instruction stringlengths 12 30k |
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I need to prove or disprove whether non-singular matrices commute. Can you help me and tell me how? WHat should i use cos i really don't know, i am beginner in this.
Thanks in advance.
|
According to the definition of stochastic process in [Wikipedia][1]:
> Given a probability space $(\Omega, \mathcal{F}, P)$, a stochastic process (or random process) with state space $X$ is a collection of $X$-valued random variables indexed by a set $T$ ("time"). That is, a stochastic process $F$ is a collection
>... |
Is there a general formula to determine the probability of unbounded, cumulative dice rolls hitting a specified number?
For Example, with a D6 and 14:
5 + 2 + 3 + 4 = 14 : success
1 + 1 + 1 + 6 + 5 + 4 = 17 : failure |
To better understand closed forms, you may want to familiarize yourself with what's called Differential Algebra. Just as number theory relies on abstract structures such as rings, fields, ideals, etc to express roots algebraic equations using elementary numbers, similarly there is a parallel apparatus for expressing... |
Reading the Wikipedia page for [Random Walk][1], I was wondering what is the definition for a general random walk as a random process, so that all the concepts such as [random walk on $\mathbb{Z}^d$][2], [Gaussian random walk][3] and [random walk on a graph][4] can fall into this general definition?
(1) In Ross's ... |
According to the definition of stochastic process in [Wikipedia][1]:
> Given a probability space $(\Omega, \mathcal{F}, P)$, a stochastic process (or random process) with state space $X$ is a collection of $X$-valued random variables indexed by a set $T$ ("time"). That is, a stochastic process $F$ is a collection
>... |
Could someone provide, please, a proof of the theorem below?
"Being **x** and **b** integers greater than 1, which can not be represented as powers of the same basis(positive integer) and **integer exponent**, then the logarithm of **x**, in base **b**, is an irrational number."
Thank you!
Paulo
|
I need to prove or disprove whether non-singular matrices commute. Can you help me and tell me how? What should i use cos i really don't know, I am beginner in this.
Thanks in advance.
|
Could someone provide, please, a proof of the theorem below?
"Being **x** and **b** integers greater than 1, which can not be represented as powers of the same basis(positive integer) and **integer exponent**, then the logarithm of **x**, in base **b**, is an irrational number."
Well .. I know I can write **x^q*... |
Could someone provide, please, a proof of the theorem below?
"Being **x** and **b** integers greater than 1, which can not be represented as powers of the same basis(positive integer) and **integer exponent**, then the logarithm of **x**, in base **b**, is an irrational number."
Well... Assuming that the logarit... |
I'm having trouble with the following exercise in do Carmo's Riemannian geometry.
Let $X$ and $Y$ be differentiable vector fields on a Riemannian manifold $M$. Let $p \in M$ and let $c: I \to M$ be an integral curve of $X$ through $p$, i.e. $c(t_0) = p$ and $\frac{dc}{dt} = X(c(t))$.
Prove that the Riemannian conn... |
The table of integrals says that
$\int \frac{dx}{a^{2}+x^{2}}=\frac{1}{a}\arctan\frac{x}{a}+C$
What's wrong with my proof?
$
\begin{align*}
y &= \arctan\frac{x}{a} \newline
a\tan y&=x \newline
\tan y &= \frac{x}{a} \newline
\frac{d}{dx} \tan y &= \frac{dy}{dx} \frac{x}{a} \newline
\sec^{2}y \frac{dy}{dx} &... |
The table of integrals says that
$\int \frac{dx}{a^{2}+x^{2}}=\frac{1}{a}\arctan\frac{x}{a}+C$
What's wrong with my proof?
