40 lysis buffer (100 mM Tris pH 8.0, 1% NP-40, 150 mM NaCl) containing protease inhibitor cocktail (Roche) for 20 min with shaking on ice. Lysates were then centrifuged at 16000 g for 2 min and supernatant was collected. Protein concentration was determined using Pierce BCA Protein Assay Kit (Thermo Fisher Scientific). SDS was then added to lysates and incubated at 95°C for 5 min. Equal amounts of total protein were loaded into a 4–15% gradient gel (Bio-Rad Laboratories). Proteins following electrophoresis were then transferred to a nitrocellulose membrane (Bio-Rad Laboratories). Nitrocellulose membranes were incubated with specific primary antibodies overnight at 4°C on a rocker. Secondary antibodies conjugated with horseradish peroxidase (Thermo Fisher Scientific) were added for 1 h at room temperature on a rocker. Chemiluminescence was detected using ProteinSimple imaging system (FluorChem E).
Quantitative PCR (qPCR)
Cells were infected with equal reverse transcriptase units as determined by SG-PERT assay. qPCR was used to determine reverse transcription, 2-LTR circle formation, and integration. DNA was isolated from cells using DNeasy Blood and Tissue Kit protocol (QIAGEN). DNA concentration was determined using NanoDrop 2000 spectrophotometer and diluted to equal concentrations. The following primers were used for 2-LTR circle qPCRs: 2-LTR Forward, 5’-AACTAGGGAACCCACTGCTTAAG-3’; 2-LTR Reverse, 5’-TCCACAGATCAAGGATATCTTGTC-3’ . Primers used to measure reverse transcription were specific for the Firefly Luciferase gene present in our virus. Firefly Luc Fwd: 5’-CGGAAAGACGATGACGGAAA-3’. Firefly Luc Rev: 5’-CGGTACTTCGTCCACAAACA-3’. Human beta actin was used as housekeeping gene for normalization: HuBeta Actin Fwd, 5’-TCACCCACACTGTGCCCATCT-3’; HuBeta Actin Rev, 5’-CAGCGGAACCGCTCATTGCCAATGG-3’.
The Alu-gag PCR assay for HIV-1 integration was performed as described previously [67,74,85]. Due to lentiviral transduction to generate cell lines, modifications were made to primer sets used. Lambda T sequence was added to the reverse primer of reaction 1 to add specificity and reduce background for reaction 2. Primers used for reaction 1 were as follows. Alu Forward: 5’-GCCTCCCAAAGTGCTGGGATTACAG-3’. Gag-LambdaT Reverse: 5’-AGTTTCGCTTACGTGGCATGTTCCTGCTATGTCACTTCC-3’. Second reaction primers for qPCR were as follows. HIV-1 Gag Forward: 5’-TCAGCCCAGAAGTAATAC-3’. Lambda T Reverse: 5’-AGTTTCGCTTACGTGGCAT-3’.
Infectivity assays
Cells were plated at equal densities and growth arrested (where indicated) in 5 μg/ml aphidicolin overnight. Infection was synchronized as described above. Infectivity was measured at 48 h post infection by either luciferase assay or flow cytometry. Following infection with luciferase reporter viruses, cells were lysed in Passive Lysis Buffer (Promega) and luciferase activity was measured using a luminometer (Promega, GloMax Navigator). To monitor infection from the mCherry reporter, the percent of mCherry positive cells were determined using flow cytometry (BD Bioscience, BD FACSCanto II cytometer).
Microscopy
Images were acquired using a DeltaVision widefield fluorescence microscope (Applied Precision) with a digital camera (CoolSNAP HQ; Photometrics) and 1.4-numerical aperture 100X objective lens. Excitatory light was created by Insight SSI solid-state illumination module (Applied Precision). Images were acquired in several Z-stacks and deconvolution was performed using the softWoRx software v.7.0.0 (Applied Precision, GE Healthcare). Deconvolved images were analyzed using Imaris x64 v.7.6.4 (Bitplane). The surfaces or spots function of Imaris was used to analyze the signal of interest. The same algorithm was used for all images of an experiment. ImageJ was used to determine raw integrated density to quantify percentage of staining within the nucleus.
Statistical analysis
Prism 10.0 (GraphPad) was used to generate graphs and for statistical analysis. Statistical significance was determined using a one-way or two-way ANOVA analysis, where multiple comparisons were measured against full-length CPSF6 (CPSF6-FL). Data are graphed as mean ± standard error mean, except when indicated otherwise in figure legend. Kruskal-Wallis test was performed where indicated in figure legends. Significant differences are indicated: P<0.05, P<0.01, P<0.001, P<0.0001.
Integration site analysis
Infections were synchronously performed with U3-modified HIV-1 at ~1 MOI to enable LTR-specific PCR amplification in cells containing pre-existing CPSF6-358 NLS lentiviral vectors. Genomic DNA was isolated from infected cells 72 h post infection using Quick-DNA Miniprep kit (Zymo Research, catalogue no. D3024).
Integration site sequencing libraries were prepared essentially as described previously [69,70]. Briefly, 5 μg genomic DNA was digested overnight with a restriction enzyme cocktail containing 100 U each of NheI-HF, AvrII, SpeI-HF, and BamHI-HF (New England Biolabs). Digested DNA was purified using a GeneJet PCR purification kit. Freshly annealed asymmetric linkers were subsequently ligated to the fragmented gDNA in four parallel overnight reactions. These reaction products were then purified and subjected to two rounds of ligation mediated (LM)-PCR. The LM-PCR products were then purified and yield was quantified by DNA fluorimetry (Qubit). LM-PCR products were pooled at 10 nM total concentration, ΦX174 control DNA was added to the pooled libraries at 30% molar concentration, and the mixture was loaded onto a P2 300 cycle cartridge at 650 pM concentration for sequencing on an Illumina NextSeq2000 sequencer. Resulting raw FASTQ files were demultiplexed with Sabre ( the demultiplexed reads were trimmed to remove HIV-1 LTR and linker sequences, and the trimmed reads were aligned to human genome build hg19 as described . Uniquely mapped integration site annotations were used in downstream analyses. SPAD annotations used here were the same as previously described [33,35]. LAD annotations were downloaded from the 4D Nucleome data portal (Experiment Set 4DNESTAJJM3X).
Random integration control (RIC) datasets were generated by randomly placing integration sites within hg19 in silico. Following integration site placement, the genome was cleaved at the nearest downstream NheI, AvrII, SpeI, or BamHI recognition site. Generated fragments were filtered to be between 14 bp and 1200 bp and paired-end sequencing reads were simulated by taking up to 150 bp from both fragment ends to generate R1 and R2 fasta files. These fasta files were then aligned to hg19 and the location of uniquely mappable random integration sites were obtained. LINE1 and alphoid repeat annotations were obtained from the hg19 RepeatMasker track annotations provided by UCSC (
Integration site overlap with genomic features (genes, SPADs, LADs, LINE1 repeats, and alphoid repeats) and gene density within 1 Mb of integration sites were quantified with the GenomicRanges Bioconductor package . For comparison of feature overlap between conditions, IS datasets from 2 or more independent replicates for each condition were pooled and the combined data was used for statistical comparisons. The fraction overlap values and the corresponding standard deviations reported in S1 Table reflect the mean and standard deviation between replicates weighted according to dataset size. Statistical comparisons of feature overlap were made using the χ 2 test for equal proportions. Student’s t-test was used to assess statistical significance of gene density around integration sites between conditions.
Supporting information
Chimeric construct expression and localization in T cells and macrophages.
Showing 1/7: ppat.1012354.s001.tiff
Skip to figshare navigation
1 / 7
Share Download
figshare
S1 Fig.Chimeric construct expression and localization in T cells and macrophages.
Western blot analysis of CPSF6-depleted stably-transduced SupT1 (A) and THP-1 (B) cell lines using anti-CPSF6 antibody to detect CPSF6-NLS construct expression following 48 h of doxycycline induction. Anti-GAPDH antibody used as loading control. Immunofluorescent imaging of SupT1 cell lines (C) and THP-1 cell lines differentiated into macrophages (D) using anti-HA antibody. Partial cytoplasmic staining of CPSF6-358 is evident in both cell types (highlighted in SupT1 cells by white arrow).
(TIFF)
S2 Fig.2-LTR circle formation in RAL-treated versus untreated HIV-1 infected cells.
qPCR analysis was performed on genomic DNA isolated from WT HIV-1 infected cells in the presence or absence of raltegravir (RAL). No significant differences in 2-LTR circles were noted among tested cell lines relative to untreated HIV-1 infected cells. Results (mean ± SEM) are representative from 3 independent experiments with at least technical duplicates.
(TIFF)
S3 Fig.Nuclear speckle numbers and total volume following infection.
WT HIV-1 infected HeLa cells were fixed 24 h post-infection and stained for the nuclear speckle marker SC35. (A) IMARIS software was used to form surfaces around SC35 staining in the nucleus. Total SC35 surfaces per cell was determined and no significant differences were observed among the tested cell lines relative to uninfected control cells. (B) IMARIS software was utilized to determine the total volume of SC35 surfaces per cell. No significant differences among the tested HIV-1 infected cell lines relative to uninfected control cells were observed. At least 10 images were taken for each condition across 3 independent experiments.
(TIFF)
S4 Fig.HIVGKO infection.
HeLa cells infected with dual reporter HIVGKO virus. mKO2 reporter is constitutive and GFP reporter is under control of the HIV LTR promoter. Comparable levels of mKO2 and GFP in the tested cell lines indicates that relative integration levels determined viral gene expression.
(TIFF)
S1 Table.Integration distributions in CPSF6-NLS chimera HeLa cells a.
a Results (weighted mean ± weighted SD) are from ≥ two independent infection experiments. b RIC–random integration control.
(DOCX)
S2 Table.Integration site statistics in CPSF6-NLS chimera HeLa cells.
a C6FL–CPSF6-FL. RIC–random integration control.
(DOCX)
S3 Table.Power analysis of integration site statistics a.
a Comparisons highlighted in red are underpowered. b C6FL–CPSF6-FL. RIC–random integration control. c Required sample sizes to achieve Power = 0.8 with Alpha = 0.05 for left (n1) and right (n2) side of comparison as stated in comparison column.
(DOCX)
References
Ori A, Banterle N, Iskar M, Andres-Pons A, Escher C, Khanh Bui H, et al. Cell type-specific nuclear pores: a case in point for context-dependent stoichiometry of molecular machines. Mol Syst Biol. 2013;9:648. pmid:23511206
View Article
PubMed/NCBI
Google Scholar
Ribbeck K, Gorlich D. Kinetic analysis of translocation through nuclear pore complexes. EMBO J. 2001;20(6):1320–30. pmid:11250898
View Article
PubMed/NCBI
Google Scholar
Yang W, Gelles J, Musser SM. Imaging of single-molecule translocation through nuclear pore complexes. Proc Natl Acad Sci U S A. 2004;101(35):12887–92. pmid:15306682
View Article
PubMed/NCBI
Google Scholar
Bonner WM. Protein migration into nuclei. I. Frog oocyte nuclei in vivo accumulate microinjected histones, allow entry to small proteins, and exclude large proteins. J Cell Biol. 1975;64(2):421–30. pmid:46868
View Article
PubMed/NCBI
Google Scholar
Paine PL, Moore LC, Horowitz SB. Nuclear envelope permeability. Nature. 1975;254(5496):109–14. pmid:1117994
View Article
PubMed/NCBI
Google Scholar
Adam SA, Lobl TJ, Mitchell MA, Gerace L. Identification of specific binding proteins for a nuclear location sequence. Nature. 1989;337(6204):276–9. pmid:2911368
View Article
PubMed/NCBI
Google Scholar
Miyamoto Y, Yamada K, Yoneda Y. Importin alpha: a key molecule in nuclear transport and non-transport functions. J Biochem. 2016;160(2):69–75.
View Article
Google Scholar
Pornillos O, Ganser-Pornillos BK, Kelly BN, Hua Y, Whitby FG, Stout CD, et al. X-ray structures of the hexameric building block of the HIV capsid. Cell. 2009;137(7):1282–92. pmid:19523676
View Article
PubMed/NCBI
Google Scholar
Gres AT, Kirby KA, KewalRamani VN, Tanner JJ, Pornillos O, Sarafianos SG. STRUCTURAL VIROLOGY. X-ray crystal structures of native HIV-1 capsid protein reveal conformational variability. Science. 2015;349(6243):99–103. pmid:26044298
View Article
PubMed/NCBI
Google Scholar
Ganser BK, Li S, Klishko VY, Finch JT, Sundquist WI. Assembly and analysis of conical models for the HIV-1 core. Science. 1999;283(5398):80–3. pmid:9872746
View Article
PubMed/NCBI
Google Scholar
Mattei S, Glass B, Hagen WJ, Krausslich HG, Briggs JA. The structure and flexibility of conical HIV-1 capsids determined within intact virions. Science. 2016;354(6318):1434–7. pmid:27980210
View Article
PubMed/NCBI
Google Scholar
Zhao G, Perilla JR, Yufenyuy EL, Meng X, Chen B, Ning J, et al. Mature HIV-1 capsid structure by cryo-electron microscopy and all-atom molecular dynamics. Nature. 2013;497(7451):643–6. pmid:23719463
View Article
PubMed/NCBI
Google Scholar
Yamashita M, Emerman M. Capsid is a dominant determinant of retrovirus infectivity in nondividing cells. J Virol. 2004;78(11):5670–8. pmid:15140964
View Article
PubMed/NCBI
Google Scholar
Bichel K, Price AJ, Schaller T, Towers GJ, Freund SM, James LC. HIV-1 capsid undergoes coupled binding and isomerization by the nuclear pore protein NUP358. Retrovirology. 2013;10:81. pmid:23902822
View Article
PubMed/NCBI
Google Scholar
Fu L, Weiskopf EN, Akkermans O, Swanson NA, Cheng S, Schwartz TU, et al. HIV-1 capsids enter the FG phase of nuclear pores like a transport receptor. Nature. 2024;626(8000):843–51. pmid:38267583
View Article
PubMed/NCBI
Google Scholar
Schaller T, Ocwieja KE, Rasaiyaah J, Price AJ, Brady TL, Roth SL, et al. HIV-1 capsid-cyclophilin interactions determine nuclear import pathway, integration targeting and replication efficiency. PLoS Pathog. 2011;7(12):e1002439. pmid:22174692
View Article
PubMed/NCBI
Google Scholar
Bhattacharya A, Alam SL, Fricke T, Zadrozny K, Sedzicki J, Taylor AB, et al. Structural basis of HIV-1 capsid recognition by PF74 and CPSF6. Proc Natl Acad Sci U S A. 2014;111(52):18625–30. pmid:25518861
View Article
PubMed/NCBI
Google Scholar
Matreyek KA, Yucel SS, Li X, Engelman A. Nucleoporin NUP153 phenylalanine-glycine motifs engage a common binding pocket within the HIV-1 capsid protein to mediate lentiviral infectivity. PLoS Pathog. 2013;9(10):e1003693. pmid:24130490
View Article
PubMed/NCBI
Google Scholar
Price AJ, Jacques DA, McEwan WA, Fletcher AJ, Essig S, Chin JW, et al. Host cofactors and pharmacologic ligands share an essential interface in HIV-1 capsid that is lost upon disassembly. PLoS Pathog. 2014;10(10):e1004459. pmid:25356722
View Article
PubMed/NCBI
Google Scholar
Shen Q, Feng Q, Wu C, Xiong Q, Tian T, Yuan S, et al. Modeling HIV-1 nuclear entry with nucleoporin-gated DNA-origami channels. Nat Struct Mol Biol. 2023;30(4):425–35. pmid:36807645
View Article
PubMed/NCBI
Google Scholar
Dickson CF, Hertel S, Tuckwell AJ, Li N, Ruan J, Al-Izzi SC, et al. The HIV capsid mimics karyopherin engagement of FG-nucleoporins. Nature. 2024;626(8000):836–42. pmid:38267582
View Article
PubMed/NCBI
Google Scholar
Shen Q, Kumari S, Xu C, Jang S, Shi J, Burdick RC, et al. The capsid lattice engages a bipartite NUP153 motif to mediate nuclear entry of HIV-1 cores. Proc Natl Acad Sci U S A. 2023;120(13):e2202815120. pmid:36943880
View Article
PubMed/NCBI
Google Scholar
Kane M, Rebensburg SV, Takata MA, Zang TM, Yamashita M, Kvaratskhelia M, et al. Nuclear pore heterogeneity influences HIV-1 infection and the antiviral activity of MX2. Elife. 2018;7:e35738. pmid:30084827
View Article
PubMed/NCBI
Google Scholar
Price AJ, Fletcher AJ, Schaller T, Elliott T, Lee K, KewalRamani VN, et al. CPSF6 defines a conserved capsid interface that modulates HIV-1 replication. PLoS Pathog. 2012;8(8):e1002896. pmid:22956906
View Article
PubMed/NCBI
Google Scholar
Wei G, Iqbal N, Courouble VV, Francis AC, Singh PK, Hudait A, et al. Prion-like low complexity regions enable avid virus-host interactions during HIV-1 infection. Nat Commun. 2022;13(1):5879. pmid:36202818
View Article
PubMed/NCBI
Google Scholar
Lee K, Ambrose Z, Martin TD, Oztop I, Mulky A, Julias JG, et al. Flexible use of nuclear import pathways by HIV-1. Cell Host Microbe. 2010;7(3):221–33. pmid:20227665
View Article
PubMed/NCBI
Google Scholar
Gruber AR, Martin G, Keller W, Zavolan M. Cleavage factor Im is a key regulator of 3’ UTR length. RNA Biol. 2012;9(12):1405–12. pmid:23187700
View Article
PubMed/NCBI
Google Scholar
Di Nunzio F, Danckaert A, Fricke T, Perez P, Fernandez J, Perret E, et al. Human nucleoporins promote HIV-1 docking at the nuclear pore, nuclear import and integration. PLoS One. 2012;7(9):e46037. pmid:23049930
View Article
PubMed/NCBI
Google Scholar
Zhang R, Mehla R, Chauhan A. Perturbation of host nuclear membrane component RanBP2 impairs the nuclear import of human immunodeficiency virus -1 preintegration complex (DNA). PLoS One. 2010;5(12):e15620. pmid:21179483
View Article
PubMed/NCBI
Google Scholar
Achuthan V, Perreira JM, Sowd GA, Puray-Chavez M, McDougall WM, Paulucci-Holthauzen A, et al. Capsid-CPSF6 Interaction Licenses Nuclear HIV-1 Trafficking to Sites of Viral DNA Integration. Cell Host Microbe. 2018;24(3):392–404 e8. pmid:30173955
View Article
PubMed/NCBI
Google Scholar
Bejarano DA, Peng K, Laketa V, Borner K, Jost KL, Lucic B, et al. HIV-1 nuclear import in macrophages is regulated by CPSF6-capsid interactions at the nuclear pore complex. Elife. 2019;8:e41800. pmid:30672737
View Article
PubMed/NCBI
Google Scholar
Selyutina A, Persaud M, Lee K, KewalRamani V, Diaz-Griffero F. Nuclear Import of the HIV-1 Core Precedes Reverse Transcription and Uncoating. Cell Rep. 2020;32(13):108201. pmid:32997983
View Article
PubMed/NCBI
Google Scholar
Francis AC, Marin M, Singh PK, Achuthan V, Prellberg MJ, Palermino-Rowland K, et al. HIV-1 replication complexes accumulate in nuclear speckles and integrate into speckle-associated genomic domains. Nat Commun. 2020;11(1):3505. pmid:32665593
View Article
PubMed/NCBI
Google Scholar
Guedan A, Donaldson CD, Caroe ER, Cosnefroy O, Taylor IA, Bishop KN. HIV-1 requires capsid remodelling at the nuclear pore for nuclear entry and integration. PLoS Pathog. 2021;17(9):e1009484. pmid:34543344
View Article
PubMed/NCBI
Google Scholar
Li W, Singh PK, Sowd GA, Bedwell GJ, Jang S, Achuthan V, et al. CPSF6-Dependent Targeting of Speckle-Associated Domains Distinguishes Primate from Nonprimate Lentiviral Integration. mBio. 2020;11(5):e02254–20. pmid:32994325
View Article
PubMed/NCBI
Google Scholar
Koh Y, Wu X, Ferris AL, Matreyek KA, Smith SJ, Lee K, et al. Differential effects of human immunodeficiency virus type 1 capsid and cellular factors nucleoporin 153 and LEDGF/p75 on the efficiency and specificity of viral DNA integration. J Virol. 2013;87(1):648–58. pmid:23097450
View Article
PubMed/NCBI
Google Scholar
Dettwiler S, Aringhieri C, Cardinale S, Keller W, Barabino SM. Distinct sequence motifs within the 68-kDa subunit of cleavage factor Im mediate RNA binding, protein-protein interactions, and subcellular localization. J Biol Chem. 2004;279(34):35788–97. pmid:15169763
View Article
PubMed/NCBI
Google Scholar
Jang S, Cook NJ, Pye VE, Bedwell GJ, Dudek AM, Singh PK, et al. Differential role for phosphorylation in alternative polyadenylation function versus nuclear import of SR-like protein CPSF6. Nucleic Acids Res. 2019;47(9):4663–83. pmid:30916345
View Article
PubMed/NCBI
Google Scholar
Matreyek KA, Engelman A. The requirement for nucleoporin NUP153 during human immunodeficiency virus type 1 infection is determined by the viral capsid. J Virol. 2011;85(15):7818–27. pmid:21593146
View Article
PubMed/NCBI
Google Scholar
Dharan A, Bachmann N, Talley S, Zwikelmaier V, Campbell EM. Nuclear pore blockade reveals that HIV-1 completes reverse transcription and uncoating in the nucleus. Nat Microbiol. 2020;5(9):1088–95. pmid:32483230
View Article
PubMed/NCBI
Google Scholar
Chai K, Wang Z, Pan Q, Tan J, Qiao W, Liang C. Effect of Different Nuclear Localization Signals on the Subcellular Localization and Anti-HIV-1 Function of the MxB Protein. Front Microbiol. 2021;12:675201. pmid:34093497
View Article
PubMed/NCBI
Google Scholar
Cazalla D, Zhu J, Manche L, Huber E, Krainer AR, Caceres JF. Nuclear export and retention signals in the RS domain of SR proteins. Mol Cell Biol. 2002;22(19):6871–82. pmid:12215544
View Article
PubMed/NCBI
Google Scholar
Greig JA, Nguyen TA, Lee M, Holehouse AS, Posey AE, Pappu RV, et al. Arginine-Enriched Mixed-Charge Domains Provide Cohesion for Nuclear Speckle Condensation. Mol Cell. 2020;77(6):1237–50 e4. pmid:32048997
View Article
PubMed/NCBI
Google Scholar
Hori T, Takeuchi H, Saito H, Sakuma R, Inagaki Y, Yamaoka S. A carboxy-terminally truncated human CPSF6 lacking residues encoded by exon 6 inhibits HIV-1 cDNA synthesis and promotes capsid disassembly. J Virol. 2013;87(13):7726–36. pmid:23658440
View Article
PubMed/NCBI
Google Scholar
Tian B, Manley JL. Alternative polyadenylation of mRNA precursors. Nat Rev Mol Cell Biol. 2017;18(1):18–30. pmid:27677860
View Article
PubMed/NCBI
Google Scholar
Mattaj IW, Englmeier L. Nucleocytoplasmic transport: the soluble phase. Annu Rev Biochem. 1998;67:265–306. pmid:9759490
View Article
PubMed/NCBI
Google Scholar
Kosugi S, Hasebe M, Matsumura N, Takashima H, Miyamoto-Sato E, Tomita M, et al. Six classes of nuclear localization signals specific to different binding grooves of importin alpha. J Biol Chem. 2009;284(1):478–85. pmid:19001369
View Article
PubMed/NCBI
Google Scholar
Cherezova L, Burnside KL, Rose TM. Conservation of complex nuclear localization signals utilizing classical and non-classical nuclear import pathways in LANA homologs of KSHV and RFHV. PLoS One. 2011;6(4):e18920. pmid:21559489
View Article
PubMed/NCBI
Google Scholar
Wang L, Li M, Cai M, Xing J, Wang S, Zheng C. A PY-nuclear localization signal is required for nuclear accumulation of HCMV UL79 protein. Med Microbiol Immunol. 2012;201(3):381–7. pmid:22628116
View Article
PubMed/NCBI
Google Scholar
Lange A, Mills RE, Devine SE, Corbett AH. A PY-NLS nuclear targeting signal is required for nuclear localization and function of the Saccharomyces cerevisiae mRNA-binding protein Hrp1. J Biol Chem. 2008;283(19):12926–34. pmid:18343812
View Article
PubMed/NCBI
Google Scholar
Michael WM, Eder PS, Dreyfuss G. The K nuclear shuttling domain: a novel signal for nuclear import and nuclear export in the hnRNP K protein. EMBO J. 1997;16(12):3587–98. pmid:9218800
View Article
PubMed/NCBI
Google Scholar
Siomi H, Shida H, Nam SH, Nosaka T, Maki M, Hatanaka M. Sequence requirements for nucleolar localization of human T cell leukemia virus type I pX protein, which regulates viral RNA processing. Cell. 1988;55(2):197–209. pmid:3048703
View Article
PubMed/NCBI
Google Scholar
Melen K, Keskinen P, Ronni T, Sareneva T, Lounatmaa K, Julkunen I. Human MxB protein, an interferon-alpha-inducible GTPase, contains a nuclear targeting signal and is localized in the heterochromatin region beneath the nuclear envelope. J Biol Chem. 1996;271(38):23478–86. pmid:8798556
View Article
PubMed/NCBI
Google Scholar
Sowd GA, Serrao E, Wang H, Wang W, Fadel HJ, Poeschla EM, et al. A critical role for alternative polyadenylation factor CPSF6 in targeting HIV-1 integration to transcriptionally active chromatin. Proc Natl Acad Sci U S A. 2016;113(8):E1054–63. pmid:26858452
View Article
PubMed/NCBI
Google Scholar
Lu J, Wu T, Zhang B, Liu S, Song W, Qiao J, et al. Types of nuclear localization signals and mechanisms of protein import into the nucleus. Cell Commun Signal. 2021;19(1):60. pmid:34022911
View Article
PubMed/NCBI
Google Scholar
Franke EK, Luban J. Inhibition of HIV-1 replication by cyclosporine A or related compounds correlates with the ability to disrupt the Gag-cyclophilin A interaction. Virology. 1996;222(1):279–82. pmid:8806510
View Article
PubMed/NCBI
Google Scholar
Luban J, Bossolt KL, Franke EK, Kalpana GV, Goff SP. Human immunodeficiency virus type 1 Gag protein binds to cyclophilins A and B. Cell. 1993;73(6):1067–78. pmid:8513493
View Article
PubMed/NCBI
Google Scholar
Franke EK, Yuan HE, Luban J. Specific incorporation of cyclophilin A into HIV-1 virions. Nature. 1994;372(6504):359–62. pmid:7969494
View Article
PubMed/NCBI
Google Scholar
De Iaco A, Luban J. Cyclophilin A promotes HIV-1 reverse transcription but its effect on transduction correlates best with its effect on nuclear entry of viral cDNA. Retrovirology. 2014;11:11. pmid:24479545
View Article
PubMed/NCBI
Google Scholar
Li L, Olvera JM, Yoder KE, Mitchell RS, Butler SL, Lieber M, et al. Role of the non-homologous DNA end joining pathway in the early steps of retroviral infection. EMBO J. 2001;20(12):3272–81. pmid:11406603
View Article
PubMed/NCBI
Google Scholar
Chin CR, Perreira JM, Savidis G, Portmann JM, Aker AM, Feeley EM, et al. Direct Visualization of HIV-1 Replication Intermediates Shows that Capsid and CPSF6 Modulate HIV-1 Intra-nuclear Invasion and Integration. Cell Rep. 2015;13(8):1717–31. pmid:26586435
View Article
PubMed/NCBI
Google Scholar
Burdick RC, Li C, Munshi M, Rawson JMO, Nagashima K, Hu WS, et al. HIV-1 uncoats in the nucleus near sites of integration. Proc Natl Acad Sci U S A. 2020;117(10):5486–5493. pmid:32094182
View Article
PubMed/NCBI
Google Scholar
Luchsinger C, Lee K, Mardones GA, KewalRamani VN, Diaz-Griffero F. Formation of nuclear CPSF6/CPSF5 biomolecular condensates upon HIV-1 entry into the nucleus is important for productive infection. Sci Rep. 2023;13(1):10974. pmid:37414787
View Article
PubMed/NCBI
Google Scholar
Rensen E, Mueller F, Scoca V, Parmar JJ, Souque P, Zimmer C, et al. Clustering and reverse transcription of HIV-1 genomes in nuclear niches of macrophages. EMBO J. 2021;40(1):e105247. pmid:33270250
View Article
PubMed/NCBI
Google Scholar
Scoca V, Morin R, Collard M, Tinevez JY, Di Nunzio F. HIV-induced membraneless organelles orchestrate post-nuclear entry steps. J Mol Cell Biol. 2023;14(11). pmid:36314049
View Article
PubMed/NCBI
Google Scholar
Liszewski MK, Yu JJ, O’Doherty U. Detecting HIV-1 integration by repetitive-sampling Alu-gag PCR. Methods. 2009;47(4):254–60. pmid:19195495
View Article
PubMed/NCBI
Google Scholar
Mbisa JL, Delviks-Frankenberry KA, Thomas JA, Gorelick RJ, Pathak VK. Real-time PCR analysis of HIV-1 replication post-entry events. Methods Mol Biol. 2009;485:55–72. pmid:19020818
View Article
PubMed/NCBI
Google Scholar
Battivelli E, Dahabieh MS, Abdel-Mohsen M, Svensson JP, Tojal Da Silva I, Cohn LB, et al. Distinct chromatin functional states correlate with HIV latency reactivation in infected primary CD4(+) T cells. Elife. 2018;7:e34655. pmid:29714165
View Article
PubMed/NCBI
Google Scholar
Serrao E, Cherepanov P, Engelman AN. Amplification, Next-generation Sequencing, and Genomic DNA Mapping of Retroviral Integration Sites. J Vis Exp. 2016(109). pmid:27023428
View Article
PubMed/NCBI
Google Scholar
Anderson-Daniels J, Singh PK, Sowd GA, Li W, Engelman AN, Aiken C. Dominant Negative MA-CA Fusion Protein Is Incorporated into HIV-1 Cores and Inhibits Nuclear Entry of Viral Preintegration Complexes. J Virol. 2019;93(21):e01118–19. pmid:31413124
View Article
PubMed/NCBI
Google Scholar
Singh PK, Bedwell GJ, Engelman AN. Spatial and Genomic Correlates of HIV-1 Integration Site Targeting. Cells. 2022;11(4):655. pmid:35203306
View Article
PubMed/NCBI
Google Scholar
Bedwell GJ, Jang S, Li W, Singh PK, Engelman AN. rigrag: high-resolution mapping of genic targeting preferences during HIV-1 integration in vitro and in vivo. Nucleic Acids Res. 2021;49(13):7330–46.
View Article
Google Scholar
Carteau S, Hoffmann C, Bushman F. Chromosome structure and human immunodeficiency virus type 1 cDNA integration: centromeric alphoid repeats are a disfavored target. J Virol. 1998;72(5):4005–14. pmid:9557688
View Article
PubMed/NCBI
Google Scholar
Winans S, Yu HJ, de Los Santos K, Wang GZ, KewalRamani VN, Goff SP. A point mutation in HIV-1 integrase redirects proviral integration into centromeric repeats. Nat Commun. 2022;13(1):1474. pmid:35304442
View Article
PubMed/NCBI
Google Scholar
Stevens SW, Griffith JD. Human immunodeficiency virus type 1 may preferentially integrate into chromatin occupied by L1Hs repetitive elements. Proc Natl Acad Sci U S A. 1994;91(12):5557–61. pmid:8202527
View Article
PubMed/NCBI
Google Scholar
Schroder AR, Shinn P, Chen H, Berry C, Ecker JR, Bushman F. HIV-1 integration in the human genome favors active genes and local hotspots. Cell. 2002;110(4):521–9. pmid:12202041
View Article
PubMed/NCBI
Google Scholar
Gupta K, Brady T, Dyer BM, Malani N, Hwang Y, Male F, et al. Allosteric inhibition of human immunodeficiency virus integrase: late block during viral replication and abnormal multimerization involving specific protein domains. J Biol Chem. 2014;289(30):20477–88. pmid:24904063
View Article
PubMed/NCBI
Google Scholar
Marquis KA, Everett J, Cantu A, McFarland A, Sherrill-Mix S, Krystal M, et al. The HIV-1 Capsid-Targeted Inhibitor GSK878 Alters Selection of Target Sites for HIV DNA Integration. AIDS Res Hum Retroviruses. 2024;40(2):114–26. pmid:37125442
View Article
PubMed/NCBI
Google Scholar
Kawachi T, Masuda A, Yamashita Y, Takeda JI, Ohkawara B, Ito M, et al. Regulated splicing of large exons is linked to phase-separation of vertebrate transcription factors. EMBO J. 2021;40(22):e107485. pmid:34605568
View Article
PubMed/NCBI
Google Scholar
Jang S, Bedwell GJ, Singh SP, Yu HJ, Arnarson B, Singh PK, et al. HIV-1 usurps mixed-charge domain-dependent CPSF6 phase separation for higher-order capsid binding, nuclear entry and viral DNA integration. Nucleic Acids Res. 2024;52(18):11060–82. pmid:39258548
View Article
PubMed/NCBI
Google Scholar
Munk C, Brandt SM, Lucero G, Landau NR. A dominant block to HIV-1 replication at reverse transcription in simian cells. Proc Natl Acad Sci U S A. 2002;99(21):13843–8. pmid:12368468
View Article
PubMed/NCBI
Google Scholar
Vermeire J, Naessens E, Vanderstraeten H, Landi A, Iannucci V, Van Nuffel A, et al. Quantification of reverse transcriptase activity by real-time PCR as a fast and accurate method for titration of HIV, lenti- and retroviral vectors. PLoS One. 2012;7(12):e50859. pmid:23227216
View Article
PubMed/NCBI
Google Scholar
Bulli L, Apolonia L, Kutzner J, Pollpeter D, Goujon C, Herold N, et al. Complex Interplay between HIV-1 Capsid and MX2-Independent Alpha Interferon-Induced Antiviral Factors. J Virol. 2016;90(16):7469–80. pmid:27279606
View Article
PubMed/NCBI
Google Scholar
Butler SL, Hansen MS, Bushman FD. A quantitative assay for HIV DNA integration in vivo. Nat Med. 2001;7(5):631–4. pmid:11329067
View Article
PubMed/NCBI
Google Scholar
Brussel A, Delelis O, Sonigo P. Alu-LTR real-time nested PCR assay for quantifying integrated HIV-1 DNA. Methods Mol Biol. 2005;304:139–54. pmid:16061972
View Article
PubMed/NCBI
Google Scholar
Lawrence M, Huber W, Pages H, Aboyoun P, Carlson M, Gentleman R, et al. Software for computing and annotating genomic ranges. PLoS Comput Biol. 2013;9(8):e1003118. pmid:23950696
View Article
PubMed/NCBI
Google Scholar
Download PDF
Citation
XML
Print
Share
Reddit
Facebook
LinkedIn
Mendeley
Bluesky
Email
Advertisement
Subject Areas
?
For more information about PLOS Subject Areas, click here.
We want your feedback. Do these Subject Areas make sense for this article? Click the target next to the incorrect Subject Area and let us know. Thanks for your help!
HIV-1Is the Subject Area "HIV-1" applicable to this article? Yes No
Thanks for your feedback.
Nuclear importIs the Subject Area "Nuclear import" applicable to this article? Yes No
Thanks for your feedback.
HeLa cellsIs the Subject Area "HeLa cells" applicable to this article? Yes No
Thanks for your feedback.
Human genomicsIs the Subject Area "Human genomics" applicable to this article? Yes No
Thanks for your feedback.
SV40Is the Subject Area "SV40" applicable to this article? Yes No
Thanks for your feedback.
Viral packagingIs the Subject Area "Viral packaging" applicable to this article? Yes No
Thanks for your feedback.
LuciferaseIs the Subject Area "Luciferase" applicable to this article? Yes No
Thanks for your feedback.
Viral gene expressionIs the Subject Area "Viral gene expression" applicable to this article? Yes No
Thanks for your feedback.
Publications
PLOS Biology
PLOS Climate
PLOS Complex Systems
PLOS Computational Biology
PLOS Digital Health
PLOS Genetics
PLOS Global Public Health
PLOS Medicine
PLOS Mental Health
PLOS Neglected Tropical Diseases
PLOS One
PLOS Pathogens
PLOS Sustainability and Transformation
PLOS Water
Home
Blogs
Collections
Give feedback
LOCKSS
Privacy Policy
Terms of Use
Advertise
Media Inquiries
Contact
PLOS is a nonprofit 501(c)(3) corporation, #C2354500, based in California, US
Cookie Preference Center
Our website uses different types of cookies. Optional cookies will only be set with your consent and you may withdraw this consent at any time. Below you can learn more about the types of cookies PLOS uses and register your cookie preferences.
Accept All Cookies
Customize Your Cookie Preference
+Strictly Necessary
Always On
+Functional
[x] On Off
+Performance and Analytics
[x] On Off
+Marketing
[x] On Off
Save Selected Preferences and Close
For more information about the cookies and other technologies used by us, please read our Cookie Policy. |
8655 | https://www.ijnrd.org/papers/IJNRD2305026.pdf | © 2023 IJNRD | Volume 8, Issue 5 May 2023 | ISSN: 2456-4184 | IJNRD.ORG IJNRD2305026 International Journal of Novel Research and Development (www.ijnrd.org) a204 A Critical Analysis of Good Governance and Enough Good Governance in Bangladesh. Ziaur Rahman Deputy Secretary & Consultant Aspire to Innovate (a2i) Program, ICT Division, Dhaka, Bangladesh. Abstract: Governance and Good Governance are two important terms in development process for both developed and developing countries. Governance is the manner in which power is exercised in the management of a country’s economic and social resources for development. Good governance means an ideal governing system. The basic traits of good governance are: participation, accountability, transparency, rule of law, effectiveness and efficiency, responsiveness, and equity and inclusiveness. The concept of good enough governance is a practical approach to the ground realities of a country. Instead of bringing changes in all aspects of public sector, it suggests that actions aiming changes should be prioritized taking into consideration the realities of the context. It advocates gradual changes in the public sector. Although good enough governance may appear feasible for the sake of economic and political developments in short run, the target should always be established good governance for the country. By analysis of good governance in Bangladesh, we see that all the symptoms of an under developed polity in Bangladesh. In order to establish good governance in Bangladesh, the government of Bangladesh needs to address issues like corruption, political interference in administration, misuse of power and resources, lack of political and bureaucratic accountability, non-transparent administration, less participative decision-making process and so on. Key words: Governance, Good governance, Bangladesh. 1. Introduction The terms ‘governance’ and ‘good governance’ have made recurrent appearances in recent times in development literature. Good governance has swept public attention in the arena of modern globalization. The term ‘governance’ refers to the manner in which power is exercised in the management of a country’s economic, political and social resources for long term development. In today's modern world political and economic, the word "good and quality governance" has gained popularity. According to ADB (2005) ‘governance is about the institutional environment in which citizens interact among themselves and with government agencies and officials. In order to solve various issues and advance socioeconomic development in emerging nations, good governance is seen as a necessary condition. The term ‘good governance’ usually requires an accountable government, an independent judicial system, freedom of thought and expression, predictable policy making, a people-oriented bureaucracy, rule of law and desirable state affairs. According to the international aid front, the concept of ‘good governance’ has been a general guiding principle for donor agencies, policy makers, development practitioners for over a decade. This concept emerged in the 90’s and became the well-known term in the both developed and developing countries. Now it is considered as a significant pillar in the consideration © 2023 IJNRD | Volume 8, Issue 5 May 2023 | ISSN: 2456-4184 | IJNRD.ORG IJNRD2305026 International Journal of Novel Research and Development (www.ijnrd.org) a205 of a state’s ability to conform to universally acceptable democratic standards (Ara and Khan, 2006). Good governance is deemed as an imperative for development because it encompasses the mechanisms, processes, and institutions through which citizens and groups articulate their interests, exercise their legal rights, meet their obligations and resolve their differences (Ahmed, 2007). Bangladesh Government recognizes the importance of good governance and adopted it as a strategy to accelerate poverty reduction in the Poverty Reduction Strategy Paper (PRSP) (Ahmed, 2007). Bangladesh shares many of the same problems as other developing nations, including a leadership crisis, a lack of transparency and accountability in the administration, ineffective political institutions, and a public sector that is unable to meet the demands of its citizens. Public services are also ineffective, and the environment is not conducive to the proper growth of the private sector. Effective demographic governance continues to be the elusive golden deer that the nation assiduously sought but could not obtain due to such poor performance in governance systems. Examining how Bangladesh compares to other countries in terms of good governance is important given this context. The purpose of the current article is to critically assess the current status of good governance in Bangladesh and to identify the barriers standing in the way of effectively implementation of good governance. This paper consists of following sections. Section one introductory section, section two conceptual framework on governance, good governance and poor governance, major traits of good governance and enough good governance. Section three states the good governance scenario in Bangladesh. Section four analyses the major obstacles of good governance system in Bangladesh. Section five shows the prospects of implementation of good governance in Bangladesh and reforms taken by government for ensuring good governance system in Bangladesh. Section six provides recommendations and concluding remarks for good governance system in Bangladesh. 1.1 Objectives of the Study This study based on the following objectives: 1. To develop conceptual framework regarding perspectives, issues and concerns of good governance. 2. To analyze the current scenario of good governance and enough good governance in Bangladesh. 3. To identify the main obstacles for proper implementation of good governance in Bangladesh. 4. To propose some policy suggestions for proper implementation of good governance system in Bangladesh. 2. Conceptual Framework: 2.1 Concept of Governance ‘Governance’ is the manner in which power is exercised in the management of a country’s economic and social resources for development in transparent and an efficient manner. According to Oxford Dictionary ‘Governance is the action or manner of governing a state, organization, etc.’ According to UNESCAP, governance is the process of decision-making and the process by which decisions are implemented. WB defines ‘Governance is the method through which the power is exercised in the management of a country’s political, economic and social resources for development. UNDP defines ‘governance is the exercise of economic, political and administrative authority to manage a country’s affairs at all levels (UNDP). According to UNDP (1997), three key actors in governance are: state/ public sector; business sector; and civil society. It has now become an important term recent global politics and economy that ensure goodness and quality of governance. Basically, the term governance refers to the development of government styles in which boundaries between and within public and private sectors have become blurred. It is not only occupying central stage in the development discourse but it is considered as the crucial element to be incorporated in the development strategy. It also may be termed as a mechanism through which relationship between those who govern and those who are governed is expressed. The concept of governance can be applied in different contexts such as global, national and international. Governance has three legs such as economic, political and administrative. Economic governance includes decision making processes that affect a country’s economic activities and its relationships with other economics. It clearly has major implications for equity, poverty and quality of life. Political governance is the process of decision making to formulate policy. Administrative governance is the system of policy implementation. According to Institute on Governance (IOG, 2006) © 2023 IJNRD | Volume 8, Issue 5 May 2023 | ISSN: 2456-4184 | IJNRD.ORG IJNRD2305026 International Journal of Novel Research and Development (www.ijnrd.org) a206 ‘governance as the process whereby societies or organizations make their important decisions, determine who has voice, who is engaged in the process and how account is rendered. Therefore, we say that governance includes: the process by which governments are selected, held accountable, monitored, and replaced; the capacity of governments to manage resources efficiently and formulate, implement, and enforce sound policies and regulations; and the extent of participation of the citizens in the affairs of the state. The four pillars of governance are accountability, transparency, predictability and participation which can be expressed in the following table -1: Accountability (Building government capacity) Transparency (Information Openness) Predictability (Legal framework) Participation (Participatory Development Process) Public Sector management Public Enterprise Management and reform Public Financial management Civil service reform Disclosure of information Law and development Legal frameworks for private sector development Participation of beneficiaries and affected groups Public sector/Private sector interface Decentralization of public and service delivery functions. Empowerment of local governments Cooperation with non-government organizations Table-1: The four pillars of governance 2.2. Concept of Good Governance The concept of “Good Governance” arose in the 1980s in the International Finance Institutions (IFI’s) such as the World Bank, the IMF, UNDP, and OECD as a positive extrapolation from the negative experiences that these organizations had in the “developing countries” by observing that their financial aid seemed to have had no effect. ‘Good governance’ means an ideal governing system. Good governance is sound governance that works. The concept centers on the responsibility of governments and governing bodies to meet the needs of the masses as opposed to select groups in society. According to Kofi Anan, Former UN Secretary General ‘Good governance is ensuring respect for human rights and the rule of law; strengthening democracy; promoting transparency; and capacity in public administration.’ Good governance is, among other things, participatory, transparent and accountable. It is also effective and equitable and it promotes the rule of law. Good governance ensures that political, social and economic priorities are based on broad consensus in society and the voices of the poorest and the most vulnerable are heard in decision making over the allocation of development resources. It is epitomized by predictable, open and enlightened policy making, a bureaucracy imbued with professional ethos acting in promotion of public good, the rule of law, transparent process and a strong civil society participating in public affairs. It also implies a democratic structure, human rights and freedom of press and media which ensures sustainable economic and social development and institutional growth. In broad sense, Good Governance is defined as the existence within states of political accountability, bureaucratic transparency, the exercise of legitimate power, freedom of association and participation, freedom of information and expression, sound fiscal management and public financial accountability, respect for rule of law, a predictable legal framework encompassing an interdependent and credible judicial system, respect for human rights, an active legislature, enhanced opportunities for the development of pluralistic forces including civil society and capacity development. According to World Bank, ‘good governance’ involves “an efficient public service; an independent judicial system and legal framework to enforce contracts; the accountable administration of public funds; an independent public auditor, responsible to representative legislature; respect for the law and human rights at all levels of governments; a pluralistic institutional structure, and a free press” (Rhodes,1996). World Bank (1992) identified a number of parameters of good governance such as: © 2023 IJNRD | Volume 8, Issue 5 May 2023 | ISSN: 2456-4184 | IJNRD.ORG IJNRD2305026 International Journal of Novel Research and Development (www.ijnrd.org) a207 1. The legitimacy of a political system which can best be achieved through regular elections and political accountability; 2. Freedom of association and participation by various socioeconomic, religious, cultural and professional groups in the governance; 3. An established legal framework based on rule of law and independent of judiciary to protect human rights and secure social justice; 4. Bureaucratic accountability including transparency in administration; 5. Freedom of information and expression; 6. A sound administration system leading to efficiency and effectiveness; and 7. Cooperation between the government and Civil Society Organizations (CSOs), which have significant implications for implantation good governance in developed and developing countries. United Nations (UN) has identified eight basic elements of good governance: participatory, consensus oriented, accountable, transparent, responsive, effective and efficient, equitable and inclusive, and rule of law (Ara and Khan, 2006, Grindle, 2007). Different multilateral organizations and donor’s agency like The United Nations Development Programme (UNDP), The Organization for Economic Cooperation and Development (OECD) equate good governance with better economic management emphasizing on accountability, transparency, participation and predictability. OECD identifies a set of criteria for good governance such as: Legitimacy of government; Accountability of political and official elements of government; Competence of government to make policy and deliver services; and Respect of human rights and rule of law. The Department for International development (DFID) identifies seven key governance capabilities for good governance such as: To operate political system which provide opportunities for all people; To influence government policy and practice; To provide macroeconomic stability; To promote the growth necessary to reduce poverty; To implement pro-poor policy; To guarantee the equitable and universal provision of effective basic services; and To ensure personal safety and security; to manage national security arrangements accountably; and to develop honest and accountable government (Grindle, 2007) Hyden et al (2004) proposed five dimensions of good governance namely participation, fairness, decency, efficiency, accountability and transparency in each of the following six arenas: civil society, political society, economic society, government, bureaucracy and judiciary for measuring good governance (Grindle, 2007). Sir Keneth Stowe defines ‘good governance’ with following traits: political freedom, including free speech and a freely elected parliament; constitutional and judicial protection for the rights of the individual; maintenance of the rule of law by an independent judiciary; maintenance of stable currency; development of a society as a whole by education and health care; and executive accountability to a freely elected legislature (Ara and Khan, 2006). D. Bandyopadhyay defines ‘good governance’ with following main elements: Accountability both financial and political; Transparency both financial and political; Easy access to information; Popular participation in decision making and implementation; Responsiveness; © 2023 IJNRD | Volume 8, Issue 5 May 2023 | ISSN: 2456-4184 | IJNRD.ORG IJNRD2305026 International Journal of Novel Research and Development (www.ijnrd.org) a208 Efficient delivery system of services and goods; enforcement of rule of law, supplanting the rule of whims and caprices of rulers whether stipendiary or elected; and Client/citizen satisfaction; and an overall caring and humane ambience promoting an equalitarian and equities social and economic order. Thus, good governance means an ideal governing system that is imperative for political, economic, social and cultural development of a country. 2.3. Poor Governance The opposite of good governance is poor governance which is characterized by arbitrary policy making, unaccountable bureaucracy, unforced or unjust legal system, the abuse of political power, a civil society unchanged in public affairs and wide spread corruption. It also undermines all efforts to improve policy making and create durable institutions. 2.4 Basic Traits of Good Governance The basic traits of good governance are: participation, accountability, transparency, rule of law, effectiveness and efficiency, responsiveness, equity and inclusiveness and consensus oriented. Figure-1: Basic Traits of Good Governance, Source: Governance for Sustainable Human Development: A UNDP Policy Document, 1997 2.4.1 Participation In a democracy participation means involvement in the decision-making process either directly or through legitimate intermediate institutions or representatives. Participation by both men and women is a key cornerstone of good governance. According to Muhith (2000), good governance by definition enshrines participation. He also states that good governance must reflect the will of the people and at the same time articulate and mould this will as well. This can be achieved through mass participation in governance. Without common people’s participation in the governmental system good governance cannot be ensured. Rowe and Frewer (2004) define public participation as the practice of involving the people in the agenda-setting, decision-making and policy-forming activities. According to (Paul 1990), participation is an active process by which interests groups influence the decision which affect their well-being. Wolfe (1985) states that participation is the way of generating popular control of issue-agenda, decision-making and implementation. Participation is closely linked with the concept of empowerment. Participation truly means empowering the marginalized and excluded. Gran (1993), Oakley and Marsden (1984) and Oakley (1987) put forward that © 2023 IJNRD | Volume 8, Issue 5 May 2023 | ISSN: 2456-4184 | IJNRD.ORG IJNRD2305026 International Journal of Novel Research and Development (www.ijnrd.org) a209 people’s is the process of empowerment of deprived and excluded (Samad, 2002). Participation is also partnership (Cornwall, 2002). Khan (1998) summarizes the definition of participation are as follows: participation is an organized effort to increase control over resources and regulative institutions; people’s involvement in decision-making, implementation, benefit-sharing and in evaluation of programs; people’s capacity to take initiative in development, to become “subjects” rather than “objects” of their own destiny; this can only be achieved through a de-professionalization in all domains of life in order to make “ordinary people” responsible for their own well-being; Participation involves a reversal of role playing: people should be the primary actors and government agencies and outsiders should particitipate in people’s activities. Whereas ‘deep participation’ involves participations in all stages of decision-making, from problem identification to decision making, ‘narrow participation’ involves only a handful of people or particular interests’ groups. However, participation can be wide engaging all walks of people of a society and shallow when the stakeholders are only informed or consulted (Cornwall, 2002). Participation can take various forms; but all types of participation are not real or genuine. As far as good governance is concerned, participation does not include those which have the show of participation; rather it means participation that transforms ‘objects’ into ‘subjects’, ‘mute recipients’ in ‘active decision-makers’. The aim of participatory governance is the recognize voice of each and everyone, regardless of their social status or economic condition or cultural diversity. 2.4.2. Accountability Accountability is the pre-requisite condition for good governance. It must be ensured in order to improve governance scenario in any country. Paul (1992) states that accountability means “holding individuals and organizations responsible for performance measured as objectively as possible. Samaratunge et al (2008) point out two key elements of accountability: answerability and enforceability whereas ‘answerability’ means fixing liability for the individuals as well as organizations and enforceability’ means to the mechanisms as well as capacity of subjecting the individuals as well as organizations to account for respective performances. Accountability has various forms: political, administrative, legal and professional. 2.4.2.1. Political Accountability Political accountability refers to democratic pressures imposed by the legislature on politicians and bureaucratic/ public managers. It is an essential precondition for the functioning of a democratic system. It demands that the persons politically in charge of public offices must be accountable to the citizens for all their actions. 2.4.2.2. Administrative/Bureaucratic Accountability Administrative accountability focuses on the application of supervisory control processes to a wide range of agency activities. It denotes that the civil servants are fully aware of their responsibilities in a multiparty democratic system. It requires effective performance of public agencies and officials in their relations with the public in a transparent manner (Ara and Khan, 2006). Bureaucratic accountability can only be ensured if political accountability is in place. 2.4.2.3. Legal Accountability Legal accountability involves the enforcement of legal control over the activities of public sector organizations. 2.4.2.4. Professional Accountability Professional accountability refers to the placement of control over organizational activities in the hands of the employee with the expertise or special skills to get the job done. © 2023 IJNRD | Volume 8, Issue 5 May 2023 | ISSN: 2456-4184 | IJNRD.ORG IJNRD2305026 International Journal of Novel Research and Development (www.ijnrd.org) a210 Accountability mechanisms are embedded in regulatory and institutional arrangements. Without appropriate and effective accountability mechanisms good governance cannot be ensured. Haque (2000) enumerates various means of accountability which are as follows: external-formal mechanisms: for example, legislative instruments (legislative committees and parliamentary questions), executive means (controls exercised by political executives over public agencies), and judicial or quasi-judicial processes (administrative courts and ombudsmen); external-informal mechanisms, such as public hearings, interest groups, opinion polls and media security; internal-formal means including official rules, codes of conducts, official hierarchies, and performance reviews; and internal-informal mechanisms such as organizational culture, professional ethics and peer pressure. Good governance requires that most of accountability mechanisms are effectively in operation in ensuring accountability of those who are in power as well as those who exercise power. It is repeatedly underlined that political neutrality of the civil servants is a precondition of ensuring accountability in a country. In addition, free and fair media and an unbiased, neutral as well as functional civil society are deemed indispensable for good governance. They play paramount roles in not only ensuring accountability but also ensuring transparency. 2.4.3. Transparency Transparency means that decisions taken and their enforcement are done in a manner that follows rules and regulations. It entails free flow of information and direct accessibility to that information by those who are affected by such decisions and their enforcement. The culture of secrecy help breed corruption in government which is probably the ‘single biggest factor’ hindering development. Corruption not only hinders development but also perpetuates poverty and inequality. Because of corruption about 40% of the population in Asia and Africa live below the poverty line they do not get enough to eat preserve their body weight are physically stunned before they become adults and unable to lead healthy productive lives (Rizvi, 2008). The culture of transparency, accountability and the rule of law are essential to eliminating corruption. 2.4.4. Rule of Law Good governance requires that the state is governed by laws fairly and impartially enforced ensuring protection of human rights. Establishment of rule of law through impartial enforcement of laws is an essential precondition for good governance. It entails that there should be full protection of human rights, especially those of minorities, women, and ultra poor. To establish good governance an independent judiciary and impartial and incorruptible law enforcing agencies are necessary pre-conditions. A sound judicial system is an imperative for good governance. In a democracy as the custodian of the fundamental rights of its citizens enumerated in the constitution and as the medium of checking abuses of power committed by government functionaries the role of judiciary in the establishment of good governance is immense and indispensable. It is through judiciary that people truly experience the working of democracy (Ara and Khan, 2006). 2.4.5. Effectiveness and Efficiency Making the best use of resources, sustainable use of natural resources and the protection of the environment are prerequisites for good governance. Efficient management of economy is considered as a hallmark of good governance. An efficient and effective bureaucracy, which is accountable, is of vital importance in ensuring good governance. A professional and well organized bureaucracy is generally considered as a means of achieving development. So, the concept of efficiency in the context of good governance also covers the sustainable use of natural resources and the protection of the environment. 2.4.6. Responsiveness Good governance requires that institutions and processes try to serve all stakeholders within a reasonable timeframe. It requires that services are delivered to the citizens, who are clients in the modern concepts of © 2023 IJNRD | Volume 8, Issue 5 May 2023 | ISSN: 2456-4184 | IJNRD.ORG IJNRD2305026 International Journal of Novel Research and Development (www.ijnrd.org) a211 governance within reasonable time limit. Only an efficient, effective and accountable democratic government is responsive to the citizens wishes (Rizvi, 2008). Corruption, nepotism, mismanagement of resources, red-tapism, complex bureaucratic procedures, plethora of rules and regulations and excessive adherence to them, misuse of law in favor of the privileged, bypassing established norms for personal gains, lack of will to perform better, political interference in administration-all these are not only impede the development process but also render the public institutions low responsive. 2.4.7. Equity and Inclusiveness Equity and inclusiveness mean the assurance that everyone has a stake in the society and not excluded from it. It requires that all groups, particularly the vulnerable and minority have opportunities to improve or maintain their wellbeing. 2.4.8 Consensus Oriented In a certain culture, there exists a variety of actors and viewpoints. In order to develop a broad social consensus on what is desirable for the entire community and how this might be done, good governance necessitates mediation of the various interests in society. Furthermore, it necessitates a long-term, comprehensive viewpoint on what is required for sustainable human development and how to accomplish its objectives. Understanding the past, present, and future of a particular civilization or group requires knowledge of its historical, cultural, and social settings. Finally, the parameters of good governance can be classified as institutional parameters such as independent election commission, effectiveness of parliament, efficient bureaucracy , independent judiciary, strong and autonomous local government, freedom of press media, strong and vigorous civil servant, independent and effective anticorruption agencies and non-institutional parameters such as legitimacy of political system, mass participation in the polity, dynamic and efficient leadership, responsiveness of government, accountability and transparency of government, efficient administrative system, respect for the rule of law, decentralization, equal distribution of the state’s wealth and resource, respect for human rights, women empowerment, sound education system and combating corruption. 2.5. Good Enough Government The concept of ‘Good Governance’ denotes a utopian society, which, according to many can hardly achieved. It actually alludes to a radical change in every sphere of the public sector. Many even argue that a magic wand is needed in order to achieve good governance. In this regard, Grindle ( 2004) states that “Getting good governance calls for improvements that touch virtually all aspects of the public sector from institutions that set the rules of the game for economic and political interaction, to decision-making structures that determine priorities among public problems and allocate resources to respond to them, to organizations that manage administrative systems and deliver goods and services to citizens, to human resources that staff government bureaucracies, to the interface of officials and citizens in political and bureaucratic arenas ---Not surprisingly, advocating good governance raises a host of questions about what needs to be done, when it needs to be done , and how needs to be done.” As a matter of fact, it is often contended that good governance agenda, particularly in a developing country, is unrealistically long and growing longer over time (Grindle 2004). Hence emerges the concept of ‘Good Enough governance’. The concept calls of actions towards betterment of governance scenario taking into considerations the contextual factors or realities of a country. Good enough governance suggests not all governance deficits need to be tackled at once and those institutions and capacity buildings are products of time; governance achievements can also be reserved. Good enough governance means that interventions thought to contribute to the ends of economic and political development need to be questioned, prioritized, and made relevant to the conditions of individual countries. They need to be assessed in the light of historical evidence, sequence and timing and they should be selected carefully in terms of their contributions to particular end such poverty reduction and democracy (Grindle, 2007). The concept of good enough governance is a practical approach to the ground realities of a country. Instead of bringing changes in all aspects of public sector, it suggests that actions aiming changes should be prioritized taking into consideration the realities of the context. It advocates gradual changes in the public sector. However, though good enough governance may appear feasible for the sake © 2023 IJNRD | Volume 8, Issue 5 May 2023 | ISSN: 2456-4184 | IJNRD.ORG IJNRD2305026 International Journal of Novel Research and Development (www.ijnrd.org) a212 of economic and political developments in the short run, the target should always be established good governance in the country. 3. Good Governance Scenario in Bangladesh The Bangladesh constitution (Article:10 ensure the attainment of a just and egalitarian society, free from exploitation of man by man ; Article 11: guarantee fundamental human rights and freedom and respect human dignity and worth of the human person ; Article 19: create equal opportunity for all citizens ; remove social and economic inequality between man and man and ensure equitable distribution of wealth and opportunities among citizens ; ensure participation of women in all spheres of national life; Article 20: pay everyone for his work on the basis of the principle “from each according to his abilities to each according to his work”; and allow none to be able to enjoy unearned incomes ) provides for all the elements necessary for good governance. Bangladesh is yet to fully develop right policies, environment, institutions, capacity and practice of good governance. However, the scope of good governance in Bangladesh is broad and encompasses a wide range of areas that are essential for building a strong, sustainable, and prosperous society. The scope of good governance in Bangladesh can be divided the following broad areas: Political Governance: Good governance in Bangladesh includes having a strong and independent judiciary, a well-functioning parliament, and an effective executive branch that is accountable to the people. This also includes ensuring a level playing field for all political parties and maintaining a free and fair electoral process. Administrative Governance: Good governance in Bangladesh requires having a professional, competent, and transparent bureaucracy that is responsive to the needs of the citizens. This includes promoting merit-based recruitment, ensuring accountability, and providing opportunities for citizen participation in decision-making. Economic Governance: Good governance in Bangladesh also includes promoting economic growth and development through sound macroeconomic policies, promoting private sector investment, and ensuring equitable distribution of resources. Social Governance: Good governance in Bangladesh involves providing quality education, healthcare, and social services to all citizens, promoting gender equality and social inclusion, and ensuring the protection of human rights. Environmental Governance: Good governance in Bangladesh also involves addressing environmental challenges such as pollution, climate change, and deforestation through effective policies and regulations, promoting sustainable development practices, and raising public awareness about environmental issues. Bangladesh has made significant progress in improving its governance system in recent years, but there are still challenges and areas for improvement. The overview of the good governance scenario in Bangladesh are stated below: Stable Economic Growth: Bangladesh has achieved remarkable economic growth in recent years, driven by robust private sector investment, a growing export sector, and significant improvements in infrastructure development. The government has implemented a range of policies to promote economic growth, including reducing red tape, improving the business environment, and investing in infrastructure development. Political Stability in Bangladesh: Bangladesh has made significant progress in terms of political stability in recent years. The government has taken different measures to address political polarization and promote democratic institutions. However, there are still challenges, including the lack of an effective opposition, which undermines the democratic process. Anti-Corruption Measures in Bangladesh: The government of Bangladesh has taken measures to combat corruption, including enacting anti-corruption laws, establishing an independent anti-corruption commission, and promoting transparency and accountability in governance. However, corruption remains a significant challenge, particularly in the public sector. Enhance Citizen Participation: The government has taken measures to promote citizen participation in governance, including through decentralization and local government reforms. However, there are still challenges in this area, including limited citizen engagement in decision-making processes and a lack of awareness of citizens' rights. Development of Socio-economic infrastructure: The government has made significant investments in infrastructure development, including in transportation, energy, and telecommunications. This has helped to © 2023 IJNRD | Volume 8, Issue 5 May 2023 | ISSN: 2456-4184 | IJNRD.ORG IJNRD2305026 International Journal of Novel Research and Development (www.ijnrd.org) a213 improve connectivity and access to services, particularly in rural areas. However, there are still significant infrastructure gaps that need to be addressed. Expansion of International Cooperation: Bangladesh has benefited from international cooperation and partnerships to address governance challenges, including through technical assistance, knowledge sharing, and financial support. Evaluated from the concept of good governance in Bangladesh shows all the symptoms of an underdeveloped polity. Democracy is still fragile and has not been institutionalized. Politics is still a property of a few belongings to affluent section of society. Elections are still expensive and complex affairs and common people are none but passive voters. Power and authority are concentrated at the top echelons of government. Corruption has become so endemic that it has engulfed the entire society. A weak legislature, limited rule of parliament, the dominant executive, dysfunctional bureaucracy, a poorly performed judiciary, politicization of public offices, unsatisfactory law and order situation, lack of accountability and transparency in the state affairs have been impediments to effective governance in Bangladesh. According to Hossain Zillur Rahman, the basic problems of good governance in Bangladesh are as follows: A tendency towards encyclopedic wish list in agenda formulation; An insufficient appreciation of the institutional and political realities through which reform initiatives have to be carried forward; Over focusing on what does not work while ignoring which does work; and A lack of clarity as to where the governance agenda best interface with poverty reduction goals. Although Bangladesh has made significant progress in improving its governance system, there are still challenges and areas for improvement in governance system in Bangladesh. In this perspective, government of Bangladesh needs to continue to invest in institutional strengthening, capacity-building, and reforms to promote transparency, accountability, and citizen participation in governance system. 1. Problems of Good governance in Bangladesh Bangladesh has made significant progress in improving its governance system. Despite significant there are still several challenges and problems for proper implementation of good governance that need to be addressed. In Bangladesh we have faced a lot of problems in the path of good governance such as lack of free and fair election; problem of consensus building; ineffectiveness of parliament; lack of accountability and transparency; absence of rule of law; absence of real independence of judiciary ; lack of institutionalization of political parties; fierce elite competition; absence of worthy leadership; confrontational political culture; inefficient bureaucracy and public administration; biased press and media; fragile local government system; social inequalities and cleavages; political violence and rampant corruption. In briefly, we can identify the main obstacles of good governance in Bangladesh are as follows: ignorance; corruption; politicization of administration; weak democratic institutions; ineffective parliament; absence of the rule of law and absence of meritocracy etc. However, the key problems in good governance in Bangladesh can be classified as: Political Polarization: Political polarization is a significant problem in Bangladesh, which has resulted in a lack of consensus-building and constructive dialogue among the political parties. This has led to frequent political violence, strikes, and demonstrations that undermine stability and disrupt the economy as well as overall progress and image of the country. Corruption: Corruption remains a significant problem in Bangladesh, particularly in the public sector. Despite efforts to combat corruption, it remains a major obstacle to good governance and sustainable development in Bangladesh. Corruption not only hinders economic growth, but also weakens public trust in government institutions, and undermines social cohesion respectively. Weak Judicial System: Bangladesh's judicial system is often criticized for its slow pace, lack of transparency, and political interference. This undermines citizens' faith in the rule of law and impairs access to justice. There is also a shortage of qualified judges and legal professionals in the country respectively. © 2023 IJNRD | Volume 8, Issue 5 May 2023 | ISSN: 2456-4184 | IJNRD.ORG IJNRD2305026 International Journal of Novel Research and Development (www.ijnrd.org) a214 Limited Citizen Participation: While the government has made efforts to encourage citizen participation in governance, including through decentralization and local government reforms, there are still significant challenges in this area. Many citizens remain uninformed about their rights and how to engage with the government, and there are limited opportunities for citizens to provide feedback on government policies and programs in Bangladesh. Limited Capacity: There are several areas of governance where Bangladesh lacks sufficient capacity, including in public financial management, infrastructure development, and public service delivery. This limits the effectiveness of government institutions and their ability to address the needs of citizens. In order to proper implementation of good governance in Bangladesh, we need to addressing these problems which will require a sustained effort from all stakeholders, including the government, civil society, private sector, and citizens. It will also require strong political will, effective policies and regulations, and investment in capacity-building and infrastructure development. 5. Prospects of Good Governance in Bangladesh Despite certain issues with the good governance system in Bangladesh, there are great opportunities for its effective implementation in our nation. The government of Bangladesh has undertaken a number of reforms initiatives to improve governance and ensure transparency, accountability, and citizen participation in governance. By taking above initiative, Bangladesh has made significant advancements in social development, women's empowerment, raising the standard of life, and guaranteeing inclusive and equitable growth. The key reforms initiatives for ensuring good governance in Bangladesh are as follows: Digital / E- Governance: The government has introduced various digital governance initiatives to improve public service delivery and reduce corruption. This includes the introduction of e-governance, digital payment systems, and online service delivery. Decentralization and Local Government Reforms initiative: The government has undertaken decentralization and local government reforms initiative to promote citizen participation in governance and improve public service delivery at the local level. This includes the devolution of power to local government bodies and the introduction of participatory budgeting system in Bangladesh. Anti-Corruption Measures for ensuring good governance: The government has taken strong measures to combat against corruption, including enacting and enforcing anti-corruption laws, establishing independent anti-corruption commission, and promoting transparency and accountability in governance system in Bangladesh. Judicial Reforms initiative for Bangladesh: The government has undertaken judicial reforms to strengthen the judicial system in Bangladesh, including increasing the number of qualified judges, improving access to justice, and reducing political interference in the judiciary system. Electoral Reforms initiative in Bangladesh: The government has introduced various electoral reforms to improve the fairness and transparency of elections. This includes the introduction of electronic voting machines (EVM), measures to reduce electoral violence, and the establishment of an independent election commission. Public Financial Management Reforms: The government has undertaken public financial management reforms to improve budget formulation, execution, and monitoring. This includes the introduction of a medium-term budget framework (MTBF), the establishment of a treasury single account, and the adoption of international accounting standards. Education Reforms initiative: The government has introduced various educational reforms to improve the quality of education and ensure greater access to education for all. This includes the introduction of a national education policy, the expansion of primary education, and the establishment of new universities. Besides the above reforms initiative, government of Bangladesh has taken following administrative reforms for ensuring good governance such as: National Integrity Strategy (NIS); Government Performance Measurement System(GPMS); Annual Performance Agreement (APA); Right to Information Act 2009 and RTI implementation; Citizens Charter; Grievance Redress System (GRS); E-governance and Public Service Innovation; Public Hearing; Service Process Simplification(SPS) and Use of social media such as Facebook. Overall, these reforms initiatives are helping to promote good governance in Bangladesh and ensure that © 2023 IJNRD | Volume 8, Issue 5 May 2023 | ISSN: 2456-4184 | IJNRD.ORG IJNRD2305026 International Journal of Novel Research and Development (www.ijnrd.org) a215 government policies and programs are responsive to citizens' needs and priorities basis. However, there is still more work to be done, and the government needs to continue to invest in institutional strengthening, capacity-building, and reforms to promote transparency, accountability, and citizen participation in governance. 6. Policy Suggestions for Ensuring Good Governance in Bangladesh In order to solve the existing problems of good governance in Bangladesh, we need collective effort from all stakeholders, including the government, civil society, private sector, and citizens. To address the problems of good governance in Bangladesh we can take the following steps: Political Reforms: Political reforms are needed to strengthen democratic institutions, promote consensus-building, and reduce political polarization. This could include electoral reforms, political party financing regulations, and reforms to the parliamentary system in Bangladesh. Judicial Reforms: Reforms are needed to strengthen the judicial system, including increasing the number of qualified judges, improving access to justice, and reducing political interference in the judiciary. Anti-Corruption Measures: The government needs to take strong measures to combat corruption, including enacting and enforcing anti-corruption laws, establishing independent anti-corruption agencies, and promoting transparency and accountability in governance. Citizen Participation: The government needs to promote greater citizen participation in governance, including through promoting access to information, creating opportunities for public consultation, and engaging with civil society organizations. Capacity-Building: Capacity-building is needed to improve public financial management, infrastructure development, and service delivery. This could include improving the skills and competencies of government officials, promoting innovation and technology adoption, and investing in infrastructure development. International Cooperation: Bangladesh could benefit from international cooperation and partnerships to address governance challenges, including through promoting knowledge sharing, technical assistance, and financial support. Besides the above measures, the following policies are recommended for ensuring good governance in Bangladesh: Law and order situation should be improved further Administrative neutrality should be protected and improved Administration should be kept free from political pressure Size of government should be reduced and private sector should be expanded Government should be sincere in removing the political deadlock from the parliament Political unrest should be removed to attract foreign direct investors Rules of business should be amended giving more administrative and financial powers to department and corporations Poor people should be brought under the safety net programmed and implementation life cycle based social safety net program Public servants should be trained adequately in administrative ethics and morality Administration should be decentralized and making district as the centre of the development and planning Rule of law should be applied strictly There should be vision for a patriotic and incorruptible bureaucracy Nationalism, religious and cultural values should be protected All promotions should be merit and quality based Measures should be taken to remove and sustainable natural disaster management NGO programmed should be closely monitored Education should be given highest priority Poverty alleviation programmed like Grameen model should be encouraged Strict accountability will be ensured everywhere with transparency in all dealings Finally, solving the problems of good governance in Bangladesh will require a multi steps approach that addresses the root causes of the problems and promotes sustainable solutions for the country. For these reasons © 2023 IJNRD | Volume 8, Issue 5 May 2023 | ISSN: 2456-4184 | IJNRD.ORG IJNRD2305026 International Journal of Novel Research and Development (www.ijnrd.org) a216 we need to build a strong commitment from all stakeholders to work together towards a common goal of building a more democratic, transparent, and prosperous society. Conclusion Good governance is a very important for the overall development of a poor and developing country. But establishing good governance is not easy task. In fact, the desire for good governance is a craving for an ideal situation. Very few countries and societies have come close to achieving good governance in its totally. There are a host of social, cultural, economic, administrative and legal variables which determine the nature and shape of governance in a country. The government of Bangladesh, in order to establish good governance, needs to address issues like corruption, political interference in administration, misuse of power and resources, lack of political and bureaucratic accountability, non-transparent administration, less participative decision-making process and so on. Good governance in Bangladesh cannot be achieved unless there is an urgent, focused and clearly scheduled cultural shift in both political and bureaucratic thinking and doing. To ensure socio-economic emancipation of any poor nation, all major political parties, civil servants, academics and private sector have to come forward to take decisions to ensure good governance. Finally, in order to proper implementation of good governance in Bangladesh, we need to addressing these problems which will require a sustained effort from all stakeholders, including the government, civil society, private sector, and citizens. It will also require strong political will, effective policies and regulations, and investment in capacity-building and infrastructure development. References Ahmed Imtiaz. (2007). Governance and the international development community: Making sense of the Bangladesh experience. Contemporary South Asia Vol. No 8: 295-309. Ara, Fardaus and Khan, Md. Mostafizur Rahman. (2006). Good Governance: Bangladesh Perspective. The Social Sciences 1(2), Dept. of Public Administration, Rajshahi University, Bangladesh. Cornwall, Andrea. (2002). Beneficiary, Consumer, and Citizen: Perspective on Poverty Reduction. Sida Studies no.2. Grindle, Merilee S. (2004). Good Enough Governance: Poverty Reduction and Reform in Developing Countries in Governance. An International Journal of Policy, Administration, and Institution, 17(4): 525-548. Grindle, Merilee S. (2007). Good Enough Governance Revisited. Development Policy Review, 25(5): 553-574. Hyden, G., Court, J. and Mease, K. (2004). Making Sense of Governance: Empirical Evidence from Sixteen Developing Countries. Boulder, CO: Lynne Rienner. Edgar, Laura, Claire Marshall, and Michael Bassett. (2006). Partnerships: Putting good governance principles in practice. Institute on Governance (IOG) Ottawa, Ontario, Canada. Haque, M. Shamsul. (2000). Significance of Accountability under the New Approach to Public Governance. International Review of Administrative Sciences, 66(4): 599-617. Khan, Mizanur R. (1998). Sustainable Development: What does it mean to the world’s poor? Bangladesh Institute of International and Strategic Studies, 19(1). Muhith, A.M.A. (2000). Issues of Governance in Bangladesh. Mowla Brothers, Dhaka. Paul, Samuel. (1992). Accountability in Public Services: Exit, Voice and Control. World Development, 20(7):1047-1060. Paul, Samuel. (1990). Poverty Alleviation and Participation: The Case of Government Exit, Voice and Control. World Development, 20(7):1047-1060. Rhodes, R.A.W. (1996). The New Governance: Governing without Government. Political Studies, XLV. Richardson, A. (1993). Participation, Poutledge and Kegan Paul, London. Rizvi, Gowher. (2008). Innovations in Government. Ash Institute for Governance and Innovation, Kennedy School of Government, Harvard University. Rowe, Gene and Frewer, Lynn J. (2004). Evaluating Public-Participation Exercises: A Research Agenda. Science, Technology and Human Values, 29(4) , Sage Publications, Inc. Samad, Muhammad. (2000). Participation of the Rural Poor in Government and NGO Programs. Mowla Brothers, Dhaka. Samaratunge, Ramanie, Alam, Quamrul and Teicher, Julian. (2008). Public sector reforms, 10(1):101-126. Governance for Sustainable Human Development: A UNDP Policy Document, 1997 Wolfe, Joel D. (1985). A Defense of Participatory Democracy. The Review of Politics,47(3), Cambridge Press for the University of Notre Dame du lac. |
8656 | https://math.mit.edu/classes/18.783/2022/lectures.html | 18.783 - Elliptic Curves
Home | Syllabus | Lectures | Problem Sets | Links
SCHEDULE
| # | Date | Topic (references) | Materials |
--- --- |
| 1 | 1/31 | Introduction | notes, slides |
| 2 | 2/2 | The group law, Weierstrass and Edwards equations (Washington 2.1-3, 2.6.3, Bernstein-Lange 2007, Bernstein-Lange 2010) | notes, worksheet 1, worksheet 2 |
| 3 | 2/7 | Finite field arithmetic (Modern Computer Algebra 3.2, 8.1-4, 9.1, 11.1, 14.2-6, HEHCC 9.1-2, Rabin 1980, Cantor-Zassenhaus 1981) | notes, slides, worksheet,bonus video |
| 4 | 2/9 | Isogenies (Washington 2.9, Silverman III.4) | notes, slides |
| 5 | 2/14 | Isogeny kernels and division polynomials (Washington 3.2, 12.3, Silverman III.4) | notes, slides, worksheet |
| 6 | 2/16 | Endomorphism rings (Washington 4.2, , Silverman III.6) | notes, slides |
| 7 | 2/22 | Hasse's Theorem, point counting (Washington 4.3) | notes, slides |
| 8 | 2/23 | Schoof's algorithm (Washington 4.2, 4.5, Schoof 1985) | notes, slides, worksheet |
| 9 | 2/28 | Generic algorithms for the discrete logarithm problem (Washington 5.2, Pohlig-Hellman 1978, Pollard 1978, Shor 1994, Shoup 1997) | notes, slides |
| 10 | 3/2 | Index calculus, smooth numbers, factoring integers (Washington 5.1, 7.1, Granville, Lenstra) | notes, slides, worksheet 1, worksheet 2, worksheet 3 |
| 11 | 3/7 | Elliptic curve primality proving (ECPP) (Washington 7.2, Goldwasser-Kilian, Pomerance) | notes, slides |
| 12 | 3/9 | Endomorphism algebras (Silverman III.9) | notes, slides |
| 13 | 3/14 | Ordinary and supersingular curves (Silverman III.1,V, Washington 2.7, 4.6) | notes, slides |
| 14 | 3/16 | Elliptic curves over C (part I) (Cox Sec. 10, Silverman VI.2-3, Washington 9.1-2) | notes, slides |
| 15 | 3/28 | Elliptic curves over C (part II) (Cox Sec. 10-11, Silverman VI.4-5, Washington 9.2-3) | notes, slides |
| 16 | 3/30 | Complex multiplication (CM) (Cox Sec. 11, Silverman VI.5, Washington 9.3) | notes |
| 17 | 4/4 | The CM torsor (Cox Sec. 7, Silverman (advanced topics) II.1.1) | notes |
| 18 | 4/11 | Riemann surfaces and modular curves (Silverman (advanced topics) I.2, Milne V.1) | slides, notes |
| 19 | 4/13 | The modular equation (Cox Sec. 11, Milne V.2, Washington pp. 273-274) | slides, notes |
| 20 | 4/20 | The Hilbert class polynomial (Cox Sec. 8, 11) | slides, notes |
| 21 | 4/25 | Ring class fields and the CM method (Cox 8, 11) | slides, notes |
| 22 | 4/25 | Isogeny volcanoes (Sutherland) | slides, notes |
| 23 | 5/2 | The Weil pairing (Miller, Washington 11, Silverman III.8) | slides, notes |
| 24 | 5/4 | Modular forms and L-functions (Milne V.3-4) | slides, notes |
| 25 | 5/9 | Fermat's last theorem (Milne V.7-9, Washington 15, Cornell-Stevens-Silverman I) | slides, notes |
Massachusetts Institute of Technology
Department of Mathematics |
8657 | https://physics.stackexchange.com/questions/395851/left-or-right-hand-rule | electromagnetism - Left or Right hand rule? - Physics Stack Exchange
Join Physics
By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy.
Sign up with Google
OR
Email
Password
Sign up
Already have an account? Log in
Skip to main content
Stack Exchange Network
Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
Visit Stack Exchange
Loading…
Tour Start here for a quick overview of the site
Help Center Detailed answers to any questions you might have
Meta Discuss the workings and policies of this site
About Us Learn more about Stack Overflow the company, and our products
current community
Physics helpchat
Physics Meta
your communities
Sign up or log in to customize your list.
more stack exchange communities
company blog
Log in
Sign up
Home
Questions
Unanswered
AI Assist Labs
Tags
Chat
Users
Teams
Ask questions, find answers and collaborate at work with Stack Overflow for Teams.
Try Teams for freeExplore Teams
3. Teams
4. Ask questions, find answers and collaborate at work with Stack Overflow for Teams. Explore Teams
Teams
Q&A for work
Connect and share knowledge within a single location that is structured and easy to search.
Learn more about Teams
Hang on, you can't upvote just yet.
You'll need to complete a few actions and gain 15 reputation points before being able to upvote. Upvoting indicates when questions and answers are useful. What's reputation and how do I get it?
Instead, you can save this post to reference later.
Save this post for later Not now
Thanks for your vote!
You now have 5 free votes weekly.
Free votes
count toward the total vote score
does not give reputation to the author
Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation.
Got it!Go to help center to learn more
Left or Right hand rule?
Ask Question
Asked 7 years, 6 months ago
Modified7 years, 6 months ago
Viewed 29k times
This question shows research effort; it is useful and clear
2
Save this question.
Show activity on this post.
I'm currently looking at the right hand rule for magnetism, and have got myself a bit mixed up. I was originally taught to use the left hand rule, with the acronym FBI: Force for the thumb, B field for the first finger, and current for the middle finger.
Since current flows in the opposite direction to electrons, shouldn't this work for magnetic forces on moving partciles? Yet all my textbooks use the right hand rule.
A summary of which rule is appropriate for which case would be ideal!
electromagnetism
electricity
magnetic-fields
Share
Share a link to this question
Copy linkCC BY-SA 3.0
Cite
Improve this question
Follow
Follow this question to receive notifications
asked Mar 26, 2018 at 17:02
Noah PNoah P
1,838 3 3 gold badges 19 19 silver badges 32 32 bronze badges
1
1 Your left hand rule works for negatively charged particles. For positively charged particles or conventional current use the right hand rule.garyp –garyp 2018-03-26 17:14:50 +00:00 Commented Mar 26, 2018 at 17:14
Add a comment|
3 Answers 3
Sorted by: Reset to default
This answer is useful
4
Save this answer.
Show activity on this post.
Fleming's Right Hand Rule:
It is for electromagnetic induction, that is, current being induced by a varying magnetic flux. An application is an AC generator.
Your "FBI"acronym will work appropriately. Just remember that I I is the induced current.
Fleming's Left Hand Rule:
It is for Force on a charged particle (or conductor) placed in a magnetic field. An application is a DC motor.
"FBI" will work here too. F F is the force on the charged particle or the conductor.
Right Hand Palm Rule:
Used for finding direction of magnetic field around a conductor.
If you point your thumb towards the direction of flow of current, then the direction of curl of the other fingers will indicate the direction of magnetic field lines.
Share
Share a link to this answer
Copy linkCC BY-SA 3.0
Cite
Improve this answer
Follow
Follow this answer to receive notifications
edited Mar 26, 2018 at 19:40
answered Mar 26, 2018 at 19:34
Wrichik BasuWrichik Basu
2,930 6 6 gold badges 30 30 silver badges 41 41 bronze badges
Add a comment|
This answer is useful
0
Save this answer.
Show activity on this post.
Both your left hand rule and the more common right hand rule are correct.
Share
Share a link to this answer
Copy linkCC BY-SA 3.0
Cite
Improve this answer
Follow
Follow this answer to receive notifications
answered Mar 26, 2018 at 17:09
my2ctsmy2cts
27.9k 2 2 gold badges 23 23 silver badges 79 79 bronze badges
Add a comment|
This answer is useful
0
Save this answer.
Show activity on this post.
You can use any of the rules.The Right Hand Rule is generally applied on a charge moving in magnetic field while Flemings Left Hand Rule is used in case of motors and current carrying conductors in magnetic field.
Share
Share a link to this answer
Copy linkCC BY-SA 3.0
Cite
Improve this answer
Follow
Follow this answer to receive notifications
answered Mar 26, 2018 at 17:15
Jasgeet SinghJasgeet Singh
77 4 4 silver badges 14 14 bronze badges
Add a comment|
Protected question. To answer this question, you need to have at least 10 reputation on this site (not counting the association bonus). The reputation requirement helps protect this question from spam and non-answer activity.
Start asking to get answers
Find the answer to your question by asking.
Ask question
Explore related questions
electromagnetism
electricity
magnetic-fields
See similar questions with these tags.
Featured on Meta
Introducing a new proactive anti-spam measure
Spevacus has joined us as a Community Manager
stackoverflow.ai - rebuilt for attribution
Community Asks Sprint Announcement - September 2025
Related
9Which mathematical operation does the right hand rule for current come from?
3About Right-Hand-Rule and Cross Projuct
0Right hand rule confusion
0Am I wrong about the right hand grip rule?
1What's the difference between flemming's left hand rule and the right hand rule of magnetism?
0Direction of B B-field from right hand rule for electromagnetic wave
2Misunderstanding the right hand screw rule for magnetic fields
2When to use the different formats of the right hand rule?
Hot Network Questions
"Unexpected"-type comic story. Aboard a space ark/colony ship. Everyone's a vampire/werewolf
With with auto-generated local variables
How to use \zcref to get black text Equation?
Exchange a file in a zip file quickly
Direct train from Rotterdam to Lille Europe
в ответе meaning in context
Where is the first repetition in the cumulative hierarchy up to elementary equivalence?
Bypassing C64's PETSCII to screen code mapping
What happens if you miss cruise ship deadline at private island?
Calculating the node voltage
How exactly are random assignments of cases to US Federal Judges implemented? Who ensures randomness? Are there laws regulating how it should be done?
How to locate a leak in an irrigation system?
How many stars is possible to obtain in your savefile?
Is existence always locational?
The rule of necessitation seems utterly unreasonable
For every second-order formula, is there a first-order formula equivalent to it by reification?
Does a Linux console change color when it crashes?
Lingering odor presumably from bad chicken
Identifying a movie where a man relives the same day
Origin of Australian slang exclamation "struth" meaning greatly surprised
Passengers on a flight vote on the destination, "It's democracy!"
What "real mistakes" exist in the Messier catalog?
Non-degeneracy of wedge product in cohomology
What is a "non-reversible filter"?
Question feed
Subscribe to RSS
Question feed
To subscribe to this RSS feed, copy and paste this URL into your RSS reader.
Why are you flagging this comment?
It contains harassment, bigotry or abuse.
This comment attacks a person or group. Learn more in our Code of Conduct.
It's unfriendly or unkind.
This comment is rude or condescending. Learn more in our Code of Conduct.
Not needed.
This comment is not relevant to the post.
Enter at least 6 characters
Something else.
A problem not listed above. Try to be as specific as possible.
Enter at least 6 characters
Flag comment Cancel
You have 0 flags left today
Physics
Tour
Help
Chat
Contact
Feedback
Company
Stack Overflow
Teams
Advertising
Talent
About
Press
Legal
Privacy Policy
Terms of Service
Your Privacy Choices
Cookie Policy
Stack Exchange Network
Technology
Culture & recreation
Life & arts
Science
Professional
Business
API
Data
Blog
Facebook
Twitter
LinkedIn
Instagram
Site design / logo © 2025 Stack Exchange Inc; user contributions licensed under CC BY-SA. rev 2025.9.26.34547
By clicking “Accept all cookies”, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy.
Accept all cookies Necessary cookies only
Customize settings
Cookie Consent Preference Center
When you visit any of our websites, it may store or retrieve information on your browser, mostly in the form of cookies. This information might be about you, your preferences, or your device and is mostly used to make the site work as you expect it to. The information does not usually directly identify you, but it can give you a more personalized experience. Because we respect your right to privacy, you can choose not to allow some types of cookies. Click on the different category headings to find out more and manage your preferences. Please note, blocking some types of cookies may impact your experience of the site and the services we are able to offer.
Cookie Policy
Accept all cookies
Manage Consent Preferences
Strictly Necessary Cookies
Always Active
These cookies are necessary for the website to function and cannot be switched off in our systems. They are usually only set in response to actions made by you which amount to a request for services, such as setting your privacy preferences, logging in or filling in forms. You can set your browser to block or alert you about these cookies, but some parts of the site will not then work. These cookies do not store any personally identifiable information.
Cookies Details
Performance Cookies
[x] Performance Cookies
These cookies allow us to count visits and traffic sources so we can measure and improve the performance of our site. They help us to know which pages are the most and least popular and see how visitors move around the site. All information these cookies collect is aggregated and therefore anonymous. If you do not allow these cookies we will not know when you have visited our site, and will not be able to monitor its performance.
Cookies Details
Functional Cookies
[x] Functional Cookies
These cookies enable the website to provide enhanced functionality and personalisation. They may be set by us or by third party providers whose services we have added to our pages. If you do not allow these cookies then some or all of these services may not function properly.
Cookies Details
Targeting Cookies
[x] Targeting Cookies
These cookies are used to make advertising messages more relevant to you and may be set through our site by us or by our advertising partners. They may be used to build a profile of your interests and show you relevant advertising on our site or on other sites. They do not store directly personal information, but are based on uniquely identifying your browser and internet device.
