qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
2,969,195 | <p>I have a cube with rotations <span class="math-container">$\{r, r^2, s, t\}$</span>. <strong>I want to find the cardinality of the conjugacy classes for these elements.</strong> (I know they are 6, 3, 8 and 6 respectively) I couldn't find any formula or anything so I tried to do it by hand for <span class="math-cont... | Baran Zadeoglu | 575,644 | <p>Hint: Maybe instead of thinking about the cube you can think about the diagonals. Can you recover the same information from the diagonals? If so, which group does this look like? With diagonals I mean line segments between pairs of points with distance sqrt(3) assuming the cube has edge length 1. <br/>
On a note, I ... |
2,148,631 | <p>$a_{n+1} = 1-\frac{1}{a_n + 1}$<br>
$a_1 = \sqrt 2$</p>
<p>I need to prove:<br>
1. $a_n$ is irrational for every $n$.<br>
2. $a_n$ convregres</p>
<p>My ideas: </p>
<ol>
<li>Induction - as I know that $a_1 $ is irrational, I'm assuming $a_n$ is also irrational, and then: $a_{n+1} = 1-\frac{1}{a_n + 1} \implies a... | Laray | 396,534 | <p>Show convergence by proving that $a_n$ is bounded below by $0$, as you already did.
Then you can show, that the sequence is drecreasing by starting with $a_{n+1}<a_n$ and going each step of the constructiong for the next element (keep track of the direction of the sign) to show, that $a_{n+2}<a_{n+1}$.
Since... |
177,091 | <p>What would $\int\limits_{-\infty}^\infty e^{ikx}dx$ be equal to where $i$ refers to imaginary unit? What steps should I go over to solve this integral? </p>
<p>I saw this in the Fourier transform, and am unsure how to solve this.</p>
| robjohn | 13,854 | <p>$$
\begin{align}
\int_{-\infty}^\infty e^{ixy}\overbrace{\ \ \ e^{-\epsilon x^2}\ \ \ }^{\to1}\,\mathrm{d}x
&=e^{-\frac{y^2}{4\epsilon}}\color{#C00}{\int_{-\infty}^\infty e^{-\epsilon\left(x-\frac{iy}{2\epsilon}\right)^2}\,\mathrm{d}x}\tag{1}\\
&=\underbrace{\color{#C00}{\sqrt{\frac\pi\epsilon}}\,e^{-\frac{y... |
177,091 | <p>What would $\int\limits_{-\infty}^\infty e^{ikx}dx$ be equal to where $i$ refers to imaginary unit? What steps should I go over to solve this integral? </p>
<p>I saw this in the Fourier transform, and am unsure how to solve this.</p>
| felipeh | 73,723 | <p>I just want to clarify what is meant by saying that "as a distribution",
$$
2\pi\delta(k) = \int_{-\infty}^\infty e^{ikx}\,dx.
$$
The right hand side certainly doesn't make sense, and the left hand side can only make sense for $k\not=0$. What we mean though is that for any smooth function $\phi$ with rapid decay,
... |
1,036,964 | <blockquote>
<p>Let $\{e_1, e_2\}$ and $\{f_1, f_2, f_3\}$ the canonical ordered bases of $\mathbb{R}^2$ and $\mathbb{R}^3$ respectively. Find the coordinates of $x \otimes y$ with respect to the basis $\{e_i\otimes f_j\}$ of $\mathbb{R}^2\otimes\mathbb{R}^3$ where $x = (1, 1)$ and $y = (1, -2, 1)$.</p>
</blockquote>... | Brian M. Scott | 12,042 | <p>Clearly $0\in f[A]$. Suppose now that $t\in(0,8)$. Choose any $u\in(t,8)$, and set $y=\frac{u}4$; then $0<y<2$. Moreover, $0<\frac{t}y=\frac{4t}u<4$, so $\sqrt{\frac{t}y}<2$. Let $x=-\sqrt\frac{t}y$; then $\langle x,y\rangle\in A$, and $f(x,y)=x^2y=t$.</p>
<p>The case $t\in(-8,0)$ is easier, since yo... |
3,932,883 | <p>Can someone suggest how to find the infinite series sum for</p>
<p><span class="math-container">$$\frac{k(k+1)3^k}{k!}$$</span> where k goes from <span class="math-container">$1$</span> to infinity.</p>
<p>I know that <span class="math-container">$\sum_0\frac{3^k}{k!}=e^3$</span> but I'm not sure if that helps here.... | Albus Dumbledore | 769,226 | <p><strong>Hint</strong>:let <span class="math-container">$$f(x)=xe^x=\sum_{k=0}^{\infty}\frac{x^{k+1}}{k!}$$</span> what is <span class="math-container">$3f''(3)?$</span></p>
|
527,665 | <p>I need to bound a sum of the form $\sum_{j=1}^m |z+j|^{-n}$ with $\Re z\ge1$ and $n \ge 2$. I am searching for a bound of the form
$$\sum_{j=1}^m |z+j|^{-n} \le C |z|^{-n+1}.$$</p>
<p>It is easy to see that</p>
<p>$$\sum_{j=1}^m |z+j|^{-n} \le \int_{0}^{m+1} |z+x|^{-n}dx \le \int_{0}^{\infty} |z+x|^{-n}dx.$$</p>
... | Felix Marin | 85,343 | <p>$\newcommand{\angles}[1]{\left\langle #1 \right\rangle}%
\newcommand{\braces}[1]{\left\lbrace #1 \right\rbrace}%
\newcommand{\bracks}[1]{\left\lbrack #1 \right\rbrack}%
\newcommand{\dd}{{\rm d}}%
\newcommand{\ds}[1]{\displaystyle{#1}}%
\newcommand{\expo}[1]{{\rm e}^{#1}}%
\newcommand{\ic}{{\rm i}}%
\newcomman... |
523,074 | <p>I have never done integration in my life and I am in first year of university. Is it harder than taking the derivative? I've heard its just going backwards. Also, my high school taught me only differentiation, I don't know why we never touched on integration. I'm going to be starting it next week and I want to know ... | Aravindan Sai Narayanan | 1,038,658 | <p>Integration is the reverse of differentiation through the Fundamental Theorem of Calculus.Integration is generally tougher because you will learn and use techniques like partial fractions, trigonometric substitutions, algebraic expression substitutions,integration by parts, etc.Certain real valued integrals can only... |
18,090 | <p>I am wondering if anyone could prove the following equivalent definition of recurrent/persistent state for Markov chains:</p>
<p>1) $P(X_n=i,n\ge1|X_0=i)=1$<p>
2) Let $T_{ij}=min\{n: X_n=j|X_0=i\}$, a state $i$ is recurrent if for $\forall j$ such that $P(T_{ij}<\infty)>0$, one has $P(T_{ij}<\infty)=1$</p>... | Community | -1 | <p>A recurrent state will satisfy property (2), but not the other way around. Take a two state Markov chain that jumps from $i$ to $j$ with probability one, then stays there forever. Then $i$ is transient, but $P(T_{ij}<\infty)=1$. </p>
<p><strong>Edit:</strong>
Here's a sketch of (1) implies (2) in terms of "tabo... |
4,159,451 | <p>Kindly see the boldened sentence below. What probability concept does "the most likely value" refer to? How do I symbolize it?</p>
<p>I don't think the author's referring to Expected Value, because it's additive.</p>
<blockquote>
<p> The principle of additivity is so intuitively appealing that it’s ea... | A rural reader | 874,661 | <p>It sounds like you're trying to describe a situation in which <span class="math-container">$\operatorname{mode}(X_1 + \cdots +X_n) = \operatorname{mode} X_1 + \cdots + \operatorname{mode} X_n$</span>.</p>
|
1,361,517 | <p>$K_1 K_2 \dotsb K_{11}$ is a regular $11$-gon inscribed in a circle, which has a radius of $2$. Let $L$ be a point, where the distance from $L$ to the circle's center is $3$. Find
$LK_1^2 + LK_2^2 + \dots + LK_{11}^2$.</p>
<p>Any suggestions as to how to solve this problem? I'm unsure what method to use. </p>
| JimmyK4542 | 155,509 | <p>Use complex coordinates. Let $L = 0$ (the origin), and let $K_n = 3 + 2e^{i\tfrac{2\pi n}{11}}$ for $n = 1,\ldots,11$. </p>
<p>Clearly $K_1,\ldots,K_{11}$ are $11$ equally spaced points on the circle of radius $2$ centered at $3$.</p>
<p>Then for each $n$, you have $LK_n^2 = \left|3 + 2e^{i\tfrac{2\pi n}{11}}\righ... |
466,909 | <p>How many ways can you split the numbers 1 to 5 into two groups of varying size?