$$
\begin{align*}
y &= \arctan\frac{x}{a} \\\
a\tan y&=x \\\
\tan y &= \frac{x}{a} \\\
\frac{d}{dx} \tan y &= \frac{d}{dx} \frac{x}{a} \\\
\sec^{2}y \frac{dy}{dx} &= \frac{1}{a} \\\
\... |
Other than having unnecessary steps (why did you move $a$ to the left hand side in the second step, only to bring it back to the right hand side in the next step?), and missing the constant of integration on the left hand side in the last equation (as well as the missing "$dx$" which I fixed), nothing wrong so far. Now... |
I would say it very much depends on the context, and what tools are at your disposal. For instance, telling a student who's just mastered the usual tricks of integrating elementary functions that
$$\int\frac{\exp{u}-1}{u}\mathrm{d}u$$
and
$$\int\sqrt{(u+1)(u^2+1)}\mathrm{d}u$$
have no closed form solutions ... |
What are good resources for learning predicate logic / predicate calculus? |
> ON a small island there are 25
> inhabitants. One of these inhabitants,
> named Jack, starts a rumor which
> spreads around the isle. Any person
> who hears the rumor continues
> spreading it until he or she meets
> someone who has heard the story
> before. At that point, the person
> stops spreading it, sinc... |
This is a cultural comment, rather than an answer; but it is a bit long for the comment box, which is why I am writing it here.
My comment is the following: giving a connection, and giving the associated parallel transport, are essentially the same thing, and so it should be possible to establish the formula you are... |
> On a small island there are 25
> inhabitants. One of these inhabitants,
> named Jack, starts a rumor which
> spreads around the isle. Any person
> who hears the rumor continues
> spreading it until he or she meets
> someone who has heard the story
> before. At that point, the person
> stops spreading it, sinc... |
I see people like Terry Tao and others take extensive notes. But is this really necessary? When I do this I feel like I am rewriting a textbook. |
What are examples of mathematicians who dont take many notes? |
I've read a few articles explaining the way to use the Minkowski reduced basis in a lattice in order to measure the uniformity of the output of a random number generator. However, I can't prove a conclusion drawn from the following statement.
Given a set of linearly independent vectors $\\{v_{1}, v_{2}, ... ,v_{d}\\... |
I've read a few articles explaining the way to use the Minkowski reduced basis in a lattice in order to measure the uniformity of the output of a random number generator. However, I can't prove a conclusion drawn from the following statement.
Given a set of linearly independent vectors $\\{v_{1}, v_{2}, ... ,v_{d}\\... |
After thinking about it for a while and consulting other students, noone seems to be able to find an example of the following.
Given the PDE
$\dfrac{\partial f}{\partial x} = 0 \quad $ on $U = \{ (x,y) \in \mathbb R^2 ; x>0, 1 < x^2 + y^2 < 4\}$
I am looking for a solution $f$ that does not only depend o... |
After thinking about it for a while and consulting other students, no one seems to be able to find an example of the following:
Given the PDE
$\dfrac{\partial f}{\partial x} = 0 \quad $ on $U = \{ (x,y) \in \mathbb R^2 ; x>0, 1 < x^2 + y^2 < 4\}$
I am looking for a solution $f$ that does not only depend ... |
Imagine that we have something like the rubik's cube on $U=I^{3}=[0,1]^{3}$, and that we say that all the points on the xy, yz, and zx faces are black, and that all the points on the other three faces are white.
Define a legal move to be a rotation by an integral multiple of $\frac{\pi}{4}$ around one of the lines $... |
Imagine that we have something like the rubik's cube on $U=I^{3}=[0,1]^{3}$, and that we say that all the points on the xy, yz, and zx faces are black, and that all the points on the other three faces are white.
Define a legal move to be a rotation by an integral multiple of $\frac{\pi}{4}$ around one of the lines g... |
Imagine that we have something like the rubik's cube on $U=I^{3}=[0,1]^{3}$, and that we say that all the points on the xy, yz, and zx faces are black, and that all the points on the other three faces are white.
Define a legal move to be a rotation by an integral multiple of $\frac{\pi}{4}$ around one of the lines g... |
Imagine that we have something like the rubik's cube on $U=I^{3}=[0,1]^{3}$, and that we say that all the points on the xy, yz, and zx faces are black, and that all the points on the other three faces are white.
Define a legal move to be a rotation by an integral multiple of $\frac{\pi}{4}$ around one of the lines g... |
Could someone provide, please, a proof of the theorem below?
"Being **x** and **b** integers greater than 1, which can not be represented as powers of the same basis(positive integer) and **integer exponent**, then the logarithm of **x**, in base **b**, is an irrational number."
Well... Assuming that the logarit... |
Could someone provide, please, a proof of the theorem below?