Cookies Details
Cookie List
Clear
[x] checkbox label label
Apply Cancel
Consent Leg.Interest
[x] checkbox label label
[x] checkbox label label
[x] checkbox label label
Necessary cookies only Confirm my choices |
8658 | https://www.mathbits.com/MathBits/TISection/Statistics2/normaldistribution.html | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |
--- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- ---
| Normal Probability Distribution | | | Chart prepared by the NY State Education Department | | | | A chart, such as that seen above, is often used when dealing with normal distribution questions. Understand that this chart shows only percentages that correspond to subdivisions up to one-half of one standard deviation. Percentages for other subdivisions require a statistical mathematical table or a graphing calculator. (See example 4) | | | | | --- | | | | --- | | The Normal Probability Distribution menu for the TI-84+ is found under DISTR (2nd VARS). NOTE: A mean of zero and a standard deviation of one are considered to be the default values for a normal distribution on the calculator, if you choose not to set these values. | | | | The Normal Distribution functions: #1: normalpdf pdf = Probability Density Function This function returns the probability of a single value of the random variable x. Use this to graph a normal curve.Using this function returns the y-coordinates of the normal curve. Syntax: normalpdf (x, mean, standard deviation) #2: normalcdf cdf = Cumulative Distribution Function This function returns the cumulative probability from zero up to some input value of the random variable x. Technically, it returns the percentage of area under a continuous distribution curve from negative infinity to the x. You can, however, set the lower bound. Syntax: normalcdf (lower bound, upper bound, mean, standard deviation) #3: invNorm( inv = Inverse Normal Probability Distribution FunctionThis function returns the x-value given the probability region to the left of the x-value. (0 < area < 1 must be true.) The inverse normal probability distribution function will find the precise value at a given percent based upon the mean and standard deviation. Syntax: invNorm (probability, mean, standard deviation) | | | | Example 1: Given a normal distribution of values for which the mean is 70 and the standard deviation is 4.5. Find: a) the probability that a value is between 65 and 80, inclusive. b) the probability that a value is greater than or equal to 75. c) the probability that a value is less than 62. d) the 90th percentile for this distribution. (answers will be rounded to the nearest thousandth) | | | | --- | | 1a: Find the probability that a value is between 65 and 80, inclusive. (This is accomplished by finding the probability of the cumulative interval from 65 to 80.) Syntax: normalcdf(lower bound, upper bound, mean, standard deviation) Answer: The probability is 85.361%. (The "PASTE" command simply means that the values that you typed after the template prompts will be "pasted" into the normalcdf() function and will appear on the home screen, as shown.) | | | 1b: Find the probability that a value is greater than or equal to 75. (The upper boundary in this problem will be positive infinity. The largest value the calculator can handle is 1 x 1099 . Type 10^99 to represent postive infinity, [or type 1 EE 99. Enter the EE by pressing 2nd, comma -- only one E will show on the screen.] Answer: The probability is 13.326%. | | | 1c: Find the probability that a value is less than 62. (The lower boundary in this problem will be negative infinity. The smallest value the calculator can handle is -1 x 1099 Type -10 ^ 99 [or type -1 EE 99. Enter the EE by pressing 2nd, comma -- only one E will show on the screen.] Answer: The probability is 3.772%. | | | 1d: Find the 90th percentile for this distribution. (Given a probability region to the left of a value (i.e., a percentile), determine the value using invNorm.) Answer: The x-value is 75.767. | | | | | Example 2: Graph and investigate the normal distribution curve where the mean is 0 and the standard deviation is 1. | | | | --- | | For graphing the normal distribution, choose normalpdf. The normalpdf (normal probability density function) is found under DISTR (2nd VARS) #1normalpdf(. | | | Go to the Y = menu. The parameters will be (variable, mean, standard deviation). | Adjust the WINDOW. You will have to set your own window. | | | Guideline is: Xmin = mean - 3 SD Xmax = mean + 3 SD Xscl = SD Ymin = 0 Ymax = 1/(2 SD) Yscl = 0 | | GRAPH. Using TRACE, simply type the desired x value and the point will be plotted. | | Investigate: What happens to the curve as the standard deviation increases? | | Double the standard deviation and see what happens to the graph. | When graphing 2 normal curves, the window will need to be adjusted. | | | Xmin = mean - 3 (largest SD)Xmax = mean + 3 (largest SD)Xscl = largest SDYmin = 0Ymax = 1/(2 smallest SD)Yscl = 0 | | Observe that as the standard deviation increases, the more spread out the graph becomes. | | Now, the area under the curve between particular values represents the probabilities of events occurring within that specific range. This area can be seen using the command ShadeNorm(. Before attempting ShadeNorm, be sure that Y1 = normalpdf(x, mean, standard deviation) is active and that the appropriate window has been set as indicated in the Window Guidelines in the previous section. To find ShadeNorm(, go to DISTR and right arrow to DRAW. Choose #1:ShadeNorm(. ShadeNorm (lower bound, upperbound, mean, standard deviation) Notice that the calculator defaults to a mean of 0 and a standard deviation of 1 unless changed by the user. Remember that there is approximately a 68% probability of a score falling within 1 standard deviation from the mean in a normally distributed set of values. The "area" value in this graph is indicating 0.682689 or approximately 68%. | | | Notice how this answer supports the percentage listed in the chart at the top of this page. | | | | | Example 3: Graph and examine a situation where the mean score is 46 and the standard deviation is 8.5 for a normally distributed set of data. | | | Go to Y= . | Adjust the window. | GRAPH. | | Examine: What is the probability of a value falling between the mean and the first standard deviation to the right? Answer: approximately 34% Notice how this percentage supports the information found in the chart at the top of this page for the percentage of information falling within one standard deviation above the mean. | |
| | | | Example 4: The lifetime of a battery is normally distributed with a mean life of 40 hours and a standard deviation of 1.2 hours. Find the probability that a randomly selected battery lasts longer than 42 hours. | |
| | |
--- |
| | | | The most accurate answer to this problem cannot be obtained by using the chart at the top of this page. One standard deviation above the mean would be located at 41.2 hours, 2 standard deviations would be at 42.4, and one and one-half standard deviations would be at 41.8 standard deviations. None of these locations corresponds to the needed 42 hours. We need more power than we have in the chart. Calculator to the rescue!! | |
| Let's get a visual look at the situation by examining the graph. The location of 42 hours indicates that our answer is going to be quite small. | Go to Y= . | GRAPH. |
| NOTE: If you do not graph first, you may not see the normal curve displayed in the answer due to the previously set window on the calculator. |
| Now: What is the probability of a value falling to the right of 42 hours (between 42 hours and infinity)? Answer: approximately 4.8% ShadeNorm( go to DISTR and right arrow to DRAW. Choose #1:ShadeNorm(. ShadeNorm (lower bound, upperbound, mean, standard deviation) ENTER. The percentage is read from the Area =. |
Finding Your Way Around TABLE of CONTENTS |
8659 | https://study.com/academy/lesson/analytic-geometry-definition-proofs-help.html | Analytic Geometry Origin, Formulas & Examples | Study.com
Log In
Sign Up
Menu
Plans
Courses
By Subject
College Courses
High School Courses
Middle School Courses
Elementary School Courses
By Subject
Arts
Business
Computer Science
Education & Teaching
English (ELA)
Foreign Language
Health & Medicine
History
Humanities
Math
Psychology
Science
Social Science
Subjects
Art
Business
Computer Science
Education & Teaching
English
Health & Medicine
History
Humanities
Math
Psychology
Science
Social Science
Art
Architecture
Art History
Design
Performing Arts
Visual Arts
Business
Accounting
Business Administration
Business Communication
Business Ethics
Business Intelligence
Business Law
Economics
Finance
Healthcare Administration
Human Resources
Information Technology
International Business
Operations Management
Real Estate
Sales & Marketing
Computer Science
Computer Engineering
Computer Programming
Cybersecurity
Data Science
Software
Education & Teaching
Education Law & Policy
Pedagogy & Teaching Strategies
Special & Specialized Education
Student Support in Education
Teaching English Language Learners
English
Grammar
Literature
Public Speaking
Reading
Vocabulary
Writing & Composition
Health & Medicine
Counseling & Therapy
Health
Medicine
Nursing
Nutrition
History
US History
World History
Humanities
Communication
Ethics
Foreign Languages
Philosophy
Religious Studies
Math
Algebra
Basic Math
Calculus
Geometry
Statistics
Trigonometry
Psychology
Clinical & Abnormal Psychology
Cognitive Science
Developmental Psychology
Educational Psychology
Organizational Psychology
Social Psychology
Science
Anatomy & Physiology
Astronomy
Biology
Chemistry
Earth Science
Engineering
Environmental Science
Physics
Scientific Research
Social Science
Anthropology
Criminal Justice
Geography
Law
Linguistics
Political Science
Sociology
Teachers
Teacher Certification
Teaching Resources and Curriculum
Skills Practice
Lesson Plans
Teacher Professional Development
For schools & districts
Certifications
Teacher Certification Exams
Nursing Exams
Real Estate Exams
Military Exams
Finance Exams
Human Resources Exams
Counseling & Social Work Exams
Allied Health & Medicine Exams
All Test Prep
Teacher Certification Exams
Praxis Test Prep
FTCE Test Prep
TExES Test Prep
CSET & CBEST Test Prep
All Teacher Certification Test Prep
Nursing Exams
NCLEX Test Prep
TEAS Test Prep
HESI Test Prep
All Nursing Test Prep
Real Estate Exams
Real Estate Sales
Real Estate Brokers
Real Estate Appraisals
All Real Estate Test Prep
Military Exams
ASVAB Test Prep
AFOQT Test Prep
All Military Test Prep
Finance Exams
SIE Test Prep
Series 6 Test Prep
Series 65 Test Prep
Series 66 Test Prep
Series 7 Test Prep
CPP Test Prep
CMA Test Prep
All Finance Test Prep
Human Resources Exams
SHRM Test Prep
PHR Test Prep
aPHR Test Prep
PHRi Test Prep
SPHR Test Prep
All HR Test Prep
Counseling & Social Work Exams
NCE Test Prep
NCMHCE Test Prep
CPCE Test Prep
ASWB Test Prep
CRC Test Prep
All Counseling & Social Work Test Prep
Allied Health & Medicine Exams
ASCP Test Prep
CNA Test Prep
CNS Test Prep
All Medical Test Prep
College Degrees
College Credit Courses
Partner Schools
Success Stories
Earn credit
Sign Up
Math/Geometry/Coordinate and Analytic Geometry
Analytic Geometry Origin, Formulas & Examples
Lesson
Quiz
Course
Lesson
Additional Info
Contributors: Danilo Marco, Shaun Ault, Kathryn Boddie
Instructors Danilo MarcoView bio
Shaun AultView bio
Kathryn BoddieView bio
Learn about analytic geometry. Trace the history of analytic geometry, identify key topics and formulas for analytic geometry, and discover examples of proofs. Updated: 11/21/2023
Create an account to begin studying today
Used by over 30 million students worldwide
Create an account
What is Analytic Geometry -------------------------
Analytic geometry is the algebraic study of geometry using a coordinate plane. Employing the concept of the Cartesian coordinate system, it established the relationship between geometry and algebra making way for the development of mathematics. It provides the identity of points, lines, and curves by defining their relationship through algebraic equations, describing measurements and locations of geometric figures, and providing proofs to certain existing theorems. Thus, analytic geometry is useful in both Euclidian and non-Euclidian geometry.
Origin of Analytic Geometry
Going back around 300 B.C. when Euclidian geometry was written by a Greek mathematician named Euclid of Alexandria. He is considered the father of Geometry. His studies focused on planes and solid geometrical shapes based on different theorems and axioms or postulates. In fact, there were a number of them developed at the time but they were not considered of great significance in the practical applications in various fields such as science, astronomical navigation, and studies, including earth exploration, military activities, and other applications requiring mathematical support in the quantitative analysis. In the 17th century, two French mathematicians named Rene Descartes and Pierre de Fermat developed significant studies in geometry. Their independent studies have remarkable contributions to the foundation of analytic geometry.
Rene Descartes
Rene Descartes is the one responsible for the initial development of coordinate geometry. It started from his curiosity to know the position of a fly on the tile-patterned ceiling of his room while looking up and resting in bed. Using one of the corners of the ceiling as the reference point and counting the tile pattern, he successfully described the position of the fly. By getting the horizontal and vertical distances of a point from the reference point described its location, which he called coordinates.
In 1637, Rene Descartes published his study entitled La Geometrie in an appendix of his book Discours de la Methode, earlier than the Introduction to Loci of Pierre de Fermat in 1679. Analytic geometry combined the analytical tools of algebra and powerful visual representations of geometry. This concept is also known as the coordinate system or the Cartesian coordinate system in honor of Rene Descartes. Thus, analytic geometry is also known as Cartesian geometry or coordinate geometry.
Lesson
Quiz
Course
25K views
Main Topics in Analytic Geometry --------------------------------
The application of analytic geometry with the aid of a Cartesian coordinate system starts with giving a unique identity to each point on a cartesian plane. As shown in Figure 1, the cartesian plane consists of two perpendicular lines called the axes; one is horizontal or the x-axis, and the other is vertical or the y-axis. The point at which these axes intersect at (0, 0), is called the origin. It is where the horizontal and vertical distances of a point are referenced. The horizontal distance from the origin is the x-coordinate while the vertical distance from the origin is the y-coordinate, collectively known as the coordinates of the point, represented by an ordered pair (x, y). The cartesian plane is divided into four quadrants. A point located to the right and above the origin has both positive values of x and y coordinates and can be found on the 1st Quadrant. A point to the left and above the origin has a negative x-coordinate and positive y-coordinate and can be found on the 2nd Quadrant. A point to the left and below of the origin has both negative values of x and y coordinates, and can be found on the 3rd Quadrant. A point to the right and below the origin has a positive x-coordinate and negative y-coordinate, and can be found on the 4th Quadrant. The x-coordinate is also called the abscissa while the y-coordinate is called the ordinate.
Figure 1. Cartesian coordinate system
Among the notable studies involving the concept of analytic geometry are the analysis of vectors, projections, and conics. A vector is a quantity with magnitude and direction. Vector analysis may be 2-dimensional or 3-dimensional in nature. 2-dimensional only utilizes the cartesian coordinate system involving x and y axes. While 3-dimensional utilizes a 3rd plane defined by the z-axis on a z-plane. Using analytic geometry, the relationship between two points identified by position vectors is defined by a 3rd vector called a displacement vector. Projection geometry involves geometric figures or images on a certain position and projected into another surface as used in conventional television and in the method of perspective drawings. The conic geometry includes the analysis of curves derived from the intersection of a cutting plane and a cone or a double cone. Conics describe the locus of points that varies in constant proportion. Examples of conics include the circle, ellipse, parabola, and hyperbola. Because of analytic geometry, each of these is given a unique identity and representation called an algebraic equation.
Formulas for Analytic Geometry ------------------------------
The main formulas used in analytic geometry give appropriate importance to the properties of geometric figures. Among these are the distance formula, midpoint formula, and slope formula. Other formulas in analytic geometry can be derived with the aid of these formulas.
Distance formula - This is useful in defining the separation of two given points. It can also be used in identifying the location of a point given the coordinate of the other point along with the distance between them. As shown in Figure 2, given the coordinates of two points A and B, the distance can be solved with this formula:
d=(x 2−x 1)2+(y 2−y 1)2
Figure 2. Distance between 2 points.
Midpoint formula - This is useful in identifying the position of a point in the middle of two given points as shown in Figure 3. Midpoint is noted as M (x, y). Here is the midpoint formula:
M=(x 1+x 2 2,y 1+y 2 2)
Figure 3. The midpoint of a line with the given endpoints.
Slope formula - Slope is the measure of the inclination of a line with respect to the x-axis or any horizontal line, as shown in Figure 4. Basically, the slope (m) of a line is defined as the ratio of the vertical displacement to the horizontal displacement, or simply the rise over the run. A horizontal line has a slope equal to zero. A vertical line has an infinite slope. Two lines are perpendicular lines when the product of their slopes is equal to -1, or their slopes are negative reciprocal to each other. Two lines are parallel if their slopes are equal. A line leaning to the right has a positive slope. While a line leaning to the left has a negative slope. Slope can be computed using this formula:
m=Δ(y)Δ(x)=(y 2−y 1 x 2−x 1)
Figure 4. The slope of a line.
Analytic Geometry Proof Examples --------------------------------
Let us try using these formulas in the following examples.
Example 1: Prove that triangle ABC in Figure 5 is an isosceles right triangle.
Figure 5. A triangle.
Proof: A triangle is isosceles if two of its sides are equal. Using the distance formula, we can compare the length of the sides of the triangle whose vertices have the coordinates A (-1, 1), B (1, 3), and C (3, 1).
A B=(1−(−1))2+(3−1)2=8=2.282 B C=(3−1)2+(1−3)2=8=2.282 A C=(3−(−1))2+(1−1)2=16=4
Since AB and BC are equal, triangle ABC is an isosceles triangle.
Proof: A right triangle has two sides perpendicular to each other. Using the slope formula, we can get the slope of each side. Recall that two lines are perpendicular if the product of the slopes is equal to -1 or the slopes are negative reciprocal to each other.
m A B=(y 2−y 1)(x 2−x 1)=(3−1)(1−(−1))=2 2=1 m B C=(y 2−y 1)(x 2−x 1)=(1−3)(3−1)=−2 2=−1 m A C=(y 2−y 1)(x 2−x 1)=(1−1)(3−(−1))=0 4=0
Since the product of the slopes of AB and BC is equal to -1, AB and BC are perpendicular to each other.
Therefore, triangle ABC is an isosceles right triangle.
Example 2: Prove that line AB is tangent to the circle at point C in Figure 6.
Figure 5. Tangent line to a circle.
Proof: A line is a tangent to a circle at a point if the line is perpendicular to the radius of the circle at that point. Using the slope formula, we can evaluate the slopes of the line AB and radius CO.
m A B=(y 2−y 1)(x 2−x 1)=(2−0)(1−(−1))=2 2=1 m C O=(y 2−y 1)(x 2−x 1)=(0−1)(1−0)=−1 1=−1
Since the slope of the radius (CO) is a negative reciprocal of the slope of line AB, and also, the product of the slopes is equal to -1, they are perpendicular at point C.
Therefore, line AB is tangent to the circle at point C.
Example 3: Prove that the diagonals of the polygon in Figure 6 are perpendicular to each other and bisect each other.
Figure 6. 2 diagonals of a given polygon.
Proof: Using the slope formula, evaluate if the lines are perpendicular to each other.
m A C=(y 2−y 1)(x 2−x 1)=(−2−0)(3−(−1))=−2 4=−1 2 m D B=(y 2−y 1)(x 2−x 1)=(1−(−3))(2−0)=4 2=2
Since the slopes of AC and DB are negative reciprocal to each other, they are perpendicular lines.
Proof: A line is bisected if it is divided into two equal segments. Using the midpoint formula, we can evaluate if point M is the midpoint of diagonals AC and DB.
M A C=(x 1+x 2 2,y 1+y 2 2)=((−1)+3 2,0+(−2)2)=(1,−1)M D B=(x 1+x 2 2,y 1+y 2 2)=(0+2 2,(−3)+1 2)=(1,−1)
Since the resulting coordinates using the midpoint formula are the same, (1, -1), and match the coordinates of M in Figure 6, point M is indeed the midpoint of diagonals AB and DB.
Therefore, the diagonals AC and DB are perpendicular to each other and bisect each other at point M.
Lesson Summary --------------
Analytic geometry is the algebraic study of geometry using the coordinate plane. It was discovered and developed by a French mathematician and philosopher named Rene Descartes. In 1637, he published La Geometrie in his book Discours de la Methode. His idea uses a coordinate system that combined the analytical tools of algebra and powerful visual representations of geometry, making it useful in both Euclidian and non-Euclidian geometry. In honor of him, it was called the Cartesian coordinate system. Thus, analytic geometry is also called Cartesian geometry or coordinate geometry. Its discovery is an important development in mathematics. It provided the foundations of modern mathematics.
There are important formulas developed and utilized in analytic geometry such as the distance formula which can be used to compute the distance between two points given the coordinates; the midpoint formula which can be used to locate a point in the middle of two given points; and, the slope formula which can be used to identify the inclination of the line with respect to the x-axis or any horizontal line. These formulas are very useful in the analysis of notable studies in geometry including vector analysis, projection, and conics. Vector analysis deals with quantities with both magnitude and direction known as vectors. Projection geometry is the study involving the projection of geometric figures such as a point on a line into another line or another plane. The conic geometry is the analysis of curves called conics which are derived from the intersection of a cutting plane and a cone or a double cone. It includes the circle, ellipse, parabola, and hyperbola.
Additional Info
The Main Idea of Analytic Geometry
Analytic geometry is the study of geometry on a grid called the coordinate plane, or xy-plane. Certain aspects of geometry can be handled very easily in the coordinate plane such as distance between points, slopes of lines, and finding midpoints.
A mathematician and philosopher by the name of René Descartes (1596-1650) was one of the first to come up with the idea of using a grid to study geometry. As the story goes, one night René was lying awake in bed looking up at the ceiling, which had a tile pattern. He observed a fly crawling along the tiles and realized that the position of the bug could be described uniquely by counting the number of tiles from the west wall (x coordinate), and the number of tiles from the south wall (y coordinate). Thus was born the coordinate plane, also known as the Cartesian plane after the Latinization of his name, Cartesius.
Rene Descartes
The Coordinate Plane
Imagine an empty field, stretching as far as the eye can see. Now suppose your friend hid a treasure somewhere in the field. The only clue he left is this: From your position, the treasure is 400 feet east and 200 feet north. With only two coordinates (and a compass!) you can easily find the treasure. On the coordinate plane, there is a point called the origin (often labeled O), which is much like your initial position in the field. Directions like east, west, north, and south are handled by x or y coordinates.
In the coordinate plane there are two perpendicular axes (lines) going through the origin: the x-axis (horizontal, or east-west), and the y-axis (vertical, or north-south). Every point P in the coordinate plane can now be located using two numbers, or coordinates, an x-coordinate and a y-coordinate. These coordinates are usually written as an ordered pair, (x, y). The value of x tells how far to go to the right of the origin, if x> 0, or left if x< 0. The value of y tells how far to go up (y> 0) or down (y< 0). The numbers x and y don't have to be whole numbers; they could be negative, fractional, or in general any real number.
If a point P has coordinates (x, y), then you can identify the point by the notation P(x, y).
Coordinate plane with a few points plotted
The Distance Formula
The distance between points P(x 1, y 1) and Q(x 2, y 2) on the coordinate plane can be found by the formula below.
Distance formula
For example, let's find the distance between P(-5, 1) and Q(3, -2).
The Midpoint Formula
Recall from geometry that the midpoint of a segment PQ is the point M halfway between P and Q. In analytic geometry, P and Q are specified by coordinates, and so the midpoint formula identifies point M by a pair of coordinates as well:
Let's find the coordinates of the midpoint between between P(-5, 1) and Q(3, -2).
M is the midpoint of segment PQ.
Slope
The slope of a line is a measure of how tilted the line is. Traditionally we use the lowercase m for slope. A line of slope m = 0 is completely horizontal. A line tilting upwards (as you look at it from left to right) has positive slope (m> 0), while a line tilting downward has negative slope (m< 0). The slope of a vertical line is undefined (or m = ∞).
Slope formula
For example, let's find the slope of line PQ, where P and Q are the same points as in the previous examples, P(-5, 1) and Q(3, -2).
Slope is important in geometry for identifying parallel lines or line segments. The line through PQ is parallel to the line through RS if and only if the slopes of PQ and RS are the same.
An Analytic Geometry Proof
We end this lesson with a couple short proofs incorporating formulas from analytic geometry.
Example proof 1
Prove that triangle ABC is isosceles.
Triangle ABC
Proof: A triangle is isosceles if two of the sides share the same length. We can use the distance formula to find the lengths of AB, AC, and BC. The coordinates of the points are A(0,4), B(7,5), and C(4,1).
Since AC = BC = 5, the triangle ABC is isosceles.
Example proof 2
Prove that PQRS is a parallelogram.
Is PQRS a parallelogram?
Proof: A quadrilateral is a parallelogram if and only if each pair of opposite sides is parallel. This is a job for slope! The coordinates of the points are P(-1, 2), Q(2, 0), R(-1, -1), and S(-4, 1).
Since the slopes of PQ and RS are the same, we know that PQ is parallel to RS. Similarly, since the slopes of QR and SP are the same, sides QR and SP are parallel. Thus, PQRS is indeed a parallelogram.
Lesson Summary
Analytic geometry uses the coordinate plane to study geometric concepts such as distance, midpoint, and slope. Every point on the plane is specified by two coordinates, (x, y). The formulas from analytic geometry may be used in proofs when the coordinates of points are given.
Register to view this lesson
Are you a student or a teacher?
I am a student I am a teacher
Unlock Your Education
See for yourself why 30 million people use Study.com
Become a Study.com member and start learning now.
Become a Member
Already a member? Log In
Back
Resources created by teachers for teachers
Over 30,000 video lessons& teaching resources‐all in one place.
Video lessons
Quizzes & Worksheets
Classroom Integration
Lesson Plans
I would definitely recommend Study.com to my colleagues. It’s like a teacher waved a magic wand and did the work for me. I feel like it’s a lifeline.
Jennifer B.
Teacher
Try it now
Back
Analytic Geometry Topics
Slope
Conic Sections
FAQ
What is the basic tool in analytic geometry?
The basic tool of Analytic geometry is the combination of analytical tools of algebra and the powerful visual representations of geometry which includes the simplest geometric figures like points, lines, planes, and curves.
How do you solve proofs in analytic geometry?
To solve proofs in analytic geometry, one must know the properties of the elements involved in the problem in order to properly use the appropriate formula.
What is one of the main formulas used in analytical geometry?
One of the main formulas used in analytic geometry is the distance formula. It can be used to identify the distance between points, between a point and a line, or between lines.
More Analytic Geometry Lessons
Geometry Assignment - Practicing Analytical Geometry
Identifying Lines in Space
Graphing in Three Dimensions: Method & Formula
Calculating the Angle Formed From Intersecting Lines
Coordinate Graph | Definition, Characteristics & Examples
Coordinate Plane Lesson Plan
Cylindrical & Spherical Coordinates | Conversion & Examples
Distance Between Perpendicular Lines | Overview & Examples
Distance Between Two Points | Formula, Calculation & Examples
Distance Formula Activities
Distance From a Point to a Line | Formula & Examples
Equidistant | Definition, Formula & Examples
Euclidean Distance | Calculation, Formula & Examples
Finding Perimeter & Areas Using Coordinates & Distance Formula
Finding the Distance Between Two Points in a Three Dimensional Space
Finding the Plane Equation From 3 Points | Overview & Examples
Geometric Boundary & Boundary Lines | Definition & Examples
Graphing Points & Lines in Three Dimensions
Graphing Polygons on the Coordinate Plane
How to Create a Coordinate Plane
How to Find the Distance Between Parallel Lines
How to Find the Distance Between Points on a Solid
How to Find the Distance between Two Planes
Locating Points on the Surface of the Earth
Locus of Points in Math | Definition, Formula & Examples
Math Coordinates Lesson Plan
Math Grid Overview, Uses & Examples
Midpoint Formula Activities & Games
Midpoint Formula Lesson Plan
Midpoint, Distance & Slope on Coordinated Plane | Formula & Graph
Ordered Pairs on the Coordinate Plane
Origin in Math | Definition, Graph & Examples
Overview of Graphing Shapes on the Coordinate Plane
Partitioning a Line Segment by a Ratio
Partitioning a Line Segment | Definition, Formula & Examples
Perpendicular Lines | Definition, Equation & Graph
Plotting Simple Figures on Coordinate Graphs
Slope Criteria for Parallel & Perpendicular Lines: Proof & Problems
Slopes of Parallel & Perpendicular Lines | Overview, Criteria & Examples
The Distance Formula Lesson Plan
Triangles in Coordinate Planes | Classifications & Examples
Using & Graphing Parallelograms in the Coordinate Plane
Using Slope to Prove or Disprove a Quadrilateral
Using Slopes to Prove Lines Are Parallel or Perpendicular
Vertical Line | Slope & Equation
View all (30 more)
Analytic Geometry Courses
Geometry MHT-CET: Practice & Study Guide View course
Test Prep MTEL Mathematics/Science (Middle School)(77) Study Guide and Test Prep View course
Geometry Indiana Core Assessments Mathematics: Test Prep & Study Guide View course
Test Prep OSAT Advanced Mathematics (111) Study Guide and Test Prep View course
Test Prep OSAT Middle Level Intermediate Mathematics (225) Study Guide and Test Prep View course
Create an account to start this course today
Used by over 30 million students worldwide
Create an account
All Coordinate and Analytic Geometry Topics
Coordinate Geometry
Start today. Try it now
Geometry: High School
14 chapters | 145 lessons
Ch 1. High School Geometry: Foundations of Geometry
Geometry Definition, History & Branches 4:36
Inductive vs. Deductive Reasoning in Geometry | Definition & Uses 4:59
Thales & Pythagoras: Early Contributions to Geometry 5:14
The Axiomatic System: Definition & Properties 5:17
Euclid's Axiomatic Geometry: Developments & Postulates 5:58
Undefined Terms in Geometry | Definition & Examples 5:23
Properties and Postulates of Geometric Figures 4:53
Algebraic Laws and Geometric Postulates 5:37
Ch 2. High School Geometry: Logic in...
Ch 3. High School Geometry: Introduction to...
Ch 4. High School Geometry: Properties of...
Ch 5. High School Geometry: Triangles,...
Ch 6. High School Geometry: Parallel Lines...
Ch 7. High School Geometry: Similar...
Ch 8. High School Geometry:...
Ch 9. High School Geometry: Circular Arcs and...
Ch 10. High School Geometry: Conic...
Ch 11. High School Geometry: Geometric...
Ch 12. High School Geometry: Analytical...
Ch 13. High School Geometry:...
Ch 14. High School Geometry: Introduction to...
Analytic Geometry Topics
Slope
Conic Sections
More Analytic Geometry Lessons
Geometry Assignment - Practicing Analytical Geometry
Identifying Lines in Space
Graphing in Three Dimensions: Method & Formula
Calculating the Angle Formed From Intersecting Lines
Coordinate Graph | Definition, Characteristics & Examples
Coordinate Plane Lesson Plan
Cylindrical & Spherical Coordinates | Conversion & Examples
Distance Between Perpendicular Lines | Overview & Examples
Distance Between Two Points | Formula, Calculation & Examples
Distance Formula Activities
Distance From a Point to a Line | Formula & Examples
Equidistant | Definition, Formula & Examples
Euclidean Distance | Calculation, Formula & Examples
Finding Perimeter & Areas Using Coordinates & Distance Formula
Finding the Distance Between Two Points in a Three Dimensional Space
Finding the Plane Equation From 3 Points | Overview & Examples
Geometric Boundary & Boundary Lines | Definition & Examples
Graphing Points & Lines in Three Dimensions
Graphing Polygons on the Coordinate Plane
How to Create a Coordinate Plane
How to Find the Distance Between Parallel Lines
How to Find the Distance Between Points on a Solid
How to Find the Distance between Two Planes
Locating Points on the Surface of the Earth
Locus of Points in Math | Definition, Formula & Examples
Math Coordinates Lesson Plan
Math Grid Overview, Uses & Examples
Midpoint Formula Activities & Games
Midpoint Formula Lesson Plan
Midpoint, Distance & Slope on Coordinated Plane | Formula & Graph
Ordered Pairs on the Coordinate Plane
Origin in Math | Definition, Graph & Examples
Overview of Graphing Shapes on the Coordinate Plane
Partitioning a Line Segment by a Ratio
Partitioning a Line Segment | Definition, Formula & Examples
Perpendicular Lines | Definition, Equation & Graph
Plotting Simple Figures on Coordinate Graphs
Slope Criteria for Parallel & Perpendicular Lines: Proof & Problems
Slopes of Parallel & Perpendicular Lines | Overview, Criteria & Examples
The Distance Formula Lesson Plan
Triangles in Coordinate Planes | Classifications & Examples
Using & Graphing Parallelograms in the Coordinate Plane
Using Slope to Prove or Disprove a Quadrilateral
Using Slopes to Prove Lines Are Parallel or Perpendicular
Vertical Line | Slope & Equation
View all (29 more)
Analytic Geometry Courses
Geometry MHT-CET: Practice & Study Guide View course
Test Prep MTEL Mathematics/Science (Middle School)(77) Study Guide and Test Prep View course
Geometry Indiana Core Assessments Mathematics: Test Prep & Study Guide View course
Test Prep OSAT Advanced Mathematics (111) Study Guide and Test Prep View course
Test Prep OSAT Middle Level Intermediate Mathematics (225) Study Guide and Test Prep View course
Like this lessonShare
Explore our library of over 88,000 lessons
Search
Browse
Browse by subject
College Courses
Business
English
Foreign Language
History
Humanities
Math
Science
Social Science
See All College Courses
High School Courses
AP
Common Core
GED
High School
See All High School Courses
Other Courses
College & Career Guidance Courses
College Placement Exams
Entrance Exams
General Test Prep
K-8 Courses
Skills Courses
Teacher Certification Exams
See All Other Courses
Upgrade to enroll×
Upgrade to Premium to enroll in Geometry: High School
Enrolling in a course lets you earn progress by passing quizzes and exams.
Track course progress
Take quizzes and exams
Earn certificates of completion
You will also be able to:
Create a Goal
Create custom courses
Get your questions answered
Upgrade to Premium to add all these features to your account!
Upgrade Now
Upgrade to Premium to add all these features to your account!
Plans
Study help
Test prep
College credit
Teacher resources
Working Scholars®
School group plans
Online tutoring
About us
Blog
Careers
Teach for us
Press Center
Ambassador
Scholarships
Support
Contact support
FAQ
Site feedback
Resources and Guides
Start now
Create an account
Already a member? Log in
Download the app
Study.com on Facebook Study.com on YouTube Study.com on Instagram Study.com on Twitter Study.com on LinkedIn
© Copyright 2025 Study.com. All other trademarks and copyrights are the property of their respective owners. All rights reserved.
Contact us by phone at (877)266-4919, or by mail at 100 View Street#202, Mountain View, CA 94041.
About Us
Terms of Use
Privacy Policy
DMCA Notice
ADA Compliance
Honor Code For Students |
8660 | https://www.youtube.com/watch?v=WwKLA-6S-e0 | How to Add Decimals | Math with Mr. J
Math with Mr. J
1730000 subscribers
6633 likes
Description
691392 views
Posted: 29 Jul 2021
Welcome to How to Add Decimals with Mr. J! Need help with adding decimals? You're in the right place!
Whether you're just starting out, or need a quick refresher, this is the video for you if you're looking for help with how to add decimals. Mr. J will go through adding decimals examples and explain the steps of how to add decimals.
✅ MORE ADDING AND SUBTRACTING DECIMALS VIDEOS:
How to Subtract Decimals -
How to Add a Decimal to a Whole Number -
How to Subtract a Decimal from a Whole Number -
How to Add Three Decimals -
About Math with Mr. J: This channel offers instructional videos that are directly aligned with math standards. Teachers, parents/guardians, and students from around the world have used this channel to help with math content in many different ways. All material is absolutely free.
Click Here to Subscribe to the Greatest Math Channel On Earth:
Follow Mr. J on Twitter: @MrJMath5
Email: math5.mrj@gmail.com
Music:
Hopefully this video is what you're looking for when it comes to adding decimals.
Have a great rest of your day and thanks again for watching!
455 comments
Transcript:
[Music] welcome to math with mr j [Music] in this video i'm going to cover how to add decimals now if you're able to add whole numbers you're going to be able to add decimals just a couple of different things as far as setting up the problems go if you get how to set up the problems you're going to be good to go let's jump into our examples and start with number one where we have eighteen and four tenths plus six and seventy two hundredths so when we set these problems up we wanna make sure that we line our decimals up that's going to line up all of our places so number one we're going to do eighteen and four tenths plus six and seventy two hundredths notice how the decimals are lined up now it may look a little offset there but we can use placeholder zeros to better align this and make it look a little more organized meaning eighteen and four-tenths we can put a zero in the hundredths place that way we look a little more lined up remember zeros to the right of a decimal do not change the value that's an equivalent decimal so we are able to do that we're not changing the value of the problem once we have that we add start with the hundredths place so zero plus two is two then we move to the tenths place four plus seven is eleven so put our one and regroup or carry there now bring our decimal straight down it's lined up throughout the entire problem now we add the ones 1 plus 8 is 9 plus 6 is 15. regroup or carry the one now we add the tens place one plus one is two so we get twenty five and twelve hundredths to recap as far as where we put the decimal in our answer the decimal should be lined up throughout the entire problem so we bring it straight down now before i move on to number two i do want to mention a common mistake and that's going to be lining the digits up meaning for number one it would look like this 18 and four tenths plus six and seventy two hundredths so three digits lined up with three digits so you can see that the decimals are not lined up therefore our places are not lined up and we're not going to place the decimal correctly in our answer so this is going to give us something completely incorrect and is a common mistake so avoid just lining the digits up make sure you have the decimals lined up now it is possible that you line the decimals up and then the digits line up as well most important part being the decimals are lined up let's try another example and move on to number two where we have 55 and 909 thousandths plus 23 and 3 tenths so first step line up our decimals so 55 and 909 thousandths plus 23 and three-tenths so we can use placeholder zeros to line this up a little better so we can place some zeros to the right of 23 and three-tenths remember zeros to the right of a decimal do not change the value that's an equivalent decimal so we can do that our last step we add so nine plus zero is nine zero plus zero zero nine plus three is twelve regroup or carry the one our decimal comes straight down then we add the ones one plus five is six plus three is nine and then we end with the tens so five plus two is seven so our final answer 79 and 209 thousandths so there you have it there's how you add decimals i hope that helped thanks so much for watching until next time peace |
8661 | https://pmc.ncbi.nlm.nih.gov/articles/PMC9909396/ | Medial artery calcification in peripheral artery disease - PMC
Skip to main content
An official website of the United States government
Here's how you know
Here's how you know
Official websites use .gov
A .gov website belongs to an official government organization in the United States.
Secure .gov websites use HTTPS
A lock ( ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites.
PMC Search Update
PMC Beta search will replace the current PMC search the week of September 7, 2025. Try out PMC Beta search now and give us your feedback. Learn more
Search
Log in
Dashboard
Publications
Account settings
Log out
Search… Search NCBI
Primary site navigation
Search
Logged in as:
Dashboard
Publications
Account settings
Log in
Search PMC Full-Text Archive
Search in PMC
Journal List
User Guide
New Try this search in PMC Beta Search
View on publisher site
Download PDF
Add to Collections
Cite
Permalink PERMALINK
Copy
As a library, NLM provides access to scientific literature. Inclusion in an NLM database does not imply endorsement of, or agreement with, the contents by NLM or the National Institutes of Health.
Learn more: PMC Disclaimer | PMC Copyright Notice
Front Cardiovasc Med
. 2023 Jan 26;10:1093355. doi: 10.3389/fcvm.2023.1093355
Search in PMC
Search in PubMed
View in NLM Catalog
Add to search
Medial artery calcification in peripheral artery disease
Tanner I Kim
Tanner I Kim
1 Deparment of Surgery, John A. Burns School of Medicine, University of Hawaii, Honolulu, HI, United States
2 The Queen’s Health Systems, Honolulu, HI, United States
Find articles by Tanner I Kim
1,2, Raul J Guzman
Raul J Guzman
3 Division of Vascular Surgery and Endovascular Therapy, Department of Surgery, Yale School of Medicine, New Haven, CT, United States
Find articles by Raul J Guzman
3,
Author information
Article notes
Copyright and License information
1 Deparment of Surgery, John A. Burns School of Medicine, University of Hawaii, Honolulu, HI, United States
2 The Queen’s Health Systems, Honolulu, HI, United States
3 Division of Vascular Surgery and Endovascular Therapy, Department of Surgery, Yale School of Medicine, New Haven, CT, United States
Edited by: Dwight A. Towler, University of Texas Southwestern Medical Center, United States
Reviewed by: Rolando A. Cuevas, University of Pittsburgh, United States; Manfred Boehm, National Heart, Lung, and Blood Institute (NIH), United States
✉
Correspondence: Raul J. Guzman, Raul.guzman@yale.edu
This article was submitted to Atherosclerosis and Vascular Medicine, a section of the journal Frontiers in Cardiovascular Medicine
Received 2022 Nov 8; Accepted 2023 Jan 13; Collection date 2023.
Copyright © 2023 Kim and Guzman.
This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
PMC Copyright notice
PMCID: PMC9909396 PMID: 36776265
Abstract
Medial artery calcification (MAC) is a distinct, highly regulated process that is often identified in small and mid-sized arteries of the lower extremities. It is associated with advanced age, diabetes, and chronic kidney disease. MAC often occurs in conjunction with atherosclerotic occlusive disease in lower extremity arteries, and when seen together or in isolation, long-term limb outcomes are negatively affected. In patients with peripheral artery disease (PAD), the extent of MAC independently correlates with major amputation and mortality rates, and it predicts poor outcomes after endovascular interventions. It is associated with increased arterial stiffness and decreased pedal perfusion. New endovascular methods aimed at treating calcified lower-extremity lesions may improve our ability to treat patients with limb-threatening ischemia. Although recent developments have increased our understanding of the mechanisms contributing to MAC, further investigations are needed to understand the role of medial calcification in PAD, and to develop strategies aimed at improving patient outcomes.
Keywords: medial artery calcification, peripheral artery (occlusive) disease, arterial calcification, arterial stiffness, chronic limb-threatening ischemia
Introduction
Medial artery calcification (MAC), previously known as Monckeberg’s sclerosis, affects the peripheral arteries and is most strongly associated with diabetes and chronic kidney disease. It develops through a complex process involving the transformation of vascular smooth muscle cells (VSMC) into an osteogenic phenotype that culminates in the formation of calcium hydroxyapatite crystals between the intimal and adventitial layers (1). Recent work has demonstrated that this highly regulated process may develop independent of atherosclerosis and has features similar to that of developing bone (1–3). The process is distinct from intimal calcification that develops on the luminal side of the internal elastic lamina. The intimal form is commonly found in coronary and larger arteries throughout the body (Table 1). Based on recent studies, medial calcification is now thought to be an independent contributor to the complex pathophysiology of PAD (4–9). In this review, we discuss the relationship between MAC and PAD and its effects on long-term limb outcomes.
Table 1.
Characteristics of intimal and medial calcification.
| Intimal calcification | Medial calcification |
:--- |
| Associated with atherosclerosis and traditional cardiovascular risk factors | Associated with diabetes, chronic kidney disease and metabolic dysfunction |
| Characterized by subintimal lipid deposition and macrophages | Characterized by elastin layer disruption and calcification |
| Can be associated with inflammatory cell infiltrate and bone formation within atherosclerotic plaque | Associated with osteogenic transformation of medial smooth muscle cells. Bone formation less common. |
| Appears as patchy or spotty areas of calcification on the luminal side of internal elastic lamina | Appears as smooth, continuous areas of calcification, typically seen as parallel lines or “railroad” tracks on plain x-rays. |
| Commonly affect coronary, carotid, and larger arteries | More commonly seen in peripheral small and medium-sized arteries |
| Advanced form leads to vessel stenosis or occlusion | Advanced form associated with increased arterial stiffness |
Open in a new tab
Epidemiology and risk factors
Although MAC can develop in the setting of typical atherosclerosis risk factors, it is most commonly identified in association with diabetes and chronic kidney disease (10–14). The medial calcification that arises in patients with diabetes is thought to result from multiple overlapping mechanisms including prolonged hyperglycemia, inflammation, advanced glycation end products, circulation of osteoprogenitor cells, and decreased matrix gamma-carboxyglutamic acid protein (MGP) levels (15–18). We previously reported on the risk factors for MAC in patients without renal disease (19). Increased tibial artery calcium scores were associated with age, male sex, diabetes, and tobacco use, but interestingly, not with hypertension, hyperlipidemia, or body mass index.
In patients with chronic kidney disease, medial calcification is thought to develop in part through dysregulated calcium and phosphorus metabolism as well as increased oxidative stress (1, 15). The role of hyperphosphatemia in the development of arterial calcification has been investigated (1, 20–22). Hyperphosphatemia leads to a variety of changes that affect VSMCs including calcium-phosphate deposits, promotion of an osteogenic phenotype, and apoptosis (23–26). We previously demonstrated that high serum phosphate levels are independently associated with worse amputation-free survival following intervention for chronic limb-threatening ischemia (CLTI) (22). In 941 patients with CLTI undergoing lower extremity intervention, 14% were found to have elevated phosphate levels. Patients with elevated phosphate levels had significantly higher mortality rates and decreased amputation-free survival. These results persisted after adjustment for multiple demographic and cardiovascular risk factors including age, diabetes, and chronic kidney disease.
Differing risk factor profiles have also been observed between intimal and medial artery calcification. Zwakenberg et al. pooled together two cohort studies to assess risk factors associated with lower extremity intimal and medial calcification (27). Patients with the medial form had higher rates of diabetes and elevated ankle-brachial indices, and lower rates of smoking compared to those with intimal calcification. Interestingly, the inactive form of matrix-Gla protein was associated with medial calcification, while osteonectin was associated with the intimal type. Matrix Gla protein is an inhibitor of arterial calcification, and medications that lead to decreased carboxylation of matrix Gla protein such as warfarin have also been found to promote MAC (28, 29). Alappan et al. demonstrated that warfarin use was associated with a four-fold increase in MAC progression (29). Progression of MAC was further accelerated among warfarin users with end-stage renal disease, and slowed following cessation of warfarin therapy.
Inflammatory disorders such as rheumatoid arthritis have been associated with MAC formation. This is thought to occur secondary to chronic inflammation, as elevated IL-6 levels are associated with osteogenic differentiation (30–32). The presence of MAC in patients with rheumatoid arthritis has also been shown to be a risk factor for worse all-cause mortality (33). The role of specific genotypes in the development medial calcification has recently been highlighted. St. Hilaire et al. reported on families with symptomatic arterial and joint calcification with identified mutations in the gene NT5E (34). This gene encodes for CD73, which is an enzyme involved in the adenosine triphosphate (ATP) metabolic pathway that has been since found to influences inorganic pyrophosphate and phosphate metabolism. Pseudoxanthoma elasticum is a rare disease that is also well associated with MAC formation (35, 36). The gene involved in this disease is ABCC6, which encodes for an adenosine triphosphate-binding cassette transporter primarily expressed in the liver and kidney. However, the underlying pathophysiology leading to MAC formation remains unclear (35). Generalized arterial calcification of infancy (GACI) is a rare disorder most commonly associated with mutations in the ENPP1 gene (Ectonucleotide pyrophosphatase/phosphodiesterase family member one or less commonly the ABCC6 gene). It is characterized by calcification and disruption of the media that can lead to stenosis or rupture (37).
These studies emphasize the multiple, overlapping risk factors that contribute to the complex pathophysiology of MAC, including inflammation, mineral metabolism, and osteogenic pathways. The development of relevant treatments will require continued investigations into its shared and distinct pathways with plaque calcification, and further efforts to understand its role in the pathogenesis of PAD.
Medial artery calcification in peripheral arteries
The histopathology of medial calcification is distinct from that of the intimal form (Figure 1). In a study performed on atherectomy specimens from symptomatic PAD patients, Krishnan et al. found that the combined presence of diabetes and chronic kidney disease was more strongly associated with MAC than individual risk factors (38). Other histological studies have demonstrated a correlation between age and MAC. Kamenskiy et al. analyzed femoropopliteal arteries from 431 donors and found that over 60% of patients over age 60 years had evidence of femoropopliteal calcification and at least half had the medial form (39). Interestingly, small calcium deposits were observed in femoropopliteal arteries of patients as young as 18 years of age.
Figure 1.
Open in a new tab
H&E stain of tibial arteries from patient with intimal (A) and medial (B) calcification. L denotes lumen, I denotes intima, M denotes media.
The association of MAC with peripheral atherosclerosis is well known. In 1988, Everhart et al. published their longitudinal findings on 4,553 Pima Indians who were followed over a period of 20 years (40). Patients with diabetes and MAC as defined by a linear appearance of calcifications on x-rays of the hands, feet, calf regions, and thighs had a 5.5-fold increased risk of major amputation, and 1.5-fold mortality rate compared to diabetes patients without it (Figure 2). More recent studies have confirmed a predictive role for MAC on limb outcomes, and several investigators have shown an independent association between it and the severity of PAD as well as the risk of major amputation (5–7). It is noteworthy that traditional tests for PAD such as the ankle-brachial index may be inaccurate in patients with calcified arteries, making it difficult to assess the true severity of ischemia using standard measures. In a study involving 116 patients with symptomatic PAD, we showed that increasing calcification scores correlated with the extent of ischemia, and that this relationship was maintained after adjustment for cardiovascular risk factors and the extent of occlusive disease (6).
Figure 2.
Open in a new tab
(A) Foot x-rays showing typical “railroad track” appearance of dorsalis pedis artery signifying medial calcification (long yellow arrow) and “patchy” appearance of intimal calcification in plantar artery (short white arrow). (B) Digital subtraction arteriogram from same patient showing a patent DP (long yellow arrow) as is typical of medial calcification and high-grade stenosis of plantar arteries associated with intimal calcification (short white arrow).
Lower extremity calcification scores also correlate with the presence of foot ulceration in the setting of PAD. In a study involving 162 patients with diabetes, we showed that those with foot ulcers were more likely to have a higher tibial calcification score, and that this relationship persisted after adjustment for cardiovascular risk factors and the peripheral artery occlusion index. These findings suggest that medial calcification contributes to decreased pedal perfusion in a manner that is independent of atherosclerotic burden, and possibly via mechanisms related to alterations in blood vessel structure.
Tibial artery calcification is also a significant prognostic marker for PAD severity and worse limb-related outcomes. We previously investigated the relationship between tibial artery calcification, severity of limb ischemia, and risk of major amputation among 229 patients with symptomatic PAD (5). The tibial artery calcification score was found to correlate with the severity of limb ischemia at presentation, and higher levels were associated with major amputation and were better predictors of amputation than the ABI. A recent study from Liu et al. created a simple scoring system utilizing foot x-rays to score pedal artery calcification in 250 patients who underwent infra-inguinal revascularization for CLTI (41). After a mean follow up of 759 days, higher pedal calcification scores were associated with an elevated risk of major amputation. Furthermore, the calcification score was able to further stratify the risk of major amputation among high-risk patients with diabetes. These studies demonstrate a clear relationship between MAC and worse limb-related outcomes. One potential mechanistic link between medial calcification and poor limb outcomes is through its effects on arterial stiffness, which can lead to dysfunctional blood flow dynamics as discussed in the following section.
Several limitations continue to affect current clinical studies on the epidemiology and consequences of medial calcification. Importantly, it remains difficult to confidently distinguish medial from intimal calcification on x-rays and CT scans. The resolution of these commonly used methods does not provide sufficient resolution to specify which form is involved in a specific vessel segment with the most common protocols used. Additionally, peripheral arteries are commonly affected by both medial and intimal calcification in varying ratios. As such, efforts to isolate the effects of one form over the other are either not possible or severely limited. Finally, unlike measurement of calcification in the coronaries, different calcium measuring methodologies are currently in use by various investigators, and this has complicated the comparison of results. Future efforts to identify the best methods for identifying and quantifying medial artery calcification are needed.
Medial artery calcification and arterial stiffness
MAC leads to significant structural changes within the vessel wall. The accumulation of medial calcification reduces vessel compliance and alters blood flow dynamics (42, 43). In larger arteries, the so-called Windkessel effect, an imprecise term used to describe the dampening effect of large elastic arteries on the pulse waveforms (44), causes increased pulse pressure and cardiac afterload which have been associated with left ventricular hypertrophy and heart failure (43, 45, 46). The loss of vessel wall compliance within smaller arteries has been associated with stasis and an attenuated autoregulatory response which may contribute to decreased perfusion (42). This suggests one mechanism whereby MAC can contribute to an increased risk of cardiovascular morbidity and mortality (11, 47, 48).
Arterial stiffness has also been associated with worse outcomes following lower extremity interventions for limb-threatening ischemia. We previously investigated the association between pulse pressure, as a surrogate for arterial stiffness, and endovascular interventions for CLTI (49). Among 371 patients who underwent tibial artery interventions, 185 patients had a preoperative pulse pressure of 80 mm Hg or greater signifying increased arterial stiffness. Patients with a higher pulse pressure were more likely to suffer from procedural complications that included arterial dissection, spasm, embolism, rupture, or arteriovenous fistula. Although short terms outcomes were not worse in the high pulse pressure group, five-year mortality was higher among these patients. Similarly, a pulse pressure of 80 mm Hg or greater was also found to be independently associated with worse amputation-free survival at 1 year after lower extremity bypass (50). Interestingly, histologic studies of the superficial femoral artery have shown that patients with MAC have a thinner elastic lamina with discontinuous elastic fibers potentially explaining the higher risk of procedural complications (39). Calcification is also associated with increased stiffness in both the longitudinal and circumferential directions (39, 51).
Another potential explanation for the worse outcomes seen in patients with MAC is its association with an impaired ability to recover from ischemic events. We have investigated the relationship between MAC and limb ischemia in rodent models of disease (52). Animals with MAC induced by vitamin D overload exhibited an impaired ability to recover from ischemia caused by femoral artery ligation. Animals with calcified arteries demonstrated more severe ischemia and delayed functional recovery compared with controls. Interestingly, capillary density and muscle morphology were not different between the two groups, suggesting that differences in angiogenic responses were not the primary mechanistic explanation. However, arterial stiffness was increased in animals with calcified arteries and was associated with medial elastin degradation, suggesting that MAC may lead to worse outcomes in part due to its effects on arterial compliance. Other studies have also demonstrated that decreased arterial compliance is associated with increased peripheral vascular resistance and a decreased vasodilatory response, which may contribute to decreased perfusion (53, 54). However, it is likely that other factors also contribute to the complex mechanisms behind MAC and decreased perfusion, warranting further investigation.
Interventions for peripheral artery disease with medial artery calcification
While intimal calcification within dense atherosclerotic plaques has been known to affect technical aspects of endovascular interventions, emerging data also suggests a negative impact of MAC. Kang et al. investigated the relationship between calcium burden and tibial endovascular interventions among patients with CLTI (55). Patients with extensive tibial artery calcification had the lowest rates of technical success, and at 2 years had significantly higher rates of unplanned major amputation. Similarly, Skolnik et al. found that an increased burden of MAC in the foot and ankle by x-ray was predictive of an increased risk of major adverse limb events following tibial artery angioplasty (56). One accepted explanation for the worse outcomes is that severely calcified arteries have higher rates of elastic recoil following balloon angioplasty and limit stent expansion leading to poor patency rates (57–59). However, data in this regard are limited by the current inability to clearly distinguish medial from intimal calcification on plain X-rays and CT scans as well as the high degree of overlap between the two forms in many anatomic regions.
An additional concern is that the presence of MAC can promote the development of neointimal hyperplasia following angioplasty leading to increased levels of in-stent restenosis. Using a rodent model, we previously showed that neointimal hyperplasia was significantly increased in calcified arteries (60). Interestingly, calcified arteries also had higher levels of the bone morphogenetic protein-2 (BMP-2) and increased BMP-induced smooth muscle cell proliferation compared with non-calcified arteries. Increased BMP-2 expression and BMP-responsiveness of SMC from calcified arteries suggests that controlling the restenosis response may require different therapeutic targets in calcified arteries.
The state of current therapies and future directions
Mechanical methods to address calcified arteries in patients with PAD are under development. Intravascular lithotripsy has recently been introduced as a novel endovascular technique to address severe calcification. This method utilizes a specialized angioplasty balloon inflated to a low atmospheric pressure with emitters that generate pulsatile mechanical energy to fracture surrounding arterial calcium while minimizing trauma to the surrounding vessel (61). Initial single arm studies of patients with heavily calcified lesions in the femoral and tibial arteries have demonstrated high rates of technical success with low complication rates and bailout stenting (61–63). Randomized studies are ongoing, with encouraging mid-term results at 1 year (64). While most attention has been focused on treatment of atherosclerotic intimal disease, studies by Brodmann et al. have shown good early results when used in tibial lesions where calcification is more likely to be medial (64).
Despite advances in our understanding of its central mechanisms, no therapies have proven effective in slowing or reversing peripheral artery calcification. Many of the underlying causes of MAC are still unknown, and it is unclear whether the expression of calcification-related genes is a driver of the adverse pathologic responses or the result of it (42). At this time, many studies involving various medications including Vitamin K, sodium thiosulfate, magnesium, calcimimetics, and bisphosphonates among others are ongoing (65–69). Investigating the genetics associated with MAC will also improve our understanding of the underlying pathophysiology and may lead to new therapeutics. MAC formation is well-described in several rare monogenetic disorders (31, 35, 36, 70). Genome-wide association studies (GWAS) may also provide further insight in the future. Several studies have already used GWAS to identify genetic associations with PAD (71, 72). Zekavat et al. performed GWAS using the United Kingdom Biobank to identify whether predisposition to increased arterial stiffness was associated with hypertension and coronary artery disease (73). Rare diseases and new technologies provide an opportunity to further our understanding of the genetic and pathophysiologic mechanisms underlying MAC and will hopefully lead to new treatment options in the future.
Conclusion
MAC is a common yet complex process that is often seen in patients with PAD. Increased medial calcification levels correlate with worse limb and cardiovascular outcomes in a manner that appears to be independent of atherosclerosis. It is associated with increased arterial stiffness and decreased pedal perfusion and it limits the durability of endovascular interventions. New therapies are currently under development for the treatment of calcified vessels. Further research is needed to better understand the role of MAC in PAD, and to identify newer treatments aimed at preventing it and improving outcomes for our vascular patients.
Author contributions
TK and RG contributed to the writing of the manuscript. All authors contributed to the article and approved the submitted version.
Funding
This work was supported by R01 HL138357 and R01 HL157111.
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Publisher’s note
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
References
1.Ho CY, Shanahan CM. Medial arterial calcification: an overlooked player in peripheral arterial disease. Arterioscler Thromb Vasc Biol. (2016) 36:1475–82. doi: 10.1161/ATVBAHA.116.306717 [DOI] [PubMed] [Google Scholar]
2.Shao J-S, Cai J, Towler DA. Molecular mechanisms of vascular calcification: lessons learned from the aorta. Arterioscler Thromb Vasc Biol. (2006) 26:1423–30. doi: 10.1161/01.ATV.0000220441.42041.20 [DOI] [PubMed] [Google Scholar]
3.Guzman RJ. Clinical, cellular, and molecular aspects of arterial calcification. J Vasc Surg. (2007) 45:A57–63. doi: 10.1016/j.jvs.2007.02.049, PMID: [DOI] [PMC free article] [PubMed] [Google Scholar]
4.Norgren L, Hiatt WR, Dormandy JA, Nehler MR, Harris KA, FGR F, et al. Inter-society consensus for the Management of Peripheral Arterial Disease (TASC II). J Vasc Surg. x (2007) 45:S5–S67. doi: 10.1016/j.jvs.2006.12.037 [DOI] [PubMed] [Google Scholar]
5.Guzman RJ, Brinkley DM, Schumacher PM, Donahue RMJ, Beavers H, Qin X. Tibial artery calcification as a marker of amputation risk in patients with peripheral arterial disease. J Am Coll Cardiol. (2008) 51:1967–74. doi: 10.1016/j.jacc.2007.12.058, PMID: [DOI] [PMC free article] [PubMed] [Google Scholar]
6.Zettervall SL, Marshall AP, Fleser P, Guzman RJ. Association of arterial calcification with chronic limb ischemia in patients with peripheral artery disease. J Vasc Surg. (2018) 67:507–13. doi: 10.1016/j.jvs.2017.06.086, PMID: [DOI] [PMC free article] [PubMed] [Google Scholar]
7.Huang C-L, Wu I-H, Wu Y-W, Hwang J-J, Wang S-S, Chen W-J, et al. Association of Lower Extremity Arterial Calcification with amputation and mortality in patients with symptomatic peripheral artery disease. PLoS One. (2014) 9:e90201. doi: 10.1371/journal.pone.0090201, PMID: [DOI] [PMC free article] [PubMed] [Google Scholar]
8.Costacou T, Huskey ND, Edmundowicz D, Stolk R, Orchard TJ. Lower-extremity arterial calcification as a correlate of coronary artery calcification. Metabolism. (2006) 55:1689–96. doi: 10.1016/j.metabol.2006.08.012, PMID: [DOI] [PubMed] [Google Scholar]
9.Allison MA, Hsi S, Wassel CL, Morgan C, Ix JH, Wright CM, et al. Calcified atherosclerosis in different vascular beds and the risk of mortality. Arterioscler Thromb Vasc Biol. (2012) 32:140–6. doi: 10.1161/ATVBAHA.111.235234, PMID: [DOI] [PubMed] [Google Scholar]
10.Stabley JN, Towler DA. Arterial calcification in diabetes mellitus: preclinical models and translational implications. Arterioscler Thromb Vasc Biol. (2017) 37:205–17. doi: 10.1161/ATVBAHA.116.306258, PMID: [DOI] [PMC free article] [PubMed] [Google Scholar]
11.London GM, Guérin AP, Marchais SJ, Métivier F, Pannier B, Adda H. Arterial media calcification in end-stage renal disease: impact on all-cause and cardiovascular mortality. Nephrol Dial Transplant. (2003) 18:1731–40. doi: 10.1093/ndt/gfg414, PMID: [DOI] [PubMed] [Google Scholar]
12.Ohtake T, Oka M, Ikee R, Mochida Y, Ishioka K, Moriya H, et al. Impact of lower limbs’ arterial calcification on the prevalence and severity of PAD in patients on hemodialysis. J Vasc Surg. (2011) 53:676–83. doi: 10.1016/j.jvs.2010.09.070, PMID: [DOI] [PubMed] [Google Scholar]
13.Chistiakov DA, Sobenin IA, Orekhov AN, Bobryshev YV. Mechanisms of medial arterial calcification in diabetes. Curr Pharm Des. (2014) 20:5870–83. doi: 10.2174/1381612820666140212210451 [DOI] [PubMed] [Google Scholar]
14.Chen NX, Moe SM. Arterial calcification in diabetes. Curr Diab Rep. (2003) 3:28–32. doi: 10.1007/s11892-003-0049-2 [DOI] [PubMed] [Google Scholar]
15.Demer LL, Tintut Y. Inflammatory, metabolic, and genetic mechanisms of vascular calcification. Arterioscler Thromb Vasc Biol. (2014) 34:715–23. doi: 10.1161/ATVBAHA.113.302070, PMID: [DOI] [PMC free article] [PubMed] [Google Scholar]
16.Menini S, Iacobini C, Ricci C, Blasetti Fantauzzi C, Salvi L, Pesce CM, et al. The galectin-3/RAGE dyad modulates vascular osteogenesis in atherosclerosis. Cardiovasc Res. (2013) 100:472–80. doi: 10.1093/cvr/cvt206, PMID: [DOI] [PubMed] [Google Scholar]
17.Fadini GP, Albiero M, Menegazzo L, Boscaro E, Agostini C, de Kreutzenberg SV, et al. Procalcific phenotypic drift of circulating progenitor cells in type 2 diabetes with coronary artery disease. Exp Diabetes Res. (2012) 2012:921685. doi: 10.1155/2012/921685, PMID: [DOI] [PMC free article] [PubMed] [Google Scholar]
18.Doyon M, Mathieu P, Moreau P. Decreased expression of γ-carboxylase in diabetes-associated arterial stiffness: impact on matrix Gla protein. Cardiovasc Res. (2013) 97:331–8. doi: 10.1093/cvr/cvs325, PMID: [DOI] [PubMed] [Google Scholar]
19.Deas DS, Marshall AP, Bian A, Shintani A, Guzman RJ. Association of cardiovascular and biochemical risk factors with tibial artery calcification. Vasc Med. (2015) 20:326–31. doi: 10.1177/1358863X15581448, PMID: [DOI] [PMC free article] [PubMed] [Google Scholar]
20.Kestenbaum B, Sampson JN, Rudser KD, Patterson DJ, Seliger SL, Young B, et al. Serum phosphate levels and mortality risk among people with chronic kidney disease. J Am Soc Nephrol. (2005) 16:520–8. doi: 10.1681/ASN.2004070602 [DOI] [PubMed] [Google Scholar]
21.Tonelli M, Sacks F, Pfeffer M, Gao Z, Curhan G. Cholesterol and recurrent events trial investigators. Relation between serum phosphate level and cardiovascular event rate in people with coronary disease. Circulation. (2005) 112:2627–33. doi: 10.1161/CIRCULATIONAHA.105.553198, PMID: [DOI] [PubMed] [Google Scholar]
22.Zettervall SL, Soden PA, Ultee KHJ, Seldon C, Oh J, McGann K, et al. Elevated serum phosphate levels are associated with decreased amputation-free survival after interventions for critical limb ischemia. J Vasc Surg. (2017) 65:431–7. doi: 10.1016/j.jvs.2016.06.097, PMID: [DOI] [PMC free article] [PubMed] [Google Scholar]
23.Jono S, McKee MD, Murry CE, Shioi A, Nishizawa Y, Mori K, et al. Phosphate regulation of vascular smooth muscle cell calcification. Circ Res. (2000) 87:E10–7. doi: 10.1161/01.res.87.7.e10 [DOI] [PubMed] [Google Scholar]
24.Villa-Bellosta R. Vascular calcification: key roles of phosphate and pyrophosphate. Int J Mol Sci. (2021) 22:13536. doi: 10.3390/ijms222413536, PMID: [DOI] [PMC free article] [PubMed] [Google Scholar]
25.Steitz SA, Speer MY, Curinga G, Yang HY, Haynes P, Aebersold R, et al. Smooth muscle cell phenotypic transition associated with calcification: upregulation of Cbfa1 and downregulation of smooth muscle lineage markers. Circ Res. (2001) 89:1147–54. doi: 10.1161/hh2401.101070 [DOI] [PubMed] [Google Scholar]
26.Proudfoot D, Skepper JN, Hegyi L, Bennett MR, Shanahan CM, Weissberg PL. Apoptosis regulates human vascular calcification in vitro: evidence for initiation of vascular calcification by apoptotic bodies. Circ Res. (2000) 87:1055–62. doi: 10.1161/01.res.87.11.1055 [DOI] [PubMed] [Google Scholar]
27.Zwakenberg SR, De JPA, Hendriks EJ, Westerink J, Spiering W, De BGJ, et al. Intimal and medial calcification in relation to cardiovascular risk factors. PLoS One. (2020) 15:e0235228. doi: 10.1371/journal.pone.0235228, PMID: [DOI] [PMC free article] [PubMed] [Google Scholar]
28.Lomashvili KA, Wang X, Wallin R, O’Neill WC. Matrix Gla protein metabolism in vascular smooth muscle and role in uremic vascular calcification. J Biol Chem. (2011) 286:28715–22. doi: 10.1074/jbc.M111.251462, PMID: [DOI] [PMC free article] [PubMed] [Google Scholar]
29.Alappan HR, Kaur G, Manzoor S, Navarrete J, O’Neill WC. Warfarin accelerates medial arterial calcification in humans. Arterioscler Thromb Vasc Biol. (2020) 40:1413–9. doi: 10.1161/ATVBAHA.119.313879, PMID: [DOI] [PubMed] [Google Scholar]
30.Sun M, Chang Q, Xin M, Wang Q, Li H, Qian J. Endogenous bone morphogenetic protein 2 plays a role in vascular smooth muscle cell calcification induced by interleukin 6 in vitro. Int J Immunopathol Pharmacol. (2017) 30:227–37. doi: 10.1177/0394632016689571, PMID: [DOI] [PMC free article] [PubMed] [Google Scholar]
31.del Rincón I, Haas RW, Pogosian S, Escalante A. Lower limb arterial incompressibility and obstruction in rheumatoid arthritis. Ann Rheum Dis. (2005) 64:425–32. doi: 10.1136/ard.2003.018671, PMID: [DOI] [PMC free article] [PubMed] [Google Scholar]
32.Paccou J, Brazier M, Mentaverri R, Kamel S, Fardellone P, Massy ZA. Vascular calcification in rheumatoid arthritis: prevalence, pathophysiological aspects and potential targets. Atherosclerosis. (2012) 224:283–90. doi: 10.1016/j.atherosclerosis.2012.04.008, PMID: [DOI] [PubMed] [Google Scholar]
33.Solow EB, Yu F, Thiele GM, Sokolove J, Robinson WH, Pruhs ZM, et al. Vascular calcifications on hand radiographs in rheumatoid arthritis and associations with autoantibodies, cardiovascular risk factors and mortality. Rheumatology. (2015) 54:1587–95. doi: 10.1093/rheumatology/kev027, PMID: [DOI] [PMC free article] [PubMed] [Google Scholar]
34.St. Hilaire C, Ziegler SG, Markello TC, Brusco A, Groden C, Gill F, et al. NT5E mutations and arterial calcifications. N Engl J Med. (2011) 364:432–42. doi: 10.1056/NEJMoa0912923, PMID: [DOI] [PMC free article] [PubMed] [Google Scholar]
35.Rutsch F, Buers I, Nitschke Y. Hereditary disorders of cardiovascular calcification. Arterioscler Thromb Vasc Biol. (2021) 41:35–47. doi: 10.1161/ATVBAHA.120.315577 [DOI] [PubMed] [Google Scholar]
36.Kranenburg G, de Jong PA, Mali WP, Attrach M, Visseren FLJ, Spiering W. Prevalence and severity of arterial calcifications in pseudoxanthoma elasticum (PXE) compared to hospital controls. Novel insights into the vascular phenotype of PXE. Atherosclerosis. (2017) 256:7–14. doi: 10.1016/j.atherosclerosis.2016.11.012, PMID: [DOI] [PubMed] [Google Scholar]
37.Kawai K, Sato Y, Kawakami R, Sakamoto A, Cornelissen A, Mori M, et al. Generalized arterial calcification of infancy (GACI): optimizing care with a multidisciplinary approach. J Multidiscip Healthc. (2022) 15:1261–76. doi: 10.2147/JMDH.S251861, PMID: [DOI] [PMC free article] [PubMed] [Google Scholar]
38.Krishnan P, Moreno PR, Turnbull IC, Purushothaman M, Zafar U, Tarricone A, et al. Incremental effects of diabetes mellitus and chronic kidney disease in medial arterial calcification: synergistic pathways for peripheral artery disease progression. Vasc Med. (2019) 24:383–94. doi: 10.1177/1358863X19842276, PMID: [DOI] [PMC free article] [PubMed] [Google Scholar]
39.Kamenskiy A, Poulson W, Sim S, Reilly A, Luo J, MacTaggart J. Prevalence of calcification in human FEMOROPOPLITEAL arteries and its ASSOCIATION with demographics, risk factors, and arterial stiffness. Arterioscler Thromb Vasc Biol. (2018) 38:e48–57. doi: 10.1161/ATVBAHA.117.310490, PMID: [DOI] [PMC free article] [PubMed] [Google Scholar]
40.Everhart JE, Pettitt DJ, Knowler WC, Rose FA, Bennett PH. Medial arterial calcification and its association with mortality and complications of diabetes. Diabetologia. (1988) 31:16–23. doi: 10.1007/BF00279127 [DOI] [PubMed] [Google Scholar]
41.Liu IH, Wu B, Krepkiy V, Ferraresi R, Reyzelman AM, Hiramoto JS, et al. Pedal arterial calcification score is associated with the risk of major amputation in chronic limb-threatening ischemia. J Vasc Surg. (2022) 75:270–278.e3. doi: 10.1016/j.jvs.2021.07.235, PMID: [DOI] [PubMed] [Google Scholar]
42.Lanzer P, Boehm M, Sorribas V, Thiriet M, Janzen J, Zeller T, et al. Medial vascular calcification revisited: review and perspectives. Eur Heart J. (2014) 35:1515–25. doi: 10.1093/eurheartj/ehu163, PMID: [DOI] [PMC free article] [PubMed] [Google Scholar]
43.Soldatos G, Jandeleit-Dahm K, Thomson H, Formosa M, D’orsa K, Calkin AC, et al. Large artery biomechanics and diastolic dysfunctionin patients with type 2 diabetes. Diabet Med. (2011) 28:54–60. doi: 10.1111/j.1464-5491.2010.03146.x, PMID: [DOI] [PubMed] [Google Scholar]
44.Segers P, O’Rourke MF, Parker K, Westerhof N, Hughes A. Towards a consensus on the understanding and analysis of the pulse waveform: results from the 2016 workshop on arterial hemodynamics: past, present and future. Artery Res. (2017) 18:75–80. doi: 10.1016/j.artres.2017.03.004, PMID: [DOI] [PMC free article] [PubMed] [Google Scholar]
45.London GM. Cardiovascular disease in chronic renal failure: pathophysiologic aspects. Semin Dial. (2003) 16:85–94. doi: 10.1046/j.1525-139x.2003.16023.x, PMID: [DOI] [PubMed] [Google Scholar]
46.Paulus WJ, Tschöpe C. A novel paradigm for heart failure with preserved ejection fraction: comorbidities drive myocardial dysfunction and remodeling through coronary microvascular endothelial inflammation. J Am Coll Cardiol. (2013) 62:263–71. doi: 10.1016/j.jacc.2013.02.092 [DOI] [PubMed] [Google Scholar]
47.Vlachopoulos C, Aznaouridis K, Stefanadis C. Prediction of cardiovascular events and all-cause mortality with arterial stiffness: a systematic review and meta-analysis. J Am Coll Cardiol. (2010) 55:1318–27. doi: 10.1016/j.jacc.2009.10.061 [DOI] [PubMed] [Google Scholar]
48.Kouis P, Kousios A, Kanari A, Kleopa D, Papatheodorou SI, Panayiotou AG. Association of non-invasive measures of subclinical atherosclerosis and arterial stiffness with mortality and major cardiovascular events in chronic kidney disease: systematic review and meta-analysis of cohort studies. Clin Kidney J. (2020) 13:842–54. doi: 10.1093/ckj/sfz095, PMID: [DOI] [PMC free article] [PubMed] [Google Scholar]
49.Zettervall SL, Buck DB, Darling JD, Lee V, Schermerhorn ML, Guzman RJ. Increased preoperative pulse pressure predicts procedural complications and mortality in patients who undergo tibial interventions for critical limb ischemia. J Vasc Surg. (2016) 63:673–7. doi: 10.1016/j.jvs.2015.09.033, PMID: [DOI] [PMC free article] [PubMed] [Google Scholar]
50.Wise ES, Wergin JE, Mace EH, Kallos JA, Muhlestein WE, Shelburne NJ, et al. Upper extremity pulse pressure predicts amputation-free survival after lower extremity bypass. Am Surg. (2017) 83:804–11. doi: 10.1177/000313481708300742, PMID: [DOI] [PubMed] [Google Scholar]
51.Desyatova A, MacTaggart J, Kamenskiy A. Constitutive modeling of human femoropopliteal artery biaxial stiffening due to aging and diabetes. Acta Biomater. (2017) 64:50–8. doi: 10.1016/j.actbio.2017.09.042, PMID: [DOI] [PMC free article] [PubMed] [Google Scholar]
52.Zettervall SL, Wang X-L, Monk S, Lin T, Cai Y, Guzman RJ. Recovery of limb perfusion and function after hindlimb ischemia is impaired by arterial calcification. Physiol Rep. (2021) 9:e15008. doi: 10.14814/phy2.15008, PMID: [DOI] [PMC free article] [PubMed] [Google Scholar]
53.Guo X, Kassab GS. Increased serum klotho with age-related aortic stiffness and peripheral vascular resistance in Young and middle-aged swine. Front Physiol. (2020) 11:591. doi: 10.3389/fphys.2020.00591, PMID: [DOI] [PMC free article] [PubMed] [Google Scholar]
54.Wang L, Jerosch-Herold M, Jacobs DR, Shahar E, Detrano R, Folsom AR, et al. Coronary artery calcification and myocardial perfusion in asymptomatic adults: the MESA (multi-ethnic study of atherosclerosis). J Am Coll Cardiol. (2006) 48:1018–26. doi: 10.1016/j.jacc.2006.04.089, PMID: [DOI] [PMC free article] [PubMed] [Google Scholar]
55.Kang IS, Lee W, Choi BW, Choi D, Hong M-K, Jang Y, et al. Semiquantitative assessment of tibial artery calcification by computed tomography angiography and its ability to predict infrapopliteal angioplasty outcomes. J Vasc Surg. (2016) 64:1335–43. doi: 10.1016/j.jvs.2016.04.047, PMID: [DOI] [PubMed] [Google Scholar]
56.Skolnik J, Weiss R, Meyr AJ, Dhanisetty R, Choi ET, Cunningham-Hill M, et al. Evaluating the impact of medial arterial calcification on outcomes of Infrageniculate endovascular interventions for treatment of diabetic foot ulcers. Vasc Endovasc Surg. (2021) 55:382–8. doi: 10.1177/1538574421993314, PMID: [DOI] [PubMed] [Google Scholar]
57.Rocha-Singh KJ, Zeller T, Jaff MR. Peripheral arterial calcification: prevalence, mechanism, detection, and clinical implications. Catheter Cardiovasc Interv. (2014) 83:E212–20. doi: 10.1002/ccd.25387, PMID: [DOI] [PMC free article] [PubMed] [Google Scholar]
58.Capek P, McLean GK, Berkowitz HD. Femoropopliteal angioplasty. Factors influencing long-term success. Circulation. (1991) 83:I70–80. PMID: [PubMed] [Google Scholar]
59.Bausback Y, Botsios S, Flux J, Werner M, Schuster J, Aithal J, et al. Outback catheter for femoropopliteal occlusions: immediate and long-term results. J Endovasc Ther. (2011) 18:13–21. doi: 10.1583/10-3248.1, PMID: [DOI] [PubMed] [Google Scholar]
60.Marshall AP, Luo W, Wang X-L, Lin T, Cai Y, Guzman RJ. Medial artery calcification increases neointimal hyperplasia after balloon injury. Sci Rep. (2019) 9:8193. doi: 10.1038/s41598-019-44668-4, PMID: [DOI] [PMC free article] [PubMed] [Google Scholar]
61.Brodmann M, Werner M, Brinton TJ, Illindala U, Lansky A, Jaff MR, et al. Safety and performance of Lithoplasty for treatment of calcified peripheral artery lesions. J Am Coll Cardiol. (2017) 70:908–10. doi: 10.1016/j.jacc.2017.06.022, PMID: [DOI] [PubMed] [Google Scholar]
62.Wong CP, Chan LP, Au DM, Chan HWC, Chan YC. Efficacy and safety of intravascular lithotripsy in lower extremity peripheral artery disease: a systematic review and meta-analysis. Eur J Vasc Endovasc Surg. (2022) 63:446–56. doi: 10.1016/j.ejvs.2021.10.035, PMID: [DOI] [PubMed] [Google Scholar]
63.Brodmann M, Holden A, Zeller T. Safety and feasibility of intravascular lithotripsy for treatment of below-the-knee arterial Stenoses. J Endovasc Ther. (2018) 25:499–503. doi: 10.1177/1526602818783989, PMID: [DOI] [PMC free article] [PubMed] [Google Scholar]
64.Tepe G, Brodmann M, Bachinsky W, Holden A, Zeller T, Mangalmurti S, et al. Intravascular lithotripsy for peripheral artery calcification: mid-term outcomes from the randomized disrupt PAD III trial. J Soc Cardiovasc Angiogr Interv. (2022) 1:100341. doi: 10.1016/j.jscai.2022.100341 [DOI] [PMC free article] [PubMed] [Google Scholar]
65.Hoogervorst R, van Overhagen H, de Jong PA, Spiering W, de Borst GJ, Veger HTC, et al. Treatment of arterial calcification in patients with chronic limb threatening ischemia with etidronate: protocol of an investigator-initiated multicenter, double blind, placebo-controlled, randomized clinical trial. CVIR Endovasc. (2022) 5:26. doi: 10.1186/s42155-022-00298-y, PMID: [DOI] [PMC free article] [PubMed] [Google Scholar]
66.Chen N-C, Hsu C-Y, Chen C-L. The strategy to prevent and regress the vascular calcification in dialysis patients. Biomed Res Int. (2017) 2017:9035193–11. doi: 10.1155/2017/9035193, PMID: [DOI] [PMC free article] [PubMed] [Google Scholar]
67.Raggi P, Chertow GM, Torres PU, Csiky B, Naso A, Nossuli K, et al. The ADVANCE study: a randomized study to evaluate the effects of cinacalcet plus low-dose vitamin D on vascular calcification in patients on hemodialysis. Nephrol Dial Transplant. (2011) 26:1327–39. doi: 10.1093/ndt/gfq725, PMID: [DOI] [PubMed] [Google Scholar]
68.Schantl AE, Ivarsson ME, Leroux J-C. Investigational pharmacological treatments for vascular calcification. Adv Ther. (2019) 2:1800094. doi: 10.1002/adtp.201800094 [DOI] [Google Scholar]
69.Karamched SR, Nosoudi N, Moreland HE, Chowdhury A, Vyavahare NR. Site-specific chelation therapy with EDTA-loaded albumin nanoparticles reverses arterial calcification in a rat model of chronic kidney disease. Sci Rep. (2019) 9:2629. doi: 10.1038/s41598-019-39639-8, PMID: [DOI] [PMC free article] [PubMed] [Google Scholar]
70.Lefthériotis G, Omarjee L, Le Saux O, Henrion D, Abraham P, Prunier F, et al. The vascular phenotype in pseudoxanthoma elasticum and related disorders: contribution of a genetic disease to the understanding of vascular calcification. Front Genet. (2013) 4:4. doi: 10.3389/fgene.2013.00004, PMID: [DOI] [PMC free article] [PubMed] [Google Scholar]
71.van Zuydam NR, Stiby A, Abdalla M, Austin E, Dahlström EH, McLachlan S, et al. Genome-wide association study of peripheral artery disease. Circulation. (2021) 14:e002862. doi: 10.1161/CIRCGEN.119.002862, PMID: [DOI] [PMC free article] [PubMed] [Google Scholar]
72.Klarin D, Lynch J, Aragam K, Chaffin M, Assimes TL, Huang J, et al. Genome-wide association study of peripheral artery disease in the million veteran program. Nat Med. (2019) 25:1274–9. doi: 10.1038/s41591-019-0492-5, PMID: [DOI] [PMC free article] [PubMed] [Google Scholar]
73.Zekavat SM, Aragam K, Emdin C, Khera AV, Klarin D, Zhao H, et al. Genetic association of finger photoplethysmography-derived arterial stiffness index with blood pressure and coronary artery disease. Arterioscler Thromb Vasc Biol. (2019) 39:1253–61. doi: 10.1161/ATVBAHA.119.312626, PMID: [DOI] [PMC free article] [PubMed] [Google Scholar]
Articles from Frontiers in Cardiovascular Medicine are provided here courtesy of Frontiers Media SA
ACTIONS
View on publisher site
PDF (788.5 KB)
Cite
Collections
Permalink PERMALINK
Copy
RESOURCES
Similar articles
Cited by other articles
Links to NCBI Databases
On this page
Abstract
Introduction
Conclusion
Author contributions
Funding
Conflict of interest
Publisher’s note
References
Cite
Copy
Download .nbib.nbib
Format:
Add to Collections
Create a new collection
Add to an existing collection
Name your collection
Choose a collection
Unable to load your collection due to an error
Please try again
Add Cancel
Follow NCBI
NCBI on X (formerly known as Twitter)NCBI on FacebookNCBI on LinkedInNCBI on GitHubNCBI RSS feed
Connect with NLM
NLM on X (formerly known as Twitter)NLM on FacebookNLM on YouTube
National Library of Medicine 8600 Rockville Pike Bethesda, MD 20894
Web Policies
FOIA
HHS Vulnerability Disclosure
Help
Accessibility
Careers
NLM
NIH
HHS
USA.gov
Back to Top |
8662 | https://www.pediatricsurgical.com/orchiopexy/ | Orchiopexy
Orchiopexy is a surgical procedure to move testicles from the abdomen or groin into the scrotum. Before birth, the testicle moves from the pelvic area down into the scrotum. In some boys, this does not occur and it is called undescended testicles. This issue may require more than one operation depending on the position of the testes and any other associated problems. An Orchiopexy is usually done in early childhood. If undescended testicles are left too long, changes occur which can affect growth and function.
The physicians of Pediatric Surgical Associates perform this surgery more than any other pediatric surgical practice in the Midwest. If you would like to inquire about the services provided through Pediatric Surgical Associates, please give our office a call at (612) 813-8000.
Additional Resources |
8663 | https://www.energy.gov/science/doe-explainsprotons | DOE Explains...Protons | Department of Energy
Skip to main content
An official website of the United States government
Here's how you know
Here's how you know
Official websites use .gov
A .gov website belongs to an official government organization in the United States.
Secure .gov websites use HTTPS
A lock ( ) or https:// means you’ve safely connected to the .gov website. Share sensitive information only on official, secure websites.
About
About
Our Agency
Our Agency
History
Our Leadership & Offices
The National Nuclear Security Administration
The Energy Information Administration
National Laboratories
Power Marketing Administrations
Our Outreach
Our Outreach
Newsroom
Digital Engagement and Media
Brand Guides
Our Mission
Our Mission
Security & Safety
Security & Safety
Nuclear Security
Energy Security
Cybersecurity
Environmental & Legacy Management
Research, Technology, & Economic Security
Emergency Response
Scientific Excellence
Scientific Excellence
Scientific Research Database (OSTI)
Nobel Prize Laureates
National Science Bowl
Environment
Environment
Earth Systems Modeling
Extreme Weather Resiliency
Energy Access
Energy Access
Grid Deployment & Transmission
Puerto Rico Grid Resilience & Transitions (PR 100)
Tribal Energy Access
Economic Growth
Economic Growth
Energy Workforce
American Manufacturing
Technology Transitions & Early Investments
Commercial Implementation
Global Diplomacy & Leadership
Global Diplomacy & Leadership
Arctic Cooperation
International Market Development
New Horizons
New Horizons
Fusion
Supercomputing
Quantum Science
Space Exploration & The Universe
Cancer Research
Biotechnology
Artificial Intelligence
Topics
Topics
View All Topics
Consumer Savings
Consumer Savings
Tax Credits & Rebates
Energy Saving Tips
Weatherization Assistance Program
Services & Opportunities
Services & Opportunities
Career Pathways
Career Pathways
Internships & Fellowships
Apprenticeships & Workforce Development
Work at DOE
Breadcrumb
Office of Science
DOE Explains...Protons
DOE Explains...Protons
Quark and gluons inside a proton. Two up quarts, one down quark, gluons holding them together.
Image courtesy of Brookhaven National Laboratory
The proton is a subatomic particle with a positive electrical charge. They are found in every atomic nucleus of every element.
In almost every element, protons are accompanied by neutrons. The only exception is the nucleus of the simplest element, hydrogen. Hydrogen contains only a single proton and no neutrons.
The number of protons in an atom determines its atomic number in the Periodic Table of Elements. The number of protons plus the number of neutrons in a nucleus determines an element’s “baryon number,” which is nearly equal to the atomic mass of that element. Protons have a slightly smaller mass than neutrons. That’s why atomic mass numbers aren’t whole numbers. For example, carbon has six protons and six neutrons for an atomic mass of 12.011 “atomic mass units.”
Because they are part of the nucleus, scientists sometimes refer to protons and neutrons as nucleons. Scientists also refer to protons and neutrons as hadrons. Hadron is a term for a composite subatomic particle – a particle that consists of even smaller particles. Specifically, hadrons are made of two or more quarks held together by the strong interaction force, one of the fundamental forces in the universe. Protons contain two up quarks and one down quark, while neutrons contain one up quark and two down quarks. These quarks are called “valence” quarks to contrast them with the “sea” quarks, which constantly pop in and out of existence inside protons and neutrons.
Unlike neutrons, protons are stable. By stable, scientists mean that free protons—protons that are not connected to neutrons in the nucleus—do not break down, or decay, on their own. This is different from neutrons, which are also made up of smaller particles but break down due to radioactive decay. In fact, protons are the only stable type of subatomic particle that is made of even smaller particles.
DOE Office of Science: Contributions to Subatomic Particle Research
The DOE Office of Nuclear Physics in the Office of Science supports research to understand all forms of nuclear matter and the subatomic particles that make up atomic nuclei. This research includes unraveling previously unknown properties of atoms and the subatomic particles they are composed of in their natural state. This information could have important applications in medicine, commerce, and national defense. Another area of study is understanding precisely how nuclei are structured depending on the number of protons and neutrons inside them. Other research focuses on heating nuclei to the temperature of the early universe to understand how they condensed out of the quark-gluon soup that existed just after the Big Bang.
Fast Facts
Learn more with this Interactive Periodic Table of Elements.
Scientists are getting closer to measuring the radius of the proton.
Many things in nature are symmetrical, but not protons. Learn more about proton asymmetry and what it means.
Learn the technical details of the proton and other subatomic particles at the interactive Particle Data Group site.
Resources
The Department of Energy Office of Science’s Nuclear Physics program
The U.S. Particle Physics research consortium
Science research highlight: How stiff is the proton?
Science research highlight: The proton antimatter imbalance
Scientific terms can be confusing.DOE Explainsoffers straightforward explanations of key words and concepts in fundamental science. It also describes how these concepts apply to the work that the Department of Energy’s Office of Science conducts as it helps the United States excel in research across the scientific spectrum.
Committed to Restoring America’s Energy Dominance.
Follow Us
Link to FacebookLink to InstagramLink to LinkedinLink to YoutubeLink to XLink to Flickr
Quick Links
Leadership & Offices
Mission
Newsroom
Contact Us
Careers
Resources
Budget & Performance
Directives, Delegations, & Requirements
Freedom of Information Act (FOIA)
Inspector General
Privacy Program
Federal Government
USA.gov
The White House
Open Gov
Accessibility
Privacy
Information Quality
Web Policies
Vulnerability Disclosure Program
Whistleblower Protection
Equal Employment Opportunity |
8664 | https://www.wikitechy.com/technology/backtracking-set-8-solving-cryptarithmetic-puzzles/ | C Programming - Backtracking Set 8 Solving Cryptarithmetic Puzzles - Wikitechy
TUTORIALS
TECHNOLOGY
ARTICLES
FULL FORMS
ERRORS & FIXES
INTERVIEW
PROJECTS
VIDEOS
QUANTITATIVE APTITUDE
NON VERBAL
GROUP DISCUSSION
COUNTRY WISE INTERVIEW QUESTIONS
ENGINEERING
CAREERS
HR INTERVIEW QUESTIONS & ANSWERS
GD INTERVIEW
RESUME SAMPLES
LETTERS
ENGINEERING
APTITUDE
REASONING
COMPANY QUESTIONS
COUNTRY WISE VISA
VISA DRESS CODE
BUSINESS
TOP WEB HOSTING
TOP CAR ACCIDENT LAWYERS
TOP CAR INSURANCE
TOP SOFTWARE TOOLS
TOP 10
LUNG CANCER
Wikitechy
TUTORIALS
TECHNOLOGY
ARTICLES
FULL FORMS
ERRORS & FIXES
INTERVIEW
PROJECTS
VIDEOS
QUANTITATIVE APTITUDE
NON VERBAL
GROUP DISCUSSION
COUNTRY WISE INTERVIEW QUESTIONS
ENGINEERING
CAREERS
HR INTERVIEW QUESTIONS & ANSWERS
GD INTERVIEW
RESUME SAMPLES
LETTERS
ENGINEERING
APTITUDE
REASONING
COMPANY QUESTIONS
COUNTRY WISE VISA
VISA DRESS CODE
BUSINESS
TOP WEB HOSTING
TOP CAR ACCIDENT LAWYERS
TOP CAR INSURANCE
TOP SOFTWARE TOOLS
TOP 10
LUNG CANCER
Press ESC to close
Or check our Popular Categories...
ANDROIDMOBILECodingPCAlgorithmINTERNET
C
C Programming – Backtracking Set 8 Solving Cryptarithmetic Puzzles
Newspapers and magazines often have crypt-arithmetic puzzles of the form:
SEND
+ MORE
MONEY
The goal here is to assign each letter a digit from 0 to 9 so that the arithmetic works out correctly. The rules are that all occurrences of a letter must be assigned the same digit, and no digit can be assigned to more than one letter.
First, create a list of all the characters that need assigning to pass to Solve
If all characters are assigned, return true if puzzle is solved, false otherwise
Otherwise, consider the first unassigned character
for (every possible choice among the digits not in use)
make that choice and then recursively try to assign the rest of the characters
if recursion sucessful, return true
if !successful, unmake assignment and try another digit
[ad type=”banner”]
If all digits have been tried and nothing worked, return false to trigger backtracking C Program
c
bool ExhaustiveSolve(puzzleT puzzle, string lettersToAssign)
{
if (lettersToAssign.empty()) // no more choices to make
return PuzzleSolved(puzzle); // checks arithmetic to see if works
for (int digit = 0; digit <= 9; digit++) // try all digits
{
if (AssignLetterToDigit(lettersToAssign, digit))
{
if (ExhaustiveSolve(puzzle, lettersToAssign.substr(1)))
return true;
UnassignLetterFromDigit(lettersToAssign, digit);
}
}
return false; // nothing worked, need to backtrack
}
The algorithm above actually has a lot in common with the permutations algorithm, it pretty much just creates all arrangements of the mapping from characters to digits and tries each until one works or all have been successfully tried. For a large puzzle, this could take a while.
A smarter algorithm could take into account the structure of the puzzle and avoid going down dead-end paths. For example, if we assign the characters starting from the ones place and moving to the left, at each stage, we can verify the correctness of what we have so far before we continue onwards. This definitely complicates the code but leads to a tremendous improvement in efficiency, making it much more feasible to solve large puzzles.Below pseudocode in this case has more special cases, but the same general design
Start by examining the rightmost digit of the topmost row, with a carry of 0
If we are beyond the leftmost digit of the puzzle, return true if no carry, false otherwise
If we are currently trying to assign a char in one of the addends
If char already assigned, just recur on row beneath this one, adding value into sum
If not assigned, then
for (every possible choice among the digits not in use)
make that choice and then on row beneath this one, if successful, return true
if !successful, unmake assignment and try another digit
return false if no assignment worked to trigger backtracking
Else if trying to assign a char in the sum
If char assigned & matches correct,
recur on next column to the left with carry, if success return true,
If char assigned & doesn’t match, return false
If char unassigned & correct digit already used, return false
If char unassigned & correct digit unused,
assign it and recur on next column to left with carry, if success return true
return false to trigger backtracking
[ad type=”banner”]
Categorized in:
AlgorithmBacktrackingC ProgrammingCoding
Tagged in:
4 queens problem using backtracking algorithm, arithmetic problems with solutions, artificial intelligence ppt slides, artificial intelligence problems and their solutions pdf, artificial intelligence problems ppt, backtracking algorithm pdf, backtracking general method, base games, concentrated solar power ppt, constraint satisfaction problem in artificial intelligence, constraint satisfaction problem in artificial intelligence pdf, cp program, cryptarithmetic, cryptarithmetic addition, cryptarithmetic addition problems with solutions pdf, cryptarithmetic multiplication problem, cryptarithmetic multiplication questions, cryptarithmetic problem, cryptarithmetic problem in artificial intelligence, cryptarithmetic problem in artificial intelligence examples, cryptarithmetic problem solved examples, cryptarithmetic problem solver, cryptarithmetic problems for elitmus, cryptarithmetic problems with solutions pdf, cryptarithmetic questions, cryptarithmetic solver, csp, csp full form, domain constraints, fast track objective arithmetic pdf, hill climbing algorithm example, hill climbing algorithm in artificial intelligence, how to solve cryptarithmetic problem, how to solve cryptarithmetic problem in artificial intelligence examples, how to solve cryptarithmetic problems easily, map coloring problem, n queens, problem solving in artificial intelligence pdf, the strain, what is constrain
Share Article:
Wikitechy Editor
Wikitechy Founder, Author, International Speaker, and Job Consultant. My role as the CEO of Wikitechy, I help businesses build their next generation digital platforms and help with their product innovation and growth strategy. I'm a frequent speaker at tech conferences and events.
View All Articles
Leave a Reply
[x] Save my name, email, and website in this browser for the next time I comment.
Related Articles
Msafely Review 2025: Is it Worth It? Features, Performance, and More
May 16, 2025
888starz App Review
April 28, 2024
ChainX (PCX): Empowering Individuals to Achieve Financial Self-Determination
September 9, 2023
Determining the Right Entry and Exit Points in Digital Coin Trading
February 14, 2023
Other Stories
R
Randomized Algorithms | Set 0 (Mathematical Background)
Next Story
C
Cpp Algorithm – Ford-Fulkerson Algorithm for Maximum Flow Problem
Previous Story
Recent posts
The best slot games of 2023
December 14, 2023
Download Samsung Galaxy S8 Stock Apps and System Dump
May 5, 2017
S
Search in an almost sorted array
October 15, 2017
The Growing Popularity of Cryptocurrency in Online Casinos
April 9, 2023
How To Learn About Camcorder Video File Formats
March 31, 2018
INTERVIEW TIPS
Final Year Projects
GD Interview
Resume Samples
Engineering
Aptitude
Reasoning
Company Questions
Country wise visa
Interview Dress Code
CAREER GUIDANCE
Entrance Exam
Colleges
Admission Alerts
ScholarShip
Education Loans
Letters
Learn Languages
Summer Offline Internship
Internship for cse students
Internship for it students
Internship for ece students
Internship for eee students
Internship for mechanical engineering students
Internship for aeronautical engineering students
Internship for civil engineering students
Internship for bcom students
Internship for mcom students
Internship for bca students
Internship for mca students
Internship for biotechnology students
Internship for biomedical engineering students
Internship for bsc students
Internship for msc students
Internship for bba students
Internship for mba students
Summer Online Internship
Online internship for cse students
Online internship for ece students
Online internship for eee students
Online internship for it students
Online internship for mechanical engineering students
Online internship for aeronautical engineering students
Online internship for civil engineering students
Online internship for bcom students
Online internship for mcom students
Online internship for bca students
Online internship for mca students
Online internship for biotechnology students
Online internship for biomedical engineering students
Online internship for bsc students
Online internship for msc students
Online internship for bba students
Online internship for mba students
Internship in Chennai
Intenship in Chennai
Intenship in Chennai for CSE Students
Internship in Chennai for IT Students
Internship in Chennai for ECE Students
Internship in Chennai for EEE Students
Internship in Chennai for EIE Students
Internship in Chennai for MECH Students
Internship in Chennai for CIVIL Students
Internship in Chennai for BIOTECH Students
Internship in Chennai for AERO Students
Internship in Chennai for BBA Students
Internship in Chennai for MBA Students
Internship in Chennai for MBA HR Students
Internship in Chennai for B.Sc Students
Internship in Chennai for M.Sc Students
Internship in Chennai for BCA Students
Internship in Chennai for MCA Students
Internship in Chennai for B.Com Students
Internship in Chennai for M.Com Students
Programming / Technology Internship in Chennai
Data Science Internship in Chennai
Artificial Intelligence Internship in Chennai
Web Development Internship in Chennai
Android Internship in Chennai
Android Internship in Chennai
Cloud Computing Internship in Chennai
.Net Internship in Chennai
JAVA Internship in Chennai
Ethical Hacking Internship in Chennai
IOT Internship in Chennai
Machine Learning Internship in Chennai
Networking Internship in Chennai
Robotics Internship in Chennai
Matlab Internship in Chennai
World's No 1 Animated self learning Website with Informative tutorials explaining the code and the choices behind it all.
WIKITECHY
About Us
Terms Of Use
Privacy Policy
Contact Us
Wikitechy
WordPress Theme by EstudioPatagon |
8665 | https://www.cerritos.edu/dford/SitePages/Math_140/Math140Lecture7_sp07.pdf | 2.4 Trigonometric Functions of General Angles So far we have been talking about trig functions of acute angles.
We employ the rectangular coordinate system for trig functions of all angles (acute or not). We place the angle in standard position (vertex at the origin, initial side on positive x-axis). We’ll let the point (a,b) be on the terminal side of the angle.
