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The question comes down to computing what's called the "derivative of", the "differential of", or the "tangent map to", the exponential map from $M_n(\mathbb R)$ into itself at a given matrix $A$ (not necessarily the zero matrix). There is a classical formula for this. Here is the first reference I found: pages 1 and 2... |
The question comes down to computing what's called the "derivative of", the "differential of", or the "tangent map to", the exponential map from $M_n(\mathbb R)$ into itself at a given matrix $A$ (not necessarily the zero matrix). There is a classical formula for this. Here is the first reference I found: pages 1 and 2... |
I recently read a very good inequality concerning the no of primes $\pi(x)$:
$$\pi(n)>\frac{1}{6}\frac{n}{\log n}\mathrm{\ for\ }n\ge 2$$
Are any other such elementary inequalities concerning the primes? |
Recently I found [this post][1] on Reddit. It describes the following algorithm to find e:
> Here is an example of e turning up
> unexpectedly. Select a random number
> between 0 and 1. Now select another
> and add it to the first. Keep doing
> this, piling on random numbers. How
> many random numbers, on aver... |
Explanation of numeric experiment that approximates e? |
Recently I found [this post][1] on Reddit. It describes the following algorithm to find e:
> Here is an example of e turning up
> unexpectedly. Select a random number
> between 0 and 1. Now select another
> and add it to the first. Keep doing
> this, piling on random numbers. How
> many random numbers, on aver... |
You probably figured it out already, but for posterity -- this <a href="http://www-diglib.stanford.edu/~klein/lagrange-multipliers.pdf">document</a> gives intuition which I didn't see in Boyd book, esp. Figure 3 on page 3.
To solve equality constrained problem you look for points where gradient of objective is in th... |
Recently I found [this post][1] on Reddit. It describes the following algorithm to find e:
> Here is an example of e turning up
> unexpectedly. Select a random number
> between 0 and 1. Now select another
> and add it to the first. Keep doing
> this, piling on random numbers. How
> many random numbers, on aver... |
For regression, least sum of squares criterion tried to fit your function through the mean of the data. In other words, with enough degrees of freedom, value of your fitted function for x will be the average of all observed values at x. Least sum of absolute values instead produces a function that'll go through the med... |
Have you tried reducing the matrix to row echelon form? Maybe my answer to this question:
http://math.stackexchange.com/questions/1822/is-reducing-a-matrix-to-row-echelon-form-useful-at-all/1831#1831 could be helpful. |
The question comes down to computing what's called the "derivative of", the "differential of", or the "tangent map to", the exponential map from $M_n(\mathbb R)$ into itself at a given matrix $A$ (not necessarily the zero matrix). There is a classical formula for this. Here is the first reference I found: pages 1 and 2... |
The question comes down to computing what's called the "derivative of", the "differential of", or the "tangent map to", the exponential map from $M_n(\mathbb R)$ into itself at a given matrix $A$ (not necessarily the zero matrix). There is a classical formula for this. Here is the first reference I found: pages 1 and 2... |
I know that if I have n items and I want to choose k of them, I use the binomial coefficient and there are n!/(n - k)!k! ways. Between each of these choices, there is an overlap of at most k - 1 items.
How can I calculate the number of ways I can choose k items out of n, so there is an overlap of at most m items be... |
How can I determine the size of the largest collection of k-element subsets of an n-element set such that each pair of subsets has at most m elements in common?
|
How is moduli of curves relevant in physics? |
If there were no repeats, what would the degree sequence look like? Why could it not look like that?
Gerhard "Ask Me About System Design" Paseman, 2010.08.10 |
Here is a time sample: Q = {(t(i), x(i)) | 0 <= x(i) <= x(i+1), i = 1..n}
and rules:
(1) T1 <= t(i+1) - t(i) < T2 where T1, T2 > 0
(2) x(i) comes with error:
x(i) = [x(t(i))] where [...] - hole part of a number, x(t(i)) - voluntary unknown law of object motion.