"Being **x** and **b** integers greater than 1, which can not be represented as powers of the same basis(positive integer) and **integer exponent**, then the logarithm of **x**, in base **b**, is an irrational number."
Well... Assuming that the logarit... |
How can I render the wedge sum of $n$ topological spaces (or other quite strange topological operations) in a "product-like" manner? What I'm looking for is writing, for example
$$
{\Large{*}}_{i=1}^n G_i
$$
for the free product of n groups. How can I render this with the # operation in topological spaces? (Yes, I'... |
Prove that, where a,b,...,e are real numbers and a≠0,
if ax + by = c
has the same solution set as
ax+dy=e
then they are the same equation. What if a=0?
Note: If a≠0 then the solution set of the first equation is {(x,y)|x=c-by/a}
|
What a great question!
Here is my solution. I am aiming for a black circle on a
white face (where the boundary edge points are invisible,
as mentioned in the comments).
Pick one of the white faces to be the desired final face F.
Now, consider any given horizontal row on that face, which
is all white. These ro... |
How to solve this problem? |
What a great question!
Here is my solution. I am aiming for a black circle on a
white face (where the boundary edge points are invisible,
as mentioned in the comments).
Pick one of the white faces to be the desired final face F.
Now, consider any given horizontal row on that face, which
is all white. We want ... |
I have a simple formula for the n-th derivative of some expression. I'm suppose to prove that it's true for all N, by using induction. I set n=1, and show that the base case works.
The next step, were I assume that it works a K is fine. But how is that shown on paper? And the that I don't understand at all, is how ... |
Induction, not what, but how it's done? |
What a great question!
Here is my solution. I am aiming for a black circle inscribed on a
white face (where the boundary edge points are invisible,
as mentioned in the comments).
Pick one of the white faces to be the desired final face F.
Now, consider any given horizontal row on that face, which
is all white... |
How can I render the *connected* sum of $n$ topological spaces (or other quite strange topological operations) in a "product-like" manner? What I'm looking for is writing, for example
$$
{\Large{*}}_{i=1}^n G_i
$$
for the free product of n groups. How can I render this with the # operation in topological spaces? (Y... |
I understand that we use the chain rule to differentiate the y part of it. Like if it stood x*y, I would use the product rule, that say that the derivative of y is dy/dx. But what if it's just y alone or with a constant in front off it?
Thank you! |
If I have a quadratic form like $y^2 - x^2 - x = k$ none of the techniques I know work because of the nasty $x$. Note that homogenizing doesn't work because a solution of $Y^2 - X^2 - X Z = k Z^(2)$ does not lead to a solution of the original equation in integers, at least as far as I have been able to determine.
Ho... |
How to solve inhomogeneous quadratic forms in integers? |
If I have a quadratic form like $y^2 - x^2 - x = k$ none of the techniques I know work because of the nasty $x$. Note that homogenizing doesn't work because a solution of $Y^2 - X^2 - X Z = k Z^{(2)}$ does not lead to a solution of the original equation in integers, at least as far as I have been able to determine.
... |
After thinking about it for a while and consulting other students, no one seems to be able to find an example of the following:
Given the PDE
$\dfrac{\partial f}{\partial x} = 0 \quad $ on $U = \{ (x,y) \in \mathbb R^2 ; y>0, 1 < x^2 + y^2 < 4\}$
I am looking for a solution $f$ that does not only depend ... |
This is motivated by [this question][1] and the fact that I have no access to the Timothy Chow's paper [What Is a Closed-Form Number?][2] indicated there by
Qiaochu Yuan.
If an equation $f(x)=0$ has no closed form solution, what does it normally
mean? Added: $f$ may depend (and normally does) on parameters.
To... |
I need to find an isomorphism between [0,1] to (0,1) can you help me with this please?
thanks. benny
|
What do you mean by "isomorphism"? Are you considering these sets with a given group or ring structure (if then, which one?), or as topological spaces (in which case you'd be looking for a "homeomorphism"), or just as sets?
I don't know of a canonical group structure on either set, but if you want your map between t... |
I need to find an isomorphism between [0,1] to (0,1) can you help me with this please?
thanks. benny
|
I have $X_1,...$ Bernoulli variables with probability $p$ of success.
