Mathematica

Absolute Minimum

2007-02-08T04:58:00.000-05:00

Well, so I'm back here to this abandoned place. I am uncertain if I should post this. But maybe just for the record. I guess I have reached the lowest point that I can remember in recent memory. Glad to have friends that helped me through the night. I guess things just happened so quickly that I cannot follow. Looking back though, I think I did the right thing. Perhaps its the most noble thing I could do. I certainly don't like it. Nonetheless, it was a rational decision. As much as I have lost, I hope I still upheld my principles.
I hope one day, when I visit this abandoned place again. I could look back to this whole experience not as a minimum point in my life, but as an inflection point towards greater good.

State of Disorder

2006-11-08T02:51:00.000-05:00

It seems to be an unusual time for me to revisit this abandon place. I guess I just need a moment to write a couple of lines. Haha without going into the detail, I really need to find method in this chaos. Blah. =|

3 * 7 = 21

2006-07-05T23:01:00.000-04:00

So i'm 21 now. Yeah.

March Madness!

2006-03-26T00:52:00.000-05:00

Haha well it's march madness time. Really have been spending too much time watching NCAA and playing computer games. =|
With 3 weeks left only, I better gather myself and stay focus again. Another school year already... =)

A monotonically decreasing function

2006-01-25T23:04:00.000-05:00

Haha, well that's probably the best description of my current state. I don't know if I am putting things in the right perspective, it seems to be quite a difficult time. I guess it's probably good to face some difficulties. But whatever the case, I can't dwell on this too much, there are still much to be done. Hopefully I can turn things around soon. =)

As epsilon goes to zero

2006-01-06T19:20:00.000-05:00

Well, about time to conclude my two weeks break and perhaps the semester that just passed. I actually know not of what to write. I wonder if I have really put myself to the test or the test is flawed itself. Perhaps I will never know. It takes a lot of confidence to stay in my condition after so long, but perhaps I have too much of it. Like the irony of larger stars burns faster for example.
I do not know of what the next step will be, but there are always choices to be made. There are times where I have doubts of my choice of path thus far. But maybe I have moved to0 far, or maybe I just deep down wants to stay on course. Whatever the case, I offer no resolution.
It seems I have reached the point where this entry shall end meaninglessly. Well I hope I can add some meaning to it in the near future. For now, as long as I know epsilon is not zero, that's all it matters.

Theorem of the Day -- 13/11/2005

2005-11-13T10:33:00.000-05:00

Roth's Theorem

For algebraic a
|a - p/q| < 1/p^(2+epsilon)

with epsilon > 0, has finitely many solutions. Klaus Roth received a Fields medal for this result.
(Excerpt from Mathworld)

Comments:
Well, so that's what it takes to win a Fields Medal. Haha, first I have to understand what the above really means. Blah =|

Theorem of the Day -- 30/10/2005

2005-10-30T01:38:00.000-05:00

Frivolous Theorem of Arithmetic

Almost all natural numbers are very, very, very large. (Excerpt from Mathworld)

Comments:
Haha, this is funny. Funnier that it is actually a "theorem". =)

Theorem of the Day -- 25/10/2005

2005-10-25T17:35:00.000-04:00

Banach Fixed Point Theorem

Let f be a contraction mapping from a closed subset F of a Banach space E into F . Then there exists a unique z in F such that f(z) = z.
(Excerpt from Mathworld)

Comments:
In the spirit of communication test tomorrow, I decided to post something on Banach spaces. Unfortunately, it is doubtful many communication students have any idea about Banach spaces. As the old jokes goes: "What's yellow normed and complete?" "A bananach space". Sorry the lack of post lately. This is not going to be daily.

Theorem of the Day -- 19/10/2005

2005-10-19T23:06:00.000-04:00

Odd Number Theorem

The sum of the first n odd numbers is a square number,

sum of (k=1 to n) of 2k-1 = n^2

A sort of converse also exists, namely the difference of the nth and (n-1)st square numbers is the nth odd number, which follows from

n^2 - (n-1)^2 = 2n - 1

(Excerpt from Mathworld)

Comments:
This is a very easy yet interesting result about numbers. Can be extremely useful when it comes to problem solving. Apologies of the lack of post lately.

Theorem of the Day -- 13/10/2005

2005-10-13T23:44:00.000-04:00

Isoperimetric Theorem

Of all convex n-gons of a given perimeter, the one which maximizes area is the regular n-gon.

(Excerpt from Mathworld)

Comments:
This is one of those results that people with a little geometry background will intuitively expect. I guess it isn't theorem of the day anymore, since Im' starting to miss days. =|

Theorem of the Day -- 9/10/2005

2005-10-09T14:28:00.000-04:00

Urysohn's Metrization Theorem
For every topological T1-space X, the following conditions are equivalent.

1. X is regular and second countable,

2. X is separable and metrizable.

3. X is homeomorphic to a subspace of the Hilbert cube.

(Excerpt from Mathworld)

Comments:
This theorem is important in topology. I should be learning this formally soon. Topology is really interesting, and seems to be easy to understand intuitatively (so far at least).

Theorem of the Day -- 8/10/2005

2005-10-08T23:52:00.000-04:00

Steiner-Lehmus Theorem
Any triangle that has two equal angle bisectors (each measured from a polygon vertex to the opposite sides) is an isosceles triangle. This theorem is also called the "internal bisectors problem" and "Lehmus' theorem."

(Excerpt from Mathworld)

Comments:
Hmm this seems to be a very neat result in Euclidean geometry. The name of the theorem, however, suggest that it is pretty recent. The proof perhaps is considerably difficult.

Theorem of the Day -- 7/10/2005

2005-10-07T14:47:00.000-04:00

Kronecker's Basis Theorem
A generalization of the Kronecker decomposition theorem which states that every finitely generated Abelian group is isomorphic to the group direct sum of a finite number of groups, each of which is either cyclic of prime power order or isomorphic to Z. This decomposition is unique, and the number of direct summands is equal to the group rank of the Abelian group.

(Excerpt from Mathworld)

Comments:
This is a very neat theorem in group theory. Quite an advanced one as well. But any theorem on basis is worth knowing. Apologies for the lack of post yesterday.

Theorem of the Day -- 5/10/2005

2005-10-05T17:21:00.000-04:00

Uniqueness Theorem
A theorem, also called a unicity theorem, stating the uniqueness of a mathematical object, which usually means that there is only one object fulfilling given properties, or that all objects of a given class are equivalent (i.e., they can be represented by the same model). This is often expressed by saying that the object is uniquely determined by a certain set of data. The word unique is sometimes replaced by essentially unique, whenever one wants to stress that the uniqueness is only referred to the underlying structure, whereas the form may vary in all ways that do not affect the mathematical content.

