Episode 44 - James Propp

Evelyn Lamb: Hello, and welcome to My Favorite Theorem, a math podcast. I'm Evelyn Lamb. I'm a freelance math and science writer in Salt Lake City, Utah. And this is your other host.

Kevin Knudson: I’m Kevin Knudson, professor of mathematics at the University of Florida. How's it going?

EL: All right, yeah. Up early today for me. You know, you’re on the East Coast, and I'm in the Mountain Time Zone. And actually, when my husband is on math trips — sorry, if I'm on math trips on the East Coast, and he's in the mountain time zone, then we have, like, the same schedule, and we can talk to each other before we go to bed. I'm sort of a night owl. So yeah, it's early today. And I always complain about that the whole time.

KK: Sure. Is he a morning person?

EL: Yes, very much.

KK: So Ellen, and I are decidedly not. I mean, I'd still be in bed, really, if I if I had my way. But you know, now that I'm a responsible adult chair of the department, I have to—even in the summer—get in here to make sure that things are running smoothly.

EL: But yeah, other than other than the ungodly hour (it’s 8am, so everyone can laugh at me), everything is great.

KK: Right. Cool. All right, I’m excited for this episode.

EL: Yes. And today, we're very happy to have Jim Propp join us. Hi, Jim, can you tell us a little bit about yourself?

Jim Propp: Yeah, I'm a math professor at UMass Lowell. My research is in combinatorics, probability, and dynamical systems. And I also blog and tweet about mathematics.

KK: You do. Your blog’s great, actually.

EL: Yeah.

KK: I really enjoy it, and you know, you're smart. Once a month.

EL: Yes. That that was a wise choice. Most months, I think on the 17th, Jim has an excellent post at Math Enchantments.

KK: Right.

EL: So that’s a big treat. I think somehow I didn't realize that you did some things with dynamical systems too. I feel like I'm familiar with you in, like, the combinatorics kind of world. So I learned something new already.

KK: Yup.

JP: Yeah, I actually did my PhD work in ergodic theory. And after a few years of doing postdoc in that field, I thought, “No, I'm going to go back to combinatorics, which was sort of my first love. And then some probability mixed into that.

KK: Right. And actually, we had some job candidates this year in combinatorics, and one of them was talking about—you have a list of problems, apparently, that's famous. I don't know.

JP: Oh, yes. Tilings. Enumeration of tilings.

KK: That’s right. It was it was a talk about tilings. Really interesting stuff.

JP: Yeah, actually, I should say, I have gone back to dynamical systems a little bit, combining it with combinatorics. And that's a big part of what I do these days, but I won't be talking about that at all.

EL: Okay. And what is your favorite theorem?

JP: Ah, well, I've actually been sort of leading you on a bit because I'm not going to tell you my favorite theorem, partly because I don't have a favorite theorem.

KK: Sure.

JP: And if I did, I wouldn't tell you about it on this podcast, because it would probably have a heavy visual component, like most of my favorite things in math, and it probably wouldn't be suited to the purely auditory podcast medium.

KK: Okay, so what are you gonna tell us?

JP: Well, I could tell you about one theorem that I like that doesn't have much geometric content. But I'm not going to do that either.

EL: Okay, so what bottom of the barrel…

JP: I’m going to tell you about two theorems that I like, okay, they’re sort of like twins. One is in continuous mathematics, and one is in discrete mathematics.

KK: Great.

JP: The first one, the one in continuous mathematics, is pretty obscure. And the second one, in discrete mathematics, is incredibly obscure. Like nobody’s named it. And I've only found it actually referred to, stated as a result in the literature once. But I feel it's kind of underneath the surface, making a lot of things work, and also showing resemblances between discrete and continuous mathematics. So these are, like, my two favorite underappreciated theorems.

EL: Okay.

KK: Oh, excellent. Okay, great. So what do we got?

JP: Okay, so for both of these theorems, the underlying principle, and this is going to sound kind of stupid, is if something doesn't change, it’s constant.

EL: Okay. Yes, that is a good principle.

JP: Yeah. Well, it sounds like a tautology, because, you know, doesn't “not changing” and “being constant” mean the same thing? Or it sounds like a garbled version of “change is the only constant.” But no, this is actually a mathematical idea. So in the continuous realm, when I say “something,” what I mean is some differentiable function. And when I say “doesn't change,” I mean, has derivative zero.

