text:
Generalized solutions of first-order PDEs. The dynamical optimization perspective.(English)Zbl 0820.35003

Basel: Birkhäuser,. xi, 312 p. (1994).
In the author’s own words, “this book is intended as a self-contained exposition of the theory of minimax solutions” for some classes of first-order PDE of the form: $F(x,u (x),Du(x)) = 0, \;\;x \in G = \text{Int} (G) \subset \mathbb{R}^ n, \quad u(x) = u_ 0(x)\;\forall x \in G_ 0 \subset \mathbb{R}^ n.\tag{1}$ The functions $$f(x,.,p): \mathbb{R} \to \mathbb{R}$$, $$(x,p) \in G \times \mathbb{R}^ n$$, are assumed to be nonincreasing and in most of the results, for any $$(x,z) \in G \times \mathbb{R}$$, the function $$F(x,z,.)$$ is assumed to be Lipschitzian of rank $$\rho (x,z)$$ where $$\rho (.,.)$$ is continuous.
The first definition of the author’s “minimax solution”, as a continuous function, $$u(.) : G \to \mathbb{R}$$, such that for any $$(x_ 0,z_ 0) \in \text{graph} u(.)$$ and any $$p \in \mathbb{R}^ n$$ there exist $$r>0$$ and a Lipschitzian mapping $$(x(.), z(.)): [0,r] \to \text{graph} u(.)$$ such that: $$(x(0), z(0)) = (x_ 0,z_ 0)$$, $$z'(t) = \langle x'(t)$$, $$p \rangle - F(x(t), z(t),p)$$ a.e. in $$([0,r])$$, suggests its interpretation as a generalization of the classical solutions obtained by Cauchy’s method of characteristics.
An equivalent definition, according to which a continuous function $$u(.)$$ is a minimax solution iff its graph is weakly invariant with respect to the family of differential inclusions: $$(x',z') \in E(x,z,p)$$, $$p \in \mathbb{R}^ n$$, where $$E(x,z,p) : = \{(v,w) \in \mathbb{R}^ n \times \mathbb{R}$$; $$\| v \| \leq \rho (x,z)$$, $$w= \langle v,p \rangle - F(x,z,p)\}$$, leads to other equivalent definitions expressed in terms of tangential criteria for weakly invariant sets with respect to convex-valued differential inclusions.
The main result of Chapter I is Theorem 4.3 containing 7 equivalent definitions of the minimax solutions, in addition to those already mentioned, being the celebrated viscosity solutions for the equation $- F(x,u(x),Du(x)) = 0, \qquad x \in G \tag $$1'$$$ [e.g. M. G. Crandall and P.-L. Lions, Trans. Am. Math. Soc. 277, 1-42 (1983; Zbl 0599.35024)], and “contingent solutions”, defined in terms of differential inequalities verified by the extreme contingent directional derivatives of the solution.
The basic results, concerning uniqueness, existence, Hopf’s formulas and numerical methods for minimax solutions, are presented in Ch. II for Cauchy problems of the form: $D_ tu + H(t,x,u,D_ xu) = 0,\;(t,x) \in G : = (0, \theta) \times \mathbb{R}^ n, \quad u(\theta,x) = u_ 0(x)\;\forall x \in \mathbb{R}^ n, \tag{2}$ and in Ch. IV for Cauchy problems with additional conditions of the form: $$u(t,x) \leq w(t,x)$$ $$\forall (t,x) \in G$$, and also for Dirichlet problems of the form: $H(x,Du) - u = 0, \quad x \in G, \qquad u(x) = u_ 0(x)\quad \forall x \in \partial G. \tag{3}$ A large part of the book (Ch. III and part of Ch. IV) contains basic concepts and results from the theory of feedback (positional) differential games which is at the origin of the author’s concept of minimax and contingent solutions; in fact, the results of N. N. Krasovskij and the author published in the early 70s [e.g., Dokl. Akad. Nauk SSSR 190, 523-526 (1970; Zbl 0213.159), Positional differential games (in Russian) (Nauka 1974, Zbl 0298.90067)], concerning $$u$$-stable and $$v$$-stable solutions of the Bellman-Isaacs equation, anticipated with more than 10 years the theory of viscosity (super- and sub-) solutions.
The book also contains a 28 page Appendix presenting the necessary notations, definitions and auxiliary results from nonsmooth analysis and differential inclusions, and a large bibliographical list with 231 items.
In a brief but, of course, incomplete characterization, one may say that the present book provides several new equivalent definitions, new insight and an apparently more constructive approach to the theory of viscosity solutions and their applications to differential games.
Unfortunately, as in many other books and papers, any possible limits of the elaborated theory are either ignored or overlooked and lightly passed by; in this case (as in that of the theory of viscosity solutions) certain limits of the theory may be produced by the fact that the effective domain of the function $$F(.,.,.)$$ in problem (1) is the set $$G \times \mathbb{R} \times \mathbb{R}^ n$$. To illustrate these limits we give only two classical examples: as it is well known [e.g. L. Cesari, Optimization – theory and applications (Springer, N. Y., 1983; Zbl 0506.49001), the value function of the celebrated Brachistochrone problem is a classical (hence viscosity, minimax, etc.) solution of the problem $H(x, Du(x))=0, \quad x \in G \subset \mathbb{R}^ 2, \qquad u(x) = u_ 0(x) \quad \forall x \in G_ 0 \tag{4}$ defined by: $$G_ 0 = \{(0,0)\}$$, $$u_ 0(0,0) = 0$$, $$G = (0, \infty) \times (0, \infty)$$, $$H(x,p) = 0$$ if $$x = (x_ 1, x_ 2) \in G$$, $$p \in \mathbb{R}^ 2$$, $$\| p \| \leq 1/(x_ 2 + k)^{1/2}$$ and $$H(x,p) = - \infty$$ otherwise, where $$k \geq 0$$ is a given constant; similarly, the value function of Plateau’s problem for surfaces of revolution is a locally-Lipschitz (viscosity, minimax, contingent) solution of the problem (4) defined by: $$G_ 0 = \{(0,y)\}$$, $$y > 0$$, $$u_ 0(0,y) = 0$$, $$G = (0, \infty) \times [0, \infty)$$, $$H(x,p) = 0$$ if $$x = (x_ 1,x_ 2) \in G$$, $$p \in \mathbb{R}^ 2$$, $$\| p \| \leq x_ 2$$ and $$H(x,p) = - \infty$$ otherwise. Obviously, the two Hamiltonians above do not satisfy the hypotheses under which the results in the book are obtained; in particualar, problem (4) does not enjoy the uniqueness property since $$u(x) \equiv 0$$ is a classical solution in both cases.

