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Are there homeomorphisms from $\mathbb{R}^{m\times n} / S_n$ into $\mathbb R^k$ where $k$ grows more slowly than $k\sim O(m^2 n)$? |
This is a one-liner if you use the fact that $\lfloor x+a \rfloor = \lfloor x\rfloor +a$ for $x \in \Bbb R$ and $a \in \Bbb Z$. Then the inequality is equivalent to
$$\frac{l}{2^{b+1}} \le \left \lfloor \frac{l+2^{b+1} - 1}{2^{b+1}} \right \rfloor$$
which is trivially true. To see why, pull out and cancel the quotien... |
In the group $\mathbb R^*$ can we find elements a and b such that $|a| = \infty$, $|b| = \infty$ and $ |ab| = 2$?
My attempt:
The group $\mathbb R^*$ means the set of real numbers with zero omitted, forms a group under multiplication.
1 is the identity.
And we can't find such a & b such that, $|a| = \infty$, $|... |
How do you solve for $3$ positive integers that satisfy the following:
> Let $r=\sqrt{\frac{\sqrt{53}}{2}+\frac{3}{2}}$. There is a unique triple of positive integers($a, b, c$) such that:
$r^{100}=2r^{98}+14r^{96}+11r^{94}-r^{50}+ar^{46}+br^{44}+cr^{40}$.
What is the value of $a^{2}+b^{2}+c^{2}$?
What
![en... |
I am trying to prove the limit of $$\lim\limits_{n\to\infty} \frac{\sin{(n\pi/4)}}{\sqrt{n+2}-1}= 0$$ I have to use the limit definition to do this, but i get stuck with the $\sin(n\pi/4)$ part. I know how to do the $1/(\sqrt{n+2}-1)$ part, any help? |
Across all [additive bases][1] $A\subset \mathbb{N}$ of $\mathbb{N}\setminus\{1\}$ of order $2$, what is the maximum possible value of the $n-$th term of $A?$
For example, across all additive bases of $\mathbb{N}\setminus\{1\},$ the maximum possible value the $1$st term of $A$ is $1,$ because $1$ must be the $1$st t... |
Across all additive bases $A$ of $\mathbb{N}\setminus\{1\}$ of order $2$, what is the maximum possible value of the $n-$th term of $A?$ |
I am trying to prove the limit of $$\lim_{n\to\infty} \frac{\sin{(n\pi/4)}}{\sqrt{n+2}-1}= 0$$ I have to use the limit definition to do this, but i get stuck with the $\sin(n\pi/4)$ part. I know how to do the $1/(\sqrt{n+2}-1)$ part, any help? |
I am writing a boid simulation in a toroidal space. The boids wrap around from the left to right edge and the top to bottom edge. To calculate what boids are visible, I need to find the minimum distance between those two boids' positions. I have been doing the following
Given: $$P_1 = (x_1, y_1), P_2 = (x_2, y_2)$$
... |
Across all [additive bases][1] $A\subset \mathbb{N}$ of $\mathbb{N}\setminus\{1\}$ of order $2$, what is the maximum possible value of the $n-$th term of $A?$
For example, across all additive bases of $\mathbb{N}\setminus\{1\},$ the maximum possible value the $1$st term of $A$ is $1,$ because $1$ must be the $1$st t... |
I am reading complex analysis by Gamelin and I am having trouble understanding the square root function.
The principal branch of $\sqrt{z}$ is $f_1(z)$ is defined as $|z|^{\frac 1 2} e^{\frac{i \operatorname{Arg}(z)}{2}}$ for $z \in \mathbb{C} - (-\infty,0]$ and $f_2(z)$ is defined as $-f_1(z)$
By this defini... |
I'll quote this from the Wikipedia on [Scott-Potter][1] set theory:
> **Reflection:** Let $\Phi$ denote a first-order formula in which any number of free variables are present. Let $\Phi^{(V)}$ denote $\Phi$ with these free variables all quantified, with the quantified variables restricted to the level $V$. Then: $\... |
Scott-Potter reflection? |
Across all [additive bases][1] $A\subset \mathbb{N}$ of $\mathbb{N}\setminus\{1\}$ of order $2$, what is the maximum possible value of the $n-$th term of $A?$
For example, across all additive bases of $\mathbb{N}\setminus\{1\},$ the maximum possible value of the $1$st term of $A$ is $1,$ because $1$ must be the $1$s... |
How to evaluate an integral involving incomplete gamma , exponential and power functions? |
I have to show the following thing:
1) Every two maps $ \varphi, \psi: \Delta_{\star}(X\times Y) \to \Delta_{\star}(X) \otimes \Delta_{\star}(Y)$ are chain homotopic (which are trivial on degree 0).