KK: Sure.

JP: Derivatives are the way you measure change for differentiable functions. So if you’ve got a differentiable function whose derivative is zero—let’s assume it's a function on the real line, so its derivative is zero everywhere—then it's just a constant function.

KK: Yes. And this is a corollary of the mean value theorem, correct?

JP: Yes! I should mention that the converse is very different. The converse is almost a triviality. The converse says if you've got a constant function, then it's derivative is zero.

KK: Sure.

JP: And that just follows immediately from the definition of the derivative. But the constant value theorem, as you say, is a consequence of the mean value theorem, which is not a triviality to prove.

KK: No.

JP: In fact, we'll come back later to the chain of implications that lead you to the constant valnue theorem, because it's surprisingly long in most developments.

KK: Yes.

JP: But anyway, I want to point out that it's kind of everywhere, this result, at least in log tables— I mean, not log tables, but anti-differentiation tables. If you look up anti-derivatives, you'll always see this “+C” in the anti-derivative in any responsible, mathematically rigorous table of integrals.

EL: Right.

JP: Because for anti derivatives, there's always this ambiguity of a constant. And those are the only anti-derivatives of a function that's defined on the whole real line. You know, you just add a constant to it, no other way of modifying the function will leave its derivative alone. And more generally, when you've got a theorem that says what all the solutions to some differential equation are, the theorem that guarantees there aren't any other solutions you aren't expecting is usually proved by appealing to the constant value theorem at some level. You show that something has derivative zero, you say, “Oh, it must be constant. “

KK: Right.

JP: Okay. So before I talk about how the constant value theorem gets proved, I want to talk about how it gets used, especially in Newtonian physics, because that's sort of where calculus comes from. So Newtonian physics says that if you know the initial state of a system, you know, of a bunch of objects—you know their positions, you know their velocities—and you know the forces that act on those objects as the system evolves, then you can predict where the objects will be later on, by solving a differential equation. And if you know the initial state and the differential equation, then you can predict exactly what's going to happen, the future of the system is uniquely determined.

KK: Right.

JP: Okay. So for instance, take a simple case: you’ve got an object moving at a constant velocity. And let's say there are no forces acting on it all. Okay? Since there are no forces, the acceleration is zero. The acceleration is the rate of change of the velocity, so the velocity has derivative zero everywhere. So that means the velocity will be constant. And the object will just keep on moving at the same speed. If the constant value theorem were false, you wouldn't really be able to make that assertion that, you know, the object continues traveling at constant velocity just because there are no forces acting on it.

KK: Sure.

JP: So, kind of, pillars of Newtonian physics are that when you know the derivative, then you really know the function up to an ambiguity that can be resolved by appealing to initial conditions.

EL: Yeah.

KK: Sure.

JP: Okay. So this is actually telling us something deep about the real numbers, which Newton didn't realize, but which came out, like, in the 19th century, when people began to try to make rigorous sense of Newton's ideas. And there's actually a kind of deformed version of Newton's physics that's crazy, in the sense that you can't really predict things from their derivatives and from their initial conditions, which no responsible physicist has ever proposed, because it's so unrealistic. But there are some kind of crazy mathematicians who don't like irrational numbers. I won't name names. But they think we should purge mathematics of the real number system and all of these horrible numbers that are in it. And we should just do things with rational numbers. And if these people tried to do physics just using rational numbers, they would run into trouble.

EL: Right.

JP: Because you can have a function from the rational numbers to itself, whose derivative is zero everywhere—with derivative being defined, you know, in the natural way for functions from the rational to itself—that isn't a constant function.

KK: Okay.

JP: So I don't know if you guys have heard this story before.

KK: This is making my head hurt a little, but okay. Yeah.

EL: Yeah, I feel like I have heard this, but I cannot recall any details. So please tell me.

JP: Okay, so we know that the square root of two is irrational, so every rational number, if you square it, is either going to be less than two, or greater than two.

KK: Yes.

JP: So we could define a function from the rational numbers to itself that takes the value zero if the input value x satisfies the inequality x2 is less than 2 and takes the value 1 if x2 is bigger than two.

EL: Yes.

JP: Okay. So this is not a constant function.

KK: No.