-4 -3 -2 -1 0 1 2 3 -4 -3 -2 -1 0 1 2 3 4 x- axis y- axis origin (0,0) (-2,2) θ Adjacent side Opposite side Hypotenuse We can form a right triangle from any point (a,b) and the origin (0,0). In the example above, (a,b) = (-2,2).
The adjacent side = a = -2 Using the actual value of the x-coordinate, a, instead of the distance identifies the quadrant of this angle.
The opposite side = b = 2 The hypotenuse, c, is the distance from (a,b) to the origin c = 2 2 8 ) 2 ( ) 2 ( ) 0 ( ) 0 ( 2 2 2 2 2 2 = = + − = + = − + − b a b a a/b= -1/1 = -1 a/b=adj/opp cot(θ) c/a= c/a=hyp/adj sec(θ) c/b= c/b=hype/opp csc(θ) b/a= 1/-1 = -1 b/a=opp/adj tan(θ) a/c= a/c=adj/hyp cos(θ) b/c= b/c=opp/hyp sin(θ) Exact Value Definition Trig Fctn 2 2 2 2 2 = 2 2 2 2 2 − = − 2 2 2 2 = 2 2 2 2 − = − PDF created with pdfFactory Pro trial version www.pdffactory.com Finding the Exact Value of the Six Trig Functions of Quadrantal Angles Example 2 on p.154 a) θ = 0rad = 0° b) θ = π/2rad = 90° c) θ = πrad = 180° d) θ = 3π/2rad = 270° a) 0° is on the positive x-axis, so choose a point on the positive x-axis. (a,b)=(1,0) c = square root of (a2 + b2)=square root of (12 + 02) = 1.
a/b= 0/1 = 1 a/b= =1/0 undefined a/b= 0/1 = 0 a/b= 1/0 undefined a/b= adj/opp cot(θ) c/a=1/0 undefined c/a=1/-1= -1 c/a=1/0 undefined c/a=1/1=1 c/a= hyp/adj sec(θ) c/b=1/-1= -1 c/b=1/0 undefined c/b=1/1=1 c/b=1/0 undefined c/b= hyp/opp csc(θ) b/a= -1/0 undefined b/a= 0/-1 = 0 b/a= 1/0 undefined b/a= 0/1 = 0 b/a= opp/adj tan(θ) a/c= 0/1 = 0 a/c= -1/1 = -1 a/c= 0/1 = 0 a/c= 1/1 = 1 a/c= adj/hyp cos(θ) b/c=-1/1 = 1 b/c=0/1 = 0 b/c=1/1 = 1 b/c=0/1 = 0 b/c= opp/hyp sin(θ) Exact Value for θ = 3π/2rad = 270° [use (a,b)=(0,-1)] Exact Value for θ = πrad = 180° [use (a,b)=(-1,0)] Exact Value for θ = π/2rad = 90° [use (a,b)=(0,1)] Exact Value for θ = 0rad = 0° [use (a,b)=(1,0)] Def.
Trig Funct.
-4 -3 -2 -1 0 1 2 3 -4 -3 -2 -1 0 1 2 3 4 y δ= 0rad = 0° β= πrad = 180° α= π/2rad = 90° Φ= 3π/2rad = 270° α β φ δ PDF created with pdfFactory Pro trial version www.pdffactory.com Coterminal Angles If θ is an angle measured in degrees, then θ ±360°(k) , where k is any integer, is an angle coterminal with θ.
If θ is an angle measured in radians, then θ ±2πk , where k is any integer, is an angle coterminal with θ.
Since coterminal angles are basically equal, all their trigonometry functions are equal.
Example 3 on p.156 Find the exact value of each of the following: a) sin 390° Reduce 390° to an angle less than 360° by subtracting the highest multiple of 360° that is less than 390°. That would be 360x1 = 360 sin(390°) = sin(390° - 360°) = sin(30°) = ½ c) tan 9π/4 Reduce 9π/4 to an angle less than 2π by subtracting the highest multiple of 2π that is less than 9π/4.
2π = 8π/4 tan 9π/4 = tan (8π/4 + π/4) = tan π/4 = 1 d) sec(-7π/4) Change -7π/4 to an angle between 0 and 2π by adding a multiple of 2π.
-7π/4 + 8π/4 = π/4 sec(-7π/4) = sec(-7π/4 + 8π/4) = sec(π/4) = square root of 2.
PDF created with pdfFactory Pro trial version www.pdffactory.com Signs of Trigonometric Functions -+ -IV + --III --+ II + + + I tan, cot cos, sec sin, csc Quadrant -4 -3 -2 -1 0 1 2 3 -4 -3 -2 -1 0 1 2 3 4 δ= 0rad = 0° β= πrad = 180° α= π/2rad = 90° Φ= 3π/2rad = 270° I (+,+) All trig functions >0 II (-,+) Sin, csc >0 cos,sec,tan,cot <0 III (-,-) Tan,cot > 0 sin, cos, sec, csc <0 IV (+,-) Cos,sec > 0 sin, csc, tan,cot <0 A S T C ASTC = “All Surviving Trig take Calculus” PDF created with pdfFactory Pro trial version www.pdffactory.com 2.5 Finding the Value of the Trig Functions Using a Point on the Unit Circle A unit circle is a circle with its center at the origin (0,0) and a radius of 1. The equation for this circle is x2+y2 = 1.
Example 1 on p. 165-166 Find the value of sin t, cos t, tan t, csc t, sec t, and cot t for a point P = (a,b) = on a unit circle.
− 2 3 , 2 1 (0,1) − = 2 3 , 2 1 ) , ( b a trad a b θ r=1 cos t = a/r = a sin t = b/r = b tan t = b/a cot t = a/b sec t = r/a csc t = r/b (0,r) ) , ( b a trad a b θ r cos t = a/1 = a= -½ sin t = b/1 = b= tan t = b/a = cot t = a/b = sec t = 1/a = 1 / -½ = -2 csc t = 1/b = Now let’s look at a circle with center (0,0) and radius r. The equation for this circle is x2+y2 = r2.
Notice for any point (a,b) on the circle, a2+b2 = r2.
So r = 2 2 b a + 2 3 3 3 3 1 2 3 2 1 − = − = − 3 1 3 2 1 2 3 − = − = − 3 3 2 3 2 2 3 1 = = Now you try #15 on p.172 PDF created with pdfFactory Pro trial version www.pdffactory.com Domain and Range of the Trigonometric Functions (0,1) ) , ( b a trad a b θ r Notice that for every right triangle formed inside a unit circle with one corner at the center and another corner on the circle, the hypotenuse is the radius, r.
The hypotenuse is always the longest side of a right triangle. Therefore |a| ≤r, and |b| ≤r.
This mains that the trigonometric functions that are ratios with the hypotenuse as the denominator are always ≤1 and ≥-1. This gives us the following ranges for sin and cos: sin (θ) = opp/hyp -1 ≤sin(θ) ≤1 sin (θ) = adj/hyp -1 ≤cos(θ) ≤1 Reciprocally: csc(θ) ≥1 or csc(θ) ≤-1 sec(θ) ≥1 or sec(θ) ≤-1 The domain of sin and cos is the set of all real numbers. Any number can be used as θ. θ > 360°or 2πrad is just an angle that has made more than 1 full revolution around the circle.
θ < 0 is just an angle that has rotated clockwise around the circle.
Since csc(θ) is the reciprocal of sin, that is, 1/ sin(θ), then csc(θ) is undefined when sin(θ) = 0. This occurs when b=0 (on the x-axis). So the domain of csc(θ) is all real numbers except for integral (integer valued) multples of π. The domain of sin and cos is the set of all real numbers. Any number can be used as θ. θ > 360°or 2πrad is just an angle that has made more than 1 full revolution around the circle.
θ < 0 is just an angle that has rotated clockwise around the circle.
PDF created with pdfFactory Pro trial version www.pdffactory.com Domain and Range of Tan and Cot (a,b) =(0,1) π/2 What about tan and cot?
tan(θ) = opp/adj = b/a If a = 0, then tan(θ) is undefined.
Therefore the domain of tan does not include angles at points where a=0. Those are all odd multiples of π/2 (i.e. π/2, 3π/2, 5π/2, etc..) The range of tan(θ) is all real numbers, because when b is large and a is small, tan(θ) can increase infinitely; and when b is small and a is large, tan(θ) can decrease infinitely.
cot(θ) = adj/opp = a/b If b = 0, then cot(θ) is undefined.
Therefore the domain of cot does not include angles at points where b=0. Those are all integral (integer valued) multiples of π. (i.e. 0, π, 2π, 3π, 4π, etc..) Similarly to tan, the range of cot is all real numbers.
PDF created with pdfFactory Pro trial version www.pdffactory.com Period of Trig Functions Review of Coterminal Angles: If θ is an angle measured in degrees, then θ ±360°(k) , where k is any integer, is an angle coterminal with θ.
If θ is an angle measured in radians, then θ ± 2πk , where k is any integer, is an angle coterminal with θ.
Since coterminal angles are basically equal, all their trigonometry functions are equal.
Therefore sin(θ + 2π) = sin(θ) and cos(θ + 2π) = cos(θ) Functions that exhibit this type of behavior are called periodic functions.
A function f is periodic if there is a positive number p such that, whenever θ is in the domain of f, so is θ+p, and f(θ+p) = f(θ).
Let’s revisit our “ASTC” graph: -4 -1 0 1 -1 0 1 δ= 0rad = 0° β= πrad = 180° α= π/2rad = 90° Φ= 3π/2rad = 270° I (+,+) All trig functions >0 II (-,+) Sin, csc >0 cos,sec,tan,cot <0 III (-,-) Tan,cot > 0 sin, cos, sec, csc <0 IV (+,-) Cos,sec > 0 sin, csc, tan,cot <0 θ π+θ Going from 0 to 2π, sin(θ) starts out with the value 0, then rises to 1 at π/2, then goes back to 0 at π. At θ> π, sin(θ) goes from 0 to -1 at 3π/2, then back to 0 at 2π. At this point the sin values repeat. The period of the sine function is 2π.
(Likewise for cosine).
Going from 0 to 2π, tan(θ) starts out with the value 0, then rises to infinity as θ approaches π/2. In Quadrant II, tan(θ) rises from negative infinity (since the y-coordinates are + and the x-coordinates are -), then goes back to 0 at π. At θ> π, tan (θ) repeats the same process. The period of the tangent function is π. (Likewise for cotangent).
tan(θ + π) = tan(θ) and cot(θ + π) = cot(θ) (a,b) (-a,-b) PDF created with pdfFactory Pro trial version www.pdffactory.com Using Periodic Properties to find exact trig function values; Even-Odd Properties Example 2 on p.171 Find the exact value of b) tan 5π/4 Since the period of tan is π, we just need to subtract π from 5π/4 to get a familiar angle for which we know that exact value of the tan function.
Tan (5π/4) = tan(5π/4 – 4π/4) = tan(π/4).
Recall that π/4 is the same as 45°. At 45°, the point (a,b) on the unit circle has a=b, so tan(π/4) = a/b = 1.
Now you do #21 on p.173 Even-Odd Properties A function is even if f(-θ) = f(θ) for all θ in the domain of f.
A function is odd if f(-θ) = - f(θ) for all θ in the domain of f.
Even Trig Functions cos(-θ) = cos(θ) sec(-θ) = sec(θ) Odd Trig Functions sin(-θ) = -sin(θ) csc(-θ) = -csc(θ) tan(-θ) = -tan(θ) cot(-θ) = -cot(θ) θ -θ (a,b) (a,-b) PDF created with pdfFactory Pro trial version www.pdffactory.com Example 3 on p.172 Find the exact value of d) 4 37π − The tangent function repeats its values for any integral multiple of π. Therefore, let’s add (since the angle is negative) a multiple of π to that will give us a reference angle.
Since tan is an ODD function, we can use the property that tan(-θ) = -tan(θ).
Now you do #37 and #73 on p.173 π π 9 4 36 = − 4 37 tan π − = + − = − 4 tan 4 36 4 37 tan 4 37 tan π π π π 1 4 tan 4 tan − = − = − π π PDF created with pdfFactory Pro trial version www.pdffactory.com Homework Ch 2.4 p. 163 #1-117 EOO Ch. 2.5 #9-84 ETP PDF created with pdfFactory Pro trial version www.pdffactory.com |
8666 | https://www.youtube.com/watch?v=B8iucxDm1DU | Find Horizontal Asymptote of a Rational Function
Mario's Math Tutoring
458000 subscribers
212 likes
Description
25349 views
Posted: 1 Apr 2020
Learn how to find the equation of the horizontal asymptote of a rational function in this video math tutorial by Mario's Math Tutoring. We discuss the 3 scenarios: when the ratio of the highest powered term in the numerator is the same as the highest powered term in the denominator, when the denominator's power is higher than the numerator, and thirdly when the numerator's highest power is higher than the denominator. We also discuss how to recognize when there is a slant asymptote.
Related Videos to Help You Succeed!:
Graphing Rational Functions
Organized List of My Video Lessons to Help You Raise Your Scores & Pass Your Class. Videos Arranged by Math Subject as well as by Chapter/Topic. (Bookmark the Link Below)
➡️JOIN the channel as a CHANNEL MEMBER at the "ADDITIONAL VIDEOS" level to get access to my math video courses(Algebra 1, Algebra 2/College Algebra, Geometry, and PreCalculus), midterm & final exam reviews, ACT and SAT prep videos and more! (Over 390+ videos)
6 comments
Transcript:
so we're given these three rational functions and we want to find you know the equation of the horizontal asymptote okay and what we want to do is we want to consider these three cases here you want to look at the highest powered term in the numerator and the highest powered term in the denominator so that's the one with a highest exponent and what you do is you look and you say well if the exponents are the same meaning in the M equals n in this example here then the horizontal asymptote is going to be the ratio of those two coefficients that's going to be a divided by B but see that this exponent in the numerator is smaller than the one in the denominator or you could say the one in the denominator is greater than the one in the numerator either way then the horizontal asymptote is going to be y equals zero it can be right at the x axis and the third scenario is when M is greater than n meaning the numerators exponent is higher than the denominators exponent then we have no horizontal asymptote or if it's just one degree larger than the denominator then it's a slant asymptote okay so I'll show you a three examples so the first one here what do you think well you can see these are both X to the first okay so it's a tie so what that means is we end up with the ratio of the coefficients which is y equals two fourths which reduces to 1/2 so your horizontal asymptote would look something like that for number two you can see that the denominators degree is two the numerators degree is only one the denominators degree is higher so our horizontal asymptote is going to be y equals zero that's going to be right at the x-axis okay and then the third one you can see that the highest degree term here x squared compared to the highest degree term the denominator 2 acts this is a 2 for an exponent that's only a 1 here you can see that because this is larger in the numerator by one degree you're actually going to have a slant asymptote and in order to find that slant asymptote what you would do is you would do long division polynomial long division to find the equation that slant asymptote if you want to find out more about graphing how to actually graph these entire rational functions follow me over to that video right there and we'll discuss how to do all the different aspects of graphing rational functions I'll see you over in that video |
8667 | https://www.linkedin.com/posts/scientisthub_mathematics-numbertheory-science-activity-7299637051044282368-OMiS | 320,160 followers
Report this post
Ever noticed a pattern in the cubes of natural numbers? 'Nicomachus' Theorem reveals a fascinating trutheach cube of a natural number is the sum of consecutive odd numbers! 𧮠For example, 1³ = 1, 2³ = 3+5, 3³ = 7+9+11¦ and so on! This simple yet powerful theorem shows the beauty of number sequences in an elegant way. ð¢ Did you ever come across this theorem in school, or is this your first time discovering it? ð¤ Follow @Science for more! ð #mathematics #numbertheory #science
1,158 37 Comments
Like Comment
HASSAN B.
Physics and Mathematics Teacher | ICT enthusiast
7mo
Report this comment
Its my first time
Like Reply 1 Reaction
Briana Milam
--
7mo
Report this comment
This is a valuable piece of information.
Like Reply 1 Reaction 2 Reactions
Daniel Hynes, MBA
Business Development | Account Management | Client Service | Sales | AI Associate | ERP Solutions | RevOps
6mo
Report this comment
If you continue the calculation, 4 cubed is NOT equal to 13+15+17. How do you explain that?
Like Reply 1 Reaction
Owen Thompson
Experienced Senior Software Engineer/Analyst, Android Designer/Developer.
7mo
Report this comment
Very informative
Like Reply 1 Reaction
Steven Major
Java developer with a strong background in statistics
7mo
Report this comment
Every square number is the sum of consecutive odd numbers starting from 1; n^2 is the sum of the first n odd numbers (Can easily verify using Gausss rule for summing consecutive numbers and replace i in the sum with 2i - 1). So an interesting correlary is that every cube can be written as the difference of 2 squares, e.g. 64 = 13 + 15 + 17 + 19 = sum of odds 1 through 19 minus sum of odds 1 through 11 = 10^2 - 6^2
Like Reply 2 Reactions 3 Reactions
Creedence Dengre
Chair/CEO | Family Office | Wealth Preservation in a New Era of Money | Finance & Economics | Private Equity | Engineering, Science & Technology | Real Estate
7mo
Report this comment
This make me want to go buy unnecessary quantities of acrylic sheets
Like Reply 3 Reactions 4 Reactions
Prerna Tripathi
Writer/ Advisor / Consultant/ Analyst
7mo
Report this comment
Insightful
Like Reply 1 Reaction
Jahangir Alam
Global #Affiliated Marketing #Digital Marketing #Deputy General Manager(S&M), Astha Feed Industries Ltd.
7mo
Report this comment
Insightful
Like Reply 1 Reaction 2 Reactions
Sérgio Morais
Medium Voltage Product Manager | Product Specialist | Business Development Manager
7mo
Report this comment
For cube of 3, why no Uber 5 was not considered? Is it a gap on this theorem?
Like Reply 1 Reaction 2 Reactions
Morten Vaaben Refsgaard
QA & Compliance Leader | Food & Pharma | ISO 9001 | HACCP | Audits
7mo
Report this comment
Nice first time I see it. So are the squares all double in size every second time?
Like Reply 2 Reactions 3 Reactions
See more comments
To view or add a comment, sign in
320,160 followers
View Profile Follow
Explore content categories
Career
Productivity
Finance
Soft Skills & Emotional Intelligence
Project Management
Education
Technology
Leadership
Ecommerce
Sign in to view more content
Create your free account or sign in to continue your search
Welcome back
By clicking Continue to join or sign in, you agree to LinkedIns User Agreement, Privacy Policy, and Cookie Policy.
New to LinkedIn? Join now
or
New to LinkedIn? Join now
By clicking Continue to join or sign in, you agree to LinkedIns User Agreement, Privacy Policy, and Cookie Policy.
Never miss a beat on the app
Dont have the app? Get it in the Microsoft Store.
Open the app |
8668 | https://www.cnblogs.com/zhgmaths/p/16948575.html | 5.4.2 正弦函数、余弦函数的性质2 (单调性与最值)
({\color {Red}{欢迎到学科网下载资料学习 }})
【基础过关系列】2022-2023 学年高一数学上学期同步知识点剖析精品讲义 (人教 A 版 2019）
({\color {Red}{ 跟贵哥学数学，so \quad easy！}})
必修第一册同步巩固，难度 2 颗星！
基础知识
正弦函数，余弦函数的图像与性质
注 表中的 (k∈Z)
| | (y=\sin x) | (y=\cos x) |
---
| 图像 | | |
| 定义域 | (R) | (R) |
| 值域 | (\left[-1 ,1\right]) | (\left[-1 ,1\right]) |
| 周期性 | (2π) | (2π) |
| 对称中心 | ((kπ ,0)) | (\left(kπ+\dfrac{\pi}{2},0\right)) |
| 对称轴 | (x=kπ+\dfrac{\pi}{2}) | (x=kπ) |
| 单调性 | 在(\left[-\dfrac{\pi}{2}+2kπ ,\dfrac{\pi}{2}+2kπ\right])上是增函数； 在(\left[\dfrac{\pi}{2}+2kπ ,3\dfrac{\pi}{2}+2kπ\right])上是减函数. | 在(\left[-π+2kπ ,2kπ\right])上是增函数； 在(\left[2kπ ,π+2kπ\right])上是减函数. |
| 最值 | 当(x=\dfrac{\pi}{2}+2kπ)时，(y_{\max}=1)； 当(x=-\dfrac{\pi}{2}+2kπ)时，(y_{\min}=-1). | 当(x=2kπ)时，(y_{\max}=1)； 当(x=π+2kπ)时，(y_{\min}=-1). |
解释
如何理解三角函数的单调性、最值？
主要是结合图象及其周期性，比如如何理解正弦函数 (f(x)=\sin x) 在 (\left[-\dfrac{\pi}{2}+2kπ ,\dfrac{\pi}{2}+2kπ\right]) 单调递增？
① 在一个周期 (\left[-\dfrac{\pi}{2},\dfrac{3\pi}{2}\right)) 内，找到一个单调增区间 (\left[-\dfrac{\pi}{2},\dfrac{\pi}{2}\right])；
② 接着每隔一个周期 (2π) 个单位就有一个增区间，则 (\left[-\dfrac{\pi}{2}+2kπ ,\dfrac{\pi}{2}+2kπ\right]) 是 (f(x)=\sin x) 的增区间.
类似可得到正弦函数的减区间与最值，余弦函数的单调性与最值.
(也可以利用单位圆的性质研究正弦函数、余弦函数的性质)
【例】 求正弦函数 (f(x)=\sin x)、余弦函数 (f(x)=\cos x) 在 (\left[0,2π\right]) 的单调性.
解 结合图象可得，正弦函数 (f(x)=\sin x) 的增区间是 (\left[0,\dfrac{\pi}{2}\right))，(\left[\dfrac{3\pi}{2},2π\right])，减区间是 (\left[\dfrac{\pi}{2},\dfrac{3\pi}{2}\right])；
余弦函数 (f(x)=\cos x) 的增区间是 (\left(π,2π\right])，减区间是 ([0,π)).
基本方法
【题型1】 单调性
【典题 1】 求下列函数的单调递减区间．
(1) (y=\cos \left(2x+\dfrac{\pi}{3}\right))；$\qquad \qquad $ (2)(y=2\sin \left(\dfrac{\pi}{4}-x\right))．
解析 (1) 令 (z=2x+\dfrac{\pi}{3})，而函数 (y=\cos z) 的递减区间是 (\left2kπ,2kπ+π\right)．
(\therefore) 原函数递减时，可得 (2kπ≤2x+\dfrac{\pi}{3}≤2kπ+π(k∈Z))，
解得 (kπ-\dfrac{\pi}{6}≤x≤kπ+\dfrac{\pi}{3}(k∈Z))．
(\therefore) 原函数的递减区间是 (\leftkπ-\dfrac{\pi}{6},kπ+\dfrac{\pi}{3}\right)．
(2)(y=2\sin \left(\dfrac{\pi}{4}-x \right)=-2\sin \left(x-\dfrac{\pi}{4} \right))，
令 (z=x-\dfrac{\pi}{4})，
而函数 (y=-2\sin z) 的递减区间是 (\left2kπ-\dfrac{\pi}{2},2kπ+\dfrac{\pi}{2}\right)．
(\therefore) 原函数递减时，得 (2kπ-\dfrac{\pi}{2}≤x-\dfrac{\pi}{4}≤2kπ+\dfrac{\pi}{2}(k∈Z))，
得 (2kπ-\dfrac{\pi}{4}≤x≤2kπ+\dfrac{3\pi}{4}(k∈Z))．
(\therefore) 原函数的递减区间是 (\left2kπ-\dfrac{\pi}{4},2kπ+\dfrac{3\pi}{4}\right)．
点拨 求三角函数 (f(x)=A\sin (ωx+φ)(ω>0)) 的单调性，利用换元法 (z=ωx+φ)，把 (ωx+φ) 代入 (y=\sin z) 对应的单调区间；而由复合函数的单调性 “同增异减”，(y=2\sin \left(\dfrac{\pi}{4}-x \right)) 与 (y=2\sin \left(x-\dfrac{\pi}{4} \right)) 的单调性相反.
【典题 2】 下列关系式中正确的是 ( )
A．(\sin 11°<\cos 10°<\sin 168°) (\qquad \qquad \qquad \qquad) B．(\sin 168°<\sin 11°<\cos 10°) (\qquad \qquad \qquad \qquad)
C．(\sin 11°<\sin 168°<\cos 10°) (\qquad \qquad \qquad \qquad) D．(\sin 168°<\cos 10°<\sin 11°)
解析 (\cos 10°=\sin 80°，\sin 168°=\sin 12°)．
因为 (y=\sin x) 在 (\left[0,\dfrac{\pi}{2}\right]) 上递增，所以 (\sin 11°<\sin 12°<\sin 80°)，
即 (\sin 11°<\sin 168°<\cos 10°)，故选 (C).
点拨 本题利用函数的单调性比较大小，思路是统一函数名，把角度限制在一范围.
【典题 3】 已知函数 (y=\sin \left(ωx+\dfrac{\pi}{3} \right)(ω>0)) 在区间 (\left(-\dfrac{\pi}{6},\dfrac{\pi}{3} \right)) 上单调递增，则 (ω) 的取值范围是 ( )
A．(\left(0,\dfrac{1}{2}\right]) (\qquad \qquad \qquad \qquad) B．(\left[\dfrac{1}{2},1\right]) (\qquad \qquad \qquad \qquad) C．(\left(\dfrac{1}{3}, \dfrac{3}{2} \right]) $\qquad \qquad \qquad \qquad $ D．(\left[\dfrac{3}{2} ,2\right])
解析 由 (2kπ-\dfrac{\pi}{2}≤ωx+\dfrac{\pi}{3}≤2kπ+\dfrac{\pi}{2}),(k∈Z)，
得 (2kπ-\dfrac{5\pi}{6}≤ωx≤2kπ+\dfrac{\pi}{6}),(k∈Z)，
即 (\dfrac{2 k \pi-\dfrac{5 \pi}{6}}{\omega} \leq x \leq \dfrac{2 k \pi+\dfrac{\pi}{6}}{\omega}),(k∈Z)，
(\because f(x)) 在区间 (\left(-\dfrac{\pi}{6},\dfrac{\pi}{3} \right)) 上单调递增，(\therefore) 此时函数单调递增区间经过原点，
则当 (k=0) 时，增区间为 (\left[-\dfrac{5\pi}{6ω},\dfrac{\pi}{6ω}\right])．
此时满足 (\left{\begin{array}{l}
-\dfrac{5 \pi}{6 \omega} \leq-\dfrac{\pi}{6} \
\dfrac{\pi}{6 \omega} \geq \dfrac{\pi}{3}
\end{array}\right.)，得 (\left{\begin{array}{l}
\omega \leq 5 \
\omega \leq \dfrac{1}{2}
\end{array}\right.)，解得 (0<ω≤\dfrac{1}{2})，
即 (ω) 的取值范围是 (\left(0,\dfrac{1}{2}\right])，
故选：(A)．
【巩固练习】
函数 (f(x)=\sin \left(2x+\dfrac{\pi}{6} \right)) 的单调递增区间是 ( )
A．(\left[kπ+\dfrac{\pi}{6},kπ+2\dfrac{\pi}{3}\right]),((k∈Z)) (\qquad \qquad \qquad \qquad) B．(\left[kπ,kπ+\dfrac{\pi}{2}\right]),((k∈Z))
C．(\left[kπ-\dfrac{\pi}{3},kπ+\dfrac{\pi}{6}\right]),((k∈Z)) (\qquad \qquad \qquad \qquad) D．(\left[kπ-\dfrac{\pi}{2},kπ\right]),((k∈Z))
已知函数 (f(x)=\cos (ωx+φ)(ω>0,0<φ<π)) 为奇函数，(f(x)=f(2-x))，当 (ω) 取最小值时，(f(x)) 的一个单调递减区间是 ( )
A．(\left[-1,1\right]) $\qquad \qquad \qquad $ B．(\left[- \dfrac{3}{2},\dfrac{1}{3}\right]) (\qquad \qquad \qquad) C．(\left[\dfrac{\pi}{3},\dfrac{5\pi}{6}\right]) (\qquad \qquad \qquad) D．(\left[0,\dfrac{\pi}{3}\right])
若 (f(x)=\sin \left(2x-\dfrac{\pi}{4} \right))，则 ( )
A．(f(1)>f(2)>f(3)) (\qquad) B．(f(3)>f(2)>f(1)) (\qquad) C．(f(2)>f(1)>f(3)) (\qquad) D．(f(1)>f(3)>f(2))
设 (f(x)=3\sin \left(ωx-\dfrac{\pi}{12} \right)+1)，若 (f(x)) 在 (\left[-\dfrac{\pi}{3},\dfrac{\pi}{6}\right]) 上为增函数，则 (ω) 的取值范围是 ( )
A．(\left[ \dfrac{5}{12}， \dfrac{7}{2}\right]) (\qquad \qquad) B．(\left[ \dfrac{5}{4}， \dfrac{7}{2}\right]) (\qquad \qquad) C．(\left(0， \dfrac{7}{4}\right]) (\qquad \qquad) D．(\left(0， \dfrac{5}{4}\right])
设函数 (f(x)=\sin (ωx+ϕ)),(A>0),(ω>0), 若 (f(x)) 在区间 (\left[\dfrac{\pi}{6},\dfrac{\pi}{2}\right]) 上单调，且 (f\left(\dfrac{\pi}{2} \right)=f\left(\dfrac{2\pi}{3} \right)=-f\left(\dfrac{\pi}{6} \right))，
则 (f(x)) 的最小正周期为 (\underline{\quad \quad})．
参考答案
答案 (C)
解析 对于函数 (f(x)=\sin \left(2x+\dfrac{\pi}{6} \right))，令 (2kπ-\dfrac{\pi}{2}≤2x+\dfrac{\pi}{6}≤2kπ+\dfrac{\pi}{2})，
求得 (kπ-\dfrac{\pi}{3}≤x≤kπ+\dfrac{\pi}{6})，
故函数的单调增区间为 (\left[kπ-\dfrac{\pi}{3},kπ+\dfrac{\pi}{6}\right]，k∈Z)，
故选：(C)．
答案 (A)
解析 函数 (f(x)=\cos (ωx+φ)(ω>0,0<φ<π)) 为奇函数，
则 (φ=\dfrac{\pi}{2})，(f(x)=-\sin ωx)，(ω>0)；
由 (f(x)=f(2-x)) 知函数 (f(x)) 的图象关于直线 (x=1) 对称，
由 (f(x)) 是 (R) 上的奇函数知 (f(2-x)=-f(x-2))，(f(x-4)=-f(4-x))
在 (f(2-x)=f(x)) 中，以 (x-2) 代 (x) 得：(f(2-(x-2))=f(x-2))
即 (f(4-x)=f(x-2))，
所以 (f(x)=f(2-x)=-f(4-x)=f(x-4)) , 即 (f(x+4)=f(x))，
所以 (f(x)) 是以 (4) 为周期的周期函数；
所以 (T=\dfrac{2 \pi}{\omega}=4)，解得 (ω=\dfrac{\pi}{2})；
当 (ω) 取最小值 (\dfrac{\pi}{2}) 时，(f(x)=-\sin \dfrac{\pi}{2} x)；
令 (-\dfrac{\pi}{2}+2kπ≤\dfrac{\pi}{2} x≤\dfrac{\pi}{2}+2kπ)，(k∈Z)；解得 (-1+4k≤x≤1+4k)，(k∈Z)；
令 (k=0)，得 (-1≤x≤1)，
所以 (f(x)) 的一个单调递减区间是 (\left[-1,1\right])．
故选：(A)．
答案 (A)
解析 利用函数的单调性：由于函数 (f(x)) 在区间 (\left[\dfrac{\pi}{2},\dfrac{3\pi}{2}\right]) 上单调递减，
故：(\dfrac{\pi}{2}≤2x-\dfrac{\pi}{4}≤\dfrac{3\pi}{2})，解得：(\dfrac{3\pi}{8}≤x≤\dfrac{7\pi}{8})，
所以 (f(1)) 的值在 (x=\dfrac{3\pi}{8}) 的左边且离得比较近，接近于最大值，故 (f(1)) 最大，
由于 (\dfrac{3\pi}{8}≤2<x<3≤\dfrac{7\pi}{8})，故：(f(2)>f(3))，
所以 (f(1)>f(2)>f(3))．
故选： (A)．
答案 (D)
解析 设 (f(x)=3\sin \left(ωx-\dfrac{\pi}{12} \right)+1)，在 (\left[-\dfrac{\pi}{3},\dfrac{\pi}{6}\right]) 上，(\omega x-\dfrac{\pi}{12} \in\left[-\dfrac{\omega \pi}{3}-\dfrac{\pi}{12}, \dfrac{\omega \pi}{6}-\dfrac{\pi}{12}\right])，
由于 (f(x)) 为增函数， (\therefore\left{\begin{array}{l}
-\dfrac{\omega \pi}{3}-\dfrac{\pi}{12} \geq-\dfrac{\pi}{2} \
\dfrac{\omega \pi}{6}-\dfrac{\pi}{12} \leq \dfrac{\pi}{2}
\end{array}\right.)，即 (\left{\begin{array}{l}
\omega \leq \dfrac{5}{4} \
\omega \leq \dfrac{7}{2}
\end{array}\right.)，求得 (0<\omega \leq \dfrac{5}{4})，
故选：(D)．
答案 (π)
解析 函数 (f(x)=\sin (ωx+ϕ)),(A>0),(ω>0), 若 (f(x)) 在区间 (\left[\dfrac{\pi}{6},\dfrac{\pi}{2}\right]) 上单调，
则 (\dfrac{T}{2}=\dfrac{\pi}{\omega} \geq \dfrac{\pi}{2}-\dfrac{\pi}{6})，(\therefore 0<ω≤3)．
(\because f\left(\dfrac{\pi}{2} \right)=f\left(\dfrac{2\pi}{3} \right)=-f\left(\dfrac{\pi}{6} \right))，
(\therefore x=\dfrac{\dfrac{\pi}{2}+\dfrac{2 \pi}{3}}{2}=\dfrac{7 \pi}{12})，为 (f(x)=\sin (ωx+φ)) 的一条对称轴，
且 (\left(\dfrac{\frac{\pi}{6}+\frac{\pi}{2}}{2}, 0\right)) 即 ((\dfrac{\pi}{3},0)) 为 (f(x)=\sin (ωx+φ)) 的一个对称中心，
(\therefore \dfrac{T}{4}=\dfrac{1}{4} \cdot \dfrac{2 \pi}{\omega}=\dfrac{7 \pi}{12}-\dfrac{\pi}{3}=\dfrac{\pi}{4})，解得 (ω=2∈\left(0,3\right])，
(\therefore T=\dfrac{2\pi}{2}=π)，
故答案为：(π)．
【题型2】 最值
【典题 1】 求使下列函数取得最大值、最小值的自变量 (x) 的集合，并分别写出最大值、最小值：
(1)(y=3-2\sin x)；(\qquad \qquad) (2)(y=\cos \left(2x-\dfrac{\pi}{3} \right))，(x∈\left[\dfrac{\pi}{12},\dfrac{\pi}{2}\right])．
解析 (1)(\because -1≤\sin x≤1)，(\therefore) 当 (\ \sin x=-1)，
即 (x=2kπ+\dfrac{3\pi}{2})，(k∈Z) 时，
(y) 有最大值 (5)，相应 (x) 的集合为 ({x∣x=2kπ+\dfrac{3\pi}{2},k∈Z})．
当 (\sin x=1)，即 (x=2kπ+\dfrac{\pi}{2})，(k∈Z) 时，
(y) 有最小值 (1)，相应 (x) 的集合为 ({x∣x=2kπ+\dfrac{\pi}{2},k∈Z})．
(2) 令 (z=2x-\dfrac{\pi}{3})，
(\because \dfrac{\pi}{12}≤x≤\dfrac{\pi}{2})，
(\therefore -\dfrac{\pi}{6}≤2x-\dfrac{\pi}{3}≤\dfrac{2\pi}{3})，即 (-\dfrac{\pi}{6}≤z≤\dfrac{2\pi}{3})，
(\therefore -\dfrac{1}{2}≤\cos z≤1)，(\therefore y=\cos \left(2x-\dfrac{\pi}{3} \right))，(x∈\left[\dfrac{\pi}{12},\dfrac{\pi}{2}\right]) 的最大值为 (1)，最小值为 (-\dfrac{1}{2})；
当 (z=0) 时，(y=\cos z) 取得最大值；当 (z=\dfrac{2\pi}{3}) 时，(y=\cos z) 取得最小值；
(\therefore) 当 (2x-\dfrac{\pi}{3}=0)，即 (x=\dfrac{\pi}{6}) 时， (y=\cos \left(2x-\dfrac{\pi}{3} \right)) 取得最大值；
当 (2x-\dfrac{\pi}{3}=\dfrac{2\pi}{3})，即 (x=\dfrac{\pi}{2}) 时，(y=\cos \left(2x-\dfrac{\pi}{3} \right)) 取得最小值.
【典题 2】 求函数 (y=\cos ^2 x+2\sin x) 的值域．
解析 (y=\cos ^2 x+2\sin x=-\sin ^2 x+2\sin x+1=-(\sin x-1)^2+2)．
令 (t=\sin x)，
则 (y=-(t-1)^2+2,(-1≤t≤1))，
(\therefore -2≤y≤2)，
即函数 (y=\cos ^2 x+2\sin x) 的值域为 (y∈\left[-2,2\right])．
点拨 利用 (\sin ^2 x+\cos ^2 x=1)，把函数化为仅含 (\sin x)，再利用换元法 (注意 (-1≤\sin x≤1) 取值范围) 转化为二次函数的值域问题.
【巩固练习】
函数 (y=2 \sin \left(\dfrac{\pi x}{6}-\dfrac{\pi}{3}\right)(0 \leq x \leq 9)) 的最大值与最小值之和为 ( )
A．(2-\sqrt{3}) (\qquad \qquad \qquad \qquad) B．(0) (\qquad \qquad \qquad \qquad) C．(-1) (\qquad \qquad \qquad \qquad) D．(-1-\sqrt{3})
函数 (y=\cos \left(x+\dfrac{\pi}{6} \right)),(x∈\left[0,\dfrac{\pi}{2}\right]) 的值域是 (\underline{\quad \quad}) .
求函数 (y=\cos ^2 x-\sin x),(x∈\left[-\dfrac{\pi}{4},\dfrac{\pi}{4}\right]) 的值域 (\underline{\quad \quad})．
已知函数 (f(x)=\cos \left(ωx+\dfrac{\pi}{6} \right)(ω>0)) 在区间 (\left[0,π\right]) 上的值域为 (\left[-1, \dfrac{\sqrt{3}}{2}\right])，则 (ω) 的取值范围为 (\underline{\quad \quad})．
参考答案
答案 (A)
解析 由 (0≤x≤9) 可得，(-\dfrac{\pi}{3}≤\dfrac{\pi}{6} x-\dfrac{\pi}{3}≤7\dfrac{\pi}{6})，
所以 (-\sqrt{3}≤2\sin \left(\dfrac{\pi}{6} x-\dfrac{\pi}{3} \right)≤2)，
所以最大值为 (2)，最小值为 (-\sqrt{3})，最大值与最小值 之差为 (2-\sqrt{3})．
答案 (\left[-\dfrac{1}{2}, \dfrac{\sqrt{3}}{2}\right]).
解析 由 (x∈\left[0,\dfrac{\pi}{2}\right])，可得 (x+\dfrac{\pi}{6}∈\left[\dfrac{\pi}{6},\dfrac{2\pi}{3}\right])，(\therefore) 函数 (y=\cos \left(x+\dfrac{\pi}{6} \right)∈\left[-\dfrac{1}{2},\dfrac{\sqrt{3}}{2}\right]).
答案 (\left[\dfrac{1}{2}-\dfrac{\sqrt{2}}{2}, \dfrac{5}{4}\right])
解析 (y=\cos ^2 x-\sin x=1-\sin ^2 x-\sin x=-(\sin x+\dfrac{1}{2})^2+\dfrac{5}{4})．
(\because -\dfrac{\pi}{4}≤x≤\dfrac{\pi}{4})，
(\therefore) 当 (x=-\dfrac{\pi}{6})，即 (\sin x=-\dfrac{1}{2}) 时，(y_{\max}=\dfrac{5}{4})；
当 (x=\dfrac{\pi}{4})，即 (\sin x=\dfrac{\sqrt{2}}{2}) 时， (y_{\min }=\dfrac{1}{2}-\dfrac{\sqrt{2}}{2})．
故函数 (y=\cos ^2 x-\sin x)，(x∈\left[-\dfrac{\pi}{4},\dfrac{\pi}{4}\right]) 的值域为 (\left[\dfrac{1}{2}-\dfrac{\sqrt{2}}{2}, \dfrac{5}{4}\right]).
答案 (\left[\dfrac{5}{6}, \dfrac{5}{3}\right]).
解析 在区间 (\left[0,π\right]) 上，(ωx+\dfrac{\pi}{6}∈\left[\dfrac{\pi}{6},ωπ+\dfrac{\pi}{6}\right])，
(f(x)=\cos \left(ωx+\dfrac{\pi}{6} \right)) 的值域为 (\left[-1, \dfrac{\sqrt{3}}{2}\right])，
(\therefore ωπ+\dfrac{\pi}{6}∈\left[π，11\dfrac{\pi}{6}\right])，(\therefore ωπ∈\left[\dfrac{5\pi}{6},\dfrac{5\pi}{3}\right])，(\therefore ω∈\left[\dfrac{5}{6}, \dfrac{5}{3}\right]).
【题型3】三角函数综合
【典题 1】 设函数 (f(x)=\cos \left(2x-\dfrac{\pi}{3} \right))，则下列结论错误的是 ( )
A．(f(x)) 的一个周期为 (-π) (\qquad \qquad \qquad \qquad \qquad) B．(f(x)) 的图象关于直线 (x=\dfrac{2\pi}{3}) 对称
C．(f\left(x+\dfrac{\pi}{2} \right)) 的一个零点为 (x=-\dfrac{\pi}{3}) (\qquad \qquad \qquad) D．(f(x)) 在区间 (\left[\dfrac{\pi}{3},\dfrac{\pi}{2}\right]) 上单调递减
解析 根据题意，依次分析选项：
对于 (A)、(f(x)=\cos \left(2x-\dfrac{\pi}{3} \right))，其周期 (T=\dfrac{2\pi}{2}=π)， (A) 正确；
对于 (B)、(f(x)=\cos \left(2x-\dfrac{\pi}{3} \right))，令 (2x-\dfrac{\pi}{3}=kπ)，解可得 (x=\dfrac{k\pi}{2}+\dfrac{\pi}{6})，
即 (y=f(x)) 的对称轴为 (x=\dfrac{k\pi}{2}+\dfrac{\pi}{6})，
当 (k=1) 时，(x=\dfrac{2\pi}{3})，即 (y=f(x)) 的图象关于直线 (x=\dfrac{2\pi}{3}) 对称，(B) 正确；
对于 (C)、(f\left(x+\dfrac{\pi}{2} \right)=\cos \left(2x+π-\dfrac{\pi}{3} \right)=\cos \left(2x+\dfrac{2\pi}{3} \right))，
当 (x=-\dfrac{\pi}{3}) 时，(f\left(x+\dfrac{\pi}{2} \right)=\cos 0=1)，
则 (x=-\dfrac{\pi}{3}) 不是 (f\left(x+\dfrac{\pi}{2} \right)) 的零点，(C) 错误；
对于 (D)、(f(x)=\cos \left(2x-\dfrac{\pi}{3} \right))，(2kπ≤2x-\dfrac{\pi}{3}≤2kπ+π)，
解可得 (kπ+\dfrac{\pi}{6}≤x≤kπ+\dfrac{2\pi}{3})，即函数 (f(x)) 的递减区间为 (\left[kπ+\dfrac{\pi}{6},kπ+\dfrac{2\pi}{3}\right])，
则函数在 (\left[\dfrac{\pi}{6},\dfrac{2\pi}{3}\right]) 上递减，
又由 (\left[\dfrac{\pi}{3},\dfrac{\pi}{2}\right]∈\left[\dfrac{\pi}{6},\dfrac{2\pi}{3}\right])，则 (f(x)) 在区间 (\left[\dfrac{\pi}{3},\dfrac{\pi}{2}\right]) 上递减，(D) 正确；
故选：(C)．
【典题 2】 已知 (f(x)=-\dfrac{\sqrt{2}}{2} \sin \left(2 x+\dfrac{\pi}{4}\right)+2)，求：
(1)(f(x)) 的最小正周期及对称轴方程；
(2)(f(x)) 的单调递增区间；
(3) 若方程 (f(x)-m+1=0) 在 (x∈\left[0,\dfrac{\pi}{2}\right]) 上有解，求实数 (m) 的取值范围．
解析 (1) 由于 (f(x)=-\dfrac{\sqrt{2}}{2} \sin \left(2 x+\dfrac{\pi}{4}\right)+2)，它的最小正周期为 (\dfrac{2\pi}{2}=π)，
令 (2x+\dfrac{\pi}{4}=kπ+\dfrac{\pi}{2})，求得 (x=\dfrac{k\pi}{2}+\dfrac{\pi}{8})，(k∈Z)，
故函数 (f(x)) 的图象的对称轴方程为 (x=\dfrac{k\pi}{2}+\dfrac{\pi}{8})，(k∈Z)．
(2) 令 (2kπ+\dfrac{\pi}{2}≤2x+\dfrac{\pi}{4}≤2kπ+\dfrac{3\pi}{2})，求得 (kπ+\dfrac{\pi}{8}≤x≤kπ+\dfrac{5\pi}{8})，
可得函数 (f(x)) 的增区间为 (\left[kπ+\dfrac{\pi}{8},kπ+\dfrac{5\pi}{8}\right])，(k∈Z)．
(3) 若方程 (f(x)-m+1=0) 在 (x∈\left[0,\dfrac{\pi}{2}\right]) 上有解，
则函数 (f(x)) 的图象和直线 (y=m－1) 在 (x∈\left[0,\dfrac{\pi}{2}\right]) 上有交点．
(\because x∈\left[0,\dfrac{\pi}{2}\right])，(\therefore 2x+\dfrac{\pi}{4}∈\left[\dfrac{\pi}{4},\dfrac{5\pi}{4}\right])，(\sin \left(2x+\dfrac{\pi}{4} \right)∈\left[-\dfrac{\sqrt{2}}{2},1\right])，
(\therefore f(x) \in\left[2-\dfrac{\sqrt{2}}{2}, \dfrac{5}{2}\right])，
故 (m-1∈\left[2-\dfrac{\sqrt{2}}{2}, \dfrac{5}{2}\right])，
(\therefore m \in\left[3-\dfrac{\sqrt{2}}{2}, \dfrac{7}{2}\right])．
【巩固练习】
已知函数 (f(x)=3\sin \left(2x-\dfrac{\pi}{3} \right))，下列结论中正确的是 ( )
A．函数 (f(x)) 的最小正周期为 (2π) (\qquad \qquad \qquad \qquad) B．函数 (f(x)) 的图象关于直线 (x=\dfrac{\pi}{6}) 对称
C．函数 (f(x)) 的图象关于点 (\left(-\dfrac{\pi}{6},0 \right)) 对称 (\qquad \qquad) D．函数 (f(x)) 在 (\left(-\dfrac{\pi}{12},\dfrac{5\pi}{12} \right)) 内是增函数
同时具有以下性质：①最小正周期是 (π)；②图象关于直线 (x=\dfrac{\pi}{3}) 对称；③在 (\left[-\dfrac{\pi}{6},\dfrac{\pi}{3}\right]) 上是增函数；
④一个对称中心为 (\left(\dfrac{\pi}{12},0 \right)) 的一个函数是 ( )
A．(y=\sin \left(\dfrac{x}{2}+\dfrac{\pi}{6} \right)) (\qquad \qquad \qquad \qquad \qquad) B．(y=\sin \left(2x+\dfrac{\pi}{3} \right))
C．(y=\sin \left(2x-\dfrac{\pi}{6} \right)) (\qquad \qquad \qquad \qquad \qquad) D．(y=\sin \left(2x-\dfrac{\pi}{3} \right))
设函数 (f(x)=\sin \left(2x+\dfrac{2\pi}{3} \right))，则下列结论中正确的是 ( )
A．(f(x)) 的图象关于点 (\left(\dfrac{\pi}{3},0 \right)) 对称 (\qquad \qquad \qquad) B．(f(x)) 的图象关于直线 (x=\dfrac{\pi}{3}) 对称
C．(f(x)) 在 (\left[0,\dfrac{\pi}{3}\right]) 上单调递减 (\qquad \qquad \qquad \qquad) D．(f(x)) 在 (\left[-\dfrac{\pi}{6},0\right]) 上的最小值为 (0)
若函数 (f(x)=\cos (ωx+φ)\left(ω>0,|φ|<\dfrac{\pi}{2} \right)) 的一个零点和与之相邻的对称轴之间的距离为 (\dfrac{\pi}{4})，
且当 (x=\dfrac{2\pi}{3}) 时，(f(x)) 取得最小值．
(1) 求 (f(x)) 的解析式及其单调递减区间；
(2) 若 (x∈\left[\dfrac{\pi}{4},\dfrac{5\pi}{6}\right])，求 (f(x)) 的值域．
参考答案
答案 (D)
解析 (A) 错，最小正周期为 (π)，当 (x=\dfrac{\pi}{6}) 时，(f(x)≠1)，(B) 错，
(f\left(-\dfrac{\pi}{6} \right)=3\sin \left(-\dfrac{2\pi}{3} \right)≠0)，(C) 错，
当 (x∈\left(-\dfrac{\pi}{12},\dfrac{5\pi}{12} \right)) 时，(2x-\dfrac{\pi}{3}∈(-\dfrac{\pi}{2},\dfrac{\pi}{2}))，(f(x)) 单调递增，(D) 成立，
故选：(D)．
答案 (C)
解析 由 “①最小正周期是 (π)，可得 (ω=2)，排除 (A)；
②图象关于直线 (x=\dfrac{\pi}{3}) 对称；可得：(\dfrac{2\pi}{3}+φ=\dfrac{\pi}{3}+kπ)，(k∈Z)．
对于 (D) 选项：(φ=-\dfrac{\pi}{3})，不满足，排除 (D)；
④一个对称中心为 (\left(\dfrac{\pi}{12},0 \right)) 带入函数 y 中，(B) 选项不满足．排除 (B)；
故选：(C)．
答案 (C)
解析 对于函数 (f(x)=\sin \left(2x+\dfrac{2\pi}{3} \right))，令 (x=\dfrac{\pi}{3})，求得 (f(x)=-\sqrt{3})，不是最值，
可得 (y=f(x)) 的图象不关于点 (\left(\dfrac{\pi}{3},0 \right)) 对称，也不关于直线 (x=\dfrac{\pi}{3}) 对称，故 (A)、(B) 都不正确；
在 (\left[0,\dfrac{\pi}{3}\right]) 上，(2x+\dfrac{2\pi}{3}∈\left[\dfrac{2\pi}{3},\dfrac{4\pi}{3}\right])，故 (f(x)) 在 (\left[0,\dfrac{\pi}{3}\right]) 上单调递减，故 (C) 正确；
在 (\left[-\dfrac{\pi}{6},0\right]) 上，(2x+\dfrac{2\pi}{3}∈\left[\dfrac{\pi}{3},\dfrac{2\pi}{3}\right])，
故 (f(x)) 在 (\left[0,\dfrac{\pi}{3}\right]) 上没有单调性，最小值为 (f(-\dfrac{\pi}{6})=f(0)=\dfrac{\sqrt{3}}{2})，故 (D) 不正确，
故选：(C)．
答案 (1)(f(x)=\cos \left(2x-\dfrac{\pi}{3} \right))，(\left[kπ+\dfrac{\pi}{6},kπ+\dfrac{2\pi}{3}\right]),(k∈Z); (2) (\left[-1,\dfrac{\sqrt{3}}{2}\right]).
解析 (1) 由题意，函数 (f(x)) 的一个零点和与之相邻的对称轴之间的距离为 (\dfrac{\pi}{4})，
可得 (f(x)) 的周期 (T=π)，即 (\dfrac{2 \pi}{\omega}=\pi)，解得 (ω=2)，
又因为当 (x=\dfrac{2\pi}{3}) 时，(f(x)) 取得最小值，
所以 (f\left(\dfrac{2\pi}{3} \right)=\cos \left(\dfrac{4\pi}{3}+φ \right)=-1)，
所以 (\dfrac{4\pi}{3}+φ=2kπ+π),(k∈Z)，解 (φ=2kπ-\dfrac{\pi}{3}),(k∈Z)，
因为 (|φ|<\dfrac{\pi}{2})，所以 (φ=-\dfrac{\pi}{3})，所以 (f(x)=\cos (2x-\dfrac{\pi}{3}))．
令 (2kπ≤2x-\dfrac{\pi}{3}≤2kπ+π),(k∈Z)，解得 (kπ+\dfrac{\pi}{6}≤x≤kπ+\dfrac{2\pi}{3}),(k∈Z)，
所以 (f(x)) 的单调递减区间为 (\left[kπ+\dfrac{\pi}{6},kπ+\dfrac{2\pi}{3}\right]),(k∈Z)；
(2) 因为 (x∈\left[\dfrac{\pi}{4},\dfrac{5\pi}{6}\right])，可得 (\dfrac{\pi}{6}≤2x-\dfrac{\pi}{3}≤\dfrac{4\pi}{3})，
所以当 (2x-\dfrac{\pi}{3}=π) 时，(f(x)) 取得最小值 (-1)，
当 (2x-\dfrac{\pi}{3}=\dfrac{\pi}{6}) 时，(f(x)) 取得最大值为 (\dfrac{\sqrt{3}}{2})，
所以函数 (f(x)) 的值域是 (\left[-1,\dfrac{\sqrt{3}}{2}\right])．
分层练习
【A组---基础题】
下列区间中是函数 (y=\sin \left(x+\dfrac{\pi}{4} \right)) 的单调递增区间的是 ( )
A．(\left[\dfrac{\pi}{2},π\right]) $\qquad \qquad \qquad \qquad $ B．(\left[0,\dfrac{\pi}{4}\right]) $\qquad \qquad \qquad \qquad $ C．(\left[-π,0\right]) $\qquad \qquad \qquad \qquad $ D．(\left[\dfrac{\pi}{4}, \dfrac{\pi}{2}\right])
函数 (y=2\sin 2x)，(x∈\left[0,\dfrac{\pi}{6}\right]) 的值域为 ( )
A．(\left[-2,2\right]) $\qquad \qquad \qquad \qquad $ B．(\left[-1,0\right]) $\qquad \qquad \qquad \qquad $ C．(\left[0,\sqrt{3}\right]) $\qquad \qquad \qquad \qquad $ D．(\left[0,1\right])
函数 (f(x)=\cos \left(2x-\dfrac{\pi}{3} \right)) 在 (\left[0,\dfrac{\pi}{2}\right]) 上的最小值是 ( )
A．(-\dfrac{1}{2}) (\qquad \qquad \qquad \qquad) B．(\dfrac{1}{2}) (\qquad \qquad \qquad \qquad) C． (\dfrac{\sqrt{3}}{2}) (\qquad \qquad \qquad \qquad) D．(1)
下列函数中，以 (π) 为周期且在区间 (\left(\dfrac{\pi}{2},π \right)) 单调递增的是 ( )
A．(f(x)=|\cos 2x|) (\qquad \qquad) B．(f(x)=|\sin 2x|) (\qquad \qquad) C．(f(x)=|\cos x|) (\qquad \qquad) D．(f(x)=|\sin x|)
已知函数 (f(x)=\sin x+|\sin x|)，则下列结论正确的是 ( )
A．(f(x+π)=f(x)) (\qquad \qquad \qquad \qquad \qquad \qquad) B．(f(x)) 的值域为 (\left[0,1\right])
C．(f(x)) 在 (\left[\dfrac{\pi}{2},π\right]) 上单调递减 (\qquad \qquad \qquad \qquad) D．(f(x)) 的图象关于点 ((π,0)) 对称
函数 (f(x)=\cos \left(ωx+\dfrac{\pi}{6} \right)(ω>0)) 的最小正周期为 (π)，则 (f(x)) 满足 ( )
A．在 (\left(0,\dfrac{\pi}{3} \right)) 上单调递增 (\qquad \qquad \qquad \qquad) B．图象关于直线 (x=\dfrac{\pi}{6}) 对称
C．(f\left(\dfrac{\pi}{3} \right)=\dfrac{\sqrt{3}}{2}) (\qquad \qquad \qquad \qquad \qquad) D．当 (x=\dfrac{5\pi}{12}) 时有最小值 (-1)
关于函数 (f(x)=|\sin x|+\cos x) 有下述四个结论：
①(f(x)) 是周期函数；(\qquad \qquad \qquad) ②(f(x)) 的最小值为 (-\sqrt{2})；
③(f(x)) 的图象关于 (y) 轴对称；(\qquad \qquad) ④(f(x)) 在区间 (\left(\dfrac{\pi}{4},\dfrac{\pi}{2} \right)) 单调递增．
其中所有正确结论的编号是 ( )
A．①② (\qquad \qquad \qquad \qquad) B．①③ (\qquad \qquad \qquad \qquad) C．②③ (\qquad \qquad \qquad \qquad) D．②④
已知函数 (f(x)=\sin (ωx+φ)\left(ω>0,0<φ<\dfrac{\pi}{2} \right)) 的最小正周期为 (π)，且关于 (\left(-\dfrac{\pi}{8},0 \right)) 中心对称，则下列结论正确的是 ( )
A．(f(1)<f(0)<f(2)) (\qquad \qquad \qquad \qquad) B．(f(0)<f(2)<f(1))
C．(f(2)<f(0)<f(1)) (\qquad \qquad \qquad \qquad) D．(f(2)<f(1)<f(0))
已知函数 (f(x)=\sin ωx) 的图象关于点 (\left(\dfrac{2\pi}{3},0 \right)) 对称，且 (f(x)) 在 (\left[0,\dfrac{\pi}{4}\right]) 上为增函数，则 (ω=)(\underline{\quad \quad}).
若 (f(x)=2\sin (2x+φ)(φ>0)) 的图象关于直线 (x=\dfrac{\pi}{12}) 对称，且当 (φ) 取最小值时，(∃x_0∈\left(0,\dfrac{\pi}{2} \right))，使得 (f(x_0)=a)，则 (a) 的取值范围是 (\underline{\quad \quad})．
已知点 (A\left(\dfrac{\pi}{24},0 \right)) 在函数 (f(x)=\cos (2ωx+φ)(ω>0) 且 (ω∈N^,0<φ<π)) 的图象上，直线 (x=\dfrac{\pi}{6}) 是函数 (f(x)) 的图象的一条对称轴．若 (f(x)) 在区间 (\left(\dfrac{\pi}{6},\dfrac{\pi}{3} \right)) 内单调，则 (φ=)(\underline{\quad \quad})．
已知 (f(x)=\sqrt{2} \sin \left(2x+\dfrac{\pi}{4} \right))．
(1) 求函数 (f(x)) 图象的对称轴方程．
(2) 求 (f(x)) 的单调递增区间．
(3) 当 (x∈\left[\dfrac{\pi}{4},\dfrac{3\pi}{4}\right]) 时，求函数 (f(x)) 的最大值和最小值．
参考答案
答案 (B)
解析 (\because) 函数 (y=\sin x) 的单调递增区间为 (\left2kπ-\dfrac{\pi}{2},2kπ+\dfrac{\pi}{2}\right)，
由 (2kπ-\dfrac{\pi}{2}≤x+\dfrac{\pi}{4}≤2kπ+\dfrac{\pi}{2})，得 (2kπ-\dfrac{3\pi}{4}≤x≤2kπ+\dfrac{\pi}{4}(k∈Z))．
令 (k=0)，得 (-\dfrac{3\pi}{4}≤x≤\dfrac{\pi}{4})，
(\therefore) 只有 (\left[0,\dfrac{\pi}{4}\right]⊆\left[-\dfrac{3\pi}{4},\dfrac{\pi}{4}\right])，故选 (B)．
答案 (C)
解析 (\because 0≤x≤\dfrac{\pi}{6})，(\therefore 0≤2x≤\dfrac{\pi}{3})，(\therefore 0≤\sin 2x≤\dfrac{\sqrt{3}}{2})，(\therefore y∈\left[0,\sqrt{3}\right])．
答案 (A)
解析 (x∈\left[0,\dfrac{\pi}{2}\right]) 时，(2x-\dfrac{\pi}{3}∈\left[-\dfrac{\pi}{3},\dfrac{2\pi}{3}\right])，
(\therefore x=\dfrac{\pi}{2}) 时，(\cos \left(2x-\dfrac{\pi}{3} \right)=\cos \dfrac{2\pi}{3}=-\dfrac{1}{2})；
(\therefore) 函数 (f(x)=\cos \left(2x-\dfrac{\pi}{3} \right)) 在 (\left[0,\dfrac{\pi}{2}\right]) 上的最小值是 (-\dfrac{1}{2})．
故选： (A)．
答案 (C)
解析 由于 (f(x)=|\cos 2x|) 的周期为 (\dfrac{1}{2}\cdot \dfrac{2\pi}{2}=\dfrac{\pi}{2})，故 (A) 不满足条件；
由于 (f(x)=|\sin 2x|) 的周期为 (\dfrac{1}{2}\cdot \dfrac{2\pi}{2}=\dfrac{\pi}{2})，故 (B) 不满足条件；
由于 (f(x)=|\cos x|) 的最小正周期为 (\dfrac{1}{2}\cdot 2π=π)，
在区间 (\left(\dfrac{\pi}{2},π \right)) 上，(f(x)=|\cos x|=-\cos x) 单调递增，故 (C) 满足条件；
由于 (f(x)=|\sin x|) 的最小正周期为 (\dfrac{1}{2}\cdot 2π=π)，
在区间 (\left(\dfrac{\pi}{2},π \right)) 上，(f(x)=\sin x) 单调递减，故 (D) 不满足条件，
故选：(C)．
答案 (C)
解析 函数 (f(x)=\sin x+|\sin x|=\left{\begin{array}{l}
2 \sin x, x \in[0, \pi] \
0, x \in(\pi, 2 \pi]
\end{array}\right.)，
故函数的周期为 (2π)，即 (f(x+2π)=f(x))，故排除 (A)；
显然，函数的值域为 (\left[0,2\right])，故排除 (B)；
在 (\left[\dfrac{\pi}{2},π\right]) 上，函数 (t=\sin x) 单调递减，故函数 (y=2\sin x) 单调递减，故 (C) 正确；
根据故函 (f(x)) 的图象特征，可得它的图象不关于点 ((π,0)) 对称，故排除 (D)，
故选：(C)．
答案 (D)
解析 函数 (f(x)=\cos \left(ωx+\dfrac{\pi}{6} \right)(ω>0)) 的最小正周期为 (T=\dfrac{2 \pi}{\omega}=\pi)，(\therefore ω=2)，
(\therefore f(x)=\cos (2x+\dfrac{\pi}{6}))；
当 (x∈\left(0,\dfrac{\pi}{3} \right)) 时，(2x+\dfrac{\pi}{6}∈\left(\dfrac{\pi}{6},\dfrac{5\pi}{6} \right))，(f(x)) 单调递减，(\therefore A) 错误；
(x=\dfrac{\pi}{6}) 时，(2x+\dfrac{\pi}{6}=\dfrac{\pi}{2})，(f\left(\dfrac{\pi}{6} \right)=0)，其图象不关于直线 (x=\dfrac{\pi}{6}) 对称，(B) 错误；
(f\left(\dfrac{\pi}{3} \right)=\cos \left(2×\dfrac{\pi}{3}+\dfrac{\pi}{6} \right)=-\dfrac{\sqrt{3}}{2})，(C) 错误；
(x=\dfrac{5\pi}{12}) 时，(f(x)=\cos (2×\dfrac{5\pi}{12}+\dfrac{\pi}{6})=-1)，(D) 正确．
故选：(D)．
答案 (B)
解析 函数 (f(x)=|\sin x|+\cos x)，其中 (|\sin x|) 的周期为 (π)，(\cos 2x) 的周期为 (2π)，
所以函数的最小正周期为 (2π)，故函数为周期函数．①(f(x)) 是周期函数；正确．
②函数的最小值为 (－1)，所以：(f(x)) 的最小值为 (-\sqrt{2})；错误．
③由于 (f(－x)=f(x))，(f(x)) 的图象关于 (y) 轴对称；
④(f(x)) 在区间 (\left(\dfrac{\pi}{4},\dfrac{\pi}{2} \right)) 单调递减．故错误．
故选：(B)．
答案 (D)
解析 (\because) 函数的最小周期是 (π)， (\therefore \dfrac{2 \pi}{\omega}=\pi)，得 (ω=2)，则 (f(x)=\sin (2x+φ))，
(\because f(x)) 关于 (\left(-\dfrac{\pi}{8},0 \right)) 中心对称，
(\therefore 2×\left(-\dfrac{\pi}{8} \right)+φ=kπ)，(k∈Z)，即 (φ=kπ+\dfrac{\pi}{4})，(k∈Z)，
(\because 0<φ<\dfrac{\pi}{2})，(\therefore) 当 (k=0) 时，(φ=\dfrac{\pi}{4})，即 (f(x)=\sin \left(2x+\dfrac{\pi}{4} \right))，
则函数在 (\left[-\dfrac{\pi}{8}, \dfrac{\pi}{8}\right]) 上递增，在 (\left[\dfrac{\pi}{8}，\dfrac{5\pi}{8}\right]) 上递减，(f(0)=f\left(\dfrac{\pi}{4} \right))，
(\because \dfrac{\pi}{4}<1<2)，(\therefore f\left(\dfrac{\pi}{4} \right)>f(1)>f(2))，即 (f(2)<f(1)<f(0))，
故选：(D)．
答案 (\dfrac{3}{2})
解析 依题意：(f\left(\dfrac{2\pi}{3} \right)=\sin \left(\dfrac{2\pi}{3} ω \right)=0)，(\therefore \dfrac{2\pi}{3} ω=kπ)，(k∈Z)，
(\therefore \omega=\dfrac{3 k}{2})，(k∈Z)，
又 (f(x)=\sin ωx) 在 (\left[0,\dfrac{\pi}{4}\right]) 上递增，
(\therefore 0<ω⋅\dfrac{\pi}{4}≤\dfrac{\pi}{2})，即 (0<ω≤2)，
(\therefore k=1)，(ω=\dfrac{3}{2}).
答案 (\left(-\sqrt{3},2\right])
解析 (f(x)=2\sin (2x+φ)(φ>0)) 的图象关于直线 (x=\dfrac{\pi}{12}) 对称，
所以 (2×\dfrac{\pi}{12}+φ=kπ+\dfrac{\pi}{2}(k∈Z))，解得 (φ=kπ+\dfrac{\pi}{3})，
当 (k=0) 时，(φ=\dfrac{\pi}{3})．所以 (f(x)=2\sin \left(2x+\dfrac{\pi}{3} \right))．
由于 (∃x_0∈\left(0,\dfrac{\pi}{2} \right))，所以 (\dfrac{\pi}{3}<2x_0+\dfrac{\pi}{3}<\dfrac{4\pi}{3})，
所以 (-\sqrt{3}<f(x_0)≤2)，即 (a) 的范围为 (\left(-\sqrt{3},2\right])．
答案 (\dfrac{\pi}{3})
解析 由题意得， (\dfrac{\pi}{6}-\dfrac{\pi}{24}=\dfrac{\pi}{8} \geq \dfrac{T}{4})，得 (\dfrac{1}{4} \times \dfrac{2 \pi}{2 \omega} \leq \dfrac{\pi}{8})，得 (ω≥2)，
又因为 (f(x)) 在区间 (\left(\dfrac{\pi}{6},\dfrac{\pi}{3} \right)) 内单调，
所以 (\dfrac{\pi}{3}-\dfrac{\pi}{6} \leq \dfrac{T}{2})，得 (\dfrac{1}{2} \times \dfrac{2 \pi}{2 \omega} \geq \dfrac{\pi}{6})，得 (ω≤3)．所以 (2≤ω≤3)．
又因为 (ω∈N^)，所以 (ω=2) 或 (3)．
当 (ω=2) 时，(\cos \left(4×\dfrac{\pi}{24}+φ \right)=0)，得 (φ=kπ+\dfrac{\pi}{3})，
又 (0<φ<π)，所以 (φ=\dfrac{\pi}{3})，
此时直线 (x=\dfrac{\pi}{6}) 是函数 (f(x)) 的图象的一条对称轴，且 (f(x)) 在区间 (\left(\dfrac{\pi}{6},\dfrac{\pi}{3} \right)) 内单调．
所以 (φ=\dfrac{\pi}{3})．
当 (ω=3) 时，(\cos \left(6×\dfrac{\pi}{24}+φ \right)=0)，得 (φ=kπ+\dfrac{\pi}{4})，
又 (0<φ<π)，所以 (φ=\dfrac{\pi}{4})，此时 (\cos \left(6×\dfrac{\pi}{6}+\dfrac{\pi}{4} \right)=-\dfrac{\sqrt{2}}{2}≠±1)，
所以直线 (x=\dfrac{\pi}{6}) 不是函数 (f(x)) 的图象的一条对称轴．
所以 (ω=2)，(φ=\dfrac{\pi}{3}).
答案 (1) (x=\dfrac{k\pi}{2}+\dfrac{\pi}{8})，(k∈Z)； (2) (\left[kπ－\dfrac{3\pi}{8},kπ+\dfrac{\pi}{8}\right])，(k∈Z)；
(3) 最大值为 (1)，最小值为 (-\sqrt{2}).
解析 (1) 由 (f(x)=\sqrt{2} \sin \left(2x+\dfrac{\pi}{4} \right))，
令 (2x+\dfrac{\pi}{4}=kπ+\dfrac{\pi}{2})，(k∈Z)，解得 (x=\dfrac{k\pi}{2}+\dfrac{\pi}{8})，(k∈Z)；
所以函数 (f(x)) 图象的对称轴方程是 (x=\dfrac{k\pi}{2}+\dfrac{\pi}{8})，(k∈Z)；
(2) 令 (2kπ-\dfrac{\pi}{2}≤2x+\dfrac{\pi}{4}≤2kπ+\dfrac{\pi}{2})，(k∈Z)，解得 (kπ-\dfrac{3\pi}{8}≤x≤kπ+\dfrac{\pi}{8})，(k∈Z)；
所以 (f(x)) 的单调递增区间为 (\left[kπ－\dfrac{3\pi}{8},kπ+\dfrac{\pi}{8}\right])，(k∈Z)；
(3) 当 (x∈\left[\dfrac{\pi}{4},\dfrac{3\pi}{4}\right]) 时，(\dfrac{3\pi}{4}≤2x+\dfrac{\pi}{4}≤\dfrac{7\pi}{4})，
所以 (-1≤\sin \left(2x+\dfrac{\pi}{4} \right)≤\dfrac{\sqrt{2}}{2})，所以 (-\sqrt{2}≤f(x)≤1)；
所以当 (x∈\left[\dfrac{\pi}{4},\dfrac{3\pi}{4}\right]) 时，函数 (f(x)) 的最大值为 (1)，最小值为 (-\sqrt{2})．
【B组---提高题】
已知函数 (f(x)=\left|\cos \left(ω x+\dfrac{\pi}{3} \right) \right|(ω>0)) 在区间 (\left[-\dfrac{\pi}{3},\dfrac{5\pi}{6}\right]) 上单调，则 (ω) 的取值范围为 ( )
A.(\left(0,\dfrac{12}{15} \right]) (\qquad \qquad \qquad \qquad) B.(\left(0,\dfrac{1}{5} \right]) (\qquad \qquad \qquad \qquad) C.(\left[\dfrac{1}{5} ,\dfrac{12}{15} \right]) (\qquad \qquad \qquad \qquad) D.(\left[\dfrac{12}{15} ,1\right])
已知函数 (f(x)=\sin \left(ωx+\dfrac{\pi}{3} \right)，(ω>0)) 在区间 (\left[-\dfrac{2\pi}{3},\dfrac{5\pi}{6}\right]) 上是增函数，且在区间 (\left[0,π\right]) 上恰好取得一次最大值 (1)，则 (ω) 的取值范围是 ( )
A． (\left(0,\dfrac{1}{5} \right]) (\qquad \qquad \qquad \qquad) B．(\left[\dfrac{1}{2},\dfrac{3}{5}\right]) (\qquad \qquad \qquad \qquad) C．(\left[\dfrac{1}{6},\dfrac{1}{5}\right]) (\qquad \qquad \qquad \qquad) D．(\left[\dfrac{1}{2},\dfrac{5}{2}\right))
若方程 (\sin \left(2x-\dfrac{\pi}{6} \right)=\dfrac{3}{5}) 在 ((0,π)) 上的解为 (x_1) 、(x_2)，且 (x_1>x_2)，则 (\sin (x_1-x_2)=) (\underline{\quad \quad})．
参考答案
答案 (B)
解析 (y=|\cos x|) 的单调递减区间为 (\left[kπ,kπ+\dfrac{\pi}{2}\right]),(k∈Z)，
由 (kπ≤ωx+\dfrac{\pi}{3}≤kπ+\dfrac{\pi}{2}),(k∈Z)，得 (\dfrac{k \pi-\frac{\pi}{3}}{\omega} \leq x \leq \dfrac{k \pi+\frac{\pi}{6}}{\omega})，
即函数的单调递减区间为 (\left[\dfrac{k \pi-\frac{\pi}{3}}{\omega}, \dfrac{k \pi+\frac{\pi}{6}}{\omega}\right])，(k∈Z)，
若 (f(x)) 在区间 (\left[-\dfrac{\pi}{3},\dfrac{5\pi}{6}\right]) 上单调递减，
则 (\dfrac{k \pi-\dfrac{\pi}{3}}{\omega} \leq-\dfrac{\pi}{3}) 且 (\dfrac{k \pi+\dfrac{\pi}{6}}{\omega} \geq \dfrac{5 \pi}{6})，得 (\left{\begin{array}{l}
\omega \leq \dfrac{6}{5} k+\dfrac{1}{5} \
\omega \leq-3 k+1
\end{array}\right.)，(k∈Z)，
当 (k=0) 时， (\left{\begin{array}{l}
\omega \leq \dfrac{1}{5} \
\omega \leq 1
\end{array}\right.)，即 (0<ω≤\dfrac{1}{5})，即 (ω) 的取值范围是 (\left(0,\dfrac{1}{5} \right])．
当 (f(x)) 在区间 (\left[-\dfrac{\pi}{3},\dfrac{5\pi}{6}\right]) 上单调递增时，(ω) 无解，
故选：(B)．
答案 (C)
解析 方法一 (复合函数法)：令 (X=ωx+\dfrac{\pi}{3})，(-\dfrac{2\pi}{3}≤x≤\dfrac{5\pi}{6})，
则 (-\dfrac{2 \pi \omega}{3}+\dfrac{\pi}{3} \leq X \leq \dfrac{5 \pi \omega}{6}+\dfrac{\pi}{3})．
(\therefore) 函数 (y=\sin X) 在区间 上 (\left[-\dfrac{2 \pi \omega}{3}+\dfrac{\pi}{3}, \dfrac{5 \pi \omega}{6}+\dfrac{\pi}{3}\right]) 单调递增，
(\therefore\left[-\dfrac{2 \pi \omega}{3}+\dfrac{\pi}{3}, \dfrac{5 \pi \omega}{6}+\dfrac{\pi}{3}\right] \subseteq\left[-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right])，(\therefore ω≤\dfrac{1}{5})．
当 (0≤x≤π) 时，(\dfrac{\pi}{3}≤X≤πω+\dfrac{\pi}{3})，
(\therefore) 函数 (y=\sin X) 在区间 (\left[\dfrac{\pi}{3},πω+\dfrac{\pi}{3}\right]) 恰好取一次最大值 (1)，
(\therefore \dfrac{\pi}{2}≤πω+\dfrac{\pi}{3}<\dfrac{5\pi}{2})， (\therefore \dfrac{1}{6} \leq \omega \leq \dfrac{13}{6}),
综上所知 (\dfrac{1}{6} \leq \omega \leq \dfrac{1}{5})，故选 (C)．
方法二 (特殊值法)：当 (ω=\dfrac{1}{2}) 时，令 (X=\dfrac{x}{2}+\dfrac{\pi}{3})， (-\dfrac{2 \pi}{3} \leq x \leq \dfrac{5 \pi}{6})，
则 (0≤X≤\dfrac{3\pi}{4})，则函数 (y=\sin X) 在区间 (\left[0,\dfrac{3\pi}{4}\right]) 上不单调，
(\therefore ω=\dfrac{1}{2}) 不合题意，排除 (BD)．
当 (\omega=\dfrac{1}{12}) 时，令 (X=\dfrac{x}{12}+\dfrac{\pi}{3})，(0≤x≤π),
则 (\dfrac{\pi}{3}≤X≤\dfrac{5\pi}{12})，则函数 (y=\sin X) 在区间 (\left[\dfrac{\pi}{3},\dfrac{5\pi}{12}\right]) 取不到最大值 (1)，
(\therefore \omega=\dfrac{1}{12}) 不合题意，排除 (A)．
故选：(C)．
答案 (\left[2-\sqrt{2},2\sqrt{2}\right])
解析 令 (2x-\dfrac{\pi}{6}=\dfrac{1}{2} π+kπ) 可得，(x=\dfrac{1}{3} π+\dfrac{1}{2} kπ)，
由于 (y=\sin \left(2x-\dfrac{\pi}{6} \right)=\dfrac{3}{5}) 在 ((0,π)) 上的解为 (x_1) 、(x_2)，且 (x_1>x_2)，
故 (x_1),(x_2) 关于 (x=\dfrac{1}{3} π) 对称，即 (x_1+x_2=\dfrac{2\pi}{3})，(\therefore x_2=\dfrac{2\pi}{3}-x_1)，
(\because x_1>x_2)，(\therefore π>x_1>\dfrac{2\pi}{3}-x_1>0)，解可得 (\dfrac{1}{3} π<x_1<\dfrac{2\pi}{3})，
又因为 (\sin (2x_1-\dfrac{\pi}{6})=\dfrac{3}{5}) 且 (2 x_1-\dfrac{1}{6} \pi \in\left(\dfrac{1}{2} \pi, \dfrac{7 \pi}{6}\right))，
故 (\cos (2x_1-\dfrac{\pi}{6})=-\dfrac{4}{5})，
则 (\sin (x_1-x_2 )=\sin \left(2x_1-\dfrac{2\pi}{3}\right)=\sin \left(2x_1-\dfrac{\pi}{6}-\dfrac{1}{2} π\right))(=-\cos \left(2x_1-\dfrac{\pi}{6}\right)=\dfrac{4}{5})．
故答案为： (\dfrac{4}{5})．
【C组---拓展题】
已知函数 (f(x)=2\cos \left(2x-\dfrac{\pi}{3}\right)) 在 (\lefta-\dfrac{\pi}{4},a\right) 上的最大值为 (y_1)，最小值为 (y_2)，则 (y_1-y_2) 的取值范围是 (\underline{\quad \quad}) .
已知函数 (f(x)=\cos \left(ωx+\dfrac{\pi}{3}\right)(ω>0))，图象上任意两条相邻对称轴间的距离为 (\dfrac{\pi}{2})．
(1) 求函数的单调区间和对称中心．
(2) 若关于 (x) 的方程 (2 \sin ^2x-m\cos x-4=0) 在 (x∈\left(0 ,\dfrac{\pi}{2}\right)) 上有实数解，求实数 (m) 的取值范围．
参考答案
答案 (\left[2-\sqrt{2},2\sqrt{2}\right])
解析 函数 (f(x)=2\cos \left(2x-\dfrac{\pi}{3}\right)) 的周期为 (π)，
且对称轴为 (x=\dfrac{\pi}{6}+\dfrac{k\pi}{2})，对称中心 (\left(\dfrac{5\pi}{12}+kπ,0\right))，(k∈Z)，
(f(x)) 的图象大致如图所示；
区间 (\left[a-\dfrac{\pi}{4},a\right]) 正好是函数 (\dfrac{1}{4}) 个周期，在一个周期内讨论就行，
设 (\left[a-\dfrac{\pi}{4},a\right]) 的中点为 (P)，
由图可知，当点 (P) 落在对称轴上，即 (a-\dfrac{\pi}{8}=\dfrac{\pi}{6}) 时，(y_1=2)，(y_2=\sqrt{2})，
此时 (y_1-y_2) 取得最小值为 (2-\sqrt{2})；
当点 (P) 落在对称中心上，即 (a-\dfrac{\pi}{8}=\dfrac{5\pi}{12}) 时， (y_1=\sqrt{2})，(y_2=-\sqrt{2})，
此时 (y_1-y_2) 的值为 (2\sqrt{2})；
(\therefore y_1-y_2) 的取值范围是 (\left[2-\sqrt{2},2\sqrt{2}\right])．
2. 答案 (1) 单调递增区间 (\left[kπ-\dfrac{2\pi}{3} ,kπ-\dfrac{\pi}{6}\right]) , 单调递减区间 (\left[kπ-\dfrac{\pi}{6} ,kπ+\dfrac{\pi}{3}\right]) , 对称中心为 (\left(\dfrac{1}{2} kπ+\dfrac{\pi}{12} ,0\right)), (k∈Z)， (2)({m|m<-4}.)
解析 (1) 函数 (f(x)=\cos \left(ωx+\dfrac{\pi}{3}\right)(ω>0))，图象上任意两条相邻对称轴间的距离为 (\dfrac{\pi}{2})．
(\therefore) 周期 (\dfrac{1}{2} T=\dfrac{\pi}{2})，即 (T=π)，那么 (\dfrac{2 \pi}{\omega}=\pi)，可得 (ω=2)．
(\therefore f(x)=\cos (2x+\dfrac{\pi}{3})),
令 (2kπ－π≤2x+\dfrac{\pi}{3}≤2kπ)，(k∈Z)，可得 (kπ－\dfrac{2\pi}{3}≤x≤kπ-\dfrac{\pi}{6})，
(\therefore) 可得函数的单调递增区间 (\left[kπ－\dfrac{2\pi}{3},kπ－\dfrac{\pi}{6}\right]),(k∈Z)，
令 (2kπ≤2x+\dfrac{\pi}{3}≤2kπ+π)，(k∈Z)，
可得 (kπ－\dfrac{\pi}{6}≤x≤kπ+\dfrac{\pi}{3})，
(\therefore) 可得函数的单调递减区间 (\left[kπ－\dfrac{\pi}{6},kπ+\dfrac{\pi}{3}\right]),(k∈Z)，
令 (2x+\dfrac{\pi}{3}=\dfrac{\pi}{2}+kπ)，可得 (x=\dfrac{1}{2} kπ+\dfrac{\pi}{12})，
可得函数的对称中心为 (\left(\dfrac{1}{2} kπ+\dfrac{\pi}{12},0 \right)), (k∈Z)，
(2) 方程 (2 \sin ^2x-m\cos x-4=0) 在 (x∈\left(0,\dfrac{\pi}{2}\right)) 上有实数解，
(\because \sin ^2x=1-\cos ^2x)，(\therefore 2(1-\cos ^2x )-m\cos x－4=0)，
即 (2 \cos ^2x+m\cos x+2=0)，
令 (t=\cos x)，
(\because x∈(0,\dfrac{\pi}{2})) 上，
(\therefore t∈(0,1))，则 (2t^2+mt+2=0) 在 ((0,1)) 上有解， (m=-2\left(t+\dfrac{1}{t}\right))，
令 (f(t)=t+\dfrac{1}{t} \geq 2 \sqrt{t \cdot \dfrac{1}{t}}=2)，当且仅当 (t=1) 时，取等号．
即 (-2\left(t+\dfrac{1}{t}\right) \leq-4)．
任取 (0<t_1<t_2<1)，有 (f\left(t_1\right)-f\left(t_2\right)=\left(t_1-t_2\right)\left(1-\dfrac{1}{t_1 \cdot t_2}\right)>0)．
因此 (f(t)) 在 ((0,1)) 上单调递减，因此 (m<-2k(1)=-4)，
所以 (m) 范围 ({m|m<-4})．
作者：ZhaoGui, 广东湛江
出处: 贵哥讲数学
声明：本博客版权归 [贵哥讲数学] 所有，欢迎转载，但未经作者同意必须保留此段声明，且在文章页面明显位置给出！
//更改网页ico
// 实现数学符号与汉字间有间隙
//文章页加大页面，隐藏侧边栏 |
8669 | https://www.reddit.com/r/learnmath/comments/jwg351/how_to_get_the_angle_of_arctan3/ | how to get the Angle of Arctan(3)? : r/learnmath
Skip to main contenthow to get the Angle of Arctan(3)? : r/learnmath
Open menu Open navigationGo to Reddit Home
r/learnmath A chip A close button
Log InLog in to Reddit
Expand user menu Open settings menu
Go to learnmath
r/learnmath
r/learnmath
Post all of your math-learning resources here. Questions, no matter how basic, will be answered (to the best ability of the online subscribers).
403K Members Online
•5 yr. ago
elmartifromArg
how to get the Angle of Arctan(3)?
i know that Arctan(3) can be thought like tan x =3 and tan x = sen x/cosx but i cannot get the angle without the calculator. Any help pls?
Read more
Share
Related Answers Section
Related Answers
arctan sqrt3 value and explanation
arctan 3/4 angle in degrees
arcsin 4/5 angle calculation
tan pi/12 value and simplification
arctan 1 value and explanation
New to Reddit?
Create your account and connect with a world of communities.
Continue with Email
Continue With Phone Number
By continuing, you agree to ourUser Agreementand acknowledge that you understand thePrivacy Policy.
Public
Anyone can view, post, and comment to this community
0 0
Top Posts
Reddit reReddit: Top posts of November 18, 2020
Reddit reReddit: Top posts of November 2020
Reddit reReddit: Top posts of 2020
Reddit RulesPrivacy PolicyUser AgreementAccessibilityReddit, Inc. © 2025. All rights reserved.
Expand Navigation Collapse Navigation |
8670 | https://pmc.ncbi.nlm.nih.gov/articles/PMC3786473/ | Thromboangiitis Obliterans (Buerger's Disease)—Current Practices - PMC
Skip to main content
An official website of the United States government
Here's how you know
Here's how you know
Official websites use .gov
A .gov website belongs to an official government organization in the United States.
Secure .gov websites use HTTPS
A lock ( ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites.
Search
Log in
Dashboard
Publications
Account settings
Log out
Search… Search NCBI
Primary site navigation
Search
Logged in as:
Dashboard
Publications
Account settings
Log in
Search PMC Full-Text Archive
Search in PMC
Journal List
User Guide
View on publisher site
Download PDF
Add to Collections
Cite
Permalink PERMALINK
Copy
As a library, NLM provides access to scientific literature. Inclusion in an NLM database does not imply endorsement of, or agreement with, the contents by NLM or the National Institutes of Health.
Learn more: PMC Disclaimer | PMC Copyright Notice
Int J Inflam
. 2013 Sep 11;2013:156905. doi: 10.1155/2013/156905
Search in PMC
Search in PubMed
View in NLM Catalog
Add to search
Thromboangiitis Obliterans (Buerger's Disease)—Current Practices
Abhishek Vijayakumar
Abhishek Vijayakumar
1 Department of General Surgery Victoria Hospital, Bangalore Medical College and Research Institute, Bangalore 560002, India
Find articles by Abhishek Vijayakumar
1,, Rahul Tiwari
Rahul Tiwari
1 Department of General Surgery Victoria Hospital, Bangalore Medical College and Research Institute, Bangalore 560002, India
Find articles by Rahul Tiwari
1, Vinod Kumar Prabhuswamy
Vinod Kumar Prabhuswamy
1 Department of General Surgery Victoria Hospital, Bangalore Medical College and Research Institute, Bangalore 560002, India
Find articles by Vinod Kumar Prabhuswamy
1
Author information
Article notes
Copyright and License information
1 Department of General Surgery Victoria Hospital, Bangalore Medical College and Research Institute, Bangalore 560002, India
✉
Abhishek Vijayakumar: abhishekbmc@yahoo.co.in
Academic Editor: Juan Carlos Kaski
Received 2013 Jun 18; Accepted 2013 Aug 12; Issue date 2013.
Copyright © 2013 Abhishek Vijayakumar et al.
This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
PMC Copyright notice
PMCID: PMC3786473 PMID: 24102033
Abstract
Thromboangiitis obliterans (TAO) is a nonatherosclerotic, segmental inflammatory disease that most commonly affects the small and medium-sized arteries and veins in the upper and lower extremities. Cigarette smoking has been implicated as the main etiology of the disease. In eastern parts of the world TAO forms 40–60% of peripheral vascular diseases. Clinical features and angiographic finding are the basis of early diagnosis of TAO. Abstinence from smoking is the only definitive treatment to prevent disease progression. Medical management in form of aspirin, pentoxyfylline, cilostazol, and verapamil increase pain-free walking distance in intermittent claudication, but long term usage fails to prevent disease progression in patients who continue to smoke. Surgical treatment in form of revascularization, lumbar sympathectomy, omentopexy, and Ilizarov techniques help reduce pain and promote healing of trophic changes. Newer treatment modalities like spinal cord stimulation, prostacyclin, bosentan, VEGF, and stem cell therapy have shown promising results. Latest treatment options include peripheral mononuclear stem cell, and adipose tissue derived mononuclear stem cells have been shown to be effective in preventing disease progression, decrease major amputation rates, and improving quality of life.
1. Introduction
Thromboangiitis obliterans (TAO) is a nonatherosclerotic, segmental inflammatory disease that most commonly affects the small and medium-sized arteries and veins in the upper and lower extremities. In the characteristic acute-phase lesion, in association with occlusive cellular thrombosis, the acute inflammation involving all layers of the vessel wall led TAO to be classified as a vasculitis. TAO can be distinguished from other types of vasculitis based on its tendency to occur in young male subjects, its close association with tobacco consumption, the rarity of systemic signs and symptoms, a highly cellular thrombus with relative sparing of the blood vessel wall, and the absence of elevated acute-phase reactants and of immunological markers.
Thromboangiitis obliterans (TAO) was first described in 1879, when Felix von Winiwarter, an Austrian surgeon who was an associate of Theodor Billroth, reported in the German Archives of Clinical Surgery a single case of what he described as presenile spontaneous gangrene . Buerger related the cellular nature of arterial thrombosis, as had von Winiwarter, and described the absence of large vessel involvement. It was Buerger who named the disorder “thromboangiitis obliterans”, and only briefly mentioned its relationship with smoking. In 1924, Buerger reported that tobacco use was probably a predisposing factor .
2. Epidemiology
The prevalence of the disease among all patients with peripheral arterial disease ranges from values as low as 0.5 to 5.6% in Western Europe to values as high as 45 to 63% in India, 16 to 66% in Korea and Japan, and 80% among Jews of Ashkenazi ancestry living in Israel . TAO was initially thought to affect almost exclusively men, since less than 1% of those affected were women. In the most recent studies, the proportion of female TAO patients varies between 11% and 23% . This increase may be due to an increase in smoking among women.
3. Etiology and Pathogenesis
The etiology of Buerger's disease is unknown. Although TAO is a type of vasculitis it is distinct from other vasculitis. Pathologically, the thrombus in TAO is highly cellular, with much less intense cellular activity in the wall of the blood vessel and a preserved internal elastic lamina. In addition, TAO differs from many other types of vasculitis in that the usual immunologic markers—elevation of acute-phase reactants such as erythrocyte sedimentation rate (ESR) and C-reactive protein (CRP), circulating immune complexes, and autoantibodies such as antinuclear antibody, rheumatoid factor, and complement levels are usually normal or negative.
3.1. Smoking
Use or exposure to tobacco plays a central role in the initiation and progression of the disease. By using an antigen-sensitive thymidine-incorporation assay, Adar et al. showed that patients with TAO have an increased cellular sensitivity to types I and III collagen compared to that in patients with arteriosclerosis obliterans or healthy males. It is possible that there is an abnormal sensitivity or allergy to some components of tobacco and that this sensitivity in some way leads to an inflammatory small vessel occlusive disease. Purified tobacco glycoprotein (TGP) could be related to changes in vascular reactivity that may occur in cigarette smokers . Matsushita et al. showed a very close relationship between active smoking and an active course of Buerger's disease, using the urine level of cotinine (a metabolite of nicotine) as a measurement of active smoking.
3.2. Genetics
There may be a predisposition to development of TAO, although no gene has been identified to date. There is no consistent pattern in HLA haplotypes among patients with Buerger's disease. In the United Kingdom, there was a preponderance of HLA-A9 and HLA-B5 antigens.
3.3. Hypercoagulability
Though role of hypercoagulability in pathogenesis of TAO has been proposed, some studies have failed to demonstrate any correlation. Choudhury and colleagues demonstrated that the level of urokinase-plasminogen activator was twofold higher and that of free plasminogen activator inhibitor I was 40% lower in patients with TAO than in healthy volunteers. An increased platelet response to serotonin is described in patients with Buerger's disease .
Elevated plasma homocysteine has been reported in patients with TAO . This rise may be related to the high prevalence of heavy cigarette smoking or may in some way be directly connected to the disease itself. Patients with TAO who have elevations of homocysteine may also have a higher amputation rate than those who have normal homocysteine levels . There have been several reports of increased anticardiolipin antibody titers in patients with Buerger's disease. Patients with TAO and high antibody titers tend to be younger and to suffer a significantly higher rate of major amputations than those without the antibody .
3.4. Endothelial Dysfunction
Eichhorn et al. in a study of 28 patient with TAO showed that antiendothelial cell antibody was elevated in 25% of the cases and antiendothelin antibody titers corresponded to disease severity. Makita et al. demonstrated impaired endothelium-dependent vasorelaxation in the peripheral vasculature of patients with Buerger's disease.
3.5. Infection
Initial studies by Buerger and Allen and Brown from Mayo clinic suggested an infectious etiology to TAO linking poor oral hygiene to development of TAO [15, 16]. However, they could not find out any pathogen from the lesions when using the classic orthodox bacterial culture method. Iwai et al. found oral bacteria (periodontal) DNA in the arterial specimens of Buerger disease in 93% of the cases. The oral specimen was the same as that detected in the vessels . The phlebitic lesions of TAO show also oral bacterial DNA by the PCR method . The bacterial transportation system was almost clear in the transmission electron microscopic examination . The platelets engulf the oral bacteria and are carried to terminal vessels within the platelets. An embolic occlusion with platelet aggregated mass including the oral bacteria is one possible explanation of the mechanism of TAO . The genetic relationship was also studied, and the conclusion was that HLA and the CD14 gene polymorphism which lead to defective immunity towards bacterial lipopolysaccharides increases susceptibility to TAO in southeast Asian population .
