instruction stringlengths 12 30k |
|---|
Solving a heat equation with central symmetry i got the following result:
The problem is to find a sphere center temperature vs. time given that the surface of the sphere is kept constant at $$T_1$$ and at initial moment t=0 the temperature inside the sphere is distributed uniformly with value $$T_0$$
A solut... |
Let V be a k-vector space of dimension n and T: V -> V a linear map of rank 1. Show that either T^2 = 0 or that T is diagonalisable? |
I'm trying to prove that $\frac{\prod \mathbb{Z_p}}{\bigoplus \mathbb{Z_p}}$ is a divisible $\mathbb{Z}$-module (p is prime, and the direct sum and direct product are taken over the set of all primes). It is an exercise from Rotman, An Introduction to Homological Algebra. Here's what I've done so far:
$\bigoplus \ma... |
what about the direct approach?
Suppose you have $(a_i)\in \prod \mathbb{Z}_p$ and an integer n, then modulo $\bigoplus \mathbb{Z_p}$ you may assume that $a_p=0$ for all the primes dividing n. n is invertible in all the rest of $\mathbb{Z}_p$ so you can find $\frac{a_p}{n}$ in them.
So define $b_p = 0 $ for $p\mi... |
what about the direct approach?
Suppose you have $(a_i)\in \prod \mathbb{Z}_p$ and an integer n, then modulo $\bigoplus \mathbb{Z_p}$ you may assume that $a_p=0$ for all the primes dividing n. n is invertible in all the rest of $\mathbb{Z}_p$ so you can find $\frac{a_p}{n}$ in them.
So define $b_p = 0 $ for $p\mi... |
The standard way (other than generating up-to N) is to check if $(5*N*N + 4)$ or $(5*N*N - 4)$ is a perfect square.What is the mathematical logic behind this ? Also is there any other way for checking the same ? |
Finding out if a number is a fibonacci or not ? |
Checking if a number is a fibonacci or not ? |
First, if the rank of T is 1, then the dimension of its kernel is n-1. That means 0 is an eigenvalue of order n-1.
Now, as Prometheus said, you just have to find out whether you will find that last eigenvalue or not. To find it out, just take a vector $v \notin Ker(T)$ (thus, $v \in Im(T)$), and look at $T(v)$. Eith... |
This question is a follow-up to [this one][1].
Now I'm trying to prove that the torsion submodule of $\prod \mathbb{Z}_p$ is not a direct summand of $\prod \mathbb{Z}_p$. I found this interesting because it's an example that over a PID, a non-finitely generated module may fail to be decomposed as torsion $\oplus$ so... |
I have a problem that I cannot solve regarding interest payments, and payment holidays.
The scenario is this
If I purchase a product which is paid for over a 12 month period, I need to calculate the monthly interest payments. However, the complexity here, is that I have the option to take one month payment holiday ... |
EDITED after Prometheus's comment to use the characterization using sum of dimension of eigenspaces instead of counting eigenvalues. This argument should be correct.
Obviously $\dim V = n \geq 1$. Consider $T$'s restriction to its image. This is a linear map on a 1-dimensional space, so we can write $T\vert_{T(V)}(v... |
Prometheus' answer is essentialy correct. I'd just like to point something out to you so that you can gain a little intuition.
We know that the rank of $T$ is one-dimensional, which means that there exists ($n-1$)-dimensional subspace $W \subset V$ such that $T(W) = 0$ (imagine this in three dimensions as a plane th... |
Ask yourself this question: if $x \in \prod_p \mathbb{Z}_p$ is such that $x \in n \prod_p \mathbb{Z}_p$ for each $n \in \mathbb{Z} \setminus \{0\}$, then what can we say about $x$?