2) Every two maps $\varphi,\psi: \Delta_{\star}(X\times Y) \to \Delta_{\star}(X) \otimes \Delta_{\star}(Y)$ are chain... |
I am writing a boid simulation in a toroidal space. The boids wrap around from the left to right edge and the top to bottom edge. To calculate what boids are visible, I need to find the minimum distance between those two boids' positions. I have been doing the following
Given: $$P_1 = (x_1, y_1), P_2 = (x_2, y_2)$$
... |
If I consider $z_0 \in \mathbb{C}$ and denote $g(z)=\bar{z}$, then I can verify that:
- For $\lambda \in \mathbb{R^*}$:
$$lim_{\lambda \to 0} \frac{g(z_0+\lambda)-g(z_0)}{\lambda}=\frac{\lambda}{\lambda}=1$$
- In other hand, if we consider $w \in i\mathbb{R^*}$:
$$lim_{w \to 0} \frac{g(z_0+w)-g(z_0)}{w}=\... |
Recently i got interested at a [Relaxation of Collatz Conjecture](https://mathoverflow.net/questions/216358/relaxed-collatz-3x1-conjecture#:~:text=The%20Collatz%203x%2B1,2%20whenever%20x%20is%20even.). This goes just like the normal Collatz procedure, except when $x$ is even, you have the choice of applying either $x \... |
Given a collection of sets $\{Y_\alpha: \alpha \in A \}$ for an index set $A$. For each $\alpha \in A$, let $f_\alpha: X \to Y_\alpha$ be a function. The map $f: X \to \prod_{\alpha \in A} Y_\alpha$ defined by
$$
f(x) = \prod_{\alpha \in A} f_\alpha(x) = (f_\alpha(x))_{\alpha \in A}
$$
is called the `evaluati... |
If I have fair coin, is there any way to obtain $\frac{m}{n}$ probability with it? Is there any algorithm or sequence of actions? |
I am trying to understand the pmf $p(y|\theta_1,\dots,\theta_c)=\Pi_{k=1}^c\theta_k^{y_k}$ of the categorical distribution but I do not understand why there aren't any $1-\theta_k$ terms, like in the Bernoulli $p(x)=\theta^x(1-\theta)^{1-x}$. Is it to do with encoding - that it is the probability of being classified a... |
When trying to show this I thought of the typewriter sequence but I couldn't find a reference for a full proof, just partial statements. Here is my attempt.
We are working with the lebesgue measure on $[0,1]$. Define
$$ f_n(x) = \mathbb 1_{\left[\frac{n-2^{k}}{2^k}, \frac{n-2^{k}+1}{2^k}\right]} \text{, where } 2^... |
Given positive integers **p**, **q**, and **r**, we have $p \cdot q \cdot r$ unit cubes. Each cube has a hole drilled along one of its space diagonals. These cubes are then strung onto a very thin thread of length $p \cdot q \cdot r \cdot \sqrt{3}$, resembling beads on a string. The task is to arrange these unit cubes ... |
I read the book Measure and Integration : An introduction to Real Analysis by Richard L.Wheeden, Antoni Zygmund ,Second edition . In chapter 5, the authors give a definition of a distribution function (in page 97):
It turns out that there is a remarkably simple and useful representation of Lebesgue integrals over su... |
3. Sylow's Theorem: Let $|G| = p^n · m$ with $p$ prime, m coprime to $p$, and $n ≥ 1$. The number $α(p)$ of $p$-Sylow groups of $G$ is a divisor of $m$ and of the form $α(p) = 1+kp$ for a $k ≥ 0$.
