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 |
|---|---|---|---|---|
1,625,292 | <p>Define a class $K$ of ordinals inductively as follows:</p>
<ul>
<li><p>$0=\emptyset\in K$.</p></li>
<li><p>For all $\alpha\in K$, the succesor of $\alpha$ is also an element of $K$.</p></li>
<li><p>For every function $f\colon \mathbb N\to K$, the ordinal that immediately follows after all ordinals $f(0),f(1),\dots ... | Noah Schweber | 28,111 | <p>Yes, such ordinals exist - for example, $\omega_1$, the first uncountable ordinal. </p>
<p>The crucial issue here is <em>cofinality</em>: the cofinality of an ordinal $\alpha$ is the least $\beta$ such that there is a function $f:\beta\rightarrow\alpha$ whose range is unbounded in $\alpha$. It's easy to show that $... |
1,625,292 | <p>Define a class $K$ of ordinals inductively as follows:</p>
<ul>
<li><p>$0=\emptyset\in K$.</p></li>
<li><p>For all $\alpha\in K$, the succesor of $\alpha$ is also an element of $K$.</p></li>
<li><p>For every function $f\colon \mathbb N\to K$, the ordinal that immediately follows after all ordinals $f(0),f(1),\dots ... | Asaf Karagila | 622 | <p>First of all, note that the class of <em>all</em> the ordinals satisfies this definition. You want to say that $K$ is the smallest class of ordinals satisfying that the three requirements hold.</p>
<p>And I claim that at least under the axiom of choice $\omega_1$ is such a class of ordinals, and therefore $\omega_1... |
4,032,767 | <blockquote>
<p>Prove that there exists no bijective function <span class="math-container">$f: \Bbb{N} \to \Bbb{N}$</span> such that <span class="math-container">$$f(mn)=f(m)+f(n)+3f(m)f(n)$$</span> for <span class="math-container">$m,n \geqslant1.$</span></p>
</blockquote>
<p>This was a problem from a Putnam practice ... | peters onyilo Agnes | 889,933 | <p>1.Show that the function is well defined
2.Prove that there exist a kernel.
3. Prove that identity maps to element of the kernel
If you can't prove 1 and 2 the function or mapping is not bijective.
In other words, you can prove that the function is both injective and surjective( i.e if the inverse of the function ma... |
1,821,411 | <p>$f:[a,b]\rightarrow R$ that is integrable on [a,b]</p>
<p>So we need to prove:</p>
<p>$$\int_{-b}^{-a}f(-x)dx=\int_{a}^{b}f(x)dx$$</p>
<p>1.) So we'll use a property of definite integrals: (homogeny I think it's called?)</p>
<p>$$\int_{-b}^{-a}f(-x)dx=-1\int_{-b}^{-a}f(x)dx$$</p>
<p>2.) Great, now using the fun... | Vineet Mangal | 346,869 | <p>There is a small error in your first step, you didn't change the sign of the limits. Always keep in mind that after making any substitution, don't forget to change the limits. Since you have made a substitution $x=-t$, so change the limits accordingly. So corrected step 1 is $$-\int_{b}^af(x)dx$$
Now you can reverse... |
3,888,146 | <p>When we give a proof that the tangent is the sine to cosine ratio of an oriented angle,</p>
<p><span class="math-container">$$\bbox[5px,border:2px solid #C0A000]{\tan \alpha=\frac{\sin\alpha}{\cos \alpha}}$$</span>
with <span class="math-container">$\cos \alpha \neq 0$</span>, we take the tangent <span class="math-c... | user247327 | 247,327 | <p>Saying that the diameter lies on a given line segment means that the center is on that line. If A and B are points on the circle then the line segment between them is a chord and its perpendicular bisector also passes through the center of the circle.</p>
<p>So construct the perpendicular bisector of AB. The point... |
3,755,288 | <p>I'm trying to solve this:</p>
<blockquote>
<p>Which of the following is the closest to the value of this integral?</p>
<p><span class="math-container">$$\int_{0}^{1}\sqrt {1 + \frac{1}{3x}} \ dx$$</span></p>
<p>(A) 1</p>
<p>(B) 1.2</p>
<p>(C) 1.6</p>
<p>(D) 2</p>
<p>(E) The integral doesn't converge.</p>
</blockquot... | Jack D'Aurizio | 44,121 | <p>Since <span class="math-container">$\sqrt{t^2+1/3}$</span> is a convex function on <span class="math-container">$[0,1]$</span>, you may simply use the Hermite-Hadamard inequality to derive that</p>
<p><span class="math-container">$$ \sqrt{2+\frac{1}{3}}\leq 2\int_{0}^{1}\sqrt{t^2+1/3}\,dt \leq \sqrt{3} $$</span>
so ... |
4,095,831 | <p>If <span class="math-container">$\frac{dy}{dx}=y\sec^2x$</span> and <span class="math-container">$y=5$</span> when <span class="math-container">$x=0$</span>, then <span class="math-container">$y=$</span>?