For example: '1 and 2,3,4,5' or '1,2 and 3,4,5' or '1,2,3 and 4,5'. How many combinations are there like this? What is the formula?</p>
| wonderwalrus | 673,760 | <p>Here's another way of looking at the problem of splitting the numbers into group A and group B:</p>
<p>For every number, there are two choices: into group A or group B. If there are n numbers, then there are <span class="math-container">$2^n$</span> combinations.</p>
<p>We know that the order of the groups isn't i... |
1,778,601 | <p>Let $G$, $H$ be finite groups. </p>
<p>For a homomorphism $\varphi \colon$ $G \to H$ we learned that for $g \in G$, $ord(\varphi(g))$ must divide $|G|$ and $|H|$. </p>
<p>So then, (in essence), what we did was look at the common factors of $|G|$ and $|H|$ and set $\varphi(x) = \gamma $ where $x$ is a generator of ... | lulu | 252,071 | <p>To elaborate on the discussion in the comments: Indicator variables can be very helpful for problems like these. Accordingly, let $X_i$ be the indicator variable for the $i^{th}$ value. Thus $X_i=1$ if your draw of $x$ elements gets one of value $i$, and $X_i=0$ otherwise. It is easy to compute $E[X_i]$...if $p_... |
309,872 | <blockquote>
<p>Let $a$ and $b$ be elements of a group $G$. If $|a| = 12, |b| = 22$ and $\langle a \rangle \: \cap \: \langle b\rangle \ne e$, prove that $a^6 = b^{11}$.</p>
</blockquote>
<p>Any ideas where to start would be helpful.</p>
<p>Thanks.</p>
| Potato | 18,240 | <p>I sketch a solution; you should be able to fill in the details.</p>
<p>The intersection of the groups generated by $a$ and $b$ must have order dividing each, by Lagrange's theorem. Since this order is not 1, looking at the prime factorization tells us it must be 2. Hence it contains a single non-identity element $c... |
4,531,998 | <p>As you increase the value of n, you will generate all pythagorean triples whose first square is even. Is there any visual proof of the following explicit formula and where does it come from or how to derive it?</p>
<p><span class="math-container">$(2n)^2 + (n^2 - 1)^2 = (n^2 + 1)^2$</span></p>
<div class="s-table-co... | Mark Bennet | 2,906 | <p>This formula can be derived in various ways, and with connections to different areas of mathematics. One way of seeing why it generates all the triples is to note that the unit circle <span class="math-container">$$x^2+y^2=1$$</span> in polar co-ordinates has <span class="math-container">$y=\sin \theta, x=\cos \thet... |
2,010,768 | <p>I have the question </p>
<p>Solve the simultaneous equations </p>
<p>$$\begin{cases}
3^{x-1} = 9^{2y} \\
8^{x-2} = 4^{1+y}
\end{cases}$$</p>
<p><a href="https://i.stack.imgur.com/xN9Ny.jpg" rel="nofollow noreferrer">source image</a></p>
<p>I know that $x-1=4y$ and $3X-6=2+2y$ .</p>
<p>However when I checked th... | Dr. Sonnhard Graubner | 175,066 | <p>write the system in the form
$$3^{x-1}=3^{4y}$$ and
$$2^{3(x-2)}=2^{2(1+y)}$$
and you will get
$$x-1=4y$$
and
$$3(x-2)=2(1+y)$$
can you proceed?</p>
|
6,424 | <p>Hello Mathematica users,</p>
<p>I have problems trying to eliminate variable $Q(x,y)$ froma PDE, but Mathematica output is just <code>True</code>. The problem is that I already found how to eliminate it. We can apply $d/dx$ on the second equation and then from the first equation we can determine and substitute $dQ... | Vitaliy Kaurov | 13 | <p>Generally if you have n variables you need to have at least n+1 equations to get meaningful elimination and end up with at least one equation. With m equations and k variables you get m-k resulting equations after elimination (in a well defined system). Your case is two variables and two equations which results of c... |
6,424 | <p>Hello Mathematica users,</p>
<p>I have problems trying to eliminate variable $Q(x,y)$ froma PDE, but Mathematica output is just <code>True</code>. The problem is that I already found how to eliminate it. We can apply $d/dx$ on the second equation and then from the first equation we can determine and substitute $dQ... | Daniel Lichtblau | 51 | <p><code>Eliminate</code>, and its cousin <code>GroebnerBasis</code>, work with algebraic equations. If you require differential elimination you will need to take derivatives (prolongations, that is). Here is a blind approach: just take some derivatives, sort the variables into two sets, and eliminate all the <code>Q</... |
2,251,112 | <p>A small college has 1095 students. What is the approximate probability that more than five students were born on Christmas day? Assume that the birthrates are constant throughout the year and that each year has 365 days.</p>
<p>I tried doing<br>
$X \sim Pn(3) $ and calculating $P(X\gt5)$. My calculation turned out ... | Cato | 357,838 | <p>poisson probability distribution $\lambda = 1095 / 365$
then calculate </p>
<p>using</p>
<p>$P(N) = \frac{\lambda^n}{n!} \exp(-\lambda)$</p>
<p>1 - P(0) - P(1) - P(2) - P(3) - P(4) - P(5) = </p>
<p>$1 - \exp(-\lambda)(1 + \lambda + \lambda^2 / 2! + \lambda^3 / 3! + \lambda^4/4! + \lambda^5/5!) = .084$</p>
<p>n... |
79,868 | <p>Given a function $f: \mathbb{R}^+ \rightarrow \mathbb{C}$ satisfying suitable conditions (exponential decay at infinity, continuous, and bounded variation) is good enough, its <em>Mellin transform</em> is defined by the function</p>
<p>$$M(f)(s) = \int_0^{\infty} f(y) y^s \frac{dy}{y},$$</p>
<p>and $f(y)$ can be r... | paul garrett | 15,629 | <p>[Some next-day edits in response to comments] As counterpoint to other viewpoints, one can say that Mellin inversion is "simply" Fourier inversion in other coordinates. Depending on one's temperament, this "other coordinates" thing ranges from irrelevancy to substance... The question about moral imperatives for Four... |
79,868 | <p>Given a function $f: \mathbb{R}^+ \rightarrow \mathbb{C}$ satisfying suitable conditions (exponential decay at infinity, continuous, and bounded variation) is good enough, its <em>Mellin transform</em> is defined by the function</p>
<p>$$M(f)(s) = \int_0^{\infty} f(y) y^s \frac{dy}{y},$$</p>
<p>and $f(y)$ can be r... | Phil Isett | 7,193 | <p>As others have pointed out, the Mellin inversion theorem is just the Fourier inversion theorem in disguise for the particular group ${\mathbb R_+}$ with invariant measure $\frac{dx}{x}$. The goal of the Fourier transform is to express a general function as a linear combination (i.e. integral) of the characters of t... |
79,868 | <p>Given a function $f: \mathbb{R}^+ \rightarrow \mathbb{C}$ satisfying suitable conditions (exponential decay at infinity, continuous, and bounded variation) is good enough, its <em>Mellin transform</em> is defined by the function</p>
<p>$$M(f)(s) = \int_0^{\infty} f(y) y^s \frac{dy}{y},$$</p>
<p>and $f(y)$ can be r... | Community | -1 | <p>Another property: The (inverse) Mellin transform interchanges $q$-expansions of modular forms and Dirichlet $L$-series.</p>
|
76,747 | <pre><code>Plot[D[Abs[x], x], {x, -10, 10}, Exclusions -> {0}]
</code></pre>
<p>gives out several lines of error messages and an empty plot.</p>
<pre><code>Plot[Derivative[1][Abs[#1] & ][x], {x, -10, 10}, Exclusions -> {0}]
</code></pre>
<p>just gives out an empty plot.</p>
<p>How do I plot $\left(\left|x... | ubpdqn | 1,997 | <p>I suspect the simplest way is just define the absolute value function yourself, e.g.</p>
<pre><code>f[x_] := Piecewise[{{x, x > 0}, {-x, x < 0}}]
Plot[Evaluate[D[f[x], x]], {x, -1, 1}, Exclusions -> {0}]
</code></pre>
|
3,851,399 | <p>I am attempting to prove the following statement</p>
<blockquote>
<p>Prove that if n is any integer then 4 either divides n^2 or n^2 − 1</p>
</blockquote>
<p>I have started with the case of n = 2k</p>
<pre><code>Consider the case n = 2k
n = 2k
n^2 = 4k^2
⇒ n = 4k
∴ 4 divides n^2 as there is some integer k in w... | Alessio K | 702,692 | <p>Consider the case <span class="math-container">$n=2k$</span>, then <span class="math-container">$$n^2=4k^2$$</span>
<span class="math-container">$$\therefore \space 4|n^2 \space \text{that is if $n$ is even then $n^2$ is divisible by $4$}$$</span></p>
<p>For the case <span class="math-container">$n=2k+1$</span> we h... |
1,879,346 | <p>Ok, so I have this question from my math book.</p>
<blockquote>
<p>The vertices A,B,C of a triangle are (3,5),(2,6) and (-4,-2)
respectively. Find the coordinates of the circum-centre and also the
radius of the circum-circle of the triangle.</p>
</blockquote>
<p>How can we solve this? Can we use the distance... | David Quinn | 187,299 | <p>HINT...You could find the equations of the perpendicular bisectors of two of the sides and where they meet will be the circumcentre. Then use the distance formula to work out the circumradius.</p>
|
2,046,773 | <p>Given a function $f$ that is differentiable at a point $x_0$, if we define (using the Riemann integral)</p>
<p>$$F(x) = \int_a^x f$$</p>
<p>Can we necessarily say that $F^{\prime}(x)$ is continuous at $x_0$? Going back and forth between $f$ and $F$ confuses me a bit. I <em>think</em> that the Fundamental Theorem o... | user251257 | 251,257 | <p>The statement is true in the sense, that <span class="math-container">$F'$</span> is continuous at <span class="math-container">$x_0$</span> on its domain, which need not to be the entire interval <span class="math-container">$(a, b)$</span>. We may drop that <span class="math-container">$f$</span> is differentiable... |
2,385,338 | <p>In a bag, there are 6 white disks, 6 black disks and 8 red disks. A disk is drawn at random from the bag.The colour is recorded and the disc is returned to the bag This process is repeated 10 times.