You should be able to conclude that there is no embedding from $\frac{\prod_p\mathbb{Z}_p}{\otimes_p \mathbb{Z}_p}$ to $\prod_p \mathbb{... |
There will not be a closed form, as it will vary with the month of the holiday. If you want to come out exactly even you have to choose in advance the month of the holiday. One approach is to assume the holiday is the first month (which will lead to the highest payment-do you see why?). Then you can apply one month'... |
Let $V$ be a $k$-vector space of dimension $n$ and $T: V \to V$ a linear map of rank 1. Show that either $T^2 = 0$ or that $T$ is diagonalisable? |
I understand that for a number $a$ to have a multiplicative inverse in mod $n$, it must be coprime to $n$; therefore, all numbers that do not have a multiplicative inverse mod $n$ must share a factor with $n$.
Now, my prediction was that if $n$ is even, then the sum of all numbers that do not have a multiplicative i... |
Ok, so this might not be a permutation or combination but I'm curious as to what the answer is.
Question: There are 3 groups and 5 Objects in each group. How many combinations are there if your choosing one from each group?
Yes this is in reference to Call of Duty: Black Ops perks lol. |
You presumably want $|f'(x)|\le 5/6$. It's not the case that $a_{n+1}$
need lie in the interval between $a_{n-1}$ and $a_n$. What can you say
about $|a_n-a_m|$? (The value of $a_0$ isn't relevant). |
How many combinations are there if there is only one group? For each of those, how many ways can you choose from the second group? And the third? Multiplication is your friend. |
For what values of $\beta$ does $\omega_i \mid \omega \mid^\beta$ not blow up at the origin? |
We are playing a game with my friend and trying to determine who is the better player.
The game has an luck element to it.
The current score is 9-7.
The following questions have risen:
1) How many games we should play to get statistically significant result that one is better than the other? We can assume p... |
How many games should we play to determine the better player statistically significantly? |
For what values of $\beta$ does $\omega_i^2 \mid \omega \mid^{-\beta}$ not blow up at the origin? |
$L$ is also a subsequential limit of $a_{n}$.
I came up with the following: Let $\epsilon\gt0$; if $(x_{k})\to L$ then we simply pick $x_{i}\in(L-\epsilon, L+\epsilon)$ and because $x_{i}$ is a subsequential limit of $a_{n}$ we know that in every neighborhood of $L$ there are infinite elements of $a_{n}$ and we conc... |
I want to prove that:
> If $(x_{k})\to L$ and $\forall x_{i}\in (x_{k})$, $x_{i}$ is a subsequential limit of $a_{n}$ then $L$ is also a subsequential limit of $a_{n}$.
I came up with the following: Let $\epsilon\gt0$; if $(x_{k})\to L$ then we simply pick $x_{i}\in(L-\epsilon, L+\epsilon)$ and because $x_{i}$ is a... |
I'm as mystified as Jonas Meyer on why you think this is "clumsy". It follows exactly along the intuition: I can get arbitrarily close to $L$ using the $x_i$, and I can find subsequence of $(a_i)$ that gets arbitrarily close to the $x_i$, so I can find subsequences that get arbitrarity close to things that get arbitrar... |
I along with one of my friends were just discussing some basic things in group theory, when this question came up:
- What are the *fundamental results* in group theory?
We happened to list out some:
- **Fundamental Theorem of Group Homomorphism:**
- **Cayley's Theorem**
- **Sylow's Theorem**
There may... |
In Stein-Shakarchi's book Fourier Analysis (p. 170), the solution of the Black-Scholes equation
$$\frac{\partial V}{\partial t}+rs\frac{\partial V}{\partial s}+\frac{\sigma^2s^2}{2}\frac{\partial^2 V}{\partial s^2}-rV=0$$ for
$0\lt t\lt T$
with boundary condition $V(s,T)=F(s)$, is given as
$$V(s,t)=\frac{e^{-r... |
I'm as mystified as Jonas Meyer on why you think this is "clumsy". It follows exactly along the intuition: I can get arbitrarily close to $L$ using the $x_i$, and I can find subsequence of $(a_i)$ that gets arbitrarily close to the $x_i$, so I can find subsequences that get arbitrarity close to things that get arbitrar... |
One proof of the halting problem goes by contradiction like this :
Assume there is a Turing machine $H$ that can decide the halting problem, then construct a Turing machine $Q$ that takes as input a Turing machine $M$ and an input $I$ to $M$, such that if $H$ decides that $M(I)$ halts, then $Q$ doesn't halt, and vic... |
Does this proof for the undecidability of the halting problem violate the axiom of regularity ? |
(Background for other readers: the derivation of the solution can be found
[on Google books](http://books.google.com/books?id=eHANhZwVouYC&pg=PA89),
see equations (6.18) and (6.19).)