The object of many uniqueness theorems is the solution to a problem or an equation; in such cases, a uniqueness theorem is normally combined with an existence theorem.
(Excerpt from Mathworld)

Comments:
This is just a definition. But it's worth mentioning. Uniqueness is very important in mathematics, especially in analysis. Sorry for the lack of posts for the past few days. Internet was cut off. Now I'm back! =)

Theorem of the Day -- 1/10/2005

2005-10-01T17:48:00.000-04:00

Brianchon's Theorem
The dual of Pascal's theorem (Casey 1888, p. 146). It states that, given a hexagon circumscribed on a conic section, the lines joining opposite polygon vertices (polygon diagonals) meet in a single point.

In 1847, Möbius (1885) gave a statement which generalizes Brianchon's theorem: if all lines (except possibly one) connecting two opposite vertices of a ()-gon circumscribed on a conic section meet in one point, then the same is true for the remaining line.

(Excerpt from Mathworld)

Comments:
A wonderful theorem indeed. The number of amazing theorems arises from Euclidean geometry never fails to amaze me.
On another note, Oct 1st is an important day in my family. =)

Theorem of the Day -- 30/09/2005

2005-09-30T21:01:00.000-04:00

Tychonoff Theorem
A product space (product)(i in I) Xi is compact iff Xi is compact for all i. In other words, the topological product of any number of compact spaces is compact. In particular, compactness is a productive property. As a consequence, every Hilbert cube is compact.

This statement implies the axiom of choice, as proven by Kelley (1950).

(Excerpt from Mathworld)

Comments:
Haha only heard abt his name a few days ago in topology, and hence posted his theorem. Axiom of choice is certainly a wonderful topic. Just shows there are so much we don't know.

Theorem of the Day -- 29/09/2005

2005-09-29T17:16:00.000-04:00

Fermat's 4n+1 Theorem
Fermat's 4n+1 theorem, sometimes called Fermat's two-square theorem or simply "Fermat's theorem," states that a prime number p can be represented in an essentially unique manner (up to the order of addends) in the form x^2 + x^2 for integer x and y iff p=1 (mod 4) or p = 2 (which is a degenerate case with x =1 y = 1 ). The theorem was stated by Fermat, but the first published proof was by Euler.
(Excerpt from Mathworld)

Comments:
Giving credit to Euler, it seems that he proved a lot of theorems regarding different categories of primes. I wonder if there is a chance that he would have proven the Riemann Hypothesis given it was formulated at his time.

Theorem of the Day -- 28/09/2005

2005-09-28T17:25:00.000-04:00

Deep Theorem
Qualitatively, a deep theorem is a theorem whose proof is long, complicated, difficult, or appears to involve branches of mathematics which are not obviously related to the theorem itself (Shanks 1993). Shanks (1993) cites the quadratic reciprocity theorem as an example of a deep theorem.
(Excerpt from Mathworld)

Comments:
This theorem is actually not a theorem, more like a definition, I wonder if it's actually possible to rigorously prove if a theorem is a deep theorem.

Theorem of the Day -- 27/09/2005

2005-09-27T17:30:00.000-04:00

Euler's 6n+1 theorem
Euler's theorem states that every prime of the form 6n+1, (i.e., 7, 13, 19, 31, 37, 43, 61, 67, ..., which are also the primes of the form 3n+1; Sloane's A002476) can be written in the form x^2 + 3^2 with x and y positive integers.
(Transcribed from Mathworld)

Comments:
One of the many theorem's on prime numbers, also one of the many theorems to be named after Euler. His mathematical insight is truly marvelous.

Theorem of the Day -- 26/09/2005

2005-09-27T17:25:00.000-04:00

Archimedes' Hat-Box Theorem
Enclose a sphere in a cylinder and cut out a spherical segment by slicing twice perpendicularly to the cylinder's axis. Then the lateral surface area of the spherical segment S1 is equal to the lateral surface area cut out of the cylinder S2 by the same slicing planes, i.e.,

S== S1 = S2 = 2(pi)Rh

where R is the radius of the cylinder (and tangent sphere) and h is the height of the cylindrical (and spherical) segment. Where == denotes definition of S.
(Transcribed from Mathworld)

Comments:
This theorem is probably proven by Archimedes in aniquity. It is a very neat geometrical theorem, relating two important geometrical objects. Apologies for the late post.

Theorem of the Day -- 25/09/2005

2005-09-25T19:04:00.000-04:00

Ptolemy's Theorem
For a cyclic quadrilateral, the sum of the products of the two pairs of opposite sides equals the product of the diagonals
AB*CD + BC*DA = AC*BD.
(Excerpt from Mathworld)

Comments:
This theorem has a very elegant geometrical proof. It can also be proven using complex cross ratios, which I'm currently studying. A very neat theorem

Theorem of the Day -- 24/09/2005

2005-09-24T23:51:00.000-04:00

Fundamental Theorem of Algebra
Every polynomial equation having complex coefficients and degree >=1 has at least one complex root. This theorem was first proven by Gauss. It is equivalent to the statement that a polynomial P(z) of degree n has n values zi (some of them possibly degenerate) for which P(zi)=0 . Such values are called polynomial roots.
(Excerpt from Mathworld)

Comments:
This is one of the most important results in mathematics. Gauss proved this as his Ph.D thesis. In his lifetime he produced 4 different proofs of this fact. The consequence of this theorem is used everywhere in mathematics. Gauss was truly the prince of mathematics.

Theorem of the Day -- 23/09/2005

2005-09-24T02:20:00.000-04:00

Lindemann-Weierstrass Theorem
If algebraic integers a1, ...,an are linearly independent over Q, then e^a1, ...,e^an are algebraically independent over . The Lindemann-Weierstrass theorem is implied by Schanuel's conjecture (Chow 1999).
(Excerpt from Mathworld)

Comments:
I believe this theorem is useful in algebra. I do not know sufficient material to give more comments on this theorem. Apologies for the late post.

Theorem of the Day -- 22/09/2005

2005-09-22T23:01:00.000-04:00

Jung's Theorem
Every finite set of points with geometric span d has an enclosing circle with radius no greater than sqrt(3)d/3.

In three dimensions, a generalization of the theorem states that every set of points with geometric span d has an enclosing sphere with radius no greater than sqrt(6)d/4 (Smarandache 1992, 1996).

(Excerpt from Mathworld)

Comments:
Jung a mathematician? I need to search up on him. Well, this theorem is probably useful in some type of contest problem, once again it places a bound, like an estimate.