3.6. Immunologic Mechanisms
The immune system seems to play a critical role in the etiology of TAO. However, knowledge about immunological aspects involved in the progression of vascular tissue inflammation, and consequently the evolution of this disease, is still limited. TAO may actually be an autoimmune disorder with antibodies directed towards vascular endothelium in response to antigens in tobacco. The presence of different antibodies, such as antinuclear, antielastin, anticollagens I and III, and antinicotine antibodies, as well as identification of deposits of immunoglobulin (Ig) G, IgC3, and IgC4 in the blood vessels of patients, provided evidence to the theory of the immune character of TAO . Maslowski et al. suggested that anticardiolipin antibodies are important for the pathogenesis of TAO. However, patients with generalized periodontitis had significantly greater titres of IgG or IgM anticardiolipin antibodies, and these levels were significantly higher in smokers than in nonsmokers. In addition, elevated titres of IgG against periodontal pathogens are related to development of TAO . In a study by Halacheva et al. biopsy specimen of arteries of TAO patients were studied with immunohistochemistry for presence of tumor necrosis factor-α (TNF-α) and expression of intercellular adhesion molecule-1 (ICAM-1), VCAM-1, and E-selectin. It was seen that endothelial cells are activated in TAO, and that vascular lesions are associated with TNF-α secretion by tissue-infiltrating inflammatory cells, ICAM-1, VCAM-1, and E-selectin expression on endothelial cells and leukocyte adhesion.
4. Pathology
Acute-phase lesion is characterized by acute inflammation involving all coats of the vessel wall, especially of the veins, in association with occlusive thrombosis. Around the periphery of the thrombus, there are often polymorphonuclear leukocytes with karyorrhexis, the so-called microabscesses.
Intermediate phase in which there is progressive organization of the occlusive thrombus in the arteries and veins. At this stage, there is often a prominent inflammatory cell infiltrate within the thrombus and much less inflammation in the vessel wall.
Chronic phase or end-stage lesion is characterized by organization of the occlusive thrombus with extensive recanalization, prominent vascularization of the media, and adventitial and perivascular fibrosis.
In all three phases, the normal architecture of the vessel wall subjacent to the occlusive thrombus and including the internal elastic lamina remains essentially intact. These findings distinguish TAO from arteriosclerosis and from other systemic vasculitides, in which there is usually more striking disruption of the internal elastic lamina, and the media, disproportional to the disruptions attributable to aging change. Buerger's disease is segmental in distribution; “skip” areas of normal vessel between diseased segments are common, and the intensity of the periadventitial reaction may be quite variable in different segments of the same vessel.
5. Clinical Features
Classic presentation of Buerger's disease is in a young male smoker with the onset of symptoms before the age of 40 to 45 years. TAO manifests as migratory thrombophlebitis or signs of arterial insufficiency in the extremities. In a study by Sasaki et al. the subjects were 749 men and 76 women, with a mean age of 50.8 ± 0.4 years. In 42 patients (5.1%) involvement was limited to upper extremity arteries, in 616 patients (74.7%) disease was limited to the lower extremity, 167 patients (20.2%) showed involvement of both upper and lower extremities. The most frequently affected arteries were the anterior (41.4%) or posterior (40.4%) tibial arteries in the lower extremities and the ulnar artery (11.5%) in the upper extremities. In this report, approximately 25% of the patients had upper extremity involvement.
Two or more limbs are almost always involved in Buerger's disease. In the series reported by Shionoya and Rutherford , two limbs were affected in 16% of patients, three limbs in 41%, and all four limbs in 43% of patients. In a study by Barlas et al. patients with TAO (2468 total; 94.5% men and 5.5% women) at the time of admission, 78.2% had rest pain, 58% had intermittent claudication, 17.6% had supericial thrombophlebitis, 10.5% had Raynaud's phenomenon, and 68.9% had ischemic ulcers. A staging system for clinical symptoms was proposed by Rutherford as described in Table 1. A clinico-pathological classification was proposed by Lerich et al. and later modified by Fontaine as described in Table 2.
Table 1.
Rutherford classification.
| Grade | Category | Clinical |
:---:
| 0 | 0 | Asymptomatic |
| I | 1 | Mild claudication |
| I | 2 | Moderate claudication |
| I | 3 | Severe claudication |
| II | 4 | Rest pain |
| III | 5 | Ischemic ulcer not exceeding digits |
| IV | 6 | Severe ischemic ulcer or gangrene |
Open in a new tab
Adapted from .
Table 2.
Leriche-fontaine classification.
| Stages | Symptoms | Pathophysiology | Pathophysiological classification |
:--- :--- |
| I | Asymptomatic or effort pain. | Relative hypoxia | Silent arteriopathy |
| II A | Effort pain/pain-free walking distance >200 m. | Relative hypoxia | Stabilized arteriopathy, noninvalidant claudication |
| II B | Pain-free walking distance <200 m. | Relative hypoxia | Instable arteriopathy, invalidant claudication |
| III A | Rest pain, ankle arterial pressure >50 mm Hg. | Cutaneous hypoxia, tissue acidosis, ischemic neuritis | Instable arteriopathy, invalidant claudication |
| III B | Rest pain, ankle arterial pressure <50 mm Hg. | Cutaneous hypoxia, tissue acidosis, ischemic neuritis | Instable arteriopathy, invalidant claudication |
| IV | Trophic lesions, necrosis or gangrene. | Cutaneous hypoxia, tissue acidosis, necrosis | Evolutive arteriopathy |
Open in a new tab
Adapted from .
5.1. Extremity Involvement
Claudication in the arch of the foot is an early sign and is suggestive of, or even specific to, TAO. This condition is a manifestation of infrapopliteal vessel occlusive disease. As the disease progresses, typical calf claudication and eventually ischemic pain at rest and ischemic ulcerations on the toes, feet, or fingers may develop. Ischemia of the upper limbs is clinically evident in 40–50% of patients, but may be detected in 63% of patients by Allen's test and in 91% of patients by arteriogram of the hand and forearm . In the Allen's test, the examiner places the thumbs to occlude the radial and ulnar arteries of one hand of the patient. The patient opens the fist and the examiner then releases pressure from the radial artery but not the ulnar artery. If the radial artery distal to the wrist is patent, there is prompt return to colour to the hand (negative test). If the artery is occluded, the hand will remain pale (positive test). The maneuver is repeated with the pressure released from the ulnar artery but not the radial artery.
5.2. Superficial Thrombophlebitis
Superficial thrombophlebitis is observed in 40–60% of cases. Deep vein thrombophlebitis is unusual and suggestive of an alternative diagnosis, such as Behcet's disease. This superficial thrombophlebitis is migratory and recurrent and affects the arms and legs. Migrating phlebitis (phlebitis saltans) in young patients is therefore highly suggestive of TAO .
5.3. Systemic Signs and Symptoms
Systemic signs and symptoms are very rare in patients with TAO. There has been occasional reports involvement of visceral vessels. Digestive ischemia may manifest as abdominal pain, diarrhea, weight loss, or melena. Intestinal perforation and mesenteric infarction may occur. There have been reports of TAO initially presenting as small bowel ischemia and colonic obstruction [32, 33].
In some of these cases, visceral artery damage results more likely from atherosclerosis, favored by or associated with TAO. Thus, when TAO occurs in an unusual location, diagnosis should be made only after the identification of typical inflammatory vascular lesions on histopathological examinations.
Central nervous system involvement has been reported in TAO which may present as transient ischaemic attacks or ischaemic stroke. Postmortem histological examinations have demonstrated inflammation of the small- and medium-sized arteries of the leptomeninges or even of the meninges or veins .
Coronary artery involvement is extremely rare . There have been reports of coronary artery involvement presenting as acute myocardial infarction .
TAO may present with joint manifestations . On careful history taking about 12.5% may report joint problems before the preocclusive phase . Patients present recurrent episodes of arthritis of the large joints, with transient, migratory single-joint episodes accompanied by local signs of inflammation. The wrists and knees are the most frequently involved joints. The duration of signs and symptoms ranges from 2 to 14 days. The arthritis is nonerosive. Joint problems precede the diagnosis of TAO by about 10 yrs on average.
6. Laboratory Tests and Imaging
There are no specific laboratory tests to aid in the diagnosis of TAO. A complete serologic proile to exclude other diseases that may mimic TAO should be obtained as in Table 3. A proximal source of emboli should be excluded with echocardiography (two-dimensional and/or transesophageal) and arteriography. Proximal arteries should show no evidence of atherosclerosis, aneurysm, or other source of proximal emboli. A pathologic specimen is needed to diagnose Buerger's disease in cases of proximal artery involvement or in unusual locations.
Table 3.
Diagnostic investigation for Buerger's disease.
Blood count
Liver function
Renal function
Fasting blood sugar
Erythrocyte sedimentation rate
C-reactive protein
Antinuclear antibodies
Rheumatoid factor
Complementary measurements
Anticentromere antibodies (for CREST)
Anti-Scl-70 antibodies (for scleroderma)
Antiphospholipid antibodies
Lipid profile
Urinalysis
Toxicology screen for cocaine and cannabis
Cryoproteins
Segmental arterial Doppler pressures
Arteriography
Echocardiography (to exclude source of emboli)
Computed tomography (to exclude potential source of emboli)
Biopsy (In proximal artery involvement or unusual locations)
Complete thrombophilia screen: proteins G and S, antithrombin III, factor V Leiden, prothrombin 20210, and homocysteinemia
Hand radiographs (to exclude calcinosis)
Open in a new tab
CREST Calcinosis, Raynaud's phenomenon, esophageal dysmotility, sclerodactyly, and telangiectasia. Adapted from .
7. Differential Diagnosis
Differential diagnosis of TAO includes atherosclerosis, emboli, autoimmune diseases scleroderma or the CREST syndrome, systemic lupus erythematosus, rheumatoid arthritis, mixed connective tissue disease, antiphospholipid antibody syndrome, and other types of vasculitis. In the presence of lower extremity involvement, the possibility of popliteal artery entrapment syndrome or cystic adventitial disease should be considered, both of which should be readily apparent on arteriography, computed tomography, or magnetic resonance imaging. If there is isolated involvement of the upper extremity, occupational hazards such as use of vibratory tools and hypothenar hammer syndrome should be considered. Two diagnostic criteria have been proposed by Shionoya and Olin as described in Table 4.
Table 4.
Diagnostic criteria.
| SHIONOYA criteria | OLIN criteria |
:--- |
| Onset before age 50 | Onset before age 45 |
| Smoking history | Current (or recent past) tobacco use |
| Infrapopliteal arterial occlusions upper limb involvement or phlebitis migrans | Distal extremity ischemia (infrapopliteal and/or intrabrachial), such as claudication, rest pain, ischemic ulcers, and gangrene documented with noninvasive testing |
| Absence of atherosclerotic risk factors other than smoking | Laboratory tests to exclude autoimmune or connective tissue diseases and diabetes mellitus |
| Exclude a proximal source of emboli with echocardiography and arteriography |
| Demonstrate consistent arteriographic findings in the involved and clinically noninvolved limbs |
Open in a new tab
A biopsy is rarely needed to make the diagnosis unless the patient presents with an unusual characteristic, such as large artery involvement or age greater than 45 years.
8. Treatment
The most effective treatment for Buerger's disease is smoking cessation. All possible means should be used to encourage patients to give up the use of tobacco, in all its forms. Patient education is important, but only 43–70% of cases manage to give up smoking . Psychological help may be useful in certain cases, but patients should be reassured that if they manage to give up smoking completely, the disease will go into remission and amputation can be avoided. Selective cannabinoid receptor antagonists, such as rimonabant, have shown good results in helping patient quit smoking .
8.1. Medical Treatment of Intermittent Claudication
8.1.1. Platelet Inhibitors
Aspirin. Aspirin is effective in preventing secondary events and should be considered in all patients with PAD. Aspirin is not currently indicated, however, for the treatment of the symptoms of intermittent claudication.
Clopidogrel. Clopidogrel is an antiplatelet agent that has been shown to be more potent than aspirin in reducing secondary events in patients with atherosclerotic disease. There is no evidence, however, to suggest that the symptoms of claudication are reduced by long-term treatment with clopidogrel.
8.1.2. Vasodilators
When vasodilator therapy is given, vessels proximal to the stenotic or occlusive lesion and vessels parallel to the lesion dilate and improve blood flow to that neighboring vascular bed. This improvement leads to a steal proximal to the stenotic or occlusive lesion, reducing blood flow from the already ischemic distal tissue. Vasodilators also have the capacity to reduce overall systemic vascular resistance, leading to a reduction in perfusion pressure. This reduction in perfusion pressure in conjunction with the steal phenomenon increases the ischemic insult to the underperfused extremity. This concept of enhancing blood flow by giving vasodilators systemically is probably incorrect.
A dihydropyridine calcium channel blocker, such as amlodipine or nifedipine, seems to be effective if vasospasm is present. In a study by Bagger et al. increasing doses of verapamil was used in 44 patient of TAO; it was seen that there was an increased mean pain-free walking distance by 29% from 44.9 to 57.8 meters. There was no change in ankle/brachial index, suggesting that it was not purely secondary to blood flow. A theory that has evolved from this study is that the calcium channel blocker has a secondary effect—that of changing the oxygen extraction/utilization capacity. Calcium channel blockers may improve the efficiency of oxygen use in the extremity. A dose of verapamil up to 480 mg/day can be given as an adjuvant therapy to patients.
Pentoxyfylline. Pentoxyfylline (Trental) is a methylxanthine derivative that has numerous effects. Its primary effect was thought to be an improvement in red blood cell deformability. Other effects include a decrease in blood viscosity, platelet aggregation inhibition, and a reduction in fibrinogen levels. Though usage of pentoxyfylline may increase the pain-free walking distance in many, the long-term benefit and improvement in quality of life is limited.
Cilostazol. Cilostazol (Pletal) is a phosphodiesterase type III inhibitor which inhibits cyclic adenosine monophosphate (cAMP) phosphodiesterase. By increasing the levels of cAMP in platelets and blood vessels, there is inhibition of platelet aggregation and a promotion of smooth muscle cell relaxation. Numerous side effects occur with the long-term use of cilostazol the most common side effect is headache. Headache is probably secondary to the drug's vasodilatory properties. Patient have to be informed beforehand and possibly starting with a lower dose, such as 50 mg once a day, then after approximately 1 week increasing to 50 mg twice a day and then increasing to the recommended dosage of 100 mg twice a day may alleviate most of these headaches. Gastrointestinal side effects like diarrhea and bulky stools are also common. Another side effect is palpitations, and patients on long-term treatment must be evaluated for cardiovascular status and drug discontinued if patient develops congestive heart failure. Other drugs that have been proven beneficial in TAO patients with intermittent claudication are naftidrofuryl (Praxilene), levocarnitine, arginine, buflomedil, ketanserin, niacin, and lovastatin.
Surgical revascularization is rarely possible for patients with Buerger's disease due to the diffused vascular damage and the distal nature of the disease. Sasajima et al. reported a five-year rate of primary patency of 49% and a secondary patency rate of 62% in 61 patients following infrainguinal bypass. The patency rates were 67% in those who discontinued smoking and 35% in those who continued to smoke. In situ bypass should be considered in patients with severe ischemia who have target vessels .
Sympathectomy may be performed to decrease arterial spasm in patients with Buerger's disease. A laparoscopic method for sympathectomy has also been used . Sympathectomy has been shown to provide short-term pain relief and to promote ulcer healing in some patients with Buerger's disease, but no long-term benefit has been confirmed .
Omentopexy is an attractive option, but it needs proper mobilization of omentum by experts and more surgical time, increasing complications. Prolonged ileus, wound infection, closure difficulties, and hernia have been reported .
Ilizavor's technique is very effective to induce neoangiogenesis in TAO . According to Ilizarov, gradual traction on living tissues can stimulate and maintain regeneration and active growth of tissues (bone, muscle, fascia, nerve, vessels, skin, and its appendages). This is called the “law of tension stress”. In a study by Patwa and Krishnan who used Ilizaro's technique for TAO patient, a vertical tibial osteotomy with horizontal distraction was performed. It was seen that in 60 patients followed up for 5 yrs there was significant improvement in 53 patient in terms of ulcer healing, decrease in major amputation, rest pain, and claudication distance. Ilizavor's method is an excellent and cheap procedure in treatment of Buerger's disease.
Spinal cord stimulators (SCS) are used extensively in refractory peripheral atherosclerotic disease. SCS may modulate painful stimuli through several mechanisms. Inhibition of sympathetic vasoconstriction improves the peripheral microcirculation. Nitric oxide and γ-aminobutyric acid systems in the spinal cord may be important intermediaries in SCS-induced pain relief . Initial studies mainly aimed at pain relief in severe TAO. A study by Donas et al. in TAO patients, it was seen that regional perfusion index improved significantly after SCS though patient continued to smoke. It was shown that it not only helps relieve pain but also has a role in increasing peripheral microcirculation, thus, increasing limb survival, healing of trophic ulcers, and avoidance of amputation . A study by Fabregat et al. concluded that SCS should not only be considered as a last resort strategy for pain control, but also as a valid therapeutic option to improve perfusion of the limbs in the initial stages of the disease. Further large scale RCT are required to document advantage of SCS in early stages of TAO.
Prostacyclin derivatives have been evaluated in several studies and have been shown to be more effective than placebo in Buerger's disease. It is known that prostaglandin analogues facilitate relaxation of vascular smooth cells, inhibit platelet aggregation, and inhibit chemotaxis and cell proliferation. In a randomized study, 152 patients with Buerger's disease presenting with rest pain, with or without trophic changes, received intravenous iloprost or placebo. After 21–28 days of perfusion, the trophic lesions had healed or the pain had disappeared in 85% of the patients on iloprost and 17% of the patients on aspirin. At 6 months, amputation was required in 18% of the patients in the aspirin group and only 6% of the patients in the iloprost group . Results of randomized trials comparing oral prostacyclin derivatives with placebo in peripheral arterial disease have been less impressive. The European thromboangiitis obliterans (Buerger's disease) study shows no significant difference was found between groups for the total healing of lesions . Mohler et al. reported another double-blinded, randomized, controlled trial using beraprost sodium, an orally active prostaglandin I2 analogue for the treatment of intermittent claudication. Beraprost did not improve symptoms of intermittent claudication in patients with peripheral arterial disease .
This difference in effectiveness of oral and intravenous prostaglandin may be partially attributed to the fact that hospitalized patient adhere more to cessation of smoking as compared to home-based oral therapy which infact may change the progression of disease. A recent study by Bozkurt et al. shows that in intravenous iloprost at dose 1 ng/kg/min the complete healing rate without pain or major amputation was 60.23% at 24 weeks. It is clearly evident that iloprost, when used intravenously, alleviates rest pain, improves ulcer healing, and decreases the rate of amputation in Buerger's disease.
There have been studies on role of endothelin 1 in pathogenesis of TAO . It is shown that patients with TAO have elevated serum levels of endothelin 1. In a study by De Haro et al. 13 patients received oral endothelin antagonist bosentan at dose 65 mg twice daily for a month followed by 125 mg twice daily. It was seen that though patients continued to smoke, an overall 92% of patients showed clinical improvement with only 2 patients requiring minor amputations. Ten out of twelve patients showed an increase in distal blood flow demonstrated by digital angiography. Bosentan treatment may result in an improvement of clinical, angiographic, and endothelial function outcomes. Bosentan should be investigated further in the management of TAO patients .
There have been studies reporting use of gene transfer to induce therapeutic angiogenesis in TAO. Isner et al. showed encouraging results in patients receiving intramuscular injections of vascular endothelial growth factor (VEGF). In a study by Kim et al. he evaluated the safety of intramuscular VEGF gene transfer, using naked plasmid DNA in seven patients with TAO. The injections were well tolerated. Ischaemic ulcers healed or improved in four of six patients. Five of seven patients showed an increase in collateral vessels around the injection sites.
Endothelial progenitor cells (EPCs) belong to an immature cell population that is capable of differentiating into mature endothelial cells. In adults, EPCs mainly reside in bone marrow (BM) and are more proliferative and migrative than terminally differentiated endothelial cells. EPCs can be clinically isolated as CD34 þ or AC133 þ mononuclear cells (MNCs) from adult BM or peripheral blood (PB) . Tissue ischemia or systemic administration of G-CSF, GM-CSF, vascular endothelial growth factor, or estrogen enhances mobilization of EPCs from BM into PB, and the mobilized EPCs specifically home to sites of nascent neovascularization, thereby contributing to vascular repair.
Many studies have shown efficiency of autologous bone marrow cell injections into ischemic limbs . In recent times, there is a trend towards usage of peripheral blood stem cells as alternate to BM stem cells. In a study by Moriya et al. in 42 patients of TAO treated with peripheral blood mononuclear stem cells, improvement of ischemic symptoms was observed in 60% to 70% of the patients. The annual rate of major amputation was decreased markedly by treatment.
In a study by Kawamoto et al. in 17 patients with TAO who were treated with intramuscular injection of G CSF mobilized CD34+ cells from peripheral blood. During the 12-week observation after cell therapy, the Wong-Baker FACES pain rating scale, transcutaneous partial oxygen pressure, total or pain-free walking distance, and ulcer size serially improved in all patients.
Cell therapies using bone marrow mononuclear cells (BM-MNCs) and peripheral blood mononuclear cells (PBMNCs) have effective outcomes in patients with peripheral artery disease and TAO. The adipose tissue is abundant in the human body and is consistently replenished. Therefore, this tissue is an ideal source of MSCs. It has been shown that adipose tissue derived MSCs (ATMSCs) have characteristics similar to those of bone marrow stromal cells (BMSCs) . ATMSCs differentiate into endothelial cells and have a proangiogenic effect. In a study by Lee et al. 15 patients with TAO were treated with intramuscular injections of ATMSCs; it was seen that clinical improvement occurred in 66.7% of patients. Five patients required minor amputation during followup, and all amputation sites healed completely. At 6 months, significant improvement was noted on pain rating scales and in claudication walking distance. Digital subtraction angiography before and 6 months after ATMSCs implantation showed formation of numerous vascular collateral networks across affected arteries. Further large-scale randomized trials are required to assess the long-term benefits of stem cell therapy.
9. Conclusion
TAO is a distinct form of systemic vasculitis of unknown etiology though strongly linked to cigarette smoking. Clinical features and angiography form the main basis of diagnosis. Abstinence from smoking is the only definitive treatment to prevent disease progression. Medical line of treatment with vasodilators, pentoxyfylline, and cilostazol may help improve pain-free walking distance but cannot prevent disease progression. Surgical treatment in form of revascularization, sympathectomy, Ilizarov, and omentopexy increases peripheral blood flow and decreases the rate of amputations. Newer therapy with prostaglandins, bosentan, and stem cell therapy has shown promising results. Early diagnosis and aggressive therapy can decrease patient symptoms and chances of major amputations.
Conflict of Interests
The authors declare that they have no conflict of interests.
References
1.Lie JT, Mann RJ, Ludwig J. The brothers von Winiwarter, Alexander (1848–1917) and Felix (1852–1931), and thromboangiitis obliterans. Mayo Clinic Proceedings. 1979;54(12):802–807. [PubMed] [Google Scholar]
2.Buerger L. The Circulatory Disturbances of the Extremities. Philadelphia, Pa, USA: WB Saunders; 1924. [Google Scholar]
3.Cachovan M. Epidemiologic und geographisches Verteilungsmuster der Thromboangiitis obliterans. In: Stuttgart HH, editor. Thromboangiitis Obliterans Morbus Winiwarter-Buerger. Germany Georg Thieme; 1988. pp. 31–36. [Google Scholar]
4.Mills JL, Taylor LM, Jr., Porter JM. Buerger’s disease in the modern era. American Journal of Surgery. 1987;154(1):123–129. doi: 10.1016/0002-9610(87)90301-1. [DOI] [PubMed] [Google Scholar]
5.Adar R, Papa MZ, Halpern Z. Cellular sensitivity to collagen in thromboangiitis obliterans. The New England Journal of Medicine. 1983;308(19):1113–1116. doi: 10.1056/NEJM198305123081901. [DOI] [PubMed] [Google Scholar]
6.Papa M, Bass A, Adar R, et al. Autoimmune mechanisms in thromboangiitis obliterans (Buerger’s disease): the role of tobacco antigen and the major histocompatibility complex. Surgery. 1992;111(5):527–531. [PubMed] [Google Scholar]
7.Matsushita M, Shionoya S, Matsumoto T. Urinary cotinine measurement in patients with Buerger’s disease—effects of active and passive smoking on the disease process. Journal of Vascular Surgery. 1991;14(1):53–58. [PubMed] [Google Scholar]
8.Choudhury NA, Pietraszek MH, Hachiya T, et al. Plasminogen activators and plasminogen activator inhibitor 1 before and after venous occlusion of the upper limb in thromboangiitis obliterans (Buerger’s disease) Thrombosis Research. 1992;66(4):321–329. doi: 10.1016/0049-3848(92)90282-f. [DOI] [PubMed] [Google Scholar]
9.Pietraszek MH, Choudhury NA, Koyano K, et al. Enhanced platelet response to serotonin in Buerger’s disease. Thrombosis Research. 1990;60(3):241–246. doi: 10.1016/0049-3848(90)90186-g. [DOI] [PubMed] [Google Scholar]
10.Caramaschi P, Biasi D, Carletto A, et al. Three cases of Buerger’s disease associated with hyperhomocysteinemia. Clinical and Experimental Rheumatology. 2000;18(2):264–265. [PubMed] [Google Scholar]
11.Olin JW, Childs MB, Bartholomew JR, et al. Anticardiolipin antibodies and homocysteine levels in patients with thromboangiitis obliterans. Arthritis & Rheumatism. 1996;39(supplement 9):p. 47. [Google Scholar]
12.Olin JW. Are anticardiolipin antibodies really important in thromboangiitis obliterans (Buerger’s disease)? Vascular Medicine. 2002;7(4):257–258. doi: 10.1191/1358863x02vm457ed. [DOI] [PubMed] [Google Scholar]
13.Eichhorn J, Sima D, Lindschau G, et al. Antiendothelial cell antibodies in thromboangiitis obliterans. American Journal of the Medical Sciences. 1998;315(1):17–23. doi: 10.1097/00000441-199801000-00004. [DOI] [PubMed] [Google Scholar]
14.Makita S, Nakamura M, Murakami H, Komoda K, Kawazoe K, Hiramori K. Impaired endothelium-dependent vasorelaxation in peripheral vasculature of patients with thromboangiitis obliterans (Buerger’s disease) Circulation. 1996;94(9):II211–II215. [PubMed] [Google Scholar]
15.Buerger L. Is thromboangiitis an infection disease? The Journal of Surgery, Gynecology and Obstetrics. 1914;19:582–588. [Google Scholar]
16.Allen EV, Brown GE. Thromboangiitis obliterans. A clinical study of 200 cases. Annals of Internal Medicine. 1928;1:535–549. [Google Scholar]
17.Iwai T, Inoue Y, Umeda M, et al. Oral bacteria in the occluded arteries of patients with Buerger disease. Journal of Vascular Surgery. 2005;42(1):107–115. doi: 10.1016/j.jvs.2005.03.016. [DOI] [PubMed] [Google Scholar]
18.Iwai T, Sato S, Kume H, et al. Clinical study of phlebitis migrans and incompetence of the leg's superficial vein in Buerger disease. Annals of Vascular Surgery. 2012;5(1):45–51. doi: 10.3400/avd.oa.11.00044. [DOI] [PMC free article] [PubMed] [Google Scholar]
19.Li X, Iwai T, Nakamura H, et al. An ultrastructural study of Porphyromonas gingivalis-induced platelet aggregation. Thrombosis Research. 2008;122(6):810–819. doi: 10.1016/j.thromres.2008.03.011. [DOI] [PubMed] [Google Scholar]
20.Iwai T. Buerger Disease: New Evidence. Oxford, UK: Health Press; 2009. (Vascular and endovascular surgery highlights). Edited by: Davies A.H., Mitchell AWM. [Google Scholar]
21.Chen Z, Takahashi M, Naruse T, et al. Synergistic contribution of CD14 and HLA loci in the susceptibility to Buerger disease. Human Genetics. 2007;122(3-4):367–372. doi: 10.1007/s00439-007-0408-1. [DOI] [PubMed] [Google Scholar]
22.Roncon de Albuquerque R, Delgado L, Correia P, Fleming Torrinha J, Serrao D, Braga A. Circulating immune complexes in Buerger’s disease. Endarteritis obliterans in young men. Journal of Cardiovascular Surgery. 1989;30(5):821–825. [PubMed] [Google Scholar]
23.Maslowski L, McBane R, Alexewicz P, Wysokinski WE. Antiphospholipid antibodies in thromboangiitis obliterans. Vascular Medicine. 2002;7(4):259–264. doi: 10.1191/1358863x02vm452oa. [DOI] [PubMed] [Google Scholar]
24.Chen Y-W, Iwai T, Umeda M, et al. Elevated IgG titers to periodontal pathogens related to Buerger disease. International Journal of Cardiology. 2007;122(1):79–81. doi: 10.1016/j.ijcard.2006.11.032. [DOI] [PubMed] [Google Scholar]
25.Halacheva K, Gulubova MV, Manolova I, Petkov D. Expression of ICAM-1, VCAM-1, E-selectin and TNF-α on the endothelium of femoral and iliac arteries in thromboangiitis obliterans. Acta Histochemica. 2002;104(2):177–184. doi: 10.1078/0065-1281-00621. [DOI] [PubMed] [Google Scholar]
26.Sasaki S, Sakuma M, Kunihara T, Yasuda K. Distribution of arterial involvement in thromboangiitis obliterans (Buerger’s disease): results of a study conducted by the intractable vascalitis syndromes research group in Japan. Surgery Today. 2000;30(7):600–605. doi: 10.1007/s005950070099. [DOI] [PubMed] [Google Scholar]
27.Shionoya S. Buerger's disease (thromboangiitis obliterans) In: Rutherford RB, editor. Vascular Surgery. 3rd edition. Philadelphia, Pa, USA: WB Saunders; 1989. pp. 207–217. [Google Scholar]
28.Barlas S, Elmaci T, Dayioğlu E, et al. Has the clinical definition of thromboangiitis obliterans changed indeed? International Journal of Angiology. 1997;6(1):49–55. [Google Scholar]
29.Olin JW, Young JR, Graor RA, Ruschhaupt WF, Bartholomew JR. The changing clinical spectrum of thromboangiitis obliterans (Buerger’s disease) Circulation. 1990;82(5):V-3–V-8. [PubMed] [Google Scholar]
30.Hirai M, Shionoya S. Arterial obstruction of the upper limb in Buerger’s disease: its incidence and primary lesion. British Journal of Surgery. 1979;66(2):124–128. doi: 10.1002/bjs.1800660213. [DOI] [PubMed] [Google Scholar]
31.Juergens JL. Thromboangiitis obliterans, (Buerger's disease, TAO) In: Juergens JL, Spittel JA, Fairbairn JF, editors. Peripheral Vascular Diseases. Philadelphia, Pa, USA: WB Saunders; 1980. pp. 469–491. [Google Scholar]
32.Kobayashi M, Kurose K, Kobata T, Hida K, Sakamoto S, Matsubara J. Ischemic intestinal involvement in a patient with Buerger disease: case report and literature review. Journal of Vascular Surgery. 2003;38(1):170–174. doi: 10.1016/s0741-5214(02)75469-4. [DOI] [PubMed] [Google Scholar]
33.Sauvaget F, Debray M, De Sigalony J-PH, et al. Colonic ischemia reveals thromboangiitis obliterans (Buerger’s disease) Gastroenterology. 1996;110(3):900–903. doi: 10.1053/gast.1996.v110.pm8608901. [DOI] [PubMed] [Google Scholar]
34.No YJ, Lee EM, Lee DH, Kim JS. Cerebral angiographic findings in thromboangiitis obliterans. Neuroradiology. 2005;47(12):912–915. doi: 10.1007/s00234-005-1445-x. [DOI] [PubMed] [Google Scholar]
35.Hong TE, Faxon DP. Coronary artery disease in patients with Buerger's disease. Reviews in Cardiovascular Medicine. 2005;6:222–226. [PubMed] [Google Scholar]
36.Ohno H, Matsuda Y, Takashiba K. Acute myocardial infarction in Buerger’s disease. American Journal of Cardiology. 1986;57(8):690–691. doi: 10.1016/0002-9149(86)90863-5. [DOI] [PubMed] [Google Scholar]
37.Lambotte O, Chazerain P, Vinciguerra C, Meyer O, Ziza J-M. Thromboangiitis obliterans with inaugural rheumatic manifestations. A report of three cases. Revue du Rhumatisme. 1997;64(5):334–338. [PubMed] [Google Scholar]
38.Puéchal X, Fiessinger J-N, Kahan A, Menkès CJ. Rheumatic manifestations in patients with thromboangiitis obliterans (Buerger’s disease) Journal of Rheumatology. 1999;26(8):1764–1768. [PubMed] [Google Scholar]
39.Boyd ST, Fremming BA. Rimonabant—a selective CB1 antagonist. Annals of Pharmacotherapy. 2005;39(4):684–690. doi: 10.1345/aph.1E499. [DOI] [PubMed] [Google Scholar]
40.Dormandy JA, Rutherford RB. Management of peripheralarterial disease (PAD). TASC Working Group. TransAtlantic Inter-Society Concensus (TASC) Journal of Vascular Surgery. 2000;31:S1–S296. [PubMed] [Google Scholar]
41.Novo S, Coppola G, Milio G. Critical limb ischemia: definition and natural history. Current Drug Targets. 2004;4(3):219–225. doi: 10.2174/1568006043335989. [DOI] [PubMed] [Google Scholar]
42.Bagger JP, Helligsoe P, Randsbaek F, Kimose HH, Jensen BS. Effect of verapamil in intermittent claudication: a randomized, double- blind, placebo-controlled, cross-over study after individual dose-response assessment. Circulation. 1997;95(2):411–414. doi: 10.1161/01.cir.95.2.411. [DOI] [PubMed] [Google Scholar]
43.Lazarides MK, Georgiadis GS, Papas TT, Nikolopoulos ES. Diagnostic criteria and treatment of Buerger’s disease: a review. International Journal of Lower Extremity Wounds. 2006;5(2):89–95. doi: 10.1177/1534734606288817. [DOI] [PubMed] [Google Scholar]
44.Shionoya S. Diagnostic criteria of Buerger’s disease. International Journal of Cardiology. 1998;66(1):S243–S245. doi: 10.1016/s0167-5273(98)00175-2. [DOI] [PubMed] [Google Scholar]
45.Olin JW. Thromboangiitis obliterans (Buerger’s disease) The New England Journal of Medicine. 2000;343(12):864–869. doi: 10.1056/NEJM200009213431207. [DOI] [PubMed] [Google Scholar]
46.Sasajima T, Kubo Y, Inaba M, Goh K, Azuma N. Role of infrainguinal bypass in Buerger’s disease: an eighteen-year experience. European Journal of Vascular and Endovascular Surgery. 1997;13(2):186–192. doi: 10.1016/s1078-5884(97)80017-2. [DOI] [PubMed] [Google Scholar]
47.Bozkurt AK, Besirli K, Koksal C, et al. Surgical treatment of Buerger's disease. Vascular. 2004;12:192–197. doi: 10.1258/rsmvasc.12.3.192. [DOI] [PubMed] [Google Scholar]
48.Chander J, Singh L, Lal P, Jain A, Lal P, Ramteke VK. Retroperitoneoscopic lumbar sympathectomy for buerger’s disease: a novel technique. Journal of the Society of Laparoendoscopic Surgeons. 2004;8(3):291–296. [PMC free article] [PubMed] [Google Scholar]
49.Nesargikar PN, Ajit MK, Eyers PS, Nichols BJ, Chester JF. Lumbar chemical sympathectomy in peripheral vascular disease: does it still have a role? International Journal of Surgery. 2009;7(2):145–149. doi: 10.1016/j.ijsu.2009.01.004. [DOI] [PubMed] [Google Scholar]
50.Hoshino S, Nakayama K, Igari T, Honda K. Long-term results of omental transplantation for chronic occlusive arterial diseases. International Surgery. 1983;68(1):47–50. [PubMed] [Google Scholar]
51.Chaudhary M, Chaudhary P, Chaudhary MR. Buerger's disease—its management by bone widening technique of Ilizarov. Recent Advances in Surgery, Roshan Lal Gupta, Jaypee Brothers, New Delhi. 2006;8:259–271. [Google Scholar]
52.Patwa JJ, Krishnan A. Buerger’s disease (thromboangiitis obliterans)—management by ilizarov’s technique of horizontal distraction. A retrospective study of 60 cases. Indian Journal of Surgery. 2011;73(1):40–47. doi: 10.1007/s12262-010-0186-1. [DOI] [PMC free article] [PubMed] [Google Scholar]
53.Stanton-Hicks M, Salamon J. Stimulation of the central and peripheral nervous system for the control of pain. Journal of Clinical Neurophysiology. 1997;14(1):46–62. doi: 10.1097/00004691-199701000-00004. [DOI] [PubMed] [Google Scholar]
54.Donas KP, Schulte S, Ktenidis K, Horsch S. The role of epidural spinal cord stimulation in the treatment of Buerger’s disease. Journal of Vascular Surgery. 2005;41(5):830–836. doi: 10.1016/j.jvs.2005.01.044. [DOI] [PubMed] [Google Scholar]
55.Fabregat G, Villanueva VL, Asensio JM, De Andrés J, López D. Spinal cord stimulation for the treatment of buerger disease: a report on 3 cases. Clinical Journal of Pain. 2011;27(9):819–823. doi: 10.1097/AJP.0b013e31821d8eff. [DOI] [PubMed] [Google Scholar]
56.Fiessinger JN, Schäfer M. Trial of iloprost versus aspirin treatment for critical limb ischaemia of thromboangiitis obliterans. The Lancet. 1990;335(8689):555–557. doi: 10.1016/0140-6736(90)90346-7. [DOI] [PubMed] [Google Scholar]
57.Verstraete M. Oral iloprost in the treatment of thromboangiitis obliterans (Buerger’s disease): a double-blind, randomised, placebo-controlled trial. European Journal of Vascular and Endovascular Surgery. 1998;15(4):300–307. doi: 10.1016/s1078-5884(98)80032-4. [DOI] [PubMed] [Google Scholar]
58.Mohler ER, III, Hiatt WR, Olin JW, Wade M, Jeffs R, Hirsch AT. Treatment of intermittent claudication with beraprost sodium, an orally active prostaglandin I2 analogue: double-blinded, randomized, controlled trial. Journal of the American College of Cardiology. 2003;41(10):1679–1686. doi: 10.1016/s0735-1097(03)00299-7. [DOI] [PubMed] [Google Scholar]
59.Bozkurt AK, Cengiz K, Arslan C, et al. A Stable prostacyclin analogue (iloprost) in the treatment of Buerger's disease: a prospective analysis of 150 patients. Annals of Thoracic and Cardiovascular Surgery. 2013;19(2):120–150. doi: 10.5761/atcs.oa.11.01868. [DOI] [PubMed] [Google Scholar]
60.Czarnacki M, Gacka M, Adamiec R. A role of endothelin 1 in the pathogenesis of thromboangiitis obliterans (initital news) Przeglad Lekarski. 2004;61(12):1346–1350. [PubMed] [Google Scholar]
61.De Haro J, Acin F, Bleda S, Varela C, Esparza L. Treatment of thromboangiitis obliterans (Buerger’s disease) with bosentan. BMC Cardiovascular Disorders. 2012;12, article 5 doi: 10.1186/1471-2261-12-5. [DOI] [PMC free article] [PubMed] [Google Scholar]
62.Isner JM, Baumgartner I, Rauh G, et al. Treatment of thromboangiitis obliterans (Buerger’s disease) by intramuscular gene transfer of vascular endothelial growth factor: preliminary clinical results. Journal of Vascular Surgery. 1998;28(6):964–975. doi: 10.1016/s0741-5214(98)70022-9. [DOI] [PubMed] [Google Scholar]
63.Kim HJ, Jang SY, Park JI, et al. Vascular endothelial growth factor-induced angiogenic gene therapy in patients with peripheral arterydisease. Experimental & Molecular Medicine. 2004;36:336–344. doi: 10.1038/emm.2004.44. [DOI] [PubMed] [Google Scholar]
64.Friedrich EB, Walenta K, Scharlau J, Nickenig G, Werner N. CD34−/CD133+/VEGFR-2+ endothelial progenitor cell subpopulation with potent vasoregenerative capacities. Circulation Research. 2006;98(3):e20–e25. doi: 10.1161/01.RES.0000205765.28940.93. [DOI] [PubMed] [Google Scholar]
65.Tateishi-Yuyama E, Matsubara H, Murohara T, et al. Therapeutic angiogenesis for patients with limb ischaemia by autologous transplantation of bone-marrow cells: a pilot study and a randomised controlled trial. The Lancet. 2002;360(9331):427–435. doi: 10.1016/S0140-6736(02)09670-8. [DOI] [PubMed] [Google Scholar]
66.Moriya J, Minamino T, Tateno K, et al. Long-term outcome of therapeutic neovascularization using peripheral blood mononuclear cells for limb ischemia. Circulation. 2009;2(3):245–254. doi: 10.1161/CIRCINTERVENTIONS.108.799361. [DOI] [PubMed] [Google Scholar]
67.Kawamoto A, Katayama M, Handa N, et al. Intramuscular transplantation of G-CSF-mobilized CD34+ cells in patients with critical limb ischemia: a phase I/IIa, multicenter, single-blinded, dose-escalation clinical trial. Stem Cells. 2009;27(11):2857–2864. doi: 10.1002/stem.207. [DOI] [PubMed] [Google Scholar]
68.Lee RH, Kim B, Choi I, et al. Characterization and expression analysis of mesenchymal stem cells from human bone marrow and adipose tissue. Cellular Physiology and Biochemistry. 2004;14(4-6):311–324. doi: 10.1159/000080341. [DOI] [PubMed] [Google Scholar]
69.Lee HC, An SG, Park JS, et al. Safety and effect of adipose tissue-derived stem cell implantation in patients with critical limb ischemia. Circulation Journal. 2012;76(7):1750–1760. doi: 10.1253/circj.cj-11-1135. [DOI] [PubMed] [Google Scholar]
Articles from International Journal of Inflammation are provided here courtesy of Wiley
ACTIONS
View on publisher site
PDF (547.2 KB)
Cite
Collections
Permalink PERMALINK
Copy
RESOURCES
Similar articles
Cited by other articles
Links to NCBI Databases
On this page
Abstract
1. Introduction
2. Epidemiology
3. Etiology and Pathogenesis
4. Pathology
5. Clinical Features
6. Laboratory Tests and Imaging
7. Differential Diagnosis
8. Treatment
9. Conclusion
Conflict of Interests
References
Cite
Copy
Download .nbib.nbib
Format:
Add to Collections
Create a new collection
Add to an existing collection
Name your collection
Choose a collection
Unable to load your collection due to an error
Please try again
Add Cancel
Follow NCBI
NCBI on X (formerly known as Twitter)NCBI on FacebookNCBI on LinkedInNCBI on GitHubNCBI RSS feed
Connect with NLM
NLM on X (formerly known as Twitter)NLM on FacebookNLM on YouTube
National Library of Medicine 8600 Rockville Pike Bethesda, MD 20894
Web Policies
FOIA
HHS Vulnerability Disclosure
Help
Accessibility
Careers
NLM
NIH
HHS
USA.gov
Back to Top |
8671 | https://www.ck12.org/flexi/cbse-math/higher-order-metric-conversions/how-do-you-move-decimals-in-scientific-notation/ | How do you move decimals in scientific notation?
Flexi Says:
To move decimals in scientific notation, you need to adjust the exponent of the base 10 while keeping the number in the form of a × 10^n, where 1 ≤ |a| < 10 and n is an integer. Here's a step-by-step guide:
Step 1: Identify the number you want to convert to scientific notation.
Example: 0.0056
Step 2: Move the decimal point to create a number between 1 and 10 (not including 10).
Example: Move the decimal point 3 places to the right: 5.6
Step 3: Count the number of places you moved the decimal point. This will be the exponent of 10.
Example: We moved the decimal point 3 places to the right, so the exponent is -3 (negative because we moved the decimal to the right).
Step 4: Write the number in scientific notation using the new decimal value and the exponent of 10.
Example: 5.6 × 10^(-3)
When converting a number from scientific notation back to standard notation, you'll move the decimal point in the opposite direction of the exponent. If the exponent is positive, move the decimal point to the right; if it's negative, move it to the left.
To learn more click here!
Try Asking:
Convert 400 minutes to seconds.Convert 38 minutes to seconds.Convert 5 minutes to seconds.
By messaging Flexi, you agree to our Terms and Privacy Policy |
8672 | https://ocw.mit.edu/courses/10-626-electrochemical-energy-systems-spring-2014/resources/mit10_626s14_s11lec08/ | II. Equilibrium Thermodynamics
Lecture 8: The Nernst Equation
Notes by MIT Student, re-written by MZB 2014
1 Chemical Activity
The fundamental thermodynamic quantity controlling transport and reac tions is the (electro)chemical potential of species i,
δG
μi = (1)
δN i Nj ,T,P, Φ
where G is the Gibbs free energy (for an open system at constant tempera ture and pressure). which we decompose into three parts:
μi = μΘ + kB T ln( ai) + zieφ (per particle) (2) i
Θ
the chemical potential μi in an arbitrarily defined standard state , a correc tion kB T ln ai where ai is the (dimensionless) chemical activity relative to
ai = 1 in the standard state, and the electrostatic energy zieφ for a charge
zie in the electrostatic potential φ.
In chemistry it is more common to define chemical potential and other
thermodynamic quantities per mole, in which case:
μi = μΘ + RT ln( ai) + ziF Φ (per mole) (3) i
R = universal gas constant = kB NA (4)
= 8 .31 J (NA = 6 × 10 23 particles)
K mol
F = Faraday’s constant = NAe = 96 , 487 C (5)
kB T RT
= = thermal voltage =at 25 ◦C 26mV (6)
e F
1
February 27, 2014 Lecture 8: Nernst equation 10.626 (2014) Bazant
Since we focus on microscopic mechanisms, however, we will typically de fine all thermodynamic properties per particle (which is more common in
physics).
As shown in the example below, in the limit of an infinitely dilute so lution, the activity is proportional to the concentration. In a non-dilute
“concentrated solution” where particles interact with each other, there are
corrections related to all other contributions to the chemical potential, such
as short-range forces between molecules (contributing to enthalpy h) or
temperature-dependent crowding effects (contributing to the entropy). These
non-ideal contributions to the chemical potential can be expressed via an
activity coefficient γi, whose precise definition depends on the way of mea suring concentration (per volume, per mass, per site, etc.) and is typically
scaled such that γi = 1 in the standard state of a single species (in a dilute
solution):
ai = γi
c
c˜i = γi
m
m˜ i = γi
x
xi = γi
p
pi = . . . (7)
where ˜ci = ci/c ref is the dimensionless species concentration with cref = 1M
= 1 mole/liter = 1 molar, m˜ i = mi/m ref is the dimensionless molality in
solution with mref = 1 mol/kg solvent = 1 molal, xi = Ni/N site is the mole
fraction or filling fraction of available solid sites (dimensionless), ˜pi = pi/p ref
is the dimensionless gas partial pressure with pref = 1 atm, etc.
Example: Lattice gas, or ideal solid solution. In the previous
lecture, we used statistical mechanics to derive an explicit expression for the
chemical potential of an ideal solid solution (or lattice gas) of non-interacting
particles on a lattice:
x
μ = kB T ln = kB T ln( γx ) (8)
1 − x
which comes from the Gibbs free energy of mixing (only entropy). In this
case, we have 1
γ = (9)
1 − x
The activity coefficient goes to infinity as x → 1 due to crowding effects
(reduced entropy of vacancies in the lattice). Increased activity means larger
chemical potential, and thus greater tendency to have reactions or transport
that lower the concentration by removing particles. At the simplest level,
this is what stops battery discharge when a reactant is fully consumed. Later
we will apply this model to Li-ion batteries. •
2Lecture 8: Nernst equation 10.626 (2014) Bazant electrode (usually metal) electrolyte "half cell" +z iφe
2 Faradaic reactions in equilibrium
Any Faradaic “half-cell” charge-transfer reaction at an electrode can be ex pressed in the following standard form, producing n electrons:
−
siM zi → ne (10) i
i
where
si = stoichiometric coefficients (11)
Mi = chemical symbols (12)
zi = charge numbers (13)
n = number of electrons transferred (14)
Charge conservation requires
sizi = −n (15)
i
The forward direction of the Faradaic reaction represents oxidation , or
increase of the charge state of the molecules (on the left hand side) while
liberating electrons. In a galvanic cell, electrons flow spontaneously from
the anode to the cathode, so the forward direction of the reaction is also
called the anodic reaction . At the anode, si > 0 for reactants, and si < 0 for
products, while the signs are reversed for a cathodic reaction which consumes
electrodes and leads to reduction of the charge state of the molecules.
For example, the hydrogen oxidation reaction can be written as
−
H2 − 2H+ → 2e
(sH2 = 1 , s H+ = −2, z H+ = 1 , n = 2)
The Faradaic reaction can also be written with only positive coefficients
as −
R = siM zi → |si|M zi + ne − = O + ne (16) i i
si>0si<0
3Lecture 8: Nernst equation 10.626 (2014) Bazant (a)$ OzOs
(s)
RzRs
(s)
ne−
φ
φe electrode$$|$$solu/on$
OzOe
(e)
RzRe
(e)
(b)$ OzO
RzR
(c)$ OzO
RzR
ne−
ne−
Figure 1: Types of Faradaic reactions O + ne − → R. (a) General mixed
ion-electron conductor electrode/electrolyte interface. (b) Redox in solution.
(c) Ion intercalation or electrodeposition.
where R is the “reduced state” and O is the ”oxidized state”.
The figure shows different Faradaic reactions at an electrode/electrolyte
interface. Let us assume that ions only exist in the electrolyte (neglecting
mixed ion-electron conductors). For redox reactions (1(b)), e.g. Fe 3+ +e− →
Fe 2+ , the reduced state is in the solution at the same potential, φR = φO =
φ. For electrodeposition (1(c)), e.g. Cu 2+ + 2 e− → Cu, or ion intercalation
as a neutral polaron (ion/electron pair), e.g. CoO 2+Li + + e− → LiCoO 2,
the reduced state is uncharged, zR = 0, so we can also set φR = φ, even
though it is in the electrode. For this broad class of Faradaic reactions,
we can thus let φ be the electrostatic potential of each ion, and φe be the
potential of the electrons in the electrode.
The Nernst potential or interfacial voltage drop (electrode minus elec trolyte) is the
Δφ = φe − φ (17)
and we will now derive its value in equilibrium.
The total electrochemical potential of each side of the reaction must be
4Lecture 8: Nernst equation 10.626 (2014) Bazant
equal in equilibrium,
siμi = nμ e (18)
where the electrochemical potential of the electron μe is the Fermi energy
of the highest occupied electronic quantum state.
Inserting the definitions above,
μi = μΘ + kB T ln ai + zieφ (19) i
Θ
μe = μ + kB T ln ae − eφ e (20) e
and using charge conservation
sizi = −n (21)
we arrive at the Nernst equation :
_ si
kB T i ai
Δφ = Δ φ0 − ln (22)
anne e
where Θ Θ
ei
siμi
ΔφΘ = μ − (23)
ne
is the standard potential, or interfacial voltage when all reactants are in their
standard states. The Nernst equation shows how the interfacial voltage of
the half-cell reaction varies with reactant activities away from the standard
state. We can also write
n
kB T aOae
Δφ = Δ φΘ + ln (24)
ne aR
where si |si|
aR = a , aO = ai (25) i
si>0si<0
Redox Equilibrium Constants
For a general chemical reaction, the equilibrium constant K is the ratio of
the activities of the products to those of the reactants, in chemical equilib rium. If K > 1 (or K < 1), the forward (or backward) reaction is ther modynamically favored under standard conditions and leads to more (or
5
3
∑
i
∑
i
Lecture 8: Nernst equation 10.626 (2014) Bazant
less) products compared to reactants. In the case of the general oxidation
reaction, R → O + ne −, the equilibrium constant is
n
= K−1 = aOae (26) Kox red aR
whose inverse Kred is for the backward reduction reaction. (The term redox
is used to refer to reversible charge transfer reactions, and combinations of
oxidation and reduction half reactions are called redox couples .)
The physical significance of the Nernst equation becomes more clear in
terms of the redox equilibrium constants,
(27)
Now we see that by increasing the Nernst equilibrium potential, Δφeq , we
increase Kox so as to favor the forward oxidation reaction that produces
electrons. This makes sense because electrons are drawn to the more positive
potential of the electrode as its voltage relative to the solution is increased.
The opposite trend holds for the standard Nernst potential. If we in crease ΔφΘ, then Kox decreases so as to favor the reduction reaction that
consumes electrons. In galvanic cells, therefore, half-cell Faradaic reactions
with larger ΔφΘ act as cathodes (consuming electrons) when coupled with
those having smaller ΔφΘ, which act as anodes (producing electrons).
4 Standard Cell Potential
In order to measure the potential of an electrode φe, we must form a complete
electrochemical cell with two electrodes, and two half-cell Faradaic reactions,
since only differences in potential are well defined. As such, the standard
cell potential ,
EΘ = Δ φΘ − ΔφΘ = φe (28) ref − φref
is defined relative to a standard reference electrode . An electrode reaction
with EΘ < 0 acts as an anode (negative battery terminal, producing elec trons), and one with EΘ > 0 as the cathode (positive terminal, consuming
electrons), when coupled to the standard reference electrode.
Comment on notation: In electrochemistry, it is common to denote cell
voltage (potential difference) with E to denote “electro-motive force” or
“emf”. This can cause confusion with the electric field amplitude, also de noted E (in all fields of science), which is the local gradient of the potential!
6
Kox = K−1
red
= exp
( ne (∆ φeq − ∆φΘ)
kB T
)Lecture 8: Nernst equation 10.626 (2014) Bazant
Faradaic reactions in aqueous electrolytes EΘ (V)
Li metal → Li +
(aq)
e−
P b + SO 2−
4
→ P bSO 4 + 2e−
H2 → 2H+ − 2e−
2H2O → O2 + 4H+ + 4e−
P bSO 4 + 2H2O → P bO 2 + SO 2−
4
4H+ + 2e−
-3.045
-0.356
0
1.229
1.685
To avoid confusion, we use φ to denote the electrostatic potential, EE = Vφ
for the electric field, Δφ for potential differences (e.g. across interfaces), V
for cell voltage (Δ φ between two electrodes), and V Θ for its reference value
under standard conditions. However, for consistency with electrochemistry
tables, we use EΘ for the standard cell potential, relative to a given refer ence.
For aqueous electrolytes, the usual reference electrode is the “standard
(or normal) hydrogen electrode” (SHE), which consists of H2 gas at 1 atm
partial pressure at room temperature undergoing fast, reversible oxidation,
typically at a platinum electrode, in acidic solution (aH+ = 1, pH=0). As
shown in the figure, one can measure the potential of another electrode
relative to SHE by preparing two aqueous solutions connected by a low-resistance salt bridge, which allows ions to pass and equilibrate across the
two chambers, while a very small current is drawn to measure the open cir cuit voltage of the cell. If the test system is at standard conditions (1atm,
25 ◦C) then we write EΘ = standard cell potential relative to SHE. Some
examples are shown in the table: lithium oxidation, lead-acid battery reac tions, and water electrolysis.
5 Open Circuit Voltage
In equilibrium, the open circuit voltage (OCV) of a two-electrode electro chemical cell can be obtained as the difference of the two half-cell voltages.
Under standard conditions, this is just given by the difference of standard
potentials, which can be looked up in a table or measured experimentally.
For example, we can create a galvanic cell with lithium metal oxidation
(electro-dissolution) occurring at the anode, and oxygen gas reduction to
produce water at the cathode, as shown in the figure. The open circuit
voltage under standard conditions would be VO = 1 .229 − (−3.045) = 4.3V .
7Lecture 8: Nernst equation 10.626 (2014) Bazant Li Li HPEM OH O 22++water
More generally, we use the Nernst equation to describe how the OCV
changes with the activities of the reactants away from standard conditions.
The cell voltage is generally the difference
V = φc − φa (29)
between the potentials of the cathode and anode, or the positive and negative
terminals, respectively, in the case of a galvanic cell. Under open circuit
conditions in equilibrium, the electrolyte potential is constant, so we can
write
VO = Δ φc − Δφa (30)
and use the Nernst equation,
_ s
kT _ j a j
j
(anode reaction)
= V Θ
VO + ln si (31)
ne i ai (cathode reaction)
where the standard OCV with all reactants in the standard states is
V Θ = Δ φΘ − ΔφΘ = EΘ − EΘ
caca
(32) By charge conservation, net cell reaction involves only neutral reactants
and products, so we can also write the OCV as
V Θ kT ln
( acell reactants
VO = +
)
(33)
ne acell products
If we write out the redox reactions,
Ra → Oa + ne − (anode half cell)
−
Oc + ne → Rc (cathode half cell) (34)
Ra + Oc → Oa + Rc (full cell)
then
Θ
kT aR aO
V c
O
= V + ln a (35)
ne aOa aRc
86 Cell Equilibrium Constant
The equilibrium constant of the net a
cell reaction, K = products , is measured areactants
under short circuit conditions at equilibrium, VO = 0, when there is no
spontaneous current flow, and thus no chemical driving force between the
anode and the cathode. The Nernst equation relates the cell equilibrium
constant to the standard cell voltage,
a
neV Θ
K = e /kT K
= ox (36)
Kc
red
which is also related to the ratio of redox equilibrium constants of the two
electrodes. Because this value can be very large or small, it is convenient to
refer to the “pK” value defined by
˜nV Θ
pK = − log 10 K = − (37) log 10 e
although since the historical convention is to use base ten, we must keep
track of the factor (log 10 e)−1 = 2 .302.
From the definition of cell voltage as the Gibbs free energy change of the
system per charge passing through the external circuit, V = Δ G/ (−ne ), we
also recover the fundamental relation
−ΔGΘ/kT
K = e (38)
where ΔGΘ is the free energy of reaction (of products minus reactants) for
the net cell reaction under standard conditions.
Example: Short-circuited hydrogen-oxygen fuel cell. The net
reaction is water production: H2 + O 2 → H2O.
V Θ = 1 .229V (39)
n = 2 (40)
kB T = 26mV (room temperature) (41)
e
∴ K ≈ e94 .5 = 1 .0 × 10 41 » 1, −pK = 41 » 1 (42)
We see that with only one volt standard potential, the equilibrium con stant is astronomically large, almost as large as Avogadro’s number squared
(N 23
A
= 6 .022 × 10 particles per mole), implying that essentially every sin gle molecule of H2 and O2 is converted to H2O in equilibrium, if no other
Lecture 8: Nernst equation 10.626 (2014) Bazant
9Lecture 8: Nernst equation 10.626 (2014) Bazant
reactions or kinetic limitations can occur. This is the typical situation when
an electrochemical cell is short-circuited. Because Faradaic reactions involve
breaking and reforming stable chemical bonds (which are not destroyed by
thermal fluctuations), the free energy of reaction is usually much larger than
the thermal voltage, leading to a very large equilibrium constant for the net
reaction.
Away from short circuit conditions, the OCV can be expressed in terms
of the cell equilibrium constant,
kT Ka reactants
VO = ln (43)
ne aproducts
or in terms of the redox equilibrium constants,
KakT ox aRa aOc
VO = ln (44)
Kcne aOa aRcred
where it is clear that VO = 0 in equilibrium.
10 MIT OpenCourseWare
10.626 Electrochemical Energy Systems
Spring 20 14
For information about citing these materials or our Terms of Use, visit: |
8673 | https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Electro-Optics/Electromagnetic_Field_Theory%3A_A_Problem_Solving_Approach_(Zahn)/09%3A_Radiation/9.02%3A_Radiation_from_Point_Dipole | 9.2: Radiation from Point Dipole - Engineering LibreTexts
Skip to main content
Table of Contents menu
search Search build_circle Toolbar fact_check Homework cancel Exit Reader Mode
school Campus Bookshelves
menu_book Bookshelves
perm_media Learning Objects
login Login
how_to_reg Request Instructor Account
hub Instructor Commons
Search
Search this book
Submit Search
x
Text Color
Reset
Bright Blues Gray Inverted
Text Size
Reset
+-
Margin Size
Reset
+-
Font Type
Enable Dyslexic Font - [x]
Downloads expand_more
Download Page (PDF)
Download Full Book (PDF)
Resources expand_more
Periodic Table
Physics Constants
Scientific Calculator
Reference expand_more
Reference & Cite
Tools expand_more
Help expand_more
Get Help
Feedback
Readability
x
selected template will load here
Error
This action is not available.
chrome_reader_mode Enter Reader Mode
9: Radiation
Electromagnetic Field Theory: A Problem Solving Approach (Zahn)
{ }
{ "9.01:_The_Retarded_Potentials" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "9.02:_Radiation_from_Point_Dipole" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "9.03:_Point_Dipole_Arrays" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "9.04:_Long_Dipole_Antennas" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "9.05:_Problems" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1" }
{ "00:_Front_Matter" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "01:_Review_of_Vector_Analysis" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "02:_The_Electric_Field" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "03:_Polarization_and_Conduction" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "04:_Electric_Field_Boundary_Value_Problems" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "05:_The_Magnetic_Field" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "06:_Electromagnetic_Induction" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "07:_Electrodynamics-fields_and_Waves" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "08:_Guided_Electromagnetic_Waves" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "09:_Radiation" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "zz:_Back_Matter" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1" }
Tue, 03 Oct 2023 23:59:29 GMT
9.2: Radiation from Point Dipole
48177
48177
Joshua Halpern
{ }
Anonymous
Anonymous User
2
false
false
[ "article:topic", "license:ccbyncsa", "program:mitocw", "authorname:mzahn", "licenseversion:40", "source@ ]
[ "article:topic", "license:ccbyncsa", "program:mitocw", "authorname:mzahn", "licenseversion:40", "source@ ]
Search site Search Search Go back to previous article
Sign in
Username Password Sign in
Sign in
Sign in
Forgot password
Expand/collapse global hierarchy
1. Home
2. Bookshelves
3. Electrical Engineering
4. Electro-Optics
5. Electromagnetic Field Theory: A Problem Solving Approach (Zahn)
6. 9: Radiation
7. 9.2: Radiation from Point Dipole
Expand/collapse global location
9.2: Radiation from Point Dipole
Last updated Oct 3, 2023
Save as PDF
9.1: The Retarded Potentials
9.3: Point Dipole Arrays
Page ID 48177
Markus Zahn
Massachusetts Institute of Technology via MIT OpenCourseWare
( \newcommand{\kernel}{\mathrm{null}\,})
Table of contents
1. The Electric Dipole
2. Alternate Derivation Using the Scalar Potential
3. The Electric and Magnetic Fields
4. Electric Field Lines
5. Radiation Resistance
6. Rayleigh Scattering (or why is the sky blue?)
7. Radiation from a Point Magnetic Dipole
The Electric Dipole
The simplest building block for a transmitting antenna is that of a uniform current flowing along a conductor of incremental length dl as shown in Figure 9-1. We assume that this current varies sinusoidally with time as
(9.2.1)i(t)=Re(I^e jωt)
Because the current is discontinuous at the ends, charge must be deposited there being of opposite sign at each end [q(t)=Re(Q^e jωt)]:
(9.2.2)i(t)=±dq dt⇒I^=±jωQ^,z=±dl 2
This forms an electric dipole with moment
(9.2.3)p=q dl i z
If we can find the potentials and fields from this simple element, the solution for any current distribution is easily found by superposition.
Figure 9-1 A point dipole antenna is composed of a very short uniformly distributed current-carrying wire. Because the current is discontinuous at the ends, equal magnitude but opposite polarity charges accumulate there forming an electric dipole.
By symmetry, the vector potential cannot depend on the angle ϕ,
(9.2.4)A z=Re[A^z(r,θ)e jωt]
and must be in the same direction as the current:
(9.2.5)A z(r,t)=Re[∫−dl/2+dl/2 μI^e j[ω(t−r QP/c)]4πr QP d z]
Because the dipole is of infinitesimal length, the distance from the dipole to any field point is just the spherical radial distance r and is constant for all points on the short wire. Then the integral in (5) reduces to a pure multiplication to yield
(9.2.6)A^z=μI^dl 4πre−jkr,A z(r,t)=Re[A^z(r)e jωt]
where we again introduce the wavenumber k=ω/c and neglect writing the sinusoidal time dependence present in all field and source quantities. The spherical components of A^z, are (i z=i rcosθ−i θsinθ):
(9.2.7)A^r=A^zcosθ,A^θ=−A^zsinθ,A^ϕ=0
Once the vector potential is known, the electric and magnetic fields are most easily found from
(9.2.8)H^=1 μ∇×A^,H(r,t)=Re[H^(r,θ)e jωt]E^=1 jωε∇×H^,E(r,t)=Re[E^(r,θ)e jωt]
Before we find these fields, let's examine an alternate approach.
Alternate Derivation Using the Scalar Potential
It was easiest to find the vector potential for the point electric dipole because the integration in (5) reduced to a simple multiplication. The scalar potential is due solely to the opposite point charges at each end of the dipole,
(9.2.9)V^=Q 4πε(e−jkr+r+−e−jkr−r−)
where r+ and r− are the distances from the respective dipole charges to any field point, as shown in Figure 9-1. Just as we found for the quasi-static electric dipole in Section 3-1-1, we cannot let r+and r−equal r as a zero potential would result. As we showed in Section 3-1-1, a first-order correction must be made, where
(9.2.10)r+≈r−dl 2cosθ r−≈r+dl 2cosθ
so that (9) becomes
(9.2.11)V^≈Q 4πεre−jkr(e jk(dl/2)cosθ(1−dl 2rcosθ)−e−jk(dl/2)cosθ(1+dl 2rcosθ))
Because the dipole length dl is assumed much smaller than the field distance r and the wavelength, the phase factors in the exponentials are small so they and the 1/r dependence in the denominators can be expanded in a first-order Taylor series to result in:
(9.2.12)lim k dl≪1 dl/r≪1V^≈Q 4πεre−jkr[(1+jk dl 2cosθ)(1+dl 2rcosθ)−(1−jkdl 2cosθ)(1−dl 2rcosθ)]=Qdl 4πεr 2e−jkrcosθ(1+jkr)
When the frequency becomes very low so that the wavenumber also becomes small, (12) reduces to the quasi-static electric dipole potential found in Section 3-1-1 with dipole moment p^=Q^dl. However, we see that the radiation correction terms in (12) dominate at higher frequencies (large k) far from the dipole (kr≫1) so that the potential only dies off as 1/r rather than the quasi-static 1/r 2. Using the relationships Q^=I^/jω and c=1/εμ, (12) could have been obtained immediately from (6) and (7) with the Lorentz gauge condition of Eq. (13) in Section 9-1-1:
V^=−c 2 jω∇⋅A^=−c 2 jω(1 r 2∂∂r(r 2A^r)+1 rsinθ∂∂θ(A^θsinθ))=μI^dlc 2 4πjω(1+jkr)r 2e−jkrcosθ=Q^dl 4πεr 2(1+jkr)e−jkrcosθ
The Electric and Magnetic Fields
Using (6), the fields are directly found from (8) as
(9.2.13)H^=1 μ×A^=i ϕ1 μr(∂∂r(rA^θ)−∂A^r∂θ)=−i ϕI^dl 4πk 2sinθ(1 jkr+1(jkr)2)e−jkr
(9.2.14)E^=1 jωε∇×H^=1 jωε(1 rsinθ∂∂θ(H^ϕsinθ)i r−1 r∂∂r(rH^ϕ)i θ)=−I^dlk 2 4πμ ε{i r[2cosθ(1(jkr)2+1(jkr)3)]+i θ[sinθ(1 jkr+1(jkr)2+1(jkr)3)]}e−jkr
Note that even this simple source generates a fairly complicated electromagnetic field. The magnetic field in (14) points purely in the ϕ direction as expected by the right-hand rule for a z-directed current. The term that varies as 1/r 2 is called the induction field or near field for it predominates at distances close to the dipole and exists even at zero frequency. The new term, which varies as 1/r, is called the radiation field since it dominates at distances far from the dipole and will be shown to be responsible for time-average power flow away from the source. The near field term does not contribute to power flow but is due to the stored energy in the magnetic field and thus results in reactive power.
The 1/r 3 terms in (15) are just the electric dipole field terms present even at zero frequency and so are often called the electrostatic solution. They predominate at distances close to the dipole and thus are the near fields. The electric field also has an intermediate field that varies as 1/r 2, but more important is the radiation field term in the io component, which varies as 1/r. At large distances (kr≫1) this term dominates.
In the far field limit (kr≫1), the electric and magnetic fields are related to each other in the same way as for plane waves:
(9.2.15)lim kr≫1E^θ=μ εH^ϕ=E^0 jkrsinθe−jkr,E^0=−I^dlk 2 4πμ ε
The electric and magnetic fields are perpendicular and their ratio is equal to the wave impedance η=μ/ε. This is because in the far field limit the spherical wavefronts approximate a plane.
Electric Field Lines
Outside the dipole the volume charge density is zero, which allows us to define an electric vector potential C:
(9.2.16)∇⋅E=0⇒E=∇×C
Because the electric field in (15) only has r and θ components, C must only have a ϕ component, C ϕ(r,θ):
(9.2.17)E=∇×C=1 rsinθ∂∂θ(sinθC ϕ)i r−1 r∂∂r(rC ϕ)i θ
We follow the same procedure developed in Section 4-4-3b, where the electric field lines are given by
(9.2.18)dr rdθ=E r E θ=−∂∂θ(sinθC ϕ)sinθ∂∂r(rC ϕ)
which can be rewritten as an exact differential,
(9.2.19)∂∂r(rsinθC ϕ)dr+∂∂θ(rsinθC ϕ)dθ=0⇒d(rsinθC ϕ)=0
so that the field lines are just lines of constant stream-function rsinθC ϕ. C ϕ is found by equating each vector component in (18) to the solution in (15):
(9.2.20)1 rsinθ∂∂θ(sinθC ϕ^)=E^r=−I^dlk 2 4πμ ε[2cosθ(1(jkr)2+1(jkr)3)]e−jkr−1 r∂∂r(rC ϕ^)=E^θ=−I^dlk 2 4πμ ε[sinθ(1(jkr)+1(jkr)2+1(jkr)3)]e−jkr
which integrates to
(9.2.21)C ϕ^=I^dl 4πμ εsinθ r(1−j(kr))e−jkr
Then assuming I^ is real, the instantaneous value of C ϕ is
(9.2.22)C ϕ=Re(C^ϕe jωt)=I^dl 4πμ εsinθ r(cos(ωt−kr)+sin(ωt−kr)kr)
so that, omitting the constant amplitude factor in (23), the field lines are
(9.2.23)rC ϕsinθ=const⇒sin 2θ(cos(ωt−kr)+sin(ωt−kr)kr)=const
Figure 9-2 The electric field lines for a point electric dipole at ωt=0 and ωt=π/2.
These field lines are plotted in Figure 9-2 at two values of time. We can check our result with the static field lines for a dipole given in Section 3-1-1. Remembering that k=ω/c, at low frequencies,
(9.2.24)lim ω→0{cos(ωt−kr)≈1 sin(ωt−kr)kr≈(t−r/c)r/c≈t r/c−1
so that, in the low-frequency limit at a fixed time, (24) approaches the result of Eq. (6) of Section 3-1-1:
(9.2.25)lim ω→0sin 2θ(ct r)=const
Note that the field lines near the dipole are those of a static dipole field, as drawn in Figure 3-2. In the far field limit
(9.2.26)lim kr≫1sin 2θcos(ωt−kr)=const
the field lines repeat with period λ=2π/k.
Radiation Resistance
Using the electric and magnetic fields of Section 9-2-3, the time-average power density is
(9.2.27)=1 2Re(E^×E^∗)=Re{|I^dl|2ηk 4 2(4π)2[−i θsin2θ(−1(jkr)3+1(jkr)5)]+i rsin 2θ(−1(jkr)2+1(jkr)5)}=1 2|I^dl|2(k 4π)2η r 2sin 2θi r=1 2|E^0|2 ηsin 2θ(kr)2i r
where E^0 is defined in (16).
Only the far fields contributed to the time-average power flow. The near and intermediate fields contributed only imaginary terms in (28) representing reactive power.
The power density varies with the angle θ, being zero along the electric dipole's axis (θ=0,π) and maximum at right angles to it (θ=π/2), illustrated by the radiation power pattern in Fig. 9-3. The strength of the power density is proportional to the length of the vector from the origin to the
Figure 9-3 The strength of the electric field and power density due to a z-directed point dipole as a function of angle θ is proportional to the length of the vector from the origin to the radiation pattern.
radiation pattern. These directional properties are useful in beam steering, where the directions of power Row can be controlled.
The total time-average power radiated by the electric dipole is found by integrating the Poynting vector over a spherical surface at any radius r:
(9.2.28)=∫θ=0 π∫ϕ=0 2πr 2sinθdθdϕ=1 2|I^dl|2(k 4π)2η2π∫θ=0 π sin 3θdθ=|I^dl|2 16πηk 2[−1 3cosθ(sin 2θ+2)]|0 π=|I^dl|2 12πηk 2
As far as the dipole is concerned, this radiated power is lost in the same way as if it were dissipated in a resistance R,
(9.2.29)=1 2|I^|2R
where this equivalent resistance is called the radiation resistance:
(9.2.30)R=η 6π(k dl)2=2πη 3(dl λ)2,k=2π λ
In free space η 0=μ 0/ε 0≈120π, the radiation resistance is
(9.2.31)R 0=80π 2(dl λ)2(free space)
These results are only true for point dipoles, where dl is much less than a wavelength (dl/λ≪1). This verifies the validity of the quasi-static approximation for geometries much smaller than a radiated wavelength, as the radiated power is then negligible.
If the current on a dipole is not constant but rather varies with z over the length, the only term that varies with z for the vector potential in (5) is I^(z):
(9.2.32)A^z(r)=Re[∫−dl/2+dl/2 μI^(z)e−jkr QP 4πr QP d z]≈Re[μe−jkr QP 4πr QP∫−dl/2+dl/2 I^(z)d z]
where, because the dipole is of infinitesimal length, the distance r QP from any point on the dipole to any field point far from the dipole is essentially r, independent of z. Then, all further results for the electric and magnetic fields are the same as in Section 9-2-3 if we replace the actual dipole length dl by its effective length,
(9.2.33)dl eff=1 I^0∫−dl/2+dl/2 I^(z)d z
where I^0 is the terminal current feeding the center of the dipole.
Generally the current is zero at the open circuited ends, as for the linear distribution shown in Figure 9-4,
(9.2.34)I^(z)={I 0(1−2z/dl),0≤z≤dl/2 I 0(1−2z/dl),−dl/2≤z≤0
so that the effective length is half the actual length:
(9.2.35)dl eff=1 I 0∫−dl/2+dl/2 I^(z)d z=dl 2
Figure 9-4 (a) If a point electric dipole has a nonuniform current distribution, the solutions are of the same form if we replace the actual dipole length dl by an effective length dl eff. (b) For a triangular current distribution the effective length is half the true length.
Because the fields are reduced by half, the radiation resistance is then reduced by :
(9.2.36)R=2πη 3(dl eff λ)2=20π 2μ r ε r(dl λ)2
In free space the relative permeability μ r and relative permittivity ε r are unity.
Note also that with a spatially dependent current distribution, a line charge distribution is found over the whole length of the dipole and not just on the ends:
(9.2.37)λ^=−1 jωdI^dz
For the linear current distribution described by (35), we see that:
(9.2.38)λ^=±2I 0 jωdl{0≤z≤dl/2−dl/2≤z≤0
Rayleigh Scattering (or why is the sky blue?)
If a plane wave electric field Re[E 0e jωti x]is incident upon an atom that is much smaller than the wavelength, the induced dipole moment also contributes to the resultant field, as illustrated in Figure 9-5. The scattered power is perpendicular to the induced dipole moment. Using the dipole model developed in Section 3-1-4, where a negative spherical electron cloud of radius R 0 with total charge −Q surrounds a fixed
Figure 9-5 An incident electric field polarizes dipoles that then re-radiate their energy primarily perpendicular to the polarizing electric field. The time-average scattered power increases with the fourth power of frequency so shorter wavelengths of light are scattered more than longer wavelengths. (a) During the daytime an earth observer sees more of the blue scattered light so the sky looks blue (short wavelengths). (b) Near sunset the light reaching the observer lacks blue so the sky appears reddish (long wavelength).
positive point nucleus, Newton's law for the charged cloud with mass m is:
(9.2.39)d 2x dt 2+ω 0 2x=Re(QE 0 me jωt),ω 0 2=Q 2 4πεmR 0 3
The resulting dipole moment is then
(9.2.40)p^=Qx^=Q 2E 0/m ω 0 2−ω 2
where we neglect damping effects. This dipole then re-radiates with solutions given in Sections 9-2-1-9-2-5 using the dipole moment of (41) (I^dl→jωp^). The total time-average power radiated is then found from (29) as
(9.2.41)=ω 4|p^|2η 12πc 2=ω 4η(Q 2E 0/m)2 12πc 2(ω 0 2−ω 2)2
To approximately compute ω 0, we use the approximate radius of the electron found in Section 3-8-2 by equating the energy stored in Einstein's relativistic formula relating mass to energy:
(9.2.42)mc 2=3Q 2 20πεR 0⇒R 0=3Q 2 20πεmc 2≈1.69×10−15m
Then from (40)
(9.2.43)ω 0=5/320πεmc 2 3Q 2≈2.3×10 23 radian/sec
is much greater than light frequencies (ω≈10 15) so that (42) becomes approximately
(9.2.44)lim ω 0≫ω≈η 12π(Q 2E 0ω 2 mcω 0 2)2
This result was originally derived by Rayleigh to explain the blueness of the sky. Since the scattered power is proportional to ω 4, shorter wavelength light dominates. However, near sunset the light is scattered parallel to the earth rather than towards it. The blue light received by an observer at the earth is diminished so that the longer wavelengths dominate and the sky appears reddish.
Radiation from a Point Magnetic Dipole
A closed sinusoidally varying current loop of very small size flowing in the z=0 plane also generates radiating waves. Because the loop is closed, the current has no divergence so that there is no charge and the scalar potential is zero. The vector potential phasor amplitude is then
(9.2.45)A^(r)=∫μI^e−jkr QP 4πr QP d l
We assume the dipole to be much smaller than a wavelength, k(r QP−r)≪1, so that the exponential factor in (46) can be linearized to
(9.2.46)lim k(r QP−r)≪1e−jkr QP=e−jkre−jk(r QP−r)≈e−jkr[1−jk(r QP−r)]
Then (46) reduces to
(9.2.47)A^(r)=∫μI^4πe−jkr(1+jkr r QP−jk)d l=e−jkr∫μI^4π(1+jkr r QP−jk)d l=μ 4πe−jkr((1+jkr)∫I^d l r QP−jk∫I^d l)
where all terms that depend on r can be taken outside the integrals because r is independent of dl. The second integral is zero because the vector current has constant magnitude and flows in a closed loop so that its average direction integrated over the loop is zero. This is most easily seen with a rectangular loop where opposite sides of the loop contribute equal magnitude but opposite signs to the integral, which thus sums to zero. If the loop is circular with radius a,
(9.2.48)I^dl=I^i ϕadϕ⇒∫0 2π i ϕ d ϕ=∫0 2π(−sinϕi x+cosϕi y)d ϕ=0
the integral is again zero as the average value of the unit vector i ϕ around the loop is zero.
The remaining integral is the same as for quasi-statics except that it is multiplied by the factor (1+jkr)e−jkr. Using the results of Section 5-5-1, the quasi-static vector potential is also multiplied by this quantity:
(9.2.49)A^=μm^4πr 2sinθ(1+jkr)e−jkri ϕ,m^=I^dS
The electric and magnetic fields are then
(9.2.50)H^=1 μ∇×A^=−m^4πjk 3e−jkr{i r[2cosθ(1(jkr)2+1(jkr)3)]+i θ[sinθ(1 jkr+1(jkr)2+1(jkr)3)]}E^=1 jωε∇×H^=m^jk 3 4πηe−jkrsinθ(1 jkr+1(jkr)2)i ϕ
The magnetic dipole field solutions are the dual to those of the electric dipole where the electric and magnetic fields reverse roles if we replace the electric dipole moment with the magnetic dipole moment:
(9.2.51)p ε=qdl ε=Idl jωε→m
This page titled 9.2: Radiation from Point Dipole is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Markus Zahn (MIT OpenCourseWare) via source content that was edited to the style and standards of the LibreTexts platform.
Back to top
9.1: The Retarded Potentials
9.3: Point Dipole Arrays
Was this article helpful?
Yes
No
Recommended articles
9.1: The Retarded PotentialsMaxwell's equations in complete generality are
9.3: Point Dipole ArraysThe power density for a point electric dipole varies with the broad angular distribution sin 2θ. Often it is desired that the power patt...
9.4: Long Dipole AntennasThe radiated power, proportional to (dl/λ)2, is small for point dipole antennas where the dipole's length dl is. much...
9.5: Problems
Article typeSection or PageAuthorMarkus ZahnLicenseCC BY-NC-SALicense Version4.0OER program or PublisherMIT OpenCourseWare
Tags
source@
© Copyright 2025 Engineering LibreTexts
Powered by CXone Expert ®
?
The LibreTexts libraries arePowered by NICE CXone Expertand are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Privacy Policy. Terms & Conditions. Accessibility Statement.For more information contact us atinfo@libretexts.org.
Support Center
How can we help?
Contact Support Search the Insight Knowledge Base Check System Status×
contents readability resources tools
☰ |
8674 | https://arxiv.org/html/2102.02546v2 | Order parameter conditions from mutual information and symmetry conditions
1 Introduction
2 Vacuum degeneracy and spontaneous symmetry breaking
2.1 Example: Ferromagnetism
3 The mutual information method and the construction of a local order operator
3.1 The local order operator
4 Additional conditions: General broken symmetries of the system
4.1 Mapping one vacuum into another one with the same energy level via broken symmetry
5 Applications
5.1 Ferromagnetic long-range order
5.2 One dimensional ferromagnetic
6 Conclusions
Order parameter conditions from mutual information and symmetry conditions
Ivan Arraut
Institute of Science and Environment and FBL,
University of Saint Joseph
Estrada Marginal da Ilha Verde, 14-17, Macao, China
ivan.arraut@usj.edu.mo
\And Wing Chi Yu
Department of Physics,
City University of Hong Kong,
Kowloon, Hong Kong.
wingcyu@cityu.edu.hk
Abstract
The mutual information method has demonstrated to be very useful for deriving the potential order parameter of a system. Although the method suggests some constraints which help to define this quantity, there is still some freedom in the definition. The method then results inefficient for cases where we have order parameters with a large number of constants in the expansion, which happens when we have many degenerate vacuums. Here we introduce some additional constraints based on the existence of broken symmetries, which help us to reduce the arbitrariness in the definitions of the order parameter in the proposed mutual information method.
1 Introduction
When the symmetry of a system is spontaneously broken, a vacuum degeneracy emerges naturally . This is a consequence of the fact that the vacuum itself does not respect some of the symmetries of the Lagrangian (Hamiltonian). These are called broken symmetries and their generators are called broken generators. There are many systems in the nature able to develop this condition called spontaneous symmetry breaking [2, 3]. The phenomena has several applications in condensed matter systems as well as in high-energy physics. Tied to this phenomena, is the Nambu-Goldstone theorem, which establishes that the number of broken symmetries are equal to the number of gapless particles appearing in the system and called Nambu-Goldstone bosons [1, 2]. When the system experience spontaneous symmetry breaking, an order parameter appears. The order parameter is just a field showing some long correlation in the system. One way to find whether a system is able to develop long range order or not, is by using the concept of mutual information [4, 5, 6, 7, 8]. Once it is established that the system develops long range order, then an order parameter field should be associated to this phenomena. This field can be taken as an expansion of n 𝑛 n italic_n-terms with n 𝑛 n italic_n arbitrary constants. The constants are partially fixed by the trace-less conditions of the order parameter with respect to the degenerate vacuum . This constraint is good for the cases where there are only a few constants to fix in the expansion. However, for the cases where we have several constants, the trace-less condition alone is of limited scope. In , besides the traceless condition, an arbitrarily imposed normalization’s condition over the constants is assumed. Although in it is assumed no-knowledge about the symmetries in the system, we can still assume their existence. It is known that the combination of the order-parameter plus broken generators, can create some additional constraints over the order-parameters of the system [9, 10, 11]. In this paper, by assuming the general existence of broken symmetries, we add two important (and additional) constraints. The first one corresponds to the fact that for every m 𝑚 m italic_m-set of vacuums |0⟩m subscript ket 0 𝑚\ket{0}{m}| start_ARG 0 end_ARG ⟩ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT connected by a set of broken symmetries Q m subscript 𝑄 𝑚 Q{m}italic_Q start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT, we have the condition |0⟩=e−iω|0⟩2=e−2iω|0⟩3=…=e−i(m−1)ω|0⟩m ket 0 superscript 𝑒 𝑖 𝜔 subscript ket 0 2 superscript 𝑒 2 𝑖 𝜔 subscript ket 0 3…superscript 𝑒 𝑖 𝑚 1 𝜔 subscript ket 0 𝑚\ket{0}=e^{-i\omega}\ket{0}{2}=e^{-2i\omega}\ket{0}{3}=...=e^{-i(m-1)\omega}% \ket{0}_{m}| start_ARG 0 end_ARG ⟩ = italic_e start_POSTSUPERSCRIPT - italic_i italic_ω end_POSTSUPERSCRIPT | start_ARG 0 end_ARG ⟩ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_e start_POSTSUPERSCRIPT - 2 italic_i italic_ω end_POSTSUPERSCRIPT | start_ARG 0 end_ARG ⟩ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = … = italic_e start_POSTSUPERSCRIPT - italic_i ( italic_m - 1 ) italic_ω end_POSTSUPERSCRIPT | start_ARG 0 end_ARG ⟩ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT, with ω 𝜔\omega italic_ω defining the phase difference between the degenerate vacuums. This means that each vacuum differs at most by a phase. The phase difference between neighbors vacuums will be defined in this paper. The second condition corresponds to the cyclic property of vacuums connected by broken symmetries, which suggests that the successive action of a broken generator over a series of vacuums, connected each other through the action of the same generator, will bring back the initial vacuum after n 𝑛 n italic_n successive applications of the broken generator. These two conditions complement correctly the mutual information method and they are enough for determining the form of the order parameter and they automatically include the traceless condition. The paper is organized as follows: In Sec. (3), we analyze the relation between mutual information and spontaneous symmetry breaking; in addition we analyze the previously proposed constraints which partially fix the order parameter of a system. In Sec. (4), we introduce one extra-constraint related to the existence of broken symmetries and we demonstrate that the traceless condition is just a special case of the constraints. In Sec. (5), we verify a few systems where we are able to predict the form of the order parameter by using the method proposed in this paper. Finally, in Sec. (6), we conclude.
2 Vacuum degeneracy and spontaneous symmetry breaking
For understanding the notions of vacuum degeneracy and spontaneous symmetry breaking, we can use as a starting point the σ 𝜎\sigma italic_σ-model, which contains all the necessary ingredients for understanding these concepts . The model has N scalar fields defined by ϕ i(x)superscript italic-ϕ 𝑖 𝑥\phi^{i}(x)italic_ϕ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ( italic_x ) and we can then define its Lagrangian as
£=1 2(∂μ ϕ i)2+1 2μ 2(ϕ i)2−λ 4[(ϕ i)2]2.£1 2 superscript subscript 𝜇 superscript italic-ϕ 𝑖 2 1 2 superscript 𝜇 2 superscript superscript italic-ϕ 𝑖 2 𝜆 4 superscript delimited-[]superscript superscript italic-ϕ 𝑖 2 2\pounds=\frac{1}{2}(\partial_{\mu}\phi^{i})^{2}+\frac{1}{2}\mu^{2}(\phi^{i})^{% 2}-\frac{\lambda}{4}[(\phi^{i})^{2}]^{2}.£ = divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( ∂ start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT italic_ϕ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_μ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_ϕ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - divide start_ARG italic_λ end_ARG start_ARG 4 end_ARG [ ( italic_ϕ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ] start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT .(1)
This Lagrangian is symmetric with respect to the transformations
ϕ i→R ijϕ j.→superscript italic-ϕ 𝑖 superscript 𝑅 𝑖 𝑗 superscript italic-ϕ 𝑗\phi^{i}\to R^{ij}\phi^{j}.italic_ϕ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT → italic_R start_POSTSUPERSCRIPT italic_i italic_j end_POSTSUPERSCRIPT italic_ϕ start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT .(2)
This is a representation of the group O(N)𝑂 𝑁 O(N)italic_O ( italic_N ) of orthogonal matrices in N 𝑁 N italic_N-dimensions. For understanding the dynamics of the system, we can focus on its potential term, which from eq. (1) is defined as
V(ϕ i)=−1 2μ 2(ϕ i)2+λ 4[(ϕ i)2]2.𝑉 superscript italic-ϕ 𝑖 1 2 superscript 𝜇 2 superscript superscript italic-ϕ 𝑖 2 𝜆 4 superscript delimited-[]superscript superscript italic-ϕ 𝑖 2 2 V(\phi^{i})=-\frac{1}{2}\mu^{2}(\phi^{i})^{2}+\frac{\lambda}{4}[(\phi^{i})^{2}% ]^{2}.italic_V ( italic_ϕ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ) = - divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_μ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_ϕ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + divide start_ARG italic_λ end_ARG start_ARG 4 end_ARG [ ( italic_ϕ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ] start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT .(3)
The minimum value for this potential can be obtained from the condition ∂V(ϕ i)/∂ϕ i=0 𝑉 superscript italic-ϕ 𝑖 superscript italic-ϕ 𝑖 0\partial V(\phi^{i})/\partial\phi^{i}=0∂ italic_V ( italic_ϕ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ) / ∂ italic_ϕ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT = 0. The non-trivial ground state is given by
(ϕ 0 i)2=μ 2 λ.superscript subscript superscript italic-ϕ 𝑖 0 2 superscript 𝜇 2 𝜆(\phi^{i}_{0})^{2}=\frac{\mu^{2}}{\lambda}.( italic_ϕ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = divide start_ARG italic_μ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_λ end_ARG .(4)
It is evident that this result gives us the magnitude of ϕ 0 i subscript superscript italic-ϕ 𝑖 0\phi^{i}_{0}italic_ϕ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT. Its direction is still arbitrary but the system selects one among all the possible directions. Then for example, the system could select the ground state as follows
ϕ 0 i=(0,0,…,v),subscript superscript italic-ϕ 𝑖 0 0 0…𝑣\phi^{i}_{0}=(0,0,...,v),italic_ϕ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = ( 0 , 0 , … , italic_v ) ,(5)
with v=μ/λ 𝑣 𝜇 𝜆 v=\mu/\sqrt{\lambda}italic_v = italic_μ / square-root start_ARG italic_λ end_ARG. This is a basic example of spontaneous symmetry breaking, where the system selects arbitrarily a vacuum state among the infinite possibilities it has. The figure (1) illustrates the form of the potential (3) for the ground state defined in eq. (4). In what follows we will show an example of this phenomena, applied to ferromagnetism.
Figure 1: Typical potential of the σ 𝜎\sigma italic_σ-model. For certain combination of parameters, the potential develops a multiplicity of ground states (vacuums).