This is a good and very natural question, but your reasoning is flawed.
The solution formula that you quote for $T(t)$ is only va... |
Take projective real space $\mathbb P_n (\mathbb R)$ of ODD dimension. It is easy to proof that all his Stiefel-Whitney numbers are zero . So according Thom theorem
there must exists manifold $M$ with boundary such that boundary is
$\partial M= \mathbb P_n (\mathbb R)$. I should like to see directly such $M$, witho... |
For what manifold is boundary given odd-dimensional projective space? |
For what value of $\alpha$ is $W_t^{\alpha}$ a martingale where $\alpha \in \mathbb{R}$? ($W_t$ standard Brownian motion)
I can prove for $\alpha = 0$ and $\alpha = 1$, it is a martingale. (rather it is obvious). But my question is:
How to prove that $W_t^{\alpha}$ is not a martingale for $\alpha \in \mathbb{R}\c... |
For what value of $\alpha$ is $W_t^{\alpha}$ a martingale? |
In exercise 36 *Miscellaneous Taylor Coefficients using Bernoulli
numbers* on pages 88-89 of Louis Comtet's *[Advanced Combinatorics][1]*, 1974,
one is asked to obtain the following explicit formula for the Bernoulli numbers:
$B_{2n}=(-1)^{n-1}\dfrac{1+\left[ \varphi _{n}\right] }{2(2^{2n}-1)}$,
where
$\va... |
Let $S$ be a sheaf over $X$ and $r$ an element in $S_x$ for some $x$ in $X$. Must there exist a section $s$ in $S(X)$ such that such that $s$ equals $r$ when mapped to $S_x$ by the canonical map? |
Can every element in the stalk be represented by a section in the top space? |
For what value of $\alpha$ is $W_t^{\alpha}$ a martingale where $\alpha \in \mathbb{Z}$? ($W_t$ standard Brownian motion)
I can prove for $\alpha = 0$ and $\alpha = 1$, it is a martingale. (rather it is obvious). But my question is:
How to prove that $W_t^{\alpha}$ is not a martingale for $\alpha \in \mathbb{Z}\c... |
What are some good ways to motivate the material on infinite series that appears at the end of a typical American Calculus II course?
My students in this course are generally from biochemistry, computer science, economics, business, and physics (with a few humanities folks taking the course for fun) - not just math ... |
This is essentially Raphael's answer expanded, and taking into account what I was (probably unsuccessfully) trying to communicate in the comments. As I recall, the reason the Universal Machine is important is that it is through the UTM that we can actually explain how the "construction" of the modified machines come ab... |
What is the largest prime less than 2^31? |
Say we have the function $$G(z) = \sum_{n \geq 0} g_n z^n.$$
Is there a name for the transform T defined so that $$(T(G))(z) = \sum_{n \geq 0} g_n z^{n^2}?$$
Is there anything known about this kind of transformation? |
What is known about the transformation of a power series in which $z^n$ is replaced with $z^{n^2}$? |
what could this notation mean? |
Did you check page 34 (which I got by checking the List of Symbols on page 506)?
http://books.google.com/books?id=seH8dMrEgggC&pg=PA34
![alt text][1]
[1]: https://i.stack.imgur.com/QlpEf.png |
This is essentially Raphael's answer expanded, and taking into account what I was (probably unsuccessfully) trying to communicate in the comments. As I recall, the reason the Universal Machine is important is that it is through the UTM that we can actually explain how the "construction" of the modified machines come ab... |
If I have a distribution mass function p(X), and values of $X_0,X_1,...,X_i$, I want to assign values to $X_{i+1},...,X_n$ such that X follows p(X) distribution. Is there any technique to do this? |
A farmer ties a cow with a rope to a pole at the edge of a round field of radius **r**. How long must the rope be compared to **r** if the farmer wants to insure that the cow can roam exactly half of the field?