How we can find (approximately but sufficie... |
The reason I ask this is because difficult mathematical proofs are just not plain self-evident. You would need a few years of intensive study before you can get to the point of understanding the topics and the proofs. The problem is that for a difficult math theorem, maybe only a handful of mathematicians have studied ... |
Is It True that We Can Never Be Sure of Validity of a Mathematical Proof? |
What is the probability of getting [yahtzee][1] using N dices with X sides in Y throws in a single round?
Which side of the dice one gets yahtzee with doesn't matter (e.g. it doesn't matter if it is ones or twos etc.). I also assume perfect strategy is used, that is, after each throw one saves the number it is the m... |
**Problem**: Let $n$ be an integer and $p$ a prime dividing $5(n^2-n+\frac{3}{2})^2-\frac{1}{4}$. Prove that $p \equiv 1 \pmod{10}$.
The polynomial can be re-written as $(\sqrt{5}(n^2-n+\frac{3}{2})-\frac{1}{2})(\sqrt{5}(n^2-n+\frac{3}{2})+\frac{1}{2})$. If this vanishes mod $p$ then $5$ is a quadratic residue mod $... |
Does there exist an $f:\mathbb{N} \to \mathbb{N}$ such that $\sum f(n)/n^2$ converges? |
Sometimes one encounter the requirement that the morphisms of a category needs to be a set. What if it was not, could you provide examples of what could go wrong? (One example per answer).
|
What's so special with small categories? |
Is possible to construct infinitely differentiable functions that interpolate through arbitrary points, the way polynomial splines do? If so, do they have a name and is there an algorithm for constructing the curves.
I suppose a infinite degree polynomial spline might qualify but I was hoping there might be somethin... |
I have been convinced for years of the following claim: *Every compact topological space is sequentially compact.* (The converse is trivially false.) I'm positive I have even been taught this in my topology course as an undergraudate, and I can provide at least two wrong, but very subtly so, "proofs".
So this is one... |
The question comes down to computing what's called the "derivative of", the "differential of", or the "tangent map to", the exponential map from $M_n(\mathbb R)$ into itself at a given matrix $A$ (not necessarily the zero matrix). There is a classical formula for this. Here is the first reference I found: pages 1 and 2... |
Most philosophers have agreed for centuries that we can be *completely sure* of very little, if anything. Descartes famously argued [cogito ergo sum][1], i.e., that the action of contemplating the possibility of his nonexistence confirms his existence (a nice early example of a "self-defeating proof by contradiction")... |
In quantum field theory many quantities are calculated as (formal) Feynman path integrals, that is, integrals "over the space of all paths". To make sense of this in dimensions higher than 2, one uses a perturbation expansion (in powers of Planck's constant) of the integral which leads to a sequence of finite dimensio... |
It's a bit disingenuous to say polynomial splines are infinitely differentiable; a spline with nth order pieces can only have at most C^{n-1} continuity. That being said, a different basis set (e.g. the Fourier basis or rational functions) would in principle be infinitely differentiable, but might be a very "wiggly" wa... |
Sometimes one encounter the requirement that the objects of a category needs to be a set. What if it was not, could you provide examples of what could go wrong? (One example per answer).
|
I have a partition of a positive integer (p). How can I prove that the factorial of p can always be divided by the product of the factorials of the parts?
As a quick example 9!/(2!3!4!) = 1260 (no remainder), where 9=2+3+4.
I can *nearly* see it by looking at factors, but I can't see a way to guarantee it. |
A curious compactness confusion: space filling curves in the hilbert cube that contradict a bona fide theorem?!?!? |
These quotients are integers since they solve counting problems.
For instance, how many nine-letter words are there with 2 As, 3 Bs and 4 Cs?
For the full story see the [multinomial theorem][1].
[1]: http://en.wikipedia.org/wiki/Multinomial_theorem |
[Bertrand's postulate](http://en.wikipedia.org/wiki/Bertrand%27s_postulate) is
> if n > 3 is an integer, then there always exists at least one prime number p with n < p < 2n − 2.
Thus ε < 2 for n > 3. What if n ≤ 3?
