Check out the LEGO Difference Machine here.
Thursday, December 28, 2006
Sunday, December 10, 2006
Math + JavaScript = Cool
Check out ASCIIMathML and ASCIIsvg to learn how to display math equations and graphs in your website using nothing but JavaScript. You've probably seen ASCIIMathML in action in some of my other posts.
Saturday, December 02, 2006
Golomb Ruler
In mathematics, the term "Golomb Ruler" refers to a set of non-negative integers such that no two distinct pairs of numbers from the set have the same difference. Conceptually, this is similar to a ruler constructed in such a way that no two pairs of marks measure the same distance. An Optimal Golomb Ruler (OGR) is the shortest Golomb Ruler possible for a given number of marks. However, finding (and proving) OGR's becomes exponentially more difficult as the number of marks increases, and it is for this reason that we have turned to the web for help in finding the OGR's with 24 and more marks.
For more information, check out the distributed.net: Project OGR page and more info at MathWorld and Wikipedia.
For more information, check out the distributed.net: Project OGR page and more info at MathWorld and Wikipedia.
Thursday, November 30, 2006
Antikythera Mechanism
The Antikythera mechanism is an ancient mechanical analog computer (as opposed to digital computer) designed to calculate astronomical positions. It was discovered in the Antikythera wreck off the Greek island of Antikythera, between Kythera and Crete, and has been dated to about 150-100 BC. Click here for more information.
Sunday, November 05, 2006
InstaCalc Online Calculator
InstaCalc is a vision for a fast, simple and powerful calculator. You can embed it in your website or blog.
Wednesday, November 01, 2006
Wednesday, September 13, 2006
44th Known Mersenne Prime Found!
Less than a year after their last discovery, the Central Missouri State University (CMSU) team, led by professors Curtis Cooper and Steven Boone, has broken their own record for the largest known prime number: 232,582,657-1.
Tuesday, August 22, 2006
Great Mathematical Books
Sunday, July 09, 2006
The Mathematics Genealogy Project
The intent of this project is to compile information about ALL the mathematicians of the world. We earnestly solicit information from all schools who participate in the development of research level mathematics and from all individuals who may know desired information.
Please notice: Throughout this project when we use the word "mathematics" or "mathematician" we mean that word in a very inclusive sense. Thus, all relevant data from statistics, or computer science or operations research is welcome.
In the following paragraphs we shall try to outline our goals and our underlying philosophy for the GENEALOGY PROJECT. It is our goal to list all individuals who have received a doctorate in mathematics. For each individual we plan to show the following:
Please notice: Throughout this project when we use the word "mathematics" or "mathematician" we mean that word in a very inclusive sense. Thus, all relevant data from statistics, or computer science or operations research is welcome.
In the following paragraphs we shall try to outline our goals and our underlying philosophy for the GENEALOGY PROJECT. It is our goal to list all individuals who have received a doctorate in mathematics. For each individual we plan to show the following:
- The complete name of the degree recipient.
- The name of the university which awarded the degree.
- The year in which the degree was awarded.
- The complete title of the dissertation.
- The complete name(s) of the advisor(s).
Hilbert's 23 Problems (#10)
Definition: Given a Diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: To devise a process according to which it can be determined by a finite number of operations whether the equation is solvable in rational integers.
What this basically says is that is there a way for us to give a boolean answer to whether any such equation is solvable or not without necessarily knowing the solution?
The answer turns out to be no. Get the book Hilbert's Tenth Problem by Yuri V. Matiyasevich to understand why.
Although I've started reading this book, I must say that it is a little above my level of understanding. I've already read the first chapter twice just to get some basic understanding of his definitions. However, the commentary at the end of Chapter 3 was absolutely remarkable.
In my number theory and abstract algebra classes in college (university doesn't have that nice ring to it), we learned that there does not exist any polynomial that returns only primes for all integer arguments, but rather only for some (Euler's $f(x) = x^2 + x + 41$).