I want to get an $n$ such that with probability $\delta$
$P(\sum_{i=1}^n X_i \ge k) \ge \delta$
This $n$ would of course depend on $k$, $p$ and $\delta$. Can I get an upper bound for that $n(k,\delta,p)$? (meaning, I want some $n(k,\delta,p)... |
can I get aupper bound on the tail of a binomial variable? |
Let $k$ be an algebraic closed field. Let $x$ be a point in $X=P_k^1$. What is $O_{X,x}$?
For example, if I have $x=(t-a)\in \text{Spec }k[t]$. Looking $x$ inside $P_k^1$, does $O_{X,x}=k[t]_{(t-a)}$? I'm confused when I have to deal with the sheaf of rings. |
can I get an upper bound on the tail of a binomial variable? |
I don't think rewriting this way helps. In the first place, how would you show that if $y$ is the optimal solution of $\min_y f(y)$, then $x = 2^y$ is the optimal solution to $\min_x f(\log x)$? |
The maps $(0,1) \to [0,1], x \mapsto x$ and $[0,1] \to (0,1), x \mapsto 1/4 + x/2$ are injective. The theorem of Cantor-Schröder-Bernstein implies that there is a bijection. It is actually possible to work out the proof of the theorem in these simple examples and get an explicit bijection, namely
$g : [0,1] \to (0,1... |
How to prove
$\int_{0}^{+\infty} e^{-x^2} dx = \frac{\sqrt \pi}{2}$
? |
Let $R\subset A\times A$ a binary relation, and note $a\sim b$ for $(a,b)\in R$. We define
$$(a,b)\in R\circ R\Leftrightarrow \exists b\in A : a\sim b, b\sim c$$
and $R^n=R\circ\dots\circ R$.
My books says that $(a,b)\in R^n$ iff there is a path of length $n$ in the graph of $R$. Isn't it false ?
For exam... |
Say I have a symmetric matrix. I have the concept of norm as defined on wikipedia. Now I want to prove (disprove?) that the norm of a symmetric matrix is maximum absolute value of its eigenvalue. I would really appreciate if this can be done only using simple concepts of linear algebra.
I am quite new to mathematics... |
Norm of a symmetric matrix? |
This is a cultural comment, rather than an answer; but it is a bit long for the comment box, which is why I am writing it here.
My comment is the following: giving a connection, and giving the associated parallel transport, are essentially the same thing, and so it should be possible to establish the formula you are... |
Say I have a binomial coefficient $y=\binom{5n+3}{n+2}$ or $y=\binom{n^2+4}{3n}$ something of the sorts in terms of the variable $n$. How can I determine $O(y)$?
Is there a general method for this sort of thing. I have only ever seen $O$-notation in a computer science setting and its treatment was not concerned wit... |
How can I determine asymptotic growth of binomial coefficients? |
By the classification of finite abelian groups, any abelian group $G$ can be written as $Z_{n_1} \oplus \cdots \oplus Z_{n_k}$ where the $n_i$ are prime powers. For odd abelian groups, none of the $n_i$ are powers of two. Write each element of the group $G$ as a $k$-tuple $(m_1,\ldots, m_k)$ where $0 \le m_i < n_i$. ... |
We were given a challenge by our calculus professor and I've been stuck on it for a while now.
Show that the set of subsequence limits of An=sin(n) is [-1, 1] (another way to phrase this would be: ∀r∈[-1,1] there exists a subsequence of An=sin(n) that converges to r).
What would be a good way to start? |
It's a good question. The naive answer is that the two concepts are different.
Random walks on ${\mathbb Z}^d$ or ${\mathbb R}$, etc. are examples of a random walk on an additive group $G$. There is a fixed distribution $\mu$ on $G$, and at each time you randomly select an element of the group (using distribution $... |
I'm reading page number 4 here. In particular the section where it deals with the case $p=\infty$, that is , showing that $L^{\infty}$ is complete.
http://www.core.org.cn/NR/rdonlyres/Mathematics/18-125Fall2003/5E3917E2-C212-463B-9EDB-671486133388/0/18125_lec15.pdf
Two questions:
1) Why is the convergence unif... |
The key is that we are working in $N^c$, where we have, morally speaking, defined $N$ to be the set of points where things go wrong. In detail, $N$ is the set of points $x$ where either 1) $f(x)$ is bigger than the limsup $\|f\|$, or 2) the series $(f_n)$ is not Cauchy at $x$.