* For n = 3, 3 < 5 < 6 - ε ⇒ ε < 1
* For n = 2, 2 < 3 < 4 - &eps... |
**HINT** Your polynomial $p(n)$ splits over ${\mathbb Q}(w), w = \zeta_5$, namely
$ 125 \; p(x) = 125 \; (5 x^4-10 x^3+20 x^2-15 x+11) $
$\quad\quad\quad\quad\quad\; = \;\; (5 x+3 w^3-4 w^2-w-3) (5 x+4 w^3+3 w^2+7 w+1)$
$\quad\quad\quad\quad\quad\quad\; * \; (5 x-3 w^3+4 w^2+w-2) (5 x-4 w^3-3 w^2-7 w-6... |
**HINT** Your polynomial $p(n)$ splits over ${\mathbb Q}(w), w = \zeta_5$, namely
$ 125 \; p(x) = 125 \; (5 x^4-10 x^3+20 x^2-15 x+11) $
$\quad\quad\quad\quad\quad\; = \;\; (5 x+3 w^3-4 w^2-w-3) (5 x+4 w^3+3 w^2+7 w+1)$
$\quad\quad\quad\quad\quad\quad\; * \; (5 x-3 w^3+4 w^2+w-2) (5 x-4 w^3-3 w^2-7 w-6... |
[Bertrand's postulate](http://en.wikipedia.org/wiki/Bertrand%27s_postulate) is
> if n > 3 is an integer, then there always exists at least one prime number p with n < p < 2n − 2.
Thus ε < 2 for n > 3. What if n ≤ 3?
* For n = 3, 3 < 5 < 6 - ε ⇒ ε < 1
* For n = 2, 2 < 3 < 4 - &eps... |
Has this problem been published? |
Though not discussed in the full generality you describe, the problem is discussed at length for n = 1, 2, and 3 in *Mathematics for High School Teachers: An Advanced Persepective* by Usiskin, Peressini, Machisotto, and Stanley (section 5.1.4; © 2003; published by Pearson Education). Specifically, the text uses t... |
a) Prove that $G$ has a path of length at least $k$.
solution: We know that
$\delta(G) = min\lbrace deg(v) | v \in V(G) \rbrace$
if $\delta(G) = k$ then there exists some $v \in V(G)$ such that $deg(v) = k$. This means all other vertices $u \in V(G)$ have $deg(u) \geq k$.
Now i know this must be part of the... |
Are there publications or references describing N dimensional space divided by N-1 dimensional boundaries? |
How can I characterize the type of solution vector that comes out of a matrix? |
Suppose $G$ is a locally compact abelian group, with Haar Measure $\mu$, then is $\mu(E)=\mu(E^{-1})$? for all subsets $E$ of $G$.
I have seen that this is true for all Borel subsets of $G$, but i am in need of a proof and also an example as to why it doesn't hold for other subsets. |
I'm optimizing a family of objectives over x1,...,xn with a single constraint that x's add up to 1, and when using method of Lagrange multipliers, the single multiplier always ends up 0 for members of this family...does this tell us anything interesting about these objectives?
Edit: the answer was -- it means the co... |
> ... octonions are the crazy old uncle nobody lets out of the attic: they are nonassociative.
Comes from a a quote by John Baez. Clearly, the sucessor to quaterions from the Caley-Dickson process is a numerical beast, but has anybody found any real-world uses for them? For example, quaterions have a nice connectio... |
What are some real-world uses of Octonions? |
The Hilbert Cube $H$ is defined to be $[0,1]^{\mathbb{N}}$, i.e., a countable product of unit intervals, topologized with the product topology.
Now, I've read that the Hilbert Cube is homogeneous. That is, given two points $p, q\in H$, there is a homeomorphism $f:H\rightarrow H$ with $f(p)=q$.
What's confusing t... |
Why is the Hilbert Cube homogeneous? |
Here's an entry point into the literature, from the introduction to Lam's [paper \[1\]][1]:
In the study of any algebraic system in which there is a notion of a direct sum, the theme of cancellation arises very naturally: if $A \oplus B \cong A\oplus C$ in the given system, can we conclude that $B \cong C$? (For a... |
It will need something clever. For n + 1 = 6, you need 6 times the lcm of 1 5 and 10 in
order to get enough powers of 2 ( and 3 ). Is there a characterization of those n + 1 where the
entire factor of n + 1 is needed, and not the lcm, for the left hand side? |
Here is a non-answer which may prove interesting and tractable.