Julia Robinson proved in 1952 that the binomial coefficients and the factorial are exponential Diophantine, and gave an exponential Diophantine representation for the set of all prime numbers. In 1960, Putnam noted that a Diophantine set is the positive part of the range of a polynomial. Thus, it became clear (to someone) that if exponentiation were established to be Diophantine, it would become possible to construct a polynomial such that the positive values it assumed would coincide precisely with the prime numbers. In 1971a, Matiyasevich gave the first upper bound of 24 variables in his Russian article, which was later reduced to 21 in the appendix of the English translation.
In 1976, Jones, Sato, Wada and Wiens exhibited the following polynomial:
$(k + 2){1 - [wz + h + j - q]^2 - [(gk + 2g + k + 1)(h + j) + h - z]^2$ - $[2n + p + q + z - e]^2$ - $[16(k + 1)^3(k + 2)(n + 1)^2 + 1 - f^2]^2$ - $[e^3(e + 2)(a + 1)^2 + 1 - o^2]^2$ - $[(a^2 - 1)y^2 + 1 - x^2]^2$ - $[n + l + v - y]^2$ - $[((a + u^2(u^2 - a))^2 - 1)(n + 4dy)^2 + 1 - (x + cu)^2]^2$ - $[(a^2 - 1)t^2 + 1 - m^2]^2$ - $[q + y(a - p -1) + s(2ap + 2a - p^2 - 2p - 2) - x]^2$ - $[z + pl(a - p) + t(2ap - p^2 - 1) - p\m]^2$ - $[ai + k + 1 - l -i]^2$ - $[p + l(a - n - 1) + b(2an +2a - n^2 - 2n - 2) - m]^2\}$
This polynomial contains 26 variables (all the letters of the English alphabet), and the set of its positive values is exactly the set of all prime numbers. Note: The polynomial written above, representing only primes, is itself the product of two polynomials.
Later, the bound was further reduced to 12 variables by Wada in 1975, and by Jones, Sato, Wada and Wiens in 1976. Currently, the record is at 10 variables, achieved by Matiyasevich in 1977a.
What this basically says is that is there a way for us to give a boolean answer to whether any such equation is solvable or not without necessarily knowing the solution?
The answer turns out to be no. Get the book Hilbert's Tenth Problem by Yuri V. Matiyasevich to understand why.
Although I've started reading this book, I must say that it is a little above my level of understanding. I've already read the first chapter twice just to get some basic understanding of his definitions. However, the commentary at the end of Chapter 3 was absolutely remarkable.
In my number theory and abstract algebra classes in college (university doesn't have that nice ring to it), we learned that there does not exist any polynomial that returns only primes for all integer arguments, but rather only for some (Euler's $f(x) = x^2 + x + 41$).
Julia Robinson proved in 1952 that the binomial coefficients and the factorial are exponential Diophantine, and gave an exponential Diophantine representation for the set of all prime numbers. In 1960, Putnam noted that a Diophantine set is the positive part of the range of a polynomial. Thus, it became clear (to someone) that if exponentiation were established to be Diophantine, it would become possible to construct a polynomial such that the positive values it assumed would coincide precisely with the prime numbers. In 1971a, Matiyasevich gave the first upper bound of 24 variables in his Russian article, which was later reduced to 21 in the appendix of the English translation.
In 1976, Jones, Sato, Wada and Wiens exhibited the following polynomial:
$(k + 2){1 - [wz + h + j - q]^2 - [(gk + 2g + k + 1)(h + j) + h - z]^2$ - $[2n + p + q + z - e]^2$ - $[16(k + 1)^3(k + 2)(n + 1)^2 + 1 - f^2]^2$ - $[e^3(e + 2)(a + 1)^2 + 1 - o^2]^2$ - $[(a^2 - 1)y^2 + 1 - x^2]^2$ - $[n + l + v - y]^2$ - $[((a + u^2(u^2 - a))^2 - 1)(n + 4dy)^2 + 1 - (x + cu)^2]^2$ - $[(a^2 - 1)t^2 + 1 - m^2]^2$ - $[q + y(a - p -1) + s(2ap + 2a - p^2 - 2p - 2) - x]^2$ - $[z + pl(a - p) + t(2ap - p^2 - 1) - p\m]^2$ - $[ai + k + 1 - l -i]^2$ - $[p + l(a - n - 1) + b(2an +2a - n^2 - 2n - 2) - m]^2\}$
This polynomial contains 26 variables (all the letters of the English alphabet), and the set of its positive values is exactly the set of all prime numbers. Note: The polynomial written above, representing only primes, is itself the product of two polynomials.