Outside of $N$, the terms $f_n$ are bou... |
Suppose that $S_1$ and $S_2$ are the vanishing sets of a system of polynomial equations in n variables over a field $\mathbb{k}$ (ideal in $\mathbb{k}[X_1,\dots,X_n]$) and a system of polynomial equations in m variables over a field $\mathbb{k}$ (ideal in $\mathbb{k}[Y_1,\dots,Y_m]$). We can give $S_1$ and $S_2$ the Za... |
Are there Zariski-continuous maps between algebraic sets that are not polynomial maps? |
I've been trying to google the answer to this question, but have had no luck, so I'm asking this question here. Let's say the origin is at (0, 0), body 1 with mass m1 is at (-r1, 0), and body 2 with mass m2 is at (r2, 0). How can I calculate the initial velocities v1 and v2 for the bodies so that they trace circular or... |
I was wondering if equilibrium distribution, steady-state distribution, stationary distribution and limiting distribution mean the same, or there are differences between?
I learn them in the context of Discrete-time Markov Chain as far as I know. Or they also appear in other situations of stochastic processes and pr... |
We were given a challenge by our calculus professor and I've been stuck on it for a while now.
Show that the set of subsequence limits of $A_n=\sin(n)$ is $[-1, 1]$ (another way to phrase this would be: $\forall r\in [-1,1]$ there exists a subsequence of $A_n=\sin(n)$ that converges to $r$).
What would be a good... |
If I understand your question correctly, then (1) yes, there are going to be lots of Zariski continuous maps that are not polynomial (although, if one asks that all higher products of the map preserve the Zariski topology on the corresponding sources and targets, this is a much stronger restriction, I think --- does an... |
Silly question, but have you read the Wikipedia page on the two-body problem (http://en.wikipedia.org/wiki/Two-body_problem)? If I understand your problem correctly, then it seems that it is fairly well treated on that page.
You say you want to determine the set of initial vectors $v_1$ and $v_2$ such that the two b... |
How to proof limsup(A U B) = limsup(A) U limsup(B)? Thanks! |
How to prove $\limsup(A \cup B) = \limsup(A) \cup \limsup(B)$? Thanks! |
Say I have the following numbers:
10
10
9
5
4
4
4
1
How do i go about calculating the percentile for each score? Is there a standard formula for figuring this out? |
How do you calculate percentile? |
I am trying to reproduce the following bound:
$\sum_{1\leq m\leq x, m\neq n}\frac{1}{|\log(m/n)|}=O(x\log(x))$,
for $x\geq 2$ and some fixed $n$, $1\leq n\leq x$ (the implicit constant shouldn't depend on $n$).
I have tried to bound each term by $\frac{1}{|\log(1\pm 1/x)|}$, but this doesn't seem to be good ... |
Say I have a binomial coefficient $y=\binom{5n+3}{n+2}$ or $y=\binom{n^2+4}{3n}$ something of the sorts in terms of the variable $n$. How can I determine $f$ so that $y = O(f)$?
Is there a general method for this sort of thing. I have only ever seen $O$-notation in a computer science setting and its treatment was n... |
I am trying to reproduce the following bound:
$\sum_{1\leq m\leq x, m\neq n}\frac{1}{|\log(m/n)|}=O(x\log(x))$,
for $x\geq 2$ and some $n$, $1\leq n\leq x$ (the implicit constant shouldn't depend on $n$).
I have tried to bound each term by $\frac{1}{|\log(1\pm 1/x)|}$, but this doesn't seem to be good enough.... |
Can saoeone simply explain to me how to calculate linear integrate linke below?