Theorem of the Day -- 21/09/2005

2005-09-21T17:16:00.000-04:00

Linnik's Theorem
Let p(d,a) be the smallest prime in the arithmetic progression (a+kd) for k an integer >0. Let
p(d) == max p(d,a) (== as defined as)
such that 1 <= a < d and (a,d) = 1. Then there exists a d0>= 2 and an L>1 such that p(d)d0 . L is known as Linnik's constant.

(Excerpt from Mathworld)

Comments:
Ah, this is one of those theorems relating prime numbers. I really do not the implication of this theorem. It probably serves as some type of estimate.

Theorem of the Day -- 20/09/2005

2005-09-20T19:27:00.000-04:00

Meteorology Theorem
Somewhere on the Earth, there is a pair of antipodal points having simultaneously the same temperature and pressure.
(Excerpt from Mathworld)

Comments:
Haha, I have no idea if there is a mathematical proof for this theorem. It's probably a physics "theorem" than a mathematical one.

Theorem of the Day -- 19/09/2005

2005-09-20T00:34:00.000-04:00

Tennis Ball Theorem
Any nontrivial, closed, simple, smooth spherical curve dividing the surface of a sphere into two parts of equal areas has at least four inflection points.(Excerpt from Mathworld)

Comments:
My apologies for the late post. I guess I will be posting twice today (in honour of my friend Maggie's 20th birthday). Haha. Anyway, this theorem involved a few mathematical terms "nontrivial", "closed", "simple", "smooth" etc. The result of this theorem is quite reasonable, in the sense that it's not a very surprising result.

Theorem of the Day -- 18/09/2005

2005-09-18T13:25:00.000-04:00

Pizza Theorem
If a circular pizza is divided into 8, 12, 16, ...slices by making cuts at equal angles from an arbitrary point, then the sums of the areas of alternate slices are equal.(Excerpt from Mathworld)

Comments:
I apologize for the lack of mathematical formality in this theorem. It's a Sunday, so we will have some fun, some pizza =)

Theorem of the Day -- 17/09/2005

2005-09-17T18:21:00.000-04:00

Ham Sandwich Theorem
The volumes of any n n-dimensional solids can always be simultaneously bisected by a (n-1)-dimensional hyperplane. Proving the theorem for n=2 (where it is known as the pancake theorem) is simple and can be found in Courant and Robbins (1978). (Excerpt from Mathworld)

Comments:
Haha I found this theorem and decided to post it simply by its name. Apparently the proof of this theorem for n = 3 is already very involved. I'm not exactly sure where is theorem is used, but it does provide an interesting result.

Theorem of the Day -- 16/09/2005

2005-09-16T13:36:00.000-04:00

Japanese Theorem
Let a convex cyclic polygon be triangulated in any manner, and draw the incircle to each triangle so constructed. Then the sum of the inradii is a constant independent of the triangulation chosen. This theorem can be proved using Carnot's theorem. (Excerpt from Mathworld)

Comments:
I probably should have stated Carnot's theorem first. But this theorem's neatness is doesn't rely on its proof. I think in mathematics, the proofs involves "independence" are quite intriguing. They essentially shows some type of invariance properties. These type of properties helps mathematicians (or humans in general) to classify objects into categories.
As for a historical note from MathWorld: According to an ancient custom of Japanese mathematicians, this theorem was a Sangaku problem inscribed on tablets hung in a Japanese temple to honor the gods and the author in 1800 (Johnson 1929).

Theorem of the Day -- 15/09/2005

2005-09-15T17:40:00.000-04:00

Tarski's Theorem
Tarski's theorem says that the first-order theory of reals with +, *, =, and > allows quantifier elimination. Algorithmic quantifier elimination implies decidability assuming that the truth values of sentences involving only constants can be computed. However, the converse is not true. For example, the first-order theory of reals with +, *, and = is decidable, but does not allow quantifier elimination. (Excerpt from Mathworld)

Comments:
This theorem arises in number theory & mathematical logic. The statement is very profound and requires quite advanced mathematical background to understand it fully. I personally just read about it a couple of days ago, and hence this post.

Theorem of the Day -- 14/09/2005

2005-09-14T14:48:00.000-04:00

Wilson's Theorem
Wilson's Theorem states that (p-1)! + 1 is a multiple of p if and only if p is prime. That is (p-1)! = -1 (mod p)(Transcribed from Mathworld)

Comments:
This theorem gives a necessary and sufficient condition for being prime. It was first stated by John Wilson and proved by Lagrange. It's one of many interesting statements about prime numbers in number theory.

Theorem of the Day -- 13/09/2005

2005-09-13T14:49:00.000-04:00

Picard's Great Theorem
Every nonconstant entire function attains every complex value with at most one exception (Henrici 1988, p. 216; Apostol 1997). Furthermore, every analytic function assumes every complex value, with possibly one exception, infinitely often in any neighborhood of an essential singularity. (Excerpt from Mathworld)

Comments:
This theorem is important in complex analysis. I'm not in the position to comment if it fits the name of "great". Certainly to be named a "great" theorem makes it all the nicer. This theorem is probably proved by Emile Picard. One of the two famous Picards in history of mathematics

Theorem of the Day -- 12/09/2005

2005-09-12T15:57:00.000-04:00

Pappus's Hexagon Theorem
If A, B, and C are three points on one line, D, E, and F are three points on another line, and AE meets BD at X, AF meets CD at Y, and BF meets CE at Z, then the three points X, Y, and Z are collinear. Pappus's hexagon theorem is self-dual. (Excerpt from Mathworld)

Comments:
This theorem probably dates by to antiquity simply by its name. It is an elegant geometric theorem. A very simple statement, and yet very neat result. The application of this theorem can be used in proving statements in Euclidean geometry.

Theorem of the Day -- 11/09/2005

2005-09-11T11:43:00.000-04:00

Sampling Theorem
In order for a band-limited (i.e., one with a zero power spectrum for frequencies v > B) baseband (v > 0) signal to be reconstructed fully, it must be sampled at a rate v>= 2B . A signal sampled at v = B is said to be Nyquist sampled, and is called the Nyquist frequency. No information is lost if a signal is sampled at the Nyquist frequency, and no additional information is gained by sampling faster than this rate. (Excerpt from Mathworld)

Comments:
This theorem has very important applications in communication, signal processing and other engineering areas. Abeit a simple mathematical theorem, it's use is far-stretching. The reasoning of this theorem is widely shown in most signals/communication courses. I have yet to seen a completely mathematically rigorous proof, although it's idea is simple and is just a matter of formality.

Theorem of the Day -- 10/09/2005

2005-09-10T15:09:00.000-04:00

Six Circles Theorem
Starting with a triangle, draw a circle touching two sides. Then draw a circle tangent to this circle and two other sides. Continue in the same direction. The result is a chain of circles in which the sixth circle is tangent to the first. (Excerpt from Mathworld)

Comments:
This theorem probably do not give much important mathematical result. But the theorem itself is still a beauty. I have yet to read a proof of this theorem myself. I think there should be a geometrical proof. Proofs in geometry are in many cases very elegant. This theorem itself exemplifies this point. You can start with any triangle first circle of any size, and this property holds! Illustrations of this theorem can be found at http://mathworld.wolfram.com/SixCirclesTheorem.html .