**The point is to find a [closed-form expression][1], not solve this with numerical analysis etc. Is that ... |
Is it possible to express the aria of the intersection of 2 circles as a closed-form expression? |
If I have a distribution mass function p(Y) , and values of $X_0,X_1,...,X_i$, I want to assign values to $X_{i+1},...,X_n$ such that the series of X follows p(Y) distribution where Y is the set of all distinct values of X. Is there any technique to do this? |
I am not a math major but more of an amateur mathematician working in computational math. So the question I ask may not make complete sense.
I am comfortable with the different sizes of infinities and Cantor's "diagonal argument" to prove that the set of all subsets of an infinite set has cardinality strictly greate... |
If my understanding is correct, naive set theory needs to be restricted in order to avoid paradoxes including the Russell paradox. Typically, the restriction is expressed in terms of size. For example, the set of all sets must be excluded.
I recall that I came across a paper in Arxiv some time ago which explained th... |
First, let me clear up a misunderstanding.
Question: Is $2^\omega = \aleph_1$? More generally is $2^{\aleph_\alpha} = \aleph_{\alpha+1}$?
The answer of "yes" to the first question is known as the continuum hypothesis (CH), while an answer of "yes" to the second is known as the generalized continuum hypthothesis... |
Why is $\aleph_0$ the smallest $\infty$? |
Is it possible to express the area of the intersection of 2 circles as a closed-form expression? |
A farmer ties a cow with a rope to a pole at the edge of a round field of radius **r**. How long must the rope be compared to **r** if the farmer wants to ensure that the cow can roam exactly half of the field?
**The point is to find a [closed-form expression][1], not solve this with numerical analysis etc. Is that ... |
Are all dense open sets in R cocountable?
That is, are all dense open sets in R such that their complements are at most countable?
It would seem like they must be since the closed sets that are uncountable are all intervals, so their complements are not dense. |
This is essentially Raphael's answer expanded, and taking into account what I was (probably unsuccessfully) trying to communicate in the comments. As I recall, the reason the Universal Machine is important is that it is through the UTM that we can actually explain how the "construction" of the modified machines come ab... |
I'm very n00b in maths, and I don't understand what I can do with a generating function.
For example, if I want to compute the firstspread polynomial, how can I do that from the generating function:
![alt text][1]
or the exponential one:
![alt text][2]
?
Thanks in advance,
rubik
[1]: https://i... |
One of the answers to <a href="http://math.stackexchange.com/questions/2502/what-are-large-cardinals-for">this question</a> indicates that large cardinals are useful for destructive testing of set theory. That aside, and not assuming GCH, are there any sets known that have a cardinality of $\aleph_{n}$, $n>0$, or that ... |
Let $f:R->R$ is a function with these special properties. $f$ is continuous everywhere. $f$ is not smooth (not infinitely differentiable). $f$ is differentiable only finitely many times everywhere. $f$ belongs to $L^p$.Is such a function feasible? I need suggestions on ways to construct such a special function. |
Let $f:\mathbb{R} \to \mathbb{R}$ is a function with these special properties. $f$ is continuous everywhere. $f$ is not smooth (not infinitely differentiable). $f$ is differentiable only finitely many times everywhere. $f$ belongs to $L^p$.Is such a function feasible? I need suggestions on ways to construct such a spec... |
Let $f:\mathbb{R} \to \mathbb{R}$ is a function with these special properties. $f$ is continuous everywhere. $f$ is not smooth (not infinitely differentiable). $f$ is differentiable only finitely many times everywhere. $f$ belongs to $L^p$. For any $k$ belongs to $\mathbb{R}$ let $E$ be the set of all points where $f$ ... |
