Wikipedia:Reference desk/Archives/Mathematics/2009 December 23

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December 23

Functional Analysis

Where can I get a very basic introduction to the current research directions in functional analysis? Also I am interested in knowing about applications of Ramsey theory to functional analysis.[1] Thanks-Shahab (talk) 04:44, 23 December 2009 (UTC)

Edit Conflict

A good (and reasonably basic) book on the subject would be "Functional Analysis" by Walter Rudin. Alternatively, if you wish to learn about the theory of C* algebras, you could read "An Invitation to C* algebras" (in the GTM series).

Prior to studying functional analysis, it would be good to have a strong background in point-set topology, the topology of metric spaces, linear algebra, and ring theory. Although I think that you already have such a background, it is especially important to have a ring-theoretic intuition (or an intuition of linear transformations); for instance, it would help to be acquainted with a result of the nature of the Jacobson density theorem (somewhat related to the Von Neumann bicommutant theorem in functional analysis). In fact, a strong background in noncommutative ring theory would help should you wish to delve deeper into the subject.

Perhaps, it would be advisable to read the articles operator algebra, operator topology, and Von Neumann algebra, for this may give you a sense of the sorts of basic notions encapsulated in functional analysis. All in all, the two books I suggested may be useful (though there are other excellent texts), but it is important to have a good feel for linear transformations. Might I also add that there are many sorts of branches of functional analysis; the one I have emphasized here does not really encapsulate mathematical physics and the geometry of Banach spaces (note also noncommutative geometry)? --PST 05:34, 23 December 2009 (UTC)

Sorry - I made the above post before you altered your inquiry to note Ramsey theory. --PST 05:34, 23 December 2009 (UTC)

Now that you have mentioned Ramsey theory, the book "Geometric Functional Analysis and Its Applications" by Richard B. Holmes, may be appropriate (it is a book in the GTM series). --PST 05:40, 23 December 2009 (UTC)

I'm a big fan of Kreyszig's Introductory Functional Analysis with Applications which is very clear and well written, though doesn't have any Ramsey theory 86.15.141.42 (talk) 12:45, 23 December 2009 (UTC)

Thank you both. I have obtained the recommended books and will start reading them.-Shahab (talk) 16:49, 23 December 2009 (UTC)

reduction algorithm

What algorithm will reduce to minimum form an equation consisting of polynary variables? 71.100.6.206 (talk) 04:45, 23 December 2009 (UTC)

120 links between five people??

I thought there would only be a network of 10 links between five people. But the man here says there are 120: http://www.ted.com/index.php/talks/bruce_bueno_de_mesquita_predicts_iran_s_future.html How does he calculate a figure of 120, not 10? 92.29.68.169 (talk) 16:03, 23 December 2009 (UTC)

Can you indicate where he said that? Anyway, $5!=120$ so he may have talked about ways to arrange 5 people in a line or something. -- Meni Rosenfeld (talk) 16:13, 23 December 2009 (UTC)

Interesting... there are 10 lines on his diagram. Either he's simply wrong (which seems unlikely, since he did include the diagram and one would hope he can count to 10!) or he means something different by "link". He talks about one person knowing what others are saying to each other, so if we count thinks like "A thinks B has said X to C" (where A-E are people and X is an idea) as a link then there are far more than 10. There are 120 different ways to order the five people (you have 5 choices for the first, 4 for the second and so on), so there are 120 links of the type "A thinks that B thinks that C thinks that D thinks that E thinks X". It could be that he's talking about that. He doesn't explain it at all well, though. --Tango (talk) 16:34, 23 December 2009 (UTC)

PS My greater concern is about the 90% accuracy claim. That is a completely meaningless number. First of all, we need to know if the predictions were made before or after the events happened - it is far easier to come up with a method that "would have" predicted the outcome once you know what the outcome was. Secondly, we need to know how well other methods predicted those outcomes (eg. just surveying experts and seeing what most of them say is likely to happen). --Tango (talk) 16:37, 23 December 2009 (UTC)

Why does a*b give the area of a rectangle? / Why does arithmetic give meaningful geometric results?

All my life I have known that a*b gives the area of a rectangle with sides a and b. It's repeated so often that I surely don't doubt it. I've realized, though, that I don't feel like I have a solid understanding of why it's true. It seems like something that needs further explanation.

For rectangles with integer sides, there's an explanation that's at least mostly satisfying:

• By definition, the area of a figure is the # of 1x1 unit squares that fit inside it
• If a rectangle has integer sides a and b, then you can fit an axb array of 1x1 squares inside it. (This seems like it could use some kind of justification of its own, but it's at least pretty intuitive to visualize.)
• We know that a*b is a good way to count an axb array of objects. (If you have any doubts there, they can be addressed in this case by thinking of multiplication as repeated addition.)
• So a*b is the # of 1x1 squares inside an axb rectangle.
• So, by definition, a*b is the area of the rectangle.

Moving beyond integers it seems more mysterious to me. One way to phrase the mystery is this; How does a*b "know" how many 1x1 squares are inside an axb rectangle? If we're talking about real numbers, we can't just count object arrays anymore, so the above justification won't work.

One possibility I've encountered is that maybe you shouldn't think of area of a figure as the # of 1x1 squares in it but rather as the ratio between its area and that of a 1x1 unit square. (See http://www.math.ubc.ca/~cass/graphics/manual/pdf/ch2.pdf) But I haven't figured whether looking at area in that different way could make the connection between multiplication and area seem less mysterious for real numbers.