Theorem of the Day -- 09/09/2005

2005-09-09T14:53:00.000-04:00

Fundamental Theorem of Arithmetic
The fundamental theorem of arithmetic states that every positive integer (except the number 1) can be represented in exactly one way apart from rearrangement as a product of one or more primes.
(Excerpt from Mathworld)

Comments:
This theorem's result is well-known to almost everyone high school math background. The extension of this theorem, however, can be quite profound. It basically states that the set of positive integers are part of a category of mathematical objects called unique factorization domains (UFD). This unique factorization property is very important in algebra.

Theorem of the Day -- 08/09/2005

2005-09-08T17:49:00.000-04:00

Fermat's Last Theorem
Fermat's Last Theorem states that there is no positive integer x, y and z such that x^n+ y^n = z^n for any natural number n> 2.

Comments:
Perhaps one of the most elusive theorem's in mathematics. This theroem, or more appropriately a conjecture has withstood efforts of mathematicians 300 years. It was finally proven by Andrew Wiles in 1995, using very advanced mathematics. According to legend, Pierre de Fermat famously wrote on the margin of his notebook "I have discovered a truly remarkable proof of this theorem that the margin of this page is too small to contain". Mathematicians are doubtful that Fermat actually had a prove. But nonetheless, this simple statement shows that perhaps the most difficult mathematical problems are in fact the simplest ones.

Theorem of the Day -- 07/09/2005

2005-09-07T10:45:00.000-04:00

Pythagorean Theorem
For a right angle triangle (in Euclidean geometry) with sides a, b and hypotenuse c. The following relationship holds:
a^2 + b^2 = c^2

Comments:
This is perhaps one of the most famous theorems in mathematics. It's statement is simple and dates back to antiquity. The geometric proof of this statement can be found in Book I of Euclid's Elements. It is definitely a wonderful reading. Hundreds of different proofs of this theorem have been found, some using very advanced mathematics. The consequence of this result is used ubiquitously in mathematics.

Theorem of the Day -- 06/09/2005

2005-09-06T23:05:00.000-04:00

Four - Colour Theorem
The four-color theorem states that any map in a plane can be colored using four-colors in such a way that regions sharing a common boundary (other than a single point) do not share the same color.
(Excerpt from Mathworld)

Comments:
This theorem was first proven using a computer to check all possible cases. An elegant, pencil & paper proof still awaits. Nonetheless, this theorem shows an amazing result. It shows that although dividing a plane into irregular figures can be very complex, only four colours suffice in it's unambiguous mapping.

Theorem of the Day

2005-09-06T23:01:00.000-04:00

It just occurred to me that it will be nice to try something different on my blog. Thus, I came up with the idea of "Theorem of the Day". Each day, I will post a new mathematical theorem and perhaps some brief comments of the theorem. Hopefully I will able to provide important theorems that shows great mathematical insight. The first one will be posted shortly. Enjoy! =)

Galois Theory

2005-08-30T17:23:00.000-04:00

Well, it's the end of another summer. It sure has been fast. In some ways I should look forward to the new school year ahead of me. Certainly have a lot of things to prove about myself. Now with a bet on top of it, hopefully I can really have the motivation to get it done.
Learning Galois theory on my subway rides, it's amazing what Galois accomplished as such a young age. Perhaps I'm not yet in the position to admire the mathematical beauty of his theory. The complexity and ingenuity of Galois theory certainly have amazed many generations of mathematicians. From this, perhaps I can draw some inspiration. =)

20 + 1/12 + 1/365

2005-08-04T00:29:00.000-04:00

Haha well, 1 month and 1 day being a 20 year old. It hasn't been the most pleasant period of time I guess. Haha well. I guess my blogs have been a little sparse. Also lack definite math content. Haha, so under-mathed this summer. Only doing algebra and being so bad at it. Really not devoting much time at all. Really should buckle up a bit. But time just flies by before I even realize anything. Anyway, better get cracking, a new school year is coming! =)

So Twenty It Is

2005-07-03T23:45:00.000-04:00

Well, so I turned 20. Never really gave it much of a thought really. Haha, not exactly going to think too much in the next few minutes. I don't really have a wish or anything. Don't have any big ambitions, expectations or anything. It does seems, however, to be a fittingly date to start afresh on some new things in life. Or perhaps just improve on the things I have been doing.
I came into this day with high expectation. But with a series of subtle disappointments, I really just wanted to turn out good. It turned out great. When I look deeper and really see how these whole series of events unfold, I realize that there are so much I can be appreciative for.
As much as I have strived for perfection, I realized, it's not perfection that makes life great. It's really the little things that makes life such a wonderful journey. With two parties in two totally different places, different friends, and one same old me. It is a really a testament of who I am and how I live my life.
Whatever future might hold for me ahead, at least I know I won't be walking down it's path alone. =)

Yoda!!!

2005-06-13T11:47:00.000-04:00

Watched too much Star Wars I have. Amazed by Yoda I was.
Like a mathematician a Jedi is. Judged not by size, but by the use of the force. To a mathematician, the mind the force is. Pencils and papers, the weapons of a mathematician are. Powerful weapons they are. Patience, rationality, diligence, creativity and conciseness, the characteristics of a mathematician are.
Impatience, irrationality and laziness, to the dark side they lead. Beware of the the dark side. The power of the dark side, underestimate not. If the path of the dark side, you follow. Never return, you shall. Never a true mathematician, you will be.

Hold on to your pencil, protect your paper, and may the force be with you.

Groups, Rings & Fields

2005-05-27T17:46:00.000-04:00

MAT347 in the summer!! haha yup, can't believe there is a math course I can take in the summer. Very interesting indeed. It's mostly self-study I guess. But it's fun nonetheless.
Anthropology is a lot more interesting that I originally thought actually.
And Scientific Revolution is well, you know, science! Haha, it's science history. Pretty sad that no one in the class responded when the Prof asked about the book Euclid wrote. Ah. ELEMENTS!!!
Okay I should really get down to work. =)

So It Begins

2005-05-16T17:49:00.000-04:00

Well, today should be called the real begining of summer. Last year today I was packing up and waiting to embark on a trip of a lifetime. Today this year, I will start on a journey that is perhaps not as enjoyable; but it is just as important, and hopefully even more fulfilling. =)

Isolated Singularity

2005-05-09T02:40:00.000-04:00

Well, with the marks out, I was obviously shocked. But then, I guess it won't have been too much better either. I haven't had all that much time to think and reflect back on the year. But one thing I know for sure is that I haven't been spending the time on things that I should spend time on. It seems that I have yet to make the necessary sacrifices to reach my goals. Too many social events is one key problem. So as this title describes, that's what I shall be -- an isolated singularity. Exactly what that means, I don't know for sure yet (Both mathematically and for myself actually).
I will start taking action tomorrow.
And I won't want to post anymore regretful entries.
It better be real this time.