Find the probability that less than 4 red disks are drawn </p>
<p>I approached this question with n= 8 and p=0.4, q=... | Jordan Abbott | 468,755 | <p>As the disc is returned to the bag it will be the <em>same</em> probability of a red disc coming out each time. As it is <em>less than</em> four red discs, we need to find the probability of $0, 1, 2$ and $3$ red discs being chosen. Your values of $p$ and $q$ are indeed correct, however I'm a bit confused as to why ... |
1,012,730 | <p>A purely imaginary number is one which contains no non-zero real component. </p>
<p>If I had a sequence of numbers, say $\{0+20i, 0-i, 0+0i\}$, could I call this purely imaginary? </p>
<p>My issue here is that because $0+0i$ belongs to multiple sets, not just purely imaginary, is there not a valid case to say that... | Seth | 31,659 | <p>A complex number is said to be purely imaginary if it's real part is zero. Zero is purely imaginary, as it's real part is zero. </p>
|
1,012,730 | <p>A purely imaginary number is one which contains no non-zero real component. </p>
<p>If I had a sequence of numbers, say $\{0+20i, 0-i, 0+0i\}$, could I call this purely imaginary? </p>
<p>My issue here is that because $0+0i$ belongs to multiple sets, not just purely imaginary, is there not a valid case to say that... | user191016 | 191,016 | <p>0 is both purely real and purely imaginary. The given set is purely imaginary. That's not a contradiction since "purely real" and "purely imaginary" are not fully incompatible. Somewhat similarly baffling is that "all members of X are even integers" and "all members of X are odd integers" is not a contradiction. ... |
3,570,943 | <p>From the comments I got, my question amonts to : can the " inverse of inverse" law be derived from the definition of division. </p>
<hr>
<p>Division is defined as : <span class="math-container">$\dfrac AB = A.\dfrac1B$</span>, that is, </p>
<p>" dividisng by A by B is, by definition, mutiplicating A by the inver... | Abhirup Adhikary | 742,851 | <p><span class="math-container">$1/(1/c)$</span> </p>
<p>Here, according to the definition, we need to multiply 1 with the multiplicative inverse of (1/c)</p>
<p>Multiplicative inverse of a number is a number which when multiplied by the given number yields 1</p>
<p>So, here multiplicative inverse of (1/c) is c (you... |
3,570,943 | <p>From the comments I got, my question amonts to : can the " inverse of inverse" law be derived from the definition of division. </p>
<hr>
<p>Division is defined as : <span class="math-container">$\dfrac AB = A.\dfrac1B$</span>, that is, </p>
<p>" dividisng by A by B is, by definition, mutiplicating A by the inver... | fleablood | 280,126 | <p>Just do it:</p>
<p><span class="math-container">$\frac a {\frac bc}$</span> will by definition be <span class="math-container">$a\cdot \frac 1{\frac bc}$</span></p>
<p>So we need to figure out what <span class="math-container">$\frac 1{\frac bc}$</span> is.</p>
<p>And it is the value <span class="math-container">... |
1,778,982 | <p>I can prove that $G$ is cyclic, but I am not sure how to prove the orders. I know I need to use the Fundamental Theorem of Cyclic Groups but I'm not sure how to apply it. Is there something obvious I am missing?</p>
| Alex Wertheim | 73,817 | <p>The subgroups of any finite cyclic group $G$ are in bijective correspondence with the divisors of $|G|$. Write $|G| = p_{1}^{k_{1}}p_{2}^{k_{2}}\cdots p_{n}^{k_{n}}$ for distinct primes $p_{1}, p_{2}, \ldots, p_{n}$ and positive integers $k_{1}, \ldots, k_{n}$. Then each divisor of $|G|$ can be specified by a tuple ... |
2,608,455 | <p>Could someone please help me with how do I calculate the sum of the $$\sum_{n=1}^{\infty}\frac{1}{4n^{2}-1}$$ infinite series? I see that $$\lim_{n\rightarrow\infty}\frac{1}{4n^{2}-1}=0$$ so the series is convergent based on the Cauchy's convergence test. But how do I calculate the sum? Thank you.</p>
| José Carlos Santos | 446,262 | <p>Since$$\frac1{4n^2-1}=\frac1{(2n-1)(2n+1)}=\frac12\left(\frac1{2n-1}-\frac1{2n+1}\right),$$your series is a telescopic series.</p>
|
3,043,319 | <p>I just begin with the 3- dimension function. I dont really understand how to begin with this problem.
Function <span class="math-container">$$f(x,y)= (x^2+4y^2-5)(x-1)$$</span>
Find where <span class="math-container">$$f(x,y)=0; f(x,y)>0; f(x,y)<0.$$</span></p>
<p>To find where <span class="math-container">$... | J.G. | 56,861 | <p>Since <span class="math-container">$f(x)=x$</span> bijects <span class="math-container">$A$</span> to <span class="math-container">$A$</span>, <span class="math-container">$\sim$</span> is reflexive. Write a similar proof <span class="math-container">$\sim$</span> is symmetric, using the fact bijections have inverse... |
616,454 | <p>How can I calculate this integral?</p>
<p>$$ \int {\exp(2x)}{\sin(3x)}\, \mathrm{d}x$$ </p>
<p>I tried using integration by parts, but it doesn't lead me any improvement. So I made an attempt through the replacement $$ \cos(3x) = t$$ and it becomes $$\frac{1}{-3}\int \exp\left(2\left(\dfrac{\arccos(t)}{3}\right)... | Mhenni Benghorbal | 35,472 | <p>You need to use integration by parts or use the identity</p>
<p>$$ \sin y = \frac{e^{iy} - e^{-iy} }{2i} $$</p>
<p>Which makes the integral easier.</p>
|
190,392 | <p>How to prove that the Torus and a Ball have the same Cardinality ?
The proof is using the Cantor Berenstein Theorem.
I now that they are subsets of $\mathbb{R}^{3}$ so I can write $\leq \aleph$ but I do not know how to prove $\geqslant \aleph$.
Thanks</p>
| Asaf Karagila | 622 | <p><strong>Hint:</strong> </p>
<p>Show that a circle is equipotent with $[0,2\pi)$ by fixing a base point, and sending each point on the circle to its angle, where $0$ is the base point.</p>
<p>Since both the torus and the ball contain a circle, both have at least $\aleph$ many elements.</p>
|
190,392 | <p>How to prove that the Torus and a Ball have the same Cardinality ?
The proof is using the Cantor Berenstein Theorem.
I now that they are subsets of $\mathbb{R}^{3}$ so I can write $\leq \aleph$ but I do not know how to prove $\geqslant \aleph$.
