Let M(m,n)[R] be the vector space of m by n matrices with elements in R and scalar field R.
Let A, B, C elements in M(m,n)[R] and x,y elements in R. Define the map
f: V x V > R by (A,B) > Tr(B^T * A).
Since
 (xA+yB,C) = x(A,C) + y(B,C)
 (A,B) = (B,A)
 (A,A) >= 0
(proofs are trivial)
f is an inner product.
Sunday, February 24, 2008
Saturday, February 23, 2008
The psychology of Charlie Eppes.
He is often shown "stimming". "Stimming" is basically selfstimulation, a repetitive sound or motion made by people who have difficulty integrating all the sensory input around them as a buffer between themselves and the sensory stimulus.
Charlie has sensory integration issues. This means he has difficulty integrating sensory input around him into a cohesive whole, something that most of us do automatically but is sometimes an exhausting task for individuals in the autism spectrum. Charlie wears enormous headphones with no music playing while figuring out a math problem.
Charlie fails to see the big picture at almost every turn and has extreme difficulty anticipating spontaneous behavior. Charlie is most comfortable with inductive reasoning, and despite his high intelligence levels, finds deductive reasoning difficult at best.
Charlie struggles to interpret social cues from others, when he picks up on them at all. Charlie rarely intuits when he annoys everyone around him, either with his singleminded preoccupation with mathematics or his inability to accurately read the nuances of social situations. He is very literal minded, often rambles past the point where anyone is listening, and in fact, sometimes DOES NOT NOTICE everyone has stopped listening.
Charlie fails to connect his concrete theorizing to abstract human reality, and in fact, doing so is deeply disturbing to him.
Charlie displays an extremely limiting and persistent social naivete that also can translate as extreme social immaturity. He is 27, doesn't date, doesn't notice that his very pretty friend is probably interested in him.
There is the compulsive and completely preoccupying nature of Charlie's obsession with discrete mathematics.
( This entry is a summary of this article on "tv makes you stupid". )
Charlie has sensory integration issues. This means he has difficulty integrating sensory input around him into a cohesive whole, something that most of us do automatically but is sometimes an exhausting task for individuals in the autism spectrum. Charlie wears enormous headphones with no music playing while figuring out a math problem.
Charlie fails to see the big picture at almost every turn and has extreme difficulty anticipating spontaneous behavior. Charlie is most comfortable with inductive reasoning, and despite his high intelligence levels, finds deductive reasoning difficult at best.
Charlie struggles to interpret social cues from others, when he picks up on them at all. Charlie rarely intuits when he annoys everyone around him, either with his singleminded preoccupation with mathematics or his inability to accurately read the nuances of social situations. He is very literal minded, often rambles past the point where anyone is listening, and in fact, sometimes DOES NOT NOTICE everyone has stopped listening.
Charlie fails to connect his concrete theorizing to abstract human reality, and in fact, doing so is deeply disturbing to him.
Charlie displays an extremely limiting and persistent social naivete that also can translate as extreme social immaturity. He is 27, doesn't date, doesn't notice that his very pretty friend is probably interested in him.
There is the compulsive and completely preoccupying nature of Charlie's obsession with discrete mathematics.
( This entry is a summary of this article on "tv makes you stupid". )
Thursday, February 21, 2008
Sunday, February 17, 2008
Henry Pitcher
92yearold to graduate from UH with a bachelor degree in math Crossing the stage to graduate from the University of Houston, summer 2007 semester culminated a quest that Henry Pitcher, 92, began 75 years ago. Full Story
Wednesday, February 13, 2008
Saturday, February 9, 2008
Math and art
Although I seriously doubt if mathematics and art are even close I added http://mathart.net/ to the 'Cool Sites' list. I do agree that math pics can be beautiful. "... Where Mathematics and Art blends into a zenlike state of peace ...".
Venn diagram
Friday, February 8, 2008
Collatz conjecture
The 3x + 1 problem or Collatz conjecture is simple in its formulation but until now no proof is known that it is in fact true for any n. A conjecture is generally believed true but a formal proof is still to be found.
The conjecture says that if you repeatedly apply the following rule to a number the end result will always be one (1):
 if even then divide by 2;
 if odd then multiply by 3 and add 1.
Example:
9
28
14
7
22
11
34
17
52
26
13
40
20
10
5
16
8
4
2
1
Take any number and try it.
What about 27?
The conjecture says that if you repeatedly apply the following rule to a number the end result will always be one (1):
 if even then divide by 2;
 if odd then multiply by 3 and add 1.
Example:
9
28
14
7
22
11
34
17
52
26
13
40
20
10
5
16
8
4
2
1
Take any number and try it.
What about 27?
Saturday, February 2, 2008
Quadratic residu
A number q is called a quadratic residue modulo n if there exists an integer x such that x^2 = q mod n.
For example
The quadratic residues mod 7 are 1,2 and 4. The set {1,2,4} is a group under multiplication mod 7:
For example
a a^2 a^2 mod 7
1 1 1
2 4 4
3 9 2
4 16 2
5 25 4
6 36 1
7 49 0
8 64 1
9 81 4
etc.
The quadratic residues mod 7 are 1,2 and 4. The set {1,2,4} is a group under multiplication mod 7:
* mod 7 1 2 4
1 1 2 4
2 2 4 1
4 4 1 2
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(Raumpatrouille – Die phantastischen Abenteuer des Raumschiffes Orion, colloquially aka Raumpatrouille Orion was the first German science fiction television series. Its seven episodes were broadcast by ARD beginning September 17, 1966. The series has since acquired cult status in Germany. Broadcast six years before Star Trek first aired in West Germany (in 1972), it became a huge success.)