Search found 1753 matches

by antonfire
Fri Jul 07, 2017 9:54 pm UTC
Forum: Mathematics
Topic: Some questions about the volume of the n-ball.
Replies: 25
Views: 6346

Re: Some questions about the volume of the n-ball.

Well, I finally got around to doing this computation and the result is pretty nice, so here we go. First, normalize the p-norm so that (1,1,...,1) has norm 1. This corresponds to making it so that (||x|| p ) p is the mean of the numbers |x 1 | p , ..., |x n | p rather than their sum. This leaves the...
by antonfire
Tue Jun 10, 2014 12:30 am UTC
Forum: Mathematics
Topic: Vectors without components
Replies: 19
Views: 4509

Re: Vectors without components

The fact that "a person with a hat on" means a different thing from "a person" shouldn't suggest to you that there are people who can't wear hats.
by antonfire
Sun Aug 11, 2013 8:12 am UTC
Forum: Mathematics
Topic: Group theory II
Replies: 46
Views: 13233

Re: Group theory II

When I have to do these things I like to draw the Cayley graph of the group. The Cayley graph of <a, b| a 2 = b 3 = 1> is a 3-regular tree with every vertex replaced by a triangle. (A truncated 3-regular tree.) Add in the relation (ab) 6 = 1 and you get a truncated hexagonal grid . So now it's an ex...
by antonfire
Sun Jun 30, 2013 5:10 am UTC
Forum: Mathematics
Topic: Negative Dimensions
Replies: 9
Views: 4434

Re: Negative Dimensions

The basic observation here is that if you scale up the integers by a factor of 2, then in some sense you get a set that's twice as small as the integers. So from that point of view Z has dimension -1. It's a jump to say that every countable set has dimension -1, though. For instance, you should say ...
by antonfire
Thu Mar 28, 2013 4:23 pm UTC
Forum: Mathematics
Topic: Informal proof that no quadratic irrational is normal?
Replies: 6
Views: 1566

Re: Informal proof that no quadratic irrational is normal?

How good is "better than the denominator would suggest"? If p/q is one of the convergents of the continued fraction for alpha, then the difference between p/q and alpha is on the order of 1/q^2, not on the order of 1/q, so it's already better than the denominator would suggest. You'd need ...
by antonfire
Sun Mar 24, 2013 7:30 am UTC
Forum: Mathematics
Topic: Number of holes in the Sierpinski triangle
Replies: 3
Views: 1878

Re: Number of holes in the Sierpinski triangle

On the other hand, its fundamental group is not simply the free group generated by one loop for each hole in the triangle. You have the same thing going on as in the Hawaiian earring: single loops which "see" infinitely many holes.
by antonfire
Sat Mar 16, 2013 6:47 am UTC
Forum: Mathematics
Topic: Groups & Their Presentations
Replies: 10
Views: 2355

Re: Groups & Their Presentations

Then, I was hoping that I could require that none of the s would fully cancel (i.e. there would be at least one letter coming from each s), and that this would be the correct group. I can't think of a counterexample (a word, representing the identity, which cannot be written without such a cancella...
by antonfire
Thu Mar 14, 2013 10:10 am UTC
Forum: Mathematics
Topic: Groups & Their Presentations
Replies: 10
Views: 2355

Re: Groups & Their Presentations

letterX wrote:Not just NP-complete, but can-be-reduced-to-the-halting-problem kind of uncomputable.

You mean can-reduce-the-halting-problem-to-it kind of uncomputable.
by antonfire
Wed Mar 06, 2013 5:18 am UTC
Forum: Mathematics
Topic: Discontinituous increasing function?
Replies: 20
Views: 3562

Re: Discontinituous increasing function?

The codomain of that function is R, not Q.
by antonfire
Fri Mar 01, 2013 5:13 pm UTC
Forum: Mathematics
Topic: Connected v. Path-connected
Replies: 9
Views: 2513

Re: Connected v. Path-connected

Maybe an even more positive definition, and definitely one more analogous to path connectedness, is this: A topological space X is connected if for any open cover of X, there's a chain of elements of that open cover connecting any two elements of that open cover. More formally, if {U i } i in I is a...
by antonfire
Fri Mar 01, 2013 5:04 pm UTC
Forum: Mathematics
Topic: Discontinituous increasing function?
Replies: 20
Views: 3562

Re: Discontinituous increasing function?