MSC:

 35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations 35F20 Nonlinear first-order PDEs 49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games 49N70 Differential games and control 49N75 Pursuit and evasion games


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## Friday, September 09, 2005

### Of distraction and potato chips

So yes, I'm way behind on math, but there are reasons for that: I've just recently started teaching, and figuring out how to do right by the 29 kids whose only instruction is me for 4.5 hours a week has occupied much of my willingness to explain math for the last several weeks (before that there was another thing, but harder to explain -- I merely assure you of its presence). We'll see when it recovers.

However, via Making Light, I've come across the blog of one Chippie, taking particular interest in her recent potato-chip making foray. I started trying to comment about it, but it got so long it seemed like at least some of my regular readers might take an interest. So, in the spirit of unsolicited advice from strangers, here's what small wisdom I have about chip making (and I must say I can make some pretty decent chips):

-Potato choice. I've only ever fried russets, and I find that they come out remarkably well -- golden brown with a serious potato flavor I've never seen on other chips. I imagine most other potatos are also ok.

-Thin is key -- chips aren't fries. Short of a mandolin or v-slicer (devices that can cut foods into very thin strips), the best way to go is to use a vegetable peeler and peel long strips off the potato. You won't get that nice chip shape, but you will get thin without buying a gadget you may not otherwise need. Also, one potato makes a decent-sized collection of chips, but you'll probably need two or three to replace that Lay's party-size bag.

-Keep the proto-chips soaking in water, but dry them as well as you can (salad spinner, towels) before frying.

-General frying principles: more oil is better. Use a large, heavy vessel, so that the heat will return more quickly to the right temperature after you put in food (it drops due to the additional mass). I use a dutch oven and keep my oil in two green (keeps the light out) wine bottles. Get a fry thermometer and watch it like a hawk -- 375 is good for chips, but going over 400 will spoil your oil (it'll start to burn). Needless to say, use vegetable/canola/corn/peanut oil, not olive or something else with flavor, as such things burn way sooner than 375. You can (and should) save your oil from one session to another: let it cool, filter, store, and reuse.

-Drop in the chips a handful at a time, say 10-15 chips total in a single batch in a reasonable-sized pot (you'll do several). Pull them out when they've almost stopped bubbling, about 3 minutes (they'll still be good even if they've stopped). Remove to draining rack (I like paper towels on a plate), salt, pepper, enjoy. These are certainly my favorite chips (nice and potato-y!).

With practice, you can even peel the strips online -- peel 15, start frying, peel 15 more, pull the others out, start the new batch, season the old, peel 15 more...

The footnote: this recipe is essentially due to Alton Brown in his book "I'm Just Here for the Food," and the chips are extra-good with his version of homemade onion dip, and extra extra good if you make it with your own homemade mayo.

## Tuesday, August 16, 2005

### Math Post the Ninth: Group Actions

Well it looks like that whole "I'll blog during the conference" idea petered out quickly; what with thinking about math enough hours a day to wear me out before blogging, there have been no posts in three weeks. So now, right back into it.

Following some extremely gentle chiding from Neil, along with a link to the following article, I've decided to again defer the promised note on factoring maps and talk instead about group actions, implicitly the subject of the piece linked (you don't need to read it yet).

One of the most fruitful ways to study something is to study its symmetries; correspondingly, one of the most fruitful ways to study groups is to study the things of which they are symmetries. So, how can a group be a symmetry of a thing? Practically every object (ok, every object) out there in the world of mathematics has what's usually called an automorphism (=isomorphism to yourself) group, denoted Aut(X). If X is a set, Aut(X) is composed of the bijections X->X; if X is a group (or other algebraic structure), the isomorphisms; if X is a polygon, the rigid self-maps (rotations and reflections); if X is some kind of bigger geometric object like the plane, symmetries are defined in close to the same way -- as invertible maps which preserve "the structure," where "the structure" is whatever structure you happen to care about at the moment. The simplest way to define a group action of G on a thing X is to say it's a homomorphism \phi:G->Aut(X). Equivalently, if X is a set with some additional structure (like all of our examples thus far), you can write it as a "multiplication" of elements of X by elements of G such (i) ex=x for all x, e the identity of G, and (ii) g(hx)=(gh)x,* as long as that multiplication "respects the structure."

Some examples of group actions include:

(1) A group G acts on the set of its elements by multiplication; just multiply in the group. This isn't a homomorphism, and in fact seems kinda dumb, but is still surprisingly important. Similarly, G acts on the (left) cosets of any subgroup in the same way.

(2) A group G acts on itself as a group by conjugation -- g acting on h is ghg^(-1). This actually is a homomorphism, since if you multiply two g-conjugates together the g's in the middle cancel; thus G acts by conjugation on its set of subgroups. Normal subgroups are precisely those stable under this action. This action is important enough that the subgroup of Aut(G) that is hit by it is called the inner automorphisms and written Inn(G); it is actually a normal subgroup (of Aut(G)!) in its own right, and Aut(G)/Inn(G)=Out(G), the outer automorphisms.

(3) Permutation groups act on the set they're permuting.

(4) Symmetries of polygons act on the polygons.