2.1 Example: Ferromagnetism
One of the best examples about a multiplicity of ground states with a subsequent spontaneous symmetry breaking, is the phenomena associated to ferromagnetism. In this case we can use the theory developed by Landau [12, 13]. The general idea is that the material develops a preferred axis of magnetization and the total magnetization M 𝑀 M italic_M is then the order parameter. This phenomena can be better perceived by defining the Gibbs-free energy G 𝐺 G italic_G as a series expansion of the total magnetization for temperatures near the critical value T≈T c 𝑇 subscript 𝑇 𝑐 T\approx T_{c}italic_T ≈ italic_T start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT (second order phase transition) [12, 13]
G(M)≈A(T)+B(T)M 2+C(T)M 4+…𝐺 𝑀 𝐴 𝑇 𝐵 𝑇 superscript 𝑀 2 𝐶 𝑇 superscript 𝑀 4…G(M)\approx A(T)+B(T)M^{2}+C(T)M^{4}+...italic_G ( italic_M ) ≈ italic_A ( italic_T ) + italic_B ( italic_T ) italic_M start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_C ( italic_T ) italic_M start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT + …(6)
The even power expansion on M 𝑀 M italic_M, is due to the symmetry under the change M→−M→𝑀 𝑀 M\to-M italic_M → - italic_M. The perturbative expansion around M≈0 𝑀 0 M\approx 0 italic_M ≈ 0 is possible because near the critical point T≈T c 𝑇 subscript 𝑇 𝑐 T\approx T_{c}italic_T ≈ italic_T start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT, we have M≈0 𝑀 0 M\approx 0 italic_M ≈ 0. The equilibrium condition corresponds to the case where ∂G/∂M=0 𝐺 𝑀 0\partial G/\partial M=0∂ italic_G / ∂ italic_M = 0. Considering the fact that the parameters B(T)𝐵 𝑇 B(T)italic_B ( italic_T ) and C(T)𝐶 𝑇 C(T)italic_C ( italic_T ) are defined as
B(T)=b(T−T c),C(T)=c,formulae-sequence 𝐵 𝑇 𝑏 𝑇 subscript 𝑇 𝑐 𝐶 𝑇 𝑐 B(T)=b(T-T_{c}),\;\;\;C(T)=c,italic_B ( italic_T ) = italic_b ( italic_T - italic_T start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ) , italic_C ( italic_T ) = italic_c ,(7)
with b 𝑏 b italic_b and c 𝑐 c italic_c being constant terms. From these results, we can see that B(T)𝐵 𝑇 B(T)italic_B ( italic_T ) changes sign when the temperature T 𝑇 T italic_T crosses the critical point T≈T c 𝑇 subscript 𝑇 𝑐 T\approx T_{c}italic_T ≈ italic_T start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT. Considering this issue, then the ground state condition for this system is
M=0,T>T C,formulae-sequence 𝑀 0 𝑇 subscript 𝑇 𝐶\displaystyle M=0,\;\;\;\;\;T>T_{C},italic_M = 0 , italic_T > italic_T start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ,
M=±(b 2c(T−T C))1/2.𝑀 plus-or-minus superscript 𝑏 2 𝑐 𝑇 subscript 𝑇 𝐶 1 2\displaystyle M=\pm\left(\frac{b}{2c}(T-T_{C})\right)^{1/2}.italic_M = ± ( divide start_ARG italic_b end_ARG start_ARG 2 italic_c end_ARG ( italic_T - italic_T start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ) ) start_POSTSUPERSCRIPT 1 / 2 end_POSTSUPERSCRIPT .(8)
This is obtained from the condition ∂G/∂M=0 𝐺 𝑀 0\partial G/\partial M=0∂ italic_G / ∂ italic_M = 0. If we now consider a non-zero magnetic field H 𝐻 H italic_H applied over the material and if the T<T C 𝑇 subscript 𝑇 𝐶 T<T_{C}italic_T < italic_T start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT, then the vacuum state is non-unique. This is the point where the system has to select one among the many possible ground states of the system, breaking in this way the symmetry under rotations spontaneously. It is important to remark that this phenomena of spontaneous symmetry breaking, is associated to long correlation lengths inside the material arrangement. For analyzing this situation, we can take the total magnetization as
M=∫𝑑 x 3s(𝐱).𝑀 differential-d superscript 𝑥 3 𝑠 𝐱 M=\int dx^{3}s({\bf x}).italic_M = ∫ italic_d italic_x start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_s ( bold_x ) .(9)
Here 𝐱 𝐱{\bf x}bold_x is a vector related to the coordinates inside the material. In this way, we can express the Gibbs free-energy in eq. (6) as follows
G=∫𝑑 x 3(1 2(∇s(𝐱))2+b(T−T c)s 2+cs 4−Hs).𝐺 differential-d superscript 𝑥 3 1 2 superscript∇𝑠 𝐱 2 𝑏 𝑇 subscript 𝑇 𝑐 superscript 𝑠 2 𝑐 superscript 𝑠 4 𝐻 𝑠 G=\int dx^{3}\left(\frac{1}{2}(\nabla s({\bf x}))^{2}+b(T-T_{c})s^{2}+cs^{4}-% Hs\right).italic_G = ∫ italic_d italic_x start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ( divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( ∇ italic_s ( bold_x ) ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_b ( italic_T - italic_T start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ) italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_c italic_s start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT - italic_H italic_s ) .(10)
In this case, we include the external magnetic field H 𝐻 H italic_H and we have also included the kinetic terms 1 2(∇s(𝐱))2 1 2 superscript∇𝑠 𝐱 2\frac{1}{2}(\nabla s({\bf x}))^{2}divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( ∇ italic_s ( bold_x ) ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT. The variational principle gives us the result
(−∇2+2b(T−T C)+4cs 2(𝐱))s(𝐱)=H(𝐱).superscript∇2 2 𝑏 𝑇 subscript 𝑇 𝐶 4 𝑐 superscript 𝑠 2 𝐱 𝑠 𝐱 𝐻 𝐱\left(-\nabla^{2}+2b(T-T_{C})+4cs^{2}({\bf x})\right)s({\bf x})=H({\bf x}).( - ∇ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + 2 italic_b ( italic_T - italic_T start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ) + 4 italic_c italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( bold_x ) ) italic_s ( bold_x ) = italic_H ( bold_x ) .(11)
Here we have taken the magnetic field as a function of the position 𝐱 𝐱{\bf x}bold_x. When T>T C 𝑇 subscript 𝑇 𝐶 T>T_{C}italic_T > italic_T start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT, the magnetization almost vanishes and then we can ignore the term 4cs 2(𝐱)4 𝑐 superscript 𝑠 2 𝐱 4cs^{2}({\bf x})4 italic_c italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( bold_x ) in eq. (11). Under this approximation, if we take H(𝐱)=H 0δ(𝐱)𝐻 𝐱 subscript 𝐻 0 𝛿 𝐱 H({\bf x})=H_{0}\delta({\bf x})italic_H ( bold_x ) = italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_δ ( bold_x ), then after doing a Fourier transformation over eq. (11), we get
D(𝐱)=H 0 4π1 re−r/ζ,𝐷 𝐱 subscript 𝐻 0 4 𝜋 1 𝑟 superscript 𝑒 𝑟 𝜁 D({\bf x})=\frac{H_{0}}{4\pi}\frac{1}{r}e^{-r/\zeta},italic_D ( bold_x ) = divide start_ARG italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG start_ARG 4 italic_π end_ARG divide start_ARG 1 end_ARG start_ARG italic_r end_ARG italic_e start_POSTSUPERSCRIPT - italic_r / italic_ζ end_POSTSUPERSCRIPT ,(12)
with ζ=(2b(T−T C))−1/2 𝜁 superscript 2 𝑏 𝑇 subscript 𝑇 𝐶 1 2\zeta=(2b(T-T_{C}))^{-1/2}italic_ζ = ( 2 italic_b ( italic_T - italic_T start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ) ) start_POSTSUPERSCRIPT - 1 / 2 end_POSTSUPERSCRIPT, which is precisely the correlation length. Then for example, when T→T C→𝑇 subscript 𝑇 𝐶 T\to T_{C}italic_T → italic_T start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT, we have ζ→∞→𝜁\zeta\to\infty italic_ζ → ∞, namely, an infinite correlation between the spins inside the material. Then there is exchange of information between distant spins inside the material when the temperature reaches the critical value. This brings us to a very important concept for analyzing the phase transitions in different materials, namely, the concept of mutual information.
3 The mutual information method and the construction of a local order operator
The mutual information method, developed in , suggests that the existence of long range order can be deduced by evaluating a quantity which depends on the von-Neumann entropy. For understanding the method, we can start by defining a Hamiltonian for a quantum many-body system as
H^=∑i h^i,^𝐻 subscript 𝑖 subscript^ℎ 𝑖\hat{H}=\sum_{i}\hat{h}_{i},over^ start_ARG italic_H end_ARG = ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT over^ start_ARG italic_h end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ,(13)
where h^i subscript^ℎ 𝑖\hat{h}_{i}over^ start_ARG italic_h end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT corresponds to the local Hamiltonian. Consider now two blocks i 𝑖 i italic_i and j 𝑗 j italic_j, separated by the distance |i−j|𝑖 𝑗|i-j|| italic_i - italic_j |. These blocks can consist of a single site or multiple sites, depending on the minimal sized of the subsystem that captures the long-range correlation. The reduced density matrix of a block i 𝑖 i italic_i is defined as
⟨μ′|ρ i|μ⟩=tr∉i(a^iμ′ρa^iμ+).bra superscript 𝜇′subscript 𝜌 𝑖 ket 𝜇 𝑡 subscript 𝑟 absent 𝑖 subscript^𝑎 𝑖 superscript 𝜇′𝜌 subscript superscript^𝑎 𝑖 𝜇\bra{\mu^{\prime}}\rho_{i}\ket{\mu}=tr_{\notin i}(\hat{a}{i\mu^{\prime}}\rho% \hat{a}^{+}{i\mu}).⟨ start_ARG italic_μ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG | italic_ρ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | start_ARG italic_μ end_ARG ⟩ = italic_t italic_r start_POSTSUBSCRIPT ∉ italic_i end_POSTSUBSCRIPT ( over^ start_ARG italic_a end_ARG start_POSTSUBSCRIPT italic_i italic_μ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_ρ over^ start_ARG italic_a end_ARG start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i italic_μ end_POSTSUBSCRIPT ) .(14)
The trace is done over all the degrees of freedom, except those corresponding to the block i 𝑖 i italic_i itself. The density matrix is defined as usual by the expression ρ=|Ψ⟩⟨Ψ|𝜌 ket Ψ bra Ψ\rho=\ket{\Psi}\bra{\Psi}italic_ρ = | start_ARG roman_Ψ end_ARG ⟩ ⟨ start_ARG roman_Ψ end_ARG |, where |Ψ⟩ket Ψ\ket{\Psi}| start_ARG roman_Ψ end_ARG ⟩ is the ground state of the Hamiltonian. Additionally, a^i subscript^𝑎 𝑖\hat{a}_{i}over^ start_ARG italic_a end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT corresponds to the annihilation operator acting over the states |μ⟩ket 𝜇\ket{\mu}| start_ARG italic_μ end_ARG ⟩ on the site i 𝑖 i italic_i and satisfying the commutation or anticommutation relations. Similarly, it is possible to define the reduced density matrix for two blocks as
⟨μ′ν′|ρ i∪j|μν⟩=tr∉i∪j(a^iμ′a^jν′ρa^jν+a^iμ+).bra superscript 𝜇′superscript 𝜈′subscript 𝜌 𝑖 𝑗 ket 𝜇 𝜈 𝑡 subscript 𝑟 absent 𝑖 𝑗 subscript^𝑎 𝑖 superscript 𝜇′subscript^𝑎 𝑗 superscript 𝜈′𝜌 superscript subscript^𝑎 𝑗 𝜈 subscript superscript^𝑎 𝑖 𝜇\bra{\mu^{\prime}\nu^{\prime}}\rho_{i\cup j}\ket{\mu\nu}=tr_{\notin i\cup j}(% \hat{a}{i\mu^{\prime}}\hat{a}{j\nu^{\prime}}\rho\hat{a}{j\nu}^{+}\hat{a}^{+% }{i\mu}).⟨ start_ARG italic_μ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_ν start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG | italic_ρ start_POSTSUBSCRIPT italic_i ∪ italic_j end_POSTSUBSCRIPT | start_ARG italic_μ italic_ν end_ARG ⟩ = italic_t italic_r start_POSTSUBSCRIPT ∉ italic_i ∪ italic_j end_POSTSUBSCRIPT ( over^ start_ARG italic_a end_ARG start_POSTSUBSCRIPT italic_i italic_μ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT over^ start_ARG italic_a end_ARG start_POSTSUBSCRIPT italic_j italic_ν start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_ρ over^ start_ARG italic_a end_ARG start_POSTSUBSCRIPT italic_j italic_ν end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT over^ start_ARG italic_a end_ARG start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i italic_μ end_POSTSUBSCRIPT ) .(15)
The trace is over all the degrees of freedom except that of i 𝑖 i italic_i and j 𝑗 j italic_j. Due to the probability interpretation of the reduced density matrix’s diagonal elements, it is a matter of convention to normalize the density matrices as follows
tr(ρ i)=tr(ρ j)=tr(ρ i∪j)=1.𝑡 𝑟 subscript 𝜌 𝑖 𝑡 𝑟 subscript 𝜌 𝑗 𝑡 𝑟 subscript 𝜌 𝑖 𝑗 1 tr(\rho_{i})=tr(\rho_{j})=tr(\rho_{i\cup j})=1.italic_t italic_r ( italic_ρ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) = italic_t italic_r ( italic_ρ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) = italic_t italic_r ( italic_ρ start_POSTSUBSCRIPT italic_i ∪ italic_j end_POSTSUBSCRIPT ) = 1 .(16)
The density matrices can be diagonalized in the following form
ρ i=∑μ p μ|ψ iμ⟩⟨ψ iμ|,ρ j=∑ν p ν|ψ iν⟩⟨ψ iν|,ρ i∪j=∑μν q μν|ϕ iν⟩⟨ϕ iν|.formulae-sequence subscript 𝜌 𝑖 subscript 𝜇 subscript 𝑝 𝜇 ket subscript 𝜓 𝑖 𝜇 bra subscript 𝜓 𝑖 𝜇 formulae-sequence subscript 𝜌 𝑗 subscript 𝜈 subscript 𝑝 𝜈 ket subscript 𝜓 𝑖 𝜈 bra subscript 𝜓 𝑖 𝜈 subscript 𝜌 𝑖 𝑗 subscript 𝜇 𝜈 subscript 𝑞 𝜇 𝜈 ket subscript italic-ϕ 𝑖 𝜈 bra subscript italic-ϕ 𝑖 𝜈\displaystyle\rho_{i}=\sum_{\mu}p_{\mu}\ket{\psi_{i\mu}}\bra{\psi_{i\mu}},\;\;% \;\;\;\rho_{j}=\sum_{\nu}p_{\nu}\ket{\psi_{i\nu}}\bra{\psi_{i\nu}},\;\;\;\;\;% \rho_{i\cup j}=\sum_{\mu\nu}q_{\mu\nu}\ket{\phi_{i\nu}}\bra{\phi_{i\nu}}.italic_ρ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT | start_ARG italic_ψ start_POSTSUBSCRIPT italic_i italic_μ end_POSTSUBSCRIPT end_ARG ⟩ ⟨ start_ARG italic_ψ start_POSTSUBSCRIPT italic_i italic_μ end_POSTSUBSCRIPT end_ARG | , italic_ρ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT | start_ARG italic_ψ start_POSTSUBSCRIPT italic_i italic_ν end_POSTSUBSCRIPT end_ARG ⟩ ⟨ start_ARG italic_ψ start_POSTSUBSCRIPT italic_i italic_ν end_POSTSUBSCRIPT end_ARG | , italic_ρ start_POSTSUBSCRIPT italic_i ∪ italic_j end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT italic_q start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT | start_ARG italic_ϕ start_POSTSUBSCRIPT italic_i italic_ν end_POSTSUBSCRIPT end_ARG ⟩ ⟨ start_ARG italic_ϕ start_POSTSUBSCRIPT italic_i italic_ν end_POSTSUBSCRIPT end_ARG | .(17)
Additionally, we can define the von-Neumann entropy as follows
S=−∑μ p μlog 2p μ.𝑆 subscript 𝜇 subscript 𝑝 𝜇 subscript 2 subscript 𝑝 𝜇 S=-\sum_{\mu}p_{\mu}\log_{2}p_{\mu}.italic_S = - ∑ start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT .(18)
This quantity measures the amount of entanglement between the block i 𝑖 i italic_i and the rest of the system. If we want to evaluate only the correlation between a pair of blocks i 𝑖 i italic_i and j 𝑗 j italic_j, then we define the mutual information as
S(i|j)=S(ρ i)+S(ρ j)−S(ρ i∪j).𝑆 conditional 𝑖 𝑗 𝑆 subscript 𝜌 𝑖 𝑆 subscript 𝜌 𝑗 𝑆 subscript 𝜌 𝑖 𝑗 S(i|j)=S(\rho_{i})+S(\rho_{j})-S(\rho_{i\cup j}).italic_S ( italic_i | italic_j ) = italic_S ( italic_ρ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) + italic_S ( italic_ρ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) - italic_S ( italic_ρ start_POSTSUBSCRIPT italic_i ∪ italic_j end_POSTSUBSCRIPT ) .(19)
It has been proved before that this quantity can be used for analyzing the critical phenomena . In fact, the non-vanishing mutual information as it is defined in eq. (19) and defined between two distant blocks, means that there exists a long-range correlation between the blocks, here defined correspondingly as i 𝑖 i italic_i and j 𝑗 j italic_j. Here we will not demonstrate it but the details can be found in [4, 5, 6].
3.1 The local order operator
It is possible to construct a local order operator from the spectra of the density matrix . We can then define the order parameter as
ϕ^=∑⟨μ,ν⟩(ω μνa^iμ+a^iν+ω μν∗a^iν+a^iμ).^italic-ϕ subscript 𝜇 𝜈 subscript 𝜔 𝜇 𝜈 subscript superscript^𝑎 𝑖 𝜇 subscript^𝑎 𝑖 𝜈 subscript superscript 𝜔 𝜇 𝜈 subscript superscript^𝑎 𝑖 𝜈 subscript^𝑎 𝑖 𝜇\hat{\phi}=\sum_{\langle\mu,\nu\rangle}\left(\omega_{\mu\nu}\hat{a}^{+}{i\mu}% \hat{a}{i\nu}+\omega^{}{\mu\nu}\hat{a}^{+}{i\nu}\hat{a}_{i\mu}\right).over^ start_ARG italic_ϕ end_ARG = ∑ start_POSTSUBSCRIPT ⟨ italic_μ , italic_ν ⟩ end_POSTSUBSCRIPT ( italic_ω start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT over^ start_ARG italic_a end_ARG start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i italic_μ end_POSTSUBSCRIPT over^ start_ARG italic_a end_ARG start_POSTSUBSCRIPT italic_i italic_ν end_POSTSUBSCRIPT + italic_ω start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT over^ start_ARG italic_a end_ARG start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i italic_ν end_POSTSUBSCRIPT over^ start_ARG italic_a end_ARG start_POSTSUBSCRIPT italic_i italic_μ end_POSTSUBSCRIPT ) .(20)
Here the coefficients ω μν subscript 𝜔 𝜇 𝜈\omega_{\mu\nu}italic_ω start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT (and its conjugate) can be either, diagonal or non-diagonal. If μ=ν 𝜇 𝜈\mu=\nu italic_μ = italic_ν, then we have diagonal coefficients and we only need the subindex μ 𝜇\mu italic_μ and in addition ω μν=ω μν∗subscript 𝜔 𝜇 𝜈 superscript subscript 𝜔 𝜇 𝜈\omega_{\mu\nu}=\omega_{\mu\nu}^{}italic_ω start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT = italic_ω start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT for this special case. The order parameter is diagonal or non-diagonal depending on the basis where it is expanded. In particular, it depends on the relation between the basis and the definition of the different vacuums in the same basis. Independent on whether or not the selected basis makes the order parameter diagonal, the order parameter has to satisfy the traceless condition
tr(ϕ^)=0=∑μ,ν p μνω μν=0 𝑡 𝑟^italic-ϕ 0 subscript 𝜇 𝜈 subscript 𝑝 𝜇 𝜈 subscript 𝜔 𝜇 𝜈 0 tr(\hat{\phi})=0=\sum_{\mu,\nu}p_{\mu\nu}\omega_{\mu\nu}=0 italic_t italic_r ( over^ start_ARG italic_ϕ end_ARG ) = 0 = ∑ start_POSTSUBSCRIPT italic_μ , italic_ν end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT italic_ω start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT = 0(21)
This condition becomes simplified when the order parameter is diagonal. In such a case, the expression is simply
∑μ ω μp μ=0.subscript 𝜇 subscript 𝜔 𝜇 subscript 𝑝 𝜇 0\sum_{\mu}\omega_{\mu}p_{\mu}=0.∑ start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT italic_ω start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT = 0 .(22)
Note that μ≤ζ 𝜇 𝜁\mu\leq\zeta italic_μ ≤ italic_ζ, where ζ 𝜁\zeta italic_ζ is the rank of the density matrix. The traceless conditions cannot be used if the range of the matrix ρ 𝜌\rho italic_ρ is equal to one. This condition, is able to fix some relation between the ω μ subscript 𝜔 𝜇\omega_{\mu}italic_ω start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT-coefficients. If we have n 𝑛 n italic_n-coefficients of this type, then the traceless condition reduces the number of independent coefficients to n−1 𝑛 1 n-1 italic_n - 1.
4 Additional conditions: General broken symmetries of the system
The traceless condition is useful when the amount of coefficients to fix is not so large. This happens when we have only a few degenerate vacuums. However, when we have several degenerate vacuums, then we will also have several coefficients ω μ subscript 𝜔 𝜇\omega_{\mu}italic_ω start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT. Here we formulate additional conditions besides the traceless one, based on the fact that the action of a broken symmetry generator is to map one vacuum into another one, taking into account that we are dealing with vacuum degeneracy.
4.1 Mapping one vacuum into another one with the same energy level via broken symmetry
Consider that we can break the symmetry of the system spontaneously by selecting one among all the possible vacuums. Consider now a broken generator defined as Q a subscript 𝑄 𝑎 Q_{a}italic_Q start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT. Then we can define the condition
Q a|0⟩=|0¯⟩.subscript 𝑄 𝑎 ket 0 ket¯0 Q_{a}\ket{0}=\ket{\bar{0}}.italic_Q start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT | start_ARG 0 end_ARG ⟩ = | start_ARG over¯ start_ARG 0 end_ARG end_ARG ⟩ .(23)
Here |0⟩≠|0¯⟩ket 0 ket¯0\ket{0}\neq\ket{\bar{0}}| start_ARG 0 end_ARG ⟩ ≠ | start_ARG over¯ start_ARG 0 end_ARG end_ARG ⟩ (inequivalent vacuums) but both vacuums are at the same energy level. In general, we can decompose the order parameter in two components as follows
ϕ^=ϕ^1+iϕ^2.^italic-ϕ subscript^italic-ϕ 1 𝑖 subscript^italic-ϕ 2\hat{\phi}=\hat{\phi}{1}+i\hat{\phi}{2}.over^ start_ARG italic_ϕ end_ARG = over^ start_ARG italic_ϕ end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_i over^ start_ARG italic_ϕ end_ARG start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT .(24)
We can conventionally define the component ϕ^2 subscript^italic-ϕ 2\hat{\phi}_{2}over^ start_ARG italic_ϕ end_ARG start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT as the one with zero vacuum expectation value for some arbitrarily selected vacuum. Let’s fix such a vacuum to be |0⟩ket 0\ket{0}| start_ARG 0 end_ARG ⟩, and then
⟨0|ϕ^1|0⟩≠0,⟨0|ϕ^2|0⟩=0.formulae-sequence bra 0 subscript^italic-ϕ 1 ket 0 0 bra 0 subscript^italic-ϕ 2 ket 0 0\bra{0}\hat{\phi}{1}\ket{0}\neq 0,\;\;\;\;\;\;\bra{0}\hat{\phi}{2}\ket{0}=0.⟨ start_ARG 0 end_ARG | over^ start_ARG italic_ϕ end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | start_ARG 0 end_ARG ⟩ ≠ 0 , ⟨ start_ARG 0 end_ARG | over^ start_ARG italic_ϕ end_ARG start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT | start_ARG 0 end_ARG ⟩ = 0 .(25)
The components can be selected such that they are related to each other through the broken symmetry generator Q a subscript 𝑄 𝑎 Q_{a}italic_Q start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT as
[Q a,ϕ^2]=iϕ^1.subscript 𝑄 𝑎 subscript^italic-ϕ 2 𝑖 subscript^italic-ϕ 1[Q_{a},\hat{\phi}{2}]=i\hat{\phi}{1}.[ italic_Q start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT , over^ start_ARG italic_ϕ end_ARG start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ] = italic_i over^ start_ARG italic_ϕ end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT .(26)
This result suggests another way to say that the effect of the broken symmetry is to map one vacuum into the other. If we have n 𝑛 n italic_n-degenerate vacuums connected by the same broken generator, then by applying n 𝑛 n italic_n-times Q a subscript 𝑄 𝑎 Q_{a}italic_Q start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT to the vacuum state, we just get the initially fixed vacuum as follows
(Q a)n|0⟩=|0⟩.superscript subscript 𝑄 𝑎 𝑛 ket 0 ket 0(Q_{a})^{n}\ket{0}=\ket{0}.( italic_Q start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT | start_ARG 0 end_ARG ⟩ = | start_ARG 0 end_ARG ⟩ .(27)
Then in general, each vacuum is connected to each other by one expression of the form
|0⟩2=e−iϕ|0⟩1,|0⟩3=e−2iϕ|0⟩1,|0⟩4=e−3iϕ|0⟩1,….,|0⟩n=e−(n−1)iϕ|0⟩1,|0⟩1=e−niϕ|0⟩1,\ket{0}{2}=e^{-i\phi}\ket{0}{1},\;\;\;\;\;\ket{0}{3}=e^{-2i\phi}\ket{0}{1}% ,\;\;\;\;\;\ket{0}{4}=e^{-3i\phi}\ket{0}{1},....,\ket{0}{n}=e^{-(n-1)i\phi}% \ket{0}{1},\;\;\;\;\;\ket{0}{1}=e^{-ni\phi}\ket{0}{1},| start_ARG 0 end_ARG ⟩ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_e start_POSTSUPERSCRIPT - italic_i italic_ϕ end_POSTSUPERSCRIPT | start_ARG 0 end_ARG ⟩ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , | start_ARG 0 end_ARG ⟩ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = italic_e start_POSTSUPERSCRIPT - 2 italic_i italic_ϕ end_POSTSUPERSCRIPT | start_ARG 0 end_ARG ⟩ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , | start_ARG 0 end_ARG ⟩ start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT = italic_e start_POSTSUPERSCRIPT - 3 italic_i italic_ϕ end_POSTSUPERSCRIPT | start_ARG 0 end_ARG ⟩ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … . , | start_ARG 0 end_ARG ⟩ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = italic_e start_POSTSUPERSCRIPT - ( italic_n - 1 ) italic_i italic_ϕ end_POSTSUPERSCRIPT | start_ARG 0 end_ARG ⟩ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , | start_ARG 0 end_ARG ⟩ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_e start_POSTSUPERSCRIPT - italic_n italic_i italic_ϕ end_POSTSUPERSCRIPT | start_ARG 0 end_ARG ⟩ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ,(28)
where ϕ=αQ a italic-ϕ 𝛼 subscript 𝑄 𝑎\phi=\alpha Q_{a}italic_ϕ = italic_α italic_Q start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT represents the effect of the broken generators over the states. If we have m 𝑚 m italic_m-broken generators defined as Q 1 subscript 𝑄 1 Q_{1}italic_Q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, Q 2 subscript 𝑄 2 Q_{2}italic_Q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT, Q 3 subscript 𝑄 3 Q_{3}italic_Q start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT,…Q m subscript 𝑄 𝑚 Q_{m}italic_Q start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT, each generator will reproduce similar mappings, as those defined in eqns. (23), (26), (27) and (28). If we have the n 𝑛 n italic_n vacuums defined previously, then we can know the phase difference between vacuums in a explicit form as
ϕ=2π n,italic-ϕ 2 𝜋 𝑛\phi=\frac{2\pi}{n},italic_ϕ = divide start_ARG 2 italic_π end_ARG start_ARG italic_n end_ARG ,(29)
for each broken generator acting over a set of vacuums. Then for the special case where we only have two vacuums, then ϕ=π italic-ϕ 𝜋\phi=\pi italic_ϕ = italic_π because n=2 𝑛 2 n=2 italic_n = 2. This situation appears for example for the case of ferromagnetism . Note that for the general expansion of the order parameter
ϕ^1=∑n ω na^in+a^in,subscript^italic-ϕ 1 subscript 𝑛 subscript 𝜔 𝑛 subscript superscript^𝑎 𝑖 𝑛 subscript^𝑎 𝑖 𝑛\hat{\phi}{1}=\sum{n}\omega_{n}\hat{a}^{+}{in}\hat{a}{in},over^ start_ARG italic_ϕ end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_ω start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT over^ start_ARG italic_a end_ARG start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i italic_n end_POSTSUBSCRIPT over^ start_ARG italic_a end_ARG start_POSTSUBSCRIPT italic_i italic_n end_POSTSUBSCRIPT ,(30)
the relation between the coefficients ω n subscript 𝜔 𝑛\omega_{n}italic_ω start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT is determined by the condition (29) because each value of n 𝑛 n italic_n corresponds to a different vacuum. Then, for n=2 𝑛 2 n=2 italic_n = 2, since ϕ=π italic-ϕ 𝜋\phi=\pi italic_ϕ = italic_π, we have
|0⟩1=−|0⟩2,subscript ket 0 1 subscript ket 0 2\ket{0}{1}=-\ket{0}{2},| start_ARG 0 end_ARG ⟩ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = - | start_ARG 0 end_ARG ⟩ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ,(31)
from the result (28). This condition applied to the standard expansion (30), means ω 1=−ω 2 subscript 𝜔 1 subscript 𝜔 2\omega_{1}=-\omega_{2}italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = - italic_ω start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT with the series expansion containing only two terms. When we have more than two vacuums, the conditions can be generalized and the complex coefficients ω n subscript 𝜔 𝑛\omega_{n}italic_ω start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT will be related to each other by a fixed phase. Note that the traceless condition defined in , corresponds to a special situation here. In addition, note that here the condition suggesting that the maximum value of ω n subscript 𝜔 𝑛\omega_{n}italic_ω start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT is one, is reduced to the condition fixing only one of the magnitudes of ω i subscript 𝜔 𝑖\omega_{i}italic_ω start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT to one, namely, |ω i|=1 subscript 𝜔 𝑖 1|\omega_{i}|=1| italic_ω start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | = 1. Then the difference between the different coefficients will be reduced to only phase differences. In this way, then instead of the condition (28), we can use
ω 2=e−iϕ ω 1,ω 3=e−2iϕ ω 1,ω 4=e−3iϕ ω 1,….,|0⟩n=e−(n−1)iϕ ω 1,ω 1=e−niϕ ω 1,\omega_{2}=e^{-i\phi}\omega_{1},\;\;\;\;\;\omega_{3}=e^{-2i\phi}\omega_{1},\;% \;\;\;\;\omega_{4}=e^{-3i\phi}\omega_{1},....,\ket{0}{n}=e^{-(n-1)i\phi}% \omega{1},\;\;\;\;\;\omega_{1}=e^{-ni\phi}\omega_{1},italic_ω start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_e start_POSTSUPERSCRIPT - italic_i italic_ϕ end_POSTSUPERSCRIPT italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_ω start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = italic_e start_POSTSUPERSCRIPT - 2 italic_i italic_ϕ end_POSTSUPERSCRIPT italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_ω start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT = italic_e start_POSTSUPERSCRIPT - 3 italic_i italic_ϕ end_POSTSUPERSCRIPT italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … . , | start_ARG 0 end_ARG ⟩ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = italic_e start_POSTSUPERSCRIPT - ( italic_n - 1 ) italic_i italic_ϕ end_POSTSUPERSCRIPT italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_e start_POSTSUPERSCRIPT - italic_n italic_i italic_ϕ end_POSTSUPERSCRIPT italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ,(32)
expressing then the fact that all the ω i subscript 𝜔 𝑖\omega_{i}italic_ω start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT coefficients can be expressed as a common magnitude |ω 1|subscript 𝜔 1|\omega_{1}|| italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT |, times a phase. The cyclic property given by the last term in this equation, which defines the result (29), gives us the following conditions
∑n|ω n|2=n|ω 1|2,subscript 𝑛 superscript subscript 𝜔 𝑛 2 𝑛 superscript subscript 𝜔 1 2\sum_{n}|\omega_{n}|^{2}=n|\omega_{1}|^{2},∑ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | italic_ω start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = italic_n | italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ,(33)
for the case of multiple vacuums, connected by broken symmetries and having the same energy level. If some vacuums are connected by broken symmetries but in addition have different energy levels, chemical potentials should appear, generating then some gap for the possible Nambu-Goldstone bosons (if any) appearing in the system. We can derive another important conditions if we sum over all the coefficients ω n subscript 𝜔 𝑛\omega_{n}italic_ω start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT related to the different definitions of vacuum as follows
∑n ω n=|ω 1|(e−iϕ+e−2iϕ+e−3iϕ+…+e−niϕ).subscript 𝑛 subscript 𝜔 𝑛 subscript 𝜔 1 superscript 𝑒 𝑖 italic-ϕ superscript 𝑒 2 𝑖 italic-ϕ superscript 𝑒 3 𝑖 italic-ϕ…superscript 𝑒 𝑛 𝑖 italic-ϕ\sum_{n}\omega_{n}=|\omega_{1}|\left(e^{-i\phi}+e^{-2i\phi}+e^{-3i\phi}+...+e^% {-ni\phi}\right).∑ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_ω start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = | italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | ( italic_e start_POSTSUPERSCRIPT - italic_i italic_ϕ end_POSTSUPERSCRIPT + italic_e start_POSTSUPERSCRIPT - 2 italic_i italic_ϕ end_POSTSUPERSCRIPT + italic_e start_POSTSUPERSCRIPT - 3 italic_i italic_ϕ end_POSTSUPERSCRIPT + … + italic_e start_POSTSUPERSCRIPT - italic_n italic_i italic_ϕ end_POSTSUPERSCRIPT ) .(34)
The previous expression can be simplified if we take into account that it represents a geometric series, and then
∑n ω n=|ω 1|(1−e−niϕ e iϕ−1)=|ω 1|(e−iϕ−e−(n+1)iϕ 1−e−iϕ).subscript 𝑛 subscript 𝜔 𝑛 subscript 𝜔 1 1 superscript 𝑒 𝑛 𝑖 italic-ϕ superscript 𝑒 𝑖 italic-ϕ 1 subscript 𝜔 1 superscript 𝑒 𝑖 italic-ϕ superscript 𝑒 𝑛 1 𝑖 italic-ϕ 1 superscript 𝑒 𝑖 italic-ϕ\sum_{n}\omega_{n}=|\omega_{1}|\left(\frac{1-e^{-ni\phi}}{e^{i\phi}-1}\right)=% |\omega_{1}|\left(\frac{e^{-i\phi}-e^{-(n+1)i\phi}}{1-e^{-i\phi}}\right).∑ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_ω start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = | italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | ( divide start_ARG 1 - italic_e start_POSTSUPERSCRIPT - italic_n italic_i italic_ϕ end_POSTSUPERSCRIPT end_ARG start_ARG italic_e start_POSTSUPERSCRIPT italic_i italic_ϕ end_POSTSUPERSCRIPT - 1 end_ARG ) = | italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | ( divide start_ARG italic_e start_POSTSUPERSCRIPT - italic_i italic_ϕ end_POSTSUPERSCRIPT - italic_e start_POSTSUPERSCRIPT - ( italic_n + 1 ) italic_i italic_ϕ end_POSTSUPERSCRIPT end_ARG start_ARG 1 - italic_e start_POSTSUPERSCRIPT - italic_i italic_ϕ end_POSTSUPERSCRIPT end_ARG ) .(35)
If the vacuums satisfy the cyclic property, then e−(n+1)iϕ=e−iϕ superscript 𝑒 𝑛 1 𝑖 italic-ϕ superscript 𝑒 𝑖 italic-ϕ e^{-(n+1)i\phi}=e^{-i\phi}italic_e start_POSTSUPERSCRIPT - ( italic_n + 1 ) italic_i italic_ϕ end_POSTSUPERSCRIPT = italic_e start_POSTSUPERSCRIPT - italic_i italic_ϕ end_POSTSUPERSCRIPT; and then we can conclude that
∑n ω n=0,subscript 𝑛 subscript 𝜔 𝑛 0\sum_{n}\omega_{n}=0,∑ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_ω start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = 0 ,(36)
which is a natural result equivalent to the traceless condition in . This demonstrates that the traceless condition is just a natural consequence of the fact that the vacuums are connected by some broken symmetry plus the cyclic property summarized in the definition e−niϕ=1 superscript 𝑒 𝑛 𝑖 italic-ϕ 1 e^{-ni\phi}=1 italic_e start_POSTSUPERSCRIPT - italic_n italic_i italic_ϕ end_POSTSUPERSCRIPT = 1. Note that once the norm of ω 1 subscript 𝜔 1\omega_{1}italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT is fixed, then the norm for all the other coefficients is also fixed as far as we consider vacuums of the same energy level. Then once we know how many vacuums are at the same energy level, for example, by analyzing the reduced density matrix spectrum, all their coefficients ω μ subscript 𝜔 𝜇\omega_{\mu}italic_ω start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT will be defined. Regarding the order of magnitude of ω μ subscript 𝜔 𝜇\omega_{\mu}italic_ω start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT, by fixing the magnitude of |ω 1|=1 subscript 𝜔 1 1|\omega_{1}|=1| italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | = 1, immediately fixes the magnitude of the others (not the sign) coefficients connected by a broken generator. For any number of vacuums, the phase ϕ italic-ϕ\phi italic_ϕ can be found from eq. (29). In this way for example, if we have three vacuums then n=3 𝑛 3 n=3 italic_n = 3 and ϕ=2π/3 italic-ϕ 2 𝜋 3\phi=2\pi/3 italic_ϕ = 2 italic_π / 3. In this way we have
ω 2=−1 2ω 1+3 2iω 1,ω 3=−1 2ω 1−3 2iω 1.formulae-sequence subscript 𝜔 2 1 2 subscript 𝜔 1 3 2 𝑖 subscript 𝜔 1 subscript 𝜔 3 1 2 subscript 𝜔 1 3 2 𝑖 subscript 𝜔 1\omega_{2}=-\frac{1}{2}\omega_{1}+\frac{\sqrt{3}}{2}i\omega_{1},\;\;\;\;\;% \omega_{3}=-\frac{1}{2}\omega_{1}-\frac{\sqrt{3}}{2}i\omega_{1}.italic_ω start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = - divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + divide start_ARG square-root start_ARG 3 end_ARG end_ARG start_ARG 2 end_ARG italic_i italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_ω start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = - divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - divide start_ARG square-root start_ARG 3 end_ARG end_ARG start_ARG 2 end_ARG italic_i italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT .(37)
This result satisfies evidently the result (36).
5 Applications
Here we will analyze different cases where we can apply our method for getting in a natural way the corresponding order parameters.
5.1 Ferromagnetic long-range order
For the case of ferromagnetism, we define the unbroken vacuum as
|Ψ⟩=1 2(|↑↑…↑↑⟩+|↓↓…↓↓⟩)\ket{\Psi}=\frac{1}{\sqrt{2}}\left(\ket{\uparrow\uparrow...\uparrow\uparrow}+% \ket{\downarrow\downarrow...\downarrow\downarrow}\right)| start_ARG roman_Ψ end_ARG ⟩ = divide start_ARG 1 end_ARG start_ARG square-root start_ARG 2 end_ARG end_ARG ( | start_ARG ↑ ↑ … ↑ ↑ end_ARG ⟩ + | start_ARG ↓ ↓ … ↓ ↓ end_ARG ⟩ )(38)
This situation corresponds to a spin chain, with all the atoms in the chain having a common spin either, up or down. We consider here these two possibilities as the two independent vacuums. If the symmetry of the system is spontaneously broken, the system will select any of the two possible vacuums. If we apply our formulation to this case, then n=2 𝑛 2 n=2 italic_n = 2 and then we get ϕ=π italic-ϕ 𝜋\phi=\pi italic_ϕ = italic_π in agreement with eq. (29), and from eq. (30), we obtain
ϕ^1=ω 1a^i1+a^i1+ω 2a^i2+a^i2,subscript^italic-ϕ 1 subscript 𝜔 1 subscript superscript^𝑎 𝑖 1 subscript^𝑎 𝑖 1 subscript 𝜔 2 subscript superscript^𝑎 𝑖 2 subscript^𝑎 𝑖 2\hat{\phi}{1}=\omega{1}\hat{a}^{+}{i1}\hat{a}{i1}+\omega_{2}\hat{a}^{+}{i% 2}\hat{a}{i2},over^ start_ARG italic_ϕ end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT over^ start_ARG italic_a end_ARG start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i 1 end_POSTSUBSCRIPT over^ start_ARG italic_a end_ARG start_POSTSUBSCRIPT italic_i 1 end_POSTSUBSCRIPT + italic_ω start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT over^ start_ARG italic_a end_ARG start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i 2 end_POSTSUBSCRIPT over^ start_ARG italic_a end_ARG start_POSTSUBSCRIPT italic_i 2 end_POSTSUBSCRIPT ,(39)
Since n=2 𝑛 2 n=2 italic_n = 2, then ω 1=−ω 2 subscript 𝜔 1 subscript 𝜔 2\omega_{1}=-\omega_{2}italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = - italic_ω start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT if we use the result (32). Since all the ω n subscript 𝜔 𝑛\omega_{n}italic_ω start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT-coefficients related to the existence of independent vacuums (but connected through a symmetry transformation) are related to each other by a phase transformation, they can be normalized to one and then we can define them unambiguosly. Then |ω 1|=|ω 2|subscript 𝜔 1 subscript 𝜔 2|\omega_{1}|=|\omega_{2}|| italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | = | italic_ω start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT |. This case corresponds to the ferromagnetic case showed in . Then the method proposed here works perfectly.
5.2 One dimensional ferromagnetic
Here we consider the Hamiltonian
H^=∑j=1 N(S j xS j+1 x+S j yS j+1 y+S j zS j+1 z).^𝐻 superscript subscript 𝑗 1 𝑁 superscript subscript 𝑆 𝑗 𝑥 superscript subscript 𝑆 𝑗 1 𝑥 superscript subscript 𝑆 𝑗 𝑦 superscript subscript 𝑆 𝑗 1 𝑦 superscript subscript 𝑆 𝑗 𝑧 superscript subscript 𝑆 𝑗 1 𝑧\hat{H}=\sum_{j=1}^{N}\left(S_{j}^{x}S_{j+1}^{x}+S_{j}^{y}S_{j+1}^{y}+S_{j}^{z% }S_{j+1}^{z}\right).over^ start_ARG italic_H end_ARG = ∑ start_POSTSUBSCRIPT italic_j = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT ( italic_S start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT italic_S start_POSTSUBSCRIPT italic_j + 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT + italic_S start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_y end_POSTSUPERSCRIPT italic_S start_POSTSUBSCRIPT italic_j + 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_y end_POSTSUPERSCRIPT + italic_S start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_z end_POSTSUPERSCRIPT italic_S start_POSTSUBSCRIPT italic_j + 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_z end_POSTSUPERSCRIPT ) .(40)
This Hamiltonian is basically the same of the Dimer case but excluding the next nearest neighbor interactions. Here S j x superscript subscript 𝑆 𝑗 𝑥 S_{j}^{x}italic_S start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT, S j y superscript subscript 𝑆 𝑗 𝑦 S_{j}^{y}italic_S start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_y end_POSTSUPERSCRIPT and S j z superscript subscript 𝑆 𝑗 𝑧 S_{j}^{z}italic_S start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_z end_POSTSUPERSCRIPT are spin-1/2 operators at the site j 𝑗 j italic_j. This model is normally solved by using the Bethe-ansatz method . This case has two different vacuums connected through the condition
ω 2=ω 1∗.subscript 𝜔 2 subscript superscript 𝜔 1\omega_{2}=\omega^{}_{1}.italic_ω start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_ω start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT .(41)
If ω 1=x+iy subscript 𝜔 1 𝑥 𝑖 𝑦\omega_{1}=x+iy italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_x + italic_i italic_y, then ω 2=x−iy subscript 𝜔 2 𝑥 𝑖 𝑦\omega_{2}=x-iy italic_ω start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_x - italic_i italic_y and then the two independent vacuums corresponding to the ground state are separated by an angle
tan(θ 2)=y x.𝜃 2 𝑦 𝑥\tan\left(\frac{\theta}{2}\right)=\frac{y}{x}.roman_tan ( divide start_ARG italic_θ end_ARG start_ARG 2 end_ARG ) = divide start_ARG italic_y end_ARG start_ARG italic_x end_ARG .(42)
For this case, we identify two independent vacuums. The order parameter is then obtained from eq. (30) as
ϕ^=ω 1a^i1+a^i2+ω 1∗a^i2+a^i1.^italic-ϕ subscript 𝜔 1 superscript subscript^𝑎 𝑖 1 subscript^𝑎 𝑖 2 superscript subscript 𝜔 1 superscript subscript^𝑎 𝑖 2 subscript^𝑎 𝑖 1\hat{\phi}=\omega_{1}\hat{a}{i1}^{+}\hat{a}{i2}+\omega_{1}^{}\hat{a}{i2}^{% +}\hat{a}{i1}.over^ start_ARG italic_ϕ end_ARG = italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT over^ start_ARG italic_a end_ARG start_POSTSUBSCRIPT italic_i 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT over^ start_ARG italic_a end_ARG start_POSTSUBSCRIPT italic_i 2 end_POSTSUBSCRIPT + italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT over^ start_ARG italic_a end_ARG start_POSTSUBSCRIPT italic_i 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT over^ start_ARG italic_a end_ARG start_POSTSUBSCRIPT italic_i 1 end_POSTSUBSCRIPT .(43)
With the definitions given for ω 1 subscript 𝜔 1\omega_{1}italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, we find that the previous expression corresponds to
ϕ^=xO^i x−yO^i y.^italic-ϕ 𝑥 superscript subscript^𝑂 𝑖 𝑥 𝑦 superscript subscript^𝑂 𝑖 𝑦\hat{\phi}=x\hat{O}{i}^{x}-y\hat{O}{i}^{y}.over^ start_ARG italic_ϕ end_ARG = italic_x over^ start_ARG italic_O end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT - italic_y over^ start_ARG italic_O end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_y end_POSTSUPERSCRIPT .(44)
Here O^i x,y superscript subscript^𝑂 𝑖 𝑥 𝑦\hat{O}{i}^{x,y}over^ start_ARG italic_O end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_x , italic_y end_POSTSUPERSCRIPT has a correspondence with the Pauli matrices σ^x,y subscript^𝜎 𝑥 𝑦\hat{\sigma}{x,y}over^ start_ARG italic_σ end_ARG start_POSTSUBSCRIPT italic_x , italic_y end_POSTSUBSCRIPT. This means that O^i x,y superscript subscript^𝑂 𝑖 𝑥 𝑦\hat{O}{i}^{x,y}over^ start_ARG italic_O end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_x , italic_y end_POSTSUPERSCRIPT represent the generators of rotations in the spin space. Then immediately we can conclude that the order parameter defined in eq. (43) has two components, corresponding to the two independent vacuums, connected each other through a combination of rotation around x 𝑥 x italic_x and rotation around y 𝑦 y italic_y. This conclusion can be obtained from the expression (28), which represents the connection between the number of coefficients ω i subscript 𝜔 𝑖\omega{i}italic_ω start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT and the number of independent vacuums. The two independent vacuums can be defined as
|0⟩1=e iθ|0⟩2,subscript ket 0 1 superscript 𝑒 𝑖 𝜃 subscript ket 0 2\ket{0}{1}=e^{i\theta}\ket{0}{2},| start_ARG 0 end_ARG ⟩ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_e start_POSTSUPERSCRIPT italic_i italic_θ end_POSTSUPERSCRIPT | start_ARG 0 end_ARG ⟩ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ,(45)
with the angle θ 𝜃\theta italic_θ defining the rotation around the z 𝑧 z italic_z-axis. The same angle is defined in eq. (42).
6 Conclusions
In this paper we have learned a different way to interpret and define the order parameters in systems where there is a degeneracy on the vacuum state and as a consequence, there exists the possibility of having spontaneous symmetry breaking once a specific vacuum condition is selected. This new way to visualize the problem helps us to find additional constraints for the definition of an order parameter. Our starting point is the expansion of the order parameter in terms of the particle number operator as it was suggested in . Here we have explained how to constraint the coefficients ω i subscript 𝜔 𝑖\omega_{i}italic_ω start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT of the expansion based on the fact that a broken symmetry for the vacuum, maps one of the vacuums toward another one. The order parameter itself can be expanded in a base with vectors defining the different vacuums. It is for this reason that the vacuum expectation value of an order parameter does not vanish when it is defined for a single vacuum. Other results, like the traceless condition for the order parameter emerges naturally from this scenario. This condition is a natural consequence of the expansion of the order parameter in a base defining all the possible vacuums of the system.
Acknowledgement
W. C. Yu acknowledges financial support from the National Natural Science Foundation of China (Grants No. 12005179) and City University of Hong Kong (Grant No.9610438).
References
S. Weinberg, The Quantum theory of Fields (Press Syndicate of the University of Cambridge, New York, 1996).
Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122, 345 (1961); Y. Nambu and G. Jona-Lasinio, Phys. Rev. 124, 246(1961).
Y. Nambu, J. Phys. Soc. Jap. 76, 111002 (2007).
M. M. Wolf, F. Verstraete, M. B. Hastings, and J. I. Cirac, Phys. Rev. Lett. 100, 070502 (2008).
S. J. Gu, C. P. Sun and H. Q. Lin, J. Phys. A: Math. Theor. 41, 025002 (2007).
S. J. Gu, W. C. Yu and H. Q. Lin, Construct order parameter from the spectra of mutual information, arXiv:1209.0062 [quant-ph].
W. C.Yu, S. J.Gu, and H. Q.Lin, Eur. Phys. J. B 89, 212 (2016).
W. C.Yu, Y. C. Li, P. D. Sacramento, and H. Q. Lin, Phys. Rev. B 94, 245123 (2016).
I. Arraut, Symmetry 11 803 (2019).
I. Arraut, Int. J. Mod. Phys. A, 32, 1750127 (2017).
I. Arraut, Int. J. Mod. Phys. A 33, 1850041 (2018).
Peskin, M. and Schroeder, D., (1995). An Introduction to Quantum Field Theory. Westview Press. p. 669.
L. D. Landau (1937), On the Theory of Phase Transitions, Zh. Eksp. Teor. Fiz. 7: 19-32.
M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, England, 2000).
C. K. Majumdar and D. K. Ghosh, J. Math. Phys. 10, 1388 (1969); J. Math. Phys. 10, 1399 (1969); C. K. Majumdar, J. Phys. C 3, 911 (1969).
H. A. Bethe, Z. Phys. 71, 205 (1931).
Generated on Wed Aug 28 03:30:11 2024 by L a T e XML |
8675 | https://edu.gcfglobal.org/en/percents/percentages-in-real-life/1/ | Percents: Percentages in Real Life
About cookies on this site
Sharing your cookies helps us improve the site, optimize your experience, and understand the extent of our impact. By closing this banner, you agree to the use of cookies on this website (This option applies to users outside the United States and Europe). Learn more about how we use cookies by clicking on Cookie Settings.
Learn more
Allow all Deny all Cookie settings
Cookie settings
close search
search menu
Topics
close
I want to...
Get started with computers
Learn Microsoft Office
Apply for a job
Improve my work skills
Design nice-looking docs
More...
Microsoft Office
Word
Excel
PowerPoint
Access
More...
Computers
Getting Started
Smartphones & Tablets
Typing Tutorial
Windows
Online Learning
More...
The Internet
Basic Internet Skills
Online Safety
Social Media
Zoom Basics
Email
More...
Google
Gmail
Google Docs
Google Sheets
More...
The Job Search
Career Planning
Resume Writing
Cover Letters
Job Search and Networking
Business Communication
Entrepreneurship 101
Careers without College
Job Hunt for Today
More...
Skills for Today
3D Printing
Freelancing 101
Personal Finance
Sharing Economy
Decision-Making
More...
Creativity & Design
Graphic Design
Creativity
Photography
Image Editing
Photoshop
Learning WordPress
More...
Core Skills
Math
Reading
Science
Grammar
Language Learning
Critical Thinking
More...
For Educators
Translations
Staff Picks
All Topics
print
English expand_more expand_less
English expand_more expand_less
Español
Português
LoginJoin for free!
Percents-
Percentages in Real Life
Back to Tutorial
Percents
Percentages in Real Life
search
person_outline arrow_drop_down
menu
All TopicsMy AccountAbout UsFAQContact Us
ENESPT
SigninSignupDashboardProfileLogout
navigate_before Tutorialsearch
Percents: Percentages in Real Life
560d89451a44f7d030aa8fd6 570fb547667fae058cdd2619
Lesson 3: Percentages in Real Life
/en/percents/calculating-percentages/content/
Sales tax
Depending on where you live, you might pay a salestaxon the things you buy. A sales tax is an extra charge added to the price of an item. The amount you pay in tax is almost always a percentageof that price.
Suppose you buy a $50 coffeemaker in an area where the sales tax is 8%. When you check out, 8% of $50 would be added to your total price. How much would you pay total?
Click through the slideshow to learn how to calculate sales tax.
View the slides below to learn how to calculate sales tax.
arrow_back_ios
Let's say you want to buy a $50 coffeemaker, and the sales tax is 8%.
Sales tax is a percentageof the price of the item. This means the sales tax is eight percent of fifty dollars.
Whenever you see the phrase "of something"in a math sentence, it's usually a hint that you'll need to multiply.
So we'll multiply 50 by 8%.
Before we can do that, we'll need to convert 8% into a decimal.
We'll move the decimal point two places to the left...
We'll move the decimal point two places to the left...so 8% becomes 0.08.
Now we can multiply: 0.08 times 50 equals 4.00.
So the sales tax is $4.00. Another way to say this is that 8% of $50.00 is $4.00.
Remember, sales tax is an extra charge added to the price of an item. So we'll add the sales tax to our original price.
$4.00 plus $50.00 equals $54.00.
Thefinal price of the coffeemaker is $54.00 after adding sales tax.
Let's try another example. Let's say you want to buy a six pack of soda that costs $4.50, and the sales tax on food is 4%.
This means the sales tax is 4% of $4.50.
So we'll multiply 4% by 4.50.
First, we need to convert 4% into a decimal, so we'll move the decimal point two places to the left.
4% is the same as 0.04.
Now we'll multiply: 4.5 times 0.04 is 0.18, so the sales tax is $0.18, or 18 cents.
We could also say that 4% of $4.50 is $0.18.
Finally, we'll add the sales tax to the original price.
4.50 plus 0.18 is 4.68.
So the final price is $4.68 after adding sales tax.
arrow_back_ios
Try This!
Find the total cost of each item after sales tax. Be sure to round your answer to the nearest cent. For example, $64.24305 would become $64.24.
You buy a pair of sunglasses for $49.00. The sales tax is 7%.
You need to purchase $326.00 worth of vinyl siding. The sales tax is 4%.
You buy $32.19 worth of groceries. The sales tax is 5.7%.
Curious about the sales tax where you live?
You can use this list of State and Local Sales Tax Rates from the Tax Foundation to look up the sales tax rate where you live.
Discounts, markdowns, and sales
Have you ever bought something on sale marked down by twentypercent? Or used a coupon to take 30% off? If so, you've used adiscount. A discount is usually a percentageof the original price. The percentage is subtracted from the original price to determine the sale price.
Let's say a shirt costs $8, but it's been marked down by 50%. When you check out, 50% of $8 will be subtracted from the original price. How much would the shirt cost after the discount?
Click through the slideshow to learn how to calculate discounts.
View the slides below to learn how to calculate discounts.
arrow_back_ios
Let's say a shirt costs eight dollars, but it's on sale for 50% off.
This means it will cost fifty percent less than the original price, or half as much.
This discount is a percentage of the original price. Here, 50% off means the discount is 50% of $8.
50%, orhalf, of 8 is 4. This means the shirt will cost $4less than the original price.
Remember, the discount is taken off of the original price. So we'll subtract the discount from the original price.
8 minus 4 is 4.
So the shirt would cost four dollars on sale.
Let's try another example. What if the same shirt was on sale for 20% off?
This means it would cost twenty percent less than the original price. This time, it'll take a few more steps to find the final price.
The discount is 20% of $8.
Whenever, you see the phrase "of something" in a math sentence, it's usually a hint that you'll need to multiply.
So we'll multiply 8 by 20%.
Before we can do that, we'll need to convert 20% into a decimal.
We'll move the decimal point two places to the left...
We'll move the decimal point two places to the left...so 20% becomes 0.20.
Now we can multiply. 8 times 0.20 equals 1.60.
Another way to say this is that 20% of $8.00 is $1.60.
The shirt will cost $1.60less than the original price of $8. So we'll subtract the discount from the original price.
8.00 minus 1.60 equals 6.40.
So the shirt would cost $6.40 on sale.
arrow_back_ios
Try This!
Find the final cost for each item.
You find a pair of shoes on sale for 35% off. They originally cost $90.00. What will they cost on sale?
You spend $42.00 on groceries, but you have a coupon for 10% off. What is the final price of your bill?
A bookstore is going out of business, so all books are 60% off. How much would you spend on a book that originally cost $12.95?
Calculating tips
If you've eaten at a restaurant, you've probably left a tip for your server. A tip is a small amount of money that you add to your bill when someone gives you service. The amount of a tip is usually a percentage of the total cost. For example, you might leave afifteen percenttip at a restaurant for average service.
Let's say you eat at a restaurant with some friends. The bill comes to $12.04, and you'd like to leave a 15% tip. This means you'd like to leave 15% of $12.04. How much money should you leave total?
Click through the slideshow to learn how to calculate tips.
View the slides below to learn how to calculate tips.
arrow_back_ios
Let's say you went out to eat, and your bill is $12.04. You want to leave a 15% percent tip.
A tip is a percentage of the total. In this case, it means the tip is fifteen percent of $12.04.
Whenever you see the phrase "of something" in a math sentence, it's usually a hint that you'll need to multiply.
So we'll multiply 12.04 by 15%.
Before we can do that, we'll need to convert 15% into a decimal.
We'll move the decimal point two places to the left...
We'll move the decimal point two places to the left...so 15% becomes 0.15.
Now we can multiply. 0.15 times 12.04 equals 1.80.
So the tip is $1.80.
We could also say that 15% of $12.04 is $1.80.
Remember, a tip is extra money added to the total. So we'll add the tip to the bill.
1.80 plus 12.04 equals 13.84.
So the final price of the meal is $13.84, after including tip.
Let's try another example. Let's say you had excellent service and would like to leave a 20% tip for a meal that cost $68.80.
This means the tip will be 20% of $68.80.
We'll multiply 20% by 68.80.
First, we'll need to convert 20% into a decimal, so we'll move the decimal point two placesto the left.
20% is the same as 0.20.
Now we'll multiply. 68.80 times 0.20 equals 13.76.
So the tip is $13.76.
Another way to say this is that 20% of $68.80 is $13.76.
Finally, we'll add the tip to the bill.
68.80 plus 13.76 equals 82.56.
The final price, after tip, is $82.56.
arrow_back_ios
Try This!
Find the final price for each item, including the tip.
You have a pizza delivered to your house. It costs $19. You want to leave the delivery driver a 15% tip. What's the total?
You eat out at a restaurant with some friends. The bill comes to $47.50. If you leave a 20% tip, what is the total cost of your bill?
Your cab fare from home to the airport is $35. You'd like to give your driver an 18% tip. What's the total?
Let's practice!
Question 1 of 7
A cab ride to the airport costs $32. If you want to leave an 18% tip, how much will your total cab ride cost?
$37.76
$34.82
$5.76
$37.60
✓That's right, great job!
×Sorry, that's not correct...
You answered out of questions correctly. Click the Retry button if you'd like to take the quiz again!
Retry
Continue
Previous: Calculating Percentages
Next:Converting Percentages, Decimals, and Fractions
/en/percents/converting-percentages-decimals-and-fractions/content/
keyboard_arrow_up keyboard_arrow_down
About Us
Who We Are
Who Uses Us + Our Partners
Educators' Resources
Teacher Guides
Resources and Tools
Help and Support
FAQ
Terms of Service
Privacy Notice
Cookie Policy
Contact Us
My Account
chatsimple |
8676 | https://www.vedantu.com/question-answer/number-of-valence-electrons-in-chlorine-ion-are-class-11-chemistry-cbse-60d481f773e290701a85395a | Courses for Kids
Free study material
Offline Centres
Talk to our experts
Question Answer
Class 11
Chemistry
Number of valence electrons in...
Question Answers for Class 12
Class 12 Biology
Class 12 Chemistry
Class 12 English
Class 12 Maths
Class 12 Physics
Class 12 Social Science
Class 12 Business Studies
Class 12 Economics
Question Answers for Class 11
Class 11 Economics
Class 11 Computer Science
Class 11 Biology
Class 11 Chemistry
Class 11 English
Class 11 Maths
Class 11 Physics
Class 11 Social Science
Class 11 Accountancy
Class 11 Business Studies
Question Answers for Class 10
Class 10 Science
Class 10 English
Class 10 Maths
Class 10 Social Science
Class 10 General Knowledge
Question Answers for Class 9
Class 9 General Knowledge
Class 9 Science
Class 9 English
Class 9 Maths
Class 9 Social Science
Question Answers for Class 8
Class 8 Science
Class 8 English
Class 8 Maths
Class 8 Social Science
Question Answers for Class 7
Class 7 Science
Class 7 English
Class 7 Maths
Class 7 Social Science
Question Answers for Class 6
Class 6 Science
Class 6 English
Class 6 Maths
Class 6 Social Science
Question Answers for Class 5
Class 5 Science
Class 5 English
Class 5 Maths
Class 5 Social Science
Question Answers for Class 4
Class 4 Science
Class 4 English
Class 4 Maths
Number of valence electrons in Chlorine ion are: (a) 16 (b) 8 (c) 17 (d) 18
Answer
Verified
468k+views
Hint: Chlorine is a chemical element with the atomic number 17 and the symbol Cl. It is the second-lightest of the halogens, appearing on the periodic table between fluorine and bromine, and its characteristics are largely midway between them. At room temperature, chlorine is a pale green gas with a yellow-green colour.Complete answer:A valence electron is an outer shell electron connected with an atom that can participate in the creation of a chemical bond if the outer shell is not closed; in a single covalent connection, both atoms contribute one valence electron to create a shared pair in chemistry and physics. The existence of valence electrons can affect an element's chemical characteristics, such as its valence—whether or not it can bind with other elements and, if so, how easily and how many times. In this sense, the reactivity of a particular element is strongly reliant on its electrical arrangement.Valence electrons are electrons in the outer shells that are not completely filled. Chlorine has an atomic number of 17. Here is the electrical configuration of chlorine.[{\mathbf{Cl}}\left( {{\mathbf{Z}} = {\text{ }}{\mathbf{17}}} \right){\text{ }} = {\text{ }}{\mathbf{1}}{{\mathbf{s}}^{\mathbf{2}}}\;{\mathbf{2}}{{\mathbf{s}}^{\mathbf{2}}}\;{\mathbf{2}}{{\mathbf{p}}^{\mathbf{6}}}\;{\mathbf{3}}{{\mathbf{s}}^{\mathbf{2}}}\;{\mathbf{3}}{{\mathbf{p}}^{\mathbf{5}}}] As a result, it possesses two electrons in its innermost shell, eight electrons in its second shell, and seven electrons in its outermost shell. As a result, the chlorine atom has seven valence electrons. The shell of the chlorine ion ($C{l^ - }$) has 18 electrons. There are eight valence electrons in all.Note:A valence electron may receive or release energy in the form of a photon, much like a core electron. Atomic excitation occurs when an energy gain causes an electron to migrate (jump) to an outer shell. Alternatively, the electron can break free from its linked atom's shell, resulting in ionisation and the formation of a positive ion. When an electron loses energy (and so emits a photon), it might migrate to an inner shell that isn't completely occupied.
Recently Updated Pages
Master Class 11 Economics: Engaging Questions & Answers for SuccessMaster Class 11 Accountancy: Engaging Questions & Answers for SuccessMaster Class 11 Business Studies: Engaging Questions & Answers for SuccessMaster Class 11 English: Engaging Questions & Answers for SuccessMaster Class 11 Social Science: Engaging Questions & Answers for SuccessMaster Class 11 Computer Science: Engaging Questions & Answers for Success
Master Class 11 Economics: Engaging Questions & Answers for SuccessMaster Class 11 Accountancy: Engaging Questions & Answers for SuccessMaster Class 11 Business Studies: Engaging Questions & Answers for Success
Master Class 11 English: Engaging Questions & Answers for SuccessMaster Class 11 Social Science: Engaging Questions & Answers for SuccessMaster Class 11 Computer Science: Engaging Questions & Answers for Success
Trending doubts
Which animal has three hearts class 11 biology CBSEAlgae used for the manufacture of iodine is A Nostoc class 11 biology CBSEList the factors affecting electrode potential class 11 chemistry CBSEWhat organs are located on the left side of your body class 11 biology CBSECompare the salient features of Monera with Protis class 11 biology CBSEHow do I convert ms to kmh Give an example class 11 physics CBSE
Which animal has three hearts class 11 biology CBSEAlgae used for the manufacture of iodine is A Nostoc class 11 biology CBSEList the factors affecting electrode potential class 11 chemistry CBSE
What organs are located on the left side of your body class 11 biology CBSECompare the salient features of Monera with Protis class 11 biology CBSEHow do I convert ms to kmh Give an example class 11 physics CBSE
Repeaters Course for NEET 2022 - 23
NEET Repeater 2023 - Aakrosh 1 Year Course |
8677 | https://www.mypcnow.org/fast-fact/management-of-refractory-gastroesophageal-reflux-disease/?print=print | FF #377 Management of Refractory Gastroesophageal Reflux Disease - Palliative Care Network of Wisconsin
FF #377 Management of Refractory Gastroesophageal Reflux Disease
BackgroundRefractory gastroesophageal reflux disease (rGERD) can be characterized as symptomatic reflux or esophagitis despite an adequate trial of twice daily proton pump inhibitor (PPI) therapy (1). Patients with rGERD might describe their symptoms as heartburn, poorly localized chest pain, or acid reflux. Non-verbal patients (e.g. patients obtunded from the dying process) may be at risk of untreated GERD. This Fast Fact reviews treatment options for rGERD in patients with life-limiting illness.
EtiologiesMechanisms that contribute to rGERD in palliative care populations are varied and include both gastrointestinal and non-gastrointestinal processes such as: increased esophageal acid exposure (i.e. secondary to gastric carcinoid tumors); opioid-induced gastroparesis/delayed emptying; and/or peristaltic deficiency associated with cirrhotic ascites; malignancy-induced bowel obstructions; and transient lower esophageal sphincter relaxation (TLESR) (1,2). Clinicians should consider rGERD in the differential diagnosis of non-verbal patients who appear to be imminently dying and are uncomfortable, especially if they have a history of GERD, cancer, and/or cirrhosis.
Symptom ManagementWhen possible, therapeutic management of rGERD should target the underlying etiology (e.g. triple therapy for helicobacter pylori). However, in palliative care patients, clinicians must factor in prognosis, the amount of symptom distress, and the clinical situation to determine the extent of diagnostic work-up (e.g. endoscopy) and specialized consultation that is appropriate. For many palliative care patients, empiric treatment of rGERD is pursued.
Traditional Acid Suppressing Agents
Adding a nocturnal histamine-2 receptor antagonist (H 2 RA)such as ranitidine for at least one month while on concurrent PPI therapy has been associated with improvements in night-time reflux symptoms, GERD-associated sleep disturbance, and overall GERD symptom management in up to 74% of patients; only 13% discontinued nocturnal H 2 RA due to tolerance issues (3). Coadministration of H 2 RAs and PPIs is felt to be safe (4), although some experts suggest separating evening PPI and bedtime H 2 RA doses for optimal effect. Solutions containing sodium alginate(i.e. Gaviscon®) have reliably decreased the severity and frequency of heartburn,especially when used post-prandially, with few side effects (5,6). Likewise, sucralfate in two-to-four daily doses may improve rGERD as well as mucosal healing for erosive disease (7).However, none of these agents have been well-studied in the seriously ill.
More Targeted Strategies to Palliate rGERD
Baclofen reduces regurgitation events and TLSER via gamma-aminobutyric acid activity (8). Although literature is scarce in palliative care patients, decreased duodenal reflux and improvement in symptom severity was observed in a small prospective study (n=16) of patients with heartburn lasting > 3 months (8). Baclofen crosses the blood brain barrier and is known to cause significant adverse CNS effects like drowsiness, confusion, and seizures (see Fast Fact #340). Dose ranges of 10-30 mg/day, divided into twice-daily or three-times-daily doses, are most commonly used, but up to 60 mg/day has been described (8). It should not be titrated more often than every three days (8).
Metoclopramide is a prokinetic agent that may improve rGERD symptoms and nausea in patients with delayed esophageal peristalsis, delayed gastric emptying (gastroparesis), and partial bowel obstruction at oral or parenteral doses of 10 mg three to four times a day. It is not recommended nor effective for patients who do not have evidence of gastroparesis or a partial bowel obstruction (9). It has been associated with adverse effects such as tardive dyskinesia and dystonia, especially if used for longer than 12 weeks (9).
Antidepressants Selective serotonin reuptake inhibitors and trazodone have been shown to reduce GERD symptoms in symptomatic patients with normal endoscopies (10-12). Citalopram and fluoxetine have both been shown to be effective in placebo-controlled randomized trials (10,11) and even showed superiority to omeprazole in controlling heartburn symptoms in one comparison trial in patients with concomitant depression (11). Evidence for trazodone has been limited to the symptomatic relief of chest pain in patients with esophageal contraction abnormalities (12).
Other Considerations
Lifestyle interventions: Although, tobacco use, and alcohol consumption may reduce lower esophageal sphincter pressure, cessation of these agents has not been clearly shown to lead to improvements in GERD symptoms (13). Some experts recommend head of bed elevation and avoidance of a late-night evening meal (within 2-3 hours of bedtime) to mitigate rGERD, although the degree to which these interventions help is not clearly established (14).
Botulinum toxin injection (see Fast Fact #324) can help patients with achalasia, which is a gastric motility disorder characterized by lower esophageal sphincter (LES) non-relaxation (15). Without confirmed LES non-relaxation, its use could increase LES relaxation and thereby worsen rGERD symptoms, however. Major side effects include anaphylaxis, voice disorders, and pharyngitis.
CYP2C19 genotype status can contribute to pharmacokinetic variability of the effectiveness of PPIs, but genotypic testing in palliative care patients is not routinely performed. The clinical utility of switching appropriately dosed PPIs in patients without genotypic testing is not well described.
Long-term PPI use, generally greater than eight weeks, without substantial clinical benefit is not generally recommended. Long-term PPI use has been linked to increased risk of bone fracture, clostridium difficile infections, hypomagnesemia, and vitamin B12 deficiency (16).
Cost:The average wholesale price of PPIs (e.g. omeprazole) is double that of H 2 RAs (e.g. ranitidine) at $0.72/tablet versus $0.34/capsule. Both classes of medications are available over-the-counter (OTC). Prescription pricing may be nearly 3 times that of OTC. The cost of baclofen, metoclopramide, and trazodone are $0.50/tablet, $0.13/tablet, and $0.44/tablet, respectively. SSRIs and botulinum toxin injection are costlier: $2.42/tablet and $721.20/100 unit injection, respectively.
References:
Sifrim D,Zerbib F. Diagnosis and management of patients with reflux symptoms refractory to proton pump inhibitors. Gut 2012; 61:1340-1354.
Dellon ES, Shaheen NJ. Persistent reflux symptoms in the proton pump inhibitor era: The changing face of gastroesophageal reflux disease.Gastroenterology.2010; 139:7–13.e3.
Rackoff A, Agrawal A, Hila A, Mainie I, Tutuian R, Castell D O. Histamine-2 receptor antagonists at night improve gastroesophageal reflux disease symptoms for patients on proton pump inhibitor therapy. Dis Esophagus 2005; 18: 370–3.
Abdul‐Hussein, M., Freeman, J. and Castell, D. (2015), Concomitant Administration of a Histamine 2 Receptor Antagonist and Proton Pump Inhibitor Enhances Gastric Acid Suppression. Pharmacotherapy, 35: 1124-1129.
Reimer, C.et al.Concentrated alginate as add-on therapy in gastro-esophageal reflux disease patients with inadequate response to once daily proton pump inhibitor: a multicenter, randomized, double-blind, placebo-controlled pilot study.Gastroenterology 2015; 148: S135–136.
Rohof WO,Bennink RJ,Smout AJ,Thomas, E et al. An alginate-antacid formulation localizes to the acid pocket to reduce acid reflux in patients with gastroesophageal reflux disease.Clin. Gastroenterol. Hepatol 2013; 11: 1585–1591.
Surdea-Blaga T, Băncilă I, Dobru D, et al. Mucosal protective compounds in the treatment of gastroesophageal reflux disease. A position paper based on evidence of the Romanian Society of Gastroenterology.J Gastrointest Liver Dis.2016; 25: 537–546.
Li S, Shi S, Chen F, Lin J. The effects of baclofen for the treatment of gastroesophageal reflux disease: a meta-analysis of randomized controlled trials.Gastroenterol Res Pract. 2014; 2014: 307-805.
Richter JE,Sabesin SM,Kogut DG,Kerr RM,et al. Omeprazole versus ranitidine or ranitidine/metoclopramide in poorly responsive symptomatic gastroesophageal reflux disease. Am J Gastroenterol.1996; 91: 1766-72.
Viazis N, Keyoglou A, Kanellopoulos AK, Karamanolis G et al. Selective serotonin reuptake inhibitors for the treatment of hypersensitive esophagus: A randomized- double-blind, placebo-controlled study. Amer J Gastroenterol 2012; 107: 1662-1667.
Ostovaneh MR, Saeidi B, Hajifathalian K, Farrokhi-Khajeh-Pasha Y et al. Comparing omeprazole with fluoxetine for treatment of patients with heartburn and normal endoscopy who failed once daily proton pump inhibitors: Double‐blind placebo‐controlled trial. Neurogastroent Motil 2014; 26: 670-678.
Clouse RE, Lustman PJ, Eckert TC, Ferney DM, et al. Low-dose trazodone for symptomatic patients with esophageal contraction abnormalities: a double-blind, placebo controlled trial. Gastroenterolgy 1987; 92: 1027-36.
Kaltenbach T, Crockett S, Gerson LB. Are Lifestyle Measures Effective in Patients With Gastroesophageal Reflux Disease?An Evidence-Based Approach.Arch Intern Med.2006; 166: 965–971.
Katz PO, Gerson LB, Vela MF. Guidelines for the diagnosis and management of gastroesophageal reflux disease. Am J Gastroenterol 2013; 108: 308.
Jung DH, Park H. Is Gastroesophageal Reflux Disease and Achalasia Coincident or Not?.J Neurogastroenterol Motil. 2017; 23: 5-8.
Farrell B, Pottie K, Thompson W, Boghossian T, Pizzola L, Rashid FJ, et al. Deprescribing proton pump inhibitors. Evidence-based clinical practice guideline. Can Fam Physician 2017; 63: 354-64.
Authors’ Affiliations: University of Maryland Medical Center, Baltimore, MD; University of Pittsburgh Medical Center; Pittsburgh PA
Conflicts of Interest: None reportedVersion History: Originally edited by Sean Marks MD; first electronically published in May 2019. |
8678 | https://www.quora.com/How-do-you-use-Cramers-rule-to-find-the-solution-to-simultaneous-equations-with-two-unknown-variables-matrix-algebra | How to use Cramer's rule to find the solution to simultaneous equations with two unknown variables (matrix, algebra) - Quora
Something went wrong. Wait a moment and try again.
Try again
Skip to content
Skip to search
Sign In
Mathematics
Cramer's Rule (linear alg...
Two Equations in Two Unkn...
Matrices (mathematics)
Consistent Equations
Linear Algebra 1
Solving Equations
Simultaneous Linear Equat...
Pair of Linear Equations ...
5
How do you use Cramer's rule to find the solution to simultaneous equations with two unknown variables (matrix, algebra)?
All related (31)
Sort
Recommended
Sunil
Former Experience:41 Years. · Author has 416 answers and 46.7K answer views
·2y
CRAMER'S rule can be used for TWO unknowns and also for THREE unknowns.Formula for two unknowns. x=DX/D, y=Dy/D.where D is the value of the Determinant, Dx can be calculated by deleting the first column and replaced by R.H.S. elements of the given equations. Also Dy can be calculated by deleting the second column of the D and replaced by R.H.S. elements of the given equations. The same trick can be used for three equations in three unknowns.
Upvote ·
9 1
Sponsored by Google Cloud
Skill up, stand out, and lead with Generative AI.
Your journey from learner to Gen AI leader starts now, with skills that shape the future.
Learn More
99 51
Related questions
More answers below
How can I solve this simultaneous equation by Cramer’s rule y=5x-10/2: 4x+5=-y?
Linear Algebra: How does Cramer's Rule work on a system of equations? Does it work on a 5 by 5 matrix?
How do I solve simultaneous equations with three unknowns?
Who discovered Cramer's rule and how?
Can you provide some real life examples of using simultaneous equations with two variables?
Prajod Chemmarathil Prasad
Mathematics · Author has 446 answers and 1M answer views
·Updated 9y
Related
Linear Algebra: How does Cramer's Rule work on a system of equations? Does it work on a 5 by 5 matrix?
Yes, it does.
Cramer's rule is just another way of solving a system of linear equations, just like the Gaussian Elimination (more complicated) and the conventional way. It can work from a [math]2 by 2[/math] matrix to a matrix with infinite columns and rows (only thing is that you need the columns and rows to be equal in number. Let me show you an example using the [math]5 by 5[/math] matrix.
[math]a+3b+x+y+5z=8[/math]
[math]2a+4b+7x+y+z=20[/math]
[math]4a+b+2x+3y+2z=22[/math]
[math]a+2b+x+y+z=5[/math]
[math]3a+5b+2x+3y+4z=25[/math]
Taking the coefficients of the variables in the equation and creating 6 ([math]n+1, n=5, no.of variables)[/math]matrices.
The coefficients of the variables, [math]a,b,x,y,z[/math] (in
Continue Reading
Yes, it does.
Cramer's rule is just another way of solving a system of linear equations, just like the Gaussian Elimination (more complicated) and the conventional way. It can work from a [math]2 by 2[/math] matrix to a matrix with infinite columns and rows (only thing is that you need the columns and rows to be equal in number. Let me show you an example using the [math]5 by 5[/math] matrix.
[math]a+3b+x+y+5z=8[/math]
[math]2a+4b+7x+y+z=20[/math]
[math]4a+b+2x+3y+2z=22[/math]
[math]a+2b+x+y+z=5[/math]
[math]3a+5b+2x+3y+4z=25[/math]
Taking the coefficients of the variables in the equation and creating 6 ([math]n+1, n=5, no.of variables)[/math]matrices.
The coefficients of the variables, [math]a,b,x,y,z[/math] (in the rows) of each equation (in the columns) (in order) are given in the matrix [math]\delta.[/math] The solutions to the equations ([math]8,20,22,5,25[/math]) are not given in any row of the matrix.
Now, the solutions to the equations are now replacing the [math]a[/math] (1st) row.
Now, the solutions are replacing the [math]b[/math] (2nd) row.
The solutions are now replacing the [math]x[/math] (3rd) row.
The solutions are now replacing the [math]y[/math] (4th) row.
The solutions are finally, replacing the [math]z[/math] row.
Now, in each of the matrices, you have to find the determinant (a specific number calculated). The determinant of the first matrix ([math]\delta[/math]) is 8 (Used the Determinant Calculator)
The determinant of [math]\delta_a[/math] is 21.
The determinant of [math]\delta_b[/math] is -288.
The determinant of [math]\delta_x[/math] is 286.
The determinant of [math]\delta_y[/math] is 1272.
The determinant of [math]\delta_z[/math] is 78.
We are almost near.
Remember, you have to divide each of the matrices (except for [math]\delta[/math]) by [math]\delta[/math] to get the solutions for the variables respectively.
[math]a=21/8=2.625[/math]
[math]b=-288/8=-36[/math]
[math]x=286/8=35.75[/math]
[math]y=1272/8=159[/math]
[math]z=78/8=9.75[/math]
Upvote ·
99 26
9 1
Terry Moore
M.Sc. in Mathematics, University of Southampton (Graduated 1968) · Author has 16.6K answers and 29.4M answer views
·11mo
Related
How do you solve the following system of linear equations using matrix inversion and Cramer's rule?
How do you solve the following system of linear equations using matrix inversion and Cramer's rule?
Are you sure you have given us all the information needed to slve the equations?
Hint: before we solve equations we need to know what they are.
Anyway, matrix inversion and Cramer’s rule are different methods.
Matrix inversion method:
Once you have inverted the matrix of coefficients of the variables, use it to multiply the vector of constants. There are many ways to invert a matrix, one is a method of elimination similar to solving equations by elimination. Another is to divide the adjoint matrix by
Continue Reading
How do you solve the following system of linear equations using matrix inversion and Cramer's rule?
Are you sure you have given us all the information needed to slve the equations?
Hint: before we solve equations we need to know what they are.
Anyway, matrix inversion and Cramer’s rule are different methods.
Matrix inversion method:
Once you have inverted the matrix of coefficients of the variables, use it to multiply the vector of constants. There are many ways to invert a matrix, one is a method of elimination similar to solving equations by elimination. Another is to divide the adjoint matrix by the determinant. Neither way is to be recommended except possibly with two equations in two unknowns (or maybe three).
Cramer’s rule method:
This also uses determinant and is not recommended either. Please look up the details yourself. It’s not hard but it’s more work than elimination (at least with more than three variables).
Upvote ·
9 4
Roger Asdale
Author has 2.3K answers and 327.8K answer views
·Jan 11
Related
What is the method for solving equations with two variables using matrix algebra?
Presumably, solve 2 equations in 2 variables using matrices…
2x + 3y = 12
4x + 5y = 20
Express all equations as matrix of coefficients (M) and column vectors:
[2 3][x] =
[4 5][y] =
Obtain determinant of M:
D = 25 - 34 = 10 - 12 = -2
IF D = 0 THEN “NO SOLUTION or INFINITE SOLUTIONS”
ELSE
Obtain inverse of M:
[5 -3]
[-4 2]
Pre-multiply on both sides:
[5 -3][2 3][x] = [5 -3]
[-4 2][4 5][y] = [-4 2]
[-2 0][x] =
[0 -2][y] = [-8]
Divide both sides by D…
[1 0][x] =
[0 1][y] =
[x] =
[y] =
Solution:
x = 0 ; y = 4
END
Upvote ·
9 3
9 1
Related questions
How can I solve this simultaneous equation by Cramer’s rule y=5x-10/2: 4x+5=-y?
Linear Algebra: How does Cramer's Rule work on a system of equations? Does it work on a 5 by 5 matrix?
How do I solve simultaneous equations with three unknowns?
Who discovered Cramer's rule and how?
Can you provide some real life examples of using simultaneous equations with two variables?
How do you solve simultaneous equations?
How can we solve simultaneous equations in two variables?
How can I solve the simultaneous equation X/2-y/3=1 and 3/X+2/y=3/2?
What is the simultaneous equation by using the Cramer method 2y+3x=12, 3x=5x-1?
What is the process for finding all the possible solutions to a system of two simultaneous equations with two unknowns?
What is the simultaneous simultaneous equation formula?
How does the use of Cramer's rule accelerate the time consumption of solving linear equations?
How do you solve simultaneous linear equations in two variables?
How do you use the Cramer's rule to solve these systems of linear equations (1) 5x + 6y + 7z = 40 (2) 2x + 4y + 2z= 34 (3) x + 3y + 5x = 30?
How do you solve simultaneous equations with two unknowns?
Related questions
How can I solve this simultaneous equation by Cramer’s rule y=5x-10/2: 4x+5=-y?
Linear Algebra: How does Cramer's Rule work on a system of equations? Does it work on a 5 by 5 matrix?
How do I solve simultaneous equations with three unknowns?
Who discovered Cramer's rule and how?
Can you provide some real life examples of using simultaneous equations with two variables?
How do you solve simultaneous equations?
Advertisement
About · Careers · Privacy · Terms · Contact · Languages · Your Ad Choices · Press ·
© Quora, Inc. 2025 |
8679 | https://wellsr.com/python/optimizing-recursive-functions-with-python-memoization/ | Optimizing Recursive Functions with Python Memoization
## The Python Tutorials Blog
Jul 16, 2021
Looking for VBA Tutorials?
Today, let’s explore how to implement memoization techniques to optimize the performance of recursive functions in Python.
A recursive function is a function which calls itself. Recursive functions are used to solve programming problems which are recursive in nature, like the tower of Hanoi. Recursive functions reduce code clutter, avoid unnecessary function calls and are commonly used for solving data structure problems.
The efficiency and speed of recursion may be effected if you have a deep hierarchy of recursive function calls. For cases like this, you can optimize your recursive functions using memoization techniques which we’ll explore in this tutorial.
Optimization Issue with Recursive functions
Let’s define a simple recursive function which returns the “Nth” number from a Fibonacci series. A Fibonacci series starts from two digits, 0 and 1, and every next number in the series is the sum of the previous two numbers. For example the first 10 numbers in the Fibonacci series (excluding 0) are 1, 1, 2, 3, 5, 8, 13, 21, 34, 55.
```
def find_fibonnaci(n):
if n == 1:
return 1
elif n == 2:
return 1
elif n > 2:
return find_fibonnaci(n - 1) + find_fibonnaci(n - 2)
```
Here’s what’s happening in the above function. Since the first 2 numbers in a Fibonacci series are 1 (excluding 0), the find_fibonnaci() function returns 1 if n is 1 or 2. Else, if n is greater than 2, recursion takes place and the sum of find_fibonnaci(n-1) and find_fibonnaci(n-2) is returned.
So how does this recursion find the Fibonacci number?
Suppose you want to find the 4th number in the Fibonacci series, you can find it as follows. We’ll call this equation 1.
find_fibonnaci(4) = find_fibonnaci(3) + find_fibonnaci(2) --- equation 1
Since, find_fibonnaci(2) will return 1, and find_fibonnaci(3) will return find_fibonnaci(2) + find_fibonnaci(1), equation 1 can be rewritten as follows:
find_fibonnaci(4) =(find_fibonnaci(2) + find_fibonnaci(1)) + 1 --- equation 2
We’ll call this equation 2.
If you replace values of find_fibonnaci(1), and find_fibonnaci(2) in equation 2, you’ll get the answer of find_fibonnaci(4):
```
find_fibonnaci(4) = 1 + 1 + 1
find_fibonnaci(4) = 3
```
Alright, let’s call our find_fibonnaci() function to see if it works:
find_fibonnaci(4)
The output shows the correct result.
Output:
3
Our function works well so let’s try printing the first 10 numbers of the Fibonacci series:
for i in range(1,11):
print(i, "-", find_fibonnaci(i))
The output shows the first 10 numbers in the Fibonacci series printed pretty quickly on the console.
Output:
1 - 1
2 - 1
3 - 2
4 - 3
5 - 5
6 - 8
7 - 13
8 - 21
9 - 34
10 - 55
Okay, so far so good. Let’s try a bigger number. For example, lets print the first 40 numbers in the Fibonacci series and we’ll time how long it takes to run.
Note: The script below is run in the Jupyter notebook. To find the cell execution time for the script, the %%time command is used. We published a full tutorial about how to find the execution time of a Python script.
%%time
for i in range(1,41):
print(i, "-", find_fibonnaci(i))
For the sake of space, only Fibonacci numbers from 30 to 40 are printed:
Output:
30 - 832040
31 - 1346269
32 - 2178309
33 - 3524578
34 - 5702887
35 - 9227465
36 - 14930352
37 - 24157817
38 - 39088169
39 - 63245986
40 - 102334155
Wall time: 1min 23s
The output shows that it took a whopping 1 minute and 23 seconds to calculate the first 40 numbers in a Fibonacci series. This is a significant amount of time and this happens as a direct result of the large number of hierarchical recursive calls.
Let me explain. To calculate find_fibonnaci(40), you have to calculate find_fibonnaci(39) and find_fibonnaci(38) upto find_fibonnaci(1). The number of function calls grow exponentially which significantly slows down the overall Python calculation.
This is where memoization comes in. With memoization, you can save previous function calls in a cached memory and then when a new calculation has to be performed the result of previous function calls can be used instead of making new function calls from scratch. Memoization can be implemented via custom Python code or via a built-in Python module. We’ll demonstrate both methods.
Code More, Distract Less: Support Our Ad-Free Site
You might have noticed we removed ads from our site - we hope this enhances your learning experience. To help sustain this, please take a look at our Python Developer Kit and our comprehensive cheat sheets. Each purchase directly supports this site, ensuring we can continue to offer you quality, distraction-free tutorials.
Get Our Python Dev Kit
Memoization via Custom Python Code
To implement custom memoization, you need to create a data structure, like a dictionary, which acts as a cache and stores results of any previous calls to a recursive function. Then when a recursive function is called, you first check if the result of that call already exists in the cache, in which case you simply return the result from the cache instead of calling the function again. Else, you call the function and store the result in the cache before returning it to the calling function, which can then be used for later function calls. It sounds complicated, but it’s not so bad.
The following script implements memoization for a function that returns numbers in the Fibonacci series.
```
my_cache = {}
def find_fibonnaci_cached(n):
# creating custom cache
if n in my_cache:
return my_cache[n]
if n == 1:
result = 1
if n == 2:
result = 1
elif n > 2:
result = find_fibonnaci_cached(n - 1) + find_fibonnaci_cached(n - 2)
my_cache[n] = result
return result
```
Let’s now see how long it takes to print the first 40 Fibonacci numbers using the find_fibonnaci_cached() function which implements memoization.
%%time
for i in range(1,41):
print(i, "-", find_fibonnaci_cached(i))
Output:
30 - 832040
31 - 1346269
32 - 2178309
33 - 3524578
34 - 5702887
35 - 9227465
36 - 14930352
37 - 24157817
38 - 39088169
39 - 63245986
40 - 102334155
Wall time: 4.99 ms
The output shows that it only took 4.99 ms which is lightning fast compared to the 1 minute and 23 seconds it took without memoization.
Let’s now try to find how long it takes to print the first 1000 Fibonacci numbers.
%%time
for i in range(1,1000):
print(find_fibonnaci_cached(i))
Output:
```
first 1000 fibonnaci numbers will be printed
Wall time: 334 ms
```
The result shows that it took only 334 ms seconds to print the first 1000 Fibonacci numbers using memoization!
Memoization via Python functools module
Instead of implementing custom memoization, you can use the lru_cache decorator from the Python functools module to cache the result of a specific number of calls to a function. The number of calls to cache are specified via the maxsize attribute of the lru_cache decorator. Here’s an example where we cache 1000 calls to the find_fibonnaci() method.
```
from functools import lru_cache
@lru_cache(maxsize = 1000)
def find_fibonnaci(n):
if n == 1:
return 1
elif n == 2:
return 1
elif n > 2:
return find_fibonnaci(n - 1) + find_fibonnaci(n - 2)
```
Let’s see how much time it takes to print the first 1000 Fibonacci numbers using Python’s built-in memoization.
%%time
for i in range(1,1000):
print(find_fibonnaci(i))
Output:
```
first 1000 fibonnaci numbers
Wall time: 68 ms
```
The output shows that it took only 68 ms seconds to print the first 1000 Fibonacci numbers using memoization, which is even faster than our custom memoization!
If you found this tutorial fascinating, join us using the form below and we’ll share a few more Python tips and tricks to help you get the most out of this powerful programming language.
Code More, Distract Less: Support Our Ad-Free Site
You might have noticed we removed ads from our site - we hope this enhances your learning experience. To help sustain this, please take a look at our Python Developer Kit and our comprehensive cheat sheets. Each purchase directly supports this site, ensuring we can continue to offer you quality, distraction-free tutorials.
Get Our Python Dev Kit
« Python Delete All Files in a Folder
Python Map, Filter and Reduce for Lambda Functions »
This article was written by Usman Malik, contributing writer for The Python Tutorials Blog.
About The Python Tutorials Blog
The Python Tutorials Blog was created by Ryan Wells, a Nuclear Engineer and professional VBA Developer. After launching his VBA Tutorials Blog in 2015, he designed some VBA Cheat Sheets, which have helped thousands learn to write better macros. He expanded in 2018 with The Python Tutorials Blog to teach people Python in a similar systematic way. Today, wellsr.com reaches over 2 million programmers each year!
New to Python? Start here
Chapter 1: Python Basics
Getting Started with Python
Python Data Types
Python Data Structures
Python Conditionals, Loops & Logical Operators
Chapter 2: Intermediate Python
Python Functions
Python Modules
Python Classes
Python File I/O
Python Namespaces
Support Ad-Free Coding
Love learning without ads? Support us by checking out our Python Developer Kit. Your help keeps our site ad-free!
Get Your Tools Now
Popular Python Tutorials
3 Ways to Calculate Python Execution Time
Implementing an Artificial Neural Network from Scratch in Python
How to Create Custom Modules in Python (with examples)
Upsampling and Downsampling Imbalanced Data in Python
Object Detection from Webcams with YOLO using Python
How to Read PDF Files with Python using PyPDF2
Drawing Multiple Plots with Matplotlib in Python
Pickling and Unpickling Objects with Python Pickle Module
Latest Python Tutorials
Building a MySQL Agent in Python LangGraph Framework
Developing a Simple Login Page with Python Django Framework
Extracting Structured Data Using LLMs via Hugging Face API
Multiagent Workflow in LangGraph with DeepSeek R1 Model
Retrieval Augmented Generation with LangGraph Agents
How to Develop Multi-agent Chatbots with LangGraph Agents
Elevate Your Python Skills
Unlock your programming potential with our Python Developer Kit. Your support helps keep our site ad-free, ensuring a focused learning environment for all.
Get Our Python Dev Kit |
8680 | https://www.intmath.com/functions-and-graphs/the-three-types-of-triangles.php | Skip to main content
On this page
Introduction to Geometry
Functions and Graphs
1. Introduction to Functions
2. Functions from Verbal Statements
3. Rectangular Coordinates
4. The Graph of a Function
4a. Domain and Range of a Function
4b. Domain and Range interactive applet
4c. Comparison calculator BMI - BAI
5. Graphing Using a Computer Algebra System
5a. Online graphing calculator (1): Plot your own graph (JSXGraph)
5b. Online graphing calculator (2): Plot your own graph (SVG)
6. Graphs of Functions Defined by Tables of Data
7. Continuous and Discontinuous Functions
8. Split Functions
9. Even and Odd Functions
Related Sections
Math Tutoring
Need help? Chat with a tutor anytime, 24/7.
Chat now
Online Math Solver
Solve your math problem step by step!
Online Math Solver
IntMath Forum
Get help with your math queries:
See Forum
Triangles
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices A, B, and C denoted △ABC is given below:
In the triangle above, the edges are AB, BC and AC. The angles are ∠A, ∠B and ∠C. Triangles can be classified according to the side lengths or its interior angles.
The Three Types of Triangles (Classified by Angles)
When classified by their angles, there are three main types of triangles: acute, right, and obtuse. An acute triangle has all three angles measuring less than 90°, a right triangle has one angle measuring 90°, and an obtuse triangle has one angle measuring more than 90°.
Acute Triangle
An acute triangle is a triangle with all three angles measuring less than 90°. In other words, all the angles of an acute triangle are acute angles.
The above figure shows an acute angled triangle ABC in which the angles A, B and C are acute angles less than 90°.
Right Triangle
A right triangle is a triangle in which one angle measures 90° and the other two angles measure less than 90°. In a right angled triangle, the side opposite to the right angle is called the hypotenuse while the other two sides are called legs or catheti (singular: cathetus). The hypotenuse is always the longest side while the other two sides are shorter. The below figure shows a right angled triangle ABC in which angle C measures 90°.
Obtuse Triangle
An obtuse triangle is a triangle with one angle measuring more than 90° while the other two angles measure less than 90° each. In other words, an obtuse angled triangle has one obtuse angle and two acute angles. The below figure shows an obtuse angled triangle ABC in which angle B measures more than 90° while angles A and C measure less than 90° each.
The Three Types of Triangles (Classified by Side Lengths)
When classified by side lengths, there are three main types of triangles: scalene, isosceles, and equilateral triangles.
Scalene Triangle
In a scalene triangle, all the side lengths are of different measures and hence, all the angles are of different measures.
The above figure shows a scalene triangle ABC in which the angles A, B and C are all of different measures.
Isosceles Triangle
An isosceles triangle is a triangle with two of its side lengths equal in length. In an isosceles triangle, the angles at the base of the congruent sides are referred to as the base angles and are of equal measures.
In triangle ABC above, the equal side lengths are AB and BC. The base angles are A and C, therefore m∠A ≅ m∠B.
Equilateral Triangle
An equilateral triangle is a triangle with all three sides equal in length. In an equilateral triangle, each of the interior angles has a measure of 60 degrees.
All equilateral triangles are similar.
Conclusion:
In general, there are 7 types of triangles when classified by a mixture of its angles and lengths. They are equilateral, right isosceles, obtuse isosceles, scalene, acute scalene, right scalene, and obtuse scalene.
There you have it! Next time you see a triangle, see if you can classify it by its angles!
FAQ
What is a triangle in geometry?
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry.
Classified by its angles, what are the 3 types of triangle?
The three types of triangle are acute, right, and obtuse.
Classified by its side lengths, what are the 3 types of triangle?
The three types of triangle are scalene, isosceles, and equilateral triangles.
What are the 7 triangles?
The seven general types of triangles are the equilateral, right isosceles, obtuse isosceles, scalene, acute scalene, right scalene, and obtuse scalene.
Functions and Graphs |
8681 | https://courses.lumenlearning.com/waymakercollegealgebra/chapter/distance-in-the-plane/ | College Algebra
Module 3: The Rectangular Coordinate System and Equations of Lines
Distance in the Coordinate Plane
Learning Outcomes
Derived from the Pythagorean Theorem, the distance formula is used to find the distance between two points in the plane. The Pythagorean Theorem, [latex]{a}^{2}+{b}^{2}={c}^{2}[/latex], is based on a right triangle where a and b are the lengths of the legs adjacent to the right angle, and c is the length of the hypotenuse.