Proof:
Let $S$ be the set of all $p$-Sylow groups of $G$. $G$ operates on this through conjugation. By Sylow's 2nd th... |
I am writing a boid simulation in a toroidal space. The boids wrap around from the left to right edge and the top to bottom edge. To calculate what boids are visible, I need to find the minimum distance between those two boids' positions. I have been doing the following
Given: $$P_1 = (x_1, y_1), P_2 = (x_2, y_2)$$
... |
I'm trying to prove $\mathbb Z [\sqrt{14}]$ is not euclidean regarding the Norm $N(a+b\sqrt{14})=|a^2-14b^2|$ using $2$ and $1+\sqrt{14}$.
I know how to prove it for the Gaussian integer case and I know that I have to find an Ideal in $\mathbb Z [\sqrt{14}]$ that is not principal using $2$ and $1+\sqrt{14}$ but I'm st... |
Let
$$a_n - 3a_{n-1} + 2a_{n-2} = 2n - 1$$
$$a_0 = a_1 = 0$$
Be a recurrence relation. To solve it, we need to find homogenous and particular solution. I found out that the homogenous solution is $b_1 + b_2 \cdot 2^n$ for some constants $b_1, b_2$.
To find the particular solution, I say that $a_n ^p = cn + d$ ... |
Across all [additive bases][1] $A\subset \mathbb{N}$ of $\mathbb{N}\setminus\{1\}$ of order $2$, what is the maximum possible value of the $n-$th term of $A?$
For example, across all additive bases of $\mathbb{N}\setminus\{1\},$ the maximum possible value of the $1$st term of $A$ is $1,$ because $1$ must be the $1$s... |
I read the book Measure and Integration : An introduction to Real Analysis by Richard L.Wheeden, Antoni Zygmund ,Second edition . In chapter 5, the authors give a definition of a distribution function (in page 97):
It turns out that there is a remarkably simple and useful representation of Lebesgue integrals over su... |
Is there any examples of the martingale $X(t)$ which is bounded in $L^{1}$ but not uniformly integrable?
I've just known that $$M_t=\mathrm{e}^{aW_t-a^2t/2}$$ with $a$ not $0$ and $W$ a Brownian Motion is a martingale but not uniformly integrable.
Is there any other examples which are more concrete and more cons... |
Across all [additive bases][1] $A\subset \mathbb{N}$ of $\mathbb{N}\setminus\{1\}$ of order $2$, what is the maximum possible value of the $n-$th term of $A?$
For example, across all additive bases of $\mathbb{N}\setminus\{1\},$ the maximum possible value of the $1$st term of $A$ is $1,$ because $1$ must be the $1$s... |
We are dealing with reals, so first lets answer the following question: what are possible orders of elements of $\mathbb{R}^*$? In order to avoid confusion with absolute value, I will use $o(x)$ to denote "order of $x$".
First, if $x>1$ then we can multiply both sides by $x$ to get $x^2>x$. And then by simple induct... |
We are dealing with reals, so first lets answer the following question: what are possible orders of elements of $\mathbb{R}^*$? In order to avoid confusion with absolute value, I will use $o(x)$ to denote "order of $x$".
First, if $x>1$ then we can multiply both sides by $x$ to get $x^2>x$. And then by simple induct... |
We are dealing with reals, so first lets answer the following question: what are possible orders of elements of $\mathbb{R}^*$? In order to avoid confusion with absolute value, I will use $o(x)$ to denote "order of $x$".
First, if $x>1$ then we can multiply both sides by $x$ to get $x^2>x$. And then by simple induct... |
The question :
What is the total number of words of length $500$ on $\{a,b\}$ such that the letter "$a$" appears more than "$b$"?