A) <span class="math-container">$e^{\tan x} + 4$</span></p>
<p>B) <span class="math-container">$e^{\tan x} + 4$<... | jjagmath | 571,433 | <p>Here is where you made a mistake</p>
<blockquote>
<p><span class="math-container">$\ln|y|= \tan x +C$</span></p>
<p><span class="math-container">$e^{\ln|y|}=e^{\tan x}+C$</span></p>
</blockquote>
<p>If <span class="math-container">$\ln|y|= \tan x +C$</span>, after exponentiation you get <span class="math-container">... |
3,752,162 | <p>I already knew that normal subgroups where important because they allow for quotient space to have a group structure.
But I was told that normal subgroups are also important in particular because they are the only subgroups that can occur as kernels of goup homomorphisms. Why is this property a big deal in algebra?<... | halrankard | 688,699 | <p>I suppose this is more of opinion-based question. If I interpret your question as "Why is it a big deal that the subgroup determined by the kernel of a homomorphism must be normal?", then the answer is "It's not really a big deal. That's trivial." Instead, if I interpret your question as "Wh... |
1,773,776 | <blockquote>
<blockquote>
<p>Question: Prove that the directrix is tangent to the circles that are drawn on a focal chord of a parabola as
diameter.</p>
<p>Here is a picture; <a href="https://i.stack.imgur.com/F8dga.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/F8dga.png" alt="en... | Nikunj | 287,774 | <p>An alternate way (probably shorter) would be to take two points as $(at_1^2,2at_1), (at_2^2,2at_2)$ as the ends of the focal chord. </p>
<p>As this chord passes through focus, we obtain $t_1t_2=-1$
Now, the equation of circle can be written in diametric form as:</p>
<p>$$(x-at_1^2)\left(x-\frac{a}{t_1^2}\right)+(y... |
1,773,776 | <blockquote>
<blockquote>
<p>Question: Prove that the directrix is tangent to the circles that are drawn on a focal chord of a parabola as
diameter.</p>
<p>Here is a picture; <a href="https://i.stack.imgur.com/F8dga.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/F8dga.png" alt="en... | Jean Marie | 305,862 | <p>Here is a simple alternative way, fully geometrical.</p>
<p>Have a look at the following picture, with $M_1,M_2$ on parabola with focus $F$ and directrix $D$, $H_1, H_2, H$ the orthogonal projections on D of $P_1, P_2, F$ resp. </p>
<p>Let $r_k:=M_kH_k=M_kF \ (k=1,2)$. Let $C$ be the midpoint of $M_1M_2$, i.e., th... |
20,942 | <p>If I had a recursive function (<code>f(n) = f(n-1) + 2*f(n-2)</code> for example), how would I derive a formula to solve this? For example, with the Fibonacci sequence, Binet's Formula can be used to find the nth term.</p>
| Klaus | 524 | <p><a href="http://en.wikipedia.org/wiki/Formal_power_series" rel="nofollow">Formal power series</a> can be also be used to solve recurrence relations.</p>
<p>Let $a_n=f(n)$ and</p>
<p>$$\begin{eqnarray}
S=\sum a_nx^n&=&a_0+a_1x+\sum_2 a_{n-1}x^n+2\sum_2 a_{n-2}x^n\\
&=&a_0+a_1x+\sum_1 a_n x^{n+1}+2... |
4,188,656 | <p><strong>Problem</strong>: How many strings are there of length <span class="math-container">$ n $</span> over <span class="math-container">$ \{ 1,2,3,4,5,6 \} $</span> s.t. the sum of all characters in the string divide by <span class="math-container">$ 3 $</span>.</p>
<p><strong>Attempt</strong>:
Initially I though... | peek-a-boo | 568,204 | <p>No, let <span class="math-container">$\Omega$</span> be any subset of <span class="math-container">$\Bbb{R}$</span> with finite, positive Lebesgue measure, and let <span class="math-container">$A=\Omega$</span>. For each <span class="math-container">$\alpha\in A$</span>, let <span class="math-container">$f_{\alpha}(... |