The relationship of sides [latex]|{x}_{2}-{x}_{1}|[/latex] and [latex]|{y}_{2}-{y}_{1}|[/latex] to side d is the same as that of sides a and b to side c. We use the absolute value symbol to indicate that the length is a positive number because the absolute value of any number is positive. (For example, [latex]|-3|=3[/latex]. ) The symbols [latex]|{x}_{2}-{x}_{1}|[/latex] and [latex]|{y}_{2}-{y}_{1}|[/latex] indicate that the lengths of the sides of the triangle are positive. To find the length c, take the square root of both sides of the Pythagorean Theorem.
It follows that the distance formula is given as
We do not have to use the absolute value symbols in this definition because any number squared is positive.
A General Note: The Distance Formula
Given endpoints [latex]\left({x}_{1},{y}_{1}\right)[/latex] and [latex]\left({x}_{2},{y}_{2}\right)[/latex], the distance between two points is given by
Example: Finding the Distance between Two Points
Find the distance between the points [latex]\left(-3,-1\right)[/latex] and [latex]\left(2,3\right)[/latex].
Let us first look at the graph of the two points. Connect the points to form a right triangle.
Then, calculate the length of d using the distance formula.
Try It
Find the distance between two points: [latex]\left(1,4\right)[/latex] and [latex]\left(11,9\right)[/latex].
[latex]\sqrt{125}=5\sqrt{5}[/latex]
In the following video, we present more worked examples of how to use the distance formula to find the distance between two points in the coordinate plane.
Example: Finding the Distance between Two Locations
Let’s return to the situation introduced at the beginning of this section.
Tracie set out from Elmhurst, IL to go to Franklin Park. On the way, she made a few stops to do errands. Each stop is indicated by a red dot. Find the total distance that Tracie traveled. Compare this with the distance between her starting and final positions.
The first thing we should do is identify ordered pairs to describe each position. If we set the starting position at the origin, we can identify each of the other points by counting units east (right) and north (up) on the grid. For example, the first stop is 1 block east and 1 block north, so it is at [latex]\left(1,1\right)[/latex]. The next stop is 5 blocks to the east so it is at [latex]\left(5,1\right)[/latex]. After that, she traveled 3 blocks east and 2 blocks north to [latex]\left(8,3\right)[/latex]. Lastly, she traveled 4 blocks north to [latex]\left(8,7\right)[/latex]. We can label these points on the grid.
Next, we can calculate the distance. Note that each grid unit represents 1,000 feet.
Next, we will add the distances listed in the table.
| From/To | Number of Feet Driven |
--- |
| [latex]\left(0,0\right)[/latex] to [latex]\left(1,1\right)[/latex] | 2,000 |
| [latex]\left(1,1\right)[/latex] to [latex]\left(5,1\right)[/latex] | 4,000 |
| [latex]\left(5,1\right)[/latex] to [latex]\left(8,3\right)[/latex] | 5,000 |
| [latex]\left(8,3\right)[/latex] to [latex]\left(8,7\right)[/latex] | 4,000 |
| Total | 15,000 |
The total distance Tracie drove is 15,000 feet or 2.84 miles. This is not, however, the actual distance between her starting and ending positions. To find this distance, we can use the distance formula between the points [latex]\left(0,0\right)[/latex] and [latex]\left(8,7\right)[/latex].
At 1,000 feet per grid unit, the distance between Elmhurst, IL to Franklin Park is 10,630.14 feet, or 2.01 miles. The distance formula results in a shorter calculation because it is based on the hypotenuse of a right triangle, a straight diagonal from the origin to the point [latex]\left(8,7\right)[/latex]. Perhaps you have heard the saying “as the crow flies,” which means the shortest distance between two points because a crow can fly in a straight line even though a person on the ground has to travel a longer distance on existing roadways.
Using the Midpoint Formula
When the endpoints of a line segment are known, we can find the point midway between them. This point is known as the midpoint and the formula is known as the midpoint formula. Given the endpoints of a line segment, [latex]\left({x}_{1},{y}_{1}\right)[/latex] and [latex]\left({x}_{2},{y}_{2}\right)[/latex], the midpoint formula states how to find the coordinates of the midpoint [latex]M[/latex].
A graphical view of a midpoint is shown below. Notice that the line segments on either side of the midpoint are congruent.
Example: Finding the Midpoint of the Line Segment
Find the midpoint of the line segment with the endpoints [latex]\left(7,-2\right)[/latex] and [latex]\left(9,5\right)[/latex].
Use the formula to find the midpoint of the line segment.
Try It
Find the midpoint of the line segment with endpoints [latex]\left(-2,-1\right)[/latex] and [latex]\left(-8,6\right)[/latex].
[latex]\left(-5,\frac{5}{2}\right)[/latex]
Example: Finding the Center of a Circle
The diameter of a circle has endpoints [latex]\left(-1,-4\right)[/latex] and [latex]\left(5,-4\right)[/latex]. Find the center of the circle.
The center of a circle is the center or midpoint of its diameter. Thus, the midpoint formula will yield the center point.
Try It
Contribute!
Improve this pageLearn More
Candela Citations |
8682 | https://resilienteducator.com/classroom-resources/10-tricks-for-doing-fast-math/ | 10 Math Tricks for Quick Calculations in Your Head | Resilient Educator
Consent
Details
[#IABV2SETTINGS#]
About
We Value Your Privacy.
We use cookies to improve your experience with the website and to help us understand your interactions with the website such as pages you have visited, content you have viewed, and advertisements you have seen. Learn more about who we are, how you can contact us, and how we process personal data in our privacy policy. Please state your consent ID and consent date (both accessible within the cookie consent preferences widget) when you contact us regarding your consent. By using our website, you consent to the use of cookies.
Consent Selection
Necessary
[x]
Preferences
[x]
Statistics
[x]
Marketing
[x]
Show details
Details
Necessary 25- [x] Necessary cookies help make a website usable by enabling basic functions like page navigation and access to secure areas of the website. The website cannot function properly without these cookies.
Adelphic 1Learn more about this providercuUsed to detect if the visitor has accepted the marketing category in the cookie banner. This cookie is necessary for GDPR-compliance of the website. Maximum Storage Duration: 1 yearType: HTTP Cookie
Cookiebot 2Learn more about this provider1.gifUsed to count the number of sessions to the website, necessary for optimizing CMP product delivery. Maximum Storage Duration: SessionType: Pixel Tracker CookieConsentStores the user's cookie consent state for the current domainMaximum Storage Duration: 1 yearType: HTTP Cookie
Google 1Learn more about this providerSome of the data collected by this provider is for the purposes of personalization and measuring advertising effectiveness.
test_cookieUsed to check if the user's browser supports cookies.Maximum Storage Duration: 1 dayType: HTTP Cookie
LinkedIn 2Learn more about this providerbcookiePendingMaximum Storage Duration: 1 yearType: HTTP Cookie li_gcPendingMaximum Storage Duration: 180 daysType: HTTP Cookie
Nativo 1Learn more about this provideropt_outUsed to detect if the visitor has accepted the marketing category in the cookie banner. This cookie is necessary for GDPR-compliance of the website. Maximum Storage Duration: 1 yearType: HTTP Cookie
RTB House 1Learn more about this providertsThis cookie is necessary for the PayPal login-function on the website.Maximum Storage Duration: 1 yearType: HTTP Cookie
Rubicon Project 1Learn more about this providerauditDetermines whether the visitor has accepted the cookie consent box. This ensures that the cookie consent box will not be presented again upon re-entry. Maximum Storage Duration: 1 yearType: HTTP Cookie
TED 1Learn more about this provider__mp_opt_in_out_#Stores the user's cookie consent state for the current domainMaximum Storage Duration: SessionType: HTTP Cookie
admedo.com creative-serving.com qd.admetricspro.com bidswitch.net 4c[x4]Used in order to detect spam and improve the website's security. Maximum Storage Duration: 1 yearType: HTTP Cookie
contextweb.com qd.admetricspro.com 3INGRESSCOOKIE[x3]This cookie is used to distinguish between humans and bots. Maximum Storage Duration: SessionType: HTTP Cookie
qd.admetricspro.com 4checkForPermissionDetermines whether the user has accepted the cookie consent box. Maximum Storage Duration: 1 dayType: HTTP Cookie optoutIdentifies if the visitor has deselected any cookies, trackers or other audience targeting tools.Maximum Storage Duration: SessionType: HTTP Cookie gdprDetermines whether the visitor has accepted the cookie consent box. This ensures that the cookie consent box will not be presented again upon re-entry. Maximum Storage Duration: SessionType: HTTP Cookie usprivacyDetects whether the user has checked the Do Not Sell My Personal Information-button on the website. This feature is a part of the CCPA. Maximum Storage Duration: 1 dayType: HTTP Cookie
resilienteducator.com 1wpEmojiSettingsSupportsThis cookie is part of a bundle of cookies which serve the purpose of content delivery and presentation. The cookies keep the correct state of font, blog/picture sliders, color themes and other website settings.Maximum Storage Duration: SessionType: HTML Local Storage
temu.com twitter.com weebly.com 3__cf_bm[x3]This cookie is used to distinguish between humans and bots. This is beneficial for the website, in order to make valid reports on the use of their website.Maximum Storage Duration: 1 dayType: HTTP Cookie
Preferences 4- [x] Preference cookies enable a website to remember information that changes the way the website behaves or looks, like your preferred language or the region that you are in.
Between Digital 2Learn more about this providerdcNecessary for the functionality of the website's chat-box function. Maximum Storage Duration: 1 yearType: HTTP Cookie ssNecessary for the functionality of the website's chat-box function. Maximum Storage Duration: 1 yearType: HTTP Cookie
LinkedIn 1Learn more about this providerlidcPendingMaximum Storage Duration: 1 dayType: HTTP Cookie
RTB House 1Learn more about this providergUsed in context with the language setting on the website. Facilitates the translation into the preferred language of the visitor.Maximum Storage Duration: 1 yearType: HTTP Cookie
Statistics 11- [x] Statistic cookies help website owners to understand how visitors interact with websites by collecting and reporting information anonymously.
Google 2Learn more about this providerSome of the data collected by this provider is for the purposes of personalization and measuring advertising effectiveness.
_gaRegisters a unique ID that is used to generate statistical data on how the visitor uses the website.Maximum Storage Duration: 2 yearsType: HTTP Cookie ga#Used by Google Analytics to collect data on the number of times a user has visited the website as well as dates for the first and most recent visit. Maximum Storage Duration: 2 yearsType: HTTP Cookie
Hotjar 2Learn more about this providerhjSession#Collects statistics on the visitor's visits to the website, such as the number of visits, average time spent on the website and what pages have been read.Maximum Storage Duration: 1 dayType: HTTP Cookie hjSessionUser#Collects statistics on the visitor's visits to the website, such as the number of visits, average time spent on the website and what pages have been read.Maximum Storage Duration: 1 yearType: HTTP Cookie
Nazwa.pl 1Learn more about this providerbrowser_idUsed to recognise the visitor's browser upon reentry on the website.Maximum Storage Duration: 7 daysType: HTTP Cookie
New Relic 1Learn more about this providerNRBA_SESSIONCollects data on the user’s navigation and behavior on the website. This is used to compile statistical reports and heatmaps for the website owner.Maximum Storage Duration: PersistentType: HTML Local Storage
Quantcast 1Learn more about this providerspRegisters statistical data on users' behaviour on the website. Used for internal analytics by the website operator. Maximum Storage Duration: 3 monthsType: HTTP Cookie
Skimlinks 1Learn more about this providerpx.gifThis cookie is used to determine if the visitor has any adblocker software in their browser – this information can be used to make website content inaccessible to visitors if the website is financed with third-party advertisement.Maximum Storage Duration: SessionType: Pixel Tracker
TED 1Learn more about this providermuxDataThis cookie is used in context with a video-player, so that if the visitor is interrupted while viewing video content, the cookie remembers where to start the video when the visitor reloads the video.Maximum Storage Duration: 1 yearType: HTTP Cookie
qd.admetricspro.com 2crumThis cookie is used to identify the frequency of visits and how long the visitor is on the website.Maximum Storage Duration: SessionType: Pixel Tracker logUsed to distinguish between internal and external visitors to the website, in order to obtain more concise statistical data regarding the use of the website. Maximum Storage Duration: SessionType: Pixel Tracker
Marketing 187- [x] Marketing cookies are used to track visitors across websites. The intention is to display ads that are relevant and engaging for the individual user and thereby more valuable for publishers and third-party advertisers.
1rx.io unrulymedia.com 2_rxuuid[x2]Sets a unique ID for the visitor, that allows third party advertisers to target the visitor with relevant advertisement. This pairing service is provided by third party advertisement hubs, which facilitates real-time bidding for advertisers.Maximum Storage Duration: 1 yearType: HTTP Cookie
3lift.com media.net rubiconproject.com 3receive-cookie-deprecation[x3]Collects information on user behaviour on multiple websites. This information is used in order to optimize the relevance of advertisement on the website.Maximum Storage Duration: 3 monthsType: HTTP Cookie
Adform 1Learn more about this providerCUsed to check if the user's browser supports cookies.Maximum Storage Duration: 30 daysType: HTTP Cookie
Adition Technologies 1Learn more about this providerUserID1This cookie is used to collect information on a visitor. This information will become an ID string with information on a specific visitor – ID information strings can be used to target groups with similar preferences, or can be used by third-party domains or ad-exchanges.Maximum Storage Duration: 3 monthsType: HTTP Cookie
Adman Media 2Learn more about this providerac_rPendingMaximum Storage Duration: 7 daysType: HTTP Cookie admtrUsed in context with video-advertisement. The cookie limits the number of times a user is shown the same advertisement. The cookie is also used to ensure relevance of the video-advertisement to the specific user. Maximum Storage Duration: 7 daysType: HTTP Cookie
Adobe Inc.3Learn more about this providerdemdexVia a unique ID that is used for semantic content analysis, the user's navigation on the website is registered and linked to offline data from surveys and similar registrations to display targeted ads.Maximum Storage Duration: 180 daysType: HTTP Cookie dpmSets a unique ID for the visitor, that allows third party advertisers to target the visitor with relevant advertisement. This pairing service is provided by third party advertisement hubs, which facilitates real-time bidding for advertisers.Maximum Storage Duration: 180 daysType: HTTP Cookie everest_g_v2Used for targeted ads and to document efficacy of each individual ad.Maximum Storage Duration: 1 yearType: HTTP Cookie
Adotmob 1Learn more about this providerpartnersPresents the user with relevant content and advertisement. The service is provided by third-party advertisement hubs, which facilitate real-time bidding for advertisers.Maximum Storage Duration: 180 daysType: HTTP Cookie
Amazon 2Learn more about this providerad-idUsed by Amazon Advertising to register user actions and target content on the website based on ad clicks on a different website.Maximum Storage Duration: 7 monthsType: HTTP Cookie ad-privacyUsed by Amazon Advertising to register user actions and target content on the website based on ad clicks on a different website.Maximum Storage Duration: 400 daysType: HTTP Cookie
AntVoice 2Learn more about this providerav-midPresents the user with relevant content and advertisement. The service is provided by third-party advertisement hubs, which facilitate real-time bidding for advertisers.Maximum Storage Duration: 400 daysType: HTTP Cookie av-tp-bswPresents the user with relevant content and advertisement. The service is provided by third-party advertisement hubs, which facilitate real-time bidding for advertisers.Maximum Storage Duration: 2 daysType: HTTP Cookie
Appier 1Learn more about this provider_auidIdentifies all devices the user uses with the purpose of displaying targeted ads on all devices.Maximum Storage Duration: 1 yearType: HTTP Cookie
Beeswax 2Learn more about this providerbitoSets a unique ID for the visitor, that allows third party advertisers to target the visitor with relevant advertisement. This pairing service is provided by third party advertisement hubs, which facilitates real-time bidding for advertisers.Maximum Storage Duration: 13 monthsType: HTTP Cookie bitoIsSecurePresents the user with relevant content and advertisement. The service is provided by third-party advertisement hubs, which facilitate real-time bidding for advertisers.Maximum Storage Duration: 13 monthsType: HTTP Cookie
Between Digital 1Learn more about this providerutCollects data on user behaviour and interaction in order to optimize the website and make advertisement on the website more relevant. Maximum Storage Duration: 1 yearType: HTTP Cookie
Bidtheatre 2Learn more about this provider__ktpctPendingMaximum Storage Duration: 1 dayType: HTTP Cookie __kuidThis cookie is used to collect information on a visitor. This information will become an ID string with information on a specific visitor – ID information strings can be used to target groups with similar preferences, or can be used by third-party domains or ad-exchanges.Maximum Storage Duration: 7 daysType: HTTP Cookie
Blismedia.com 1Learn more about this providerbCollects statistical data related to the user's website visits, such as the number of visits, average time spent on the website and what pages have been loaded. The purpose is to segment the website's users according to factors such as demographics and geographical location, in order to enable media and marketing agencies to structure and understand their target groups to enable customised online advertising.Maximum Storage Duration: 1 yearType: HTTP Cookie
Brand-display.com 1Learn more about this providerknxqDetermines when the visitor last visited the different subpages on the website, as well as sets a timestamp for when the session started.Maximum Storage Duration: 400 daysType: HTTP Cookie
Casale Media 3Learn more about this providerCMIDCollects visitor data related to the user's visits to the website, such as the number of visits, average time spent on the website and what pages have been loaded, with the purpose of displaying targeted ads.Maximum Storage Duration: 1 yearType: HTTP Cookie CMPROCollects data on visitor behaviour from multiple websites, in order to present more relevant advertisement - This also allows the website to limit the number of times that they are shown the same advertisement. Maximum Storage Duration: 3 monthsType: HTTP Cookie CMPSCollects visitor data related to the user's visits to the website, such as the number of visits, average time spent on the website and what pages have been loaded, with the purpose of displaying targeted ads.Maximum Storage Duration: 3 monthsType: HTTP Cookie
Crimtan 1Learn more about this providercid_#Collects unidentifiable data that is sent to an unidentifiable source. The source's identity is kept secret by the company, Whois Privacy Protection Service, Inc.Maximum Storage Duration: 1 yearType: HTTP Cookie
Exponential 1Learn more about this providerANON_IDCollects data on user visits to the website, such as what pages have been accessed. The registered data is used to categorise the user's interest and demographic profiles in terms of resales for targeted marketing.Maximum Storage Duration: 3 monthsType: HTTP Cookie
Freewheel 1Learn more about this provideruid-bp-#PendingMaximum Storage Duration: 30 daysType: HTTP Cookie
Google 1Learn more about this providerSome of the data collected by this provider is for the purposes of personalization and measuring advertising effectiveness.
NIDRegisters a unique ID that identifies a returning user's device. The ID is used for targeted ads.Maximum Storage Duration: 6 monthsType: HTTP Cookie
Gumgum 1Learn more about this providervstPresents the user with relevant content and advertisement. The service is provided by third-party advertisement hubs, which facilitate real-time bidding for advertisers.Maximum Storage Duration: 1 yearType: HTTP Cookie
ID5 1Learn more about this providerid5Sets a unique ID for the visitor, that allows third party advertisers to target the visitor with relevant advertisement. This pairing service is provided by third party advertisement hubs, which facilitates real-time bidding for advertisers.Maximum Storage Duration: 3 monthsType: HTTP Cookie
IQM 1Learn more about this provideriqm.retarget.uidSets a unique ID for the visitor, that allows third party advertisers to target the visitor with relevant advertisement. This pairing service is provided by third party advertisement hubs, which facilitates real-time bidding for advertisers.Maximum Storage Duration: 15 daysType: HTTP Cookie
LoopMe 1Learn more about this providerviewer_tokenUsed to track visitors on multiple websites, in order to present relevant advertisement based on the visitor's preferences. Maximum Storage Duration: 3 monthsType: HTTP Cookie
Media.net 1Learn more about this provider[empty name]PendingMaximum Storage Duration: SessionType: HTTP Cookie
N.Rich 1Learn more about this provider_nauidCollects data on visitor behaviour from multiple websites, in order to present more relevant advertisement - This also allows the website to limit the number of times that they are shown the same advertisement. Maximum Storage Duration: 400 daysType: HTTP Cookie
Outbrain 1Learn more about this providerobuidSets a unique ID for the visitor, that allows third party advertisers to target the visitor with relevant advertisement. This pairing service is provided by third party advertisement hubs, which facilitates real-time bidding for advertisers. Maximum Storage Duration: 3 monthsType: HTTP Cookie
Pubmatic 2Learn more about this providerKCCHRegisters if the PubMatic partner-cookie has been set in the user's browser.Maximum Storage Duration: 1 dayType: HTTP Cookie KTPCACOOKIERegisters a unique ID that identifies the user's device during return visits across websites that use the same ad network. The ID is used to allow targeted ads.Maximum Storage Duration: 3 monthsType: HTTP Cookie
Pulsepoint 2Learn more about this providerpb_rtb_ev_partPendingMaximum Storage Duration: 1 yearType: HTTP Cookie VPPendingMaximum Storage Duration: 1 yearType: HTTP Cookie
Quantcast 1Learn more about this providermcCollects data on the user's visits to the website, such as what pages have been loaded. The registered data is used for targeted ads.Maximum Storage Duration: 13 monthsType: HTTP Cookie
Rubicon Project 4Learn more about this provideraudit_pPendingMaximum Storage Duration: 1 yearType: HTTP Cookie khaosRegisters user data, such as IP address, geographical location, visited websites, and what ads the user has clicked, with the purpose of optimising ad display based on the user's movement on websites that use the same ad network.Maximum Storage Duration: 1 yearType: HTTP Cookie khaos_pPendingMaximum Storage Duration: 1 yearType: HTTP Cookie tap.phpRegisters data on visitor such as IP addresses, geographical location and advertisement interaction. This information is used optimize the advertisement on websites that make use of Rubicon Project-services. Maximum Storage Duration: SessionType: Pixel Tracker
Simpli.fi 1Learn more about this providersuidCollects information on user preferences and/or interaction with web-campaign content - This is used on CRM-campaign-platform used by website owners for promoting events or products.Maximum Storage Duration: 1 yearType: HTTP Cookie
Smart 2Learn more about this providercsyncOptimises ad display based on the user's movement combined and various advertiser bids for displaying user ads.Maximum Storage Duration: 30 daysType: HTTP Cookie pidRegisters a unique ID that identifies a returning user's device. The ID is used for targeted ads.Maximum Storage Duration: 3 monthsType: HTTP Cookie
Sonobi 1Learn more about this providerHAPLB8GPendingMaximum Storage Duration: SessionType: HTTP Cookie
Sovrn 4Learn more about this providerljtrtb#Collects data related to reader interests, context, demographics and other information on behalf of the Lijit platform with the purpose of finding interested users on websites with related content.Maximum Storage Duration: 1 yearType: HTTP Cookie ljt_readerCollects data related to reader interests, context, demographics and other information on behalf of the Lijit platform with the purpose of finding interested users on websites with related content.Maximum Storage Duration: 1 yearType: HTTP Cookie ljtrtbCollects data related to reader interests, context, demographics and other information on behalf of the Lijit platform with the purpose of finding interested users on websites with related content.Maximum Storage Duration: 1 yearType: HTTP Cookie mergePresents the user with relevant content and advertisement. The service is provided by third-party advertisement hubs, which facilitate real-time bidding for advertisers.Maximum Storage Duration: SessionType: Pixel Tracker
TED 2Learn more about this providermp_#_mixpanel[x2]PendingMaximum Storage Duration: SessionType: HTTP Cookie
TapTap 1Learn more about this providerSONATA_IDUsed to track the visitor across multiple devices including TV. This is done in order to re-target the visitor through multiple channels. Maximum Storage Duration: 19 daysType: HTTP Cookie
Tapad 4Learn more about this provideridsync/ex/receive/checkThis cookie is set by the audience manager of the website to determine the time and frequencies of visitor data synchronization - cookie data synchronization is used to synchronize and gather visitor data from several websites.Maximum Storage Duration: SessionType: Pixel Tracker TapAd_3WAY_SYNCSUsed for data-synchronization with advertisement networks.Maximum Storage Duration: 2 monthsType: HTTP Cookie TapAd_DIDUsed to determine what type of devices (smartphones, tablets, computers, TVs etc.) is used by a user.Maximum Storage Duration: 2 monthsType: HTTP Cookie TapAd_TSUsed to determine what type of devices (smartphones, tablets, computers, TVs etc.) is used by a user.Maximum Storage Duration: 2 monthsType: HTTP Cookie
Triplelift 2Learn more about this providertluidThis cookie is used to identify the visitor and optimize ad-relevance by collecting visitor data from multiple websites – this exchange of visitor data is normally provided by a third-party data-center or ad-exchange.Maximum Storage Duration: 3 monthsType: HTTP Cookie tluidpPendingMaximum Storage Duration: 3 monthsType: HTTP Cookie
Twitter Inc.1Learn more about this provideri/jot/embedsSets a unique ID for the visitor, that allows third party advertisers to target the visitor with relevant advertisement. This pairing service is provided by third party advertisement hubs, which facilitates real-time bidding for advertisers.Maximum Storage Duration: SessionType: Pixel Tracker
WP 1Learn more about this providerstatidSets a unique ID for the visitor, that allows third party advertisers to target the visitor with relevant advertisement. This pairing service is provided by third party advertisement hubs, which facilitates real-time bidding for advertisers.Maximum Storage Duration: 400 daysType: HTTP Cookie
Xaxis 1Learn more about this providert/v2/syncUsed for data-synchronization with advertisement networks.Maximum Storage Duration: SessionType: Pixel Tracker
Yahoo 1Learn more about this providersync/rubicon/#PendingMaximum Storage Duration: SessionType: Pixel Tracker
YouTube 23Learn more about this provider__mpq_#_#Collects data on user behaviour and interaction in order to optimize the website and make advertisement on the website more relevant. Maximum Storage Duration: PersistentType: HTML Local Storage #-#Used to track user’s interaction with embedded content.Maximum Storage Duration: SessionType: HTML Local Storage __Secure-ROLLOUT_TOKENPendingMaximum Storage Duration: 180 daysType: HTTP Cookie __Secure-YECStores the user's video player preferences using embedded YouTube videoMaximum Storage Duration: SessionType: HTTP Cookie iU5q-!O9@$Registers a unique ID to keep statistics of what videos from YouTube the user has seen.Maximum Storage Duration: SessionType: HTML Local Storage LAST_RESULT_ENTRY_KEYUsed to track user’s interaction with embedded content.Maximum Storage Duration: SessionType: HTTP Cookie LogsDatabaseV2:V#||LogsRequestsStoreUsed to track user’s interaction with embedded content.Maximum Storage Duration: PersistentType: IndexedDB nextIdUsed to track user’s interaction with embedded content.Maximum Storage Duration: SessionType: HTTP Cookie remote_sidNecessary for the implementation and functionality of YouTube video-content on the website. Maximum Storage Duration: SessionType: HTTP Cookie requestsUsed to track user’s interaction with embedded content.Maximum Storage Duration: SessionType: HTTP Cookie ServiceWorkerLogsDatabase#SWHealthLogNecessary for the implementation and functionality of YouTube video-content on the website. Maximum Storage Duration: PersistentType: IndexedDB TESTCOOKIESENABLEDUsed to track user’s interaction with embedded content.Maximum Storage Duration: 1 dayType: HTTP Cookie VISITOR_INFO1_LIVETries to estimate the users' bandwidth on pages with integrated YouTube videos.Maximum Storage Duration: 180 daysType: HTTP Cookie YSCRegisters a unique ID to keep statistics of what videos from YouTube the user has seen.Maximum Storage Duration: SessionType: HTTP Cookie ytidb::LAST_RESULT_ENTRY_KEYStores the user's video player preferences using embedded YouTube videoMaximum Storage Duration: PersistentType: HTML Local Storage YtIdbMeta#databasesUsed to track user’s interaction with embedded content.Maximum Storage Duration: PersistentType: IndexedDB yt-remote-cast-availableStores the user's video player preferences using embedded YouTube videoMaximum Storage Duration: SessionType: HTML Local Storage yt-remote-cast-installedStores the user's video player preferences using embedded YouTube videoMaximum Storage Duration: SessionType: HTML Local Storage yt-remote-connected-devicesStores the user's video player preferences using embedded YouTube videoMaximum Storage Duration: PersistentType: HTML Local Storage yt-remote-device-idStores the user's video player preferences using embedded YouTube videoMaximum Storage Duration: PersistentType: HTML Local Storage yt-remote-fast-check-periodStores the user's video player preferences using embedded YouTube videoMaximum Storage Duration: SessionType: HTML Local Storage yt-remote-session-appStores the user's video player preferences using embedded YouTube videoMaximum Storage Duration: SessionType: HTML Local Storage yt-remote-session-nameStores the user's video player preferences using embedded YouTube videoMaximum Storage Duration: SessionType: HTML Local Storage
Zemanta 1Learn more about this providerzuidSets a unique ID for the visitor, that allows third party advertisers to target the visitor with relevant advertisement. This pairing service is provided by third party advertisement hubs, which facilitates real-time bidding for advertisers.Maximum Storage Duration: 3 monthsType: HTTP Cookie
Zeta Global 3Learn more about this providereudRegisters user data, such as IP address, geographical location, visited websites, and what ads the user has clicked, with the purpose of optimising ad display based on the user's movement on websites that use the same ad network.Maximum Storage Duration: 1 yearType: HTTP Cookie rudRegisters user data, such as IP address, geographical location, visited websites, and what ads the user has clicked, with the purpose of optimising ad display based on the user's movement on websites that use the same ad network.Maximum Storage Duration: 1 yearType: HTTP Cookie rudsRegisters user data, such as IP address, geographical location, visited websites, and what ads the user has clicked, with the purpose of optimising ad display based on the user's movement on websites that use the same ad network.Maximum Storage Duration: SessionType: HTTP Cookie
adform.net adotmob.com criteo.com turn.com 4uid[x4]Registers a unique user ID that recognises the user's browser when visiting websites that use the same ad network. The purpose is to optimise display of ads based on the user's movements and various ad providers' bids for displaying user ads.Maximum Storage Duration: 2 monthsType: HTTP Cookie
admedo.com betweendigital.com bidswitch.net company-target.com creative-serving.com qd.admetricspro.com 6tuuid[x6]Collects visitor data related to the user's visits to the website, such as the number of visits, average time spent on the website and what pages have been loaded, with the purpose of displaying targeted ads.Maximum Storage Duration: 1 yearType: HTTP Cookie
admedo.com bidswitch.net company-target.com creative-serving.com qd.admetricspro.com 5tuuid_lu[x5]Contains a unique visitor ID, which allows Bidswitch.com to track the visitor across multiple websites. This allows Bidswitch to optimize advertisement relevance and ensure that the visitor does not see the same ads multiple times. Maximum Storage Duration: 1 yearType: HTTP Cookie
adotmob.com mathtag.com 2uuid[x2]This cookie is used to optimize ad relevance by collecting visitor data from multiple websites – this exchange of visitor data is normally provided by a third-party data-center or ad-exchange.Maximum Storage Duration: 180 daysType: HTTP Cookie
adx.opera.com stickyadstv.com 2UID[x2]Unique user ID that recognizes the user on returning visitsMaximum Storage Duration: 1 yearType: HTTP Cookie
aniview.com qd.admetricspro.com 3aniC[x3]Used in context with video-advertisement. The cookie limits the number of times a user is shown the same advertisement. The cookie is also used to ensure relevance of the video-advertisement to the specific user. Maximum Storage Duration: 30 daysType: HTTP Cookie
criteo.com qd.admetricspro.com 2cto_bundle[x2]Presents the user with relevant content and advertisement. The service is provided by third-party advertisement hubs, which facilitate real-time bidding for advertisers.Maximum Storage Duration: 1 yearType: HTTP Cookie
eskimi.com 1__eDIdCollects data on visitors' behaviour and interaction - This is used to optimize the website and make advertisement on the website more relevant. Maximum Storage Duration: 30 daysType: HTTP Cookie
mediago.io 1__mguid_Registers a unique ID that identifies a returning user's device. The ID is used for targeted ads.Maximum Storage Duration: 1 yearType: HTTP Cookie
omnitagjs.com 1ayl_visitorThis cookie registers data on the visitor. The information is used to optimize advertisement relevance. Maximum Storage Duration: 30 daysType: HTTP Cookie
primis.tech 1csuuidCollects information on user behaviour on multiple websites. This information is used in order to optimize the relevance of advertisement on the website.Maximum Storage Duration: 25 daysType: HTTP Cookie
qd.admetricspro.com 48dsp_match/275PendingMaximum Storage Duration: SessionType: Pixel Tracker getuidThis cookie is set by the audience manager of the website to determine the time and frequencies of visitor data synchronization - cookie data synchronization is used to synchronize and gather visitor data from several websites.Maximum Storage Duration: SessionType: Pixel Tracker sync[x4]Collects data on user behaviour and interaction in order to optimize the website and make advertisement on the website more relevant. Maximum Storage Duration: SessionType: Pixel Tracker track/cmf/genericPresents the user with relevant content and advertisement. The service is provided by third-party advertisement hubs, which facilitate real-time bidding for advertisers.Maximum Storage Duration: SessionType: Pixel Tracker track/cmf/rubiconPresents the user with relevant content and advertisement. The service is provided by third-party advertisement hubs, which facilitate real-time bidding for advertisers.Maximum Storage Duration: SessionType: Pixel Tracker track/usersyncPendingMaximum Storage Duration: SessionType: Pixel Tracker bsw_origin_initCollects visitor data related to the user's visits to the website, such as the number of visits, average time spent on the website and what pages have been loaded, with the purpose of displaying targeted ads.Maximum Storage Duration: SessionType: HTTP Cookie custom_dataCollects data on user behaviour and interaction in order to optimize the website and make advertisement on the website more relevant. Maximum Storage Duration: 1 dayType: HTTP Cookie pixel/cookiesyncDetects whether partner data synchronization is functioning and currently running - This function sends user data between third-party advertisement companies for the purpose of targeted advertisements. Maximum Storage Duration: SessionType: Pixel Tracker pixel/cookiesyncredirUsed for data-synchronization with advertisement networks.Maximum Storage Duration: SessionType: Pixel Tracker rumCollects data related to the user's visits to the website, such as the number of visits, average time spent on the website and what pages have been loaded, with the purpose of displaying targeted ads.Maximum Storage Duration: SessionType: Pixel Tracker cm/index/tp_outSets a unique ID for the visitor, that allows third party advertisers to target the visitor with relevant advertisement. This pairing service is provided by third party advertisement hubs, which facilitates real-time bidding for advertisers.Maximum Storage Duration: SessionType: Pixel Tracker demconf.jpgSets a unique ID for the visitor, that allows third party advertisers to target the visitor with relevant advertisement. This pairing service is provided by third party advertisement hubs, which facilitates real-time bidding for advertisers. Maximum Storage Duration: SessionType: Pixel Tracker ct/upi/pid/gjIEMT18PendingMaximum Storage Duration: SessionType: Pixel Tracker ct/upi/pid/ZMAwryCIPendingMaximum Storage Duration: SessionType: Pixel Tracker pagead/gen_204Collects data on visitor behaviour from multiple websites, in order to present more relevant advertisement - This also allows the website to limit the number of times that they are shown the same advertisement. Maximum Storage Duration: SessionType: Pixel Tracker csiCollects data on visitors' preferences and behaviour on the website - This information is used make content and advertisement more relevant to the specific visitor. Maximum Storage Duration: SessionType: Pixel Tracker callbackCollects data on visitor behaviour from multiple websites, in order to present more relevant advertisement - This also allows the website to limit the number of times that they are shown the same advertisement. Maximum Storage Duration: SessionType: HTTP Cookie carPresents the user with relevant content and advertisement. The service is provided by third-party advertisement hubs, which facilitate real-time bidding for advertisers.Maximum Storage Duration: SessionType: HTTP Cookie cfSets a unique ID for the visitor, that allows third party advertisers to target the visitor with relevant advertisement. This pairing service is provided by third party advertisement hubs, which facilitates real-time bidding for advertisers.Maximum Storage Duration: SessionType: HTTP Cookie cipPresents the user with relevant content and advertisement. The service is provided by third-party advertisement hubs, which facilitate real-time bidding for advertisers.Maximum Storage Duration: SessionType: HTTP Cookie cnacPresents the user with relevant content and advertisement. The service is provided by third-party advertisement hubs, which facilitate real-time bidding for advertisers.Maximum Storage Duration: SessionType: HTTP Cookie gppPendingMaximum Storage Duration: SessionType: HTTP Cookie visitor/syncThis cookie is used to assign specific visitors into segments, this segmentation is based on visitor behavior on the website - the segmentation can be used to target larger groups.Maximum Storage Duration: SessionType: Pixel Tracker w/1.0/cmPresents the user with relevant content and advertisement. The service is provided by third-party advertisement hubs, which facilitate real-time bidding for advertisers.Maximum Storage Duration: SessionType: Pixel Tracker suid/101967PendingMaximum Storage Duration: SessionType: Pixel Tracker PubMatic_USPPresents the user with relevant content and advertisement. The service is provided by third-party advertisement hubs, which facilitate real-time bidding for advertisers.Maximum Storage Duration: PersistentType: HTML Local Storage cto_bundlePresents the user with relevant content and advertisement. The service is provided by third-party advertisement hubs, which facilitate real-time bidding for advertisers.Maximum Storage Duration: PersistentType: HTML Local Storage id5id_Collects data on visitors' preferences and behaviour on the website - This information is used make content and advertisement more relevant to the specific visitor. Maximum Storage Duration: PersistentType: HTML Local Storage pixel/p-EtBqU4Lj3YbAv.gifPendingMaximum Storage Duration: SessionType: Pixel Tracker pixel/p-zLwwakwy-hZw3.gifPendingMaximum Storage Duration: SessionType: Pixel Tracker pbjs-unifiedidPresents the user with relevant content and advertisement. The service is provided by third-party advertisement hubs, which facilitate real-time bidding for advertisers.Maximum Storage Duration: 2 monthsType: HTTP Cookie [empty name]PendingMaximum Storage Duration: SessionType: Pixel Tracker fcapPresents the user with relevant content and advertisement. The service is provided by third-party advertisement hubs, which facilitate real-time bidding for advertisers.Maximum Storage Duration: PersistentType: HTML Local Storage khaosCollects data on user behaviour and interaction in order to optimize the website and make advertisement on the website more relevant. Maximum Storage Duration: PersistentType: HTML Local Storage puxCollects data on user behaviour and interaction in order to optimize the website and make advertisement on the website more relevant. Maximum Storage Duration: PersistentType: HTML Local Storage adagioPresents the user with relevant content and advertisement. The service is provided by third-party advertisement hubs, which facilitate real-time bidding for advertisers.Maximum Storage Duration: PersistentType: HTML Local Storage suid_legacyCollects information on user preferences and/or interaction with web-campaign content - This is used on CRM-campaign-platform used by website owners for promoting events or products.Maximum Storage Duration: 1 yearType: HTTP Cookie ANON_ID_oldCollects data about the user's visit to the site, such as the number of returning visits and which pages are read. The purpose is to deliver targeted ads.Maximum Storage Duration: 3 monthsType: HTTP Cookie sync/casalePendingMaximum Storage Duration: SessionType: Pixel Tracker sync/gumgumPendingMaximum Storage Duration: SessionType: Pixel Tracker sync/iponwebPendingMaximum Storage Duration: SessionType: Pixel Tracker usersync/adyoulike/PendingMaximum Storage Duration: SessionType: Pixel Tracker usersync/index/PendingMaximum Storage Duration: SessionType: Pixel Tracker usersync/smart/PendingMaximum Storage Duration: SessionType: Pixel Tracker
qd.admetricspro.com springserve.com 2usersync[x2]Sets a unique ID for the visitor, that allows third party advertisers to target the visitor with relevant advertisement. This pairing service is provided by third party advertisement hubs, which facilitates real-time bidding for advertisers.Maximum Storage Duration: SessionType: Pixel Tracker
qd.admetricspro.com stackadapt.com 12sa-user-id[x4]Used to track visitors on multiple websites, in order to present relevant advertisement based on the visitor's preferences. Maximum Storage Duration: SessionType: HTTP Cookie sa-user-id-v2[x4]Used to track visitors on multiple websites, in order to present relevant advertisement based on the visitor's preferences. Maximum Storage Duration: SessionType: HTTP Cookie sa-user-id-v3[x4]Collects data on the user across websites - This data is used to make advertisement more relevant.Maximum Storage Duration: SessionType: HTTP Cookie
servenobid.com 1pid_#Presents the user with relevant content and advertisement. The service is provided by third-party advertisement hubs, which facilitate real-time bidding for advertisers.Maximum Storage Duration: 7 daysType: HTTP Cookie
Unclassified 25 Unclassified cookies are cookies that we are in the process of classifying, together with the providers of individual cookies.
Aniview 2Learn more about this provider1_C_24[x2]PendingMaximum Storage Duration: 10 daysType: HTTP Cookie
IQM 1Learn more about this providerequativPendingMaximum Storage Duration: 30 daysType: HTTP Cookie
TED 1Learn more about this providermp_uuidPendingMaximum Storage Duration: SessionType: HTTP Cookie
YouTube 9Learn more about this providerautoplay_preferencePendingMaximum Storage Duration: PersistentType: HTML Local Storage mixpanelBrowserDb#mixpanelRecordingEventsPendingMaximum Storage Duration: PersistentType: IndexedDB mixpanelBrowserDb#mixpanelRecordingRegistryPendingMaximum Storage Duration: PersistentType: IndexedDB mp_gen_new_tab_id_mixpanel_cefc84a7a812fb3bb40d6efadb6b3163PendingMaximum Storage Duration: SessionType: HTML Local Storage mp_tab_id_mixpanel_cefc84a7a812fb3bb40d6efadb6b3163PendingMaximum Storage Duration: SessionType: HTML Local Storage muted_statePendingMaximum Storage Duration: SessionType: HTML Local Storage playback_rate_statePendingMaximum Storage Duration: SessionType: HTML Local Storage show-browser-update-bannerPendingMaximum Storage Duration: SessionType: HTML Local Storage subtitle_preferencePendingMaximum Storage Duration: PersistentType: HTML Local Storage
bedrockplatform.bid 1bedrockplatform-usersyncPendingMaximum Storage Duration: 30 daysType: HTTP Cookie
inmobi.com 1TEST-COOKIEPendingMaximum Storage Duration: 1 yearType: HTTP Cookie
public.servenobid.com 1nbxcPendingMaximum Storage Duration: 8516 monthsType: HTTP Cookie
qd.admetricspro.com 6ht/htw-pixel.gifPendingMaximum Storage Duration: SessionType: Pixel Tracker av_sd_62f53b2c7850d0786f227f64PendingMaximum Storage Duration: PersistentType: HTML Local Storage id5id_cstPendingMaximum Storage Duration: PersistentType: HTML Local Storage amp_pieogPendingMaximum Storage Duration: 7 daysType: HTTP Cookie pbjs-unifiedid_cstPendingMaximum Storage Duration: 2 monthsType: HTTP Cookie api_uidPendingMaximum Storage Duration: 1 yearType: HTTP Cookie
resilienteducator.com 2hotchalk.sourcePendingMaximum Storage Duration: SessionType: HTTP Cookie pdb-sessPendingMaximum Storage Duration: SessionType: HTTP Cookie
ymmobi.com 1ym_user_cookiePendingMaximum Storage Duration: 1 yearType: HTTP Cookie
Cross-domain consent[#BULK_CONSENT_DOMAINS_COUNT#] [#BULK_CONSENT_TITLE#]
List of domains your consent applies to: [#BULK_CONSENT_DOMAINS#]
Cookie declaration last updated on 8/31/25 by Cookiebot
[#IABV2_TITLE#]
[#IABV2_BODY_INTRO#]
[#IABV2_BODY_LEGITIMATE_INTEREST_INTRO#]
[#IABV2_BODY_PREFERENCE_INTRO#]
[#IABV2_LABEL_PURPOSES#]
[#IABV2_BODY_PURPOSES_INTRO#]
[#IABV2_BODY_PURPOSES#]
[#IABV2_LABEL_FEATURES#]
[#IABV2_BODY_FEATURES_INTRO#]
[#IABV2_BODY_FEATURES#]
[#IABV2_LABEL_PARTNERS#]
[#IABV2_BODY_PARTNERS_INTRO#]
[#IABV2_BODY_PARTNERS#]
About
Cookies are small text files that can be used by websites to make a user's experience more efficient. The law states that we can store cookies on your device if they are strictly necessary for the operation of this site. For all other types of cookies, we need your permission. This site uses different types of cookies. Some cookies are placed by third-party services that appear on our pages. You can at any time change or withdraw your consent from the cookie consent preferences widget on our website.
[x] Do not sell or share my personal information
Necessary Cookies Only Allow selection Customize Accept All Cookies
MENU
Menu
Toolkits
Toolkits dedicated to specific teaching subjects
Toolkits
#### ChatGPT for Teachers
#### Trauma-Informed Practices in Schools
#### Teacher Well-Being
#### Cultivating Diversity, Equity, & Inclusion
#### Integrating Technology in the Classroom
#### Social-Emotional Development
#### COVID-19 Resources
#### Invest in Resilience: Summer Toolkit
#### Civics & Resilience
#### All Toolkits
Careers
Discover various career paths and PD in the education industry
Careers
#### Degree Programs
#### Trauma-Informed Professional Development
#### Teacher Licensure & Certification
#### How to Become - Career Information
Blog
Articles that address various teaching and self-care topics
Blog
#### Classroom Management
#### Instructional Design
#### Lifestyle & Self-Care
#### Online Higher Ed Teaching
#### Current Events
Follow us
10 Ways to Do Fast Math: Tricks and Tips for Doing Math in Your Head
The Editorial Team November 21, 2012
Share
Article continues here
You don’t have to be a math teacher to know that a lot of students—and likely a lot of parents (it’s been awhile!)—are intimidated by math problems, especially if they involve large numbers. Learning techniques on how to do math quickly can help students develop greater confidence in math, improve math skills and understanding,and excel in advanced courses.
If it’s your job to teach those, here’s a great refresher.
Fast math tricks infographic
10 tricks for doing fast math
Here are 10 fast math strategies students (and adults!) can use to do math in their heads. Once these strategies are mastered, students should be able to accurately and confidently solve math problems that they once feared solving.
1. Adding large numbers
Adding large numbers just in your head can be difficult. This method shows how to simplify this process by making all the numbers a multiple of 10. Here is an example:
644 + 238
While these numbers are hard to contend with, rounding them up will make them more manageable. So, 644 becomes 650 and 238 becomes 240.
Now, add 650 and 240 together. The total is 890. To find the answer to the original equation, it must be determined how much we added to the numbers to round them up.
650 – 644 = 6 and 240 – 238 = 2
Now, add 6 and 2 together for a total of 8
To find the answer to the original equation, 8 must be subtracted from the 890.
890 – 8 = 882
So the answer to 644 +238 is 882.
2. Subtracting from 1,000
Here’s a basic rule to subtract a large number from 1,000: Subtract every number except the last from 9 and subtract the final number from 10
For example:
1,000 – 556
Step 1: Subtract 5 from 9 = 4
Step 2: Subtract 5 from 9 = 4
Step 3: Subtract 6 from 10 = 4
The answer is 444.
3. Multiplying 5 times any number
When multiplying the number 5 by an even number, there is a quick way to find the answer.
For example, 5 x 4 =
Step 1: Take the number being multiplied by 5 and cut it in half, this makes the number 4 become the number 2.
Step 2: Add a zero to the number to find the answer. In this case, the answer is 20.
5 x 4 = 20
When multiplying an odd number times 5, the formula is a bit different.
For instance, consider 5 x 3.
Step 1: Subtract one from the number being multiplied by 5, in this instance the number 3 becomes the number 2.
Step 2: Now halve the number 2, which makes it the number 1. Make 5 the last digit. The number produced is 15, which is the answer.
5 x 3 = 15
4. Division tricks
Here’s a quick way to know when a number can be evenly divided by these certain numbers:
10 if the number ends in 0
9 when the digits are added together and the total is evenly divisible by 9
8 if the last three digits are evenly divisible by 8 or are 000
6 if it is an even number and when the digits are added together the answer is evenly divisible by 3
5 if it ends in a 0 or 5
4 if it ends in 00 or a two digit number that is evenly divisible by 4
3 when the digits are added together and the result is evenly divisible by the number 3
2 if it ends in 0, 2, 4, 6, or 8
5. Multiplying by 9
This is an easy method that is helpful for multiplying any number by 9. Here is how it works:
Let’s use the example of 9 x 3.
Step 1: Subtract 1 from the number that is being multiplied by 9.
3 – 1 = 2
The number 2 is the first number in the answer to the equation.
Step 2: Subtract that number from the number 9.
9 – 2 = 7
The number 7 is the second number in the answer to the equation.
So, 9 x 3= 27
6. 10 and 11 times tricks
The trick to multiplying any number by 10 is to add a zero to the end of the number. For example, 62 x 10 = 620.
There is also an easy trick for multiplying any two-digit number by 11. Here it is:
11 x 25
Take the original two-digit number and put a space between the digits. In this example, that number is 25.
2_5
Now add those two numbers together and put the result in the center:
2_(2 + 5)_5
2_7_5
The answer to 11 x 25 is 275.
If the numbers in the center add up to a number with two digits, insert the second number and add 1 to the first one. Here is an example for the equation 11 x 88
8_(8 +8)_8
(8 + 1)_6_8
9_6_8
There is the answer to 11 x 88: 968
7. Percentage
Finding a percentage of a number can be somewhat tricky, but thinking about it in the right terms makes it much easier to understand. For instance, to find out what 5% of 235 is, follow this method:
Step 1: Move the decimal point over by one place, 235 becomes 23.5.
Step 2: Divide 23.5 by the number 2, the answer is 11.75. That is also the answer to the original equation.
8. Quickly square a two-digit number that ends in 5
Let’s use the number 35 as an example.
Step 1: Multiply the first digit by itself plus 1.
Step 2: Put a 25 at the end.
35 squared = [3 x (3 + 1)] & 25
[3 x (3 + 1)] = 12
12 & 25 = 1225
35 squared = 1225
9. Tough multiplication
When multiplying large numbers, if one of the numbers is even, divide the first number in half, and then double the second number. This method will solve the problem quickly. For instance, consider
20 x 120
Step 1: Divide the 20 by 2, which equals 10. Double 120, which equals 240.
Then multiply your two answers together.
10 x 240 = 2400
The answer to 20 x 120 is 2,400.
10. Multiplying numbers that end in zero
Multiplying numbers that end in zero is actually quite simple. It involves multiplying the other numbers together and then adding the zeros at the end. For instance, consider:
200 x 400
Step 1: Multiply the 2 times the 4
2 x 4 = 8
Step 2: Put all four of the zeros after the 8
80,000
200 x 400= 80,000
Practicing these fast math tricks can help both students and teachers improve their math skills and become secure in their knowledge of mathematics—and unafraid to work with numbers in the future.
You may also like to read
Research-Based Math Teaching Strategies
Tips in Teaching a Hands-On Math Curriculum
5 Tips to Help Get Students Engaged in High School Math
3 Tips for Running an Elementary School Math Workshop
Seven Everyday Online Math Resources for Teachers
Three Tips for Developing Elementary Math Tests
Share
Categorized as: Tips for Teachers and Classroom Resources
Tagged as: Math and Science,Mathematics
About
Privacy
Citation
Also of Interest
Master's in Trauma-Informed Education and Car...
Online Associate's Degree Programs in Educati...
2020 Civics Engagement & Resilience: Tools fo...
© ResilientEducator.com 2025
Notifications |
8683 | https://pubmed.ncbi.nlm.nih.gov/38793561/ | Human Papillomavirus and Associated Cancers: A Review - PubMed
Clipboard, Search History, and several other advanced features are temporarily unavailable.
Skip to main page content
An official website of the United States government
Here's how you know
The .gov means it’s official.
Federal government websites often end in .gov or .mil. Before sharing sensitive information, make sure you’re on a federal government site.
The site is secure.
The https:// ensures that you are connecting to the official website and that any information you provide is encrypted and transmitted securely.
Log inShow account info
Close
Account
Logged in as:
username
Dashboard
Publications
Account settings
Log out
Access keysNCBI HomepageMyNCBI HomepageMain ContentMain Navigation
Search: Search
AdvancedClipboard
User Guide
Save Email
Send to
Clipboard
My Bibliography
Collections
Citation manager
Display options
Display options
Format
Save citation to file
Format:
Create file Cancel
Email citation
Email address has not been verified. Go to My NCBI account settings to confirm your email and then refresh this page.
To:
Subject:
Body:
Format:
[x] MeSH and other data
Send email Cancel
Add to Collections
Create a new collection
Add to an existing collection
Name your collection:
Name must be less than 100 characters
Choose a collection:
Unable to load your collection due to an error
Please try again
Add Cancel
Add to My Bibliography
My Bibliography
Unable to load your delegates due to an error
Please try again
Add Cancel
Your saved search
Name of saved search:
Search terms:
Test search terms
Would you like email updates of new search results? Saved Search Alert Radio Buttons
Yes
No
Email: (change)
Frequency:
Which day?
Which day?
Report format:
Send at most:
[x] Send even when there aren't any new results
Optional text in email:
Save Cancel
Create a file for external citation management software
Create file Cancel
Your RSS Feed
Name of RSS Feed:
Number of items displayed:
Create RSS Cancel
RSS Link Copy
Full text links
MDPIFree PMC article
Full text links
Actions
Cite
Collections
Add to Collections
Create a new collection
Add to an existing collection
Name your collection:
Name must be less than 100 characters
Choose a collection:
Unable to load your collection due to an error
Please try again
Add Cancel
Permalink
Permalink
Copy
Display options
Display options
Format
Page navigation
Title & authors
Abstract
Conflict of interest statement
Figures
Similar articles
Cited by
References
Publication types
MeSH terms
Substances
Related information
Grants and funding
LinkOut - more resources
Review
Viruses
Actions
Search in PubMed
Search in NLM Catalog
Add to Search
. 2024 Apr 26;16(5):680.
doi: 10.3390/v16050680.
Human Papillomavirus and Associated Cancers: A Review
JaNiese E Jensen1,Greta L Becker1,J Brooks Jackson1,Mary B Rysavy2
Affiliations Expand
Affiliations
1 Department of Pathology, University of Iowa Carver College of Medicine, Iowa City, IA 52242, USA.
2 Department of Obstetrics, Gynecology & Reproductive Sciences, McGovern Medical School, University of Texas Health Science Center at Houston, Houston, TX 78030, USA.
PMID: 38793561
PMCID: PMC11125882
DOI: 10.3390/v16050680
Item in Clipboard
Review
Human Papillomavirus and Associated Cancers: A Review
JaNiese E Jensen et al. Viruses.2024.
Show details
Display options
Display options
Format
Viruses
Actions
Search in PubMed
Search in NLM Catalog
Add to Search
. 2024 Apr 26;16(5):680.
doi: 10.3390/v16050680.
Authors
JaNiese E Jensen1,Greta L Becker1,J Brooks Jackson1,Mary B Rysavy2
Affiliations
1 Department of Pathology, University of Iowa Carver College of Medicine, Iowa City, IA 52242, USA.
2 Department of Obstetrics, Gynecology & Reproductive Sciences, McGovern Medical School, University of Texas Health Science Center at Houston, Houston, TX 78030, USA.
PMID: 38793561
PMCID: PMC11125882
DOI: 10.3390/v16050680
Item in Clipboard
Full text links Cite
Display options
Display options
Format
Abstract
The human papillomavirus is the most common sexually transmitted infection in the world. Most HPV infections clear spontaneously within 2 years of infection; however, persistent infection can result in a wide array of diseases, ranging from genital warts to cancer. Most cases of cervical, anal, and oropharyngeal cancers are due to HPV infection, with cervical cancer being one of the leading causes of cancer death in women worldwide. Screening is available for HPV and cervical cancer, but is not available everywhere, particularly in lower-resource settings. HPV infection disproportionally affects individuals living with HIV, resulting in decreased clearance, increased development of cancer, and increased mortality. The development of the HPV vaccine has shown a drastic decrease in HPV-related diseases. The vaccine prevents cervical cancer with near 100% efficacy, if given prior to first sexual activity. Vaccination uptake remains low worldwide due to a lack of access and limited knowledge of HPV. Increasing awareness of HPV and access to vaccination are necessary to decrease cancer and HPV-related morbidity and mortality worldwide.
Keywords: HIV; HPV vaccine; cervical cancer; human papillomavirus.
PubMed Disclaimer
Conflict of interest statement
The authors declare no conflict of interest.
Figures
Figure 1
Depiction of HPV-related cancers comparing…
Figure 1
Depiction of HPV-related cancers comparing percentages of HPV-associated cause and other causes [9,10].
Figure 1
Depiction of HPV-related cancers comparing percentages of HPV-associated cause and other causes [9,10].
Figure 2
World map representing the percentages…
Figure 2
World map representing the percentages of HPV vaccine availability by WHO region: the…
Figure 2
World map representing the percentages of HPV vaccine availability by WHO region: the Americas, Europe, Oceania, Asia, and Africa. Map courtesy of mapchart.net under Creative Commons licensing.
See this image and copyright information in PMC
Similar articles
[Papillomavirus, could we speak about cancer and prevention?].Pernot S, Pavie J, Péré H, Menard M, Hurel S, Cochand-Priollet B, Bats AS, Badoual C.Pernot S, et al.Bull Cancer. 2018 Feb;105(2):137-139. doi: 10.1016/j.bulcan.2017.12.001. Epub 2018 Jan 17.Bull Cancer. 2018.PMID: 29373106 French.No abstract available.
Prevention of Human Papillomavirus Infection. Beyond Cervical Cancer: A Brief Review.Medeiros R, Vaz S, Rebelo T, Figueiredo-Dias M.Medeiros R, et al.Acta Med Port. 2020 Mar 2;33(3):198-201. doi: 10.20344/amp.12259. Epub 2020 Mar 2.Acta Med Port. 2020.PMID: 32130098 Review.
Comprehensive control of human papillomavirus infections and related diseases.Bosch FX, Broker TR, Forman D, Moscicki AB, Gillison ML, Doorbar J, Stern PL, Stanley M, Arbyn M, Poljak M, Cuzick J, Castle PE, Schiller JT, Markowitz LE, Fisher WA, Canfell K, Denny LA, Franco EL, Steben M, Kane MA, Schiffman M, Meijer CJ, Sankaranarayanan R, Castellsagué X, Kim JJ, Brotons M, Alemany L, Albero G, Diaz M, de Sanjosé S; authors of the ICO Monograph ‘Comprehensive Control of HPV Infections and Related Diseases’ Vaccine Volume 30, Supplement 5, 2012.Bosch FX, et al.Vaccine. 2013 Dec 29;31 Suppl 5:F1-31. doi: 10.1016/j.vaccine.2013.10.001.Vaccine. 2013.PMID: 24331745 Review.
Potential impact of a nonavalent HPV vaccine on the occurrence of HPV-related diseases in France.Riethmuller D, Jacquard AC, Lacau St Guily J, Aubin F, Carcopino X, Pradat P, Dahlab A, Prétet JL.Riethmuller D, et al.BMC Public Health. 2015 May 2;15:453. doi: 10.1186/s12889-015-1779-1.BMC Public Health. 2015.PMID: 25934423 Free PMC article.
Prevention of human papillomavirus (HPV)-related tumors in people living with human immunodeficiency virus (HIV).Poljak M, Šterbenc A, Lunar MM.Poljak M, et al.Expert Rev Anti Infect Ther. 2017 Nov;15(11):987-999. doi: 10.1080/14787210.2017.1392854. Epub 2017 Oct 20.Expert Rev Anti Infect Ther. 2017.PMID: 29027811 Review.
See all similar articles
Cited by
The dietary inflammatory index (DII®) and human papillomavirus infection: a cross-sectional study of US women.Li Q, Chen Y, Yang M, Li P.Li Q, et al.BMC Public Health. 2024 Nov 1;24(1):3031. doi: 10.1186/s12889-024-20490-0.BMC Public Health. 2024.PMID: 39482657 Free PMC article.
Papillomavirus Vaccination Programs and Knowledge Gaps as Barriers to Implementation: A Systematic Review.Cangelosi G, Sacchini F, Mancin S, Petrelli F, Amendola A, Fappani C, Sguanci M, Morales Palomares S, Gravante F, Caggianelli G.Cangelosi G, et al.Vaccines (Basel). 2025 Apr 25;13(5):460. doi: 10.3390/vaccines13050460.Vaccines (Basel). 2025.PMID: 40432072 Free PMC article.Review.
Minimally invasive diagnosis of precancerous cervical lesions using single-cell peripheral immune atlas.Xu J, Ji Q, Kong Q, Lv L, Zhu B, Huang X, Chen Z, Xu P, Li X, Yin W, Wang H.Xu J, et al.Cell Rep Med. 2025 Jun 17;6(6):102149. doi: 10.1016/j.xcrm.2025.102149. Epub 2025 May 23.Cell Rep Med. 2025.PMID: 40412381 Free PMC article.
Factors Influencing Knowledge and Acceptance of Nonavalent Human Papillomavirus Vaccine Among University Population in Southern China: A Cross-Sectional Study.Fang P, Zheng H, Liu L, Pan J, Chen M, Yu X, Chen M, Yuan W.Fang P, et al.Cancer Control. 2024 Jan-Dec;31:10732748241293989. doi: 10.1177/10732748241293989.Cancer Control. 2024.PMID: 39435865 Free PMC article.
Advancing Cervical Cancer Prevention Equity: Innovations in Self-Sampling and Digital Health Technologies Across Healthcare Settings.Gomes M, Provaggi E, Pembe AB, Olaitan A, Gentry-Maharaj A.Gomes M, et al.Diagnostics (Basel). 2025 May 6;15(9):1176. doi: 10.3390/diagnostics15091176.Diagnostics (Basel). 2025.PMID: 40361993 Free PMC article.Review.
See all "Cited by" articles
References
Mammas I.N., Sourvinos G., Spandidos D.A. The Paediatric Story of Human Papillomavirus (Review) Oncol. Lett. 2014;8:502–506. doi: 10.3892/ol.2014.2226. - DOI - PMC - PubMed
Zur Hausen H. Current Topics in Microbiology and Immunology. Volume 78. Springer; Berlin/Heidelberg, Germany: 1977. Human Papillomaviruses and Their Possible Role in Squamous Cell Carcinomas; pp. 1–30. - DOI - PubMed
de Villiers E.-M. Cross-Roads in the Classification of Papillomaviruses. Virology. 2013;445:2–10. doi: 10.1016/j.virol.2013.04.023. - DOI - PubMed
Bosch F.X., Manos M.M., Muñoz N., Sherman M., Jansen A.M., Peto J., Schiffman M.H., Moreno V., Kurman R., Shah K.V. Prevalence of Human Papillomavirus in Cervical Cancer: A Worldwide Perspective. International Biological Study on Cervical Cancer (IBSCC) Study Group. J. Natl. Cancer Inst. 1995;87:796–802. doi: 10.1093/jnci/87.11.796. - DOI - PubMed
Burd E.M. Human Papillomavirus and Cervical Cancer. Clin. Microbiol. Rev. 2003;16:1–17. doi: 10.1128/CMR.16.1.1-17.2003. - DOI - PMC - PubMed
Show all 72 references
Publication types
Review
Actions
Search in PubMed
Search in MeSH
Add to Search
MeSH terms
Anus Neoplasms / epidemiology
Actions
Search in PubMed
Search in MeSH
Add to Search
Anus Neoplasms / prevention & control
Actions
Search in PubMed
Search in MeSH
Add to Search
Anus Neoplasms / virology
Actions
Search in PubMed
Search in MeSH
Add to Search
Female
Actions
Search in PubMed
Search in MeSH
Add to Search
HIV Infections / complications
Actions
Search in PubMed
Search in MeSH
Add to Search
HIV Infections / prevention & control
Actions
Search in PubMed
Search in MeSH
Add to Search
HIV Infections / virology
Actions
Search in PubMed
Search in MeSH
Add to Search
Human Papillomavirus Viruses
Actions
Search in PubMed
Search in MeSH
Add to Search
Humans
Actions
Search in PubMed
Search in MeSH
Add to Search
Male
Actions
Search in PubMed
Search in MeSH
Add to Search
Neoplasms / virology
Actions
Search in PubMed
Search in MeSH
Add to Search
Oropharyngeal Neoplasms / prevention & control
Actions
Search in PubMed
Search in MeSH
Add to Search
Oropharyngeal Neoplasms / virology
Actions
Search in PubMed
Search in MeSH
Add to Search
Papillomaviridae / pathogenicity
Actions
Search in PubMed
Search in MeSH
Add to Search
Papillomavirus Infections / complications
Actions
Search in PubMed
Search in MeSH
Add to Search
Papillomavirus Infections / prevention & control
Actions
Search in PubMed
Search in MeSH
Add to Search
Papillomavirus Infections / virology
Actions
Search in PubMed
Search in MeSH
Add to Search
Papillomavirus Vaccines / administration & dosage
Actions
Search in PubMed
Search in MeSH
Add to Search
Uterine Cervical Neoplasms / prevention & control
Actions
Search in PubMed
Search in MeSH
Add to Search
Uterine Cervical Neoplasms / virology
Actions
Search in PubMed
Search in MeSH
Add to Search
Vaccination
Actions
Search in PubMed
Search in MeSH
Add to Search
Substances
Papillomavirus Vaccines
Actions
Search in PubMed
Search in MeSH
Add to Search
Related information
MedGen
Grants and funding
NA/Project was partly funded by the Mark Gilbert and Karen Simmonds Research gift.
LinkOut - more resources
Full Text Sources
Europe PubMed Central
MDPI
PubMed Central
Medical
MedlinePlus Consumer Health Information
MedlinePlus Health Information
Full text links[x]
MDPIFree PMC article
[x]
Cite
Copy Download .nbib.nbib
Format:
Send To
Clipboard
Email
Save
My Bibliography
Collections
Citation Manager
[x]
NCBI Literature Resources
MeSHPMCBookshelfDisclaimer
The PubMed wordmark and PubMed logo are registered trademarks of the U.S. Department of Health and Human Services (HHS). Unauthorized use of these marks is strictly prohibited.
Follow NCBI
Connect with NLM
National Library of Medicine
8600 Rockville Pike Bethesda, MD 20894
Web Policies
FOIA
HHS Vulnerability Disclosure
Help
Accessibility
Careers
NLM
NIH
HHS
USA.gov |
8684 | https://pubmed.ncbi.nlm.nih.gov/38051821/ | American Burn Association Clinical Practice Guidelines on Burn Shock Resuscitation - PubMed
Clipboard, Search History, and several other advanced features are temporarily unavailable.
Skip to main page content
An official website of the United States government
Here's how you know
The .gov means it’s official.
Federal government websites often end in .gov or .mil. Before sharing sensitive information, make sure you’re on a federal government site.
The site is secure.
The https:// ensures that you are connecting to the official website and that any information you provide is encrypted and transmitted securely.
Log inShow account info
Close
Account
Logged in as:
username
Dashboard
Publications
Account settings
Log out
Access keysNCBI HomepageMyNCBI HomepageMain ContentMain Navigation
Search: Search
AdvancedClipboard
User Guide
Save Email
Send to
Clipboard
My Bibliography
Collections
Citation manager
Display options
Display options
Format
Save citation to file
Format:
Create file Cancel
Email citation
Email address has not been verified. Go to My NCBI account settings to confirm your email and then refresh this page.
To:
Subject:
Body:
Format:
[x] MeSH and other data
Send email Cancel
Add to Collections
Create a new collection
Add to an existing collection
Name your collection:
Name must be less than 100 characters
Choose a collection:
Unable to load your collection due to an error
Please try again
Add Cancel
Add to My Bibliography
My Bibliography
Unable to load your delegates due to an error
Please try again
Add Cancel
Your saved search
Name of saved search:
Search terms:
Test search terms
Would you like email updates of new search results? Saved Search Alert Radio Buttons
Yes
No
Email: (change)
Frequency:
Which day?
Which day?
Report format:
Send at most:
[x] Send even when there aren't any new results
Optional text in email:
Save Cancel
Create a file for external citation management software
Create file Cancel
Your RSS Feed
Name of RSS Feed:
Number of items displayed:
Create RSS Cancel
RSS Link Copy
Full text links
Silverchair Information Systems
Full text links
Actions
Cite
Collections
Add to Collections
Create a new collection
Add to an existing collection
Name your collection:
Name must be less than 100 characters
Choose a collection:
Unable to load your collection due to an error
Please try again
Add Cancel
Permalink
Permalink
Copy
Display options
Display options
Format
Page navigation
Title & authors
Abstract
Similar articles
Cited by
Publication types
MeSH terms
Related information
LinkOut - more resources
Practice Guideline
J Burn Care Res
Actions
Search in PubMed
Search in NLM Catalog
Add to Search
. 2024 May 6;45(3):565-589.
doi: 10.1093/jbcr/irad125.
American Burn Association Clinical Practice Guidelines on Burn Shock Resuscitation
Robert Cartotto1,Laura S Johnson2,Alisa Savetamal3,David Greenhalgh4,John C Kubasiak5,Tam N Pham6,Julie A Rizzo78,Soman Sen9,Emilia Main10
Affiliations Expand
Affiliations
1 Department of Surgery, Ross Tilley Burn Centre, Sunnybrook Heath Sciences Centre, University of Toronto, Canada.
2 Department of Surgery, Walter L. Ingram Burn Center, Grady Memorial Hospital, Emory University, Atlanta, GAUSA.
3 Department of Surgery, Connecticut Burn Center, Bridgeport Hospital, Bridgeport, CT, USA.
4 Shriners Hospital for Children, Northern California, Sacramento, CA, USA.
5 Department of Surgery, Loyola University Medical Center, Maywood, IL, USA.
6 Department of Surgery, University of Washington Regional Burn Center, Harborview Medical Center, Seattle, WA, USA.
7 Department of Trauma, Brooke Army Medical Center, Fort Sam Houston, San Antonio, TX, USA.
8 Uniformed Services University of Health Sciences, Bethesda, MD, USA.
9 Department of Surgery, Division of Burn Surgery, University of California, Davis, CA, USA.
10 Sunnybrook Health Sciences Centre, Toronto, Canada.
PMID: 38051821
DOI: 10.1093/jbcr/irad125
Item in Clipboard
Practice Guideline
American Burn Association Clinical Practice Guidelines on Burn Shock Resuscitation
Robert Cartotto et al. J Burn Care Res.2024.
Show details
Display options
Display options
Format
J Burn Care Res
Actions
Search in PubMed
Search in NLM Catalog
Add to Search
. 2024 May 6;45(3):565-589.
doi: 10.1093/jbcr/irad125.
Authors
Robert Cartotto1,Laura S Johnson2,Alisa Savetamal3,David Greenhalgh4,John C Kubasiak5,Tam N Pham6,Julie A Rizzo78,Soman Sen9,Emilia Main10
Affiliations
1 Department of Surgery, Ross Tilley Burn Centre, Sunnybrook Heath Sciences Centre, University of Toronto, Canada.
2 Department of Surgery, Walter L. Ingram Burn Center, Grady Memorial Hospital, Emory University, Atlanta, GAUSA.
3 Department of Surgery, Connecticut Burn Center, Bridgeport Hospital, Bridgeport, CT, USA.
4 Shriners Hospital for Children, Northern California, Sacramento, CA, USA.
5 Department of Surgery, Loyola University Medical Center, Maywood, IL, USA.
6 Department of Surgery, University of Washington Regional Burn Center, Harborview Medical Center, Seattle, WA, USA.
7 Department of Trauma, Brooke Army Medical Center, Fort Sam Houston, San Antonio, TX, USA.
8 Uniformed Services University of Health Sciences, Bethesda, MD, USA.
9 Department of Surgery, Division of Burn Surgery, University of California, Davis, CA, USA.
10 Sunnybrook Health Sciences Centre, Toronto, Canada.
PMID: 38051821
DOI: 10.1093/jbcr/irad125
Item in Clipboard
Full text links Cite
Display options
Display options
Format
Abstract
This Clinical Practice Guideline (CPG) addresses the topic of acute fluid resuscitation during the first 48 hours following a burn injury for adults with burns ≥20% of the total body surface area (%TBSA). The listed authors formed an investigation panel and developed clinically relevant PICO (Population, Intervention, Comparator, Outcome) questions. A systematic literature search returned 5978 titles related to this topic and after 3 levels of screening, 24 studies met criteria to address the PICO questions and were critically reviewed. We recommend that clinicians consider the use of human albumin solution, especially in patients with larger burns, to lower resuscitation volumes and improve urine output. We recommend initiating resuscitation based on providing 2 mL/kg/% TBSA burn in order to reduce resuscitation fluid volumes. We recommend selective monitoring of intra-abdominal and intraocular pressure during burn shock resuscitation. We make a weak recommendation for clinicians to consider the use of computer decision support software to guide fluid titration and lower resuscitation fluid volumes. We do not recommend the use of transpulmonary thermodilution-derived variables to guide burn shock resuscitation. We are unable to make any recommendations on the use of high-dose vitamin C (ascorbic acid), fresh frozen plasma (FFP), early continuous renal replacement therapy, or vasopressors as adjuncts during acute burn shock resuscitation. Mortality is an important outcome in burn shock resuscitation, but it was not formally included as a PICO outcome because the available scientific literature is missing studies of sufficient population size and quality to allow us to confidently make recommendations related to the outcome of survival at this time.
Keywords: burns; fluid resuscitation; shock.
© The Author(s) 2023. Published by Oxford University Press on behalf of the American Burn Association. All rights reserved. For commercial re-use, please contact reprints@oup.com for reprints and translation rights for reprints. All other permissions can be obtained through our RightsLink service via the Permissions link on the article page on our site—for further information please contact journals.permissions@oup.com.
PubMed Disclaimer
Similar articles
[Fluid resuscitation strategy and efficacy evaluation in shock stage in severely burned children with different burn areas in different age groups].Yang M, Dai XH, Guo GH, Min DH, Liao XC, Zhang HY, Fu ZH, Liu MZ.Yang M, et al.Zhonghua Shao Shang Za Zhi. 2021 Oct 20;37(10):929-936. doi: 10.3760/cma.j.cn501120-20210408-00119.Zhonghua Shao Shang Za Zhi. 2021.PMID: 34689462 Free PMC article.Chinese.
TRALI following fresh frozen plasma resuscitation from burn shock.Jones LM, Deluga N, Bhatti P, Scrape SR, Bailey JK, Coffey RA.Jones LM, et al.Burns. 2017 Mar;43(2):397-402. doi: 10.1016/j.burns.2016.08.016. Epub 2016 Oct 28.Burns. 2017.PMID: 28029475
A clinical randomized study on the effects of invasive monitoring on burn shock resuscitation.Holm C, Mayr M, Tegeler J, Hörbrand F, Henckel von Donnersmarck G, Mühlbauer W, Pfeiffer UJ.Holm C, et al.Burns. 2004 Dec;30(8):798-807. doi: 10.1016/j.burns.2004.06.016.Burns. 2004.PMID: 15555792 Clinical Trial.
American Burn Association practice guidelines burn shock resuscitation.Pham TN, Cancio LC, Gibran NS; American Burn Association.Pham TN, et al.J Burn Care Res. 2008 Jan-Feb;29(1):257-66. doi: 10.1097/BCR.0b013e31815f3876.J Burn Care Res. 2008.PMID: 18182930 Review.No abstract available.
The use of vasopressors during acute burn resuscitation.Adibfar A, Camacho F, Rogers AD, Cartotto R.Adibfar A, et al.Burns. 2021 Feb;47(1):58-66. doi: 10.1016/j.burns.2020.09.005. Epub 2020 Oct 3.Burns. 2021.PMID: 33293152 Review.
See all similar articles
Cited by
Emergency Treatment of Burns in Adults-Characteristics of Adult Patients and Acute/Pre-Hospital Burn Management.Oprita B, Burlacu G, Ispas VM, Serban IA, Oprita R.Oprita B, et al.Eur Burn J. 2025 Apr 10;6(2):19. doi: 10.3390/ebj6020019.Eur Burn J. 2025.PMID: 40265374 Free PMC article.
The Delayed Cortisol Peak Effect After Severe Burn Injury: A Prospective Observational Study.Bamberg M, Maros ME, Menger MM, Thiel JT, Illg C, Viergutz T, Kalenka A, Fontana J.Bamberg M, et al.In Vivo. 2025 Mar-Apr;39(2):877-884. doi: 10.21873/invivo.13891.In Vivo. 2025.PMID: 40010948 Free PMC article.
Update burn surgery: overview of current multidisciplinary treatment concepts.Schlottmann F, Lorbeer L.Schlottmann F, et al.Innov Surg Sci. 2024 Aug 23;9(4):181-190. doi: 10.1515/iss-2024-0020. eCollection 2024 Dec.Innov Surg Sci. 2024.PMID: 39678122 Free PMC article.Review.
Adapted approaches to initial fluid management of patients with major burns in resource-limited settings: A systematic review.Hsiao KH, Kalanzi J, Watson SB, Murthy S, Movsisyan A, Kothari K, Salio F, Relan P.Hsiao KH, et al.Burns Open. 2024 Nov;8(4):None. doi: 10.1016/j.burnso.2024.100365.Burns Open. 2024.PMID: 39540031 Free PMC article.Review.
Lessons from the similarities and differences in fluid resuscitation between burns and sepsis: a bibliometric analysis.Zhou D, He L, Shi W, Ma P.Zhou D, et al.Front Med (Lausanne). 2025 Mar 4;12:1561619. doi: 10.3389/fmed.2025.1561619. eCollection 2025.Front Med (Lausanne). 2025.PMID: 40103790 Free PMC article.Review.
See all "Cited by" articles
Publication types
Practice Guideline
Actions
Search in PubMed
Search in MeSH
Add to Search
MeSH terms
Burns / complications
Actions
Search in PubMed
Search in MeSH
Add to Search
Burns / therapy
Actions
Search in PubMed
Search in MeSH
Add to Search
Fluid Therapy / methods
Actions
Search in PubMed
Search in MeSH
Add to Search
Humans
Actions
Search in PubMed
Search in MeSH
Add to Search
Resuscitation / methods
Actions
Search in PubMed
Search in MeSH
Add to Search
Shock / etiology
Actions
Search in PubMed
Search in MeSH
Add to Search
Shock / therapy
Actions
Search in PubMed
Search in MeSH
Add to Search
Societies, Medical
Actions
Search in PubMed
Search in MeSH
Add to Search
United States
Actions
Search in PubMed
Search in MeSH
Add to Search
Related information
Cited in Books
MedGen
LinkOut - more resources
Full Text Sources
Silverchair Information Systems
Medical
MedlinePlus Health Information
Miscellaneous
NCI CPTAC Assay Portal
Full text links[x]
Silverchair Information Systems
[x]
Cite
Copy Download .nbib.nbib
Format:
Send To
Clipboard
Email
Save
My Bibliography
Collections
Citation Manager
[x]
NCBI Literature Resources
MeSHPMCBookshelfDisclaimer
The PubMed wordmark and PubMed logo are registered trademarks of the U.S. Department of Health and Human Services (HHS). Unauthorized use of these marks is strictly prohibited.
Follow NCBI
Connect with NLM
National Library of Medicine
8600 Rockville Pike Bethesda, MD 20894
Web Policies
FOIA
HHS Vulnerability Disclosure
Help
Accessibility
Careers
NLM
NIH
HHS
USA.gov |
8685 | https://www.youtube.com/watch?v=c-VkOXIGTqU | Interpreting solutions of trigonometric equations | Trigonometry | Precalculus | Khan Academy
Khan Academy
9090000 subscribers
40 likes
Description
2619 views
Posted: 7 Jan 2022
Keep going! Check out the next lesson and practice what you’re learning:
Starting from a context represented by a trigonometric function, interpret equations based on the function.
3 comments
Transcript:
alvaro presses the treetle of a spinning wheel with his foot it moves a bar up and down making the wheel spin so just to be clear what a treadle is this is an old spinning wheel and this little pedal that is a treadle and as this goes up and down it's going to pull on this bar which is then going to spin this wheel which can then be used to essentially power the machine so it says the function b of t models the height in centimeters of the top of the bar when alvaro has pressed the treadle for t seconds so it's telling us the height of i can barely see where the top of the bar is someplace over here and this isn't exactly what they're probably talking about in this exercise here but this is just to give you a visualization of what a treadle is and what the bar is and then what the spinning wheel is alvaro has pressed those treadle for t seconds so they give us b of t right over here 90 minus 12 times sine of 5t the first question is what does the solution set to y is equal to 90 minus 12 times sine of 5 times 6 represent pause this video and see if you can think about that all right so it looks like right over here so we have the 90 90 12 12 and we're subtracting 12 sine of 5 times t 5 times t so this right over here is t so this is the solution the solution set right over here tells us what is the height because that's what b of t is so b of t is equal to y what is the height when t is equal to six remember t is in seconds so this is height height of top of bar top of bar at 6 seconds all right now we have more questions here the next question asks us what does the solution set to 95 equals 90 minus 12 sine of 5t represent pause the video and think about that all right so here they're saying that b of t is equal to 95 and so the solution set you're really solving for t so you're really solving for all of the times when our height is going to be 95 centimeters so all times t when height of top of bar of top of bar at 95 centimeters and that's going to keep happening over and over and over again as t goes forward in time so you're going to have a very large you're going to have an infinite solution set over here you're going to have an infinite number of t's at which your solution at which the top of the bar is at 95 centimeters now we have another question this one is asking us what does the solution set to y is equal to 90 minus 12 sine of pi over 2 represent so pause the video and think about that all right now this is pretty interesting we can actually evaluate what sine of pi over 2 is so sine of pi over 2 radians or sine of 90 degrees that is going to be equal to 1. and so that's the maximum value that this sine over here can take on now we're going to subtract 12 times that so this is taking on a max then when you subtract 12 times that this is actually the minimum value that you can take on you're going to have you can't get any lower than this and so this is going to be the lowest the lowest height for the top of the bar and we're done |
8686 | https://reutenauer.math.uqam.ca/wp-content/uploads/2024/05/An_order_on_circular_permutations.pdf | An order on circular permutations Antoine Abram∗ Nathan Chapelier-Laget Christophe Reutenauer D´ epartement de math´ ematiques Universit´ e du Qu´ ebec ` a Montr´ eal Qu´ ebec, Canada abram.antoine@courrier.uqam.ca nathan.chapelier@gmail.com Reutenauer.Christophe@uqam.ca Submitted: Oct 29, 2020; Accepted: May 17 2021; Published: TBD © The authors. Released under the CC BY-ND license (International 4.0).
Abstract Motivation coming from the study of affine Weyl groups, a structure of ranked poset is defined on the set of circular permutations in Sn (that is, n-cycles). It is isomorphic to the poset of so-called admitted vectors, and to an interval in the affine symmetric group ˜ Sn with the weak order. The poset is a semidistributive lattice, and the rank function, whose range is cubic in n, is computed by some special formula involving inversions.
We prove also some links with Eulerian numbers, triangulations of an n-gon, and Young’s lattice.
Mathematics Subject Classifications: 06A07, 05A05 Contents 1 Introduction 2 2 Ordering circular permutations 3 3 An isomorphism towards admitted vectors 9 4 The functions δ and triangulations of an n-gon 19 5 Properties of the poset 22 5.1 Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22 5.2 Multiplicities in the Hasse diagram and Eulerian numbers . . . . . . . . . .
23 5.3 Limiting poset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26 Partially supported by an NSERC grant.
the electronic journal of combinatorics 27 (2020), #P00 6 An interval in the affine symmetric group 27 6.1 Preliminaries on ˜ Sn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27 6.2 A poset isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29 7 Circular line diagrams 36 8 Further Questions 41 1 Introduction In , the second-named author defines for each affine Weyl group an affine variety whose integral points are in bijection with the group. The irreducible components of the variety are affine subspaces and are in bijection with the alcoves in a certain polytope; this set of alcoves is partially ordered: the covering relation of this order is determined by applying a reflection sending an alcove to a neighbouring one, further from the origin.
In the case of the affine symmetric group ˜ Sn, it turns out that the irreducible compo-nents are naturally in bijection with the set circular permutations in Sn (that is, n-cycles), so that one obtains an order on this set. It is this poset, with its three instances, that we study in the present article.
The article is self-contained and may be read independently of the motivating ar-ticle .
We begin by defining a partial order on the set of circular permutations in Sn, obtaining a ranked poset (Corollary 4), whose rank function has image the interval {0, 1, . . . , n 3 }. Note the rather unusual maximum rank, which is cubic in n. The rank function is computed using inversions in a rather subtle way: it is a signed count of in-versions (see Eq.(1)), and it is clearly not the usual length function in the symmetric group).
The edges of the Hasse diagram of this order are indexed by transpositions (i, j) with i+1 < j; the covering relation conjugates two circular permutations by this transposition, under the condition that the smaller permutation sends j onto i (we call this large circular descent, see Definition 26). The smallest element in the poset is (1, 2, . . . , n) and the largest one is its inverse. Inversion and conjugation by the longest element in Sn are anti-automorphisms of the poset. See Figures 1 and 2.
Admitted vectors, introduced by the second-named author in , are vectors v of natu-ral integers, indexed by the transpositions considered above, which satisfy the conditions vij + vjk ⩽vik ⩽vij + vjk + 1, for all i < j < k, vi,i+1 = 0 for all 1 ⩽i < n.
Admitted vectors are naturally ordered, and we show that the poset of admitted vectors is isomorphic with the previous poset of circular permutations (Theorem 12).
Using the isomorphism of posets, we derive several properties. The poset is a lattice, and we give an algorithmic construction for the supremum and the infimum of two ele-ments, using the vector incarnation of the poset (Theorem 23). The number of edges in the Hasse diagram are counted by Eulerian numbers: precisely, the number of circular the electronic journal of combinatorics 27 (2020), #P00 2 permutations in Sn+1 which is covered by k circular permutations is the Eulerian number a(n, k) (= number of permutations in Sn having k descents) (Theorem 28); in particular, the number of join-irreducible elements is 2n −n −1; the poset is however not isomorphic to Sn with the weak order: indeed the maximum rank is cubic in n, whereas the maximum rank for the weak order is quadratic. We consider also the limit poset when n →∞: that is, we show that for any k, and n large enough, the posets coincide in their elements of rank ⩽k; and the limit of the posets is Young’s lattice of partitions (Theorem 32).
We give a link with triangulations of an n-gon. We consider the functions δijk, derived from the above defining identities of admitted vectors, that is, δijk(v) = vik −vij −vjk, i < j < k.
These functions take their values in {0, 1}, and satisfy the Ptolemy-like relation δijk + δikl = δijl + δjkl, i < j < k < l.
We show that, given any triangulation of the n-gon, the component indexed by (1, n) of any admitted vector v is equal to the sum of all δijk(v), where the sum is over all triangles i, j, k of the triangulation (Theorem 22).
The third incarnation of the poset, close to the initial motivation , is an interval in the affine symmetric group ˜ Sn. We construct a special element fc in this group, and show that our poset is isomorphic with the interval [id, fc] with the left weak order (Theorem 36). The rank function is therefore the length function in this Coxeter group.
From this isomorphism we derive several consequences: by a result of Reading and Speyer, the lattice is semidistributive (but not distributive, nor modular) (Corollary 44); and by a result of Bj¨ orner and Brenti, the M¨ obius function takes values in {−1, 0, 1} (see end of Subsection 6.2).
Finally, in Section 7, we illustrate the bijection between circular permutations and admitted vectors using circular line diagrams.
Aknowledgements The authors thank Nicolas England, Christophe Hohlweg, Hugh Thomas and Richard Stanley for useful discussions and mails. We also thank the referee for his/her thorough reading of our article and for many useful suggestions. Finally we thank Darij Grinberg, Travis Schedler and David Speyer; they brought to our attention respectively (see Remark 6) and .
2 Ordering circular permutations In this section we define the poset which is the topic of the whole article. It is defined first on circular permutations, by way of the covering relation. For each circular permutation, there are as many covers as “large circular descents”. The rank function of the poset is computed.
the electronic journal of combinatorics 27 (2020), #P00 3 A conjugate of word w is a word w′ such that for some words x, y, one has w = xy, w′ = yx. A factor of a word w is a word u such that w = xuy for some words x, y; a circular factor of w is a factor of some conjugate of w. We say that the word sr of length 2 is a subword of w if w = xsyrz for some words x, y, z.
For example, the alphabet being {1, 2, 3, . . . }, the word u = 213 is a factor of the word w = 21354, the word 25 is a subword of length 2 of w and 542 is a circular factor of w.
For any permutation w ∈Sn, that we view as a word on the alphabet {1, . . . , n}, and for any i, j with 1 ⩽i < j ⩽n, define the function γij(w) = 1 if j appears before i in w, and 0 otherwise; that is, γij(w) = 1 if ji is an inversion by value in w (which means that ji is a subword of the word w), and = 0 if not. Then we define N(w) = X 1⩽k⩽n−1 k(n −k)γk,k+1(w) − X 1⩽i r + 1 and let w′′ be obtained from w by exchanging r and s. Then N(w′′) = N(w) + 1.
(iii) N(12 · · · n) = 0.
(iv) N(n · · · 21) = n 3 .
Proof. (i). It suffices to prove this in the case where w = ru, w′ = ur, for r ∈{1, . . . , n}.
By inspection of the inversions of w and w′, we see that: for any i < r, w has the inversion ri, but w′ not; for any j > r, w′ has the inversion jr, but w not; all other inversion are identical in w and w′. Finally we synthesize this for all 1 ⩽i < r < j ⩽n and k, l ̸= r by: γrj(w) = 0 and γir(w) = 1, γrj(w′) = 1 and γir(w′) = 0, γkl(w) = γkl(w′).
Thus it follows that N(w) −N(w′) = (r −1)(n −(r −1))γr−1,r(w) − r−1 X i=1 γi,r(w) − r(n −r)γr,r+1(w′) + n X j=r+1 γrj(w′) = r(n −r) −(r −1)(n −r + 1) −(n −r) + (r −1) = rn −r2 −rn + r2 −r + n −r + 1 −n + r + r −1 = 0.
the electronic journal of combinatorics 27 (2020), #P00 4 (ii). By (i), we may assume that sr is a factor of w. Since s > r + 1 it follows that k(k + 1) and (k + 1)k are simultaneously subwords of w and w′′, or not; moreover, there is one subword of the form ji, j > i, less in w′′ than in w; hence N(w′′) = N(w) + 1.
(iii). This is straightforward.
(iv). Recall that the sum of the integers from 1 to n −1 (resp. of their squares) is equal to (n−1)n/2 (resp. (n−1)n(2n−1)/6). Since γij(n...21) = 1 for all i < j, it follows that N(n...21) = n−1 X k=1 k(n −k) − X 1⩽i r + 1 and w′ is obtained by replacing in w this factor by rs.
It is easy to verify that this relation does not depend on the chosen representatives of the circular permutations. For example, with sr = 51, (12345) →(52341) is seen also by (34512) →(34152).
Corollary 4. The reflexive and transitive closure of the relation →is a partial order on the set of circular permutations. The corresponding poset is graded by N, its smallest element is (12 · · · n) and its largest one is (n · · · 21).
Proof. There is no cycle in the transitive closure of →, by Proposition 1 (ii). Thus it is an order.
Note that if for a circular permutation (w), w has the property of having no circular factor sr with s > r + 1, then (w) = (n · · · 21): indeed we may assume that w begins by n, and then the second letter must be n −1, the third one must be n −2, and so on. The element (n · · · 21) is the maximum of the poset; indeed, it is the only one having the previous property; moreover, any element σ having some circular factor sr (with s > r+1) is smaller than some other element, by the definition of →, so that by iteration, σ < (n · · · 21).
Similarly, (12 · · · n) is the smallest element of the poset.
Corollary 5. The image of the function N is {0, 1, . . . , n 3 }.
Proof. This follows from the previous corollary and from Proposition 1.
Remark 6. It turns out that our poset gives a direct answer to a question proposed in the problem 2 on the USAMO 2010 . This problem is the following one: say there are n students standing in a circle, one behind the other.
The students have heights h1 < h2 < · · · < hn. If a student with height hk is standing directly behind a student with height hk−2 or less, the two students are permitted to switch places. Prove that it is not possible to make more than n 3 n such switches before reaching a position in which no further switches are possible.
Corollary 7. The functions σ 7→σ−1 and σ 7→w0 ◦σ ◦w0 (where w0 = n · · · 21 is the longest permutation) are anti-automorphisms of the poset.
In Figures 1 and 2, the first automorphism corresponds to a central symmetry of the Hasse diagram.
Proof. 1. For inversion, this follows from the fact that the inverse of (w) is ( ˜ w), where ˜ w is the reversal of w. Therefore, cyclic factors rs and sr are interchanged in (w) and its inverse, showing that (u) →(v) if and only if (˜ v) →(˜ u).
2. Note that a cyclic factor rs gives, after applying the second function the cyclic factor (n + 1 −r)(n + 1 −s), which implies the result.
the electronic journal of combinatorics 27 (2020), #P00 6 Figure 1: Poset of circular permutations for n = 4. Red labels indicate the cover relation, which is the conjugation. Ex: (14)(1234)(14) = (1423).
Remark 8. In the Hasse diagram of the poset of circular permutations, each edge is labelled by a transposition (rs), r + 1 < s, which is defined by the fact that the largest permutation τ of this edge has the circular factor rs, and the smallest σ has the circular factor sr (see Figures 1 and 2); this transposition clearly conjugates both permutations: τ = (rs)◦σ ◦(rs). Taking any upwards path in this diagram, from a circular permutation σ to a larger one, τ say, we may perform the corresponding product of transpositions in Sn from right to left, obtaining a permutation α, which conjugates them: τ = α ◦σ ◦α−1.
This implies that the class of α modulo the centralizer of σ (which has n elements) does not depend on the chosen path. But we may be more precise.
Proposition 9. With the previous notations, α depends only on σ and τ and not on the chosen path. In particular, for a maximal path (thus σ = (12 · · · n) and τ = (n · · · 21) = σ−1), one has, as words, α = n · · · 21 if n is odd, and (n/2) · · · 21n · · · (1 + n/2) if n is even.
As examples, for n = 5 and 6 the latter permutations are 54321 and 321654.
The proof of the proposition will be given at the end of Section 3.
the electronic journal of combinatorics 27 (2020), #P00 7 Figure 2: Poset of circular permutations for n = 5. For the significance of the red labels, see Figure 1.
the electronic journal of combinatorics 27 (2020), #P00 8 3 An isomorphism towards admitted vectors Admitted vectors are vectors of natural integers, indexed by the set T of transpositions in Sn, which satisfy certain inequalities. They inherit the partial order of NT. The poset thus obtained is shown to be isomorphic to the poset of circular permutations of the previous section.