Thanks</p>
| Andrea Mori | 688 | <p>Let's assume known the fact that $\Bbb R^m$ and $\Bbb R^n$ have the same cardinality for all pairs of positive integers $(m,n)$.</p>
<p>Then the following follows easily: Let $X\subset\Bbb R^n$ such that there exists a subset $A\subset X$ homeomorphic to an open subset of $\Bbb R^m$ for some $m\leq n$. Then $|X|=|\... |
2,888,464 | <p>Let $S$ be the set of polynomials $f(x)$ with integer coefficients satisfying </p>
<p>$f(x) \equiv 1$ mod $(x-1)$</p>
<p>$f(x) \equiv 0$ mod $(x-3)$</p>
<p>Which of the following statements are true?</p>
<p>a) $S$ is empty .</p>
<p>b) $S$ is a singleton.</p>
<p>c)$S$ is a finite non-empty set.</p>
<p>d) $S$... | mfl | 148,513 | <p>We have that</p>
<p>$$f(x)\equiv 1\bmod{(x-1)}\iff \exists a(x)|f(x)=(x-1)a(x)+1.$$</p>
<p>$$f(x)\equiv 0\bmod{(x-3)}\iff \exists b(x)|f(x)=(x-3)b(x).$$</p>
<p>So, if $f$ exists then it must be </p>
<p>$$(x-1)a(x)+1=(x-3)b(x).$$ We have for $x=1:$</p>
<p>$$1=-2b(1)\implies b(1)=-\dfrac 12,$$ which is not possib... |
3,200,431 | <p>We can say, that any field <span class="math-container">$\mathbb{K}$</span> -- <span class="math-container">$1$</span>-dim vector space on itself: <span class="math-container">$\mathbb{K}_{\mathbb{K}}$</span>. So any vector of one another finite-dimensional vector space <span class="math-container">$V_{\mathbb{K}}$<... | James Silipo | 365,160 | <p>The category of fields has no product. If it had, then the projection on one factor would be injective, which is clearly false! </p>
|
2,572,564 | <p>So I have a straight line, in the classic <span class="math-container">$y = mx + b$</span>, and I'm just trying to translate the formula for the line a certain distance along its normal.</p>
<p>For example, with <a href="https://i.stack.imgur.com/r8vct.jpg" rel="nofollow noreferrer">this graph</a>, how would I tran... | Michael Rozenberg | 190,319 | <p>$$\lim_{x\rightarrow0}\frac{\ln\left(1+\frac{x^4}{3}\right)}{\sin^6x}=\lim_{x\rightarrow0}\left(\left(\frac{x}{\sin{x}}\right)^6\cdot\ln\left(1+\frac{x^4}{3}\right)^{\frac{3}{x^4}}\cdot\frac{1}{3x^2}\right)=+\infty$$</p>
|
1,368,447 | <p><img src="https://i.stack.imgur.com/TL80w.jpg" alt="enter image description here"></p>
<p>So I tried the good old Calculus 1 approach and turned this into an optimization problem. The equations got REALLY hairy, but it was okay since this was the graphing calculator section of the exam. I called the longer part of ... | Hagen von Eitzen | 39,174 | <p>The upper (and/or) lower half of the kite is a triangle. Such a triangle with two given side lengths has maximal area when it is a right triangle. Now you can find the exact lengths of the diagonals with Pythagoras & Cie.</p>
|
1,656,294 | <p>Apart from the Dirchet function ( equals 1 for rational x and 0 for irrational x), and similar constructs (equals some constant a for rational x and b for irrational , where a!=b) , what are the other functions which are defined on all real numbers and discontinuous at all points?</p>
<p>Are there infinite such fun... | Win Vineeth | 311,216 | <p>Let me try this way : <br> In $(0,$$\pi\over 4$$)$ , $\tan x <1 $ and $\cot x >1$ <br> I make a statement - <em>Anything less than one to the power of a positive number, is less than one!</em> </p>
<p>Another statement - <em>Anything greater than one to the power of a positive number, is greater than one</em>... |
1,397,016 | <p>We put
<span class="math-container">$\|f\|_{(N, \alpha)}:= \sup_{x\in \mathbb R} (1+|x|)^{\alpha} | D^{\beta}f(x)|; $</span> and he Schwartz space,</p>
<p><span class="math-container">$S(\mathbb R): = \{f\in C^{\infty}(\mathbb R): \|f\|_{(N, \alpha)}< \infty , \forall \alpha, N \in \mathbb N \cup \{0\} \}.$</spa... | David C. Ullrich | 248,223 | <p>There's certainly an approximate identity in $\mathcal S$.</p>
<p>For $\phi\in\mathcal S$ and $t>0$ define $\phi_t(x)=t^{-1}\phi(x/t)$. Then if $\int\phi=1$ it follows that $\phi_t*f\to f$ in $\mathcal S$ for every $f\in\mathcal S$.</p>
<p>Say $\psi=\hat\phi$. It's easiest to verify that you have an approximate... |
1,397,016 | <p>We put
<span class="math-container">$\|f\|_{(N, \alpha)}:= \sup_{x\in \mathbb R} (1+|x|)^{\alpha} | D^{\beta}f(x)|; $</span> and he Schwartz space,</p>
<p><span class="math-container">$S(\mathbb R): = \{f\in C^{\infty}(\mathbb R): \|f\|_{(N, \alpha)}< \infty , \forall \alpha, N \in \mathbb N \cup \{0\} \}.$</spa... | Jochen | 38,982 | <p>Concerning <em>bounded approximate identities</em>: $\mathcal S$ is a Montel spaces, that is bounded sets are relatively compact ($\mathcal S$ is even nuclear). If it had a bounded approximate identity compactness would give a limit which then would be an identity element which certainly does not exist.</p>
|
3,340,723 | <p>How can we find the total number of ways in which we can divide <span class="math-container">$n$</span> elements into two subsets such that none of them are empty and the union of both sets should be equal to the whole set?</p>
<p>Eg. If <span class="math-container">$S=\{1,2,3\}$</span>, the answer can be <span cla... | ab123 | 454,871 | <p>Choosing elements of one group determines the other group as well. So, the total number of ways needs summing over all possible ways of "choosing" elements of one group:</p>
<p>For any general <span class="math-container">$n$</span>:
<span class="math-container">$$\dfrac{{n \choose 1} + {n \choose 2} + \dots + {n... |
3,054,558 | <p>I've just learned a throrem which states that : A metric space has the structure of a topological space in which the open sets are unions of balls . </p>
<p>But the theorem only told me there "exist" one topology with respect to the metric.When we refer to topology on a metric space <span class="math-container">$S$... | Perturbative | 266,135 | <p>Yes, a metric space <span class="math-container">$(S, d)$</span> induces a topology <span class="math-container">$\mathcal{T}$</span> which is generated by the basis <span class="math-container">$$\mathcal{B} = \{B(x, r) \ | \ x \in S \ \text{ and } \ r > 0\}$$</span></p>
<p>When authors refer to the topology on... |
10,978 | <p>So, in <a href="https://math.meta.stackexchange.com/a/10083/43351">this post</a>, I suggested to replace <a href="https://math.stackexchange.com/questions/tagged/rootsystem" class="post-tag" title="show questions tagged 'rootsystem'" rel="tag">rootsystem</a> with <a href="https://math.stackexchange.com/quest... | Community | -1 | <p>Ask a moderator to perform this change by making a meta post. Moderators can also directly merge the two tags to rename the old tag, which is easier than retagging all old questions by hand.</p>
|
4,287,450 | <p>I tried solving this by finding dy/dx and letting the denominator be zero, yielding an equation whose solution can only be approximated. But that approximated value is not right as seen on Desmos.</p>
| Claude Leibovici | 82,404 | <p>As @Hosam Hajjir wrote, the slope is infinite when
<span class="math-container">$$\cos (\theta) - \theta \sin (\theta)=0$$</span> and the solutions are closer and closer to <span class="math-container">$n\pi$</span>.</p>
<p>Expanding as series around <span class="math-container">$x=n \pi$</span>,we have
<span class=... |
3,649,125 | <p>Bridge is a game of four players in which each player is dealt 13 cards from a standard 52 card deck. Bridge players (such as myself) are interested in the number of possible deals, where each player is distinct. This can be counted by</p>
<p><span class="math-container">$$\binom{52}{13}\binom{39}{13}\binom{26}{13}... | Mike Earnest | 177,399 | <p>I will answer a much more general problem:</p>
<blockquote>
<p>Suppose you have a collection of playing cards, each falling into one of <span class="math-container">$k$</span> types, so that cards of the same type are indistinguishable. For each <span class="math-container">$i\in \{1,\dots,k\}$</span>, there are ... |
3,888,805 | <p>"Six men and three women are standing in a supermarket queue. Three of the people in the queue are chosen to take part in a customer survey. How many different choices are possible if at least one woman must be included?"</p>
<p>I went about solving this question by considering 3 cases, first with a single... | Derek Luna | 567,882 | <p>Because you can choose Becky with <span class="math-container">${3 \choose 1}$</span> and Kate with <span class="math-container">${8 \choose 2}$</span>, and in another way choose Kate with <span class="math-container">${3 \choose 1}$</span> and Becky with <span class="math-container">${8 \choose 2}$</span> all other... |
3,888,805 | <p>"Six men and three women are standing in a supermarket queue. Three of the people in the queue are chosen to take part in a customer survey. How many different choices are possible if at least one woman must be included?"</p>
<p>I went about solving this question by considering 3 cases, first with a single... | Toby Mak | 285,313 | <p>There is a much simpler approach. "At least one woman" is the opposite of "zero women" or "all men" from the total, so subtracting from the total gives us:</p>
<p><span class="math-container">$${9 \choose 3} - {6 \choose 3} = 64$$</span></p>
<hr />
<p>If you are interested in another su... |
3,556,490 | <p>Can someone explain me how to solve the equation <span class="math-container">$\frac{x-n}{x+n} = e^{-x}$</span>, where <span class="math-container">$n$</span> is a non-zero natural number ? Unfortunately, I have not even an idea how to start. Any hint is much appreciated. </p>
<p>Many thanks in advance. </p>
| egreg | 62,967 | <p>You can observe that a solution must satisfy <span class="math-container">$x>n$</span>. So we can consider the equivalent equation
<span class="math-container">$$
\log(x+n)-\log(x-n)=x
$$</span>
so we can study the function <span class="math-container">$f(x)=\log(x+n)-\log(x-n)-x$</span>. We can note that
<span c... |
115,477 | <p>Let us say that a finite set $A$ in the plane is $1$-separated if:</p>
<p><strong>1)</strong> it has an even number of points;</p>
<p><strong>2)</strong> no open ball of diameter $1$ contains more than $|A|/2$ points.</p>
<p>For a $1$-separated set $A$ define $G(A)$ to be a graph where two points $x,y$ in $A$ are... | Alfred | 29,755 | <p>I think Domotorp is correct. Take a regular $(2n-1)$-gon such that its longest diagonal is 1, along with its center. Then $A$ cannot be partitioned, and $G(A)=C_{2n-1}$.</p>
|
45,800 | <p>Let's say I want to calculate the following scalar-by-matrix derivative</p>
<p>$$\frac{\partial}{\partial A} \text{tr} \left[(\vec X^T A)^T (\vec X^T A)\right],$$</p>
<p>with $\vec X$ and $A$ being a $n \times 1$ and a $n \times m$ matrix, respectively.