Let $f:\mathbb{R} \to \mathbb{R}$ is a function with these special properties. $f$ is continuous everywhere. $f$ is not smooth (not infinitely differentiable). $f$ is differentiable only finitely many times everywhere. $f$ belongs to $L^p$. For any $k$ belongs to $\mathbb{N}$ let $E$ be the set of all points where $f$ ... |
A friend of mine is a differential geometer, and keeps insisting that he doesn't need the axiom of choice for the things he does. I'm fairly certain that's not true, though I haven't dug into the details of Riemannian geometry (or the real analysis it's based on, or the topology, or the theory of vector spaces, etc...)... |
Where do we need the axiom of choice in Riemannian geometry? |
In a recent (and to me, very beautiful) paper, entitled "Relational EPR", [Smerlak and Rovelli][1] present a way of thinking about EPR which relies upon Rovelli's previously published work on relational quantum mechanics (at see arxiv.org/abs/quant-ph/9609002 ). In relational quantum mechanics, there is no non-local... |
Let $f:\mathbb{R} \to \mathbb{R}$ is a function with these special properties. $f$ is continuous everywhere. $f$ is not smooth (not infinitely differentiable). $f$ is differentiable only finitely many times everywhere. $f$ belongs to $L^p$. For any $k$ belongs to $\mathbb{N}$ let $E$ be the set of all points where $f$ ... |
Let $f:\mathbb{R} \to \mathbb{R}$ is a function with these special properties. $f$ is continuous everywhere. $f$ is not smooth (not infinitely differentiable). $f$ is differentiable only finitely many times everywhere. $f$ belongs to $L^p$. For any $k$ belongs to $\mathbb{N}$ let $E$ be the set of all points where $f$ ... |
Let any Amicable Pair be represented by (a, b), with a < b. In a search for Amicable Pairs, let (n, m) with n < m, be a candidate to be tested for being an amicable pair. Given n, is there way to calculate both the least upper bound and greatest lower bound for the m that would make (n, m) an Amicable Pair? |
Amicable Pair (a, b): given a, what are limits on size of b? |
This raises another question. If the restriction maps from $S(X)$ to $S(U)$
are surjective for all open sets $U$ (a sheaf with this
property is called *flasque* or *flabby*) then certainly the maps
from $S(X)$ to each stalk $S_x$ are surjective. So are there sheaves
$S$ with the property that each map $S(X)\to S_x$... |
When operating with sigma algebras, what does it mean when we talk about a countably infinite set? And, what are then closed countably infinite interesections? |
What's a closed countably infinite intersection? |
The thing is, since I do a job, and work on mathematics in my spare time, I do not have any connection with the academia. Secondly, the internet is the only way I can interact and reach out to other mathematicians or math enthusiasts like me.
If I need to discuss my paper with someone, where do I do so over the inte... |
I've got a solution to a economy class calculation, but I can't figure out how they came to the answer. Could you help me?
-5p + 80 = 8p +-50
.......? = ......?
120 = 12p
p = 10
Extra: do you know which rule they are using to get from the first line to the 3d?
Many thx
Math beginner,
Frank |
Suppose I have a set $\mathcal{S}$ of $N$ distinct items. Now consider the set $\mathcal{P}$ of all possible pairs that I can draw from $S$. Naturally, $|\mathcal{P}| = \binom{N}{2}$. Now when I draw $k$ items (pairs) from $\mathcal{P}$ with a uniform distribution, what is the expected number of distinct items from $S$... |
How to calculate the expected number of distinct items when drawing pairs? |
This is a follow-up to the question [Undecidability in Conway’s Game of Life][1] I posted at mathoverflow.
For some cellular automata it can be proven that they can simulate a Turing machine, normally by explicit construction. For some of them it can be proven that they can *not* simulate a Turing machine.
> Are ... |
We claim that $f(a_n) \to \xi = f(r) = f(1-r)$. If $r = \tfrac{1}{2}$ then $(a_n)$ must be convergent (why?). Since $f$ is continuous $f(a_n)$ converges too.