$(*)$ We know that the total number of words is $ 2^{500} $. At first glance it looks like half of them. But there are subsets of words which contain the same number of letters of "$a$... |
What is the total number of words of length $500$ on $\{a,b\}$ such that the letter $"a"$ appears more than $"b"$ ( without Brute force)? |
Problem :
Let :
$$\frac{\Gamma(x+1)}{(x+x^2)}=f(x),x>0$$
Then let $A=\sqrt{\frac{\pi}{4}}-1$ And $y$ be the global minimum over $x\in (0,\infty)$ of $f(x)$ then it seems we have :
$$2+A-A^4-30A^6-103A^8-76A^{10}-23A^{12}-22A^{14}-3A^{16}-\cdots=y\tag{I}$$
I cannot pursue it because of the floating point ... |
>I need maximum of $$ \frac{x^{15}(1-x)y^{15}(1-y)}{(1-xy)^{15}}$$ where $0<x<1$, $0<y<1$
Define $$ f(x,y)=\frac{x^{15}(1-x)y^{15}(1-y)}{(1-xy)^{15}}$$
So the function is positive for $0<x<1$, $0<y<1$ and vanishes on the boundary $x=0$ or $x=1$ or $y=0$ or $y=1$. Therefore it attains the maximal value inside the r... |
Let $f: X \to \mathbb{R}$ be a continuous function over a compact metric space $X$. Assume that $\mu$ and $\nu$ are two Borel probability measures on $X$.
Suppose that $\mu << \nu$. Is it true that $$f_{\ast}\mu << f_{\ast} \nu?$$
We recall $f_{\ast}\mu(A)=\mu(f^{-1} (A)).$
If measures are invariant, then ... |
https://twitter.com/martinmbauer/status/1763622278128947464?t=YObGhK4ZqjAXrwPaHxB8gw&s=19
> Many know Euler's identity, but did you also know that
>
> $|e^{i\pi}| = |\pi^{ie}| = |i^{\pi e}| = 1$
While the claim about $\pi$'s powers feels intriguing to me (for I don't know any better), the claim about $i$'s pow... |
I saw this problem:
> Find $\lim\limits_{n \to \infty }\prod\limits_{r=1}^n \frac{n^2-r}{n^2+r} $
I tried to prove this problem and got:
$$\ln(1+x) = \sum_{k=1}^ \infty \frac{(-1)^{k+1} x^k}{ k} \ \ \ \ \text{ with radius of convergence = 1 }$$
$$\ln(1-x)=- \sum_{k=1}^ \infty \frac{ x^k}{ k} \ \ \ \ \text{ wi... |
I have a question. I’m solving a problem;
Let <a_n>, <b_n> be two sequence in R. b_k=sum(a_n) from n=2^k-1 to 2(2^k-1) for each k in N. Show that if sigma(a_n)=1, then lim a_n=0 and sigma(b_n)=1. Dose the converse hold?
I solved a direction(=>). But I don’t know the converse is true or false.
My Attempt:
... |
Analysis(convergence of series) If partial sum has a convergent subsequence with lim a_n=0, then series converges? |
I saw this problem:
> Find $\lim\limits_{n \to \infty }\prod\limits_{r=1}^n \frac{n^2-r}{n^2+r} $
I tried to prove this problem and got:
$$\ln(1+x) = \sum_{k=1}^ \infty \frac{(-1)^{k+1} x^k}{ k} \ \ \ \ \text{ with radius of convergence = 1 }$$
$$\ln(1-x)=- \sum_{k=1}^ \infty \frac{ x^k}{ k} \ \ \ \ \text{ wi... |
Let $\mathcal{S}_f:\mathcal{C^\infty\rightarrow C^\infty}$ be an integral transform such that, for any $\{f,\psi\}\subseteq\mathcal{C}^\infty$, $\int_0^\infty f(x)dx=\infty$, we have that:
$$\mathcal{S}_f\{f\cdot\psi\}(t)=\int_0^\infty f(x)\psi(x)K_f(x,t)dx$$
Where $K_f$ is the kernel defined by:
$$K_f(x,t)=\exp\lef... |
What is the inverse of an integral transformation that turns second order ODEs into first order ones? |
The truth table of $p\implies q$ is as follows:
$$\begin{array}{c|c|c}
\boldsymbol{p} & \boldsymbol{q} & \boldsymbol{p\implies q} \\ \hline
\color{green}{\text{T}} & \color{green}{\text{T}} & \color{green}{\text{T}} \\ \hline
\color{green}{\text{T}} & \color{red}{\text{F}} & \color{red}{\text{F}} \\ \hline
\colo... |
What is the motivation behind the definition of the truth table of $p\implies q$? |
**Background**
In the Equation (2.45) Introduction to Econometrics, GLobal Edition by Stock and Watson, it says
$E(\overline{Y}) = \frac{1}{n}\sum_{i=1}^{n}E(Y_i) = \mu_Y$
where observations $Y_1, \cdots, Y_n$ are i.i.d. and $\mu_Y$ and $\sigma^2_Y$ denote the mean and variance of $Y_i$.