1,719,840 | <p>If there is a group $G$ with order $a$, having a subgroup $H_1$ with order $b$, and $H_2$ with order $c$, and $bc=a$, $H_1 \cap H_2 = e $. Is $H_1 H_2 =G$? </p>
| Community | -1 | <p>The answer to your question is yes: if $G$ is a finite group, and $H$ and $K$ are subgroups such that $|G| = |H||K|$ and $H \cap K = 1$, then $G = HK$.</p>
<p>The easiest way to see this is to use the identity
$$|HK| = \frac{|H||K|}{|H \cap K|}$$
Note that this identity holds even if $HK$ is merely a subset (not a ... |
1,342,747 | <p>I am studying H. L. Royden's Real Analysis which includes some introduction to Measure Theory; and I encountered $(a,\infty]$ instead of $(a,\infty)$ for the first time! </p>
<p>What is the difference(s) between $(a,\infty)$ and $(a,\infty]$?</p>
| Plutoro | 108,709 | <p>A lot of times these two mean the same thing, but it is important to consider the superset of which this is an interval. Sometimes, (especially in measure theory, which is why I mention it) it is useful to work in the extended reals, which includes a point at $\infty$, so $(a,\infty)$ means every number greater than... |
1,669,096 | <p>How do I show that <span class="math-container">$\ell^{ \infty}$</span> is a normed linear space,
where <span class="math-container">$\ell^{ \infty}$</span> is define as <span class="math-container">$$\|\{a_n\}_{n=1}^{\infty}\|_{\ell^\infty}=\sup_{1 \leq k \leq \infty} |a_k|?$$</span>
There are three properties that... | carmichael561 | 314,708 | <p>For the triangle inequality, suppose that <span class="math-container">$\{a_n\},\{b_n\}\in \ell^{\infty}$</span>. For each index <span class="math-container">$n$</span> we have
<span class="math-container">$$|a_n+b_n|\leq |a_n|+|b_n|\leq \sup_{m}|a_m|+\sup_m|b_m|=\|a\|_{\infty}+\|b\|_{\infty}.$$</span>
Then taking t... |
3,084,479 | <p><span class="math-container">$h\in \mathbb{R}$</span>, because we have defined the Trigonometric Functions only on <span class="math-container">$\mathbb{R}$</span> so far.</p>
<p>I have a look at <span class="math-container">$e^{ih}=\sum_{k=0}^{\infty}\frac{(ih)^k}{k!}=1+ih-\frac{h^2}{2}+....$</span> </p>
<p><stro... | Community | -1 | <p>Show that <span class="math-container">$\lim_{h\rightarrow 0}\frac{e^{ih}-1}{h}=i$</span>:</p>
<p>As we know: <span class="math-container">$ \cos \theta + i \sin \theta = e^{i \theta}$</span></p>
<p>So, <span class="math-container">$$\lim \limits_{h\rightarrow 0} [\frac { \cos h + i \sin h - 1 } {h} ] = \lim \limi... |
1,335,698 | <p>For this problem do I use the distance formula that I would use between two regular points? </p>
<p>$d=\sqrt{(x_2−x_1)^2+(y_2−y_1)^2}$</p>
<p>The distance between points $u$ and $v$ on the $x$-axis is given by $|u-v|$. Solve $|x-5|+|x-6|=1$ (think geometrically).</p>
| Mythomorphic | 152,277 | <p><img src="https://i.stack.imgur.com/uYnuy.png" alt="enter image description here"></p>
<p>For $x\le5$,</p>
<p>$$|x-6|-|x-5|=1$$
So by adding the original equation,
$$2|x-6|=2$$
$$x=5/7\text{ (rej.)}$$</p>
<p>Similarly </p>
<p>For $x\ge6$, we get $x=6/4\text {(rej.)}$</p>
<p>For $5<x<6$, Let $x=5+k$, where... |
306,788 | <p>I understand determinants but I cannot understand the following question, can someone explain it to me ?</p>
<p>Suppose that a $4 x 4$ matrix with rows $v_1,v_2,v_3$ and $v_4$ has determinant det A = -1. Find the following determinants: </p>
<p>$$det \begin{bmatrix}v_1\\6v_2\\v_3\\v_4 \end{bmatrix}=$$
$$det \beg... | Sean Ballentine | 62,751 | <p>Split the determinant across multiplication and notice that:</p>