Given a configuration of rectangles that properly tile a surrounding rectangle, one can
represent the tiling in a number of ways. Two that come to mind I will call C and LL.
C represents each rectangle by giving its dimensions (x-span and y-span) and... |
Here's an entry point into the literature, from the introduction to Lam's [paper \[1\]][1]:
In the study of any algebraic system in which there is a notion of a direct sum, the theme of cancellation arises very naturally: if $A \oplus B \cong A\oplus C$ in the given system, can we conclude that $B \cong C$? (For a... |
If there were no repeats, what would the degree sequence look like? Why could it not look like that? |
Yes. This is an easy consequence of the Krull-Schmidt theorem:
http://planetmath.org/encyclopedia/KrullRemakSchmidtTheorem.html
<b>Edit</b>: To be clear, what I am claiming is that if one has finite groups $H,K,G$ such that $G \times H \cong G \times K$, then $H \cong K$. As Steve D points out below, the hypoth... |
Are there publications or references describing N dimensional space divided by N-1 dimensional (hyper)planes? |
Are you missing a condition? Let $N-1$ of the bugs form the vertices of a regular $N-1$-gon, say at unit distance from the origin and with velocity vectors pointing outward, and let the last bug be at the origin with velocity vector pointing somewhere else. |
Assuming no bugs get squashed in the process:
At $\lim_{t\to\infty}$ when $t$ is time, the bugs' beginning points shrink to $\frac{P}{t} = 0$ as observed when zoomed out. This means the current position of the bugs is $\sqrt{r^2 + (vt)^2}$, or on the line of a huge circle. Thus, you can see that the $N$ number of bu... |
There are $N$ bugs in a plane. All bugs are moving at the same constant (nonzero) speed, but no two bugs are moving in the same direction (speed vectors are of the same velocity, but no two are parallel).
Prove that at some point in time $N$ bugs will form **convex** polygon.
Edit: Can you loosen up any of the c... |
There are 100 ropes in a bag. In each step, two rope ends are picked at random, tied together and put back into a bag. The process is repeated until there are no free ends.
What is the expected number of loops at the end of the process? |
Assuming no bugs get squashed in the process:
At $\lim_{t\to\infty}$ when $t$ is time, the bugs' beginning points shrink to $\frac{P}{t} = 0$ as observed when zoomed out. This means the final position of each bug lies on $\sqrt{r^2 + (vt)^2}$, or on the edge of a huge circle. Thus, you can see that all the bugs make... |
I was wondering if anyone had any experience with an undergraduate course that emphasized the building of mathematical theories or if they'd ever heard of this being done? How did the class work (did the professor list axioms and definitions each week and see what the students were able to prove, did you discuss a spec... |
undergraduate courses emphasizing theory building? |
I have benefited from more than one professor who taught more or less according to the <a href="http://en.wikipedia.org/wiki/Moore_method">Moore Method</a>. These professors happened to come out of SUNY Binghamton in the early 1980's.
The courses required lots of input from the students. One professor in particu... |
I do not have the rep to comment but what sequence of horizontal or vertical steps would lead
B W
W W
to have the number of black and white points almost even? |
What sequence of horizontal or vertical steps would lead
B W
W W
to have the number of black and white points almost even? |
Suppose that $c_k$ is an decreasing sequence of non-negative real numbers, such that $c_0=1$
and $c_{k}\leq \frac{1}{2}(c_{k-1}+c_{k+1})$.
Is it true that the generated function of $c_k$ admit an integral representation as below
$$
\sum_{k=0}^{\infty}c_kz^k=\int_{\partial\mathbb D} \frac{1}{1-\zeta z}d\mu(\zeta)... |
The Hilbert Cube $H$ is defined to be $[0,1]^{\mathbb{N}}$, i.e., a countable product of unit intervals, topologized with the product topology.
Now, I've read that the Hilbert Cube is homogeneous. That is, given two points $p, q\in H$, there is a homeomorphism $f:H\rightarrow H$ with $f(p)=q$.
What's confusing t... |
As phrased, your question invites a philosophical answer but you said that's not what you want. One way to approach this question is to ask if complex numbers correspond to anything in the real world. What can you count or measure with them?