**What I don't un... |
Having trouble confirming that these two values are equal.
$$1 - \int_{1}^{2} \ln(y)^{\frac{1}{3}} \, dy + 1 = \int_{0}^{1} e^{x^3} \, dx$$
For context,
$$\int_{1}^{2} \ln(y)^{\frac{1}{3}} \, dy$$ is the inverse of
$$\int_{0}^{1} e^{x^3} \, dx$$
I saw that there was no simple solution for f(x) and decided... |
This problem is from the 2018 admission test to Scuola Normale Superiore in Pisa, Italy. I am having trouble with the second question.
**Problem:** Let $V^1, V^2: \mathbb{R}^N \to \mathbb{R}^N$ be two Lipschitz-continuous vector fields. Given $\lambda \in ]0,1[$ consider, for $n \in \mathbb{N}$, the vector fields $... |
**Background**
In the Equation (2.45) Introduction to Econometrics, GLobal Edition by Stock and Watson, it says
$E(\overline{Y}) = \frac{1}{n}\sum_{i=1}^{n}E(Y_i) = \mu_Y$
where observations $Y_1, \cdots, Y_n$ are i.i.d., and $\mu_Y$ and $\sigma^2_Y$ denote the mean and variance of $Y_i$.
**What I don't u... |
I am reading complex analysis by Gamelin and I am having trouble understanding the square root function.
The principal branch of $\sqrt{z}$ is $f_1(z)$ is defined as $|z|^{\frac 1 2} e^{\frac{i \operatorname{Arg}(z)}{2}}$ for $z \in \mathbb{C} - (-\infty,0]$ and $f_2(z)$ is defined as $-f_1(z)$
By this defini... |
Using reflection principle, where did things go wrong? |
Having trouble confirming that these two values are equal.
$$1 - \int_{1}^{2} \ln(y)^{\frac{1}{3}} \, dy + 1 = \int_{0}^{1} e^{x^3} \, dx$$
For context,
$$\ln(y)^{\frac{1}{3}}$$ is the inverse of
$$e^{x^3}$$
I saw that there was no simple solution for f(x) and decided to try to convert to f(y) to find oth... |
The following is presented as an example to motivate my question. It's my paraphrase of the principle of recursion theorem and proof. From [Fundamentals of Mathematics Foundations of Mathematics: The Real Number System and Algebra][1]
>*Let $c$ be a number and let $F$ be a function of two arguments defined
in $\ma... |
Obviously if $\frac{n}{3}$ is an integer then $\dfrac{2n}{3}$ is an integer. But what if $\frac{n}{3}$ is not an integer? Can it be proven that $\dfrac{2n}{3}$ is not an integer (or, more specifically, an even multiple of 3) if $\frac{n}{3}$ is not an integer?