<p>$\begin{bmatrix} v_1 \\ 6v_2 \\ v_3 \\ v_4 \end{bmatrix} = \begin{bmatrix} 1 \ 0\ 0\ 0 \\ 0\ 6\ 0\ 0 \\ 0\ 0\ 1\ 0\\ 0\ 0\ 0\ 1 \end{bmatrix}$ $\begin{bmatrix} v_1 \\ v_2 \\ v_3 \\ v_4 \end{bmatrix}$</p>
<p>$\begin{bmatrix} v_2 \\ v_1 \\ v_4 \\ v_... |
2,418,916 | <blockquote>
<p>Find how many terms there are in this geometric sequence:</p>
<p><span class="math-container">$-1, 2, -4, 8, ..., -16777216$</span></p>
</blockquote>
<p>My attempt:</p>
<p><span class="math-container">$a_k=a.r^{k-1}$</span></p>
<p>And in this sequence:</p>
<p><span class="math-container">$a=-1$</span>, ... | Vasili | 469,083 | <p>$-16777216=(-1){(-2)}^{k-1}={(-1)}^k2^{k-1}=-2^{k-1}$ (exponent can't be negative so minus has to come from -1)</p>
|
2,418,916 | <blockquote>
<p>Find how many terms there are in this geometric sequence:</p>
<p><span class="math-container">$-1, 2, -4, 8, ..., -16777216$</span></p>
</blockquote>
<p>My attempt:</p>
<p><span class="math-container">$a_k=a.r^{k-1}$</span></p>
<p>And in this sequence:</p>
<p><span class="math-container">$a=-1$</span>, ... | Oscar Lanzi | 248,217 | <p>To use logarithms, take the absolute values of the terms. Then you have</p>
<p>$|-1|×|-2|^{k-1}=|-16777216|$</p>
<p>$1×2^{k-1}=16777216$</p>
<p>where all numbers are positive and the logarithms can be manipulated without trouble. When you find $k$ you must check against the original equation with the negative s... |
482,801 | <p>What does it mean that the characteristic function <span class="math-container">$f(x)=1_{[b \le x \lt \infty]}$</span> is right continuous with left limits? Here <span class="math-container">$x ,b \in \mathbb{R}$</span>.</p>
| Anthony Carapetis | 28,513 | <p>It means that at every point $x_0\in \mathbb{R}$, both one-sided limits $$ \lim_{x \nearrow x_0} f(x) \textrm{ and } \lim_{x \searrow x_0} f(x) $$ exist, and furthermore that $f(x_0) = \lim_{x \searrow x_0} f(x)$. In this example the function is fully continuous (both one-sided limits are equal to the function) ever... |
61,106 | <p>Let <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> be Poisson random variables with means <span class="math-container">$\lambda$</span> and <span class="math-container">$1$</span>, respectively. The difference of <span class="math-container">$X$</span> and <span class="math-cont... | Michael Lugo | 143 | <p>Numerically, $\lim_{\lambda \to 0} F^\prime(\lambda) = 1/e$. Heuristically this should be true because to have $X > Y$ when $\lambda$ is very small, the most likely case will be $X = 1, Y = 0$ by far; that occurs with probability $\lambda e^{-\lambda} e^{-1}$.</p>
<p>Maple gives an explicit formula for $F^\prime... |
118,074 | <p>Just based on some reading, I know that every Möbius transformation is a bijection from the Riemann sphere to itself. </p>
<p>I'm curious about the converse. For any holomorphic bijection on the sphere, why is it necessarily a Möbius transformation? Is there a proof or reference of why this converse is true? Thanks... | Mariano Suárez-Álvarez | 274 | <p>Suppose $f$ is an holomorphic bijection of the sphere to itself. There is a Moebius transformation $g$ which maps $f(\infty)$ to $\infty$. Let $h=g\circ f$, which is again an holomorphic bijective map of the sphere to itself, and which maps $\infty$ to $\infty$. It follows that $h(\mathbb C)\subseteq\mathbb C$, beca... |
3,896,817 | <blockquote>
<p>Solve <span class="math-container">$x^2 \equiv 12 \pmod {13}$</span></p>
</blockquote>
<p>By guessing I can say that the solutions are <span class="math-container">$5$</span> and <span class="math-container">$8$</span>, but is there another way to find the solution besides guessing?</p>