Is there a way in Mathematica to get the result</p>
<p>$$2 \... | dtn | 67,019 | <p>I tried the <code>NCAlgebra</code> package:</p>
<pre><code><< NC`;
<< NCAlgebra`;
SNC[X, A]
NCGrad[tr[tp[tp[X] ** A] ** (tp[X] ** A)], A]
</code></pre>
<p>And got this result:</p>
<pre><code>2 A^T ** X ** X^T
</code></pre>
<p>This is a result similar to that which can be obtained by hand.</p>
|
1,730,569 | <p>All spaces are assumed Hausdorff. We call a topological space <em>compact</em> if every open cover has a finite subcover. We call it <em>Lindelöf</em> if every open cover has a countable subcover, and <em>hereditarily Lindelöf</em> if moreover every subspace is Lindelöf. </p>
<p>It is obvious that every compact spa... | Henno Brandsma | 4,280 | <p>The OP's own answer, the one-point compactification of an uncountable discrete space, is a classical one. Another is $\{0,1\}^X$, where $X$ is uncountable, where the set of all points with exactly one $1$ is uncountable and discrete (and so not Lindelöf too). </p>
<p>Of course, every closed subspace of a Lindelöf s... |
2,993,299 | <p>If anyone can help me with how to go about solving these kind of equations i would really appreciate it. :-)</p>
<p><span class="math-container">$$\sqrt{36-2x^2} = 4$$</span></p>
<p>Solve for X</p>
| William Elliot | 426,203 | <p>From <span class="math-container">$2$</span> on ward <span class="math-container">$0 < 1 - 1/x < 1$</span><br />
Thus <span class="math-container">$$2 < x < x/(1 - 1/x) < (1 + 1/x)/(1 - 1/x)$$</span><br />
Desired conclusion follows.</p>
|
1,882,509 | <p>As we know that, according to multiplicative identity 2 to the power of 3 means,
1*2*2*2=8;but why 2 to the power -3 is equal to the 1/8,if i think it is the reverse of multiplicative identity then from where 1 is coming?
i don't want only mathematical proof,but also the intuitive story behind this?
Why it is turnin... | Aaron | 9,863 | <p>When you are starting out with arithmetic, first you have counting, then you have addition of positive numbers, and then you have repeated addition, which is multiplication. There are obvious real world ways to give meaning to these expressions in terms of counting of objects when the operations involve positive in... |
3,430,744 | <p>I'm new to fourier transform but I somewhat know fourier series. Since the fourier series of <span class="math-container">$\sin(2\pi t)$</span> is just <span class="math-container">$\sin(2\pi t)$</span>, I thought of playing with this function to better understand what fourier transform is doing. </p>
<p>As a star... | maxmilgram | 615,636 | <p>Basically this would be very hard to solve if it weren't for the specific choice of coefficients. Observe that it can be rewritten as
<span class="math-container">$$
\frac{dy}{dx}-2x=\sqrt{4x^2-4y}\\
\frac{d}{dx}(y-x^2)=\sqrt{4x^2-4y}\\
-\frac{dz}{dx}=2\sqrt{z}
$$</span>
For <span class="math-container">$z=x^2-y$</... |
3,030,317 | <p>I have four points on a rectangular grid <span class="math-container">$(x_1,y_1)$</span>, <span class="math-container">$(x_1,y_2)$</span>, <span class="math-container">$(x_2,y_1)$</span> and <span class="math-container">$(x_2,y_2)$</span>. I also have the value of a third variable <span class="math-container">$z$</... | Federico | 180,428 | <p>Here is a silly way of doing it.</p>
<p>Lets work in <span class="math-container">$[0,1]^2$</span> for simplicity.</p>
<p>Consider the functions
<span class="math-container">$$
f(x,y)=x(1-x)^2(1-y) \qquad \text{and} \qquad
g(x,y)=(1-x)^2(1+2x)(1-y)^2(1+2y) .
$$</span>
We have 12 values we are interested in: the 4 ... |
182,307 | <p>I have a long flat list that needs to be partitioned. The list is formatted so the "header" is repeated, followed by the values. Essentially, it looks something like this:</p>
<pre><code>list={a,a,1,2,3,b,b,5,6,c,c,1,5,a,a,7,8,9,1}
</code></pre>
<p>I am looking for an output of:</p>
<pre><code>{{a,1,2,3},{b,5,6... | kglr | 125 | <pre><code>Join[Most /@ Rest @ Most @ #, {Last @ #}] & @ Split[list, UnsameQ]
</code></pre>
<blockquote>
<p>{{a, 1, 2, 3}, {b, 5, 6}, {c, 1, 5}, {a, 7, 8, 9, 1}}</p>
</blockquote>
<p>You can also use use <code>Split</code> twice and reorganize the result:</p>
<pre><code>Rest /@ Flatten /@ Split[Split[list], Le... |
116,220 | <p>Say two graphs are not isomorphic but are both strongly regular with the same set of parameters. Are there any parameters (other than the usual such as order, degrees, eigenvalues and multiplicities, etc.) that are determined, e.g., independence number, chromatic number, etc.?</p>
<p>Thanks for any help</p>
| Brendan McKay | 9,025 | <p>The number of cycles of length 3,4,5 are determined. If the girth is 4, the number of 6-cycles is determined too.</p>
|
4,412,371 | <p>Would appreciate some help with proving the following statement:</p>
<p>Let <span class="math-container">$a_{n}$</span> be a sequence, and <span class="math-container">$\lim _{n\rightarrow \infty }a_{n}=1 $</span>.</p>
<p>Prove that <span class="math-container">$\lim _{n\rightarrow \infty }a_{1}+a_{2}+\ldots +a_{n}=... | José Carlos Santos | 446,262 | <p>There is some <span class="math-container">$n\in\Bbb N$</span> suche that <span class="math-container">$n\geqslant N\implies a_n\geqslant\frac12$</span>. So, if <span class="math-container">$n\geqslant N$</span>,<span class="math-container">$$a_1+a_2+\cdots+a_n\geqslant a_1+\cdots+a_{N-1}+\frac{n-N-1}2.$$</span>Can ... |
1,241,586 | <p>I want to prove that for all positive integers
$n$, $2^1+2^2+2^3+...+2^n=2^{n+1}-2$.