I feel that intuitively this must be true, but I am not ... |
Is there a simple proof for $\frac{2n}{3}$ is not an integer when $\frac{n}{3}$ is not an integer? |
It's a bit hard to relate directly to general anticommutativity, but it is fairly parallel to how $x\star y=x-y$ would work. Clearly, $r$ is the replacement for $0$, i.e. $ x \star r = x-0 =x$. And $x \star y = r$ is $x - y =0$ is $x=y$; and of course $(x \star y) \star (y \star x) = (x-y) - (y-x) = 2(x-y)$ which is z... |
I saw this problem:
> Find $\lim\limits_{n \to \infty }\prod\limits_{r=1}^n \frac{n^2-r}{n^2+r} $
I tried to prove this problem and got:
$$\ln(1+x) = \sum_{k=1}^ \infty \frac{(-1)^{k+1} x^k}{ k} \ \ \ \ \text{ with radius of convergence = 1 }$$
$$\ln(1-x)=- \sum_{k=1}^ \infty \frac{ x^k}{ k} \ \ \ \ \text{ wi... |
Let $$\alpha(x):= \sum_{n=0}^\infty a_n x^n$$
We have
$$
(1-3x+2x^2) \alpha(x)= \sum_{n=2}^\infty \left(a_n-3a_{n-1}+2a_{n-2} \right) x^n=\sum_{n=2}^\infty (2n-1) x^n=\frac{2x}{(x-1)^2}-\frac{1}{1-x}-x=\frac{(3-x)x^2}{(x-1)^2}
$$
So
$$
\alpha(x)=\frac{(3-x)x^2}{(x-1)^2(1-3x+2x^2)}
$$
Its Taylor series... |
The integer $x_n$ expressed in base $2$ is a number
$$
[x_n]_2=\underbrace{11...1}_{n}\underbrace{00...0}_{n}
$$
with $n \geq 2$.
> Calculate the sum of the binary digits of $x_n^3.$
I have no clue about this. How am I supposed to know anything about $x_n$?
Any hints? |
What would you suggest me to do for the following limit?
$$\lim_{n\to\infty}\left(\frac{1}{e}\prod_{k=1}^{n}\displaystyle\frac{n^2+k}{n^2-k}\right)^n.$$
Thanks! |
Let $$\alpha(x):= \sum_{n=0}^\infty a_n x^n$$
We have
$$
(1-3x+2x^2) \alpha(x)= \sum_{n=2}^\infty \left(a_n-3a_{n-1}+2a_{n-2} \right) x^n=\sum_{n=2}^\infty (2n-1) x^n=\frac{(3-x)x^2}{(x-1)^2}
$$
(see [here](https://www.wolframalpha.com/input?i=sum+%5B%2F%2Fmath%3A%282n-1%29x%5En%2F%2F%5D+from+%5B%2F%2Fmath%3A2... |
In my knot theory class homework I encountered the following question:
>Prove that for every link, when calculating the Wirtinger presentation of the fundamental group of its complement, you can always omit the last relation in the list.
I tried looking at the last crossing (in a given numbering of the crossings)... |
Can you extend the unit speed vector field on a circle embedded in the Möbius band to the whole Möbius band? |
My friends and I have used an [Elo-Ranking system][1] to track our relative strength in the social deduction game "Secret Hitler". So far, the formula we used was a very basic adaptation of the original Elo-formula that uses a lot of assumptions and thus gives unfair results.<br />
First, I will describe the relevant ... |
The statement "If a simple graph $G$ is cubic and has a cut edge, then $\chi'(G) =4$" has a couple of proofs on this site, namely [here][1] and [here][2]. However, I was interested in a specific way of proving the statement.
In [Bondy's book][3] he leaves this problem as an exercise, but lays out a way in which to a... |
We are dealing with reals, so first let's answer the following question: what are possible orders of elements of $\mathbb{R}^*$? In order to avoid confusion with absolute value, I will use $o(x)$ to denote "order of $x$".
First, if $x>1$ then we can multiply both sides by $x$ to get $x^2>x$. And then by simple induc... |
The setup to this question is very simple: Take the numbers $\frac{1}{1}, \frac{21}{12}, \frac{321}{123},...,\frac{987654321}{123456789}$ and plot them versus the natural numbers, as seen here: https://i.stack.imgur.com/psgiz.png.
Now, seeing that this follows a linear path, and that adding more numerals seems to in... |
What is $\sqrt{-1}$? and what is $i$? |
What's a (single-sorted) algebraic signature? |
Let $a$ be a positive integer number. Then
- The sequence of the last digits of $a, a^2, a^3, \dots$ is periodic, or at least eventually periodic.