Later, the bound was further reduced to 12 variables by Wada in 1975, and by Jones, Sato, Wada and Wiens in 1976. Currently, the record is at 10 variables, achieved by Matiyasevich in 1977a.
Sunday, June 11, 2006
Mathematical Poetry
Finite Simple Group (of order two)
A Klein Four original by Matt Salomone
URL: The Klein Four Group
Movie: Finite Simple Group (of Order Two)
The path of love is never smooth
But mine's continuous for you
You're the upper bound in the chains of my heart
You're my Axiom of Choice, you know it's true
But lately our relation's not so well-defined
And I just can't function without you
I'll prove my proposition and I'm sure you'll find
We're a finite simple group of order two
I'm losing my identity
I'm getting tensor every day
And without loss of generality
I will assume that you feel the same way
Since every time I see you, you just quotient out
The faithful image that I map into
But when we're one-to-one you'll see what I'm about
'Cause we're a finite simple group of order two
Our equivalence was stable,
A principal love bundle sitting deep inside
But then you drove a wedge between our two-forms
Now everything is so complexified
When we first met, we simply connected
My heart was open but too dense
Our system was already directed
To have a finite limit, in some sense
I'm living in the kernel of a rank-one map
From my domain, its image looks so blue,
'Cause all I see are zeros, it's a cruel trap
But we're a finite simple group of order two
I'm not the smoothest operator in my class,
But we're a mirror pair, me and you,
So let's apply forgetful functors to the past
And be a finite simple group, a finite simple group,
Let's be a finite simple group of order two
(Oughter: "Why not three?")
I've proved my proposition now, as you can see,
So let's both be associative and free
And by corollary, this shows you and I to be
Purely inseparable. Q. E. D.
My Poem
by Eve Andersson
There once was a number named pi
Who frequently liked to get high.
All he did every day
Was sit in his room and play
With his imaginary friend named i.
There once was a number named e
Who took way too much LSD.
She thought she was great.
But that fact we must debate;
We know she wasn't greater than 3.
There once was a log named Lynn
Whose life was devoted to sin.
She came from a tree
Whose base was shaped like an e.
She's the most natural log I've seen.
eve
A Young Don From Trinity
There was a young Don from Trinity
Who solved the square root of infinity
While counting the digits
He was seized by the fidgets
Dropped science and took up divinity
Anonymous.
A New Solution to an Old Problem
The Topologist's child was quite hyper
'Til she wore a Moebius diaper.
The mess on the inside
Was thus on the outside
And it was easy for someone to wipe her.
By Eleanor Ninestein.
An Analyst, Surname of Nero
An analyst, surname of Nero,
Is my mathematical hero.
Says he, "When in doubt,
I always start out,
'Given $\epsilon$ > 0 ...'."
By James R. Martino,
Department of Mathematics
The Johns Hopkins University
Baltimore, MD 21218.
Thursday, June 08, 2006
Mathematical Sites, Software & Games
Sites:
Software:
Software:
- My Homepage - Circular Prime Number Program
- Shamus Software Ltd. - Multiprecision Integer and Rational Arithmetic C/C++ Library (MIRACL)
- The Code Project - Digits to Charts
- The Code Project - Plot Graphic Library
- The Code Project - Scientific Charting Control
- The Code Project - Sudoku
- The Code Project - Symbolic Differentiation
- The Code Project - Visual Calc
- The Netron Project
- Vestris, Inc. - Expression Calculator
Thursday, June 01, 2006
Continuous, Nowhere Differentiable Functions
The Weierstrass function $f_a(x) = sum_(k=1)^infty sin(pi k^a x)/{pi k^a}$ (originally defined for a = 2) is an example of a continuous function, but differentiable only on a set of points of measure zero.