By mathematical induction:</p>
<p>1) it holds for $n=1$, since $2^1=2^2-2=4-2=2$</p>
<p>2) if $2+2^2+2^3+...+2^n=2^{n+1}-2$, then prove that $2+2^2+2^3+...+2^n+2^{n+1}=2^{n+2}-2$ holds.</p>
<p>I have no idea how to proceed wit... | marwalix | 441 | <p>It is straightforward</p>
<p>$$2+2^2+\cdots+2^n+2^{n+1}=(2+2^2+\cdots+2^n)+2^{n+1}$$</p>
<p>Use the induction assumption</p>
<p>$$\begin{align} 2+2^2+\cdots+2^n+2^{n+1}&=2^{n+1}-2+2^{n+1}\\&=2\cdot 2^{n+1}-2\\&=2^{n+2}-2\end{align}$$</p>
|
2,437,466 | <p>A point, $p$, is defined as a cluster point of a set $S$ if $\forall \epsilon > 0,$ there exists an open ball of radius $\epsilon$ centered at $p$ that contains infinitely many points in $S$.</p>
<p>We want to prove that a subset $S$ of a metric space $(E,d)$ is closed iff $S$ contains all of its cluster points.... | José Carlos Santos | 446,262 | <p>Note that $$\frac{\frac{4^{n+1}}{n+2}}{\frac{4^n}{n+1}}=4\frac{n+1}{n+2}$$ and that $$\frac{\frac{(2(n+1))!}{(n+1)!^2}}{\frac{(2n)!}{n!^2}}=\frac{(2n+2)!}{(2n)!}\cdot\frac{n!^2}{(n+1)!^2}=\frac{(2n+2)(2n+1)}{(n+1)^2}=2\frac{2n+1}{n+1}. $$So, when is it true that$$4\frac{n+1}{n+2}\leqslant2\frac{2n+1}{n+1}?$$ Well, t... |
178,828 | <p>Let $\left\{ x_n \right\}_{n\geq0}$ be a sequence of real numbers such that $$x_{n+1}=\lambda x_n+(1-\lambda)x_{n-1},\ n\geq 1,$$for some $0<\lambda<1$ </p>
<p>(a) Show that $x_n=x_0+(x_1-x_0)\sum_{k=0}^{n-1}(\lambda -1)^k$ </p>
<p>(b) Hence, or otherwise, show that $x_n$ converges and find the limit.</p>
<... | André Nicolas | 6,312 | <p><strong>Hint:</strong> The <strong>geometric series</strong> $1+r+r^2+r^3+\cdots$ converges to $\frac{1}{1-r}$ if $|r|\lt 1$.</p>
<p>This is a fact that is undoubtedly already familiar to you. It can be proved by showing that
$$1+r+r^2+\cdots +r^{n-1}=\frac{1-r^n}{1-r},\tag{$1$}$$
and then noting that $|r|\lt 1$, ... |
178,828 | <p>Let $\left\{ x_n \right\}_{n\geq0}$ be a sequence of real numbers such that $$x_{n+1}=\lambda x_n+(1-\lambda)x_{n-1},\ n\geq 1,$$for some $0<\lambda<1$ </p>
<p>(a) Show that $x_n=x_0+(x_1-x_0)\sum_{k=0}^{n-1}(\lambda -1)^k$ </p>
<p>(b) Hence, or otherwise, show that $x_n$ converges and find the limit.</p>
<... | i. m. soloveichik | 32,940 | <p>Since $x_n−x_0=(x_1−x_0)\sum_{k=0}^{n-1}(\lambda-1)^k=A\frac{1-(\lambda-1)^n}{2-\lambda}$, where $A=x_1-x_0$. then $x_n=x_0+A\frac{1-(\lambda-1)^n}{2-\lambda}$. Since $0<\lambda<1$ then $(\lambda-1)^n\rightarrow 0$ as $n\rightarrow \infty$ and thus $$\lim_{n\rightarrow \infty} x_n=x_0+\frac{x_1-x_0}{2-\lambd... |
1,426,891 | <blockquote>
<p>Why does $$\left(\int_{-\infty}^{\infty}e^{-t^2}dt\right)^2 = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(x^2 + y^2)}dx\,dy ?$$</p>
</blockquote>
<p>This came up while studying Fourier analysis. What's the underlying theorem?</p>
| Olivier Oloa | 118,798 | <p><strong>Hint.</strong> One may observe that
$$
A\int_{-\infty}^{+\infty}e^{-x^2}dx=\int_{-\infty}^{+\infty}A\:e^{-x^2}dx
$$ for any constant $A$. Then putting $A:=\int_{-\infty}^{+\infty}e^{-y^2}dy$ and using <a href="https://en.wikipedia.org/wiki/Fubini's_theorem" rel="nofollow">Fubini's theorem</a>, which is ... |
1,426,891 | <blockquote>
<p>Why does $$\left(\int_{-\infty}^{\infty}e^{-t^2}dt\right)^2 = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(x^2 + y^2)}dx\,dy ?$$</p>
</blockquote>
<p>This came up while studying Fourier analysis. What's the underlying theorem?</p>
| Salihcyilmaz | 227,487 | <p>$$\left(\int_{-\infty}^{\infty}e^{-t^2}dt\right)^2 = \left(\int_{-\infty}^{\infty}e^{-t^2}dt\right) * \left(\int_{-\infty}^{\infty}e^{-u^2}du\right) $$since $u$ and $t$ varies independently we have$$\left(\int_{-\infty}^{\infty}e^{-t^2}dt\right) * \left(\int_{-\infty}^{\infty}e^{-u^2}du\right)= \int_{-\infty}^{\in... |
2,985,962 | <p>I was asking my friends a riddle about identifying hats. Each person has to correctly identify the colour of their own hat that was put on their head randomlly.
There is no defined number of either colour. So they could be all white or all black or any combination in between.</p>
<p>They gave an answer that gets 50... | John Hughes | 114,036 | <p>Skip forward to below the "=====" for the answer that you're probably looking for. Read the first part to find out why I put it after the "=====". </p>
<p>To make sense of this, you need to think about probability spaces. And to do that right, you need more information about the meaning of the words in your questio... |
1,406,462 | <p>I've been studying an introductory book on set theory that uses the <a href="https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory">ZFC</a> set of axioms, and it's been really exciting to construct and define numbers and operations in terms of sets. But now I've seen that there are other alternatives to ... | Daniel Ranard | 263,791 | <p>If you're looking for motivation to pursue other foundations, I recommend the article "Rethinking Set Theory" by Tom Leinster: <a href="http://arxiv.org/pdf/1212.6543v1.pdf" rel="noreferrer">http://arxiv.org/pdf/1212.6543v1.pdf</a>. In particular, he's providing a gentle, well-motivated introduction to Lawvere's El... |
2,495,789 | <p>\begin{pmatrix}4&-6&2&-6\\ \:-2&3&-1&3\end{pmatrix}</p>
<p>Am I supposed to move the $0 $ to the other side of the equation and find the inverse of the matrix above or is there another way?</p>
| FredetikS | 497,201 | <p>You have 4 variables and 2 equations. This implies that you have infinitly many solutions.</p>
|
125,645 | <p>I stumbled upon the following integral:</p>
<p>$$\int e^{-\left(\frac{\gamma}{2} + ip \right)^2} H_n \left( q-\frac{\gamma}{2} \right) H_n \left( q+\frac{\gamma}{2} \right) \, d\gamma \, ,$$</p>
<p>and I really need a closed form of it. I'm pretty sure it exists, and probably in terms of $\Gamma$ functions, since<... | yarchik | 9,469 | <p>It is interesting to solve this integral without relying on <em>Gradshteyn and Ryzhik</em>. As I suggested in the comments, one can use the generating function approach. The solution consists of 7 steps. Human input is needed on step 5. All other steps are pretty much automatic.</p>
<ol>
<li><p>Substitute generati... |
3,537,549 | <p>In the game of poker, five cards from a standard deck of 52 cards are dealt to each player.</p>
<p>Assume there are four players and the cards are dealt five at a time around the table until all four players have received five cards.</p>
<p>a. What is the probability of the first player receiving a royal flush (th... | Claude Leibovici | 82,404 | <p>If you want an exact value <span class="math-container">$$\frac{R}{2\sqrt{R}-1} \int_{\pi/2}^{\pi} d\theta \, e^{R t \cos{\theta}}=\frac{R}{2\sqrt{R}-1}\frac{\pi}{2} \, (I_0(R t)-\pmb{L}_0(R t))$$</span> where appear Bessel and Struve functions</p>
|
1,506,719 | <p>Let us have measurable spaces $(S_1, \Sigma_1)$ and $(S_2, \Sigma_2)$.