JP: Right. Okay. But it actually is not only continuous, but differentiable as a function of the

EL: Of the rationals…

JP: From the rationals to itself.

KK: Right. The derivative zero but it's not constant. Okay.

JP: Yeah. Because take any rational number, okay, it's going to have a little neighborhood around it avoiding the square—avoiding the hole in the rational number line where the square root of 2 would be. And it's going to be constant on that little interval. So the derivative of that function is going to be zero.

KK: Sure.

JP: At every national number. So there you have a non-constant function whose derivative is zero everywhere. Okay. And that's not good.

KK: No.

JP: It’s not good. for math. It's terrible for physics. So you really need the completeness property of the real in a key way to know that the constant value theorem is true. Because it just fails for things like the set of rational numbers.

EL: Right.

KK: Okay.

JP: This is part of the story that Newton didn't know, but people like Cauchy figured it out, you know, long after.

KK: Right.

JP: Okay. So let's go back to the question of how you prove the constant value theorem.

EL: Yeah.

JP: Actually, I wanted to jump back, though, because I feel like I wanted to sell a bit more strongly this idea that the constant value theorem is important. Because if you couldn't predict the motions of particles from forces acting on those particles, no one would be interested in Newton's ideas, because the whole story there is that it is predictive of what things will do. It gives us a sort of clockwork universe.

KK: Sure.

JP: So Newton's laws of motion are kind of like the rails that the Newtonian universe runs on, and the constant value theorem is what keeps the universe from jumping off those rails.

KK: Okay. I like that analogy. That’s good.

JP: That’s the note I want to end on for that part of the discussion. But now getting back to the math of it. So how do you prove the constant value theorem? Well, you told me you prove it from the mean value theorem. Do you remember how you prove the mean value theorem?

KK: You use Rolle’s theorem?

EL: Just the mean value theorem turned sideways!

KK: Sort of, yeah. And then I always joke that it’s the Forrest Gump proof. Right? You draw the mean value theorem, you draw the picture on the board, and then you tilt your head, then you see that it's Rolle’s theorem. Okay, but Rolle’s theorem requires, I guess what we sometimes call in calculus books Fermat’s theorem, that if you have a differentiable function, and you're at a local max or min, the derivative is equal to zero. Right?

JP: Yup. Okay, actually, the fact that there exists even such a point at all is something.

KK: Right.

JP: So I think that's called the Extreme Value Theorem.

KK: Maybe? Well, the Extreme Value Theorem I always think of as as—well, I'm a topologist—that’s the image of a compact set is is compact.

JP: Okay.

KK: Right. Okay. So they need to know what the compact sets of the real line are.

JP: You need to know about boundedness, stuff like that, closedness.

KK: Closed and bounded, right. Okay. You're right. This is an increasingly long chain of things that we never teach in Calculus I, really.

JP: Yeah. I've tried to do this in some honors classes with, you know, varying levels of success.

KK: Sure.

JP: There’s the boundedness theorem, which says that, you know, a continuous function is bounded on a closed interval. But then how do you prove that? Well, you know, Bolzano-Weierstrass would be a natural choice if you're teaching a graduate class, maybe you prove that from the monotone convergence theorem. But ultimately, everything goes back to the least upper bound property, or something like it.

KK: Which is an axiom.

JP: Which is an axiom, that’s right. But it sort of makes sense that you'd have to ultimately appeal to some heavy-duty axiom, because like I said, for the rational numbers, the constant value theorem fails. So at some point, you really need to appeal to the the completeness of the reals.

EL: Yeah, the structure of the real numbers.

KK: This is fascinating. I've never really thought about it in this much detail. This is great.

JP: Okay. Well, I'm going to blow your mind…

KK: Good!

JP: …because this is the really cool part. Okay. The constant value theorem isn't just a consequence of the least upper bound property. It actually implies the least upper bound property.

KK: Wow. Okay.

JP: So all these facts that this, this chain of implications, actually closes up to become a loop.

KK: Okay.

JP: Each of them implies all the others.

KK: Wow. Okay.

JP: So the precise statement is that if you have an ordered field, so that’s a number system that satisfies the field axioms: you've got the four basic operations of pre-college math, as well as inequality, satisfying the usual axioms there. And it has the Archimedean property, which we don't teach at the pre-college level. But informally, it just says that nothing is infinitely bigger or infinitely smaller than anything else in our number system. Take any positive thing, add it to itself enough times, it becomes as big as you like.