(5) A subgroup of a group acting on X also acts on X.

With these actions identified, I'll give a few more words. An element x in X has an orbit under G, consisting of all the other elements G can move x to; the subgroup of G which doesn't move x at all is called the stabilizer of x, written Stab_G(x) or myriad other ways that suggest the same thing. An action of G on a set X is called transitive if there is only one orbit; that is, if some element of G can move any element of X to any other. Every action is really a collection of transitive actions (that is, a collection of orbits), and any transitive action is the same thing as the action of G on cosets of the stabilizer of any point in the orbit, which takes some thought but isn't hard to see. An action is faithful if no two elements of G act the same way on X, or the map G->Aut(X) is injective, an action is free if no non-identity element fixes a point (or all stabilizers are trivial), and an action is simply transitive if it is transitive and free; this last implies that G and X have the same number of elements by the point about stabilizers above, since there are no stabilizers in this case. This also implies that a simply transitive action is faithful.

That may seem like a giant load of terminology, but the payoff is Cayley's theorem: every finite group is a subgroup of a permutation group. And in fact with all those words out of the way the proof is almost trivial: G acts on itself by left multiplication, giving a map G->Aut(G) as a set, and Aut(G) as a set is a permutation group. The action is simply transitive, hence faithful, so the map is injective and we're done. The argument here actually generalizes to geometric cases of various kinds, but we won't have to touch those for a long time yet.

Now, go read the article I linked above; it should be easy to figure out where he's going even before he goes there. The exercise to do is this: figure out what the author means by a "golden rule" in terms of group actions. The hint is that it can be said in five words, all of which I've defined already in some math post...

*Writing actions on the left (rather than the mathematically-equivalent right -- that is as gx rather than xg) is actually a slightly bad thing to do if the group G is not Abelian; for reasons that can be avoided if you don't want to dive deeply into the notation, you often want to act by the inverse of some natural action if you're acting on the left. However, it looks slightly more natural for most people, so for informal purposes I'll elide the concern. In fact, this issue is something I've seen distinguished professors argue about at length in class (young, distinguished, argumentative commutative guy didn't care where his action went; old, distinguished, noncommutative but slightly dyslexic guy wanted them on the right), so it should be clear why I'm hoping to skip out on the whole discussion.

## Thursday, July 21, 2005

### Math Post the Eighth: Personal Interlude with Some More Definitions

So I am now officially a doctoral candidate! Heck yeah! Also I'm now (or will soon) be attending a lengthy algebraic geometry conference in Seattle, so I'm not even in Michigan anymore. Perhaps most importantly from a blogging perspective, I'm not madly studying every second of my day anymore (merely some seconds), so I should be able to get back to blogging on a more regular basis. So nyah, say I!

I've also redone my links section, doing away with links to friends' livejournals because I couldn't think of a good way to separate them from links to friends' more serious blogs or therefore from other peoples' more serious blogs. Any complaint and I'll rethink it, but it seemed like the right way to go at the moment.

However, I've decided to defer my promised discussion of isomorphism theorems and universal properties so I can, in celebration of my prelims, explain that portion of my research that I'm currently prepared to discuss (that is, that part that shouldn't be much over anybody's head for reasons other than density). So, in broad strokes, here we go:

One first kind of assumption that mathematicians like to make is a finiteness assumption. It's usually easier to deal with things that are "small" than things that are "big," and indeed strange things can happen in the world of large objects (they can be the same as several copies of themselves stuck together, for one). One finiteness assumption we've already used is precisely that: that the number of elements in a group is finite. A lesser one harks back to my discussion of generation of a group, in particular that our group has a finite set of generators, or is finitely generated. Many (though not quite all) of the examples I've mentioned in that first group post are finitely generated (an exercise for the reader to figure out which!). Making an assumption like this is more about choosing a field of study than picking out a special case.