\int_{L} 5y dL
Where L is line segment from (0;0) to (0,2;0,2). |
Can saoeone simply explain to me how to calculate linear integrate linke below?
$$\int_{L} 5y \mathrm{d}L$$
Where L is line segment from (0;0) to (0,2;0,2). |
Can someone simply explain to me how to calculate linear integrate linke below?
$$\int_{L} 5y \mathrm{d}L$$
Where L is line segment from (0;0) to (0,2;0,2). |
Is there some increasing function $f(n)$ that grows slower than $n^{c}$ for some $c > 1$ such that $\sum_{n=1}^{\infty} \frac{1}{f(n)}$ converges? |
This is a bit of a silly question, but I've been puzzled on how this can be done, so I ask here.
You are given an implicit Cartesian equation like
$$4 x^4-4 x^3+8 y^2 x^2-27 x^2+12 y^2 x+4 y^4-27 y^2+27=0$$
or
$$x^6+3 y^2 x^4-3 x^4+3 y^4 x^2+21 y^2 x^2+3 x^2+y^6-3 y^4+3 y^2-1=0$$
and are told that you ar... |
Can someone simply explain to me how to calculate linear integral linke below?
$$\int_{L} 5y \mathrm{d}L$$
Where L is line segment from (0;0) to (0,2;0,2). |
Liner integral ? |
What is special about elementary recursive arithmetic? |
Consider a random walk on an undirected, non-bipartite graph. Let $\pi$ be the stationary distribution of this process, and let the hitting time $H(s,t)$ be the expected time until a walk beginning at node $s$ reaches node $t$.
I learned from <a href="http://www.cs.yale.edu/publications/techreports/tr1029.pdf">Rand... |
Further to my understanding <a href="http://math.stackexchange.com/questions/7743/getting-better-at-proofs">What makes a proof</a>, I'm doing exercises.
The purpose of this question is to compare the proof a certain textbook provides with the one that is _logical to me_, that I made up. If my proof is weak, _why is... |
What structure is this? |
Here is a possible line of reasoning.
Say, we start with natural numbers, $\mathbb{N}$, then add $0$, making them the whole numbers $\mathbb{N}_0$, then add negative numbers, making them integers $\mathbb{Z}$, then expand them to rationals $\mathbb{Q}$, and finally to reals $\mathbb{R}$. In some respect, we keep fil... |
In a Cartesian system, I've got the slope, start point and distance of a line segment. What's the formula to find the endpoint? |
Suppose $A$ is a set of $2n$ positive integers { $a_{i}$ } with $a_{i} \geq 2$ which can be partitioned into at most $r \leq n$ coprime pairs. The following fact that seems to be true: For all $A$ and every $n \geq 1$,
\begin{eqnarray}
\prod_{i = 1}^{2n} (a_{i} - 1) \equiv 0 \mod 2^{r}.
\end{eqnarray}
For example,... |
Suppose $A$ is a set of $2n$ positive integers { $a_{i}$ } with $a_{i} \geq 2$ which can be partitioned into at most $r \leq n$ coprime pairs. The following fact that seems to be true: For all $A$ and every $n \geq 1$,
$$
\prod_{i = 1}^{2n} (a_{i} - 1) \equiv 0 \pmod{2^{r}}.
$$
For example, the set {$2,3,5,6$} can... |
Suppose $A$ is a set of $2n$ positive integers { $a_{i}$ } with $a_{i} \geq 2$ which can be partitioned into at most $r \leq n$ coprime pairs. The following fact is true: For all $A$ and every $n \geq 1$,
$$
\prod_{i = 1}^{2n} (a_{i} - 1) \equiv 0 \pmod{2^{r}}.
$$
For example, the set {$2,3,5,6$} can be partitione... |
Suppose $A$ is a set of $2n$ positive integers { $a_{i}$ } with $a_{i} \geq 2$ which can be partitioned into at most $r \leq n$ coprime pairs. The following fact is true: For all $A$ and every $n \geq 1$,
$$
\prod_{i = 1}^{2n} (a_{i} - 1) \equiv 0 \pmod{2^{r}}.
$$
For example, the set {$2,3,5,6$} can be partitione... |
In one of the programming work I am doing, I encountered a set of linear eqns. with modulo. I am putting it in simple format with only 2 variables,
(a11x + a12y) mod 8 = b1
(a21x + a22y) mod 8 = b2
In case of simple eqns. with out modulo I can use: AX=B, X=inv(A)*B. But how to handle this modulo? Any clue or p... |
how to solve linear equations involving modulo? |
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