I need to prove myself that I can

2005-05-01T23:22:00.000-04:00

Sigh, a sudden realization perhaps. I should really be studying now rather than typing this. But I think I can't get any studying done unless I type this.
Perhaps there are too many things I want to do in my life. And I am not accomplishing any of them. I never really doubted myself. But now I feel I should. I wonder if I really have what it takes to do what I want to do in life. I think it's time for me to take a stance. I never had much self-control. Without it though, I can never be successful.
I need to have control, and it has to last.
No promises, because I don't think I can make any without 100% confidence of not breaking them. Too many times has it happened. This is sad.
I will take a stance now.
It's not too late yet, but sooner or later, it might just be.
And, I don't really have a good statement to end this post. So I will just end it here.

Calculus on Manifolds

2005-05-01T19:28:00.000-04:00

Alright. One last exam to go! Can't say I am prepared for it. But I still have some last minute cramming time left. I am glad to have mathematics as the subject to put my finishing touches for the year. Well, I can't say I will get a good mark, but at least I will enjoy the exam.
Summer is next. I can't wait. Let's just hope it will be a memoriable one.
=)

Not a very comforting thought

2005-04-20T22:06:00.000-04:00

Well, it seems I picked up playing maplestory again. Haha, well so I dont' think I should continue playing. It seems that I would have to commit a lot of time to the game again.
I have been thinking again. I seems that life is too comfortable these days. I can always find excuses for my failures and use them to make myself happier. It shouldn't be the way to be really. I realized that I perform well when there is overwhelming fear of I won't. Maybe that's the type of fear I need.
It's perhaps not yet time to make the final judgement. But, certainly it has been a sub-par year. There are many things that I wish I have done, but I didn't.
I just can't wait until the summer now. I need to a comeback. I need to be the person with dreams and goals again. I need to the me I really wanted to be.
Losing is not a comforting thought, but not putting in what I should put in and lose is far worse.
We will see. =)

3 down, 5 to go =)

2005-04-17T01:21:00.000-04:00

Well, I guess I suddenly got into the "trend" of posting how many exams I have left. It's 1 am now I didn't do any studying at all today. Sigh. Oh well. It was a good day though. but I better get cracking ! =)

Positive, Strictly Increasing

2005-04-11T20:55:00.000-04:00

Well, finally some good happened. I have a job for the summer! Now just hope I can get over with the exams smoothly. I wish for a positive, strictly increasing ride ahead =)

Exam time

2005-04-11T11:23:00.000-04:00

Haha with in 3 hrs of my first exam, that seems like epsilon time! Well, got to cram I guess. The next few weeks will define this semester. Hopefully it will turn out okay =)

Reach

2005-04-03T01:50:00.000-05:00

Well talked to Qiantai and Sis today. Haha feels like a few years ago ya? When that happened more often. Also talked to Guofeng. Quite a nostalgic day actually. But one thing is for sure, when I talked to Qt about the goals we had in the past. Ya, I still remember them, and I still have that fighting spirit. Now, I can reach higher, and I will be stronger =)

Newton is right

2005-03-29T23:26:00.000-05:00

The great Sir Issac Newton stated in his Third Law of motion:
For every action, there is an equal and opposite reaction.
This statement really can't be more true. To think about every decision you make, every action you take. There is always a win and a loss. But we often do not appreciate the win enough and complain about the loss too much. The trick is to appreciate the win while realizing the loss is a necessary sacrifice for what you deem to be more important.
=)

Well

2005-03-28T17:22:00.000-05:00

Well, I guess sometimes I really have to think forward about things that I might have preceived as unimportant. I think I have been facing the consequences of such negligences for years. It seems that I have yet to make the necessary changes. Sigh. I wonder if I ever learned my lesson.
=

The "speed" of time

2005-03-26T12:31:00.000-05:00

Is it just me, or does it seems that time is passing faster and faster?
I mean when you are not noticing, does time seems to be converging like a Cauchy sequence? It that makes any sense.
=)

Analysis

2005-03-23T04:24:00.000-05:00

Done with E&M test, now I have analysis ahead of me. At least I like the subject though. Oh well, I really need to use the 3 day weekend ahead of me well. Total of 12 more days of classes. Haha, well.
Intensity! Intensity! Intensity!

Slept like a pig

2005-03-20T21:21:00.000-05:00

Sigh. Can't believe I slept so much. Such a pig. Oh well. It's working time!

Let's begin!

2005-03-19T12:57:00.000-05:00

3 more weeks of classes.
3 more weeks of exams.
3 is prime.
Shall bring out some prime intensity for the rest of the way. It's really the home stretch now.
So, let it come. =)

Logical Truth

2005-03-17T11:55:00.000-05:00

As much as we seek logical truth, we find that the most basic human instincts defies all logics that we know. We can't explain why, the only excuse we can give ourselves is that we acted on emotions. So what does emotion has to do with the real truth about the functionality of this universe? I doubt we can find out, but again, I doubt by emotions.

Fundamental Theorem of Algebra

2005-03-14T04:22:00.000-05:00

Haha wow, I think I just understood the topological approach of proving the fundamental theorem of algebra, using Stoke's theroem that is. The proof is actually quite elegant, which makes Stoke's theroem really a beauty. I wish I have more time to dwell on the subject. Can't wait until summer time =)

From now on

2005-03-11T15:48:00.000-05:00

There are overwhelming amount of convincing arguments for me to make some important changes to my living style. Honestly, I'm not 100% confident that I can do it. But I will give on less than the very best I can.
Mark my words for this.

Look Deeper

2005-03-09T11:05:00.000-05:00

When you feel nothing is going right, it's time to look deeper into yourself to find the light =)

UNBELIEVEABLE

2005-03-09T03:19:00.000-05:00

Sigh. I just realized a mistake on that stupid signals test. Totally unbelieveable. ARGH.
Oh well.

Time

2005-03-09T02:14:00.000-05:00

Well, the last post was really nothing about convolution. This isn't going to stick very much to the title either. I hope I have already reached the lowest point of this whole deal. So that I would know everything will be going upwards from now on. It seems that I'm often in a battle with time. But no, I don't think time is the problem. The battle really comes within myself. Perhaps there can't be an ultimate victory. The best I can do is to win as much as I can and face defeat with grace.