Here is a sketch of a constructive proof that every increasing function on [0,1] has a point of continuity. Define a descending sequence of subintervals I n of [0,1] as follows. I 0 is [0,1], and I n+1 is either the left half or the right half of I n , whichever has the smaller gap between the value...
by antonfire
Thu Feb 28, 2013 10:07 am UTC
Forum: Mathematics
Topic: Connected v. Path-connected
Replies: 9
Views: 2513

Re: Connected v. Path-connected

Connectedness is also a more fundamentally topological notion than path-connectedness. From the point of view of pure point-set topology, what's the big deal about paths, namely continuous functions from [0,1] to a topological space? Why do we care about [0,1] in the first place?
by antonfire
Sun Feb 03, 2013 7:33 am UTC
Forum: Mathematics
Topic: Irreducible ternary functions
Replies: 9
Views: 1798

Re: Irreducible ternary functions

If your functions are allowed to have any domain and codomain, then any ternary function is reducible. Let h be a function that takes arguments in X, in Y, and in Z, and spits out an element of W. Take p to be the function that takes arguments in X and Y, and spits out an element of XxY, given by p(...
by antonfire
Sun Dec 09, 2012 10:15 pm UTC
Forum: Mathematics
Topic: Some troubles with logarithms...
Replies: 13
Views: 2290

Re: Some troubles with logarithms...

2. 3^2x - 5(3^x) + 4 = 0 (express x as a logarithm if necessary) If I take the logarithm of both sides, do I express each individual monomial on the left side as a logarithm, or the entire polynomial as a single logarithm? And even then, I don't know where I'd go from either of those procedures. No...
by antonfire
Wed Nov 21, 2012 10:12 pm UTC
Forum: Mathematics
Topic: Matrices
Replies: 7
Views: 1268

Re: Matrices

Each of the columns is a quadratic function.
by antonfire
Wed Sep 05, 2012 10:37 am UTC
Forum: Mathematics
Topic: Anyone wanna read "Gödel Escher Bach" on Reddit?
Replies: 3
Views: 1764

Re: Anyone wanna read "Gödel Escher Bach" on Reddit?

It's definitely worth reading. However, crackpots and fanboys flock to it, and you should try to avoid being those things.
by antonfire
Fri Jul 13, 2012 7:58 pm UTC
Forum: Mathematics
Topic: Prime, odd and even numbers
Replies: 21
Views: 5421

Re: Prime, odd and even numbers

gmalivuk wrote:Yeah, 2 being so small has some problems, but they have nothing to do with its being even.
Sure they do; the problems mostly arise from the fact that you can't divide by 2 in Z/2Z.
by antonfire
Sat Jul 07, 2012 7:20 am UTC
Forum: Mathematics
Topic: Can you have matricies inside of matricies?
Replies: 11
Views: 4039

Re: Can you have matricies inside of matricies?

It is fine to look at matrices with matrices as entries. But in this case, all entries should be matrices of the same size, otherwise not even the addition would be well-defined (unless you define the addition of matrices of different sizes or matrix+scalar or whatever, too). It's sensible to talk ...
by antonfire
Thu Jun 07, 2012 2:59 am UTC
Forum: Mathematics
Topic: Hahn Series not a Field
Replies: 5
Views: 2032

Re: Hahn Series not a Field

So you handwave something about long division, conclude that 1/(1-t) = 1 + t + t2 + ... + tw, and proceed to use that statement to derive a contradiction? You'd better make that handwaving precise.
by antonfire
Wed Jun 06, 2012 8:57 am UTC
Forum: Mathematics
Topic: Hahn Series not a Field
Replies: 5
Views: 2032

Re: Hahn Series not a Field

I'm going to pretend you're using (some large subset of) the surreal numbers as your value group, or something similar. The notion that 1/(1-t) is two different things is already silly. (1-t)(1+t+t 2 +...) is 1. (1-t)(1+t+t 2 +...+t w ) is not 1, it is 1 + t w - t w+1 . (1-t)(1+t+t 2 + ... + t w + t...
by antonfire
Mon Jun 04, 2012 2:19 am UTC
Forum: Mathematics
Topic: Math: Fleeting Thoughts
Replies: 382
Views: 123336