| cosmo5 | 818,799 | <p><strong>Hint :</strong></p>
<p>As <span class="math-container">$x^2 \equiv a^2 \pmod {n}$</span> is easiest to handle, and <span class="math-container">$12\equiv25 \pmod {13}$</span>, we have</p>
<p><span class="math-container">$$ x^2 \equiv (\pm 5)^2 \pmod {13}$$</span></p>
|
3,896,817 | <blockquote>
<p>Solve <span class="math-container">$x^2 \equiv 12 \pmod {13}$</span></p>
</blockquote>
<p>By guessing I can say that the solutions are <span class="math-container">$5$</span> and <span class="math-container">$8$</span>, but is there another way to find the solution besides guessing?</p>
| Daniel Schepler | 337,888 | <p>In the special case of trying to find a square root of <span class="math-container">$-1$</span> modulo a prime <span class="math-container">$p \equiv 1 \pmod{4}$</span>, we can use the following algorithm: first, select some random integer <span class="math-container">$a$</span> with <span class="math-container">$1 ... |
2,011,003 | <p>I stumbled upon this logic question in a math class recently. </p>
<p>My teacher told us that a statement that is not tested/is empty is true. For example, that if I stated that: "if the team A wins the game, I am gonna buy you a coke", and then team B goes on and wins the game, the statement would be true, indepen... | Bananach | 70,687 | <p>Without going into a formal treatment, what your teacher means is:</p>
<p>The statement $$\forall x\in X: A (x) \Rightarrow B (x) $$ is true if $A (x) $ is false for all $ x\in X$ , no matter what $ B $ is.</p>
<p>That your teacher is right follows from the DEFINITION of the right arrow $\Rightarrow $, that is... |
264,745 | <p>When I was learning statistics I noticed that a lot of things in the textbook I was using were phrased in vague terms of "this is a function of that" e.g. a statistic is a function of a sample from a distribution. I realized that while I know the definition of a function as a relation and I have an intuitive notion ... | Ittay Weiss | 30,953 | <p>The modern approach is, as you say, to view a function as a relation. Thus $f\subseteq A\times B$ is a function if it satisfies that if $(a,b)\in f$ and $(a,b')\in f$ then $b=b'$. It is then common to write $f(a)=b$ instead of $(a,b)\in f$.</p>
<p>This is a way to formalize the notion of $f$ defining its output as ... |
3,090,448 | <p>I have the following question to complete.</p>
<p>Let <span class="math-container">$X$</span> be an inner product space. Let <span class="math-container">$(e_{j})_{j\geq1}$</span> be an orthonormal sequence in <span class="math-container">$X$</span>. Show that,
<span class="math-container">\begin{align}
\sum_{j=1}^... | Kavi Rama Murthy | 142,385 | <p>Use Cauchy -Schwraz inequlairty for sequences rather than inner product. <span class="math-container">$|\sum \langle x,e_i \rangle \langle y,e_i \rangle| \leq (\sum |\langle x,e_i \rangle|^{2})^{1/2}(\sum |\langle y,e_i \rangle|^{2})^{1/2} \leq \|x\|\|y\|$</span>.</p>
|
1,664,081 | <p>Solve the equation $2^x - 3^{x-1}=-(x+2)^2$</p>
<p>How I got this question? I created this question so I know the answer. The answer is 5. But I have no idea how to solve it. Take note that I cannot do logarithm, guess and check and modulus. Does anybody know how to solve this? I have no idea how to start.</p>
<p>... | SS_C4 | 242,290 | <p>Well, from $2^x - 3^{x-1} = -(x+2)^2$, $2^x = 3^{x-1} - (x+2)^2$. </p>
<p>LHS is always even, and $3^n$ is always odd. Therefore $(x+2)^2$ has to be odd, $\Rightarrow$ x is odd. </p>
<p>Also, from the first equation, $2^x < 3^{x-1}$. This is true for $x > 2$.