The function was published by Weierstrass but, according to lectures and writings by Kronecker and Weierstrass, Riemann seems to have claimed already in 1861 that the function f(x) is not differentiable on a set dense in the reals. However, Ullrich (1997) indicates that there is insufficient evidence to decide whether Riemann actually bothered to give a detailed proof for this claim. du Bois-Reymond (1875) stated without proof that every interval of f contains points at which f does not have a finite derivative, and Hardy (1916) proved that it does not have a finite derivative at any irrational and some of the rational points. Gerver (1970) and Smith (1972) subsequently proved that f has a finite derivative (namely, 1/2) at the set of points x = ${2A+1}/{2B+1}$ where A and B are integers. Gerver (1971) then proved that f is not differentiable at any point of the form ${2A}/{2B+1}$ or ${2A+1}/{2B}$. Together with the result of Hardy that f is not differentiable at any irrational value, this completely solved the problem of the differentiability of f.
Since then, a number of mathematicians have developed examples of nowhere continuous, differentiable (CND) functions and it has been shown that these types of functions are the norm (PDF).
One thing that should be obvious is that these functions are not lines in the geometric sense. These are all limits of other continuous functions, that are mostly differentiable, which converge to a nowhere differentiable function.
In Euclidean geometry, a polygon (or straight line) should only be drawn by pencil and straighthedge. If we were to draw any line, then the straighthedge can be omitted and a person only needs to trace the pencil from the starting point to the finish using whatever path they like. Assuming that the person isn't drunk, the line should be of finite length since we are drawing a line fron start to finish. Thus, any line segment that is a subset of that line, must have an even smaller finite length.
However, with CND functions, if you are to pick a point with a pencil and try to trace it to any other point, you will not go anywhere. The reason being is that the standard definition of a line is not met with the CND function. Since at any point, the function is not differentiable, the line would not know which direction to take. Thus, it would appear to vibrate like an electron held motionless in place.
Therein lies the beauty of such functions. They exist and are continuous, but are not real in the geometric sense.
The function was published by Weierstrass but, according to lectures and writings by Kronecker and Weierstrass, Riemann seems to have claimed already in 1861 that the function f(x) is not differentiable on a set dense in the reals. However, Ullrich (1997) indicates that there is insufficient evidence to decide whether Riemann actually bothered to give a detailed proof for this claim. du Bois-Reymond (1875) stated without proof that every interval of f contains points at which f does not have a finite derivative, and Hardy (1916) proved that it does not have a finite derivative at any irrational and some of the rational points. Gerver (1970) and Smith (1972) subsequently proved that f has a finite derivative (namely, 1/2) at the set of points x = ${2A+1}/{2B+1}$ where A and B are integers. Gerver (1971) then proved that f is not differentiable at any point of the form ${2A}/{2B+1}$ or ${2A+1}/{2B}$. Together with the result of Hardy that f is not differentiable at any irrational value, this completely solved the problem of the differentiability of f.
Since then, a number of mathematicians have developed examples of nowhere continuous, differentiable (CND) functions and it has been shown that these types of functions are the norm (PDF).
One thing that should be obvious is that these functions are not lines in the geometric sense. These are all limits of other continuous functions, that are mostly differentiable, which converge to a nowhere differentiable function.
In Euclidean geometry, a polygon (or straight line) should only be drawn by pencil and straighthedge. If we were to draw any line, then the straighthedge can be omitted and a person only needs to trace the pencil from the starting point to the finish using whatever path they like. Assuming that the person isn't drunk, the line should be of finite length since we are drawing a line fron start to finish. Thus, any line segment that is a subset of that line, must have an even smaller finite length.
However, with CND functions, if you are to pick a point with a pencil and try to trace it to any other point, you will not go anywhere. The reason being is that the standard definition of a line is not met with the CND function. Since at any point, the function is not differentiable, the line would not know which direction to take. Thus, it would appear to vibrate like an electron held motionless in place.
Therein lies the beauty of such functions. They exist and are continuous, but are not real in the geometric sense.
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