So assume that $r \neq \tfrac{1}{2}$. WLOG we can assume that $r \lt \tfrac{1}{2}$. Consider the subsequence $(a_{n_k})$ consisting of the elements of $(a_n)$ ... |
In exercise 36 *Miscellaneous Taylor Coefficients using Bernoulli
numbers* on pages 88-89 of Louis Comtet's *[Advanced Combinatorics][1]*, 1974,
one is asked to obtain the following explicit formula for the Bernoulli numbers:
$B_{2n}=(-1)^{n-1}\dfrac{1+\left[ \varphi _{n}\right] }{2(2^{2n}-1)}$,
where
$\va... |
How do I show that a monomorphism $F \rightarrow G$ of sheaves induces an injection on stalks?
When showing that monomorphism is an injection on sets one uses the maps $x \mapsto a$ and $x \mapsto b$ where $a,b$ are some fix elements such that they have the same image under the monomorphism. Then by using the monom... |
This holds for *any* irreducible finite state Markov chain. I'm not sure about a pure combinatorial proof, but it can be shown quite quickly.
Let $M(t)$ be the expected time the chain takes to return to its initial state, if started at $t$. As it spends, on average, a time $M(t)-1$ outside the state $t$ after each v... |
This holds for *any* irreducible finite state Markov chain. I'm not sure about a pure combinatorial proof, but it can be shown quite quickly.
Let $M(t)$ be the expected time the chain takes to return to its initial state, if started at $t$. As it spends, on average, a time $M(t)-1$ outside the state $t$ after each v... |
How to prove that finite additivity follows from countable additivity!!? |
I've written Haskell function, that helps me to compute $x^{-1} \mod m$.
invM x m = invMhelp x 0 m; --multiplicative inverse x^-1 (mod m) = inv x m
invMhelp x i m = if (x*i `mod` m == 1) then i else (invMhelp x (i+1) m);
For example:
Main> 13 `invM` 2109
649
That means $13^{-1} = 649 \mo... |
How do I compute x^(-2) mod m? |
How to solve this triangle? |
I've written Haskell function, that helps me to compute $x^{-1} \mod m$.
invM x m = invMhelp x 0 m; --modular multiplicative inverse x^-1 (mod m) = inv x m
invMhelp x i m = if (x*i `mod` m == 1) then i else (invMhelp x (i+1) m);
For example:
Main> 13 `invM` 2109
649
That means $13^{-1} =... |
> **Possible Duplicate:**
> [The cow in the field problem (intersecting circular areas)](http://math.stackexchange.com/questions/51/the-cow-in-the-field-problem-intersecting-circular-areas)
<!-- End of automatically inserted text -->
A farmer ties a cow with a rope to a pole at the edge of a round field of radius... |
Can somebody please explain continuity and discontinuity in fourier series? |
Continuity and discontinuity in fourier series? |
I watched the excellent interview with Richard Feynman: http://www.youtube.com/watch?v=PsgBtOVzHKI In the interview Feynman mention that he at young age re-invented operator calculus.
I have searched for "operator calculus" and have not found any accessible references that introduce the topic. Maybe operator calculu... |
What is operator calculus? |
If $Y$ is a Hausdorff topological space, then the value of a continuous function $f\colon X\to Y$ is completely determined by the value of $f$ on a dense subset $E$ of $X$; to see this, suppose $f$ and $g$ are two functions that agree on a dense subset $E$ of $X$, and let $u\in X\setminus E$. If $f(u)\neq g(u)$, then t... |
Let $g(t)$ be a real function of a real variable such that
$$\sup |p(t)g(t)|\lt \infty$$ for any polynomial $p$. Is it true that for any nonngetive
integer $k$, one has $$(\frac{|x|}{1+|y|})^k |g(x-y)|\lt \infty$$
for all real numbers $x,y$ ?
|
Hint 1: What kind of function must $g(t)$, satisfying the first inequality, be? Can you think of some simple such function yourself?
Hint 2: What can you say about the function $y \mapsto \left({1 \over 1 + |y|}\right)^k$? Can it spoil the second inequality?
Hint 3: Let $x = y + t$. Triangle inequality is your fr... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.