A second definition I'll need is the condition of a subgroup having finite index. The index of a subgroup H in a group G is the number of cosets that you can form from H with elements of G. In the integers Z, the subgroup 3Z has three cosets, the subgroup 5Z five, and so on. For finite groups, Lagrange's theorem tells us that the index of a subgroup is simply the ratio of the orders (i.e. the number of copies of the smaller group it takes to cover the large one is the number of elements in one divided by the number of elements in the other). If we consider the group of ordered pairs of integers, the subgroup generated by the pair (1,0) is of infinite index, while the subgroup generated by the elements (2,0) and (0,2) is of index 4 (consider quotienting it out and note that you get a (the) two-element group in each component).

The first big assumption I'd make is that of being just infinite. A group G is just infinite if it is infinite but has only finite quotients. I'll cut to the examples in a second, but I need to add one more assumption to get to what I'm actually studying: I don't just want G just infinite, I also want every finite-index subgroup to be just infinite as well -- a condition called being hereditarily just infinite. Our first example is the integers. Since every normal subgroup (in fact, since the integers are abelian, every subgroup) has finite index, there can be no infinite quotients. A slightly more complicated example is the group of symmetries of an infinite line with equally-spaced dots on it (like the integer points in the real numbers), written as D_\infty, where the notation means an infinity symbol. These come essentially in two kinds -- shifts some number of spaces in one direction or the other or reflection around a particular point (or combinations of the two). The group actually looks like (something I'll get to later) the semi-direct product of the integers (the line movements) with the two-element group (a reflection about some point -- you can "make" reflecting about any point by translating, reflecting, then translating back). Seeing that this group is hereditarily just infinite is most easily done by thinking hard about group actions, which I will also get to later.

Before finishing up, I need one more assumption: that the group is not something I'll call simple. A simple group has no nurmal subgroups and therefore no interesting quotients at all. The finite groups of prime order are all simple; showing any other group to be simple (that is, actually doing the proofs) is well beyond the scope of what I'll be able to consider here for some time. Computing all the finite simple groups was a major research project, popularly considered finished around 1980, and filling many thousands of pages (so many in fact that people still aren't completely sure it's all correct -- there's a major effort to write it in some comprehensible way that's still going on). Infinite simple groups satisfy the condition of being hereditarily just infinite trivially (for the mathematicians in the audience, this is because any finite index subgroup H in a group G has a finite-index subgroup H' normal in G inside it -- think about it), but aren't very interesting examples for this same reason.

(For genuine group theorists, the previous paragraph is an ersatz for simply assuming G to be residually finite, which is equivalent for just-infinite groups but harder to explain)

The last class of examples is something harder to get your hands on -- for the big-word-friendly audience, I'm talking about irreducible lattices in higher rank semisimple Lie groups -- and harder still to show in general (again for the big-words crowd, this is Margulis' stuff). For the near future, you can simply think of them as big groups of integer matrices (say 3x3 or larger). My project is to show that these are the only interesting examples. It's likely to be uphill work, but it should at least lead to some intriguing stops along the way.

## Sunday, July 17, 2005

### Meme-ification

Wow. I used to be behind two, now I'm behind like six or eight in terms of math. I beg preliminary exams on Tuesday, the last hurdle I have before my only goal in life becomes a dissertation. For now, though, I steal from my friend/cooking idol eclectician, and meme about food:

Unclear. I think I may actually have never done it until sometime during summer 2000 when I was living on my own for the first time -- and such memories are phrased purely in terms of an industrial kitchen at an MIT fraternity rather than in terms of one dish or another. Since then, more enthusiasm, more frequency, and more skill (I hope).

Again almost uselessly hard to say in terms of people I actually know. Looking back on it, my parents figured out cooking more or less as I grew up, so early memories are few. I suppose the most honest answer is my little brother, who started cooking seriously well before me and who I must always outdo.

Personality/reference-wise, definitely Alton Brown of Good Eats fame, who manages the right mix of improvisation, science, and art (though who can let me down when he cheats).

Do you have an old photo as 'evidence' of an early exposure to the culinary world, and would you like to share it?