KK: Okay.

JP: You know, enough mice added together can outweigh an elephant.

KK: Sure.

EL: Yeah.

JP: That kind of thing. So if you've got an ordered field that satisfies the Archimedean property, then each of those eight propositions is equivalent to all the others.

KK: Okay.

JP: So I really like that because, you know, we tend to think of math as being kind of linear in the sense that you have axioms, and from those you prove theorems, and from those you prove more theorems—it's a kind of a unidirectional flow of the sap of implication. But this is sort of more organic, there's sort of a two-way traffic between the axioms and the theorems. And sometimes the theorems contain the axioms hidden inside them. So I kind of like that.

KK: Excellent.

JP: Yeah.

KK: So math’s a circle, it's not a line.

JP: That’s right. Anyway, I did say I was going to talk about two theorems. So that was the continuous constant value theorem. So I want to tell you about something that I call the discrete constant value theorem that someone else may have given another name to, but I've never seen it. Which also says that if something doesn't change, its constant. But now we're talking about sequences. S the something is just going to be some sequence. And when I say doesn't change, it means each term is equal to the next, or the difference between them is zero.

EL: Okay.

JP: So how would you prove that?

EL: Yeah, it really feels like something you don't need to prove.

KK: Yeah.

JP: If you pretend for the moment that it's not obvious, then how would you convince yourself?

KK: So you're trying to show that the sequence is eventually constant?

JP: It’s constant from the get-go, every term is equal to the next.

EL: Yeah. So the definition of your sequence is—or part of the definition of your sequence is—an equals an+1.

JP: That’s right.

EL: Or minus one, right?

JP: Right.

EL: So I guess you'd have to use induction.

KK: Right.

JP: Yeah, you’d use mathematical induction.

KK: Right.

JP: Okay. So you can prove this principle, or theorem, using mathematical induction. But the reverse is also true.

KK: Sure.

JP: You can actually prove the principle of mathematical induction from the discrete constant value theorem.

EL: And maybe we should actually say what the principle of mathematical induction is.

KK: Sure.

JP: Sure.

EL: Yeah. So that would be, you know, if you want to prove that something is true for, you know, the entire set of whole numbers, you prove it for the first one—for 1—and then prove that if it's true for n, then it's true for n+1. So I always have this image in my mind of, like, someone hauling in a chain, or like a big rope on a boat or something. And they're like, you know, each each pull of the of their arm is the is the next number. And you just pull it in, and the whole thing gets into boat. Apparently, that's where you want to be. Yeah, so that's induction.

JP: Yeah. So you can use mathematical induction to prove the discrete constant value theorem, but you can also do the reverse.

EL: Okay.

JP: So just as the continuous constant value theorem could be used as an axiom of completeness for the real number system, the discrete constant value theorem could be used as an axiom for, I don't want to say completeness, but the heavy-duty axiom for doing arithmetic over the counting numbers, to replace the axiom of induction.

EL: Yeah, it has me putting in my mind, like, oh, how could I rephrase, you know, my standard induction proof—that at this point, kind of just runs itself once I decided to try to prove something by induction—like how to make that into a sequence, a statement about sequences?

JP: Yeah, for some applications, it's not so natural. But one of the applications we teach students for mathematical induction proving formulas, right? Like, the sum of the first n positive integers is n times n+1 over 2.

KK: Right.

JP: And so we do a base case. And then we do an induction step. And that's the format we usually use.

KK: Right.

JP: Okay. Well, proving formulas like that has been more or less automated these days. Not completely, but a lot of it has been. And the way computers actually prove things like that is using something more like the discrete constant value theorem.

EL: Okay.

JP: So for example, say you've got a sequence who's nth term is defined as the sum of the first n positive integers.

KK: Okay.

JP: So it’s 1, 1+2, 1+2+3,…. Then you have another sequence whose anthem is defined by the formula, n times n+1 over 2.

KK: Right.

JP: And you ask a computer to prove that those two sequences are equal to each other term by term. The way these automated systems will work, is they will show that the two sequences differ by a constant,

EL: and then show that the constant is zero.

JP: And then they’ll show that the constant is zero. So you show that the two sequences at each step increase by the same amount. So whatever the initial offset was, it’s going to be the same. And then you see what that offset is.