Convolution

2005-03-08T19:56:00.000-05:00

Okay, I'm not sure if I have reached the lowest point of this whole deal. Frankly, I don't even know how I should feel. Sigh. I need a nap.

A day that I hope I will remember.

2005-03-05T01:27:00.000-05:00

Today might seem to be the worst day ever. Missed classes unwillingly and willingly, letter of rejection and got back tests that I won't want to comment about. Forgot my key on my trip home. But all those sum up to minor things. There is one thing I did right today and that's what I'm going to remember. I played table-tennis in a way that I haven't played for a long time. I played with the intensity I was talking about. I can feel myself refreshed and looking forward to the next step.
I have choosen this path and I need to make sacrifices for it. If I fail, I fail because of me. And if I wish to be successful, it will take more than just myself.
I have a dream, and I won't want to regret not living up to it.

Intensity

2005-03-04T04:48:00.000-05:00

Hmm, I just heard from Jas that she got 4As for A level, congrats! =)
This somehow made me think more about myself. Maybe I need a A level.
It seems that I have never picked up the same type of intensity I had in primary school. I might have worked harder sometimes, stayed up until way later on hours on days. But really, I don't think ever felt the same motivation that drove me when I was kid. The academic battles with Qiantai, and the table-tennis matches against Guofeng. To reflect back now, those things have helped greatly in building up my foundations. Perhaps I have changed somewhat in my character and uses a different approach to face life. But as I have realized more and more (even through my talk with Prof. Kwong), sometimes, things are just not idealistic. In fact, most of the time, it isn't.
I am not sure if I can ever feel the same intenisty again. Now, the fight is solely between me and myself. It's perhaps time again to make changes. I can still picture myself training hard to better, working hard to prove my worth.
And if I can still picture it, it's still part of me; and I can bring it back to me again.

No title

2005-03-03T03:38:00.000-05:00

I have no idea what I am writing anyway. I guess I am just writing for the sake of writing. haha more like not wanting to get down to work. One thing I have been thinking about:
In mathematics we often define many many things. Perhaps in many cases we are just defining things to simply the notation or make a generalization to more classes of cases. But I have two problems of this:
1) By this are we making mathematics just a game that we play base on rules, or is it really a search for truth that is undisputable? We can't simply define truth...
2) Mathematics is in many ways a modelling of the real world, but of course in much higher mathematics, it's way beyond the real world we can see. Say with a set of mathematical definitions, we are able to solve real life problems. Therefore in some sense we call the mathematics legitimate or rather a "natural way" of defining things. But, when we extend to cases we can't really test with real world. Are those definitions simply justified because we want things to work a certain way? Can we actually associate it with being the truth?

I guess those are very philosopical questions that we can't really answer. I always associated mathematics with the search for truth. Maybe such an association is not valid. Maybe mathematics does lose some of it's power because we can't clearly make this association. One thing though, is for sure. That is, mathematics is self-consistent. And if we are to accept on faith or passion the definitions of mathematics, we have no choice but to accept it's consequences. And that's its beauty =)

Sigh. I didn't think I will be really sad about basketball

2005-02-24T17:28:00.000-05:00

Well, this is going to be an unmathy post. It's a weird one too. I don't really play basketball, but it seems to have been quite a big part of my life for the past few years (watching that is). Since the Kings' dramatic playoff lost to Lakers, I have been very much a Kings' fan. Perhaps not closely to as devoted as of the Arco crowds in Sacramento, but I tried to watch every Kings game available on TV and never missed their scores for the past 3 years.

Yesterday's trade of Chris Webber was really a shock to me. Already letting Divac go made the Kings core unlike what it is before. I also disapprove emotionally of trading Christie. But I do acknowledge Mobley fits wells to the Kings' system. Trading a franchise player like Webber though, is really beyond my imagination. Be it that Webber isn't as explosive as before and no longer a decent shot-blocker, but he still averages around 20,10,5 (pts, reb, ast) as well as having a few triple doubles this year.

Maybe Kings aren't the team to beat this year, and they did went through quite a bit of playoff disappointment in the past couple of years. Despite all that, I can't imagine Kings without Webber. I don't really see in anyway how this trade will benefit Sacramento (maybe easier for them to keep Peja, he is the man now, or maybe Bibby, Miller?). But even if it does benefit the Kings, I will take a rather take a few more loses and keep a integral team (and I believe Webber is consistent in playoff situations) .

A team is not all about winning the basketball game, it's also about winning the hearts of the fans. Giving out Divac, Christie and now Webber really changed the team, and to me, the image of a great team playing great basketball.

With all the displeasure though, I still hope the Kings turn things around. And as for Webber, it was great watching him play and I hope he will get his championship someday.

Positive Definite

2005-02-23T03:32:00.000-05:00

Haha well, had ODE test today, screwed up pretty bad even though it's quite easy. Have to blame myself I guess. But I did try though. Today seems to be pretty positive even though I didn't exactly have a good day. Actually not good at all. I just don't feel bad. Maybe it's a good thing. Finally got down to do some work. I am not sure how long it will last. We will see.
I wonder if I should take the path of Riemann Hypothesis someday. It seems to be a thing that I would much want to spend time on. I will find out if I have the talent and perseverence.
There are quite somethings I want to do daily, and I should really do them. Just take a few minutes each, but perhaps can improve my life quite a bit.
Anyway, very little math content I realized. =)

Sigh.

2005-02-20T16:14:00.000-05:00

Argh. feel quite disappointed now. Slept so much these days. Really have to ban myself from the bed at irregular times. History does repeat itself ya. Perhaps the saddest thing is that in mathematics, everything is about logic. And yet I can't follow the simplest logic in living my life. Now I guess it will really be a test of my willpower. Hopefully, from now on, I can make myself do the right things more and more, like a strictly increasing function (and unbounded).

Numb3rs!!!

2005-02-18T23:55:00.000-05:00

Haha well, I didn't do any work today. But I did discover something really interesting. This new CBS show called Numb3rs! haha Well it isn't exactly new since today was the 5th episode already. It's such a great show, especially when they have real mathematics! Oh ya by the way, the show is about FBI detective asking his mathematician (suppose to be a genius) brother to help him solve crimes! Well 25 million viewers for the first episode, I certainly hope this is a good show that will last. It's definitely a great way to inform the general public about mathematics. And most importantly, the show is based on true stories, as well as real mathematics! Haha, wow, the world is really appreciating math more? I certainly hope so =)

Skating! CG CG!