Since x is odd, the new condition for $x$ is $x \... |
1,712,289 | <p>If from twice the greater of two numbers 17 is subtracted, the result is half the other number. If from half the greater number 1 is subtracted, the result is two-thirds of the smaller number.</p>
<p>$$2x - 17 = \frac{ 1 }{2}y$$</p>
<p>$$\frac{ x }{2} - 1 = \frac{ 2 }{3}y$$</p>
<p>$$-17 - 4 = \frac{ 1 }{2}y - \fr... | Nikunj | 287,774 | <p><strong>Half</strong> of the greater number, this means $\frac{x}{2}$ in the second equation, rest seems all right!</p>
<p>Now, if you multiply the lower equation by $4$ and subtract it from the first, you get:
$$-13=\frac{y}{2}-\frac{8y}{3}$$
$$\implies y=6$$ you can find that $$x=10$$</p>
|
1,712,289 | <p>If from twice the greater of two numbers 17 is subtracted, the result is half the other number. If from half the greater number 1 is subtracted, the result is two-thirds of the smaller number.</p>
<p>$$2x - 17 = \frac{ 1 }{2}y$$</p>
<p>$$\frac{ x }{2} - 1 = \frac{ 2 }{3}y$$</p>
<p>$$-17 - 4 = \frac{ 1 }{2}y - \fr... | Tom | 255,814 | <h3>This can be solved using substitution</h3>
<p>Here's a step by step solution:</p>
<ol>
<li><p>Solve for x in the first equation, <br><br><span class="math-container">$x=\frac{17}{2}+\frac{y}{4}$</span></p>
</li>
<li><p>Replace all occurrences of x in the second equation,<br><br><span class="math-container">$\frac{\... |
3,016,169 | <p>I want to prove or disprove that <span class="math-container">$C^1([a,b], \mathbb{R}^n)$</span> equipped with the norm <span class="math-container">$||x||=\underset{t\in[a,b]}{\sup}|x(t)|_{\mathbb{R}^n}+\underset{t\in[a,b]}{\sup}|\dot{x}(t)|_{\mathbb{R}^n}$</span> is a reflexive Banach space. </p>
<p>I figured out ... | Robert Israel | 8,508 | <p>We can embed <span class="math-container">$Y = C([a,b],\mathbb R)$</span> into <span class="math-container">$X = C^1([a,b], \mathbb R^n)$</span> by
<span class="math-container">$T(f)(x) = \int_a^x f(t)\; dt\; {\bf u}$</span> where <span class="math-container">$\bf u$</span> is some nonzero vector in <span class="ma... |
1,292,889 | <p>I've read the paper <a href="http://web.mit.edu/leozhou/www/gauss.pdf" rel="noreferrer">Least square fitting of a Gaussian function to a histogram</a> by Leo Zhou on how to perform a Least Square Fitting of a gaussian function to a histogram.</p>
<p>The Gaussian function used to fit the data is:
$$f(y)=A\exp\left(-... | JJacquelin | 108,514 | <p>The usual methods of non-linear regression involve iterative process starting from guessed values of the parameters. </p>
<p>There is a straight forward method (not iterative, no need for guessed values) which general principle is explain in the paper : <a href="https://fr.scribd.com/doc/14674814/Regressions-et-equ... |
1,553,391 | <p>Let $E$ be a measurable set of finite measure and $1\leq p_1 < p_2 \leq \infty$ . Then $L^{p_2} (E) \subseteq L^{p_1} (E)$ Furthermore $||f||_{p_1} \leq c \cdot ||f||_{p_2}$ for all $f$ in $L^{p_2}(E)$ where $c =[m(E)]^{\frac{p_2-p_1}{p_1p_2}} $ if $p_2<\infty$ and $c=[m(E)]^{\frac{1}{p1}}$ if $p_2 =\infty$ </... | Hamit | 277,958 | <p>$L^{p}(E)$s are not comparable unless you have finite measure space for whole space. </p>
|
1,379,513 | <p>A hot dog stand has 12 different toppings available. How many different kinds of hot dogs can be made, assuming the order of the toppings does not make a difference. I believe the correct answer is 882050, with the maximum varieties per number of toppings selected being 665280 when there six toppings. I am also n... | quid | 85,306 | <p>For each topping you can decide to use it or not to use it. </p>
<p>So for each topping you have $2$ ways. Thus in total you have $2^{12}$ ways. </p>
|
1,379,513 | <p>A hot dog stand has 12 different toppings available. How many different kinds of hot dogs can be made, assuming the order of the toppings does not make a difference. I believe the correct answer is 882050, with the maximum varieties per number of toppings selected being 665280 when there six toppings. I am also n... | Luca C. | 258,065 | <p>Also, you can view it like this:
$$
\sum_{k=0}^{12} C_{12,k} = \sum_{k=0}^{12} \binom{12}{k} = 4096
$$</p>