Re: Math: Fleeting Thoughts

It's also called the degree of that (algebraic) number. Fun fact: the growth rate of this thing is an algebraic number of degree 71.
by antonfire
Tue May 01, 2012 8:56 pm UTC
Forum: Mathematics
Topic: Limit of a series
Replies: 26
Views: 5580

Re: Limit of a series

Some philosophical points, then. The probabilistic view doesn't see the counterexamples given above, because a randomly chosen sequence of signs is (almost surely) actually very well-behaved. For instance, when p=1/2, S k (the sum of the first k signs) is O(k 1/2 + e ), so S k /k 2 converges. The ba...
by antonfire
Thu Apr 26, 2012 9:18 am UTC
Forum: Mathematics
Topic: Non-Decimal Primes
Replies: 8
Views: 2708

Re: Non-Decimal Primes

Binary, octal, hexadecimal and decimal are all just different conventions for writing down natural numbers. Whether you happen to write 13 as 11012, or as 158, or as d, it's prime. You might as well ask if there are any languages in which an apple doesn't have seeds.
by antonfire
Mon Apr 16, 2012 5:10 am UTC
Forum: Mathematics
Topic: Limit of a series
Replies: 26
Views: 5580

Re: Limit of a series

Let H k = 1/k, and h k = -H k+1 + H k . Let s k be some sequence of 1s and -1s, and S_k = \sum_{i=1}^{k-1} s_i . Summation by parts gives us \sum_{k=1}^n s_k H_k = S_{n+1} H_{n+1} + \sum_{k=1}^n S_{k+1} h_k. The first term is bounded above by 1, and the summand in the second term is roughly S k /k 2...
by antonfire
Sun Apr 08, 2012 10:06 am UTC
Forum: Mathematics
Topic: Favorite number sequences?
Replies: 24
Views: 3696

Re: Favorite number sequences?

I like the look and say sequence, mostly because how fast it grows is hilarious.
by antonfire
Sun Apr 01, 2012 4:19 pm UTC
Forum: Mathematics
Topic: Improper integral question
Replies: 14
Views: 3631

Re: Improper integral question

I often tell my students to be careful with that casual paraphrase. Notice that, for example, \int \frac1{x^2+1} dx = \arctan x which has absolutely nothing to do with logarithms. Well, not nothing . 1/(x 2 +1) = i/2 (1/(i+x) + 1/(i-x)). So its integral is i/2 (log(i+x) - log(i-x)) = i/2 log((i+x)/...
by antonfire
Sat Mar 31, 2012 7:49 am UTC
Forum: Mathematics
Topic: Is there a name for this infinite series?
Replies: 8
Views: 1684

Re: Is there a name for this infinite series?

If you want to communicate about mathematics with people who are used to communicating about mathematics, you should stick to the usual conventions for communication about mathematics. This includes the order of operations. If you have a compelling reason not to (you don't), then you should explain ...
by antonfire
Sun Mar 25, 2012 1:21 am UTC
Forum: Mathematics
Topic: Infinity paradox? Help
Replies: 54
Views: 8329

Re: Infinity paradox? Help

Well-ordering and linear ordering do not mean the same thing. A set is well-ordered if every one of its (nonempty) subsets has a least element. The fact that the natural numbers are well-ordered is what allows you to prove things about all natural numbers by induction.
by antonfire
Sat Mar 17, 2012 6:46 am UTC
Forum: Mathematics
Topic: Funny mathematical terms and statements
Replies: 21
Views: 7585

Re: Funny mathematical terms and statements

When you put something in some sort of "normal form", you are actually putting it into a very specific form.
by antonfire
Thu Mar 15, 2012 7:12 pm UTC
Forum: Mathematics
Topic: Discriminant of a pretty easy polynomial
Replies: 5
Views: 2058

Re: Discriminant of a pretty easy polynomial

There are only finitely many primes that divide the discriminant of f.
by antonfire
Thu Mar 08, 2012 5:44 pm UTC
Forum: Mathematics
Topic: Partitioning arbitrary numbers using +,-,*,^
Replies: 17
Views: 2908

Re: Partitioning arbitrary numbers using +,-,*,^

If you want it in the least number of integers, 22027 can be partitioned as 1+1+1+...+1. If you want it in the least number of operations, 22027 can be partitioned as 22027.
by antonfire
Fri Mar 02, 2012 6:29 am UTC
Forum: Mathematics
Topic: Usage of matrix multiplication
Replies: 10
Views: 4118