Define S = {(i, i + 1), i = 1, . . . , n −1}, T1 = {(i, j), 1 ⩽i, i + 1 < j ⩽n} and T = S ∪T1. For a vector v in NT, we write vij := v(i,j). We say that v ∈NT is admitted if vi,i+1 = 0 for any i, and if for any (i, k) ∈T1, one has vij + vjk ⩽vik ⩽vij + vjk + 1 for all i < j < k.
(2) Note that equivalently vik −vij −vjk = 0 or 1.
We show in Figures 3 and 4 all possible admitted vectors for n = 4, 5; we draw there the admitted vectors using a triangle, where the way to place the coordinates is explained in the figures (the coordinates vi,i+1 are not drawn, since they are equal to 0 by definition).
The set of admitted vectors inherits the natural partial order of NT. See Figures 3 and 4, where are drawn the Hasse diagrams of these posets.
We define a mapping V0 from Sn into ZT as follows: let w ∈Sn and let V0(w) = v ∈ZT be defined by vij = −γij(w) + X i⩽k<j γk,k+1(w).
Proposition 10. Each V0(w), w ∈Sn, is an admitted vector; V0(w) = V0(w′) if w, w′ are conjugate; and N(w) is equal to the sum of the components of V0(w).
Proof. Let v = V0(w). Let 1 ⩽i < j ⩽n.
If (i, i + 1) ∈S, then by definition of V0, vi,i+1 = −γi,i+1(w) + γi,i+1(w) = 0.
If (i, k) ∈T, and i < j < k, then vik −vij −vjk = −γik(w) + X i⩽p<k γp,p+1(w) + γij(w) − X i⩽p<j γp,p+1(w) + γjk(w) − X j⩽p<k γp,p+1(w) = γij(w) + γjk(w) −γik(w).
By inspecting the relative positions of i, j, k in the word w, one checks that the latter quantity is equal to 0 or 1. Hence v satisfies the inequalities (2). Moreover, vi,i+1 = 0, and by induction on j −i, these inequalities imply that the vij are all nonnegative.
This proves that v = V0(w) is an admitted vector. We show now that it is invariant under conjugation of w.
Suppose that w = ru, w′ = ur, r ∈{1, . . . , n}. Let v = V0(w), v′ = V0(w′). Let i < j.
Then vij = −γij(w) + X i⩽k<j γk,k+1(w) and v′ ij = −γij(w′) + X i⩽k i + 1. Suppose that i, j ̸= r. If i < r < j, there is the subword r(r −1) in w, but not in w′, and there is the subword (r + 1)r in w′, but not in w; thus vij = v′ ij. If on the other hand, the double inequality i < r < j does not hold, then the two previous sums are identically equal and vij = v′ ij, too.
Suppose that r = i. Then γij(w) = 0, γij(w′) = 1, and γi,i+1(w) = 0, γi,i+1(w′) = 1; the other terms are identical for w and w′, therefore vij = v′ ij. The case r = j is similar.
We deduce that V0(w) = V0(w′). Thus V0(w) is invariant under conjugation of w.
Finally, we have X i<j vij = X i<j (−γij(w) + X i⩽k<j γk,k+1(w)) = − X i<j γij(w) + n X k=1 X i⩽k<j γk,k+1(w) = − X i<j γij(w) + X 1⩽k⩽n k(n −k)γk,k+1(w) = N(w).
the electronic journal of combinatorics 27 (2020), #P00 10 Figure 4: Poset of admitted vectors for n = 5.The pyramid shows the position of each coefficients but knowing the fact that by definition vi,i+1 = 0 for all i we only keep the ones in the pink pyramid.
the electronic journal of combinatorics 27 (2020), #P00 11 Definition 11. We define the functions δijk : NT →N, 1 ⩽i ⩽j ⩽k ⩽n, by δijk(v) = vik −vij −vjk, where we put vii = 0.
These functions satisfy the following relation, for any i ⩽j ⩽k ⩽l: δijk(v) + δikl(v) = δijl(v) + δjkl(v).
(3) Indeed, the left-hand side evaluated on the admitted vector v is equal to vik −vij − vjk + vil −vik −vkl, whereas the right-hand side gives vil −vij −vjl + vjl −vjk −vkl, which are both equal to vil −vij −vjk −vkl.
Theorem 12. The mapping V0 induces a graded poset isomorphism V from the set of circular permutations in Sn into the set of admitted vectors in NT, ordered componentwise.
The inverse mapping is completely determined as follows: let w = 1w′ ∈Sn, viewed as word; then V0(w) = v implies that for any 1 ⩽i < j ⩽n, one has γij(w) = δ1ij(v).
The theorem is illustrated for n = 4 in Figure 5: each red labels represents, from left to right, the coordinate which is increased in the cover relation (with the pyramidal convention of Figure 3); the conjugating transposition; the large circular descent.
Figure 5: Isomorphism from admitted vectors onto circular permutations in S4.
The statement of Theorem 12 uses implicitly the well-known result that a permutation w ∈Sn is completely determined by its sequence of inversions (γij(w))1⩽i<j⩽n.
In order to prove Theorem 12 we need several lemmas. Before stating and proving them, we note that the inequalities (2) characterizing admitted vectors are equivalent to the condition that the image of each function δijk is contained in {0, 1}.
the electronic journal of combinatorics 27 (2020), #P00 12 Lemma 13. If u, v are admitted vectors such that u < v, then for some 1 ⩽i < j ⩽n, one has i < p < j ⇒δipj(u) = 0; 1 ⩽p < i ⇒δpij(u) = 1; j < p ⩽n ⇒δijp(u) = 1.
Proof. In the whole proof we use the fact that the functions δ map u and v onto 0 or 1.
1. We show first that there exists a pair i < j such that δipj(v) = 0 for any i < p < j.
We construct below a nonempty set E, whose minimal elements, for a certain order, are such pairs.
The order on the set of pairs (i, j), i < j, is as follows: (i, j) ⩽(i′, j′) if the interval [i, j] is contained in the interval [i′, j′] (that is, if i′ ⩽i and j ⩽j′).
Consider the set E of pairs (i, j), i < j, such that uij < vij. Note that by assumption E is nonempty. Let (i, j) be a minimal element in this set, for the previous order. Then for any (i′, j′) < (i, j), one has by minimality ui′,j′ = vi′,j′; in particular, for i < p < j, uip = vip and upj = vpj. Moreover by construction, uij −vij ⩽−1. Since δipj(v) ⩽1, it follows that: δipj(u) = uij −uip −upj = uij −vij + vij −vip −vpj = (uij −vij) + δipj(v) ⩽0.
Thus δipj(u) = 0.
2. We consider now that set F of pairs (i, j), i < j, such that for any i < p < j, one has δipj(u) = 0. This set is nonempty by 1. Let (i, j) be a maximal element in this set.
We claim that for any 1 ⩽p < i, one has δpij(u) = 1 and for any n ⩾p > j, δijp(u) = 1.
The claim implies the lemma.
Suppose by contradiction that the claim is not true; then by symmetry of the two cases, we may assume that for some n ⩾l > j, one has δijl(u) = 0, and we choose l minimum.
Let j < k < l. By minimality of l, we have δijk(u) = 1. Then by Eq.(3), δijk(u) + δikl(u) = δijl(u) + δjkl(u). Since δijk(u) = 1 and δijl(u) = 0, we must have δikl(u) = 0.
Now let i < p < j. Therefore, by Eq.(3) applied to i < p < j < l, we have δipj(u) + δijl(u) = δipl(u) + δpjl(u). Since δijl(u) = 0 and δipj(u) = 0, we must have δipl(u) = 0.
We conclude that for any p with i < p < l, we have δipl(u) = 0. Since (i, j) < (i, l), this contradicts the maximality of the pair (i, j).
Lemma 14. Let v be an admitted vector, let i < j and define v′ by vij + 1 = v′ ij, whereas vrs = v′ rs for the other coordinates. Then the two following conditions are equivalent: (i) v′ is an admitted vector; the electronic journal of combinatorics 27 (2020), #P00 13 (ii) the following equations hold: i < p < j ⇒δipj(v) = 0, 1 ⩽p < i ⇒δpij(v) = 1, j < p ⩽n ⇒δijp(v) = 1.
(4) Proof. Note that i < p < j ⇒δipj(v) = vij −vip −vpj = v′ ij −1 −v′ ip −v′ pj = δipj(v′) −1; 1 ⩽p < i ⇒δpij(v) = vpj −vpi −vij = v′ pj −v′ pi −v′ ij + 1 = δpij(v′) + 1; j < p ⩽n ⇒δijp(v) = δijp(v′) + 1; In all other cases, δrst(v) = δrst(v′).
(i) implies (ii): let i < p < j; then by the above calculations δipj(v) = δipj(v′)−1; since the values of δ are 0 or 1, we must have δipj(v) = 0. Let p < i; then δpij(v) = δpij(v′) + 1; this forces δpij(v) = 1. Similarly, p > j implies δijp(v) = 1. Hence equations (4) hold.
(ii) implies (i): It is enough to prove that the functions δ take the values 0 or 1 when evaluated on v′. Let i < p < j; then δipj(v) = δipj(v′) −1, hence δipj(v′) = 1 by Eq.(4).
Let 1 ⩽p < i; then δpij(v) = δpij(v′) + 1, hence δpij(v′) = 0. Similarly, for n ⩾p > j, δijp(v′) = 0. In all other cases, δrst(v′) = δrst(v) = 0 or 1. This concludes the proof.
Corollary 15. Let u < v be admitted vectors. Then there exists an admitted vector u′ such that u < u′ ⩽v and that for some 1 ⩽i < j ⩽n one has uij + 1 = u′ ij, whereas urs = u′ rs for the other coordinates.
Note that this implies that necessarily u′ covers u for the order of admitted vectors.
The corollary could be given a geometric proof, in the spirit of , but it follows also directly from the two previous lemmas.
Lemma 16. Let w ∈Sn, viewed as a word. If 1 ⩽i < j ⩽n, then ji is a factor of w if and only if the following conditions hold: (i) γij(w) = 1; (ii) for any i < p < j, γip(w) + γpj(w) = 1; (iii) for any 1 ⩽p < i, γpi(w) = γpj(w); (iv) for any n ⩾p > j, γip(w) = γjp(w).
Proof. Condition (i) means that ji is a subword of w. If 1 ⩽p < i, then γpi(w) = γpj(w) = 0 means that p is at the left of both i and j in w, and γpi(w) = γpj(w) = 1 means that p is at their right; hence condition (iii) means that p is not between i and j in w. Condition (iv) is similar in the case n ⩾p > j. Condition (ii) means similarly that for i < p < j, p is not between i and j.
Thus the four conditions together mean that ji is a factor of w.
the electronic journal of combinatorics 27 (2020), #P00 14 To complete the picture, we give the following dual result, whose proof may be deduced from the proof of Lemma 14: Lemma 17. Let v′ be an admitted vector, let i < j and define v by vij = v′ ij −1, whereas vrs = v′ rs for the other coordinates. Then the two following conditions are equivalent: (i) v is an admitted vector; (ii) the following equations hold: i < p < j ⇒δipj(v′) = 1; 1 ⩽p < i ⇒δpij(v′) = 0; j < p ⩽n ⇒δijp(v′) = 0.
(5) Proof of Theorem 12. 1. By Proposition 10, V is well-defined on the set of circular per-mutations.
2. We show that, for w = 1w′, and V0(w) = v, for any 1 ⩽i < j ⩽n, one has γij(w) = −vij + v1j −v1i = δ1ij(v).
In the case i ̸= 1, we have indeed δ1ij(v) = −vij + v1j −v1i = γij(w) − X i⩽k<j γk,k+1(w) −γ1j(w) + X 1⩽k<j γk,k+1(w) + γ1i(w) − X 1⩽k<i γk,k+1(w).
The three summations cancel, and γ1j(w) = 0 = γ1i(w), since w = 1w′. Thus the previous sum is equal to γij(w). In case i = 1, one has δ1ij(v) = −v1j + v1j −v11 = 0 = γ1j(w).
3.
This implies that the mapping V is injective, and proves the formulas of the statement.
4.
In order to prove surjectivity, let v be an admitted vector.
Note that by the inequalities (2), for i < j the numbers −vij + v1j −v1i = δ1ij(v) are equal to 0 or 1.
Consider the set I of pairs (i, j) such that 1 < i < j ⩽n and δ1ij(v) = 1. We show that this set is the set of inversions by value of some permutation w (meaning that I is the set of (i, j), i < j, such that ji is a subword of w).
Recall that a set I of pairs (i, j), i < j, is the set of inversions by value of a permutation if and only one has the two conditions: (i) (i, j) and (j, k) ∈I implies (i, k) ∈I; and (ii) if i < j < k and (i, k) ∈I, then (i, j) or (j, k) is in I.
Therefore with I as above, it is enough to prove (i) and (ii).
By Eq.(3), we have δ1ij(v)+δ1jk(v) = δ1ik(v)+δijk(v), and δijk(v) = 0 or 1. Therefore, (i) and (ii) easily follow.
the electronic journal of combinatorics 27 (2020), #P00 15 5.
For the permutation w constructed in 4, its set of inversions by value is I = {(i, j), 1 < i < j ⩽n, δ1ij(v) = 1}. Therefore γij(w) = −vij+v1j−v1i for any 1 ⩽i < j ⩽1 (since it is also true for i = 1, because by definition of I, there is no inversion 1j). We show that V (w) = v. We have indeed −γij(w) + X i⩽k<j γk,k+1(w) = vij −v1j + v1i + X i⩽k i + 1 and if w′ is obtained from w by exchanging i and j, then v′ = V0(w′) is obtained by increasing by 1 the component ij of v = V0(w). Since V0 is invariant under conjugation of w, we may assume that ji is a factor of w.
We have vij = −γij(w) + X i⩽k<j γk,k+1(w), and v′ ij = −γij(w′) + X i⩽k k + 1 and (k, l) ̸= (i, j). Similar arguments then show that vkl = v′ kl.
7.
To conclude, we must show that if v = V0(w), v′ = V0(w′) and v < v′, then (w) < (w′) for the order of circular permutations. In view of Corollary 15, we are reduced to the case where v and v′ differ by 1 on some coordinate ij: v′ ij = vij + 1 and vkl = v′ kl if (k, l) ̸= (i, j).
We may assume that w = 1u, and we show that ji is a circular factor of w, which will suffice, by definition of the order on circular permutations and by 6.
Suppose that i > 1. We apply Lemma 16 to show that ji is a factor of w. We apply below several times Eq.(3), Lemma 14 and 2.
(i) We have γij(w) = δ1ij(v) = 1.
(ii) Let i < p < j. Then we have γip(w) + γpj(w) = δ1ip(v) + δ1pj(v) = δ1ij(v) + δipj(v) = 1.
(iii) Let p < i. Then γpi(w) −γpj(w) = δ1pi(v) −δ1pj(v) = −δ1ij(v) + δpij(v) = 0.
(iv) is similar. We conclude that ji is a factor of w.
the electronic journal of combinatorics 27 (2020), #P00 16 Suppose now that i = 1. Then we have to show that j1 is a circular factor of w = 1u, which means that j is the last letter of w. Thus we must prove that for 1 < p < j, jp is not a subword of w, that is γpj(w) = 0; and if p > j, that jp is a not a subword, that is, pj is one, equivalently γjp(w) = 1. But we have, using Lemma 14: if p < j, γpj(w) = δ1pj(v) = 0; and if p > j, γjp(w) = δ1jp(v) = 1, which concludes the proof.
The proof implies the following result, which is illustrated by the labels of the edges in the Hasse diagram of Figures 3 and 4.
Corollary 18. Let σ, σ′ be circular permutations and v, v′ be the corresponding admitted vectors. Then σ′ covers σ if and only if v′ covers v. Moreover the following conditions are equivalent, for 1 ⩽r, r + 1 < s ⩽n: (i) σ′ is obtained from σ by replacing the circular factor sr by rs; (ii) v′ rs = vrs + 1.
In the next result, we transfer to the poset of admitted vectors the anti-automorphisms of the poset of circular permutations, as they appear in Corollary 7.
Corollary 19.
1. The anti-automorphism of the poset of admitted vectors, correspond-ing to inversion in the poset of circular permutations, is defined by u 7→v, with vij = j −i −1 −uij.
2. The anti-automorphism of the set of admitted vectors, corresponding to conjugation by the longest permutation w0, is defined by u 7→v, with vij = j−i−1−un+1−j,n+1−i.
Note that the product of the two previous anti-automorphisms of the poset is the automorphism u →v with vij = un+1−j,n+1−i.
Proof. 1. Inversion of circular permutations is the mapping (w) 7→( ˜ w); the corresponding admitted vectors are respectively u and v with uij = −γij(w) + X i⩽k<j γk,k+1(w), and vij = −γij( ˜ w) + X i⩽k<j γk,k+1( ˜ w).
Since γrs( ˜ w) = 1 −γrs(w), we have vij = −(1 −γij(w)) + X i⩽k<j (1 −γk,k+1(w)) = −uij −1 + j −i.
2.
This anti-automorphism is defined, for circular permutations, by (w) 7→(w′), where w′ is obtained by replacing in w each letter by its complement n + 1 −k. Hence the electronic journal of combinatorics 27 (2020), #P00 17 γij(w′) = 1 −γn+1−j,n+1−i(w).
Therefore, denoting by u, v respectively the admitted vectors corresponding to (w) and (w′), we have vij = −γij(w′) + X i⩽k<j γk,k+1(w′) = −(1 −γn+1−j,n+1−i(w)) + X i⩽k<j (1 −γn+1−k−1,n+1−k(w)) = −1 + j −i −(−γn+1−j,n+1−i(w) + n+1−i X l=n+1−j γl,l+1(w)) = j −i −1 −un+1−j,n+1−i.
Corollary 20. The smallest admitted vector is the null vector, and the largest one, w say, satisfies wij = j −i −1; moreover, δijk(w) = 1 for any i < j < k.
Proof. Clearly, the null vector is admitted and is the smallest element in the poset of admitted vectors. Using Corollary 19, we see that w as defined in the statement is the largest element. Now δijk(w) = wik −wij −wjk = k −i −1 −(j −i −1) −(k −j −1) = 1.
Proof of Proposition 9. 1. It is enough to prove this result when σ = (12 · · · n). Consider a path from σ to τ, whose successive edges are labelled (r1, s1), . . . , (rp, sp), with ri+1 < si, and let σ = (w0), (w1), . . . , (wp−1), (wp) = τ the successive vertices in the path, written as n-cycles, so that each wi is a word. Then siri is a circular factor of the word wi−1, and wi is obtained from the latter word by exchanging ri and si in it.
Define the permutations αi = (ri, si)◦· · ·◦(r1, s1). Then (wi) = αi ◦σ ◦α−1 i . It follows that, in cycle notation, (wi) = (αi(1) · · · αi(n)) and therefore si+1ri+1, being a circular factor of (wi), is a circular factor of the word αi. We have αi+1 = (ri+1, si+1) ◦αi and thus, as words, αi+1 is obtained from αi by exchanging ri+1 and si+1, whose positions moreover must differ by 1 in both words (modulo n). Hence: α−1 i+1(ri+1) ≡α−1 i (ri+1) −1 mod n α−1 i+1(si+1) ≡α−1 i (si+1) + 1 mod n α−1 i+1(j) = α−1 i (j) if j ̸= ri+1, si+1.
It follows by induction that for any j = 1, . . . , n, the position of j in the word αi+1 is congruent modulo n to: j plus the number of (rs) on the path with s = j minus the the electronic journal of combinatorics 27 (2020), #P00 18 number of (rs) on the path with r = j. From Corollary 18, it follows that it is congruent modulo n to: j + X r<s=j vrs − X j=r<s vrs where v is the admitted vector associated to the circular permutation (wi+1).
Therefore each αi+1 depends only on v, which depends only on (wi+1); thus α = αp depends only on τ.
2. It remains to determine α for a maximal path in the poset. The previous calculations and Corollary 20 show that α−1(j) is congruent modulo n to j+P r<j (j−r−1)−P j<s (s−j−1).
Moreover, a direct computation shows that: j + X r<j (j −r −1) − X j<s (s −j −1) = j + ((j −2) + · · · + 1 + 0) −(0 + 1 + · · · + (n −j −1)) = j + (j −2)(j −1) 2 −(n −j −1)(n −j) 2 .
The cases j = 1 and j = n have to be treated separately, but the final formula is correct in these cases as well.
Suppose that n = 2k + 1. Then using the fact that the inverse of 2 mod n is −k, we find that α−1(j) is congruent modulo n to j −k(j −2)(j −1)+kj(j +1). Hence, omitting the calculations, to n + 1 −j. Thus, as words, α−1 = n · · · 21, and α = α−1, since it is an involution.
Suppose now that n = 2k. Then a calculation left to the reader shows that modulo n, α−1(j) is congruent to k + 1 −j. Hence, as word, α−1 = (n/2) · · · 21n · · · (1 + n/2) and α = α−1 since it is an involution.
4 The functions δ and triangulations of an n-gon The n-gon is a famous object whose combinatorial and geometrical structure appears in a wide variety of subjects. A striking illustration of the ubiquity of the n-gon lies for example in the theory of cluster algebras introduced by S. Fomin and A. Zelevinsky in . In this section we relate the n-gon with the functions δijk introduced in Definition 11.
We first characterize the sequences (γij(v))1⩽i<j⩽n, v an admitted vector, and show that v is completely determined by this sequence. The computation of v will lead to triangulations of an n-gon.
Theorem 21. The mapping v 7→(δijk(v))1⩽i<j<k⩽n is a bijection from the set of admit-ted vectors onto the set of sequences (aijk)1⩽i<j<k⩽n with values in {0, 1}, satisfying the relation aijk + aikl = aijl + ajkl for any i < j < k < l.
Proof. The mapping is well-defined by Eq. (3). Injectivity follows by induction on j −i from the initial conditions vi,i+1 = 0 and if j −i > 1, from the equation vij = vip + vpj + δipj(v) for any p, i < p < j.
the electronic journal of combinatorics 27 (2020), #P00 19 To prove surjectivity, let (aijk)1⩽i<j<k⩽n be a sequence, with values in {0, 1}, satisfying the relations aijk+aikl = aijl+ajkl for any i < j < k < l. Define vij by induction: vi,i+1 = 0 and for j−i > 1, choose p between i and j, and define vij by vij = vip+vpj+aipj. We verify, by induction on j −i, that this does not depend on the choice of p. So let i < p < q < j.
We have by induction viq = vip + vpq + aipq and vpj = vpq + vqj + apqj. The above relations imply that aipq + aiqj = aipj + apqj. Hence vip + vpj + aipj = vip + vpq + vqj + apqj + aipj = vip + vpq + vqj + aipq + aiqj = viq + vqj + aiqj, and so vij is well-defined. The fact that v is an admitted vector then follows from the fact that for any i < j < k, vik −vij −vjk = aijk ∈{0, 1}, implying the inequalities (2).
We consider a regular n-gon in the plane whose vertices are sequentially numbered 1, 2, . . . , n. In this n-gon, a triangle (resp. convex quadrangle) is determined by a subset of three (resp. four) vertices of the n-gon. A triangulation of the n-gon is a set of triangles whose interiors do not pairwise intersect, and who cover the whole n-gon.
Note that a 3-subset H of {1, 2, . . . , n} determines uniquely three numbers i < j < k such that H = {i, j, k}. We write δH := δijk.
Theorem 22. If v is an admitted vector, and T a triangulation of the n-gon, then v1n is equal to the sum of all δT(v) for all triangles T ∈T .
Proof. Consider the unique triangle {1, 2, p} in T (it is the triangle containing the edge 12 of the n-gon). By induction, we have v1p = P T δT(v), sum over all triangles in T contained in the sub-p-gon with vertices 1, 2, . . . , p, which form a triangulation of this p-gon. Similarly, the triangles contained in the n−p+1-gon with vertices p, p+1, . . . , n form a triangulation of it; hence by induction (after appropriate reindexing), vpn = P T δT(v), sum over all triangles of this triangulation. If we add the the triangle with vertices 1, p, n to these two triangulations we obtain the original triangulation. Now v1n = v1p+vpn+δ1pn, which concludes the proof.
More generally, for any i < j, the coefficient vij of an admitted vector v is equal to P T δT(v), where the sum is over all triangles of a triangulation of the j −i + 1-gon with vertices i, i + 1, . . . , j.
Given a triangulation, consider a convex quadrangle such that one of its diagonal determines two triangles of the triangulation. If we replace in the triangulation these two triangles by the two others of the quadrangle, determined by the other diagonal, then we obtain a new triangulation: we say that the new triangulation is obtained by mutation.
It is known that given two triangulations of the n-gon, one may transform one into the other by a sequence of mutations.
Note that Eq.(3) implies the fact that the sum P T δT(v), over all triangles in a trian-gulation, is invariant under mutation.
the electronic journal of combinatorics 27 (2020), #P00 20 Theorem 22 is illustrated in Figure 6. We choose v defined in the figure, and therefore δ345 = v35 −v34 −v45 = 1−0−0 = 1; we thus put 1 inside the triangle with vertices 3,4,5.
Similarly for the other triangles. The mutations, with respect to the red triangle sides, are indicated by arrows in the figure. One verifies that for each triangulation, the sum of the labels in the triangles is 2, thereby confirming the theorem.
We note finally that the sequence (δijk(v))1⩽i<j<k⩽n is completely defined by the subse-quence (δ1ij(v))i<j: this follows from Eq.(3), by considering the quadrangle 1, i, j, k. The latter sequence moreover appears as the inversion sequence of the permutation w = 1w′ such that V (w) = v (see the proof of Theorem 12).
Figure 6: Illustration of Theorem 22 and the discussion following the proof.
the electronic journal of combinatorics 27 (2020), #P00 21 5 Properties of the poset We establish three properties of the poset of this article: it is a lattice, its covering degrees are related to Eulerian numbers, and its limiting poset is Young’s lattice.
5.1 Lattice Recall that the set of admitted vectors in NT has been defined if the previous section.
Theorem 23. The poset of admitted vectors (or equivalently the poset of circular permu-tations) is a lattice.
We give a combinatorial proof of this result, with a simple algorithmic construction of the supremum and the infimum. This result may also be deduced from Theorem 36: see the comment before Corollary 44.
We extend the definition of admitted vector as follows: if v ∈NT, and i < k, then we say that the component vik of v is admitted if either k = i + 1 and vik = 0 or k > i + 1 and ∀j, i < j < k ⇒vij + vjk ⩽vik ⩽vij + vjk + 1.
Hence v is admitted if and only if all its components are admitted.
Lemma 24. If for v ∈NT all components vik with k −i < n −1 are admitted, then the numbers v1i + vin, i = 2, . . . , n −1 take at most two values, which are consecutive.
Proof. By hypothesis, we have for 1 < i < j < n, v1j = v1i + vij + η1ij and vin = vij + vjn + ηijn for some η1ij, ηijn ∈{0, 1}. It follows that v1i + η1ij + vin = v1j −vij + vij + vjn + ηijn = v1j + vjn + ηijn.
Since the η’s take only the values 0 or 1, it follows that any two of the numbers v1i + vin, i = 2, . . . , n −1, differ in absolute value by at most 1. It follows easily that they take at most two values, which must be consecutive.
By appropriately reindexing the vector, we obtain the following corollary.
Corollary 25. Suppose that v ∈NT and that, for some i < j, the components vrs with s −r < j −i are admitted. Then the numbers vip + vpj, p = i + 1, . . . , j −1, take at most two values, which are consecutive.
Proof of Theorem 23. Let u, v be two admitted vectors. We define X ∈NT as follows: Xij is defined by induction on j −i ⩾1; if j = i + 1, Xij = 0; if j > i + 1, then Xij = max{uij, vij, Xip + Xpj, i < p < j}.
1.
We show first that X is an admitted vector.
It is enough to show that each coefficient Xij is admitted, and we do it by induction on j −i. The case j −i = 1 is clear, so take j −i = h > 1. By induction, all coefficients Xrs with s −r < h are admitted.
Suppose that Xij = Xip + Xpj for some p with i < p < j. By the corollary, one has {Xiq + Xqj, i < q < j} ⊂{a −1, a}, and by the definition of Xij, we must have the electronic journal of combinatorics 27 (2020), #P00 22 a = Xip + Xpj. It follows that for any q, i < q < j, one has Xiq + Xqj = Xij or Xij −1 and therefore Xiq + Xqj ⩽Xij ⩽Xiq + Xqj + 1.
If Xij is not equal to Xip+Xpj for some p with i < p < j, then we must have (choosing one of the two similar cases) Xij = vij ⩾uij and vij > Xip +Xpj for any p with i < p < j.
Then Xip+Xpj < Xij; moreover v being admitted, Xij = vij ⩽vip+vpj+1 ⩽Xip+Xpj+1, since by the recursive construction of X, vrs ⩽Xrs for any rs with s −r < j −i.
We conclude that in all cases, Xij is admitted, and therefore by induction X is admit-ted.
2. We show now that X is the supremun of u and v, that is, X is the smallest element in the set of upper bounds of u, v. By construction, Xij ⩾uij, vij, so that X ⩾u, v and X is an upper bound of u and v. Now let Y be an admitted vector which is an upper bound of u, v. We show by induction on j −i that Xij ⩽Yij, which will imply that X ⩽Y , as was to be shown. We have Xij = 0 = Yij if j −i = 1. Suppose that j −i > 1. Then Yij ⩾uij, vij; and by the inequalities (2), if i < p < j, Yij ⩾Yip + Ypj which by induction is ⩾Xip + Xpj; thus Yij ⩾max{uij, vij, Xip + Xpj, i < p < j} = Xij.
3. Thus u ∨v exists in the poset. Since the poset has an anti-automorphism, u ∧v exists, too, and the poset is a lattice.
The proof allows to compute u ∨v. A direct computation of X = u ∧v is obtained recursively, by induction on j −i, as follows: Xi,i+1 = 0; if j > i + 1, then Xij = min{uij, vij, Xip + Xpj + 1, i < p < j}. The proof is left to the reader (one may use the first anti-automorphism of Corollary 19).
The lattice is not modular. Indeed, looking in Figure 2, it is seen that the circular permutations (14235) and (13425) have infimum (14253) and supremum (13542), and respective rank 4,4,3 and 6.
Since in a modular lattice, the rank function ρ satisfies ρ(x) + ρ(y) = ρ(x ∧y) + ρ(x ∨y) (see page 104), the lattice is not modular. Since each distributive lattice is modular ( p. 106), the lattice is not distributive either.
5.2 Multiplicities in the Hasse diagram and Eulerian numbers Recall that the Eulerian numbers are defined as follows: a(n, k) is the number of per-mutations in Sn whose descent set has k elements, where the descent set of σ ∈Sn is {i, 1 ⩽i ⩽n −1, σ(i) > σ(i + 1)}. These numbers satisfy the recursion a(n, k) = (k + 1)a(n −1, k) + (n −k)a(n −1, k −1) if n, k ⩾1, with the initial conditions a(0, k) = 0 for any k ⩾1, and a(n, 0) = 1 for any n ⩾0 (see for more details about Eulerian numbers).
Definition 26. Let σ = (a1 · · · an) be a circular permutation in Sn. We call large circular descent of σ a letter ai such that ai−1 > ai + 1, were the indices are taken modulo n. The number of large circular descents of σ is denoted by d(σ).
Symmetrically, we call large circular ascent of σ a letter ai such that ai < ai+1 + 1, were the indices are taken modulo n.
the electronic journal of combinatorics 27 (2020), #P00 23 In other words, b is a large circular descent if, denoting by a the letter before b in the cycle, one has a > b + 1. Note that we say that b is the large circular descent, and not a; the reason for this notational shift will appear in the proofs below. For example, the large circular descents of (1, 4, 2, 6, 5, 3) are 1 (because before 1 there is 3, cyclically speaking), 2 (because before 2 there is 4), and 3; 5 is not a large circular descent.
Theorem 27. The number of circular permutations in Sn+1 having k large circular de-scents is equal to the Eulerian number a(n, k).
One can note that the cover relations of the poset of circular permutations defined in Definition 3 correspond to the large circular descents defined in Definition 26. This gives us directly the following result.
Corollary 28. In the poset of circular permutations in Sn+1, the number of elements which are covered by k elements is a(n, k). In particular the number of inf-irreducible elements is a(n, 1) = 2n −n −1.
The next corollary is somewhat curious, since the two posets are not isomorphic, nor are their Hasse diagram as (undirected) graphs.
Corollary 29. The total number of edges in the Hasse diagram of the poset of circular permutations in Sn+1 is equal to the total number of edges in the Hasse diagram of the poset of the weak order in Sn.
Proof. This follows from the theorem since the edges in the Hasse diagram of the right weak order are in bijection with the set of descents of all permutations.
To prove the theorem, we show that the number of circular permutations in Sn+1 with k large circular descents satisfy the same recursion as the Eulerian numbers, and the same initial conditions. Finding a bijective proof is an open question.
Proof of Theorem 27. 1. Denote by C(n) the set of circular permutations in Sn+1, so that |C(n)| = n!. Denote by C(n, k) the set of elements in C(n) having k large circular descents, and c(n, k) = |C(n, k)|.
We verify first that the initial conditions are the same. Indeed, c(0, k) for k ⩾1 is the number of circular permutations in S1 having k large circular descents, hence c(0, k) = 0.
Moreover, for n ⩾0, c(n, 0) is the number of circular permutations in Sn+1 having no large circular descent; this permutation is (n, n −1, . . . , 2, 1) as we saw in the proof of Corollary 4, and therefore c(n, 0) = 1.
We show below that c(n, k) = (k + 1)c(n −1, k) + (n −k)c(n −1, k −1), which is the same recursion as the Eulerian numbers. Thus we will obtain c(n, k) = a(n, k), which proves the theorem.
2. Recall that each element in C(n) is represented by a circular word (a1, . . . , an+1) and that we do not distinguish between all its cyclic rearrangements (so that e.g.(a1, . . . , an+1) = (a2, . . . , an+1, a1)).
the electronic journal of combinatorics 27 (2020), #P00 24 3. Define a mapping φ from C(n) into C(n −1) × [n] as follows. Associate with α = (a1, . . . , an+1) ∈C(n) the circular permutation β = (b1, . . . , bn) ∈C(n −1) obtained by removing n + 1 from the cycle, and let bj denote the letter such that (a1, . . . , an+1) is obtained from β = (b1, . . . , bn) by inserting n + 1 before bj.
To illustrate the notions above we give the two following examples for n + 1 = 7: - α1 = (2, 4, 6, 5, 1, 7, 3) ∈C(6), β1 = (2, 4, 6, 5, 1, 3) ∈C(5), bj = 3 ∈; - α2 = (2, 3, 6, 5, 1, 4, 7), β2 = (2, 3, 6, 5, 1, 4), bj = 2.
Therefore, φ is defined by φ(α) = (β, bj). This is clearly a bijection, since we recover α from β by inserting n + 1 before bj, so that φ is injective, and since the two sets have the same cardinality.
4. For β ∈C(n −1), define D(β) to be the set of large circular descents of β; hence |D(β)| = d(β) (where d(β) is the number of large circular descents as defined in Definition 26).
5. Let X := X1 ⊔X2 where X1 and X2 are defined by X1 := {(β, b), β ∈C(n −1, k), b ∈D(β) ∪{n}}, and X2 := {(β, b), β ∈C(n −1, k −1), b / ∈D(β) ∪{n}}.
We claim that the restriction of the mapping φ to C(n, k) is a bijection from C(n, k) onto X. It will follow that c(n, k) = |X|.
6. To prove the claim, it is enough to show that φ(C(n, k)) = X, since φ is bijective.
Consider α and β satisfying φ(α) = (β, bj), with the notations in 3.
Then α = (b1, . . . bj−1, n + 1, bj, . . . , bn) and β = (b1, . . . bj−1, bj, . . . , bn). We remark that: - If bj ∈D(β) or if bj = n, then D(α) = D(β) and d(α) = d(β).
- If bj / ∈D(β) and bj ̸= n, then D(α) = D(β) ∪{bj} and d(α) = d(β) + 1.
Note indeed that n + 1 is not a large circular descent of α. It follows that for i ̸= j, bi is a large circular descent of α if and only bi is a large circular descent of β. Next, if bj ∈D(β), then bj ∈D(α). Moreover, if bj = n, then bj is neither in D(α) nor in D(β).
Finally, if bj / ∈D(β) and bj ̸= n, then bj ∈D(α).
(i) We show that φ(C(n, k)) ⊂X. Suppose that α ∈C(n, k).
If bj ∈D(β) or if bj = n, then by the remark, α and β have the same number k of large circular descents; hence (β, bj) ∈X1. If on the contrary, bj / ∈D(β) and bj ̸= n, then β has k −1 large circular descents and thus (β, bj) ∈X2.
(ii) We show now the inclusion X ⊂φ(C(n, k)). Suppose that (β, bj) ∈X. It is enough to show that α ∈C(n, k).
If (β, bj) ∈X1, then d(β) = k. Moreover, bj ∈D(β) or bj = n; thus by the remark, α and β have the same number of large circular descents, hence α ∈C(n, k).
If (β, bj) ∈X2, then d(β) = k −1, and similarly, α has one large circular descent more than β, α ∈C(n, k).
the electronic journal of combinatorics 27 (2020), #P00 25 7. The set C(n, k) has c(n, k) elements. Moreover, by 5, we know that c(n, k) = |X| = |X1|+|X2|. However it is clear that |X1| = c(n−1, k)(k+1) and |X2| = c(n−1, k−1)(n−k).
Thus the recursion follows.
5.3 Limiting poset In this section, we show that the limit (in some precise sense) of the posets of circular permutations in Sn is, when n tends to infinity, Young’s lattice of partitions.
Suppose that we have a family Pn, n ∈N, of ranked posets; suppose also that for any k, and for n large enough, the subposets {x ∈Pn, rk(x) ⩽k} are all isomorphic. We may then define the limit of these posets as the ranked poset P, whose elements of rank ⩽k form the poset {x ∈Pn, rk(x) ⩽k}, n large enough.
Recall that a partition of n is a multiset of positive integers whose sum is n (n is the weight of the partition); a partition of n+1 covers a partition of n if the latter is obtained from the latter by adding 1 to the multiset, or by replacing some element a by a+1. This is Young’s lattice, with rank equal to the weight function; see Definition 5.1.2.
Let us also recall how the poset of circular permutations is constructed. First we represent the smallest element (which is the canonical n-cycle) by (w0), where w0 is the word w0 := u0v0 = (⌊n/2⌋+ 1) . . . (n −1)n12 . . . ⌊n/2⌋.
For example, in the cases n = 5, 6, it is respectively 34512 and 456123.
Next each circular permutation is obtained, starting from (w0), by applying iteratively the covering relation (w) →(w′), defined by w = usrv, w′ = ursv or w = rus, w′ = sur, s > r + 1.
In the first case we say that the covering relation is inner, and the second one, it is outer. In this way, we obtain for each circular permutation a unique representative (w), as follows from Section 2.
Lemma 30. Let (w) ∈Sn be a circular permutation, represented as above. Then the following conditions are equivalent: i) We can obtain (w) from (w0) by a sequence of inner covering relations.
ii) The word w is a shuffle of u0 and v0.
Example 31. Let n = 8 and w = 56172384. Then, using inner relations from w0 we get: 56781234 →56718234 →56178234 →56172834 →56172384 Proof. i) ⇒ii). We proceed by induction on the number of inner relations p. For p = 0 there is nothing to do because w = w0. Assume that w is obtained from w0 by p+1 inner relations. Then we can write w = ursv where r, s are letters such that r + 1 < s (that is swapping r and s is the last inner relation to obtain w). Let w′ := usrv. This word is obtained from w0 using only p inner relations. Then by induction w′ is a shuffle of u0 the electronic journal of combinatorics 27 (2020), #P00 26 and v0. Therefore, if s, r are simultaneously in u0 or v0, they are consecutive and we must have s = r −1. This is in contradiction with our assumption. Thus s is a letter of u0 and r is a letter of v0. Swapping s and r does not change appearance order of the letters of both u0 and v0. Therefore, w is also a shuffle of u0 and v0. This ends the first direction.
The direction ii) ⇒i) is obvious.
Theorem 32. The posets of circular permutations have a limit poset, which is Young’s lattice.
Proof. 1. We prove first that the limit exists. Let k be an integer. Suppose that n ⩾2k.
We show that the subposet of elements of rank ⩽k is independent of n.
Let (x) be a circular permutation of Sn such that rk(x) ⩽k. Then x is obtained from w0 using at most k covering relations. Moreover, since the first covering relation is necessarily determined by the middle factor n1, and since k ⩽⌊n/2⌋, we never perform an outer covering relation. This implies that the subposet of elements of rank ⩽k is independent of n.
2. To finish the proof, it is enough to prove that if n ⩾2k, then the subposet of elements of rank ⩽k is isomorphic with the subposet of the Young lattice whose elements are the partitions of weight ⩽k.
By the previous part and because of Lemma 30, each element of rank ⩽k is of the form (w), where w a shuffle of the two words u0 and v0. Furthermore, the order is the inverse order of the right weak order on these words.
Note that each such shuffle, w say, is determined by the sequence a1, a2, a3, . . . , a⌊n/2⌋, where ai is the number of j = (⌊n/2⌋+ 1), . . . , (n −1), n, which are at the right of i in w. The rank of w is the sum of the ai’s. One has a1 ⩾a2 ⩾a3 . . ., and therefore the sequence determines a partition of some integer ⩽k.
This sequence defines the isomorphism between the two posets. We leave the details to the reader.
Example 33. We continue Example 31. Let w = 56172384. Then a1 is the number of blue letters which are on the right side of 1. That is a1 = 2. Proceeding this way for all other cases we get a2 = 1, a3 = 1, a4 = 0. Thus, the partition corresponding to w is 2, 1, 1, 0.
6 An interval in the affine symmetric group The two isomorphic posets of circular permutations and admitted vectors are isomorphic to some interval of the affine symmetric group, ordered by the left weak order. Thus we turn back to the geometric motivations of this article, following the second author’s articles [5, 6].
6.1 Preliminaries on ˜ Sn We begin by recalling some definitions and results on the affine symmetric group. Details may be found in the book by Anders Bj¨ orner and Francesco Brenti . This group is denoted ˜ Sn. It is the group of permutations f of Z satisfying the two conditions: the electronic journal of combinatorics 27 (2020), #P00 27 Figure 7: Illustration of the proof of Theorem 32. On the left hand side, the beginning of Young’s lattice, and the right hand side, the circular permutations lattice for n = 6.
the electronic journal of combinatorics 27 (2020), #P00 28 - f(x + n) = f(x) + n for any integer x; - f(1) + f(2) + · · · + f(n) = n(n + 1)/2.
An element of ˜ Sn is called an affine permutation.
Each usual permutation in Sn extends uniquely to an affine permutation, so that ˜ Sn contains the symmetric group Sn as a subgroup.
The group ˜ Sn is generated by the adjacent affine transpositions s0, s1, . . . , sn−1, defined by the condition that si exchanges i and i + 1 if i = 0, . . . , n −1.
Each affine permutation f is represented by a window [f(1), f(2), . . . , f(n)] and we write f = [f(1), f(2), . . . , f(n)].
For example in ˜ S4, we have the elements s1 = [2, 1, 3, 4], s0 = [0, 2, 3, 5], and [−3, 5, 0, 8]. The n numbers in a window are pairwise noncongruent modulo n, and their sum is n(n + 1)/2; windows are characterized by these two previous properties.
The position inversion set of f ∈˜ Sn is the set Invp(f) := {(i, j) | 1 ⩽i ⩽n, i < j, f(i) > f(j)}, which is also equal to {(i, j) | 1 ⩽i ⩽n, i < j, f(i) ⩾f(j)}, since i < j implies f(i) ̸= f(j).
The affine symmetric group is a Coxeter group, generated by the si’s, with length function ℓ(f) =| Invp(f) |. The left weak order, denoted simply ⩽, is defined on ˜ Sn by its covering relation: g covers f if for some i, one has g = sif and ℓ(g) = ℓ(f) + 1.
By definition of the product in ˜ Sn, one has the following lemma (a variant of p.
260).
Lemma 34. Let k = 0, . . . , n −1 and [a1, . . . , an] ∈˜ Sn. Define i, j in {1, 2, . . . , n} by ai ≡k+1 mod n and aj ≡k mod n. Then sk[a1, . . . , an] = [b1, . . . , bn], with bj = aj +1, bi = ai −1, and bl = al if al is noncongruent to k nor to k + 1 modulo n.
We also need the following result, which follows from Theorem 16 in (see also Theorem 5.3).
Lemma 35. Let f, g ∈˜ Sn and k = 0, . . . , n −1. Then: (i) the sets Invp(f) and Invp(skf) differ by one element; (ii) f ⩽g if and only if Invp(f) ⊂Invp(g).
6.2 A poset isomorphism We define a mapping F from the set of admitted vectors into Zn, where elements of the latter set are denoted by brackets [a1, . . . , an]. Let v be an admitted vector. Then we define F(v) = [a1, . . . , an], by ai = i + X p<i vpi − X i<p vip.
(6) the electronic journal of combinatorics 27 (2020), #P00 29 If we denote by e1, . . . , en the canonical basis of Zn, we have equivalently F(v) = [1, 2, . . . , n] + X r<s vrs(es −er).
Now, consider the element fc ∈˜ Sn defined by its window: fc = [c1, c2, . . . , cn] with c1 = −(1/2)n(n−3) and ci+1 = ci +n−1 for i = 1, . . . , n−1. Since n−1 is a generator of the additive group of integers modulo n, the ci are pairwise distinct modulo n. Moreover the sum of all ci is (c1 + 0) + (c1 + n −1) + (c1 + 2(n −1)) + · · · + (c1 + (n −1)(n −1)) = nc1 + (n −1)n(n −1) 2 = −1 2n2(n −3) + 1 2(n −1)2n = 1 2n((n −1)2 −n(n −3)) =1 2n(n2 −2n + 1 −n2 + 3n) =n(n + 1) 2 .
This shows that indeed fc ∈˜ Sn.
As an example, for n = 4, fc = [−2, 1, 4, 7] and for n = 5, fc = [−5, −1, 3, 7, 11].
Theorem 36. The mapping F is a poset isomorphism from the set of admitted vectors onto the interval [id, fc] with the left weak order.
Notice that this result has been proven by the second-named author for any affine Weyl group Wa in by using a geometrical approach. We give here a combinatorial proof and further, we construct the element fc.
See Figure 8 for this interval. The cover relation is explained in Remark 39. For example (24).[-1,1,3,5,7]=[-1,1-1,3,5+1,7]=[-1,0,3,6,7].
We first need several preliminary results.
Lemma 37. Let F(v) = [a1, . . . , an] and 1 ⩽i < j ⩽n. Then aj −ai = j −i + nvij − X i<p<j δipj(v) + X p<i δpij(v) + X j<p δijp(v).
As a consequence, aj −ai = nvij + r, with r ∈{1, . . . , n −1} and ⌊aj−ai n ⌋= vij.
the electronic journal of combinatorics 27 (2020), #P00 30 Proof. We have aj −ai = j −i + X p<j vpj − X j<p vjp − X p<i vpi + X i<p vip = j −i + X p<i vpj + vij + X i<p<j vpj − X j<p vjp − X p<i vpi + X i<p<j vip + vij + X j<p vip = j −i + 2vij + X i<p<j (vip + vpj) + X p<i (vpj −vpi) + X j<p (vip −vjp) = j −i + 2vij + X i<p<j (vij −δipj(v)) + X p<i (vij + δpij(v)) + X j<p (vij + δijp(v)) = j −i + nvij − X i<p<j δipj(v) + X p<i δpij(v) + X j<p δijp(v).
Let r = j −i −P i<p<j δipj(v) + P p<i δpij(v) + P j<p δijp(v). Then aj −ai = nvij + r.
Moreover, since the δ’s take the values 0 and 1, r is in the interval [j −i −(j −i −1), j − i + i −1 + n −j] = [1, n −1].
Proposition 38. The image of F is contained in the set of windows. If F(v) = [a1, ..., an], then a1 < · · · < an.
Proof. For the first assertion, it is enough to show that the ai as defined above are pairwise noncongruent modulo n, and that their sum is n(n + 1)/2.
Let i < j. Because of Lemma 37 we already know that ai −aj is nonzero modulo n.
Further we have n X i=1 ai = n X i=1 (i + X p<i vpi − X i<p vip) = n X i=1 i + X p<i vpi − X i<p vip = n X i=1 i = n(n + 1) 2 .
For the last assertion, let 1 ⩽i < j ⩽n. By Lemma 37, aj −ai > 0.
Remark 39. In this paragraph we recall how the cover relation works for an affine per-mutation f (see Lemma 34). More precisely we give the cover relation in terms of the the electronic journal of combinatorics 27 (2020), #P00 31 window corresponding to f. Let g be another affine permutation such that g covers f and let (i, j) be the transposition such that (i, j).f = g. Then (i, j).[a1, . . . , ai, . . . , aj, . . . , an] = [a1, . . . , ai −1, . . . , aj + 1, . . . , an].
Lemma 40. Suppose that f, g ∈˜ Sn with f = [a1, a2, . . . , an], g = [b1, b2, . . . , bn], a1 ⩽ · · · ⩽an and b1 ⩽· · · ⩽bn. Then: (i) Invp(f) is the set of pairs (i, p + kn) with 1 ⩽p < i ⩽n and k = 1, . . . , ⌊ai−ap n ⌋, (ii) f ⩽g if and only if for any i < j, ⌊aj−ai n ⌋⩽⌊bj−bi n ⌋.
Note that assertion (ii) is a particular case of Theorem 16 (3) in .
Proof. (i).
A pair as in the statement is an inversion, since i < p + kn and f(i) = ai ⩾f(p + kn) = f(p) + kn = ap + kn because k ⩽⌊ai−ap n ⌋⩽ ai−ap n .
Conversely, let (i, j) be an inversion of f.
Then 1 ⩽i ⩽n.
Since j > i ⩾1, we may write j = p + kn, 1 ⩽p ⩽n, k ⩾0. We have f(i) > f(j) = f(p) + kn. We cannot have p ⩾i, since otherwise we would have, by the increasing property, f(i) ⩽f(p) ⩽f(j), a contradiction. Hence p < i and since p + kn = j > i, we must have k ⩾1. Moreover, kn < f(i) −f(p) = ai −ap, hence k ⩽⌊ai−ap n ⌋.
(ii) follows from (i).
Lemma 41. Consider two admitted vectors v, v′ and 1 ⩽i < j ⩽n such that v′ ij = vij +1 whereas v′ rs = vrs for the other coordinates. Let k = aj mod n, where aj is the j-th coordinate of F(v). Then in ˜ Sn, one has the equality F(v′) = skF(v) and ℓ(F(v′)) = ℓ(F(v)) + 1.
Proof. Let F(v) = [a1, . . . , an] and F(v′) = [a′ 1, . . . , a′ n]. Then by Eq.(6): a′ i = i + X p<i v′ pi − X i<p v′ ip = i + X p<i vpi − X i<p vip −1 = ai −1, and similarly a′ j = aj +1, a′ l = al if l ̸= i, j. By Lemma 37 and Lemma 14, aj −ai is equal to j −i + (i −1) + (n −j) = n −1 ≡−1 mod n. Hence ai ≡k + 1 mod n.
Let skF(v) = [b1, . . . , bn]. By Lemma 34, we deduce that bi = ai −1 and bj = aj + 1, whereas bl = al for the other l. Hence bl = a′ l for all l, that is, F(v′) = skF(v).
We have aj = ai + n −1. Hence F(v) = [. . . , ai, . . . , aj, . . .] = [. . . , ai, . . . , ai + n −1, . . .], and F(v′) = [. . . , a′ i, . . . , a′ j, . . .] = [. . . , ai −1, . . . , aj + 1, . . .] = [. . . , ai −1, . . . , ai + n, . . .].
Thus F(v′) has the position inversion (j, i + n) (because F(v′) send this pair onto (ai + n, ai + n −1)), but F(v) has not this inversion (because the pair (j, i + n) is sent by F(v) onto (ai + n −1, ai + n)). It follows from Lemma 35 (i) that the length of F(v′) is one more than the length of F(v).
the electronic journal of combinatorics 27 (2020), #P00 32 Figure 8: The interval in the affine symmetric group ˜ S5.
the electronic journal of combinatorics 27 (2020), #P00 33 Lemma 42. Let [a1, . . . , an] ∈˜ Sn. Then ai = i − X i<p ⌊ap −ai n ⌋+ X p<i ⌊ai −ap n ⌋.
This formula could be proved as a corollary of Theorem 4.1, but we give a direct proof.
Proof. By definition of ˜ Sn, there exists a permutation σ ∈Sn and integers b1, . . . , bn such that ai = σ(i) + nbi and b1 + · · · + bn = 0. One has A := i − X i<p ⌊ap −ai n ⌋+ X p<i ⌊ai −ap n ⌋ = i − X i<p ⌊σ(p) + nbp −σ(i) −nbi n ⌋+ X p<i ⌊σ(i) + nbi −σ(p) −nbp n ⌋.
Note that for integers b, s, ⌊nb+s n ⌋= b + ⌊s n⌋, that −1 < σ(j)−σ(k) n < 1, and that σ(j) −σ(k) has integer part 0 or −1 depending on σ(j) −σ(k) ⩾0 or < 0. Thus A = i − X i<p (bp −bi) + X p<i (bi −bp) − X i<p,σ(p)<σ(i) (−1) + X p<i,σ(i)<σ(p) (−1).
We claim that P i<p,σ(p)<σ(i) 1 = σ(i) −i + P p<i,σ(i)<σ(p) 1. It follows that A = i + (n −1)bi − X p̸=i bp + σ(i) −i = σ(i) + nbi = ai.
because −P p̸=i bp = bi.
In order to prove the claim, note that since σ is a permutation, one has for fixed i: X i<p,σ(p)<σ(i) 1 + X i>p,σ(p)<σ(i) 1 = X σ(p)<σ(i) 1 = σ(i) −1, and X i>p,σ(p)<σ(i) 1 + X i>p,σ(p)>σ(i) 1 = X i>p 1 = i −1.
Hence X i<p,σ(p)<σ(i) 1 = σ(i) −1 − X i>p,σ(p)<σ(i) 1 = σ(i) −1 −(i −1 − X i>p,σ(p)>σ(i) 1).
We prove now Theorem 36. The mapping G = F −1, defined in the proof, appears implicitly in p.20 and Eq. (4.2.1) p. 375 in .
the electronic journal of combinatorics 27 (2020), #P00 34 Proof of Theorem 36. 1. It is easy to see that F sends the null admitted vector onto id = [1, 2, . . . , n]. We show that F(w) = fc, where w is the largest admitted vector, so that wij = j −i −1 (Corollary 20). Let F(w) = [a1, . . . , an]. Then a1 = 1 − X 1<p⩽n w1p = 1 − X 1<p⩽n (p −2) = 1 −(n −1)(n −2) 2 = 2 −n2 + 3n −2 2 = −n(n −3) 2 .
Moreover, by Lemma 37, noting that ai+1 −ai = 1 + X p<i δp,i,i+1(w) + X i+1<p δi,i+1,p(w) and vi,i+1 = 0, we have by Corollary 20 that it is equal to 1 + (i −1) + (n −i −1) = n −1. It follows that ai = ci, and F(w) = fc.
2.
Let v be an admitted vector and F(v) = f.
We have vij ⩽wij, so that by Proposition 38, Lemma 40 (ii) and Lemma 37, one has f ⩽fc. By the same results, id ⩽f. Thus f ∈[id, fc].
3. Let 0 ⩽i < j ⩽n. The injectivity of F follows from the equality vij = ⌊aj−ai n ⌋, see Lemma 37.
4. Given f ∈[id, fc], define G(f) = v ∈ZT by vij = ⌊aj−ai n ⌋. The previous paragraph shows that G ◦F = id.
We show that the image of G is contained in the set of admitted vectors. For this note that for an integer x, ⌊x/n⌋is the quotient of the Euclidean division of x by n; therefore ⌊(x + y)/n⌋= ⌊x/n⌋+ ⌊y/n⌋+ ϵ, with ϵ = 0 or 1. It follows that if i < j < k, ⌊ak−ai n ⌋−⌊ak−aj n ⌋−⌊aj−ai n ⌋is equal to 0 or 1, and vij is thus an admitted vector.
Observe that f = [a1, . . . , an] ∈˜ Sn has no inversion of the form (i, j) with 1 ⩽i < j ⩽n if and only if a1 < . . . < an. This is the case in particular for f = fc. Therefore each f ∈[id, fc] satisfies this increasing condition, because f ⩽fc implies that by Lemma 35, each inversion of f is an inversion of fc. The nonnegativity of the coefficients of G(f) follows.
5. The equality F ◦G = id follows from Lemma 42.
6. It remains to show that F and G are increasing functions. That F is increasing follows from Lemma 41 and the description in Lemma 15 of the covering relation for admitted vectors.
Suppose now that f, g ∈[id, fc] and f ⩽g. Then f, g satisfy the hypothesis of Lemma 40 and it follows by (ii) in this lemma that G(f) ⩽G(g).
By composing the bijections, we obtain a direct poset isomorphism from the interval [id, fc] into the set of circular permutations. Recall that ˜ Sn is the semidirect product of Sn and of the additive group {(k1, . . . , kn) ∈Zn, P i ki = 0}. The canonical projection the electronic journal of combinatorics 27 (2020), #P00 35 ˜ Sn →Sn is defined by f 7→σ, with σ(i) = f(i) MOD n, where x MOD n is the unique representative modulo n of x in {1, . . . , n}.
Corollary 43. The poset isomorphism V −1 ◦F −1 from [id, fc] into the set of circular permutations in Sn is defined by f 7→(σ), where σ is as above.
Proof. This follows from the definition of G and of the proof of Proposition 9 at the end of Section 3.
In a Coxeter group, an interval I is always a lattice. Indeed it is well known that the right weak order is a complete meet-semilattice (see Section 3.2). Therefore each pair of elements x, y ∈I has an infimum. Since I admits a upper bound it turns out that each pair of elements x, y ∈I has also a supremum. Further, it is known that such an interval is a semidistributive lattice (see Theorem 8.1 and the comment after it).
Consequently, the poset considered in the present article has the following property.
Corollary 44. The poset of circular permutations in Sn is a semidistributive lattice.
We leave to the reader to verify that fc is an involution in ˜ Sn if n is odd, but not when n is even.
Define the elements sijk of ˜ Sn, where 1 ⩽i < j ⩽n and k ∈Z for any x ∈{1, 2, . . . , n} by sijk(x) = j −kn if x = i i + kn if x = j x if x ̸= i, j.
Proposition 45. Let v be an admitted vector and f = F(v).
Consider the upwards path, in the Hasse diagram of the poset of admitted vectors, from the null vector to v, with successive labels (i1, j1), . . . , (ir, jr), where r is the rank of v in the poset. For p = 1, . . . , r, let kp be the number of q ⩽p such that (iq, jq) = (ip, jp).
Then in ˜ Sn, one has the factorization f = si1j1k1 ◦· · · ◦sirjrkr.
The proof is left to the reader.
Note that it follows from Theorem 36, and from Corollary 3.2.8 in , that the M¨ obius function of the poset takes its values in {−1, 0, 1}.
7 Circular line diagrams In this section, we introduce circular line diagrams. They are inspired by classical line diagrams for permutations, but instead of being on a flat plane, they are drawn on a cylinder, allowing us to use the circular properties of circular permutations.
We will see that if such diagram respects certain properties on its crossings, it links a circular permutation σ to its associated vector V (σ) (see Corollary 48). Thus, we illustrate the bijection V with a nice geometric and combinatorial object.
the electronic journal of combinatorics 27 (2020), #P00 36 We define a circular line diagram D as follows: we take a cylinder, and we put in clockwise order the numbers from 1 to n at the top and in any order at the bottom. For any 1 ⩽i ⩽n, we draw a continuous oriented line that goes from i at the top to i at the bottom, and does not cross itself.
We can represent this diagram on a 2-dimensional surface by either projecting it like in Figure 9, or unrolling it like in Figure 10.
By turning clockwise around the cylinder, we can list the numbers at the bottom start-ing with 1 and we obtain a word w = 1i1 · · · in−1. We then say that a circular line diagram D represents the circular permutation (w). For example, in Figures 9 and 10 below, we draw two circular line diagrams representing the same permutation (1, 6, 4, 2, 3, 5). 1 1 2 2 3 4 5 6 1 1 2 3 4 4 5 6 Figure 9: Projection of a circular line diagram representing the circular permutation (1, 6, 4, 2, 3, 5) Figure 10: Unrolling of a circular line diagram representing the circular permutation (1, 6, 4, 2, 3, 5) the electronic journal of combinatorics 27 (2020), #P00 37 We denote by Li(D) the line of D that goes from i at the top to i at the bottom. We define Li,j(D) to be number of times that Li(D) and Lj(D) cross each other.
For i < j, a crossing between the line Li(D) and Lj(D) is called a legal crossing if i) i + 1 < j, ii) Lj(D) crosses Li(D) from the right side of Li(D) to the left side of Li(D) when one follows Li(D) from the top to the bottom of the cylinder. See Figure 11.
We say that a circular line diagram is legal if all its crossings are legal. Note that a legal circular line diagram must satisfy Li,i+1(D) = 0 for all 1 ⩽i < n. The diagrams in Figures 9 and 10 are examples of legal diagrams.
Figure 11: With i + 1 < j, the left crossing is legal while the right one is illegal.
We call trivial circular line diagram of n, the legal circular line diagram with no crossing. The trivial circular line diagram of n represents the permutation (1, 2, ..., n).
Now, let σ and τ be two permutations such that (ij)σ(ij) = τ for i + 1 < j and Dσ a legal circular line diagram representing σ. One can build a legal diagram representing τ by interchanging i and j at the bottom of Dσ. Because ij is a large circular descent, it will only create a crossing between Li(Dσ) and Lj(Dσ), which will be a legal crossing.
Thus, this new diagram would be a legal circular line diagram representing τ.
Therefore, by starting with the trivial circular line diagram, for any circular permuta-tion σ, one can build a legal circular line diagram D representing σ. With this construc-tion, it is straightforward that D has the property that Li,j(D) = vij, where v = V (σ).
In fact, we will prove that this property holds for any legal diagrams. For example, if one counts the crossings of Figure 10, one will obtain the admitted vector of Figure 12.
We will also see, throughout the next results a links between upward path in the poset of circular permutation and legal line diagrams.
Considering two lines Li(D) and Lj(D) of a circular line diagram D that are crossing each other, we define the triangle i −j of D, denoted Ti,j(D), to be the triangle whose vertices are: the last intersection of Li(D) and Lj(D) and the points i and j on the bottom of the cylinder.
the electronic journal of combinatorics 27 (2020), #P00 38 Figure 12: Admitted vector associated to the circular permutation (1, 6, 4, 2, 3, 5) Lemma 46. Let D be a legal circular line diagram. If two lines are crossing each other, D has a triangle that does not intersect any other line than its boundary.
Proof. We say that a triangle is basic if at least one of its non-bottom edges does not cross any line. There always exist such triangles. Indeed, follow a line Li(D) and consider the triangle whose top vertex is the last crossing of Li(D). Order basic triangles by inclusion of their surfaces. Let T be a minimal element of this poset. Then T satisfies the conclusion of this lemma.
Theorem 47. Let D be a non-trivial legal circular line diagram representing a circular permutation σ. There exists a large circular ascent ah in σ such that we can create a legal diagram D′ that represent (ah, ah+1)σ(ah, ah+1) where Lk,l(D′) = Lk,l(D), for any pair k, l, except for ah, ah+1 where Lah,ah+1(D′) = Lah,ah+1(D) −1.
Proof. By Lemma 46 we know that there exists in D a pair i, j such Ti,j(D) is not crossed by any line. Therefore, i is a large circular ascent of σ, with j following i. By interchanging i and j on the bottom of the cylinder, we can untie the last crossing between Li(D) and Lj(D) and keep the rest of the diagram unchanged. This proves the result.
Corollary 48. Let σ be a circular permutation and D a legal circular line diagram rep-resenting σ. Let v = V (σ). Then Li,j(D) = vi,j for any pair i, j.
Proof. It is easy to convince ourself that the only legal circular line diagram that represents (1, 2, ..., n) is the trivial circular line diagram, which has no crossing.
The rest follows from Theorem 47 and the poset of circular permutations.
We finish this section by presenting two algorithms. The first one takes a circular permutation σ and returns a legal circular line diagram representing σ. Thus, we can deduce the admitted vector V (σ).
The second one takes an admitted vector v and returns a legal circular line diagram D that has the property that Lij(D) = vij. Thus, D represent the circular permutation V −1(v).
the electronic journal of combinatorics 27 (2020), #P00 39 For the first algorithm: take a cylinder and mark the numbers 1 to n on top in order and the circular permutation σ at the bottom; preferably with 1 at the bottom align with 1 at the top. We draw a line linking both 1. Now, for j going from 2 to n, we draw a line going from j at the top to j at the bottom without crossing j −1 and by only doing crossing if necessary. Doing so, we are only doing legal crossing, and thus, building a legal circular line diagram. The algorithm ends when we have placed the n lines.
For the second algorithm: take a cylinder and mark the numbers 1 to n on top in order. Then mark the number 1 on the bottom line below the 1 from the top line. Draw a line from 1 at the top to 1 at the bottom.
For j going from 2 to n, let rj = (a1, ..., aj−1) with ai = vi,j.
i) For k going from 1 to j −1, if ak ̸= 0 make Lj cross Lk and ak becomes ak −1.
ii) If rj ̸= (0, ..., 0) redo step i), if rj = (0, ..., 0) then make Lj drop to the bottom of the cylinder without crossing any line. Place j where Lj touches the bottom of the cylinder.
The algorithm ends when we have placed the n lines.
Note that, if we look at admitted vectors with the coordinates placed in a pyramidal shape, this algorithm is running through each diagonal a certain number of time, as shown in Figure 13.
We give an example of the second algorithm in Figure 14. The corresponding circular permutation is (1, 5, 3, 2, 4). The lines L1, L2, L3, L4 are directly drawn. We only detail the steps of the line L5.
Figure 13: Reading direction of an admitted vector v in the diagonal descent algorithm.
the electronic journal of combinatorics 27 (2020), #P00 40 Figure 14: The diagonal descent algorithm.
8 Further Questions First of all, we mention a link between our poset, and the poset in when specialized to An.
Conjecture 49. The poset of circular permutations and the poset in for the case An are isomorphic.
We do not have a proof of this isomorphism, but the sizes and the mathematical context of both posets suggest that they are isomorphic.
In their article, Gashi et al. were able to give a closed formula for the rank generating function of their poset (see , Eq.(5.7)). Thus, if the isomorphism exists, it is also the rank generating function of our poset.
An interesting open problem, linked to the work of Gashi et al., would be to look at other affine Coxeter groups and see if some nice combinatorial objects arise.
In 1927, K. Reidemeister introduced 3 moves in knot theory : the Reidemeister moves. One can observe that Reidemeister move of type III works well with legal circular line diagrams in the following way: one can pass from a legal diagram to another that represents the same permutation using such move.
the electronic journal of combinatorics 27 (2020), #P00 41 Figure 15: Reidemeister move of type III.
We think that more might occur: Conjecture 50. Let σ be a circular permutation and let Dσ, D′ σ be two legal circular line diagrams representing σ. Then we can obtain D′ σ from Dσ using only Reidemeister moves of type III.
From this conjecture comes other questions.
Question 51. In Conjecture 50, what is the number of Reidemeister moves of type III needed to obtain D′ σ from Dσ?
Question 52. What is the number of legal diagrams representing the same circular permutation?
In order to answer that last question, we think that we have to put an order on the crossings in a circular line diagram by taking their order of appearance along the cylinder from top to bottom in the same way as pseudo-line arrangements (see or ). With this order of crossings in mind, we can define the following equivalence relation: two diagrams are equivalent if they represent the same permutation and have the same order of appearance of the crossings. We then have the following conjecture: Conjecture 53. The number of circular line diagrams representing a permutation σ is exactly the number of paths in the poset of circular permutations from the permutation (1, 2, ..., n) to σ.
As Berenstein et al. explain in Section 2 of , we know that pseudo-line arrangements are in bijection with reduced expression of elements in Sn.
Question 54. Are circular line diagrams in bijection with reduced expression of circular permutations?
the electronic journal of combinatorics 27 (2020), #P00 42 Finally, an interesting object of finite poset are the maximal chains, more particularly, the number of these chains. We were able to calculate the number of maximal chains for n up to 9. This sequence does not appear in OEIS yet.
n number of maximal chains 2 1 3 1 4 2 5 60 6 279180 7 2563983592380 8 709628135986073648478240 9 105883974421582156430080943763802159752672 References Art of Problem Solving. USAMO 2010 Problem 2.
com/wiki/index.php/2010_USAMO_Problems/Problem_2.
Arkady Berenstein, Sergey Fomin, and Andrei Zelevinsky.
Parametrizations of Canonical Bases and Totally Positive Matrices. Advances in Mathematics, 122(1):49– 149, 1996.
Anders Bj¨ orner and Francesco Brenti. Affine Permutations of Type A. Electronic Journal of Combinatoics 3, (R18):35 p, 1996.
Anders Bj¨ orner and Francesco Brenti. Combinatorics of Coxeter Groups, volume 231 of Graduate Texts in Mathematics. Springer, New York, 2005.
Nathan Chapelier-Laget. Shi Variety Corresponding to an Affine Weyl Group. arXiv preprint: 2010.04310, 2020.
Nathan Chapelier-Laget. A Symmetric Group Action on the Irreducible Components of the Shi Variety Associated to W( e An). arXiv preprint: 2010.05602, 2020.
Henrik Erikson and Kimmo Erikson. Affine Weyl Groups as Infinite Permutations.
Electronic Journal of Combinatoics 5, (R18):32 p, 1998.
Sergey Fomin and Zelevinsky Andrei. Cluster Algebras I: Foundations. J. Amer.
Math. Soc., 15(2), 2002.
Q¨ endrim R Gashi, Travis Schedler, and David E Speyer. Looping of the Numbers Game and the Alcoved Hypercube.
Journal of Combinatorial Theory, Series A, 119(3):713–730, 2012.
Christian Kassel, Alain Lascoux, and Christophe Reutenauer. Factorizations in Schu-bert Cells. 2000.
Witold Kra´ skiewicz. Reduced Decompositions in Weyl Groups. European Journal of Combinatorics, 16(3):293–313, 1995.
T. Kyle Petersen. Eulerian Numbers. Birkha¨ user/Springer, 2015.
the electronic journal of combinatorics 27 (2020), #P00 43 Nathan Reading and David E Speyer. Sortable Elements in Infinite Coxeter Groups.
Transactions of the American Mathematical Society, 363:699–761, 2011.
Kurt Reidemeister. Elementare Begr¨ undung der Knotentheorie. In Abhandlungen aus dem Mathematischen Seminar der Universit¨ at Hamburg, volume 5, pages 24–32.
Springer, 1927.
Bruce Sagan. The Symmetric Group. Springer, 2000.
Jian Yi Shi. On Two Presentations of the Affine Weyl Groups of Classical Types.
Journal of Algebra, 221:360–382, 1999.
Richard P Stanley. Enumerative Combinatorics. Wadsworth and Brooks/Cole, 1, 1986.
the electronic journal of combinatorics 27 (2020), #P00 44 |
8687 | https://www.youtube.com/watch?v=0kwIGDDrlCU | Prove that a - b = 0 if and only if a = b (ILIEKMATHPHYSICS)
ILIEKMATHPHYSICS
8060 subscribers
129 likes
Description
2737 views
Posted: 13 Jan 2025
This video is part of the “Real Analysis” series I am making.
Thanks and enjoy the video!
Real Analysis Playlist:
Here is our description of the real number system using the ten axioms. This approach is similar to the one given in Apostol's Calculus book, Volume I, Second Edition.
We are given a set R, whose elements are called real numbers. We equip R with two binary operations, addition (+) and multiplication (•), so that for every pair of real numbers a and b, a + b and a • b are real numbers. In short, we may write a • b as ab.
Axiom 1 (Commutative Laws). For all a,b in R, a + b = b + a and ab = ba.
Axiom 2 (Associative Laws). For all a,b,c in R, (a + b) + c = a + (b + c) and (ab)c = a(bc).
Axiom 3 (Distributive Law). For all a,b,c in R, a(b + c) = ab + ac.
Axiom 4 (Existence of Zero). There exists an element 0 in R such that for all x in R, x + 0 = x.
Axiom 5 (Existence of Negatives). For all x in R, there exists an element -x in R such that x + (-x) = 0.
Axiom 6 (Existence of One). There exists an element 1 in R distinct from 0 such that for all x in R, 1 • x = x.
Axiom 7 (Existence of Reciprocals). For all nonzero x in R, there exists an element x^-1 in R such that x • x^-1 = 1.
Definition. For each a,b in R, we define a - b = a + (-b).
Definition. For each a,b in R where b ≠ 0, we define a/b = a • b^-1.
We equip R with a subset R+, whose elements we call positive real numbers, which satisfy the following axioms:
Axiom 8 (Closure of R+ under + and •). For all x,y in R+, x + y and xy belong to R+.
Axiom 9 (Trichotomy). For all x in R, exactly one of the following is true: x belongs to R+, x = 0, or -x belongs to R+.
Definition. Let a,b in R. We say “a ﹥ b” means a - b belongs to R+. We say “a ﹤ b” means b ﹥ a. We say “a ≥ b” means a ﹥ b or a = b. We say “a ≤ b” means b ≥ a.
Definition. We define R- = {x in R | -x in R+}. The elements of this set we call negative real numbers.
Definition. Let S be a subset of real numbers and b is an element of S. We say b is a smallest element of S if for all x in S, x ≥ b. Using properties of order, it is easy to check that b is unique.
Definition. Let S be a subset of real numbers and w is a real number.
(1) We say w is an upper bound of S if for all x in S, x ≤ w.
(2) We say w is a lower bound of S if for all x in S, x ≥ w.
(3) We say S is bounded above if S has an upper bound. We say S is bounded below if S has a lower bound. We say S is bounded if S is both bounded above and bounded below.
Axiom 10 (Completeness of R). Every nonempty subset S of real numbers that is bounded above has a least upper bound. That is, the set of all upper bounds of S has a smallest element.
14 comments
Transcript:
hello in this video we are going to prove the following theorem suppose A and B are real numbers then a minus B is equal to Z if and only if a is equal to B now in this series we are using a list of 10 axium for the real number system and I'll leave that list of axum in the description below now in this video we are going to be using the following axioms axum 1 and two are just the commutative and associative laws axm 4 tells us that there exists a real number which we call zero that has the property that x + 0 is equal x for all real numbers X and axm 5 tells us for all real numbers X there exists a real number which we called the negative X that has the property that X plus the x is equal to Z now Now using axum 5 our definition of subtraction is as follows suppose A and B are real numbers we Define a minus B to be a plus the ne B okay so now let's get into proving this theorem now we're trying to prove a minus b equals 0 if and only if a is equal to B and to prove that we're going to prove if a - b is equal 0 then a isal b and then we're going to prove if AAL B then Aus B is equal 0 so let's start off by proving the forward that is let's prove if Aus b equal 0 then a is equal to B and to prove that suppose Aus B is equal to Z from here we want to show that a is equal to B and we're going to establish that a is equal to B through a chain of equalities and through that chain of equalities we are going to be applying axi and 1 2 4 and 5 and the definition of [Music] subtraction first of all by axum 4 we know that a is equal to a + 0 then by axum 5 0 is equal to B + the Nega of B and then by axing 1 we can replace B plus the Nega of B with the Nega of B plus b and then by axum 2 we can move the parentheses around a + the B and then by definition of subtraction a plus the B is equal to a minus B and then by assumption a - b is equal 0 so we can replace a minus B with 0 and then by aing 1 0 + B is equal b + 0 and then by asking for b + 0 is equal to B so through this chain of equal we see that a is equal to B so we have shown if a minus B is equal 0 then a is equal to B now let's prove the other direction let's prove if a is equal to B then a minus B is equal to Z and to prove that suppose a is equal B from here we want to show that a minus B is equal Z and we are going to establish that a minus B is equal Z through a chain of equalities and through that chain of equalities we are going going to be using axum 5 and the definition of subtraction first of all since a is equal to B we can replace B with a and then by definition of subtraction a - A is equal to a + the negative a but then by a 5 a plus thetive a is equal to Z so through this chain of equalities we have a Lish that Aus B is equal to0 and so we have shown if a is equal to B then Aus B is equal Z and so we have proven both directions of this statement so we have proven the entire statement and so this could pleas the proof and so yeah that's pretty much it for this video |
8688 | https://www.endotext.org/chapter/evaluation-of-amenorrhea-anovulation-and-abnormal-bleeding/figure1-41/ | figure1 - Endotext
The best Clinical Endocrinology source in the world, and FREE.
Endotext requires free registration
Login or Register
Search for:
Menu
Skip to content
Home / Sections
Clinical Guidelines
Webinar Case Studies
Editorial Team
Country Associates
About
Instructions for Authors
figure1
Published May 16, 2013 at 550 × 271 in Evaluation of Amenorrhea, Anovulation, and Abnormal Bleeding. Next →
Figure 1. Flow diagram for the laboratory evaluation of amenorrhea. Such a scheme must be considered as an adjunct to the clinical evaluation of the patient. CAH -= congenital adrenal hyperplasia; DHEAS = dehydroepiandrosterone sulfate; FSH = follicle-stimulating hormone; HCA = hypothalamic chronic anovulation; PCO = polycystic ovary syndrome; PRL = prolactin; T = testosterone; TSH = thyroid-stimulating hormone. Originally from Rebar RW, The ovaries. In: Smith LH Jr, ed. Cecil textbook of medicine, ed. 18. Philadelphia. WB Saunders, 1992:1367)
Agreement on Use
Supporters
Donations
Privacy Policy
Instructions for Authors (5/03/23)
Copyright © 2000-2025, MDText.com, Inc. All rights reserved. |
8689 | https://webbook.nist.gov/cgi/inchi/InChI%3D1S/U | uranium
Jump to content
National Institute of Standards and Technology
NIST Chemistry WebBook, SRD 69
Home
Search
Name
Formula
IUPAC identifier
CAS number
More options
NIST Data
SRD Program
Science Data Portal
Office of Data and Informatics
About
FAQ
Credits
More documentation
uranium
Formula: U
Molecular weight: 238.02891
IUPAC Standard InChI:InChI=1S/U Copy
IUPAC Standard InChIKey:JFALSRSLKYAFGM-UHFFFAOYSA-N Copy
CAS Registry Number: 7440-61-1
Chemical structure:
This structure is also available as a 2d Mol file
Information on this page:
Notes
Other data available:
Gas phase thermochemistry data
Condensed phase thermochemistry data
Gas phase ion energetics data
Data at other public NIST sites:
NIST Atomic Spectra Database - Lines Holdings (on physics web site)
NIST Atomic Spectra Database - Levels Holdings (on physics web site)
NIST Atomic Spectra Database - Ground states and ionization energies (on physics web site)
X-ray Photoelectron Spectroscopy Database, version 5.0
Options:
Switch to calorie-based units
Notes
Go To:Top
Data from NIST Standard Reference Database 69: NIST Chemistry WebBook
The National Institute of Standards and Technology (NIST) uses its best efforts to deliver a high quality copy of the Database and to verify that the data contained therein have been selected on the basis of sound scientific judgment. However, NIST makes no warranties to that effect, and NIST shall not be liable for any damage that may result from errors or omissions in the Database.
Customer support for NIST Standard Reference Data products.
© 2025 by the U.S. Secretary of Commerce on behalf of the United States of America. All rights reserved.
Copyright for NIST Standard Reference Data is governed by the Standard Reference Data Act.
Privacy Statement
Privacy Policy
Security Notice
Disclaimer (Note: This site is covered by copyright.)
Accessibility Statement
FOIA
Contact Us |
8690 | https://artofproblemsolving.com/wiki/index.php/Trigonometry?srsltid=AfmBOopW39Ey9s9B-Tb4sN_-UmYmnWw6Zhmt3Jle8u4TVtYEHKGh3CzP | Art of Problem Solving
Trigonometry - AoPS Wiki
Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical CampusVisit the Virtual Campus
Sign In
Register
online school
Class ScheduleRecommendationsOlympiad CoursesFree Sessions
books tore
AoPS CurriculumBeast AcademyOnline BooksRecommendationsOther Books & GearAll ProductsGift Certificates
community
ForumsContestsSearchHelp
resources
math training & toolsAlcumusVideosFor the Win!MATHCOUNTS TrainerAoPS Practice ContestsAoPS WikiLaTeX TeXeRMIT PRIMES/CrowdMathKeep LearningAll Ten
contests on aopsPractice Math ContestsUSABO
newsAoPS BlogWebinars
view all 0
Sign In
Register
AoPS Wiki
ResourcesAops Wiki Trigonometry
Page
ArticleDiscussionView sourceHistory
Toolbox
Recent changesRandom pageHelpWhat links hereSpecial pages
Search
Trigonometry
Trigonometry is the study of relations between the side lengths and angles of triangles through the trigonometric functions. It is a fundamental branch of mathematics, and its discovery paved the way towards countless famous results.
In contest math, trigonometry is an integral subfield of both geometry and algebra. Many essential results in geometry are written in terms of the trigonometric functions, such as the Law of Sines and the Law of Cosines; many more, such as Stewart's Theorem, are most easily proven using trigonometry. In algebra, expressions involving the trigonometric functions appear frequently on contests. These are solved by clever usage of the trigonometric functions' countless identities, which can simplify otherwise unwieldy equations.
Outside of competition math, trigonometry is the backbone of much of analysis. In particular, Fourier Analysis is written almost entirely in the language of the trigonometric functions.
Contents
[hide]
1 Definitions
1.1 Right triangle definition
1.2 Unit circle definition
1.3 Taylor series definition
2 Applications in Geometry
2.1 Law of Sines
2.2 Law of cosines
3 Trigonometric identities
4 See also
Definitions
The trigonometric functions can be defined in several equivalent ways. The definition usually taught first is the right triangle definition, for its ease of access. An intermediate to olympiad geometry course usually uses the unit circle definition of trigonometry. Beyond the scope of contest math, the Taylor series definition of trigonometry is preferred in order to extend trigonometry to a complex domain.
Right triangle definition
The right triangle definition of trigonometry involves the ratios between edges of a right triangle, with respect to a given angle. The definitions below will be referring to angle , with side lengths specified in the diagram. Because angle must be less than for the triangle to stay right, these definitions only work for acute angles.
Sine: The sine of angle , denoted , is defined as the ratio of the side opposite to the hypotenuse.
Cosine: The cosine of angle , denoted , is defined as the ratio of the side adjacent to the hypotenuse.
Tangent: The tangent of angle , denoted , is defined as the ratio of the side opposite to the side adjacent to .
A common mnemonic to remember this is SOH-CAH-TOA, where Sine = Opposite / Hypotenuse, Cosine = Adjacent / Hypotenuse, and Tangent = Opposite / Adjacent
More uncommon are the reciprocals of the trigonometric functions, listed below.
Cosecant: The cosecant of angle , denoted , is defined as the reciprocal of the sine of .
Secant: The secant of angle , denoted , is defined as the reciprocal of the cosine of .
Cotangent: The cotangent of angle , denoted , is defined as the reciprocal of the tangent of .
The right triangle definition is most commonly taught in introductory geometry classes for its simplicity. However, it has its limitations. It only works if is right, which means that the trigonometric functions are only defined when angle is acute.
Even though it is defined using right triangles, trigonometry is just as useful when used on acute and obtuse triangles. The Law of Sines and Law of Cosines mentioned below generalize the right triangle definition to include all triangles.
Unit circle definition
Consider the unit circle, the circle with radius one centered at the origin. Starting at , walk a distance counterclockwise around the unit circle, as shown in the diagram. The coordinates of this point are defined to be .
As for the other trigonometric functions, is defined to be the ratio of to , and cosecant, secant, and cotangent are defined to be the reciprocals of sine, cosine, and tangent, respectively.
The benefit of this definition is that it matches the right triangle definition for acute angles, but extends their domain from acute angles to all real-valued angles. As such, this definition is usually preferred in intermediate to olympiad geometry settings.
Taylor series definition
The Taylor series for sine and cosine are used as their definitions in all higher mathematics. This meets the rigorous standards of real analysis, and gives a concrete way to extend the definitions of trigonometric functions from the real numbers to the full complex plane. The Taylor series for sine and cosine are shown below: These formulas are not used in high school math competitions. However, they do appear on the Putnam and other undergraduate competitions.
Applications in Geometry
While trigonometry is useful at any level, intermediate competitions are particularly fond of geometry problems demanding trigonometry. In addition to those mentioned, here are some highlights of the applications of trigonometry to geometry:
Law of Sines
The Law of Sines states that in any , where is the side opposite to , opposite to , opposite to , and is the circumradius of . The law of sines is particularly handy in problems involving the circumradius, seeing extremely wide usage in intermediate geometry.
Law of cosines
The Law of Cosines states that in any , where is the side opposite to , opposite to , and opposite to . It is a generalization of the Pythagorean Theorem and is used to prove several famous results, such as Heron's Formula and Stewart's Theorem. However, it sees limited applicability compared to the Law of Sines, as usage of the Law of Cosines can get algebra-heavy. It is helpful to memorize common, "nicer" values of sine and cosine as it can come in handy in contests, especially if you wish to apply either this or the Law of Sines to problems.
Trigonometric identities
Trigonometric identities are expressions true for all inputs involving the trigonometric functions. Due to the natural relationship between their definitions, these identities run numerous. In contest math, the most useful of these are:
Pythagorean identities
Angle addition identities
Double angle identities
Half angle identities
Sum-to-product identities
Product-to-sum identities
See also
Trigonometric identities
Law of Sines
Law of Cosines
Stewart's Theorem
Retrieved from "
Categories:
Trigonometry
Definition
Art of Problem Solving is an
ACS WASC Accredited School
aops programs
AoPS Online
Beast Academy
AoPS Academy
About
About AoPS
Our Team
Our History
Jobs
AoPS Blog
Site Info
Terms
Privacy
Contact Us
follow us
Subscribe for news and updates
© 2025 AoPS Incorporated
© 2025 Art of Problem Solving
About Us•Contact Us•Terms•Privacy
Copyright © 2025 Art of Problem Solving
Something appears to not have loaded correctly.
Click to refresh. |
8691 | https://www.merckmillipore.com/IN/en/product/1-Propanol,MDA_CHEM-100996 | Merck | Israel | Life Science Products & Service Solutions
Skip to Content
Products
Cart 0
IL EN
Products
Products Applications Services Documents Support
Analytical ChemistryCell Culture & AnalysisChemistry & BiochemicalsClinical & DiagnosticsFiltration
Greener Alternative Products
Industrial MicrobiologyLab AutomationLabwareMaterials ScienceMolecular Biology & Functional GenomicsmAbs Development & ManufacturingmRNA Development & ManufacturingPharma & Biopharma ManufacturingProtein BiologyWater Purification
Analytical ChemistryCell Culture & AnalysisChemistry & SynthesisClinical & DiagnosticsEnvironmental & Cannabis TestingFood & Beverage Testing & ManufacturingGenomicsMaterials Science & EngineeringMicrobiological TestingmAbs Development & ManufacturingmRNA Development & ManufacturingPharma & Biopharma ManufacturingProtein BiologyResearch & Disease AreasWater Purification
Contract ManufacturingContract TestingCustom ProductsDigital Solutions for Life ScienceIVD Development & ManufacturingProduct ServicesSupport
Safety Data Sheets (SDS)
Certificates of Analysis (COA)
Certificates of Origin (COO)
Certificates of Quality (COQ)
Customer Support
Contact Us
Get Site Smart
FAQ
Quality & Regulatory
Calculators & Apps
Webinars
Login
Order Lookup
Quick Order
Cart 0
All In 1 Cart Premier brands partner with us to benefit you Explore Now
Online Custom Media & Chemical Requests With mConfig™ Configurator get feasibility checks & expert guidance Get Access
Unlock Your Yields Discover our new CHOZN® Elite cell line, a next generation CHO host to boost your titer up to 7-13 g/L! Explore Here
Be the Base Editor Boss PURedit® Cas9 Cytosine Base Editors are your streamlined gene editing solution Get Editing
Welcome to Merck
New Arrivals
Slide 1 of 4
Sigma-Aldrich 935042 AC1 Screen Printed Electrodes
Quick View
Sigma-Aldrich 934968 Platinum Microelectrode
Quick View
Sigma-Aldrich SML3846 Motixafortide trifluoroacetate
Quick View
Supelco P-179 Phosphatidylethanol 16:0/18:1 (PEth 16:0/18:1) solution
Quick View
Special Offers & Activities
View our list of seasonal offers and specific product promotions for various research areas.
Learn More
Need Support?
Call 08-9484222 / 08-9484100
Contact Customer Service
Shop Top Categories
Slide 1 of 12
Certified Reference Materials
Microbial Filtration
Solvents
Microbial Culture Media
Cell Separation Media for Research, Diagnostic & Pharmaceutical Applications
Millex® Syringe Filters
Glassware
Pathogen Screening Kits, Instruments & Disposables
Amicon® Centrifugals
Millipore® Filter Membranes
Milli-Q® Lab Water Purification Consumables
Greener Alternative Products
Sign In to Receive Customized Recommendations Based on Your Preferences
Best Sellers
Popular Products
New Arrivals
Slide 1 of 7
1 of 2
Sigma-Aldrich D2650 Dimethyl sulfoxide
Quick View
USP 1044651 Atovaquone
Quick View
Sigma-Aldrich T0440 3,3′,5,5′-Tetramethylbenzidine (TMB) Liquid Substrate System for ELISA
Quick View
Sigma-Aldrich M2933 MES hydrate
Quick View
SAFC 24366C EX-CELL® Advanced CHO Fed-batch Medium
Quick View
Sigma-Aldrich F7524 Fetal Bovine Serum
Quick View
Sigma-Aldrich T9531 D-(+)-Trehalose dihydrate
Quick View
Slide 1 of 10
1 of 3
Sigma-Aldrich D2650 Dimethyl sulfoxide
Quick View
Sigma-Aldrich 34860 Methanol
Quick View
Sigma-Aldrich 258148 Hydrochloric acid
Quick View
Sigma-Aldrich S9888 Sodium chloride
Quick View
Sigma-Aldrich D8418 Dimethyl sulfoxide
Quick View
Sigma-Aldrich 34851 Acetonitrile
Quick View
Sigma-Aldrich 179124 Acetone
Quick View
Sigma-Aldrich 34863 2-Propanol
Quick View
Sigma-Aldrich 459844 Ethyl alcohol, Pure
Quick View
Sigma-Aldrich 472301 Dimethyl sulfoxide
Quick View
Slide 1 of 4
1 of 1
Sigma-Aldrich 935042 AC1 Screen Printed Electrodes
Quick View
Sigma-Aldrich 934968 Platinum Microelectrode
Quick View
Sigma-Aldrich SML3846 Motixafortide trifluoroacetate
Quick View
Supelco P-179 Phosphatidylethanol 16:0/18:1 (PEth 16:0/18:1) solution
Quick View
Local Office Information
Meet the team.
Learn More
Virtual Seminars
We are committed to sharing our technical expertise and supporting your science through our Virtual Seminar Program.
Learn More
CRISPR
We are trusted worldwide for innovative solutions in CRISPR/Cas9 products and services.
Learn More
Get Site Smart
Discover our latest website features and enhance your experience with Merck.
Learn More
Discover Our Brands
Merck offers consultative services to help customers optimize their workflows and overcome complex challenges in areas of life science research, biomanufacturing, and analytical testing.
Millipore®
Preparation, Separation, Filtration & Monitoring Products
Sigma-Aldrich®
Lab & Production Materials
SAFC®
Pharma & Biopharma Raw Material Solutions
Supelco®
Analytical Products
Milli-Q®
Lab Water Solutions
BioReliance®
Pharma & Biopharma Manufacturing & Testing Services
Popular Searches
EthanolDimethyl sulfoxideMethanolAcetoneSodium hydroxideAcetic acidHydrochloric acidEthyl acetateSodium chlorideAcetonitrileTolueneDichloromethaneSulfuric acidWaterHexane
Top
Sign In To Continue
To continue reading please sign in or create an account.
Sign InDon't Have An Account?Register
SupportCustomer SupportContact UsFAQSafety Data Sheets (SDS)Certificates (COA/COO)Quality & RegulatoryCalculators & AppsWebinars
OrdersQuick OrderCustom ProductseCommerce Solutions
CompanyAbout UsResponsibilityEventsPress ReleasesProgramsCareersOffices
Social Media
Merck
Research. Development. Production.
We are a leading supplier to the global Life Science industry with solutions and services for research, biotechnology development and production, and pharmaceutical drug therapy development and production.
Sigma-Aldrich® SolutionsBioReliance® SolutionsMillipore® SolutionsSAFC® SolutionsMilli-Q® SolutionsSupelco® Solutions
© 2025 Merck KGaA, Darmstadt, Germany and/or its affiliates. All Rights Reserved, including Text and Data Mining for AI training and similar technologies.
Reproduction of any materials from the site is strictly forbidden without permission.
Site Use Terms|Privacy Policy|General Terms and Conditions of Sale|Copyright Consent|Site Map
English - EN
Learn More
Cookie Disclaimer
We use cookies in our website to give you the most relevant experience. By clicking or navigating the site, you are accepting the use of all our cookies according to our Cookie Policy / Privacy Statement. You are free to decide in the Cookie Settings which categories you would like to allow. Please note that depending on what you select, the full functionality of the website may no longer be available. You may review and change your choices at any time. Further information can be found in our Privacy Policy
Cookie Settings Reject Unnecessary Cookies Accept Cookies
We recognize your browser's Opt-Out Preference Signal.
To honor your signal, Targeting & Social Media Cookies from our websites have been disabled. But you may choose to enable them by adjusting your preferences below.
Cookie Preferences
Necessary
Functional
Targeting
Social Media Cookies
Cookie Preferences
When you visit our website, we store cookies on your browser to collect information. The information collected might relate to you, your preferences or your device, and is mostly used to make the site work as you expect it to and to provide a more personalized web experience. However, you can choose not to allow certain types of cookies, which may impact your experience of the site and the services we are able to offer. Click on the different category headings to find out more and change our default settings according to your preference. You cannot opt-out of our First Party Strictly Necessary Cookies as they are deployed in order to ensure the proper functioning of our website (such as prompting the cookie banner and remembering your settings, to log into your account, to redirect you when you log out, etc.). For more information about how we use and share your data, please follow this link.
Privacy Statement
Necessary
Always Active
These cookies are necessary to provide and safeguard the functionalities of our services and for the website to operate. We set these cookies as required and in response to your actions, and they depend on how you choose to access and interact with our website as well as specific service(s) requested.
Functional
[x] Functional
These cookies enable the provision of advanced functionalities and are used for personalizing your browsing experience on our websites. We set these cookies in response to your actions, and they depend on your interactions with our website as well as any specific requests or preference(s) settings you have elected.