<p>i.e. for each number k of toppings, you get k-combinations between those elements, from the starting 12. We are using combinations without repetition because order of selection does not matter.</p>
<p>Nume... |
4,107,232 | <h2>Problem</h2>
<p>Robert is playing a game with numbers. If he has the number <span class="math-container">$x$</span>, then in the next move, he can do one of the following:</p>
<ul>
<li>Replace <span class="math-container">$x$</span> by <span class="math-container">$\lceil{\frac{x^2}{2}}\rceil$</span></li>
<li>Repla... | Calvin Lin | 54,563 | <p>This is not a valid solution.<br />
Ravi pointed out that there is an error.</p>
<hr />
<p><strong>Claim:</strong> For any integer <span class="math-container">$n$</span>, there exists integers <span class="math-container">$K , L \geq 0$</span> such that <span class="math-container">$$ n\times 3^K \leq 2 \times 10 ... |
71,608 | <p>Consider the following question:</p>
<p>Is there a family $\mathcal{F}$ of subsets of $\aleph_\omega$ that satisfies the following properties?</p>
<p>(1) $|\mathcal{F}|=\aleph_\omega$</p>
<p>(2) For all $A\in \mathcal{F}$, $|A|<\aleph_\omega$</p>
<p>(3) For all $B\subset \aleph_\omega$, if $|B|<\aleph_\om... | Andreas Blass | 6,794 | <p>There is no such family $\mathcal F$. Suppose, toward a contradiction, that you had such an $\mathcal F$ and list it in a sequence of order-type $\aleph_\omega$. For each $n\in\omega$, let $\mathcal F_n$ consist of the first $\aleph_n$ members of the sequence that have cardinality at most $\aleph_n$. Notice that ... |
3,191,402 | <p>I have tried to answer by taking change the variable <span class="math-container">$\theta$</span> to <span class="math-container">$\theta/2$</span>, so the integration is now over unit circle, then I have taken <span class="math-container">$z=e^{i\theta}$</span>. Now I tried to use residue formula for integration, b... | Doug M | 317,176 | <p><span class="math-container">$\int_0^{\pi} (a+\cos x)^n \ dx = \frac 12 \int_0^{2\pi} (a+\cos x)^n \ dx\\
\cos x = \frac 12 (e^{ix} + e^{-ix})\\
\frac 1{2^{n+1}} \int_0^{2\pi} (2a+e^{ix} + e^{-ix})^n \ dx
z = e^{ix}\\
dx = \frac {1}{iz}\ dz$</span></p>
<p><span class="math-container">$\frac 1{2^{n+1}i} \oint_{|z| =... |
3,191,402 | <p>I have tried to answer by taking change the variable <span class="math-container">$\theta$</span> to <span class="math-container">$\theta/2$</span>, so the integration is now over unit circle, then I have taken <span class="math-container">$z=e^{i\theta}$</span>. Now I tried to use residue formula for integration, b... | Robert Israel | 8,508 | <p>If your integral is <span class="math-container">$J_n$</span>, the exponential generating function of the sequence <span class="math-container">$J_n$</span> is</p>
<p><span class="math-container">$$ \eqalign{g(x) &= \sum_{n=0}^\infty \int_0^\pi \frac{(a+\cos(\theta))^n x^n}{n!}\; d\theta \cr
&= \int_0^\pi ... |
1,987,026 | <p>So I know to get a probability like $P(2\leq X\leq 4)$, you simply do $P(X\leq4) - P(X\leq1)$, but when there is a question like $P(2<X<4)$ what am I supposed to do? </p>
<p>Not just limited to in between two values, I also don't know what to do if it's just $P(X<2)$, so far all our examples have been grea... | BGM | 297,308 | <p>The complement of $\{X \geq 2\}$ is $\{X < 2\}$, e.g. we have $\Pr\{X \geq 2\} = 1 - \Pr\{X < 2\}$</p>
<p>The first example you shown is imprecise. The general version should be
$$ \Pr\{2 \leq X \leq 4\} = \Pr\{X \leq 4\} - \Pr\{X < 2\}$$
Only, when you are given that $\Pr\{1 < X < 2\} = 0$, e.g. $X$... |
1,150,805 | <p>An unfair 3-sided die is rolled twice. The probability of rolling a 3 is $0.5$, the probability of rolling a 1 is $0.25$, and the probability of rolling a 2 is $0.25$. Let $X$ be the outcome of the first roll and $Y$ the outcome of the second.</p>
<ul>
<li><p>Find the Joint Distribution of $X$ and $Y$ in a Table.</... | Mike Pierce | 167,197 | <p>Let $X_n = \{1, 2, \dotsc, n\}$ be the finite set of $n$ elements (unique up to a bijective morphism). Denote the power set of a set $X_i$ as $\mathcal{P}(X_i)$. Let's write out the first few power sets of these finite sets:
$$\begin{align}
\mathcal{P}(X_0) = \mathcal{P}(\{\}) &= \{\{\}\} \\
\mathcal{P}(X_1)... |
127,493 | <p>How many number less than $k$ contain the digit $3$?