- The sequence of the first digits of $a, a^2, a^3, \dots$ follows [Benford's Law][1] (unless $a$ is a power of the base).
What about the middle digits sequence? I... |
According to Abbott 4.2.5 "Divergence Criterion for Functional Limits",
> Let $f$ be a function defined on $A$, and $c$ be a limit point of $A$. If there exist two sequences $(x_n)$, $(y_n)$ with $x_n \neq c$ and $y_n \neq c$ and $\lim_{x \to c} x_n = \lim_{x \to c} y_n = c$ but $\lim_{x \to c} f(x_n) \neq \lim_{x ... |
Assume $R$ is an equivalence relation over $A$ and there is a bijection between $A$
and $A / R$. Does this entail $R = \left\{ (x, y ) \in A^2 : x = y \right\} $? What I thought is the following.
Assume $A$ is an infinite set. Then we can find a counter-example. For example, if $A = \mathbb{N}$ and $R$ the equival... |
If there is a bijection $F : A \mapsto A / R$, then $R = \{(x, y) \in A^2 : x = y\}$? |
If I'm given a starting set, and an operation, what would be the generated set looks like?
Here we take $S_0=\{1\}$ and $f(a,b) = \dfrac{1}{a+b}$ as an example, the following Mathematica codes shows the recursion:
```
S = {1};
S = Union[S, 1/(#1 + #2) & @@@ Tuples[S, 2]] // Sort //
DeleteDuplicates
```
... |
What is the set generate by ${1}$ and a function $1/(a+b)$? |
If I'm given a starting set, and an operation, what would the generated set looks like?
Here we take $S_0=\{1\}$ and $f(a,b) = \dfrac{1}{a+b}$ as an example, the following Mathematica codes shows the recursion:
```
S = {1};
S = Union[S, 1/(#1 + #2) & @@@ Tuples[S, 2]] // Sort //
DeleteDuplicates
```
aft... |
If I'm given a starting set, and an operation, what would the generated set looks like?
Here we take $S_0=\{1\}$ and $f(a,b) = \dfrac{1}{a+b}$ as an example, the following Mathematica codes shows the recursion:
```
S = {1};
S = Union[S, 1/(#1 + #2) & @@@ Tuples[S, 2]] // Sort //
DeleteDuplicates
```
aft... |
I'm new to connections and I'm going over the page (https://mathworld.wolfram.com/VectorBundleConnection.html) in which they state the following
> For example, the trivial bundle $E=M\times \Bbb R^k$ admits a flat connection since any bundle section $s$ corresponds to a function $s:M\to \Bbb R^k$. Then setting $\nab... |
Assume we have:
\begin{cases}
a_{n} = 2^{n} \times \left(1 - b_{{\left\lfloor \frac{n}{2} \right\rfloor}}\right) \\
b_{n} = b_{n - 1} + \frac{a_{n}}{4^{n}} \\
b_{0} = 0
\end{cases}
I tried finding the closed formula for $a_{n}$ using generator functions. But I am stuck at this point:
$$ \frac{1}{1 - x} - A(\fr... |
Your summary of the solution manual's argument is incorrect. However, this is not really your fault, as the solution manual *itself* has an obvious typo in it that makes it pretty ambiguous what the argument was intended to be in the first place - and perhaps you attempted to rectify this using your own interpretation.... |
The probability is given by
$$\mathbb P (b<-\sqrt{4ac})+\mathbb P (b>\sqrt{4ac})=\frac{2}{\pi}\color{blue}{\int_0^\infty \left (1-\text{erf}\left(2\sqrt{t} \right) \right )K_0(t)\text{d} a}=\\
\frac{4}{3\pi}\sqrt{2/π} \left (8 \Gamma(5/4)^2 {}_3F_2(3/4, 5/4, 5/4;3/2, 7/4 ;16) - 3 \Gamma(3/4)^2 {}_3F_2(1/4, 3/4, 3/... |
If it`s true that the probability of rolling a six on one die is 1/6;
and that the probability of rolling two dice and getting same value on both is 1/6;
does that actually mean that both these events are equally likely?