Idea of measurable function $f$ with respect to $\Sigma_1,\Sigma_2$ is the following. $f:$ $S_1 \to S_2$ has to be such that:
$$\forall E_2: f^{-1}(E_2) \in \Sigma_1, $$
where
$ E_2 \subset \Sigma_2, E_1 \subset \Sigma_1 $.</p>
<hr>
<p>Anoth... | John Dawkins | 189,130 | <p>In measure theory (or probability theory) to study the behavior of a function like $f$ we often look at sets of the form $\{x\in S_1:f(x)\in E_2\}$, and it is useful to have a briefer notation for such a set, whence $f^{-1}(E_2)$. And this notion is more than just notation: the set mapping $f^{-1}$ so defined maps t... |
40,473 | <p>I found this « Worm on the rubber band » problem in Concrete Mathematics book.</p>
<p>A slow worm $W$ starts at one end of a meter-long rubber band and crawls one centimetre per minute toward the other end.</p>
<p>At the end of each minute, a keeper of the band $K$ stretches it one meter.</p>
<p>Does the worm eve... | Luboš Motl | 10,599 | <p>Bonjour, Jean-Pierre, the relative position of the insect on the rubber band is
$$r(0)=x_W(0) / x_K(0)$$
and you will have to know it to solve the problem. It will be a number between 0 and 1.</p>
<p>Now, you want to find $t$ such that $x_W(t)=x_K(t)$ i.e. $r(t)=1$. To find it, you must calculate the time derivat... |
2,124,560 | <p>A pentagon ABCDE contains 5 triangles whose areas are each one. The triangles are ABC, BCD, CDE, DEA, and EAB. Find the area of ABCDE?</p>
<p>Is there a theorem for overlapping triangle areas?</p>
| Michael Lugo | 173 | <p>It was asked how to find the area algebraically.</p>
<p>Assume that the pentagon <span class="math-container">$ABCDE$</span> is regular and the area <span class="math-container">$[ABC] = 1$</span>. Let <span class="math-container">$s$</span> be its side length. The angle at B is <span class="math-container">$3\pi... |
2,018,631 | <p>Let C(x, y) : "x and y have chatted over the Internet"</p>
<p>where the domain for the variables x and y consists of all students in your class.</p>
<p>a ) There are two students in your class who have not
chatted with each other over the Internet.</p>
<p><strong>My answer:</strong> $\exists x \exists y[(x \not =... | Bram28 | 256,001 | <p>Your answer is correct. In fact, given that you use a biconditional, you can leave out the $\lnot C(x, y)$: </p>
<p>$\exists x \exists y[(x \not =y) \land \forall a \forall b(\lnot C(a, b) \iff ((a = x \land b = y)\lor(a = y \land b = x)))]$</p>
<p>(do you see why this works?)</p>
<p>Please do make sure that you... |
1,816,783 | <p>When can one conclude that a character table has non-real entries?</p>
<p>In particular, by constructing the character table for $\mathbb{Z}/3\mathbb{Z}$ or $A_4$ how does one determine that some of the entries will be nonreal?
Is the reason that the same complex values in the table for $\mathbb{Z}/3\mathbb{Z}$ als... | Patrick Stevens | 259,262 | <p>A question from the Cambridge Part II Maths course on Representation Theory:</p>
<blockquote>
<p>Let $x$ be an element of order $n$ in finite group $G$. Say, without detailed proof, why:</p>
<ol>
<li>if $\chi$ is a character of $G$, then $\chi(x)$ is a sum of $n$th roots of unity;</li>
<li>$\tau(x)$ is r... |
1,653,569 | <p>Consider, in the first-order NGB theory of sets, the following axioms:
$$\vdash\exists x\forall y(y\notin x)$$
and
$$\vdash\forall y(y\notin\varnothing)$$</p>
<p>The second one defines the constant $\varnothing$.
Moreover, from the second axiom we get the first by $\exists$I inference rule.</p>
<p>So my question i... | Noah Schweber | 28,111 | <p>Well, you can only write that second sentence if you have a constant symbol for "$\emptyset$". Since the language of NBG is $\{\in\}$, this isn't technically a sentence you can write.</p>
<p>Now, you could make an argument that maybe we should add a constant symbol to the language - but clearly we don't need it (by... |
2,354,094 | <p>I've been stumped by this problem:</p>
<blockquote>
<p>Find three non-constant, pairwise unequal functions $f,g,h:\mathbb R\to \mathbb R$ such that
$$f\circ g=h$$
$$g\circ h=f$$
$$h\circ f=g$$
or prove that no three such functions exist.</p>
</blockquote>
<p>I highly suspect, by now, that no non-trivial ... | Jim Ferry | 458,592 | <p>A simple, isometric solution exists in $\mathbb{R}^n$ for $n \ge 2$: let $f$ be a reflection in the first coordinate, and $g$ be a reflection in the second. A bijection between $\mathbb{R}^n$ and $\mathbb{R}$ then yields a solution to the stated problem, though continuity is lost.</p>
|
2,829,121 | <p>I get how to derive the ellipse equation, but I'm struggling to understand what it means intuitively. </p>
<p>You see, a circle equation can be understood very intuitively. The circle equation models how the radius of the circle can be represented using the Pythagorean theorem. But I don't understand what the ellip... | AgentS | 168,854 | <p>Key thing to keep in mind is that the length of the rope doesn't change. This means you're basically modeling all the points whose "sum of distances from the given two fixed points" doesn't change.</p>
<p>Say $A$ and $B$ are the fixed points and $L$ is the length of the rope, then the point P traces the curve given... |
2,829,121 | <p>I get how to derive the ellipse equation, but I'm struggling to understand what it means intuitively. </p>
<p>You see, a circle equation can be understood very intuitively. The circle equation models how the radius of the circle can be represented using the Pythagorean theorem. But I don't understand what the ellip... | Doug M | 317,176 | <p>$x^2+y^2 = 1$ is our unit circle</p>
<p>$\frac {x}{a}$ dilates $x$ by a factor of $a.$ That is it stretches everything horizontally by a factor of $a$</p>
<p>Simmilarly, $\frac {y}{b}$ dilates $y$ by a factor of $b.$</p>
<p>$\frac {x^2}{a^2} + \frac {y^2}{b^2} = 1\\
\left(\frac xa\right)^2 + \left(\frac yb\right... |
26,074 | <p>Okay, so this was labeled as a "fun problem", but I'm having trouble knowing how to approach it.</p>
<p>I'm given: $\lim\limits_{x \to 0^+} f(x) = A$ and $\lim\limits_{x \to 0^-} f(x) = B$.</p>
<p>I need to find (or at least know where to start):</p>
<p>a) $\lim\limits_{x \to 0^+} f(x^3 - x)$</p>
<p>b) $\lim\lim... | Community | -1 | <p>Let $z(x)=x^3-x$</p>
<p>You have
$$\lim_{x\rightarrow 0^- }f(z(x)) =\lim_{z\rightarrow 0^+ }f(z) $$ </p>
<p>i.e $z$ approaches from $0+\delta$ when x approaches from $0-\delta$ for $\delta>0$ and vice versa for the other limit. </p>
<p>So whatever limits you have for x in will be inverted in z, i.e, answer for... |
1,701,761 | <p><a href="https://i.stack.imgur.com/Qletj.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Qletj.png" alt="Trapezium "></a> </p>
<blockquote>
<p>Prove that ABED is a parallelogram </p>
</blockquote>
<p><strong>Given:</strong></p>
<ol>
<li><p>ABCD is a trapezium</p></li>
<li><p>F and G are the m... | Narasimham | 95,860 | <p>Steps 1,2,3 need not be given.</p>
<p>AD = BE and parallel to it, ABED is a parallelogram</p>
<p>So AB = DE and parallel to it, by parallelogram definition and property.</p>
|
2,788,002 | <p>If I convert, say, #FF$1919$ to binary, I can do it in groups of three bytes like:</p>
<p>FF: $1111$ $1111$</p>
<p>$19$: $0001$ $1001$</p>
<p>$19$: $0001$ $1001$</p>
<p>So can I write that #FF$1919 =$ $1111$ $1111$ $0001$ $1001$ $0001$ $1001$?</p>
<p>Or do I have to write them as separate bytes like above?</p>
| Rhys Hughes | 487,658 | <p>Yes that is absolutely fine. You could have also done it as:
$$F: 1111$$
$$F: 1111$$
$$1: 0001$$
$$9: 1001$$
$$1: 0001$$
$$9: 1001$$
and combined in the same way</p>
<p>So $FF1919: 1111\space 1111\space 0001\space 1001\space 0001\space 1001$</p>
|
66,634 | <p>Here is a quote from <em>Lectures on Ergodic Theory</em> by Halmos:</p>
<blockquote>
<p>I cannot resist the temptation of
concluding these comments with an
alternative "proof" of the ergodic
theorem. If $f$ is a complex valued
function on the nonnegative integers,
write $\int f(n)dn=\lim
> \frac{1}{n... | John Galt | 15,214 | <p>There is a short proof in Katok's book. Introduction to the modern theory of dynamical systems [Book]</p>
|
1,273,789 | <p>Problem: X has mean and variance of 20.<br>
What can be said about $P(0<X<40)$? </p>