EL: Yeah.

KK: Okay, sure.

JP: So this is this looking a lot more like what we do in differential equations classes, where, you know, if you try and solve a differential equation, you determine a solution up to some unknown real parameters, and then you solve for them from initial conditions. There's a real strong analogy between solving difference equations in discrete math, and solving differential equations in continuous math. But somehow, the way we teach the subjects hides that.

EL: Yeah.

JP: The way we teach mathematical induction, by sort of having the base case come first, and then the induction step come later, is the reverse order from what we do with differential equations. But there's a way to, you know, change the way we present things so they're both mathematically rigorous, but they're much more similar to each other.

KK: Yeah, we've got this bad habit of compartmentalizing in math, right? I mean, the lower levels of the curriculum, you know, it's like, “Okay, well, in this course, you do derivatives and optimization. And in this course, you learn how to plow through integration techniques. And this course is multi-variable stuff. And in this course, we're going to talk about differential equations.” Only later do you do the more interesting things like induction and things like that. So are you arguing that we should just, you know, scrap it all in and start with induction on day one?

JP: Start with induction? No.

KK: Sure, why not?

JP: I’ve given talks about why we should not teach mathematical induction.

KK: Really?

JP: Yeah. Well, I mean, it’s not entirely serious. But I argue that we should basically teach the difference calculus, as a sort of counterpart to the differential calculus, and give students the chance to see that these ideas of characteristic polynomials and so forth, that work in differential equations, also work with difference equations. And then like, maybe near the end, we can blow their mind with that wonderful result that Robert Ghrist talked about.

KK: Yeah.

JP: Where you say that one of these operators, the difference operator, is e to the power of the derivative operator.

KK: Right.

EL: Yeah.

JP: They’re not just parallel theories. They're linked in a profound way.

EL: Yeah, I was just thinking this episode reminded me a lot of our conversation with him, just linking those two things that, yeah they they are in very different places in in my mental map of how how I think about math.

KK: All right. So what does one pair with these theorems?

JP: Okay, I'm going pair the potato chip.

EL: Okay, great. I love potato chips.

KK: I do too.

JP: So I think potato chips sort of bridge the gap between continuous mathematics and discrete mathematics.

EL: Okay.

JP: So the potato chip as an icon of continuous mathematics comes by way of Stokes’ theorem.

KK: Sure.

JP: So if you’ve ever seen these books like Purcell’s Electromagnetism that sort of illustrate what Stokes’ theorem is telling you, you have a closed loop and a membrane spanning in it,

EL: Right.

JP: …a little like a potato chip.

KK: Sure. Right.

JP: And the potato chip as an icon of discrete mathematics comes from the way it resembles mathematical induction.

KK: You can't eat just one.

JP: That’s right. You eat a potato chip, and then you eat another, and then another, and you keep saying, “This is the last one,” but there is no last potato chip.

EL: Yeah.

JP: And if there’s no last potato chip, you just keep eating them.

KK: That’s right. You need another bag. That's it.

EL: Yeah.

JP: But the other reason I really like the potato chip as sort of a unifying theme of mathematics, is that potato chips are crisp, in the way that mathematics as a whole is crisp. You know, people complain sometimes that math is dry. But that's not really what they're complaining about. Because people love potato chips, which are also dry. What they really mean is that it’s flavorless, that the way it's being taught to them lacks flavor.

KK: That’s valid, actually. Yeah.

JP: So I think we need to do is, you know, when the math is too flavorless, sometimes we have to dip it into something.

EL: Yeah, get your onion dip.

JP: Yeah, the onion dip of applications, the salsa of biography, you know, but math itself should not be moist, you know?

EL: So, do you prefer like the plain—like, salted, obviously—potato chips, or do you like the flavors?

JP: Yeah, I don't like the flavors so much.

EL: Oh.

JP: I don’t like barbecue or anything like that. I just like salt.

KK: I like the salt and vinegar. That’s…

EL: Yeah, that's a good one. But Kettle Chips makes this salt and pepper flavor.

KK: Oh, yeah. I’ve had those. Those are good.

EL: It’s great. Their Honey Dijon is one of my favorites too. And I love barbecue. I love every—I love a lot of flavors of chips. I shouldn't say “every.”