Targeting
[x] Targeting
These cookies may be set to learn more about your interests through your interaction with our website and are analyzed to provide better website performance and experiences, but also to enhance our advertising activities. We may use service providers to support our usage of these cookies, which work by uniquely identifying your browser and/or device. By integrating these cookies, we aim to learn more about your interests and surfing behavior to place our advertising in a more effective and relevant manner.
Social Media Cookies
[x] Social Media Cookies
These cookies may be set to learn more about your interests through your interactions with our website, as well as your interactions with third parties’ websites our external marketing agencies are partnered with to show you relevant ads throughout your overall internet browsing experience. These cookies work by uniquely identifying your browser(s) and/or device(s) and sharing your browsing activity with third party advertisement companies. By integrating these cookies, we aim to learn more about our customers interests and overall internet browsing behavior to be able to place our advertising in a more effective and relevant manner.
Cookie List
Consent Leg.Interest
[x] checkbox label label
[x] checkbox label label
[x] checkbox label label
Clear
[x] checkbox label label
Apply Cancel
Confirm My Choices
Disable Unnecessary Cookies Enable All |
8692 | https://s1b0d3d77136c1679.jimcontent.com/download/version/1532278084/module/14147538830/name/diapos%20u3.pdf | 22/07/2018 1 UNIDAD III.
Teorías y criterios de falla por cargas dinámicas Dr. Roberto Carlos García Gómez 2018 Departamento de Metal-Mecánica ITTG-rcgg 1 Programa de la unidad. 3.1. Cargas dinámicas 3.2. Fatiga 3.3. Esfuerzo fluctuante 3.4. Límite de resistencia a la fatiga 3.5. Factores que modifican la resistencia a la fatiga 3.6. Sensibilidad de la muesca 3.7. Teoría de Goodman 3.8. Teoría de Soderberg 3.9. Teoría de Gerber 3.10. Resistencia a la fatiga por torsión 3.11. Análisis de cargas de impacto.
Departamento de Metal-Mecánica ITTG-rcgg 2 22/07/2018 2 Introducción Muchos de los elementos de máquinas son sometidos a cargas variables. El comportamiento de los materiales bajo este tipo de carga es diferente a aquel bajo cargas estáticas; mientras que una pieza soporta una gran carga estática, la misma puede fallar con una carga mucho menor si ésta se repite un gran número de veces. Los esfuerzos variables en un elemento tienden a producir grietas que crecen a medida que éstos se repiten, hasta que se produce la falla total; este fenómeno se denomina fatiga.
Departamento de Metal-Mecánica ITTG-rcgg 3 Por lo tanto, el diseño de elementos sometidos a cargas variables debe hacerse mediante una teoría que tenga en cuenta los factores que influyen en la aparición y desarrollo de las grietas, las cuales pueden producir la falla después de cierto número de repeticiones (ciclos) de esfuerzo.
La teoría que estudia el comportamiento de los materiales sometidos a cargas variables se conoce como teoría de fatiga.
Nosotros estudiaremos algunos métodos para este fin.
Departamento de Metal-Mecánica ITTG-rcgg 4 22/07/2018 3 3.1 Cargas dinámicas En las piezas de máquinas se pueden hallar diferentes tipos de solicitaciones (cargas), las cuales se pueden distinguir en dos tipos característicos: ESTÁTICAS y DINÁMICAS según que no varíen o que varíen con el tiempo. También se las suele llamar con otros apelativos. Así pues las cargas estáticas suelen denominarse “ESTACIONARIAS” o “MONOTONICAS” y a las cargas dinámicas se las suele denominar “CICLICAS” o “NO ESTACIONARIAS” o “TRANSITORIAS”.
Departamento de Metal-Mecánica ITTG-rcgg 5 Diseño y calculo por resistencia Diseño y calculo para la duración Departamento de Metal-Mecánica ITTG-rcgg 6 Como es lógico se pueden desprender varias configuraciones de solicitación dinámica, sin embargo dentro de ellas existe una muy característica y que por su sencillez descriptiva (desde el punto de vista matemático) será la que se utilice en los modelos de análisis de falla por fatiga.
En la sig. figura, se puede apreciar la denominada carga cíclica o periódica, que conduce a las tensiones cíclicas o periódicas. El tipo de fuerzas y/o tensiones cíclicas puede tener diferentes casos, tales como axiales (tractivas o compresivas), flexional o torsionales.
Es claro que la forma más elemental de representación de este tipo de solicitación y/o tensión puede seguir una ley sinusoidal, amén de otras que puedan ser fácilmente representables en términos matemáticos.
22/07/2018 4 Departamento de Metal-Mecánica ITTG-rcgg 7 De acuerdo a los valores relativos que tengan las expresiones anteriores se pueden presentar cuatro casos característicos: Departamento de Metal-Mecánica ITTG-rcgg 8 Completamente alternante o Invertida.
Caso General o de tensión media no nula: 22/07/2018 5 Departamento de Metal-Mecánica ITTG-rcgg 9 Pulsante tractiva Pulsante compresiva La mayoría de las fallas de los elementos de máquinas implica condiciones de carga que fluctúan con el tiempo. Se enfatiza la comprensión y prevención de la falla de un componente bajo carga cíclica y de impacto. En lugar de considerar una variación general de la carga con el tiempo (carga fluctuante), sólo se consideran las cargas cíclicas o una secuencia de carga que se repite.
Las cargas fluctuantes inducen esfuerzos de fluctuación (cíclicos), que con frecuencia resultan en la falla por daño acumulativo.
Los esfuerzos y las deformaciones debidas a cargas de impacto son mucho mayores que los causados por una carga estática. Las propiedades físicas de un material son una función de la velocidad de carga. Afortunadamente entre más rápida sea la aplicación de una carga, mayores serán las resistencias a la fluencia y a la rotura.
Departamento de Metal-Mecánica ITTG-rcgg 10 22/07/2018 6 3.2 Fatiga El mecanismo de Fatiga es uno de los más complejos fenómenos en el estudio de falla en piezas sometidas a la acción de cargas dinámicas. Este fenómeno puede aparecer súbitamente y sin aviso previo. Este fenómeno está asociado principalmente a la presencia de patrones de carga dinámicos de tipo cíclico como los vistos en el apartado anterior. El fenómeno de rotura súbita ya era conocido desde los albores de la era industrial asociado a la rotura o falla catastrófica de puentes con el advenimiento del ferrocarril se inicio en distintos países del mundo el estudio científico de dicho fenómeno, para dar la explicación racional a la superstición que la Muerte hollaba en los puentes.
Departamento de Metal-Mecánica ITTG-rcgg 11 Departamento de Metal-Mecánica ITTG-rcgg 12 Otro Ejemplo puede verse en la cubierta superior de primera clase de un avión comercial Boeing 737-200 se separó en el aire y cuya causa fue la fatiga del material asociada con micro-corrosión.
22/07/2018 7 Departamento de Metal-Mecánica ITTG-rcgg 13 Desde mediados del siglo XIX se sabe que en aquellas piezas sometidas a cargas variables, con un numero grande de aplicaciones se producía la rotura de la pieza prácticamente sin deformaciones; a este fenómeno se lo llamo “fatiga”, por semejanza al cansancio humano.
Los distintos estudios efectuados, condujeron a distintas teorías, que tomadas en su conjunto pueden dejar las siguientes conclusiones: A) Los aceros de construcción de maquinas y en general los metales, no poseen homogeneidad en su estructura, ni continuidad de resistencia (aun a pesar de la hipótesis del continuo de la elasticidad clásica) en los metales que poseen cristales de una sola fase, que variar de tamaño y orientación, hacen que la resistencia promedio sea sólo valida para solicitaciones estáticas, debido a que estas solicitaciones permiten un re-acomodamiento adaptativo de los cristales a medida que aumenta la carga.
B) A su vez las cargas variables tienen su aplicación prácticamente instantánea, lo cual no deja mucho margen temporal para el reacomodamiento elástico, siendo este el motivo de la separación de los cristales en aquellos lugares donde hay menor cohesión intercristalina, generando el inicio de una microfisura, la que por el efecto de concentración de tensiones producida por la microentalla, crea en esa zona un incremento de tensiones que va aumentando rápidamente la fisura hasta que la sección resistente no puede soportar la carga, produciéndose en ese instante la rotura súbita de la pieza.
Departamento de Metal-Mecánica ITTG-rcgg 14 22/07/2018 8 C.
Las microfisuras o grietas iniciales de fatiga comienzan sobre la superficie de las piezas en varios puntos simultáneamente y se propagan a los sustratos inferiores. Estas grietas que son normalmente muy pequeñas y difíciles de observar, pero que se propagan en conjunto ante la presencia de un defecto dominante, pueden llevar rápidamente a la catástrofe.
En consecuencia la vida o duración de una pieza se puede maximizar si se tienen en cuenta las siguientes pautas: Minimizando defectos superficiales.
Maximizando el tiempo de iniciación.
Maximizando el tiempo de propagación.
Maximizando la longitud crítica de la grieta.
Departamento de Metal-Mecánica ITTG-rcgg 15 Las roturas por fatiga tienen dos zonas características.
En la siguiente figura se muestran las diferentes zonas de la sección fallada. La (1) es la zona de rotura por fatiga neta, donde puede apreciarse un granulado liso y fino, casi aterciopelado al tacto. Por otro lado la (2) es la Zona de rotura súbita, es aquella parte de la sección resistente original que por ser menor que la sección necesaria a la carga nominal se rompe abruptamente, dejando una superficie de grano grueso y deforme con un cierto brillo en los aceros Departamento de Metal-Mecánica ITTG-rcgg 16 solicitación variable suave solicitación variable intensa 22/07/2018 9 Sin embargo la sección (2) puede presentar dos sub-zonas, una característica y apariencia superficial más bien gruesa en comparación con la zona (1), luego la zona de falla final (normalmente pequeña) puede presentar un aspecto que da la idea de una fractura frágil o bien presentar un aspecto de ligero deslizamiento fibroso que sugiere una rotura dúctil Departamento de Metal-Mecánica ITTG-rcgg 17 Carga rotativa Departamento de Metal-Mecánica ITTG-rcgg 18 22/07/2018 10 Departamento de Metal-Mecánica ITTG-rcgg 19 Carga preponderantemente torsional Departamento de Metal-Mecánica ITTG-rcgg 20 22/07/2018 11 El término “fatiga” fue usado por primera vez por Poncelet en 1839, para describir la situación de falla de los materiales sometidos a cargas variables.
Debido a que la falla por fatiga tiene apariencia frágil, se pensaba que el material se había “cansado” y hecho frágil después de soportar un cierto número de fluctuaciones de esfuerzo.
Similarmente, en 1843, Rankine publicó un estudio sobre las causas de la ruptura inesperada de los muñones de los ejes de ferrocarril, en el cual decía que el material dúctil se había cristalizado y hecho frágil debido a la fluctuación de los esfuerzos.
Departamento de Metal-Mecánica ITTG-rcgg 21 Luego, en 1870, después de 12 años de investigación sobre las fallas por fatiga, el ingeniero alemán August Wohler publicó los resultados de su estudio, en el cual se mostraba como “culpable” de la falla al número de ciclos de esfuerzo.
Al realizar pruebas sobre las mitades rotas de los ejes que habían fallado por fatiga, observó que el material tenía la misma resistencia y ductilidad bajo carga de tensión que el material original; es decir, el material no se había “cansado” ni fragilizado como se creía.
Sin embargo, el término fatiga se sigue utilizando para referirse a las fallas de elementos sometidos a cargas variables.
Departamento de Metal-Mecánica ITTG-rcgg 22 22/07/2018 12 Este ingeniero alemán encontró, además, la existencia de un límite de resistencia a la fatiga (o límite de fatiga) para los aceros.
Wohler realizó pruebas sobre probetas de acero sometidas a “flexión giratoria”, denominada así al tipo de carga que se genera en un elemento que gira sometido a un momento flector constante.
En dichas pruebas se pretendía relacionar los niveles de esfuerzo a los cuales se sometían las probetas, con el número de ciclos de carga que soportaban hasta la falla.
Wohler obtuvo un diagrama como el que se muestra a continuación, el cual es conocido como diagrama S-nc (esfuerzo -número de ciclos) o diagrama de vida-resistencia de Wohler.
Departamento de Metal-Mecánica ITTG-rcgg 23 El esfuerzo (o resistencia) S corresponde al valor del esfuerzo máximo al cual se somete la probeta, y nc es el número de ciclos de esfuerzo. Las líneas del diagrama representan aproximaciones a los puntos reales de falla obtenidos en los ensayos.
Departamento de Metal-Mecánica ITTG-rcgg 24 22/07/2018 13 Para terminar con la historia de la fatiga, consideremos los siguientes casos: en 1954, muchos años después del descubrimiento del fenómeno de la fatiga, el Comet Británico, la primera aeronave a reacción comercial para pasajeros, tuvo dos accidentes graves debidos a fallas por fatiga causadas por ciclos de presurización y despresurización de la cabina.
Más recientemente, en 1988, un Boeing 737 de Hawaiian Airlines perdió un tercio de la parte superior de la cabina en pleno vuelo; aterrizó con pérdidas mínimas de vidas.
Podemos citar muchos ejemplos mas, ¿ustedes recuerdan uno?
Departamento de Metal-Mecánica ITTG-rcgg 25 Departamento de Metal-Mecánica ITTG-rcgg 26 22/07/2018 14 A manera de resumen del estudio de la fatiga.
Departamento de Metal-Mecánica ITTG-rcgg 27 Año Investigador Contribución 1829 Albert Primero en documentar la falla por cargas repetidas 1837 Rankine Analiza la teoría de cristalización de la fatiga 1839 Poncelet Primero en utilizar el término “fatiga” 1849 Stephenson Analiza la responsabilidad del producto asociada con la falla por fatiga en los ejes de los carros del ferrocarril 1850 Braihtwaite Primero en utilizar el término fatiga en una publicación en inglés. Analiza la teoría de la cristalización 1864 Fairbairn Informa sobre los primeros experimentos sobre cargas repetidas 2002 Alves / Jones Falla por impacto de vigas usando la mecánica de daños acumulados bajo cargas estáticas y dinámicas de modelo continuo 2003 Dwivedi / Espinosa Modelación dinámica de la propagación de grieta en fibra reforzada de material compuesto incluyendo los efectos friccionales 2004 Lin Z / Lincang Modelo simplificado para la predicción de la dinámica de daños y fractura de materiales dúctiles 2005 Rusinek / Zaera Análisis de inercia y efectos escalares en la formación dinámica durante la tensión en una lámina de acero 2006 Roy Xu / Wang Análisis dinámico de la fractura mecánica del modo de transición de falla a lo largo de interfaces en sólidos elásticos Departamento de Metal-Mecánica ITTG-rcgg 28 22/07/2018 15 En la realidad, todas las cargas que actúan sobre un determinado sistema mecánico varían con el tiempo, lo que sucede es que su frecuencia de repetición es tan baja que se pueden considerar como estáticas. Un ejemplo de esto son las puertas de metal; en épocas de calor ésta se expande y entra en contacto con su marco, también metálico, y el proceso de apertura y / o cierre de esta se hace aplicando una fuerza mayor que la requerida cuando no se ha dilatado. Aunque esta dilatación-contracción causada por cambios de temperatura se produce cada vez que hay incremento de la misma, su frecuencia no es de magnitud suficiente como para considerarse una acción cíclica y por lo tanto, no conduce a la falla por fatiga. No sucede lo mismo en centrales de vapor y otros sistemas en los cuales los cambios de temperatura son bruscos y de alta repetibilidad. En estos casos, se emplean unos dispositivos llamados juntas de dilatación (en algunos pisos de viviendas existen también tales juntas) que absorben las deformaciones térmicas que a su vez generan tensiones que serán tensiones cíclicas.
El comportamiento de estas tensiones, que son las que se toman en cuenta en el estudio de la fatiga en vez de considerar las cargas, puede modelarse mediante una senoide de la siguiente forma Departamento de Metal-Mecánica ITTG-rcgg 29 Las tensiones se obtienen: Tensión media definida como: Departamento de Metal-Mecánica ITTG-rcgg 30 La amplitud de tensiones como: La rango de tensiones como: La relación de tensiones: La relación de amplitud: 22/07/2018 16 Dependiendo como se presenten las ecuaciones anteriores, el comportamiento de las tensiones puede darse de las siguientes formas: Departamento de Metal-Mecánica ITTG-rcgg 31 Esta clasificación está representada en la figura Departamento de Metal-Mecánica ITTG-rcgg 32 22/07/2018 17 3.4. Límite de resistencia a la fatiga La determinación experimental de los límites de resistencia a la fatiga es ahora un procedimiento de rutina, aunque extenso y costoso. En términos generales, se prefiere realizar ensayos de esfuerzos a ensayos de deformación para determinar límites de fatiga.
Se han analizado muchos datos de prueba reales, provenientes de varias fuentes y se ha concluido que el límite a la resistencia a la fatiga, o de fatiga, puede estar relacionado con la resistencia a la tensión Departamento de Metal-Mecánica ITTG-rcgg 33 MPa 1400 S MPa 700 Ksi 200 S Ksi 100 MPa 1400 Ksi 200 S S 504 .
0 S ut ut ut ut ' e RESISTENCIA A LA FATIGA Para desarrollar un enfoque analítico, de la ecuación de la recta S–N será Departamento de Metal-Mecánica ITTG-rcgg 34 e ut e 2 ut b f S S 9 .
0 log 3 1 b S S 9 .
0 a aN S Sf = Resistencia a la fatiga.
a = constante N = # Ciclos.
b = Constante 22/07/2018 18 Supóngase que se tiene un esfuerzo completamente invertido a el número de ciclos de duración correspondiente a este esfuerzo puede determinarse a partir de St = aNb sustituyendo a por Sf.
Departamento de Metal-Mecánica ITTG-rcgg 35 b 1 a a N TEORÍA DE LA FATIGA DURANTE LA DEFORMACIÓN UNITARIA La fatiga es el proceso de la acumulación de daño que se manifiesta mediante la propagación de grietas; aunque la propagación de grietas no es posible sin una deformación plástica en el extremo de la grieta.
Usar las propiedades de los materiales, como la resistencia a la fluencia o a la rotura, presenta dificultades porque las cargas cíclicas cambian estos valores cerca del extremo de la grieta, las cuales se incrementan o disminuyen dependiendo del material y de su proceso de manufactura.
Dadas las dificultades al expresar las resistencias del material cerca de los extremos de las grietas, se han sugerido varios enfoques para abordar la deformación unitaria en tales extremos.
Departamento de Metal-Mecánica ITTG-rcgg 36 22/07/2018 19 Dadas las dificultades al expresar las resistencias del material cerca de los extremos de las grietas, se han sugerido varios enfoques para abordar la deformación unitaria en tales extremos. Una de las ecuaciones mejor conocidas es la relación Manson-Coffin, que proporciona la amplitud total de la deformación unitaria como la suma de las amplitudes de la deformación unitaria elástica y plástica: Departamento de Metal-Mecánica ITTG-rcgg 37 ' N 2 ' N 2 E 2 ' f a ' f = Deformación unitaria total f’ = Esfuerzo a la fractura en un ciclo de esfuerzo, Pa.
E = Módulo de elasticidad del materia, Pa.
N’ = Número de ciclos antes de la falla f’ = Coeficiente de ductilidad a la fatiga (deformación unitaria verdadera que corresponde a la fractura en un ciclo de esfuerzo).
a = Exponente de resistencia a la fatiga.
= Exponente de ductilidad a la fatiga FACTORES QUE MODIFICAN EL LÍMITE A LA FATIGA En los experimentos a la fatiga, se suponen las mejores circunstancias para la obtención de vidas largas a la fatiga. No obstante, dicha situación no se garantiza en aplicaciones de diseño, así que el límite a la fatiga del componente se debe modificar o reducir a partir del límite a la fatiga del material en el mejor de los casos.
Los factores de modificación que vamos a analizar aquí serán para cargar completamente alternante (m = 0 ). El limite a la fatiga modificado se expresa como: Departamento de Metal-Mecánica ITTG-rcgg 38 ' e m t r s f e S · K · K · K · K · K S Se = Límite de fatiga modificado Kf = Factor de acabado de superficie.
Ks = Factor de tamaño Kr = Factor de confiabilidad Kt = Factor de temperatura Km = Factor diverso S’e = Límite a la fatiga 22/07/2018 20 EFECTO DE LA CONCENTRACIÓN DE ESFUERZOS Como los sitios de concentración de esfuerzo también son sitios de concentraciones de deformaciones unitarias, se consideran como los principales causantes de la iniciación y crecimiento de grietas, por fatiga.
Para carga estática se usa el factor de esfuerzos Kc, y para cargas por fatiga, el factor de esfuerzos por fatiga Kf, donde Departamento de Metal-Mecánica ITTG-rcgg 39 muesca con especimen para fatiga la a Límite muesca sin especimen para fatiga la a Límite K f Hay que recordar que el factor de concentración de esfuerzos estáticos (Kc) es solo función de la geometría de la pieza.
Ahora, el factor de concentración de esfuerzos por fatiga también es función de las propiedades del material y del tipo de carga.
La consideración del material con frecuencia se aborda usando un factor de sensibilidad a la muesca qn, el cual se define como: Departamento de Metal-Mecánica ITTG-rcgg 40 : o 1 K 1 K q c f n 1 K q 1 K c n f 22/07/2018 21 Hay que observar que el rango de qn esta entre cero y uno.
Existen figuras que muestran la sensibilidad de la muesca contra el radio de la misma, para materiales comúnmente usados con varios tipos de carga. En todos los materiales incluidos. La sensibilidad a la muesca se aproxima a cero cuando el radio de la muesca se aproxima a cero.
Así mismo, los aceros más duros y más resistentes suelen ser más sensibles a la muesca (valor grande de qn). Esto no parece sorprendente puesto que la sensibilidad a la muesca es una medida de la ductilidad del material.
El factor de modificación del límite a la fatiga, considerando las concentraciones de esfuerzos, es: Departamento de Metal-Mecánica ITTG-rcgg 41 f 0 K 1 K Para las teorías de falla El límite de fatiga y la resistencia a la fatiga para vida finita constituyen propiedades base para el diseño de elementos sometidos a cargas variables.
Sin embargo, Se’ o Sf’ no puede ser utilizado directamente en el diseño, ya que es obtenido bajo condiciones especiales de esfuerzo: probeta normalizada y pulida, girando sometida a flexión bajo condiciones ambientales favorables.
Particularmente, nos interesa referirnos aquí a las condiciones de flexión giratoria. Bajo este tipo de carga, la probeta sufre un variación sinusoidal de esfuerzo repetido invertido para la cual el esfuerzo medio es igual a cero. Se necesitan ecuaciones de diseño que sirvan no sólo para un esfuerzo repetido invertido, sino también para cualquier tipo de variación sinusoidal, donde Sm pueda ser diferente de cero. Para encontrar dichas ecuaciones fueron necesarias más pruebas experimentales, de las cuales se concluyó que, en general, si se agrega una componente media del esfuerzo, el elemento falla con una componente alternativa menor.
Departamento de Metal-Mecánica ITTG-rcgg 42 22/07/2018 22 Las figuras sig. muestran las tendencias típicas que siguen los puntos de ensayo en diagramas de esfuerzo medio - esfuerzo alternativo, Sm-Sa y Sms-Sas, respectivamente, cuando se someten probetas normalizadas y pulidas a diferentes combinaciones de estos esfuerzos.
Las cruces en los diagramas indican las combinaciones (Sm, Sa) o (Sms, Sas) para las cuales un pequeño aumento en el esfuerzo medio o en el alternativo produciría la falla de la probeta. Las combinaciones de esfuerzos que estén entre la zona de las cruces y el origen del diagrama no producirían falla, mientras que combinaciones de esfuerzos que estén por fuera de la zona de las cruces producirían la falla de la probeta de ensayo.
Departamento de Metal-Mecánica ITTG-rcgg 43 Para modelar los datos experimentales para Sm≥0, se han propuesto, entre otros, tres modelos o aproximaciones diferentes, las cuales se muestran en la fig.: -Línea o parábola de Gerber -Línea de Goodman modificada -Línea de Soderberg Departamento de Metal-Mecánica ITTG-rcgg 44 22/07/2018 23 3.7. Teoría de Goodman La línea de Goodman modificada es una recta que corta los ejes por los mismos puntos que la parábola de Gerber. Se deja al estudiante la deducción de la ecuación siguiente, partiendo de la ecuación general de una línea recta, Sa = aSm + b, utilizando los dos puntos conocidos en la recta: Departamento de Metal-Mecánica ITTG-rcgg 45 Esta aproximación es menos exacta que la de Gerber, pero tiene dos ventajas: (i) es más conservadora, ya que la mayoría de cruces quedan por encima de la línea y, por lo tanto, es más confiable; (ii) la ecuación de una línea recta es un poco más sencilla que la ecuación de una parábola. La línea de Goodman modificada es la más utilizada en el diseño por fatiga 3.8. Teoría de Soderberg La línea de Soderberg es una recta, que a diferencia de la línea de Goodman modificada, pasa por el punto (Sy, 0) en el diagrama Sm-Sa; es decir, utiliza como criterio de falla la resistencia de fluencia y no la de rotura.
Es por esto que sólo se puede aplicar a materiales que posean límite de fluencia. Esta línea de falla es muy conservadora, pero como se verá más adelante, evita la verificación de que no ocurra fluencia inmediata. La ecuación para la línea de Soderberg es similar a la de Goodman modificada Departamento de Metal-Mecánica ITTG-rcgg 46 22/07/2018 24 3.9. Teoría de Gerber La línea de Gerber es una parábola que pasa por los puntos (0, Se’) y (Su, 0), en el diagrama Sm-Sa, y está dada por: Departamento de Metal-Mecánica ITTG-rcgg 47 Esta aproximación es razonablemente fiel a los datos de ensayo y corresponde aproximadamente al 50% de confiabilidad; es decir, la mitad (50%) de los puntos de ensayo (cruces) está por encima de la parábola y la otra mitad está por debajo. Si se diseñara directamente con estas ecuaciones se tendría un 50% de probabilidad de no falla. Esta línea de falla es particularmente útil para el análisis de piezas que han fallado por fatiga, pero, Según Norton, no se usa mucho para diseñar.
ECUACIONES DE DISEÑO De las tres aproximaciones estudiadas en las secciones previas, la de Gerber es recomendada para el diseño por algunos autores, y como se dijo es útil para el análisis de piezas que han fallado por fatiga. La línea de Soderberg podría utilizarse para materiales dúctiles, los cuales poseen límite de fluencia.
Aunque es demasiado conservadora, es recomendada por algunos autores.
Según Norton, la línea de Goodman modificada es la preferida y puede aplicarse tanto para materiales dúctiles como frágiles.
Departamento de Metal-Mecánica ITTG-rcgg 48 22/07/2018 25 El polígono de la fig. sig. supone que el material posee límites de fluencia. Si no los posee, el polígono resultante está limitado por la línea de Goodman modificada y por la línea de rotura inmediata en compresión, desapareciendo las dos líneas de fluencia inmediata. Además, si se usa la aproximación de Soderberg (línea a trazos), el polígono no incluye la línea de fluencia en tracción.
Departamento de Metal-Mecánica ITTG-rcgg 49 Materiales dúctiles Para Sm ≥ 0, la línea de seguridad de Goodman modificada, es decir, aquella en la cual se tiene en cuenta el factor de seguridad, es similar a la de la línea de Goodman modificada Departamento de Metal-Mecánica ITTG-rcgg 50 22/07/2018 26 Materiales dúctiles Para Sm > 0, parte del polígono de seguridad está limitado por la línea correspondiente a la fluencia inmediata en tracción, después de usar la ecuación anterior, debe verificarse la seguridad del elemento con respecto a esta falla (ya que parte de la línea de seguridad de fluencia está por debajo de la línea de seguridad de Goodman modificada).
Departamento de Metal-Mecánica ITTG-rcgg 51 Si se usa la línea de Soderberg, no es necesario verificar la fluencia en tracción: Materiales dúctiles Para Sm < 0, se puede optar por encontrar los rangos de cada línea, para lo cual debe determinarse el valor de Sm del punto en el cual se interceptan las líneas Departamento de Metal-Mecánica ITTG-rcgg 52 22/07/2018 27 Materiales frágiles Al comparar las fig. ant.
puede concluirse que las ecuaciones de seguridad para materiales frágiles son similares a aquellas para materiales dúctiles, excepto que en vez de Syc aparecerá Suc y que no es necesario verificar la resistencia a la fluencia en tracción. Las ecuaciones ant. se adaptan para materiales frágiles: Departamento de Metal-Mecánica ITTG-rcgg 53 Ecuaciones de diseño generales Las ecuaciones estudiadas en las secciones ant.
son aplicables a elementos sin concentradores de esfuerzos. Como se dijo en la sección ant, se debe incluir el efecto de los concentradores de esfuerzos Departamento de Metal-Mecánica ITTG-rcgg 54 Esfuerzos normales en materiales dúctiles 22/07/2018 28 Esfuerzos normales en materiales frágiles Departamento de Metal-Mecánica ITTG-rcgg 55 Esfuerzos cortantes en materiales dúctiles Para este caso puede usarse la ecuación ant.
descrita, si se calculan los esfuerzos equivalentes medio y alternativo, reemplazando Sm por σme y Sa por σae. Alternativamente, puede usarse Departamento de Metal-Mecánica ITTG-rcgg 56 22/07/2018 29 Esfuerzos cortantes en materiales frágiles Según algunos autores, puede usarse la ecuación a cortante, si se calculan los esfuerzos equivalentes medio y alternativo, reemplazando Sm por σme y Sa por σae. Alternativamente, usar Departamento de Metal-Mecánica ITTG-rcgg 57 |
8693 | https://flexbooks.ck12.org/cbook/ck-12-interactive-algebra-2/section/1.2/related/lesson/adding-and-subtracting-polynomials-alg-ii/ | Adding and Subtracting Polynomials | CK-12 Foundation
AI Teacher Tools – Save Hours on Planning & Prep.
Try it out!
Skip to content
What are you looking for? Search
Math
Grade 6
Grade 7
Grade 8
Algebra 1
Geometry
Algebra 2
PreCalculus
Science
Earth Science
Life Science
Physical Science
Biology
Chemistry
Physics
Social Studies
Economics
Geography
Government
Philosophy
Sociology
Subject
Math
Elementary Math
Grade 1
Grade 2
Grade 3
Grade 4
Grade 5
Interactive
Math 6
Math 7
Math 8
Algebra I
Geometry
Algebra II
Conventional
Math 6
Math 7
Math 8
Algebra I
Geometry
Algebra II
Probability & Statistics
Trigonometry
Math Analysis
Precalculus
Calculus
What's the difference?
Science
Grade K to 5
Earth Science
Life Science
Physical Science
Biology
Chemistry
Physics
Advanced Biology
FlexLets
Math FlexLets
Science FlexLets
English
Writing
Spelling
Social Studies
Economics
Geography
Government
History
World History
Philosophy
Sociology
More
Astronomy
Engineering
Health
Photography
Technology
College
College Algebra
College Precalculus
Linear Algebra
College Human Biology
The Universe
Adult Education
Basic Education
High School Diploma
High School Equivalency
Career Technical Ed
English as 2nd Language
Country
Bhutan
Brasil
Chile
Georgia
India
Translations
Spanish
Korean
Deutsch
Chinese
Greek
Polski
Explore
EXPLORE
Flexi A FREE Digital Tutor for Every Student
FlexBooks 2.0 Customizable, digital textbooks in a new, interactive platform
FlexBooks Customizable, digital textbooks
Schools FlexBooks from schools and districts near you
Study Guides Quick review with key information for each concept
Adaptive Practice Building knowledge at each student’s skill level
Simulations Interactive Physics & Chemistry Simulations
PLIX Play. Learn. Interact. eXplore.
CCSS Math Concepts and FlexBooks aligned to Common Core
NGSS Concepts aligned to Next Generation Science Standards
Certified Educator Stand out as an educator. Become CK-12 Certified.
Webinars Live and archived sessions to learn about CK-12
Other Resources
CK-12 Resources
Concept Map
Testimonials
CK-12 Mission
Meet the Team
CK-12 Helpdesk
FlexLets
Know the essentials.
Pick a Subject
Donate
Sign InSign Up
Back To Adding & Subtracting PolynomialsBack
1.2
Adding and Subtracting Polynomials
Written by:Lori Jordan |Kate Dirga
Fact-checked by:The CK-12 Editorial Team
Last Modified: Jul 01, 2025
Rectangular prism A has a volume of x 3+2 x 2−3. Rectangular prism B has a volume of x 4+2 x 3−8 x 2. How much larger is the volume of rectangular prism B than rectangular prism A?
Adding and Subtracting Polynomials
A polynomial is an expression with multiple variable terms, such that the exponents are greater than or equal to zero. All quadratic and linear equations are polynomials. Equations with negative exponents, square roots, or variables in the denominator are not polynomials.
Now that we have established what a polynomial is, there are a few important parts. Just like with a quadratic, a polynomial can have a constant, which is a number without a variable. The degree of a polynomial is the largest exponent. For example, all quadratic equations have a degree of 2. Lastly, the leading coefficient is the coefficient in front of the variable with the degree. In the polynomial 4 x 4+5 x 3−8 x 2+12 x+24 above, the degree is 4 and the leading coefficient is also 4. Make sure that when finding the degree and leading coefficient you have the polynomial in standard form. Standard form lists all the variables in order, from greatest to least.
Let's rewrite x 3−5 x 2+12 x 4+15−8 x in standard form and find the degree and leading coefficient.
To rewrite in standard form, put each term in order, from greatest to least, according to the exponent. Always write the constant last.
x 3−5 x 2+12 x 4+15−8 x→12 x 4+x 3−5 x 2−8 x+15
Now, it is easy to see the leading coefficient, 12, and the degree, 4.
Simplify (4 x 3−2 x 2+4 x+15)+(x 4−8 x 3−9 x−6)
To add or subtract two polynomials, combine like terms. Like terms are any terms where the exponents of the variable are the same. We will regroup the polynomial to show the like terms.
(4 x 3−2 x 2+4 x+15)+(x 4−8 x 3−9 x−6)x 4+(4 x 3−8 x 3)−2 x 2+(4 x−9 x)+(15−6)x 4−4 x 3−2 x 2−5 x+9
Simplify (2 x 3+x 2−6 x−7)−(5 x 3−3 x 2+10 x−12)
When subtracting, distribute the negative sign to every term in the second polynomial, then combine like terms.
(2 x 3+x 2−6 x−7)−(5 x 3−3 x 2+10 x−12)2 x 3+x 2−6 x−7−5 x 3+3 x 2−10 x+12(2 x 3−5 x 3)+(x 2+3 x 2)+(−6 x−10 x)+(−7+12)−3 x 3+4 x 2−16 x+5
Examples
Example 1
Earlier, you were asked to find the difference in the volume of rectangular prism B compared to rectangular prism A.
We need to subtract the volume of rectangular prism A from the volume of rectangular prism B.
(x 4+2 x 3−8 x 2)−(x 3+2 x 2−3)=x 4+2 x 3−8 x 2−x 3−2 x 2+3=x 4+x 3−10 x 2+3
Therefore, the difference between the two volumes is x 4+x 3−10 x 2+3.
Example 2
Is 2 x 3−5 x+6 a polynomial? Why or why not?
No, this is not a polynomial because x is under a square root in the equation.
Example 3
Find the leading coefficient and degree of 6 x 2−3 x 5+16 x 4+10 x−24.
In standard form, this polynomial is −3 x 5+16 x 4+6 x 2+10 x−24. Therefore, the degree is 5 and the leading coefficient is -3.
Example 4
Add the following polynomials:(9 x 2+4 x 3−15 x+22)+(6 x 3−4 x 2+8 x−14).
(9 x 2+4 x 3−15 x+22)+(6 x 3−4 x 2+8 x−14)=10 x 3+5 x 2−7 x+8
Example 5
Subtract the following polynomials:(7 x 3+20 x−3)−(x 3−2 x 2+14 x−18).
(7 x 3+20 x−3)−(x 3−2 x 2+14 x−18)=6 x 3+2 x 2+6 x+15
Review
Determine if the following expressions are polynomials. If not, state why. If so, write in standard form and find the degree and leading coefficient.
1 x 2+x+5
x 3+8 x 4−15 x+14 x 2−20
x 3+8
5 x−2+9 x−1+16
x 2 2−x 6+10
x 4+8 x 2+12 3
x 2−4 x
−6 x 3+7 x 5−10 x 6+19 x 2−3 x+41
Add or subtract the following polynomials.
(x 3+8 x 2−15 x+11)+(3 x 3−5 x 2−4 x+9)
(−2 x 4+x 3+12 x 2+6 x−18)−(4 x 4−7 x 3+14 x 2+18 x−25)
(10 x 3−x 2+6 x+3)+(x 4−3 x 3+8 x 2−9 x+16)
(7 x 3−2 x 2+4 x−5)−(6 x 4+10 x 3+x 2+4 x−1)
(15 x 2+x−27)+(3 x 3−12 x+16)
(2 x 5−3 x 4+21 x 2+11 x−32)−(x 4−3 x 3−9 x 2+14 x−15)
(8 x 3−13 x 2+24)−(x 3+4 x 2−2 x+17)+(5 x 2+18 x−19)
Review (Answers)
Click HERE to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
Image Attributions
Back to Adding and Subtracting Polynomials
| Image | Reference | Attributions |
---
Ask me anything!
Mute me
CK-12 Foundation is a non-profit organization that provides free educational materials and resources.
FLEXI APPS
ABOUT
Our missionMeet the teamPartnersPressCareersSecurityBlogCK-12 usage mapTestimonials
SUPPORT
Certified Educator ProgramCK-12 trainersWebinarsCK-12 resourcesHelpContact us
BY CK-12
Common Core MathK-12 FlexBooksCollege FlexBooksTools and apps
CONNECT
TikTokInstagramYouTubeTwitterMediumFacebookLinkedIn
v2.11.9.20250725065110-fde7a5d80f
© CK-12 Foundation 2025 | FlexBook Platform®, FlexBook®, FlexLet® and FlexCard™ are registered trademarks of CK-12 Foundation.
Terms of usePrivacyAttribution guide
Student Sign Up
Are you a teacher? Sign up here
Sign in with Google
Having issues? Click here
Sign in with Microsoft
or
Sign up using email
By signing up, I confirm that I have read and agree to the Terms of use and Privacy Policy
Already have an account? Sign In
Adaptive Practice
I’m Ready to Practice!
Get 10 correct to reach your goal
Estimated time to complete: 11 min
Start Practice I need help
Save this section to your Library in order to add a Practice or Quiz to it.
Title
(Edit Title)32/ 100
Save Go Back
This lesson has been added to your library.
Got It
Searching in:
CK-12
Looks like this FlexBook 2.0 has changed since you visited it last time. We found the following sections in the book that match the one you are looking for:
Go to the Table of Contents
Ok
No Results Found
Your search did not match anything in .
Got It
Are you sure you want to restart this practice? Restarting will reset your practice score and skill level. |
8694 | https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-26/issue-2/The-addition-of-residue-classes-modulo-n/pjm/1102985893.pdf | PACIFIC JOURNAL OF MATHEMATICS Vol. 26, No. 2, 1968 THE ADDITION OF RESIDUE CLASSES MODULO n CHARLES RYAVEC In the present paper, the following is proved: THEOREM. Let au — ,am be m distinct, nonzero residues modulo n, where n is any natural number and where m ^ Svβΰ exp ίc V l o £ n X , I log log n J where c > 0 is some large constant. Then the congruence sidi + + εmam = 0 (mod n) is solvable with ε{ = 0 or 1 and not all e = 0.
The method of proof is completely elementary, in that it is based upon well-known results concerning the addition of residues modulo a natural number n and upon results from elementary number theory.
In a recent paper by Erdos and Heilbronn (see ) the following question is investigated. Let p be a prime and alf « ,α m distinct, nonzero residue classes modulo p, and N any residue class modulo p.
Let F(N) = F(N; p; aίy , am) denote the number of solutions of the congruence (1) ε.a, + + εmαm = N (mod p) , where the ε^ are restricted to the values 0 or 1. What can be said about the function F(N)1 The authors prove the following result: THEOREM 1. F(N) > 0 if m ^ They conjecture that the bound 2>VH>p in Theorem 1 is not best possible: 3τ/6p can probably be replaced by 2]/p. On the other hand, they show that the constant 2 cannot be replaced by any smaller constant, as shown by the example αx = 1 , a2 = - 1 , , am = ( -Note that if m < 2 (ι/p" - 2), F(l/2(p - 1) - 0.
The question which now arises is what can be said about F(N) if the prime p is replaced by a composite integer ni Theorem 1 is clearly false for composite n. In fact, even the bound m Ξ > — 1 + n/2 will not guarantee that F(N) > 0 for all N when n is composite.
The difficulty is that all of the a{ may have a prime factor in common with n, in which case N = 1 could not be represented in the form 367 368 CHARLES RYAVEC (1). However, this predicament does not arise when we try to represent 0 in the form (1). Therefore, it is natural to ask what condition on m will guarantee F(0) > 0 for all n. Erdos and Heilbronn conjectured that F(0) > 0 provided m > 2Vn 1 and at a conference at Ohio State University Erdos raised the question whether i^(0) > 0 could be proved if one assumed the stronger hypothesis m > K-n ill2)+ε, where ε is any positive number, and K is some absolute constant.
Since the expression exp{c'(v /logn)/(loglogn)} is O(n ε) for any ε > 0, the theorem of this paper answers Erdos' question.
2 Necessary lemmas In order to prove the theorem a number of lemmas will be needed. They are rather straightforward modifica-tions of those given in for the case when % is a prime.
LEMMA 1. Let bu • • ,6Z be I distinct residues modulo %; and let B(x) denote the number of solutions of x = bi — bj (mod n) with 1 <; i,j ^ I. Then B(x + y) ^ —I + B(x) + B(y); i.e., l-B(x + y)£(l- B{x)) + (I - B(y)) .
Proof. See , page 150.
LEMMA 2. Let 1 < ^ k < £ I <; n/2, n ^ 2 , and let dίy , dk be k distinct nonzero residues modulo n such that (dif n) = l. Let bu , bι be I distinct residues modulo n. Then there is an i, 1 ^ i ^ k, such that Bid,) 0 for every gSG if m ^ 2p — 2v /\G .
THE ADDITION OF RESIDUE CLASSES MODULO n 369 provided jA Φ G for 1 ^ j ^ r. Hence, we obtain t ^ min (n — 1, r&) distinct, nonzero residues c19 , ct modulo n which can be expressed as sums of not more than r of the d$\ and the summands need not be distinct Since Σis.s B(c9) ^ B(l) + + B(n - 1) = ί(ί - 1), there is an s such that n — 1) _ I I - 1 Z .
2 ' - 21-1 2 l - i 2 i.e., I - B(cs) > 1/2.
By using induction on the conclusion of Lemma 1, we obtain ( 2 ) I - B(x, + + xt) ^ Σ (I ~ B(Xi)) .
By construction, cs ~ Σ<=i ε^iί (mod ^) is solvable with not all ε{ = 0.
Rewrite the above expression as c8 = Σί=i^i< (modn), where we have suppressed those terms in the sum for which ε4 = 0. Applying (2) we obtain Therefore, one obtains a d€ such that i - Bid,) > J->Σ> lΛ > A V ' 2n ~ 2r ~ 2 ( j f c + 21) ~ 6 since 1 < Ξ rx ^ r.
Now let 1 ^ ώ x < d2 < < d^ < £ w — 1 be v distinct, nonzero residues modulo n such that (^, ^) = 1. For 1 ^ u ^ v/2, consider all possible subsets, Stt, of w elements from the set {du , d2J, For each subset Sω, let L(Stt) denote the number of distinct residue classes modulo n which can be obtained in the form e ^ + + ε2ud2uy where not all εt = 0 and where ε4 = 0 or 1 and ε < = 0 if d{ is not in Su. Note that determining L(SU), we do not include the residue class 0 unless it can be expressed as the sum of ^u distinct elements of Su.
Finally, put L(u) = Max (L(SU)), where the maximum is taken over all subsets, S%, of u elements from the set {dl9 , d2u}.
370 CHARLES RYAVEC LEMMA 3. Let du ,dv satisfy the properties in the above definition. If Sjdi + + £>Λ = 0 (mod n) implies that all s < = 0, then ( 3 ) L(u + 1) ^ L(u) when u^l ( 4) L(u) ^ u + 2 when u ^ 3 , /or w :> 4 .
Proof. (3) is obvious. In order to prove (4), it may be assumed without loss of generality that the maximum, L(u), is obtained from du > du, which are distinct modulo n by assumption. Also, d1 + • + du is distinct from them by the assumption that is impossible unless all e4 — 0. Now let T = {dx + dι\2 ^ i ^u}.
Each element of T is distinct from dι + + cϊtt, when w ^ 3, and from dl9 It will be shown that at least one element of T is distinct from all of du , du. This element, in addition to the u + 1 elements du ' i duy dj_ + + dn will give u + 2 distinct residues modulo w, which proves (4), provided u ^ 3.
So assume that no element of T is distinct from du ",du, and let d1 -- di = djy where j is a function of i. It is clear that since no two dj are congruent modulo ^ and none are congruent to dx. Consequently, i=2 i + d^ = Σ dy (mod n) .
i2 Therefore, (u — 1)^ s 0 (mod w), which is impossible since (dlf n) — 1, and L E M M A 4. L e i d l f « ,cίM, • • , ώ v satisfy the same conditions as in Lemma 3. For 3 ^ u ^ — 1 + u/2, either L(u) > n/2 or + 1) > L W + - ^ r ^ - .
Proof. If L(^) > Λ/2 we are finished. So assume that L(w) ^ n/2.
Now let Stt be a set for which L(tt) = L(SU). So we have L(u) distinct residue classes 6^ , bUu) modulo n which are representable as sums THE ADDITION OF RESIDUE CLASSES MODULO n 371 of distinct elements from Su. We have v — u ^ 1 + v/2 ^ u + 2 other elements d{ which are not in Su. Select u + 2 of these and, if necessary, relabel them as du , du+2. Since l ^ w + 2 ^ L(u) <: n/2, we can apply Lemma 2 to the sets {blf , δi(1()} and {dlf , dM+2}, where k = u + 2,1 = L(u). Hence, we obtain an i, 1 <L i <^ u + 2 for which B(di) < L(^) — (w + 2)/6, where 2?(tf{) is the number of representations of di in the form d{ = bό — bh (mod n) .
Putting Su+1 = Su U {d<}, we have u + 2 - Bid,)) LEMMA 5. As before, let 1 ^ d^ < < cί v ^ ^ — 1 6e v distinct, nonzero residues modulo n such that (d{, n) = 1. Then if v ^ 3i/ 6^, ί/^e congruence εjdi + + eudv Ξ 0 (mod n) is solvable with not all et = 0.
Proof. Assume that ε ^ + ... + εudv = 0 (mod w) with ε < = 0 or 1, implies that all e = 0. We will then obtain a contradiction. By Lemma 4, either L(u) > n/2 or u 2 + 3^ + 42 12 which is larger than n/2 provided u ^ VΊϊn. Therefore, with u ^ we have L(u) > nj2 in either case. But we have v ^ SV"θn distinct residues. Applying the preceding analysis to the more than 2\/~&n remaining residues, we obtain L(u) > n/2 for this set also.
Therefore, we have two, not necessarily disjoint, sets each with more than n/2 residues modulo n. Call these two sets A, B. By a well-known argument, either A + B — G or \G\ ^ \A\ + | B | > Λ / 2 + Λ/2 = Λ .
Therefore, A + B = G; and we conclude that 0 is representable as the sum of distinct elements from {du , dv}. This contradicts our original assumption that 0 is not so represented. Therefore, Mi + + £vd» = 0 (mod n) 372 CHARLES RYAVEC is solvable nontrivially.
3. Proof of theorem For each divisor d of n, let Φ(d) = {di I d = (α^, w)}. Put Φ(d) — {cl9 , ch}, where h and the cό depend on d, although this dependence is suppressed without loss of clarity.
For each cά e Φ(d), we have cs = dc'j9 where (n/d, cj) = 1. Further-more, since the c5 are distinct modulo n, the c) are distinct modulo n/d, and they satisfy Therefore, by Lemma 5, if h ^> 2>V§n/d, the congruence βicj + + eAc£ = 0 (mod n/d) is solvable nontrivially, in which case the congruence β^H hεAcΛ = (modw) is solvable nontrivially.
So if m = Σd/n I <P(d) I ^ Σd/» 3 ^ 6 ^ , then for some d, Φ(d) will contain more than iλ/Qn/d distinct elements modulo n such that {(ajd), (n/d)} = 1. Thus, the congruence exax + + ε mα m = 0 (mod n) will be solvable nontrivially.
We now obtain an upper bound for ^d/n3]/6n/d in terms of n.
Suppose p ep 1 1 n. Then we have Σ din = 3i/6w Π pin Π (i -i Pin Put f(n) — Πp/ (1 — p~ (1/2))~ 1 and choose the prime q = g(w) such that ^ = Πp^<? p ^ n < q f JJp^q p, where q' is the smallest prime greater than q. Clearly f(η) ^ /(n). Now log (f(v)) = - Σ log (1 - P~ (1/2)) - Σ P~ (1/2) + 0(1) Mog^ But log 7] — Σp^ g log p — δ(q) <^ log n. It is well known that there exist positive constants a and β such that aq ^ for all primes q. Hence, we conclude that logw ^ a-q. Also, y = η q' > n, which implies that log rf > log w. But log)/ = δ(q') ^ /3^' = βq(qΊq) ^ T ^ > f° r some constant y > 0. Therefore, log # ^ Ti log log n; THE ADDITION OF RESIDUE CLASSES MODULO n 373 and so ~ ~ ~ I log log n> ' where c > 0 is some positive constant.
BlBLIOGRAPY 1. P. Erdos, and H. Heilbronn, On the addition of residue classes mod p, Acta Arith-metika 7 (1964), 149-159.
2. H. B. Mann, The Addition Theorems of Number Theory and Group Theory, New York, Interscience Publishers, 1965.
3. H. B. Mann, and J. E. Olson, Sums of Sets in the Elementary Abelian Group of Type {p, p), Math Research Center, United States Army, Madison, Wisconsin.
Received November 27, 1967.
UNIVERSITY OF EASTERN MICHIGAN |
8695 | https://math.stackexchange.com/questions/2375977/if-my-tax-rate-is-13-why-does-multiplying-by-1-13-intuitively-make-sense | algebra precalculus - If my tax rate is 13%, why does multiplying by $1.13$ intuitively make sense? - Mathematics Stack Exchange
Join Mathematics
By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy.
Sign up with Google
OR
Email
Password
Sign up
Already have an account? Log in
Skip to main content
Stack Exchange Network
Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
Visit Stack Exchange
Loading…
Tour Start here for a quick overview of the site
Help Center Detailed answers to any questions you might have
Meta Discuss the workings and policies of this site
About Us Learn more about Stack Overflow the company, and our products
current community
Mathematics helpchat
Mathematics Meta
your communities
Sign up or log in to customize your list.
more stack exchange communities
company blog
Log in
Sign up
Home
Questions
Unanswered
AI Assist Labs
Tags
Chat
Users
Teams
Ask questions, find answers and collaborate at work with Stack Overflow for Teams.
Try Teams for freeExplore Teams
3. Teams
4. Ask questions, find answers and collaborate at work with Stack Overflow for Teams. Explore Teams
Teams
Q&A for work
Connect and share knowledge within a single location that is structured and easy to search.
Learn more about Teams
Hang on, you can't upvote just yet.
You'll need to complete a few actions and gain 15 reputation points before being able to upvote. Upvoting indicates when questions and answers are useful. What's reputation and how do I get it?
Instead, you can save this post to reference later.
Save this post for later Not now
Thanks for your vote!
You now have 5 free votes weekly.
Free votes
count toward the total vote score
does not give reputation to the author
Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation.
Got it!Go to help center to learn more
If my tax rate is 13%, why does multiplying by 1.13 1.13 intuitively make sense?
Ask Question
Asked 8 years, 2 months ago
Modified7 years, 11 months ago
Viewed 6k times
This question shows research effort; it is useful and clear
1
Save this question.
Show activity on this post.
For example, suppose I make a purchase that is $5.36$5.36. Now there is tax added. So my total price is actually
(5.36+(5.36∗0.13))=$6.06(5.36+(5.36∗0.13))=$6.06
But really, I can just do this and get the same answer:
5.36∗1.13=$6.06 5.36∗1.13=$6.06
Why does the second way work? Why does it make sense?
As mentioned in the answers section, this is very easily mathematically shown (factoring and such), but I am looking for the actual "intuition" behind it. How would you explain it to someone without using math?
algebra-precalculus
Share
Share a link to this question
Copy linkCC BY-SA 3.0
Cite
Follow
Follow this question to receive notifications
edited Oct 3, 2017 at 22:15
user9464
asked Jul 29, 2017 at 18:17
K Split XK Split X
6,671 8 8 gold badges 49 49 silver badges 97 97 bronze badges
Add a comment|
4 Answers 4
Sorted by: Reset to default
This answer is useful
3
Save this answer.
Show activity on this post.
because multiplication is distributive, 5.36 is 100% of 5.36, 13% tax is on top of that is 13%, so you have 113% after tax. which is represented by 1.13 in decimal form.
Share
Share a link to this answer
Copy linkCC BY-SA 3.0
Cite
Follow
Follow this answer to receive notifications
answered Jul 29, 2017 at 18:22
user451844 user451844
Add a comment|
This answer is useful
3
Save this answer.
Show activity on this post.
Because
5.36+(5.36×0.13)=5.36(1+0.13)=5.36×1.13 5.36+(5.36×0.13)=5.36(1+0.13)=5.36×1.13
Share
Share a link to this answer
Copy linkCC BY-SA 3.0
Cite
Follow
Follow this answer to receive notifications
answered Jul 29, 2017 at 18:18
John_dydxJohn_dydx
4,368 1 1 gold badge 20 20 silver badges 32 32 bronze badges
4
Using math it makes sense, but if you were to verbally explain to someone why it makes sense, then what would you say?K Split X –K Split X 2017-07-29 18:20:30 +00:00 Commented Jul 29, 2017 at 18:20
This answer could arguably be improved by adding another step introducing the ×1×1.Mark S. –Mark S. 2017-07-29 18:21:36 +00:00 Commented Jul 29, 2017 at 18:21
1 I see what you mean-perhaps re-word your question so its clear what you're asking?John_dydx –John_dydx 2017-07-29 18:21:42 +00:00 Commented Jul 29, 2017 at 18:21
@MarkS. This is a fine answer, and I doubt K.Split needs such obvious a step.amWhy –amWhy 2017-07-29 18:43:45 +00:00 Commented Jul 29, 2017 at 18:43
Add a comment|
This answer is useful
1
Save this answer.
Show activity on this post.
If there's 13% sales tax on something, that means you have to pay 13% extra.
In other words, you have to pay the entire price, and then another 13% on top of that.
In other words, you have to pay 100% of the price, plus another 13% of the price.
In other words, you have to pay 100% plus 13%.
In other words, you have to pay 113%.
Share
Share a link to this answer
Copy linkCC BY-SA 3.0
Cite
Follow
Follow this answer to receive notifications
answered Jul 29, 2017 at 18:33
Sophie SwettSophie Swett
10.9k 36 36 silver badges 56 56 bronze badges
Add a comment|
This answer is useful
0
Save this answer.
Show activity on this post.
The tax is the extra money you pay on top of the total. The two ways to think about this are:
Find 13%13% of the price and add it to the total, or
Multiply the price by 1.13 1.13.
Why? As others have said, there's the distributive property, but you know that the tax is going to increase the price, hence multiplying by something greater than 1 1.
Share
Share a link to this answer
Copy linkCC BY-SA 3.0
Cite
Follow
Follow this answer to receive notifications
answered Jul 29, 2017 at 18:28
Sean RobersonSean Roberson
10.3k 3 3 gold badges 23 23 silver badges 40 40 bronze badges
Add a comment|
You must log in to answer this question.
Start asking to get answers
Find the answer to your question by asking.
Ask question
Explore related questions
algebra-precalculus
See similar questions with these tags.
Featured on Meta
Introducing a new proactive anti-spam measure
Spevacus has joined us as a Community Manager
stackoverflow.ai - rebuilt for attribution
Community Asks Sprint Announcement - September 2025
Report this ad
Related
2{Thinking}: Why equivalent percentage increase of A and decrease of B is not the same end result?
2Absolute value fraction problem
3Tetrahedron Permutations
0Does this inequality make sense? 1 = |1|?
1How does this make sense?
2Does the notion of average rate of change make sense for discontinuous functions?
Hot Network Questions
Does the Mishna or Gemara ever explicitly mention the second day of Shavuot?
The geologic realities of a massive well out at Sea
Does a Linux console change color when it crashes?
Languages in the former Yugoslavia
Explain answers to Scientific American crossword clues "Éclair filling" and "Sneaky Coward"
Is existence always locational?
Exchange a file in a zip file quickly
Calculating the node voltage
Can I go in the edit mode and by pressing A select all, then press U for Smart UV Project for that table, After PBR texturing is done?
What NBA rule caused officials to reset the game clock to 0.3 seconds when a spectator caught the ball with 0.1 seconds left?
Direct train from Rotterdam to Lille Europe
Proof of every Highly Abundant Number greater than 3 is Even
alignment in a table with custom separator
Repetition is the mother of learning
"Unexpected"-type comic story. Aboard a space ark/colony ship. Everyone's a vampire/werewolf
Does the curvature engine's wake really last forever?
Why does LaTeX convert inline Python code (range(N-2)) into -NoValue-?
How to solve generalization of inequality problem using substitution?
Non-degeneracy of wedge product in cohomology
Drawing the structure of a matrix
A time-travel short fiction where a graphologist falls in love with a girl for having read letters she has not yet written… to another man
Origin of Australian slang exclamation "struth" meaning greatly surprised
What is a "non-reversible filter"?
In Dwarf Fortress, why can't I farm any crops?
Question feed
Subscribe to RSS
Question feed
To subscribe to this RSS feed, copy and paste this URL into your RSS reader.
Why are you flagging this comment?
It contains harassment, bigotry or abuse.
This comment attacks a person or group. Learn more in our Code of Conduct.
It's unfriendly or unkind.
This comment is rude or condescending. Learn more in our Code of Conduct.
Not needed.
This comment is not relevant to the post.
Enter at least 6 characters
Something else.
A problem not listed above. Try to be as specific as possible.
Enter at least 6 characters
Flag comment Cancel
You have 0 flags left today
Mathematics
Tour
Help
Chat
Contact
Feedback
Company
Stack Overflow
Teams
Advertising
Talent
About
Press
Legal
Privacy Policy
Terms of Service
Your Privacy Choices
Cookie Policy
Stack Exchange Network
Technology
Culture & recreation
Life & arts
Science
Professional
Business
API
Data
Blog
Facebook
Twitter
LinkedIn
Instagram
Site design / logo © 2025 Stack Exchange Inc; user contributions licensed under CC BY-SA. rev 2025.9.26.34547
By clicking “Accept all cookies”, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy.
Accept all cookies Necessary cookies only
Customize settings
Cookie Consent Preference Center
When you visit any of our websites, it may store or retrieve information on your browser, mostly in the form of cookies. This information might be about you, your preferences, or your device and is mostly used to make the site work as you expect it to. The information does not usually directly identify you, but it can give you a more personalized experience. Because we respect your right to privacy, you can choose not to allow some types of cookies. Click on the different category headings to find out more and manage your preferences. Please note, blocking some types of cookies may impact your experience of the site and the services we are able to offer.
Cookie Policy
Accept all cookies
Manage Consent Preferences
Strictly Necessary Cookies
Always Active
These cookies are necessary for the website to function and cannot be switched off in our systems. They are usually only set in response to actions made by you which amount to a request for services, such as setting your privacy preferences, logging in or filling in forms. You can set your browser to block or alert you about these cookies, but some parts of the site will not then work. These cookies do not store any personally identifiable information.
Cookies Details
Performance Cookies
[x] Performance Cookies
These cookies allow us to count visits and traffic sources so we can measure and improve the performance of our site. They help us to know which pages are the most and least popular and see how visitors move around the site. All information these cookies collect is aggregated and therefore anonymous. If you do not allow these cookies we will not know when you have visited our site, and will not be able to monitor its performance.
Cookies Details
Functional Cookies
[x] Functional Cookies
These cookies enable the website to provide enhanced functionality and personalisation. They may be set by us or by third party providers whose services we have added to our pages. If you do not allow these cookies then some or all of these services may not function properly.
Cookies Details
Targeting Cookies
[x] Targeting Cookies
These cookies are used to make advertising messages more relevant to you and may be set through our site by us or by our advertising partners. They may be used to build a profile of your interests and show you relevant advertising on our site or on other sites. They do not store directly personal information, but are based on uniquely identifying your browser and internet device.
Cookies Details
Cookie List
Clear
[x] checkbox label label
Apply Cancel
Consent Leg.Interest
[x] checkbox label label
[x] checkbox label label
[x] checkbox label label
Necessary cookies only Confirm my choices |
8696 | https://www.droracle.ai/articles/53498/in-acute-tubular-necrosis-what-are-typical-findings-for-sodium-fractional-excretion-of-sodium-specific-gravity-and-sediment | What are the typical findings for sodium, fractional excretion of sodium (FeNa), specific gravity, and urine sediment in acute tubular necrosis (ATN)?
Select Language▼
What are the typical findings for sodium, fractional excretion of sodium (FeNa), specific gravity, and urine sediment in acute tubular necrosis (ATN)?
Medical Advisory BoardAll articles are reviewed for accuracy by our Medical Advisory Board
Educational purpose only • Exercise caution as content is pending human review
Article Review Status
Submitted
Under Review
Approved
Last updated: April 1, 2025 • View editorial policy
From the Research
In acute tubular necrosis (ATN), typical laboratory findings include elevated urine sodium (typically >40 mEq/L) and increased fractional excretion of sodium (FENa >2%), reflecting the kidney's inability to conserve sodium due to tubular damage, as supported by the most recent study 1.
Key Laboratory Findings
Elevated urine sodium: typically >40 mEq/L
Increased fractional excretion of sodium (FENa): >2%
Urine specific gravity: typically fixed around 1.010, indicating loss of concentrating ability
Urinary sediment examination: often reveals muddy brown granular casts, renal tubular epithelial cells, and cellular debris, which are hallmarks of tubular injury These findings occur because ATN causes damage to the tubular epithelium, impairing its normal reabsorptive functions. The damaged tubules cannot properly reabsorb sodium, leading to increased sodium excretion despite the body's need to retain it. Similarly, the kidney loses its ability to concentrate urine, resulting in isosthenuria (fixed specific gravity). The presence of tubular epithelial cells and granular casts in the sediment directly reflects the ongoing tubular cell death and sloughing characteristic of ATN.
Diagnostic Utility
The diagnostic utility of FENa in differentiating ATN from prerenal azotemia is well-established, with a recent systematic review and meta-analysis 1 demonstrating high sensitivity and specificity for this purpose, particularly in oliguric patients without a history of chronic kidney disease or diuretic therapy.
Clinical Implications
These laboratory parameters help distinguish ATN from prerenal azotemia, where sodium conservation would typically be intact, and guide clinical management accordingly. The most recent and highest quality study 1 provides strong evidence for the use of FENa in the diagnosis of ATN, and its findings should be prioritized in clinical decision-making.
References
1
Research
Diagnostic Performance of Fractional Excretion of Sodium for the Differential Diagnosis of Acute Kidney Injury: A Systematic Review and Meta-Analysis.
Clinical journal of the American Society of Nephrology : CJASN, 2022
Related Questions
What is the normal value for the Fractional Excretion of Sodium (FENa)?What is the likely location of the problem in a patient with Acute Kidney Injury (AKI) and a Fractional Excretion of Sodium (FeNa) of 4%?What do fractional excretion of sodium (FeNa) values indicate about the type and management of acute kidney injury (AKI)?What does a fractional excretion of sodium (FENa) of 2.5% indicate?What is the specificity of the Blood Urea Nitrogen (BUN) to Creatinine (Cr) ratio for diagnosing prerenal Acute Kidney Injury (AKI) in patients with cirrhosis or Chronic Kidney Disease (CKD)?What is the diagnosis and treatment plan for a 58-year-old male patient with Hypertension (HTN), Hyperlipidemia (HL), presenting with progressive muscle weakness, difficulty walking, and a history of recent upper respiratory infection, with physical examination findings of mild facial muscle weakness, diminished deep tendon reflexes, and sensory abnormalities?What is the diagnosis and treatment plan for a 58-year-old male patient with Hypertension (HTN), Hyperlipidemia (HL), presenting with progressive muscle weakness, difficulty walking, and a history of recent upper respiratory infection, with physical examination findings of mild facial muscle weakness, diminished deep tendon reflexes, and sensory abnormalities?What is the diagnosis and treatment plan for a 58-year-old male patient with Hypertension (HTN), Hyperlipidemia (HL), presenting with progressive muscle weakness, difficulty walking, and a history of recent upper respiratory infection, with physical examination findings of mild facial muscle weakness, diminished deep tendon reflexes, and sensory abnormalities?Is mild word finding difficulty considered Broca's (Expressive) aphasia?What is the diagnosis and treatment plan for a 58-year-old male patient with Hypertension (HTN), Hyperlipidemia (HL), presenting with progressive muscle weakness, difficulty walking, and a history of recent upper respiratory infection, with physical examination findings of mild facial muscle weakness, diminished deep tendon reflexes, and sensory abnormalities?What is the diagnosis and treatment plan for a 58-year-old male patient with Hypertension (HTN), Hyperlipidemia (HL), presenting with progressive muscle weakness, difficulty walking, and a history of recent upper respiratory infection, with physical examination findings of mild facial muscle weakness, diminished deep tendon reflexes, and sensory abnormalities?
Professional Medical Disclaimer
This information is intended for healthcare professionals. Any medical decision-making should rely on clinical judgment and independently verified information. The content provided herein does not replace professional discretion and should be considered supplementary to established clinical guidelines. Healthcare providers should verify all information against primary literature and current practice standards before application in patient care. Dr.Oracle assumes no liability for clinical decisions based on this content.
Have a follow-up question?
Our Medical A.I. is used by practicing medical doctors at top research institutions around the world. Ask any follow up question and get world-class guideline-backed answers instantly.
Ask Question
Original text
Rate this translation
Your feedback will be used to help improve Google Translate |
8697 | https://www.ck12.org/flexi/cbse-math/types-of-fractions/convert-154-to-a-mixed-fraction./ | Flexi answers - Convert 15/4 to a mixed fraction. | CK-12 Foundation
Subjects Explore
Donate
Sign InSign Up
All Subjects
CBSE Math
Types of Fractions
Question
Convert 15/4 to a mixed fraction.
Flexi Says:
The improper fraction 15/4 can be expressed as a mixed numberwith both a whole number and a fraction. Note that the fraction here is already in its simplest form.
As a mixed fraction,15 4 is 3 3 4.
Steps to Convert Improper Fraction into a Mixed Fraction
To convert an improper fraction (simplest form) to amixed fraction (mixed number), divide the numerator by the denominator. This will tell you the number of wholes. If there is a remainder, it is the fraction part of a mixed fraction (mixed number). Here's the step-by-step process:
Let's convert the improper fraction 15 4 into a mixed fraction:
Step 1: Divide 15 by 4.
15 ÷ 4 = 3 with a remainder of 3.
Step 2: Write down the whole number quotient (3).
Step 3: Write the remainder as the new numerator (3).
Step 4: Keep the original denominator (4).
So, the mixed fraction equivalent to 15 4 is written as Q u o t i e n t R e m a i n d e r D i v i s o r, i.e. 3 3 4.
To learn more about converting improper fractions to mixed fractions (mixed numbers) and vice versa,click here!
Analogy / Example
Try Asking:
What is 14/4 as a mixed fraction?What is 11/5 as a mixed number?What is 15/14 as a mixed number?
How can Flexi help?
By messaging Flexi, you agree to our Terms and Privacy Policy |
8698 | https://pubmed.ncbi.nlm.nih.gov/24815348/ | Isopropanol poisoning - PubMed
Clipboard, Search History, and several other advanced features are temporarily unavailable.
Skip to main page content
An official website of the United States government
Here's how you know
The .gov means it’s official.
Federal government websites often end in .gov or .mil. Before sharing sensitive information, make sure you’re on a federal government site.
The site is secure.
The https:// ensures that you are connecting to the official website and that any information you provide is encrypted and transmitted securely.
Log inShow account info
Close
Account
Logged in as:
username
Dashboard
Publications
Account settings
Log out
Access keysNCBI HomepageMyNCBI HomepageMain ContentMain Navigation
Search: Search
AdvancedClipboard
User Guide
Save Email
Send to
Clipboard
My Bibliography
Collections
Citation manager
Display options
Display options
Format
Save citation to file
Format:
Create file Cancel
Email citation
Email address has not been verified. Go to My NCBI account settings to confirm your email and then refresh this page.
To:
Subject:
Body:
Format:
[x] MeSH and other data
Send email Cancel
Add to Collections
Create a new collection
Add to an existing collection
Name your collection:
Name must be less than 100 characters
Choose a collection:
Unable to load your collection due to an error
Please try again
Add Cancel
Add to My Bibliography
My Bibliography
Unable to load your delegates due to an error
Please try again
Add Cancel
Your saved search
Name of saved search:
Search terms:
Test search terms
Would you like email updates of new search results? Saved Search Alert Radio Buttons
Yes
No
Email: (change)
Frequency:
Which day?
Which day?
Report format:
Send at most:
[x] Send even when there aren't any new results
Optional text in email:
Save Cancel
Create a file for external citation management software
Create file Cancel
Your RSS Feed
Name of RSS Feed:
Number of items displayed:
Create RSS Cancel
RSS Link Copy
Full text links
Atypon
Full text links
Actions
Cite
Collections
Add to Collections
Create a new collection
Add to an existing collection
Name your collection:
Name must be less than 100 characters
Choose a collection:
Unable to load your collection due to an error
Please try again
Add Cancel
Permalink
Permalink
Copy
Display options
Display options
Format
Page navigation
Title & authors
Abstract
Similar articles
Cited by
Publication types
MeSH terms
Substances
Related information
LinkOut - more resources
Review
Clin Toxicol (Phila)
Actions
Search in PubMed
Search in NLM Catalog
Add to Search
. 2014 Jun;52(5):470-8.
doi: 10.3109/15563650.2014.914527. Epub 2014 May 9.
Isopropanol poisoning
R J Slaughter1,R W Mason,D M G Beasley,J A Vale,L J Schep
Affiliations Expand
Affiliation
1 National Poisons Centre, Department of Preventive and Social Medicine, University of Otago , Dunedin , New Zealand.
PMID: 24815348
DOI: 10.3109/15563650.2014.914527
Item in Clipboard
Review
Isopropanol poisoning
R J Slaughter et al. Clin Toxicol (Phila).2014 Jun.
Show details
Display options
Display options
Format
Clin Toxicol (Phila)
Actions
Search in PubMed
Search in NLM Catalog
Add to Search
. 2014 Jun;52(5):470-8.
doi: 10.3109/15563650.2014.914527. Epub 2014 May 9.
Authors
R J Slaughter1,R W Mason,D M G Beasley,J A Vale,L J Schep
Affiliation
1 National Poisons Centre, Department of Preventive and Social Medicine, University of Otago , Dunedin , New Zealand.
PMID: 24815348
DOI: 10.3109/15563650.2014.914527
Item in Clipboard
Full text links Cite
Display options
Display options
Format
Abstract
Introduction: Isopropanol is a clear, colorless liquid with a fruity odor and a mild bitter taste. Most commonly found domestically as rubbing alcohol, isopropanol is also found in numerous household and commercial products including cleaners, disinfectants, antifreezes, cosmetics, solvents, inks, and pharmaceuticals.
Aim: The aim of this review is to critically review the epidemiology, toxicokinetics, mechanisms of toxicity, clinical features, diagnosis, and management of isopropanol poisoning.
Methods: OVID MEDLINE and ISI Web of Science were searched to November 2013 using the words "isopropanol", "isopropyl alcohol", "2-propanol", "propan-2-ol", and "rubbing alcohol" combined with the keywords "poisoning", "poison", "toxicity", "ingestion", "adverse effects", "overdose", or "intoxication". These searches identified 232 citations, which were then screened via their abstract to identify relevant articles referring specifically to the epidemiology, toxicokinetics, mechanisms of toxicity, clinical features, diagnosis, and management of isopropanol poisoning; 102 were relevant. Further information was obtained from book chapters, relevant news reports, and internet resources. These additional searches produced eight non-duplicate relevant citations.
Epidemiology: The majority of isopropanol exposures are unintentional and occur in children less than 6 years of age. Although isopropanol poisoning appears to be a reasonably common occurrence, deaths are rare.
Toxicokinetics: Isopropanol is rapidly absorbed following ingestion with peak plasma concentrations occurring within 30 min. It can also be absorbed following inhalation or dermal exposure. Isopropanol is widely distributed with a volume of distribution of 0.45-0.55 L/kg. Isopropanol is metabolized by alcohol dehydrogenase to acetone, acetol and methylglyoxal, propylene glycol, acetate, and formate with conversion of these metabolites to glucose and other products of intermediary metabolism. The elimination of isopropanol is predominantly renal, though some pulmonary excretion of isopropanol and acetone occurs. In one case 20% of the absorbed dose was eliminated unchanged in urine, with the remainder excreted as acetone and metabolites of acetone. The elimination half-life of isopropanol is between 2.5 and 8.0 h, whereas elimination of acetone is slower with a half-life following isopropanol ingestion of between 7.7 and 27 h.
Mechanisms of toxicity: While the exact mechanism of action of isopropanol has not been fully elucidated, brain stem depression is thought to be the predominant mechanism. While the clinical effects are thought to be mostly due to isopropanol, acetone may also contribute.
Clinical features: The major features of severe poisoning are due to CNS and respiratory depression, shock, and circulatory collapse. The most common metabolic effects are an increased osmol (osmolal) gap, ketonemia, and ketonuria. Diagnosis. Poisoning can be diagnosed using the measurement of isopropanol serum concentrations, though these may not be readily available. Diagnosis is therefore more typically made on the basis of the patient's history and clinical presentation. An osmol gap, ketonemia, and/or ketonuria without metabolic acidosis, along with a fruity or sweet odor on the breath and CNS depression support the diagnosis. Management. Supportive care is the mainstay of management with primary emphasis on respiratory and cardiovascular support. Hemodialysis enhances elimination of isopropanol and acetone and should be considered in very severe poisoning.
Conclusions: Severe isopropanol poisoning results in CNS and respiratory depression and circulatory collapse. Treatment primarily consists of symptom-directed supportive care. Although hemodialysis increases the elimination of isopropanol and acetone substantially, it should only be considered in severe life-threatening poisonings. Patients usually make a full recovery provided they receive prompt supportive care.
Keywords: 2-Propanol; Coma; Hypotension; IPA; Isopropanol; Isopropyl alcohol; Propan-2-ol; Respiratory depression; Rubbing alcohol.
PubMed Disclaimer
Similar articles
The clinical toxicology of γ-hydroxybutyrate, γ-butyrolactone and 1,4-butanediol.Schep LJ, Knudsen K, Slaughter RJ, Vale JA, Mégarbane B.Schep LJ, et al.Clin Toxicol (Phila). 2012 Jul;50(6):458-70. doi: 10.3109/15563650.2012.702218.Clin Toxicol (Phila). 2012.PMID: 22746383 Review.
Diethylene glycol poisoning.Schep LJ, Slaughter RJ, Temple WA, Beasley DM.Schep LJ, et al.Clin Toxicol (Phila). 2009 Jul;47(6):525-35. doi: 10.1080/15563650903086444.Clin Toxicol (Phila). 2009.PMID: 19586352 Review.
Poisoning with 1-propanol and 2-propanol.Vujasinovic M, Kocar M, Kramer K, Bunc M, Brvar M.Vujasinovic M, et al.Hum Exp Toxicol. 2007 Dec;26(12):975-8. doi: 10.1177/0960327107087794.Hum Exp Toxicol. 2007.PMID: 18375643
Life-threatening isopropyl alcohol intoxication: is hemodialysis really necessary?Trullas JC, Aguilo S, Castro P, Nogue S.Trullas JC, et al.Vet Hum Toxicol. 2004 Oct;46(5):282-4.Vet Hum Toxicol. 2004.PMID: 15487656
The clinical toxicology of sodium hypochlorite.Slaughter RJ, Watts M, Vale JA, Grieve JR, Schep LJ.Slaughter RJ, et al.Clin Toxicol (Phila). 2019 May;57(5):303-311. doi: 10.1080/15563650.2018.1543889. Epub 2019 Jan 28.Clin Toxicol (Phila). 2019.PMID: 30689457 Review.
See all similar articles
Cited by
Drug-Induced Metabolic Acidosis.Pham AQ, Xu LH, Moe OW.Pham AQ, et al.F1000Res. 2015 Dec 16;4:F1000 Faculty Rev-1460. doi: 10.12688/f1000research.7006.1. eCollection 2015.F1000Res. 2015.PMID: 26918138 Free PMC article.Review.
NMR-based metabolomic investigation of dogs with acute flaccid paralysis due to tick paralysis.Gülersoy E, Balıkçı C, Şahan A, Günal İ, Atlı MO.Gülersoy E, et al.Vet Med Sci. 2024 Jul;10(4):e1528. doi: 10.1002/vms3.1528.Vet Med Sci. 2024.PMID: 38952268 Free PMC article.
COVID-19 and frequent use of hand sanitizers; human health and environmental hazards by exposure pathways.Mahmood A, Eqan M, Pervez S, Alghamdi HA, Tabinda AB, Yasar A, Brindhadevi K, Pugazhendhi A.Mahmood A, et al.Sci Total Environ. 2020 Nov 10;742:140561. doi: 10.1016/j.scitotenv.2020.140561. Epub 2020 Jun 27.Sci Total Environ. 2020.PMID: 32623176 Free PMC article.Review.
Investigation of biocidal efficacy of commercial disinfectants used in public, private and workplaces during the pandemic event of SARS-CoV-2.Ambrosino A, Pironti C, Dell'Annunziata F, Giugliano R, Chianese A, Moccia G, DeCaro F, Galdiero M, Franci G, Motta O.Ambrosino A, et al.Sci Rep. 2022 Mar 31;12(1):5468. doi: 10.1038/s41598-022-09575-1.Sci Rep. 2022.PMID: 35361869 Free PMC article.
Polystyrene Opals Responsive to Methanol Vapors.Burratti L, Casalboni M, De Matteis F, Pizzoferrato R, Prosposito P.Burratti L, et al.Materials (Basel). 2018 Aug 28;11(9):1547. doi: 10.3390/ma11091547.Materials (Basel). 2018.PMID: 30154304 Free PMC article.
See all "Cited by" articles
Publication types
Review
Actions
Search in PubMed
Search in MeSH
Add to Search
MeSH terms
2-Propanol / pharmacokinetics
Actions
Search in PubMed
Search in MeSH
Add to Search
2-Propanol / poisoning
Actions
Search in PubMed
Search in MeSH
Add to Search
Acetone / poisoning
Actions
Search in PubMed
Search in MeSH
Add to Search
Animals
Actions
Search in PubMed
Search in MeSH
Add to Search
Child
Actions
Search in PubMed
Search in MeSH
Add to Search
Child, Preschool
Actions
Search in PubMed
Search in MeSH
Add to Search
Half-Life
Actions
Search in PubMed
Search in MeSH
Add to Search
Humans
Actions
Search in PubMed
Search in MeSH
Add to Search
Renal Dialysis / methods
Actions
Search in PubMed
Search in MeSH
Add to Search
Solvents / pharmacokinetics
Actions
Search in PubMed
Search in MeSH
Add to Search
Solvents / poisoning
Actions
Search in PubMed
Search in MeSH
Add to Search
Tissue Distribution
Actions
Search in PubMed
Search in MeSH
Add to Search
Substances
Solvents
Actions
Search in PubMed
Search in MeSH
Add to Search
Acetone
Actions
Search in PubMed
Search in MeSH
Add to Search
2-Propanol
Actions
Search in PubMed
Search in MeSH
Add to Search
Related information
Cited in Books
MedGen
PubChem Compound (MeSH Keyword)
LinkOut - more resources
Full Text Sources
Atypon
Full text links[x]
Atypon
[x]
Cite
Copy Download .nbib.nbib
Format:
Send To
Clipboard
Email
Save
My Bibliography
Collections
Citation Manager
[x]
NCBI Literature Resources
MeSHPMCBookshelfDisclaimer
The PubMed wordmark and PubMed logo are registered trademarks of the U.S. Department of Health and Human Services (HHS). Unauthorized use of these marks is strictly prohibited.
Follow NCBI
Connect with NLM
National Library of Medicine
8600 Rockville Pike Bethesda, MD 20894
Web Policies
FOIA
HHS Vulnerability Disclosure
Help
Accessibility
Careers
NLM
NIH
HHS
USA.gov |
8699 | https://www.keysight.com/used/us/en/knowledge/calculators/vswr-return-loss-calculator | Official Used
Equipment Store
Sign inNew customer? Register
Account Benefits
Get a customised view of the Used Equipment Store
Manage your details to get quick responses by the team
Stay up-to-date on specials, prices and availability
VSWR / Return Loss Calculator
Used Keysight Equipment Tech Guides VSWR / Return Loss Calculator
Article Feedback
Give us any feedback you might have about this article.
Introduction
Definition of VSWR
TL;DR
Understanding the Basics of VSWR
The Reflection Coefficient and Impedance Mismatch
Signal Power and Return Loss
Signal Power and Mismatch Loss
Loss in Decibels (dB) and Return Loss Measurement
VSWR and Return Loss in Oscilloscopes and Signal Generators
Glossary of Terms
Why Choose Keysight for Your Wireless Communication Needs
Final Thoughts from the Keysight Used Store
Whenever You’re Ready, Here Are 5 Ways We Can Help You
Popular Used Oscilloscopes
Your browser limits use of the Used Keysight Equipment website.
Please consider switching to a current version of Chrome, Firefox, Edge or IE11 to use this site.
Definition of VSWR
"Understanding VSWR isn't just about crunching numbers; it's about shaping the future of efficient wireless communication." – Dr. Bernard Sklar, Communications Engineer and author.
Picture this: an electrical engineer is in a bustling laboratory, running complex tests on an advanced radio frequency (RF) system. Amidst a sea of figures and data, they pause on one particular metric: the Voltage Standing Wave Ratio (VSWR). This might seem like just another number to the untrained eye, but to the engineer, it's a key indicator of how efficient, reliable, and successful their system can be.
VSWR, along with return loss, is a pivotal parameter in the realm of electrical and RF engineering. They are important for evaluating the efficiency of power transfer in various applications, such as designing telecommunication networks and ensuring the reliability of radar systems.
In this highly technical and exacting field, having precise tools like a VSWR/return loss calculator at one's disposal is a necessity. A fine-tuned understanding of these metrics can guide an engineer to enhance system performance, optimize power transmission, and develop breakthrough solutions. Let's delve deeper into these core concepts, and discover how a simple calculator can significantly improve your engineering endeavors.
TL;DR
VSWR (Voltage Standing Wave Ratio) and return loss are crucial parameters in signal transmission, impacting system efficiency and power transfer. They help analyze impedance mismatches and signal reflections in an electrical system.
Understanding the concepts of reflection coefficient and impedance mismatch is key to managing reflections and improving system efficiency. These concepts are especially important in designing and optimizing RF and telecommunication systems.
Tools such as network analyzers, reflectometers, and return loss meters are used to measure VSWR and return loss, providing insights into system performance and areas for improvement.
The application of VSWR and return loss concepts is essential in the use of tools such as oscilloscopes and signal generators, leading to more accurate testing and analysis.
Keysight's tools and services, including their antenna gain calculator, Vector Network Analyzers, and Spectrum Analyzers, offer precise, user-friendly solutions for wireless communication professionals. These tools can help improve the application of VSWR and return loss knowledge, enhancing system performance and efficiency.
Understanding the Basics of VSWR
When it comes to signal transmission and electrical engineering, the role of VSWR, or Voltage Standing Wave Ratio, is paramount. This crucial metric provides a quantifiable measure of a system's efficiency in transmitting power, effectively becoming the heart of any RF and telecommunication setup.
So, what is VSWR exactly? At its core, VSWR is a ratio. It's a measure of the maximum voltage to the minimum voltage in a standing wave pattern along a transmission line, such as a coaxial cable. If you were to visualize this, you would see a wave along the transmission line with peaks (maximum voltage) and troughs (minimum voltage). The VSWR is the ratio of these peak and trough values.
In simpler terms, VSWR is a reflection of how well the impedance of the load (like an antenna) matches the impedance of the transmission line itself. A low VSWR indicates a better match, meaning less signal reflection and more power delivered to the load – and that's what we want for efficient signal transmission.
| Key Takeaway |
| VSWR is the crux of power transfer efficiency in electrical systems. Mastering the concept of VSWR is like holding a compass that guides you towards an optimal and successful RF design. |
The Reflection Coefficient and Impedance Mismatch
Let's dig a bit deeper into the intricacies of signal transmission, specifically the concept of the reflection coefficient, often symbolized as Gamma (Γ). The reflection coefficient is a complex number that describes how much of an electromagnetic wave is reflected by an impedance discontinuity in the transmission line. It's derived from the ratio of the reflected wave amplitude to the incident wave amplitude. A value of zero signifies no reflection, i.e., perfect impedance match, while a value of one denotes total reflection.
The reflection coefficient is intrinsically tied to the phenomenon of impedance mismatch. Essentially, when the load impedance doesn't exactly match the source or transmission line impedance, we have an impedance mismatch. This mismatch can lead to signal reflections, resulting in power loss and reduced system efficiency.
Factors affecting impedance mismatch and their impact on the reflection coefficient include:
Mismatched components: If components connected in a system have significantly different impedance values, it can result in a larger mismatch, increasing the reflection coefficient.
Frequency changes: Impedance values can vary with frequency. Thus, a change in operating frequency can cause a change in the reflection coefficient.
Temperature variations: Physical properties of materials can change with temperature, affecting their impedance and consequently the reflection coefficient.
Load variations: Changes in load conditions can alter the load impedance, leading to a mismatch and higher reflection coefficient.
Understanding these factors and managing impedance mismatch is crucial as it directly impacts the reflection coefficient and thereby the overall performance of the electrical or RF system.
Signal Power and Return Loss
In the world of signal transmission and electrical engineering, another key concept to grasp is return loss. This term provides a measure of how much power is lost when it's reflected back due to impedance mismatches in the transmission line. Return loss is defined in decibels (dB) and represents the ratio of incoming power to the reflected power. A higher return loss value is desirable as it indicates a lower level of reflected power and a better impedance match.
The relation between return loss, VSWR, and the reflection coefficient provides a comprehensive view of the system's efficiency. Specifically, return loss is related to the reflection coefficient (Γ) as follows: Return Loss (dB) = -20 log10|Γ|.
On the other hand, VSWR and the reflection coefficient are connected by the equation: VSWR = (1+|Γ|)/(1-|Γ|).
By understanding these relationships, you can gain insights into how much power is being effectively utilized and how much is being wasted due to reflections. This understanding can guide strategies to minimize reflections, improve impedance matching, and thereby increase the system efficiency.
| Key Takeaway |
| Return loss, in relation with VSWR and the reflection coefficient, serves as a vital benchmark in gauging system performance. A solid understanding of return loss can lead engineers to craft more efficient and reliable signal transmission systems. |
Signal Power and Mismatch Loss
An essential concept that often comes into play in signal transmission is Mismatch Loss. Simply put, Mismatch Loss quantifies the reduction in power at the output due to impedance mismatch. It's measured in decibels (dB) and signifies how much power is wasted when the load impedance doesn't match the transmission line impedance.
When an impedance mismatch occurs, a portion of the signal power gets reflected back towards the source rather than being absorbed by the load. This results in loss of power that could have otherwise been usefully employed, thereby reducing the system's overall efficiency. Consequently, the impact of Mismatch Loss is crucial in designing and optimizing RF and telecommunication systems.
Steps to minimize mismatch loss in signal transmission:
Ensure Component Compatibility: When designing a system, ensure that the components used have compatible impedance values. The impedance of every device and transmission line used should closely match the system impedance to reduce reflections and Mismatch Loss.
Impedance Matching Techniques: Use impedance matching devices or techniques, such as stubs, transformers, or tuning elements. These devices can help match the load impedance with the source or transmission line impedance.
Careful Frequency Selection: Since impedance values can change with frequency, carefully select the operating frequency to ensure impedance match.
Regular System Testing: Regularly test your system to identify any changes in impedance values due to component aging, environmental changes, or other factors. Early detection of such changes can help minimize mismatch loss.
By following these steps, engineers can effectively limit Mismatch Loss, improving the efficiency and performance of their signal transmission systems.
Loss in Decibels (dB) and Return Loss Measurement
In the realm of signal transmission, return loss is a key metric used to indicate how much power is reflected back to the source due to impedance mismatches. It's defined as the ratio of the power of the incoming signal to the power of the reflected signal, measured in decibels (dB). A higher return loss value implies less reflected power and better impedance matching, which is desirable for system efficiency.
To measure return loss, engineers typically use a variety of tools and techniques, including network analyzers, reflectometers, and dedicated return loss meters. These devices function by transmitting a known signal into the system and measuring the power of the signal that gets reflected back.
Different Tools for Measuring Return Loss and Their Accuracy:
| Tool | Accuracy |
--- |
| Network Analyzer | Highly accurate |
| Reflectometer | Moderate to high accuracy |
| Return Loss Meter | Moderate accuracy |
While a network analyzer is often seen as the gold standard due to its high accuracy and versatility, reflectometers and return loss meters also offer reliable measurements and are commonly used in many applications. The choice of tool largely depends on the specific requirements of the system, the available resources, and the level of accuracy needed.
VSWR and Return Loss in Oscilloscopes and Signal Generators
In the world of electronics, oscilloscopes and signal generators are fundamental tools that engineers rely on to test and analyze circuits. VSWR and return loss are key parameters in the context of these devices, significantly impacting their performance and efficiency.
In oscilloscopes, VSWR and return loss play crucial roles in maintaining signal integrity. They help to understand and control reflections and signal distortions that might occur due to impedance mismatches between the device under test and the oscilloscope.
In signal generators, these parameters are important for ensuring the proper transmission of the generated signal to the device under test. A well-matched system, indicated by a low VSWR and high return loss, ensures that most of the generated signal reaches the device, leading to accurate and reliable testing.
Application of VSWR and Return Loss in Oscilloscopes and Signal Generators:
| | | |
---
| Device | Use of VSWR | Use of return loss |
| Oscilloscope | Control signal reflections and distortions | Maintain signal integrity during measurements |
| Signal Generator | Ensure effective signal transmission | Maximize power delivered to the device under test |
| Key Takeaway |
| Mastering the concepts of VSWR and return loss can empower engineers to significantly enhance the performance and efficiency of oscilloscopes and signal generators, leading to more accurate testing and analysis. |
Glossary of Terms
In the discussion of VSWR and return loss, a number of terms, symbols, and units repeatedly come up. Let's take a moment to clarify these terms to enhance understanding and application of these key concepts.
VSWR (Voltage Standing Wave Ratio): A measure of the impedance mismatch between the transmission line and its load. It's a ratio of the maximum to the minimum voltage in the standing wave pattern along a transmission line.
Return Loss: This term quantifies how much power is lost due to reflections caused by impedance mismatch. It is defined as the ratio of incoming power to the reflected power, measured in decibels (dB).
Reflection Coefficient (Gamma, Γ): A complex number that indicates how much of a wave is reflected by an impedance mismatch. It is the ratio of the reflected wave amplitude to the incident wave amplitude.
Mismatch Loss: Quantifies the reduction in power at the output due to impedance mismatch. It's measured in decibels (dB).
Impedance: It's the measure of the opposition that a circuit presents to a current when a voltage is applied. In the context of signal transmission, we often discuss source impedance, load impedance, and transmission line impedance. It's measured in ohms (Ω).
Decibel (dB): A logarithmic unit used to express the ratio of two values of a physical quantity, often power or intensity.
| Key Takeaway |
| Being well-versed with the terminology and symbols associated with VSWR and return loss can facilitate a deeper understanding of these concepts and their application in real-world engineering scenarios. |
Why Choose Keysight for Your Wireless Communication Needs
Keysight's suite of tools and services offers unparalleled benefits for your wireless communication needs. Keysight's Vector Network Analyzers and Spectrum Analyzers for measuring VSWR and return loss, offer precise results, versatile functionality, and reliability.
Benefits of choosing Keysight's tools and services include:
Precision: High accuracy measurements and calculations to ensure optimal system performance.
Ease of use: User-friendly interfaces that make complex analyses simple.
Support and Training: Comprehensive customer support and training resources to facilitate smooth operation and maximize the value from Keysight tools.
Innovation: Cutting-edge technology and continuous innovation to stay ahead of evolving wireless communication needs.
Choosing Keysight for your wireless communication requirements means choosing a partner committed to delivering quality, precision, and value.
Final Thoughts from the Keysight Used Store
Understanding and applying concepts such as VSWR and return loss are fundamental for any professional working in wireless communications. These key parameters govern the efficiency of power transfer, signal integrity, and overall system performance. Mastering these concepts, therefore, directly translates into the creation of more efficient, reliable, and effective communication systems.
Keysight provides an array of equipment designed specifically to aid in the measurement and application of these parameters. Their vector network analyzers and spectrum analyzers, backed by reliable customer support and comprehensive training resources, are integral tools for the accurate measurement of VSWR and Return Loss. The user-friendly antenna gain calculator further simplifies these complex calculations, making your job easier and more efficient.
We encourage you to explore Keysight's premium equipment tailored for wireless communication professionals. The use of their precision tools, combined with your knowledge of VSWR and return loss, can enable you to create cutting-edge solutions and push the boundaries of wireless communications.
Whenever You’re Ready, Here Are 5 Ways We Can Help You
Browse our Premium Used Equipment.
Call tech support US: 1 800 829-4444 Press #, then 2. Hours: 7 am – 5 pm MT, Mon– Fri
Talk to our sales support team by clicking the icon (bottom right corner) on every offer page.
Create an account to get price alerts and access to exclusive waitlists
Talk to your account manager about your specific needs.
Did you like this article?
YesNo
Subscribe to Get Our Latest News, Updates, and Articles.
Get Email Updates
Popular Used Oscilloscopes
Keysight Used Oscilloscopes offers a wide range of industry leading pre-owened oscilloscopes, ranging from older generation Agilent oscilloscopes to the newest Keysight oscilloscope models. So whether you are a fist time buyer, replacing a model like-for-like or looking for upgrade – we have something for you.
3000 T X Series
MSOS804A Infiniium
MSOX2024A InfiniiVision
MSOX3054G / MSOX3104G InfiniiVision
See all Oscilloscopes |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.