Not to my knowledge, alas -- everything I know is lately acquired as these things go. For instance, everything I've yet posted about math I knew before I knew anything about food beyond how to eat lots of it.

Mageiricophobia - do you suffer from any cooking phobia, a dish that makes your palms sweat?

A dish, not really -- there's two Indian desserts of which I've yet to create even a remotely decent rendition (pedas and ras gulla, for anybody wondering), but I usually only make them for myself -- but for a more general phobia there's an easy answer: cooking for the current special someone. The girl's a vegetarian and I am so totally not. Also she has deeply serious taste (unlike many vegetarians) and obviously I'm loathe to disappoint. Between these two things I think that this particular fear has taught me more about cooking than anything else in recent memory.

What would be your most valued or used kitchen gadgets and/or what was the biggest let down?

I probably get most action either out of my chef's knife (ten inches, calphalon, $20) or my cast-iron skillet (twelve inches, I don't even know,$20) -- I suspect if you just left me these and a heat source I'd be able to accomplish 80% of my cooking without great difficulty. As to letdowns, I'd like to nail my ice-cream maker, but the real problem with that one is that I just don't have enough ice-cream consumption taking place in my house to support an ice-cream maker for anything but vanity. For real, I'd probably have to nail that jar-opener my roommate's aunts got us when we moved in -- they furnished the apartment with an impressive array of goods, but that one thing I've never even considered using.

Name some funny or weird food combinations/dishes you really like - and probably no one else!

I'm (alas) an impressively normal kid regarding food combinations. The one thing that people have been surprised about me enjoying is kheer: kheer with cocoa powder or kheer as breakfast with coffee, both of which seem utterly natural to me. After all, milk, sweet? Who cares about the rice component; sense is still made.

What are the three eatables or dishes you simply don’t want to live without?

Number one has got to be cow meat -- out in the midwest, nothing else is as cheap, as good, and as versatile. Number two is garlic, since whenever I manage to make a meal without adding at least a little bit I feel vaguely ashamed, as though I'd forgotten the existence of a family member. Number three is wine. Between the variety of flavors and the sheer utility of the alcohol (not to mention the intoxication value), when the roomie's in town I go through at least a bottle a week.

... that whole list, on reflection, actually sounds pretty bland until you've eaten at my place a few times. I do have a few decent tricks.

I'm tempted to go with my associate's mention of burnt caramel, but there's an institution back in Princeton called Thomas Sweet which produces a flavor called chocolate chip cookie. Not cookie dough, but the cookie itself -- tiny chips mixed into a base of salty-sweet butter. Every bite is pure tollhouse, but summery in a way cookies simply aren't.

You will probably never eat:

Many, many things, but at this point more mediated by my inability to get them than anything else.

At this point, probably the steak I stole from an old New York Times recipe and modified until my roommate was weaned off a previous sauce addiction. Lamentably, I'm a much better thief than creative mind in the kitchen.

A common ingredient you just can’t bring yourself to stomach…

I haven't yet tried tounge...

Which one culture’s food would you most like to sample on its home turf?

Gujarati food. Blatantly self-serving, but these kids are the hardest veggies to impress I know of and I'd like to have a little better idea where it's all coming from.

The people I am tagging are:

Oddly, none of the bloggers I know who read this are foodies. Any are welcome to it, but if I had my choice of the whole wide world I'd pass the baton to Maggie the Mad Wine Girl over at the Wine Offensive, whose erstwhile wares I've sampled many a time and whose punk-gourmet attitude I simply adore.

More math soon, I promise!

## Tuesday, June 28, 2005

### Update and Apology

So I've been vacationing a bit and am thus way way behind on math posts (two, I think, is the number of my behindness), and I've actually really wanted to say something about Grokster, but haven't been able to formalize my thoughts into something resembling a post. Regardless, there will likely be at least one math post will be here by tomorrow; falling behinder, however, is likely in the near future (after which I'll try to catch up lots).