KK: Well yeah, because Lay's always has this deal every year with the competition, like, with these crazy flavors. So, they had the chicken and waffles one one year.

EL: Yeah, I think there was a cappuccino one time. I didn’t try that one.

KK: Yeah, no, that’s no good.

JP: I just realized, though, potato chips have even more mathematical content than I was thinking. Because there's the whole idea of negative curvature of surfaces.

EL: Yes, the Pringles is the ur-example of negatively curved surface.

JP: Yeah. And also, there's this wacky idea of varifolds, limits of manifolds, where you have these corrugated surfaces and you make the corregations get smaller and smaller, like, I think it’s Ruffes.

EL: Yeah, right.

JP: So a varifold is, like, the limit of a Ruffles potato chip as the Ruffles shrink, and the angles go to zero. There’s probably a whole curriculum.

EL: Yeah, we need a spinoff podcast. Make this—tell tell what this potato chip says about math.

KK: Right.

EL: Just give everyone a potato chip and go for it.

KK: Excellent.

EL: Very nice. I like this pairing a lot.

KK: Yeah.

EL: Even though it's now, like, 8:30-something, I'll probably go eat some potato chips before I have breakfast, or as breakfast.

KK: I want to thank you, Evelyn, because I know it wasn't your choice to do it this early in the morning. I had childcare duties, so thank you for your flexibility.

EL: I dug deep. Well, it was a sunny day to day. So actually the light coming in helped wake me up. It's been really rainy this whole month, and that's not great for me getting out of bed before, you know, 10 or 11 in the morning.

KK: Sure. So we also like to give our guests a chance to plug things. You have some stuff coming up, right?

JP: I do. Well, there's always my Mathematical Enchantments essays. And I think my July essay, which will come out on the 17th, as always, will be about the constant value theorems. And I'll include links to stuff I have written on the subject. So anyone who wants to know more, should definitely go first to my blog. And then in early August, I'll be giving some talks in New York City. And they'll be about a theorem with some visual content called the Wall of Fire theorem, which I love and which was actually inspired by an exhibit at the museum. So it's going to be great actually give a talk right next to the exhibit that inspired it.

EL: Oh, yeah, very nice.

KK: This is at the Museum of Math, the National Museum of Math, right? Okay.

JP: Yeah, I’ll actually give a bunch of talks. So the first talk is going to be, like, a short one, 20 minutes. That's part of a conference called MOVES, which stands for the Mathematics of Various Entertaining Subjects.

KK: Yeah.

JP: It’s held every two years at the museum, and I don't know if my talk will be on the fourth, fifth or sixth of August, but it'll be somewhere in that range. And then the second talk will be a bit longer, quite a bit longer. And it's for the general public. And I'll give it twice on August 7th, first at 4pm, and then at 7pm. And it'll feature a hands on component for audience members. So it should be fun. And that's part of the museum's Math Encounters series, which is held every month. And for people who aren't able to come to the talk, there'll be a video on the math encounters website at some point.

EL: Oh, good. I've been meaning to check that because I'm on their email list, and so I get that, but obviously living in Salt Lake City, I don't end up in New York a whole lot. So yeah, I'm always like, “Oh, that would have been a nice one to go to.”

KK: Yeah.

EL: But I'll I'll have to look for the videos.

KK: So, Jim, thanks for joining us.

JP: Thank you for having me.

KK: Thanks for making me confront that things go backwards in mathematics sometimes.

EL: Yes.

KK: Thanks again.

EL: Yeah, lots of fun.

JP: Thank you very much. Have a great day.

[outro]

In this episode of My Favorite Theorem, we were happy to talk with Jim Propp, a mathematician at the University of Massachusetts Lowell. He told us about the constant value theorem and the way it unites continuous and discrete mathematics.

Here are some links you might find interesting after listening to the episode:

Propp’s blog Math Enchantments (home page, wordpress site)
His list of problems about enumeration of tilings that we mentioned
Our previous My Favorite Theorem episode with guest Robert Ghrist, who also talked about a link between continuous and discrete math
Propp’s article “Real Analysis in Reverse”
Mean Value Theorem
Rolle’s Theorem
Fermat’s Theorem
Varifold
MOVES (Mathematics of Various Entertaining Subjects), a conference he will be speaking at in August
Math Encounters, a series at the Museum of Mathematics (he will be speaking there in August)