For context, this may be part of a larger confusion of mine about how arithmetic relates to geometry: On one hand, it seems like real numbers are defined axiomatically (I know there are other approaches, but see http://en.wikipedia.org/wiki/Real_number#Axiomatic_approach), and if you derive an algorithm for multiplication, you do that from the axioms for fields and such, without consulting geometric facts in any way. And yet, having done so, you wind up with algorithms/formulas that can be used to find the area of rectangles. What is it about this abstractly defined operation of multiplication that makes it suitable for answering anything about geometry? And what makes it suitable for answering questions about area in particular?

Ryguasu (talk) 21:56, 23 December 2009 (UTC)

I think maybe the first step is for you to ask yourself just what you mean by "area", as distinct from the product of the sides of the rectangle, which is usually taken to be pretty much the definition. If it turns out that your meaning is motivated by physical reality, you might check out The Unreasonable Effectiveness of Mathematics in the Natural Sciences, which raises questions for which there are not yet any generally accepted satisfactory answers. --Trovatore (talk) 22:02, 23 December 2009 (UTC)

Let's assume we have defined the concept of 'shape', and that you are willing to accept the following axioms regarding area:

• A 1x1 square has area 1.
• The area of a shape is unchanged when you translate or rotate it (ie. move it)
• Placing two shapes adjacent to each other so there is no overlap results in a new shape whose area is the sum of the original shapes.
• If one shape can be translated/rotated to completely cover another, the area of the first is larger than the area of the second.

As you have already demonstrated, the area of an axb rectangle where a and b are integers can be established by adding together several 1x1 squares. This gives a*b as their area.

Similarly, if you have a (1/a)x(1/b) rectangle, you can show that its area must be (1/a)*(1/b) by adding the same rectangle to itself a*b times in such a way as to make a 1x1 square. If we call the area A, this means that A*a*b=1, or A=1/(a*b). It is then just as easy to show that any (a/b)x(c/d) rectangle has area (a/b)*(c/d)=(ac)/(cd).

To show that the area of a qxr rectangle is q*r, where q and r are any real numbers, you can bound the area of the qxr rectangle above and below by use of rational-sided rectangles and the forth axiom, and get the bounds arbitrarily close to q*r. Then the only possible area is q*r.

HTH. --COVIZAPIBETEFOKY (talk) 22:41, 23 December 2009 (UTC)

Area is additive. If you have two rectangles in a plane sharing a common side, so that their union is a rectangle, then the area of that larger rectangle is the sum of the two areas. That's why. Michael Hardy (talk) 07:52, 24 December 2009 (UTC)

... which determines area as a function of a and b up to a scalar constant, which is set by our choice of units. If we measure lengths in metres and areas in square metres then the constant is 1 and area = ab; if we measure lengths in picometres and areas in barns then area = 10,000ab. Gandalf61 (talk) 11:21, 24 December 2009 (UTC)

Imagine a small, unit square made of sticky paper. Think of the process of measuring area as covering the surface of an object (say, a cup) with such squares. When covering the object, try to minimize gaps and overlaps. The number of squares needed to cover the surface, is your best approximation of its area. If you repeat the process, with squares that are 1/4 of a unit square, you'll be able to do a better job of minimizing gaps and overlaps. (If that's not immediately intuitive, think of what will happen if you increase the size of the sticky paper squares). Now, your area is the number of squares, divided by four. Repeat with squares that are 1/16, 1/64, 1/256, 1/1024 ... unit squares. By doing so, you will get a closer and closer approximation of the real number that is the object's area. --NorwegianBlue ^talk 15:02, 24 December 2009 (UTC)

The additivity of area is, of course, necessary to move beyond rectangles and to consider such as triangles. I've always taken it as axiomatic, but am happy to justify it on the painting analogy of considering how much "cover" is required.→→86.155.184.27 (talk) 15:48, 24 December 2009 (UTC)

(Continuation of sticky paper post, after peeling a ton of potatoes):

Now imagine measuring the area of a rectangle of arbitrary dimensions by tiling it with unit squares. You won't have the problem of overlaps, but will have to decide whether you want to leave a small uncovered strip at (say) the right edge and bottom edge, or to cover these strips, thus covering a surface that is larger that the rectangle. Imagine doing both, getting a low estimate and a high estimate of the area of the rectangle. In both cases, your estimate of the area will be the product of the number of unit lengths that fit along each edge. When you repeat this process with squares that are tinier and tinier fractions of a unit square, you can make the difference between the estimates as small as you want. At each step, the area will be the product of the number of squares that fit along each edge, divided by 4, 16, 64, 256, 1024, ... or, equivalently, the product of the number of squares that fit along the top edge divided by 2, 4, 8, 16, 32, ... and the number of squares that fit along the left edge divided by 2, 4, etc. The number of squares that fit along an edge divided by 2, 4, 8, 16, 32 ... approaches the real number that is the length of the edge, and the product of the number of squares that fit along each edge, divided by 4, 16, 64, 256, 1024..., approaches the area. --NorwegianBlue ^talk

In answer to your second question, the best answer I can think of (and there may be a better one) is that arithmetic is, in some sense, defined with geometry in mind. The properties of addition and multiplication have geometric counterparts; for instance, the distributive property of multiplication over addition can be justified geometrically for positive real numbers by representing a(b+c) as a rectangle whose sides are a and b+c, and noticing that we can also represent the same rectangle as a juxtaposition of two rectangles, axb and axc, giving a*b+a*c.

Don't get me wrong; numbers and lengths and areas are distinct concepts. But the first application of numbers was probably to measure geometric constructs, and geometry has had a big impact on the development of numbers, so that's probably historically the best explanation. --COVIZAPIBETEFOKY (talk) 18:07, 24 December 2009 (UTC)