Re: Usage of matrix multiplication

Fine. What about the fact that you can compute all pairs shortest paths by looking at the nth power of a graph's adjacency matrix, using the min-sum semiring. That's not linear algebra (it's not a linear space) but uses matrix multiplication. There's actually a number of similar uses of matrix mult...
by antonfire
Thu Mar 01, 2012 11:08 am UTC
Forum: Mathematics
Topic: Usage of matrix multiplication
Replies: 10
Views: 4118

Re: Usage of matrix multiplication

None of these examples are really "beside" linear algebra. Rather, they're applications of linear algebra to other areas of math and science. Asking for a use of matrix multiplication besides linear algebra is a bit like asking for a use of long division besides arithmetic.
by antonfire
Wed Feb 29, 2012 3:37 am UTC
Forum: Mathematics
Topic: Erdos Conjecture
Replies: 18
Views: 7041

Re: Erdos Conjecture

All I need to do to prove the Riemann hypothesis is prove a lower bound for zeta(z) in the critical strip but away from the middle, such as |zeta(z)|>1.
by antonfire
Mon Feb 13, 2012 3:56 am UTC
Forum: Mathematics
Topic: Function Equivalence
Replies: 22
Views: 2548

Re: Function Equivalence

Writing "f + is continuous because f + (x) = (f(x)+|f(x)|)/2, which is continuous" wouldn't anger the rigor gods, but they might be angry if you didn't justify that (f(x)+|f(x)|)/2 is continuous. In particular, why is |f(x)| continuous? On that note, is it possible for two functions to tak...
by antonfire
Fri Feb 03, 2012 7:48 am UTC
Forum: Mathematics
Topic: Temporal distortion, geodesics(?), and My Little Pony
Replies: 24
Views: 2863

Re: Temporal distortion, geodesics(?), and My Little Pony

It's easy to check whether your semicircle path satisfies the Euler-Lagrange equation. If it does, it's probably a unique solution by differential equations stuff, so it's optimal. (Well, you have to be a bit more careful than that, but whatever.) If it doesn't, some nearby paths are shorter and it'...
by antonfire
Sat Jan 28, 2012 8:12 pm UTC
Forum: Mathematics
Topic: Sorting elements of a rectangular array
Replies: 4
Views: 1737

Re: Sorting elements of a rectangular array

Here's a quicker proof: Clearly it's enough to show that it works for an array with 2 columns. Consider the operation of looking at a 2x2 box of cards, and sorting the cards in each column of that box. Sorting the columns of the entire array can be achieved by finitely many of these operations. ...
by antonfire
Thu Jan 26, 2012 12:21 am UTC
Forum: Mathematics
Topic: Phase independence of trigonometric integrals
Replies: 3
Views: 1235

Re: Phase independence of trigonometric integrals

Another way to think of it is as a special case of the following fact: if u, v and u', v' are two orthonormal bases for the same space, and <.,.> is bilinear, then <f,u>2 + <f,v>2 = <f,u'>2 + <f,v'>2. Checking this is an easy linear algebra computation.
by antonfire
Tue Jan 17, 2012 8:17 am UTC
Forum: Mathematics
Topic: Nonstandard proofs for simple theorems
Replies: 34
Views: 6714

Re: Nonstandard proofs for simple theorems

What are the roots of the functions? I plotted these for the first fifty functions or so once. If I recall correctly, for high k, they seem to trace out a vaguely ellipse-shaped curve in the complex plane. At the time, I didn't know how to interpret this, and I suppose I still don't. It would be in...
by antonfire
Tue Jan 17, 2012 7:12 am UTC
Forum: Mathematics
Topic: Nonstandard proofs for simple theorems
Replies: 34
Views: 6714

Re: Nonstandard proofs for simple theorems

A fairly similar proof to the above is to note that n(n-1)(n-2)/6 = 0 + 0 + 1 + 3 + 6 + ... + (n-1)(n-2)/2. (This is easy to see by looking at n = 0, 1, 2, 3.) When the sum has 2m terms, we can group them into pairs: 2m(2m-1)(2m-2)/6 = (0+0) + (1+3) + (6+10) + ... + ( (2m-2)(2m-3)/2 + (2m-1)(2m-2)/2...

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