If I`m making a bet with a friend , and he uses 1 die, while I use two. Can I say that he ... |
Suppose that we are given some probability space $(\Omega,\mathcal{F},\mathbb{P})$ and some event $B$ with $\mathbb{P}[B]>0$. Then one can construct a new probability space $(\Omega, \mathcal{F}',\mathbb{P}')$ where every set $A\in \mathcal{F}$ is replaced with $A\cap B\in\mathcal{F}'$. Sure, there are many possible pr... |
Is conditional probability unique? |
The LHS and RHS of your equality are not equal unfortunately.
If $f(x)=e^{x^{3}}$ then $f^{-1}(x)=\ln(x)^{\frac{1}{3}}$ and you can derive, $$\int_{0}^{1}e^{x^{3}}dx=e-\int_{1}^{e}\ln\left(y\right)^{\frac{1}{3}}dy.$$ Is this what you were looking for? |
The probability is given by
$$\frac{1}{2}+\mathbb P (b<-\sqrt{4ac})+\mathbb P (b>\sqrt{4ac})=\frac{1}{2}+\frac{2}{\pi}\color{blue}{\int_0^\infty \left (1-\text{erf}\left(2\sqrt{t} \right) \right )K_0(t)\text{d} t}=\\
\frac{1}{2}+\frac{4}{3\pi}\sqrt{2/π} \left (8 \Gamma(5/4)^2 {}_3F_2(3/4, 5/4, 5/4;3/2, 7/4 ;16) \\-... |
Suppose $V$ is a complex vector space with respect to basis $v_{1},...,v_{n}$, and $T: V \rightarrow V$ is a linear transformation with matrix representation $A$.
Now, consider $V$ as a real vector space with respect to basis $v_{1}, iv_{1},...,v_{n}, iv_{n}$. What is matrix representation of $T$ with respect to thi... |
The issue is not so much about the use of the term "observations" versus "samples" or "random variables" or "realizations," but rather, a misconception about the meaning of the expectation operator $\operatorname{E}$.
Suppose I have a *single* random variable, say $X$, that follows some probability distribution. To... |
First question: the space of connections on a vector bundle is an affine space, whose translation space consists of tensors of the appropriate type. What this means, more precisely, is that:
1) If $\nabla$ is a connection in $E$ and $A\colon \mathfrak{X}(M)\times \Gamma(E)\to \Gamma(E)$ is $C^\infty(M)$-bilinear, th... |
(Before closing this question, yes, I've read [this one][1], but my question is not about separation of variables). I'm currently studying second order PDEs with variable coefficients, and after reducing them to their canonical form, my professor solves them as if they were ODEs (except, of course, for the elliptical c... |
The probability is given by
$$\frac{1}{2}+\mathbb P (b<-\sqrt{4ac})+\mathbb P (b>\sqrt{4ac})=\frac{1}{2}+\frac{2}{\pi}\color{blue}{\int_0^\infty \left (1-\text{erf}\left(2\sqrt{t} \right) \right )K_0(t)\text{d} t}=\\
\frac{1}{2}+\frac{4}{3\pi}\sqrt{2/π} \left (8 \Gamma(5/4)^2 {}_3F_2(3/4, 5/4, 5/4;3/2, 7/4 ;16) \\-... |
On average how many keystrokes will it take to randomly hit the keypad of four characters (ABCD) to write DAC for the first time? |
The key is that $y$ does not appear as a partial in the expression. Thus you can imagine plugging in a *fixed* value of $y$, call it $y_0$, into the expression. Define $f(x) = u(x,y_0)$, then the PDE becomes
$$ 2f''(x) + 5y_0 f(x) + 2y_0^2 f(x) = y_0. $$
This would not work if we had $y$ appearing as a partial as... |
I am aware of [this question](https://math.stackexchange.com/q/552436) and [this one](https://math.stackexchange.com/q/60953), the answers to which show that the natural numbers can be equipped with a compact Hausdorff topology. But what happens if one also requires the operation
$$
\max: {\Bbb N} \times {\Bbb N} \t... |
Can $({\Bbb N}, \max)$ be topologized to be a compact Hausdorff monoid? |
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