As far as I can tell, complex numbers are most directly useful for measurin... |
Using the duality between locally compact Hausdorff spaces and commutative $C^*$-algebras one can write down a vocabulary list translating topological notions regarding a locally compact Hausdorff space $X$ into algebraic notions ragarding its ring of functions $C_0(X)$ (see Wegge-Olsen's book, for instance). For examp... |
What is the commutative analogue of a $C^*$-subalgebra? |
Using the duality between locally compact Hausdorff spaces and commutative $C^*$-algebras one can write down a vocabulary list translating topological notions regarding a locally compact Hausdorff space $X$ into algebraic notions ragarding its ring of functions $C_0(X)$ (see Wegge-Olsen's book, for instance). For examp... |
Using the duality between locally compact Hausdorff spaces and commutative $C^*$-algebras one can write down a vocabulary list translating topological notions regarding a locally compact Hausdorff space $X$ into algebraic notions ragarding its ring of functions $C_0(X)$ (see Wegge-Olsen's book, for instance). For examp... |
Let $G$ be a graph such that $\delta(G) \geq k$.
a) Prove that $G$ has a path of length at least $k$.
solution: We know that
$\delta(G) = min\lbrace deg(v) | v \in V(G) \rbrace$
if $\delta(G) = k$ then there exists some $v \in V(G)$ such that $deg(v) = k$. This means all other vertices $u \in V(G)$ have $de... |
Suppose that $c_k$ is an decreasing sequence of non-negative real numbers, such that $c_0=1$
and $c_{k}\leq \frac{1}{2}(c_{k-1}+c_{k+1})$.
Is it true that the generated function of $c_k$ admit an integral representation as below
$$
\sum_{k=0}^{\infty}c_kz^k=\int_{\partial\mathbb D} \frac{1}{1-\zeta z}d\mu(\zeta)... |
sum of harmonic progression? |
As in <a href="http://mathoverflow.net/questions/31990/continuous-functions-remaining-constant">the previous proof</a> of $f$ being constant on $\mathbb{R}$, define $g(x) = f(x-1)$, so that $g(x) = g(2x)$; the domains of $f$ and $g$ are just shifted versions of each other.
Certainly, if the domain of $g$ is small en... |
The partial sums of the harmonic series are called "harmonic numbers." The difference between the nth harmonic number and ln(n) tends to a limit as n increases, and that limit is called [Euluer's constant][1] or gamma.
There's a great book about all this called [Gamma: Exploring Euler's Constant][2].
[1]: http... |
Using the duality between locally compact Hausdorff spaces and commutative $C^*$-algebras one can write down a vocabulary list translating topological notions regarding a locally compact Hausdorff space $X$ into algebraic notions ragarding its ring of functions $C_0(X)$ (see Wegge-Olsen's book, for instance). For examp... |
Using the duality between locally compact Hausdorff spaces and commutative $C^*$-algebras one can write down a vocabulary list translating topological notions regarding a locally compact Hausdorff space $X$ into algebraic notions ragarding its ring of functions $C_0(X)$ (see Wegge-Olsen's book, for instance). For examp... |
Say we have a function $\displaystyle f(z)=\sum_{n=0}^\infty a_n z^n$ with radius of convergence R>0. Why is the radius of convergence only R? Can we conclude that there must be a pole, branch cut or discontinuity for some z_0 with |z_0|=R? What does that mean for functions like
$$f(z)=\begin{cases}
0 & \text{for... |
Using the duality between locally compact Hausdorff spaces and commutative $C^*$-algebras one can write down a vocabulary list translating topological notions regarding a locally compact Hausdorff space $X$ into algebraic notions ragarding its ring of functions $C_0(X)$ (see Wegge-Olsen's book, for instance). For examp... |
I am trying to find the answer to the following question:
If I am player poker and am dealt two cards neither of which are spades, what are the chances of two or more of the flop cards (the flop is the first three community cards) being spades.
I answered this question in the following way:
As I have no spades... |
There are $N>1$ points in a plane. Consider the set of all distances between pairs of the points. Let $n$ be the number of elements of this set.
We know that $$n \leq {N \choose 2}.$$
What is the lower bound estimate on $n$?