## Tuesday, June 21, 2005

### Math Post the Seventh: Kernels and Quotients

Last time I was talking about homomorphisms, the "nice maps" in the land of groups, and I gave some of the usual map definitions (injective, surjective, and so on). One of the things we (well, mathematicians -- I can't really answer for the rest of you) always want to know about maps is what the stuff over a particular point looks like; if you had a map to the integers, what stuff goes to three? Since this concept is so common, the set of stuff going to a particular point has a name: the fiber over that point. In the case of homomorphisms, the answer to this question has a simplifying wrinkle: the answer is the same for all points. To symbolify things a bit, let f:G->H be a homomorphism of groups and let h be in H. Now, say f(g)=f(g')=h, so both g and g' are in the fiber over h. Now, since f is a homomorphism, f(g^(-1)g)=h^(-1)h=e, so the difference between any two elements of the fiber over h are in the fiber over the identity; thus if you pick a particular element g in the fiber over h, and we call the fiber over the identity K (for reasons soon to become clear), the fiber over h is exactly gK. So all the information about the fibers is contained in the set K (called the kernel of the map f).

Now, we can say some quick things about K. First, K is a subgroup of G; two elements g and g' in K both map to the identity under f, so their product maps to the product of the identity with itself, which is still the identity, as is the inverse of the identity. However, K has one more special property: the left and right cosets of K are the same, that is for any g, gK=Kg (since both of these are precisely the fiber over the point f(g) in H). This implies the interesting property that gKg^(-1)=K, since this is the fiber over the identity again; this funny equation is called "conjugation by g". Now, if your group isn't abelian, conjugation may not fix all the elements of the subgroup K, but since K is a kernel, conjugation can only move an element of K to another element of K. Subgroups with the property of being fixed under conjugation are called "normal," in one of math's most badly overloaded terms.

Normal subgroups have one extra bonus fun property: their cosets form a group. Let N be a normal subgroup of G, and consider the cosets gN and g'N; if we try to multiply them in the straightforward way, we get gNg'N=gg'NN=gg'N (since N is closed under multiplication, NN=N). If N weren't normal, the left and right cosets wouldn't conicide, but since it is they do, and the cosets inherit all the group properties from the same properties of G. This new group is written G/N, and is called the quotient group of G by N. The easiest examples of this can be realized by the clock arithmetic groups Z/n (and now you understand the notation; because the integers are so familiar we omit the extra Z on the end) for various integers n; remember the cosets of the subgroup H=nZ are H, 1+H, 2+H, and so on up to (n-1)+H, and adding cosets is just adding the numbers out front and subtracting off n's if the number gets bigger than n. Of course, since Z is abelian, all subgroups are normal (the inverse just commutes through), so you need to be more careful in general.

G/N admits a nice map from G (an element g goes to the coset gN), nice in the way called "canonical," meaning that no choice is involved in determining the map -- we don't have to pick anything to make it up, it just comes to us. If you made a version of this map up, yours would be the same as mine. There are lots of non-canonical maps, and we'll meet plenty next time, but it's important to know about the differences in the kinds of nice maps.*

Next time: isomorphism theorems and factoring.

*And it's important to be specific about words; the inventors of a branch of mathematics called category theory, which is in some sense a language in which all other mathematics should be done, gave a definition for a kind of niceness they called "naturality." Fortunately for them (and us), the natural transformations of objects called functors are important and cool; unfortunately for us as students of math, people are sadly nonspecific with the word natural, rarely specifying when they mean the technical sense. One of my more trying math experiences was realizing after finishing a problem set (from the canonical algebraic geometry textbook) that when the word natural appeared in every problem it didn't just mean pleasant or straightforward but that there was suddenly much more work to do.