2005-02-17T18:51:00.000-05:00

Haha oh man, went skating today, quite an experience I would say. Luckily didn't fall as much as I thought I would. It was fun though.
Alright, must have math content. So the important thing about skating is centre of gravity, or equivalently centroid (or for civil engineers, first moment of area) . It is quite a complex task to determine the centroid of a human being actually. Even when we simplify the problem to some nice shapes, it still involves a triple integral. Plus we cannot assume uniform density either. Finding the formula for density is hard enough. The density function is probably only piecewise continuous (if we are ever capable of finding one). All these said, we are still only dealing with static body.
To find how the centre of gravity change when a person moves, for example skating, we are actually solving some complex differential equation. If we also take into account how the body's density varies, the problem become even more complex.
Now, it's a wonder to me that we have developed mathematics to such a theoretical standpoint, and yet we cannot solve some of the most simple physical problems completely (without a great idea of assumptions and simplifications). Perhaps there are new methods of mathematics that we can explore to solve real life problems? Or are we forever confined within our computatibility powers simply because our equations are so clumsy?
Haha, wow that was a long entry =)

In H town

2005-02-14T18:38:00.000-05:00

Haha, at MacMaster now, visiting Sal, pretty nice here. The university is compact, and by that I really mean it is closed and bounded! haha
Attended one first year engineering calculus lecture, talked about polar coordinates and stuff, quite boring. But it's math nonetheless. They even have a math building here, we don't even have one in UT! Well unless you count the Fields Institute.
Anyway, oh yeah it's Valentine's day. So Happy Valentine's day everyone. Make sure you send your love a r = 1-sin(theta)
or some variation of that. =)

A Great Day

2005-02-07T23:58:00.000-05:00

Well today is by far my best day of 2005. I shall probably remember it, not going to write more about it. But I just thought of something about math I can write:
The thing abt math is, there are so many things you never have imagined before. With everything else, like engineering stuff, and even physics. u learn things that are new; but it can't really amaze u all that much, because say u can image a rocket. and seen it before, but just never learned how it works, now you do. In math, however, you learn something that has been (most of the time) beyond your imagination.
And that's what makes mathematics so special.
=)

I hope this day, this hour, this minute, is will be worth remembering

2005-02-06T01:53:00.000-05:00

Well, not exactly a post that I would like to put on. Officially quit playing Maplestory. It has brought me more grief than fun really. I guess I haven't been making good decisions of late. I hope this is a good one. Perhaps such an experience is much needed to really see deeper into things. Maybe I didn't learn the most pleasant lessons of all. Maybe the lessons weren't even worth the time spent. But what's past is over, there is no regrets. Only a better prespective for the future.
I know what my goals are, and that shall never change.

January

2005-01-27T04:45:00.000-05:00

Wow, it has been awhile since I blogged. January is almost over, haha, it has been quite an usual month I guess. Pretty unproductive and got quite hooked onto Maplestory. Got to get down to work now.
Anyway, I think this following problem is quite interesting and worth mentioning. It's the first problem from Putnam 2004. It goes
"Basketball star Shanille O’Keal’s team statisticiankeeps track of the number, S(N), of successful freethrows she has made in her first N attempts of the season.Early in the season, S(N) was less than 80% ofN, but by the end of the season, S(N) was more than80% of N. Was there necessarily a moment in betweenwhen S(N) was exactly 80% of N?"

The solution is quite simple and yet very elegant. Readers of this blog (if any haha) can probably try it out as an exercise. But the important thing is that it seems to be something we dont' really think abt. Such a discrete probability must past certain points independent of the number of trials associated with the probability. Quite amazing huh? =)

One, another one, 2005 - Reflections

2005-01-01T01:50:00.000-05:00

Well, certainly lacks mathematical creativity in that title. It has bee a nice day actually, Dec 31st 2004 I mean. Sorry to WQW that i didn't stay long enough for a nice chat. But I promise our tradition will continue ya.
2004 have been a especially great year for me. And also perhaps a year to reflect upon many different things in life. Not sure if i'm going to have any new goals, or resolutions or whatever for 2005. I think it's best just to keep doing what's right. I guess we don't need an "end of the year" to reflect upon things ya? But having such a year by year thing certainly hmm, divides our lives into eras that perhaps makes us treasure things even more.
Well, to keep up to the promise of a mathematical blog, I certainly hope that math will treat me well in the year to come, and finally, I can clearly declare my mathematical goals for next year -- The Putnam.
This is as far as I will go.
Happy New Year everyone
and my heart goes to those victims of the Asian earthquake & tsunami.

5 Squared and a Dozen

2004-12-25T19:00:00.000-05:00

Well, I did feel hesitated to post on this particular day. Not so much because of the lack of math content, but more so of what the day signifies. Anyway, it's probably not to think too much on thoughts that doesn't lead to logical resolutions. Haha
Merry Christmas =)

End of a Semester

2004-12-15T22:05:00.000-05:00

With only Algebra II left, this semester has came to an end. I can probably give the conclusion right now--really an unpleasant struggle. Not with school, but really with myself. It's perhaps a reflection on my personality. Or perhaps something for me to think beyond what's just before me. But, no matter how I look at it, it's still a failure.
I will remember this entry for quite awhile. For now, I should open my algebra books. Because, even a failure deserves a consolatory ending, especially a mathematical one.

Ah well...

2004-12-07T20:06:00.000-05:00

Well, i hate to admit that today's circuit theory exam was quite challenging, beyond the fact that i made so many stupid mistakes. Ah well, it's done and gone. Not going to think abt it anymore. Perhaps a lesson in trusting practice does make a difference. Really need this Christmas break to reassure certain things.
I always believed in myself; I still do, and always will.

Uniqueness of this entry

2004-12-05T20:45:00.000-05:00

Haha this entry is not about math. but I guess I shall still record it somewhere. so here is the place. just spent half an hr going through in my mind abt my trip in Singapore. Amazingly, I could remember every single day of it. In order too. Wow. Just glanced through my diay for confirmation. Haha, it's such a special feeling. Got to seriously read through my diary about exams are over.
For now, high in spirit, I shall get down to work.

Putnam

2004-12-05T01:26:00.000-05:00

Ah today was the Putnam contest, regrettably I didn't write it. Oh well, definitely going to spend the next summer on it and hopefully have a chance for something next year. It has been one of my key goals in university and it will continue to be.
Exams starting next week, got to really get into study mood.
=)

AH!

2004-11-24T06:31:00.000-05:00

Ah well, so unproductive these days.
Complex analysis is quite interesting. Might go for another of those lecture, even though it really screwed up my day. Ha. well, in the next few weeks. I need to prove the following theorem:
I can work under pressure and produce the results that I desire.
Proof:
Exercise for myself, will find out in a few weeks.