<p>Chebyshev formula = $P(|X-\mu|\geq k)\leq\frac{\sigma^2}{k^2}$ </p>
<p>The first step has $P(|X-20|\geq 20)\leq\frac{\sigma^2}{20\times20}$ </p>
<p>My question is: How did they get this part $P(|X-20|\geq 20)$? ... | André Nicolas | 6,312 | <p>Let $A$ be the event $0\lt X\lt 40$. Then $A$ <strong>fails to happen</strong> if either $X\ge 40$ or $X\le 0$. </p>
<p>The event "$X\ge 40$ or $X\le 0$" happens precisely if $|X-20|\ge 20$. This is clear from the geometry. We have $X\ge 40$ or $X\le 0$ if the <strong>distance</strong> of $X$ from $20$ is $\ge 20$... |
67,039 | <p>I just noticed that <code>Exit</code> and <code>Quit</code> can work without brackets i.e. a single</p>
<pre><code>Exit
</code></pre>
<p>or</p>
<pre><code>Quit
</code></pre>
<p>will quit the kernel. Quite surprising!</p>
<p>This isn't mentioned in the document of them. Is there a list for functions with this fe... | J. M.'s persistent exhaustion | 50 | <p>At the OP's request, I am adding the following:</p>
<p><code>Nothing</code> is a function with very peculiar behavior:</p>
<pre><code>Nothing === Nothing[] === Nothing[x, y]
True
</code></pre>
<p>This means that</p>
<pre><code>Nothing & /@ {{a}, {b}}
</code></pre>
<p>and</p>
<pre><code>Nothing /@ {{a}, ... |
50,441 | <p>here is the silly question of mine,</p>
<p>if <code>mathematica</code> takes <code>3</code> seconds to evaluate the function <code>f[3]</code>
then I plug the this value <code>f[3]</code> into a new function <code>g[f[3],f[3]]</code>, Let's ignore the evaluation time for the new function <code>g</code>, how long do... | Kuba | 5,478 | <p><code>With</code> keeps the code clear for longer cases but sometimes I'd do:</p>
<pre><code>g[#,#] & @ f[3]
</code></pre>
|
76,493 | <p>i'm interested in geometric interpretations of many linear algebra notions (check also related <a href="https://mathoverflow.net/questions/75241/geometric-interpretation-of-matrix-minors">geometric interpretation of matrix minors</a>). it came to me recently that geometric description of adjugate matrix (for example... | joel | 17,755 | <p>i found quite a satisfying answer to this question which gives enough trails for further enquiry. the answer involves exterior algebra and gradation (check also hodge duality).</p>
<blockquote>
<p>if <span class="math-container">$\mathrm A\colon V \to V$</span> is a linear map, then it naturally induces a graded ope... |
190,604 | <p><strong>Background</strong></p>
<p>I am currently trying to solve exercise 1.1.18 in Hatcher's Algebraic Topology. The part of the exercise I am interested in is the following:</p>
<blockquote>
<p>Using the technique in the proof of Proposition 1.14, show that if a space $X$ is obtained from a path-connected sub... | Community | -1 | <p>If you have a little knowledge of the Van-Kampen theorem then you can prove it like this: You have $A \subseteq X$ and you are attaching an $n-cell$ $e^n$. Now recall that you have a characteristic map $\Phi : D^n \to X$ that restricts to $\varphi$ on $S^{n-1}$. We know that $\Phi$ is a homeomorphism between the int... |
2,425,669 | <p><em>Having fun with the calculator, I realized that :</em></p>
<p><code>(a^c)</code> and <code>(a^b)</code></p>
<p>with <code>c > b</code> and <code>c > 4</code> and <code>b = 2</code></p>
<pre><code>a^c / a^b = a^(c-2)
</code></pre>
<p><strong><em>So, for example:</em></strong></p>
<p><code>3^5 / 3^2 = 2... | Bernard | 202,857 | <p>A fraction with five factors equal to $3$ in the numerator, and two factors $3$ can be simplified by the elementary rules on fractions.</p>
|
180,073 | <p>The function <code>FullSimplify</code> can easily reformat this expression</p>
<pre><code>FullSimplify[Cos[omegan t]^2 + Sin[omegan t]^2]
(* output: 1 *)
</code></pre>
<p>However, it cannot simplify the same expression if contained in a larger expression:</p>
<pre><code>sol = DSolve[m*(x''[t] + omegan^2 * x[t]) =... | mikado | 36,788 | <p><code>Collect</code> with its optional 3rd argument is very effective here</p>
<pre><code>Collect[x[t] /. Flatten[sol], B, FullSimplify]
(* -((B E^(I omegad t))/(m omegad^2 - m omegan^2)) + C[1] Cos[omegan t] + C[2] Sin[omegan t] *)
</code></pre>
<p>In fact, <code>Collect</code> may not be needed here.</p>
<p>Not... |
1,938,353 | <p>A detailed solution will be helpful. Given that $a_1=3$ and $a_{n+1}=3^{a_n}$, find the remainder when $a_{2004}$ is divided by 100.</p>
| Stefan4024 | 67,746 | <p>Note that $a_{2004} = 3^{3^{.^{.^{.^{3}}}}}$ where the length of the exponent tower is $2003$. Anyway all we need to apply is Euler's Totient Formula. As $\phi(100) = 40$, so now we consider: $3^{3^{.^{.^{.^{3}}}}} \bmod {40}$ where the tower length is $2002$. Continue in this manner and we will consider $\phi(40) =... |
1,938,353 | <p>A detailed solution will be helpful. Given that $a_1=3$ and $a_{n+1}=3^{a_n}$, find the remainder when $a_{2004}$ is divided by 100.</p>
| Parcly Taxel | 357,390 | <p>$$a_1=3$$
$$a_2=3^3=27$$
$$a_3=3^{27}$$
Consider $a_4=3^{3^{27}}$. By Euler's theorem this is congruent to $3^{3^{27}\bmod40}$ modulo 100. Since $3^4\equiv1\bmod40$, $3^{27}\equiv3^3\equiv27\bmod40$. Thus we have the relation
$$3^{3^{27}}\equiv3^{27}\bmod100$$
In other words, exponentiating by 3 does not change the ... |
628,254 | <p>I am searching for some good book which section is devoted to modular arithmetic. I am self learner so I strongly prefer that book has exercises best with answers or solutions. I have CS background and has taken course on discrete mathematics but besides some basic facts on modulo operation it lacked some introducti... | user119261 | 119,261 | <p>May I ask what your goals are? Modular arithmetic gets fun when you get to deal with symmetries; the study of symmetry in this context is called group theory. From there you can look at such things as public key cryptography. There is also quite a lot of fun to be had just working with mod 2. If you want to look... |
2,696,998 | <blockquote>
<p>Show that for any integer <span class="math-container">$k ≥ 0$</span> we have</p>
<p><span class="math-container">$$\frac{1}{(1-z)^{k+1}} = \sum\limits_{n=0}^{\infty}\binom {n+k}{n}z^{n}$$</span></p>
<p>and that it converges absolutely for any <span class="math-container">$|z| < 1$</span>.</p>
</bloc... | Angina Seng | 436,618 | <p>If $T$ is diagonalisable, then $E=E_1\oplus E_2\oplus\cdots\oplus E_k$
where the $E_j$ are the eigenspaces of $T$.
Then prove that any $W$ which is preserved by $T$ is a sum
$W_1+\cdots+W_k$ where $W_j\subseteq E_j$. It may help to note that
the projection maps from $E$ to $E_j$ are polynomials in $T$.</p>
|
951,414 | <p>Consider the absolute value equation:</p>
<p>|x| + |x-2| +|x-4|= 6</p>
<p>How to find the solution(s)?</p>
<p>My attempt:</p>
<p>For |x|, we got x, for x>=0 and -x, for x <0</p>
<p>For |x-2|, we got x-2, for x >= 0 and -(x-2), for x<0</p>
<p>For |x-4|, we got x-4, for x>=0 and -(x-4), for x<0</p>
<p>... | Ben Frankel | 112,897 | <p>Define $n = x - 2$, and rewrite the equation.</p>
<p>$$|n-2| + |n| + |n+2| = 6$$</p>
<p>In this form we can easily see that if $n$ is a solution, so is $-n$. Thus WLOG we will assume that $n$ is positive. We can then remove some redundant $|\cdot|$s.</p>
<p>$$|n-2| + (n) + (n + 2) = 6 \implies |n-2| + 2n = 4$$</p... |
717,991 | <p>I've been stuck on this problem for a few weeks now. Any help?</p>
<p>Prove:
$\sum_{i=1}^{n}\prod_{j=0,j\neq i}^{n}\frac{x-x_j}{x_i-x_j}=1$</p>
<p>The sum of lagrange polynomials should be one, otherwise affine combinations of with these make no sense.</p>
<p>EDIT:
Can anybody prove this by actually working out ... | JP McCarthy | 19,352 | <p>HINT: Throw in the $f(x_i)$ and what happens if they are all 1?</p>
|
2,846,928 | <blockquote>
<p>If I have the a complex number $z \in \mathbb{C}$ with absolute value $|z| = 1$, how do I show that $-i \frac {z-1}{z+1}$ is real? </p>
</blockquote>
| David K | 139,123 | <p>A geometric reason is that $1$ and $-1$ are opposite ends of a diameter of the unit circle, and $z$ is on the circumference.
This implies that $z,$ $1,$ and $-1$ are vertices of a right triangle, with the right angle at $z.$
But $z-1$ and $z+1$ have the same lengths and directions as the legs of that triangle;
I’m p... |
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