For the sake of clearity, if points are $A_1, A2, \dots, A_N$, and $S = \\{d(A_i, A_j)| 1\... |
Despite appearances, this is not a question on computational aspects of number theory. The background is as follows. I once asked a number theorist about what he considered to be the most important unsolved problems in arithmetic geometry. He told me about a few, but along with some well-known problems he told me the... |
Using the duality between locally compact Hausdorff spaces and commutative $C^*$-algebras one can write down a vocabulary list translating topological notions regarding a locally compact Hausdorff space $X$ into algebraic notions ragarding its ring of functions $C_0(X)$ (see Wegge-Olsen's book, for instance). For examp... |
There are $N>1$ points in a plane. Consider the set of all distances between pairs of the points. Let $n$ be the number of elements of this set.
We know that $$n \leq {N \choose 2}.$$
What is the lower bound estimate on $n$?
For the sake of clearity, if points are $A_1, A_2, \dots, A_N$, and $S = \\{d(A_i, A_j)| 1... |
Despite appearances, this is not a question on computational aspects of number theory. The background is as follows. I once asked a number theorist about what he considered to be the most important unsolved problems in arithmetic geometry. He told me about a few, but along with some well-known problems he told me the f... |
Bernhard Elsner, alias MathOMan, posted this exercise in plane Geometry, [*Theorem about a circle, three chords and a midpoint*][1] on January 29th, 2010.
"Let $\mathcal{C}$ be a circle, $A,B$ two distinct points on $\mathcal{C}$ and and M be the midpoint of the chord $[AB]$. Take two other chords,$[PQ]$ and $[SR]$,... |
How to prove the midpoint of a chord is also the midpoint of the line segment defined by the points of intersection of other two chords with it? |
No feasible (polynomial time) algorithm is currently known for
recognizing squarefree integers or for computing the squarefree
part of an integer. In fact it may be the case that this problem
is no easier than the general problem of integer factorization.
This problem is important because one of the main task... |
Using the duality between locally compact Hausdorff spaces and commutative $C^*$-algebras one can write down a vocabulary list translating topological notions regarding a locally compact Hausdorff space $X$ into algebraic notions ragarding its ring of functions $C_0(X)$ (see Wegge-Olsen's book, for instance). For examp... |
All the integrals I'm familiar with have the form:
> $\int f(x)\mathrm{d}x$.
And I understand these as the sum of infinite tiny rectangles with an area of: f(x_i) * dx.
Is it valid to have integrals that do not have a differential, such as dx, or that have the differential elsewhere than as a factor ? Let me g... |
Is it guaranteed that there will be some p s.t. p|2^n-1 but not p|2^m-1 for any m<n?
In other words, does each 2^x-1 introduce a *new* prime factor? |
I would like to know how to explicitly prove that Riemann Curvature,Ricci Curvature, Sectional Curvature and Scalar Curvature are left invariant under an isometry.
I can't see this explained in most books I have looked at.
They atmost explain preservation of the connection.
I guess doing an explicit proof for... |
I am trying to find the answer to the following question:
If I am playing poker and am dealt two cards neither of which are spades, what are the chances of two or more of the flop cards (the flop is the first three community cards) being spades.
I answered this question in the following way:
As I have no spade... |
If $S$ is a complete set of (left) coset representatives for $H$, then for every $x\in G$ there exists $s\in S$ such that $xH=sH$, and moreover, if $s_1,s_2\in S$ are such that $s_1H=s_2H$, then $s_1=s_2$. That is: there is one and exactly one representative from each coset of $H$ in $G$ in the set $S$. As such, your c... |
No feasible (polynomial time) algorithm is currently known for
recognizing squarefree integers or for computing the squarefree
part of an integer. In fact it may be the case that this problem
is no easier than the general problem of integer factorization.
This problem is important because one of the main task... |
Apparently 1+2+3+4+...+n = 1/2*n*(n+1).
How? What's the proof? Or maybe it is self apparent just looking at the above? Does this problem have a name and maybe a presence on the net? And what is this math called so I can add the right tags.
Thx if you can help. |
Proof for formula for sum of sequence 1+2+3+...+n? |
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