## Friday, June 17, 2005

### Math Post the Sixth: Isomorphism

This is going to verge on the philosophical, since I'll have to justify a "why" inside, but I'll kick off with the answer to the question from last time: describe all the groups of prime order. Start with the hint; pick an element g in a group G of prime order p, and say that g is not the identity. Now, g generates a subgroup, say H, and it has some nonidentity element in it, since g is such an element. Now Lagrange's theorem tells us that the order of H divides the order of G, but the order of G is prime, so H must have order either 1 or p. Since H has at least two elements (the identity and g), H has order p, so H=G. This argument shows that any group of prime order is generated by any element not the identity, has no nontrivial subgroups, and is "structurally the same" as the group where we add hours on a p-hour clock. Now the only issue is what I mean by "structurally the same."

Consider two groups: one is the group S_3 (underscore is ASCII for subscript) of permutations of three elements (i.e. all the moves the three-card-monte-guy can do) and the other is D_3, the rigid symmetries of an equilateral triangle. Both of these groups have six elements, and they have much in common in terms of structure, since they both can achieve any configuration of three things being moved around (you can reflect the triangle on an axis as well as rotate it, and think of it as acting on the corners). I want to say that they're the same, but I can't quite; after all, they're given in different ways, written with different letters, and so on. On the other hand, the situation is the same as if I took a clock and rubbed off all the numbers, replacing them with fruits -- I might now be saying that pear plus strawberrry equals pineapple, but that doesn't stop it from being a clock. So the question is how to capture this notion of "identical except in name."

The first part of the answer starts with the notion of a homomorphism. A homomorphism is a "nice"* function between two groups, and the buzzword is that it "preserves the group structure." Homomorphisms are traditionally written with Greek letters which I'm not sure how to render here, so until I find a better way I'm going to do it with LaTeX (math typesetting) notation, which should be pretty straightforward: a small phi will be written \alpha, a capital one \Alpha, and so on. The symbolic way to write a map \phi from a set G to a set H is like this: \phi : G->H, where the arrow-looking thing is in fact an arrrow. If G and H are groups, the map \phi is said to be a homomorphism if for any g and g' in G, \phi(gg')=\phi(g)\phi(g'). The trick to this simpleminded equation is that the multiplication is carried out in G on the left and H on the right. Now, say we have two homomorphisms which are inverses, say \phi:G->H and \psi:H->G such that doing \psi first then \phi gets you back to where you started in H and doing \phi first then \psi gets you back where you started in G (\phi(\psi(h))=h and \psi(\phi(g))=g for all g in G and h in H); then we can multiply in G by going over to H, multiplying there, and coming back and vice versa. This means that we can think of either G or H as just alternate names for the other group, which tells you that they're really the same.

Before concluding I'll define quickly some relevent terms. A map is said to be one-to-one or injective if no element gets hit twice; symbolically, if f(x)=f(y) implies x=y. For people who remember precalculus, this is the "horizontal line test" for a graph. A map is onto or surjective if everything gets hit. A map which is both injective and surjective is bijective. Bijective maps are invertible**, and bijective homomorphisms have inverses and are called isomorphisms.

To restate the conclusion of the first paragraph in nicer language, we classified all groups of prime order up to isomorphism: a group of order p is isomorphic to a clock arithmetic (or cyclic) group of order p, via the isomorphism defined by taking any nonidentity element to the generator 1.

Next time: what groups are homomorphic images of other groups.

*Practically every definition in math is a definition of "nice," though the connotation that goes with it can be anything from "super duper extra bonus spiffy" to "not something you made up just to annoy me." Most branches of math have a fundamental definition of "nice function" that follows shortly after the definition of the objects under discussion; in most algebraic disciplines what this definition should include is obvious to anybody who's been doing algebra for a while. Geometry and topology have it slightly harder.

**If you need to convince yourself of this, draw two collections of dots and some arrows from one side to the other to make a function (each dot on one side gets exactly one outgoing arrow). Also, there is no short phrase for "one-to-one and onto," however little sense that may seem to make.

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Question

# A polar equation of a conic is given. r = 8/1 + 2 \cos (\theta)

Conic sections
A polar equation of a conic is given. $$\displaystyle{r}=\frac{{8}}{{1}}+{2}{\cos{{\left(\theta\right)}}}$$