Dual Spaces

2004-11-20T11:03:00.000-05:00

Ah. have been quite a while since I last posted actually. Well I can't really think of anything special mathematically right now. haha so dual space will be my title. Got to read up on that soon. Let's just put the formal definition here. It's an easy concept, but turns out to be quite useful actually.
Definition
For a vector space V over F, we define the dual space of V to be the vector space L(V,F), denoted by V*.

L(V,F) is the set of all linear transformation from V to F, namely, the linear functionals on V.

Well, I will post another one sooner, probably sooner than a month haha.=)

Alexandre Grothendieck

2004-10-30T00:52:00.000-04:00

According to Professor Jim Arthur, Alexandre Grothendieck is probably the greatest mathematican of the 20th century. I might differ on that point. But after reading about him, it came quite a shame to me that I have not heard of this great mathematician before hand. Read about him at
http://www.ams.org/notices/200409/fea-grothendieck-part1.pdf
http://www.ams.org/notices/200410/fea-grothendieck-part2.pdf
After reading this, I find mathematics even more confusing than before, and yet I have gained even more appreciation of its beauty.
=)

=)

2004-10-17T04:16:00.000-04:00

Ah haven't posted for quite awhile. haha have been doing math though. My double degree thing is finally through. I guess it will just mean more math. Got so many nice books in book sale today. Lecture notes on Bannach Algebra for $2!?! haha. So nice. =)

Continuity

2004-09-23T01:33:00.000-04:00

Theorem:
The function f : A -> R^m (A a subset of R^n) is continuous iff for all open sets U a subest of R^m, there exists open set V belonging to R^n such that f^-1 (U) =V intersection A.

Proof:
Assume f is continuous: Let U a subset of R^m be open. If a belongs to f^-1(U), f(a) a subset U. Since U is open, there exist open rectangle B such that f(a) belongs to B, a subset of U. Since f is continuous at a, there exist open rectangle C such that a belongs to C and f (C intersection A) is a subset of B.
Do this for all a belonging to f^-1(U). Let V be the union of all such C. Then V a subset of R^n is open. Then f^-1(U) = V intersection A

Math =|

2004-09-12T23:23:00.000-04:00

Well, it has been awhile since i really touched math, first algebra assignment is quite challenging. Finally school is here. Well let me just type a few things on inner product spaces.
Suppose V and W aare vector spaces over F. Suppose <,> is an inner product on W.
Let T: V -> W be a one-to-one [N(T)= {0}] linear transformation
set, for x, y in V
Pt (x, y) = inner product of ( T(x), T(y) )
Check Pt defines an inner product on V.

The Reisz Representation Theorem

2004-08-12T19:47:00.000-04:00

Let X be a locally compact Hausdorff space, and let A be a positiver linear functional on Cc(X). THen there exists a sigma-algebra M in X which contains all Borel sets in X, and there exists a unique postive measure m on M which presents A in the sense that:
a) Af = Integral (over x) f dm for every f belonging to Cc(X)
and which has the following additional properties:
b) m(K) < infinity for every compact set K << X.
c) For every E belonging to M, we have
m(E) = inf {m(V): E << V, V open}
d) The relation
m(E) = sup {m(K): K << E, K compact}
holds for every open set E, and for every E belonging to M with m(E) < infinity.
e) If E belongs to M, A << E and m(E) = 0, then A belongs to M.

Note: << denotes "a subset of", and does not denote "much less than" haha

This is certainly an important theorem in analysis. It will take me days to understand the proof itself. =

Why Math?

2004-08-12T00:54:00.000-04:00

Math is good because there is always a solution, or you can prove it's not solvable or you know that if you keep working on it you will get either of the above cases, nothing else.

WOW! OPEN-CLOSED!

2004-08-11T17:29:00.000-04:00

Definition
Let f be a real (or extended-real) function on a topological space. If
{x : f(x) > a}
is open for every real a, f is said to be lower semicontinuous. If
{x : f(x) < a}
is open for every real a, f is said to be upper semicontinuous.

Now, by this definition it can be shown that:
The simplest examples of semicontinuity are furnished by characteristic functions:
a) Characteristic functions of open sets are lower semicontinuous
b) Characteristic functions of closed sets are upper semicontinuous.

Haha. wow, it took me awhile, but i finally understood a) and b)
=)

Topological Preliminaries

2004-08-11T14:42:00.000-04:00

Let X be a topological space.
a) A set E belongs to X is closed if its complement E^c is open. (Hence empty set and X are closed, finite unions of closed sets are closed, and arbitrary intersections of closed sets are closed.)
b)The closure E-hat (E with the hat, let's call it E-hat) of a set E belongs to X is the smallest closed set in X which contains E. (The folloing argument proves the existence of E-hat: The collection Omega of all closed subets of X which contain E is not empty, since X belongs to Omega; let E-hat be the intersection of all members of Omega)
c) A set K belongs to X is compact if every open cover of K contains a finite subcover. More explicitly, the requirement is that if {Va} is a collection of open sets whose union contains K, then the union of some finite subcollection of {Va} also contains K.
In particular, if X is itself compact, then X is called a compact space.
d)A neighborhood of a point p belong to X is any open subset of X which contains p.
e)X is a Hausdorff space if the following is true: If p belongs X, q belongs to X, and p not equal to q. Then p has a neighborhood U and q has a neighborhood V such that U intersection V = empty set.
f) X is locally compact if every point of X has a neighborhood whose closure is compact.

Haha ah. blogger is certainly not math friendly.

Subgroups of Finite Cyclic Groups

2004-08-10T18:48:00.000-04:00

This is really cool
Theorem:
Let G be a cyclic group with n elements and generated by a. Let b belong to G and let B = a^s. Then b generates a cyclic subgroup H of G containing n/d elements, where d is the greatest common divisor of n and s. ALso, = if and only if gcd(s,n) = gcd(t,n).

=)

"Vague" Theorem"

2004-08-10T16:57:00.000-04:00

Let {Ek} be a sequence of measurable sets in X, such that
sum (from k = 1 to infinity ) of m(Ek) < infinity.

Then almost all x belong to X lie in at most finitely many of the sets Ek.

Haha, vague?

Mathematical Pronunications

2004-08-10T10:45:00.000-04:00

Haha ever came across these mathematicians? Ever tried to pronounce their names? Here are some i found from http://www.antimoon.com/forum/posts/3806.htm

Brouwer -- rhymes with sour.
Cauchy -- rhymes with "So she."
Euler -- this one is famous, is "OIL-er" (not YOOL-er.. please)
Fubini -- "foo - BEE - nee"
Haar --rhymes with star and bar.
Jordan -- properly, it's pronounced "zhor-DON" (French)
Lebesgue -- Luh BAYG.
Mazur --MAY zer, rhymes with razor.
Radon-Nikodym --RA-don NIK-o-deem
Weierstrass --VY - er - shtross.