For instance:</p>
<p>How many number contain the digit $3$ in the following list?</p>
<pre><code>Table[n, {n, 33}]
</code></pre>
<p>$\lbrace 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, \
20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32... | Haohu Shen | 43,438 | <p>1.So if you meet such situations again which I mean you can't get the result you want in MMA,try to solve the same problem in other alternatives such as Maxima,Wolfram Alpha to see if there is a same result first.</p>
<p>2.If there is an difference among these platforms,try to debug your calculation process to make... |
4,315,283 | <p>Let <span class="math-container">$X$</span> be a continuous random variable, having pdf <span class="math-container">$f(x)$</span> and let be <span class="math-container">$Y=f(X)$</span>.
I have already proved these two following results:</p>
<ol>
<li>If <span class="math-container">$f(x)$</span> is strictly increas... | Tobsn | 414,776 | <p>Assume that <span class="math-container">$f$</span> is continuously differentiable and bijective on its support. Then by the classical transformation rule <span class="math-container">$Y$</span> has density <span class="math-container">$p$</span> with
<span class="math-container">\begin{equation}
p(x)=x|(f^{-1})'(x)... |
2,725,317 | <blockquote>
<p>For which $p,q\in \mathbb R$ is the following system stable?$$\frac{\mathrm dx}{\mathrm dt} = \begin{bmatrix} p & -q \\ q & p \end{bmatrix}x(t)$$ </p>
</blockquote>
<p>If I'm correct about this, isn't it just when the eigenvalues are $< 1?$ Or is there something more fancy to it? Any and ... | Robert Lewis | 67,071 | <p>Assuming that $p, q \in \Bbb R$ are constants, the system</p>
<p>$\dfrac{d \vec x(t)}{dt} = \begin{bmatrix} p & -q \\ q & p \end{bmatrix} \vec x(t) \tag 1$</p>
<p>has a well-known explicit solution. Suppose we set</p>
<p>$A = \begin{bmatrix} p & -q \\ q & p \end{bmatrix}, \tag 2$</p>
<p>and let<... |
1,156,907 | <p>I don't know anything about measure theory, I'm studying real analysis and this showed up in the book I'm reading as a way to characterize integrable functions. The author defined that a subset $X \subset \mathbb{R}$ has measure zero if for each $\epsilon > 0$ we can find infinitely countable open intervals $I_n$... | Community | -1 | <p>Given two sequences $x_n$ and $y_n$, we have the relation
$$
x_n\leq y_n\Rightarrow \sum_n x_n\leq \sum_n y_n.
$$
Therefore, if we set $x_n=\sum^{\infty}_{j=1}|I_{n_j}|$ and $y_n=\frac{\epsilon }{2^n}$, applying the above gives
$$
\sum_n\sum^{\infty}_{j=1}|I_{n_j}|<\sum_n\frac{\epsilon}{2^n}
$$</p>
|
1,156,907 | <p>I don't know anything about measure theory, I'm studying real analysis and this showed up in the book I'm reading as a way to characterize integrable functions. The author defined that a subset $X \subset \mathbb{R}$ has measure zero if for each $\epsilon > 0$ we can find infinitely countable open intervals $I_n$... | Anthony Peter | 58,540 | <p>We're exploiting Zeno's Paradox. Since each set has measure $0$, we can cover it by intervals whose total length is less than any positive real number. Since the union is countable, we can enumerate our sets of measure $0$ as $\{I_1, I_2, I_3, \ldots, \}$. Let $\mu(S) = (b-a)$ for $S=(a,b)$.</p>
<p>Let $\epsilon &g... |
4,350,015 | <p>I'm trying to learn calculus through self study and I happened upon the following exercise:</p>
<p><span class="math-container">$$ \int_{0}^{2} \sqrt{4 - x^2}\cdot\operatorname{sgn}(x-1) \,dx $$</span></p>
<p>Seeing the sgn I thought: well this is easy and concluded that since <span class="math-container">$ \int_{0}... | Taladris | 70,123 | <p>To expand on @José Carlos Santos' answer, you can compute <span class="math-container">$\int \sqrt{4-x^2}\; dx$</span> by trigonometric substitution: let <span class="math-container">$x=2\sin(t)$</span> with <span class="math-container">$-\frac{\pi}{2}\le t\le \frac{\pi}{2}$</span> . Then <span class="math-container... |
1,793,854 | <p>I am messed up on solving this question. What should I do first in order to get the answer ?</p>
<p><a href="https://i.stack.imgur.com/hE4rG.png" rel="nofollow noreferrer">This is the trigonometric function</a></p>
<p>$$ \lim \limits_{x \rightarrow 0} \frac{(a+x)\sec(a+x) - a \sec(a)}{x} $$</p>
| Claude Leibovici | 82,404 | <p>One solution, if allowed, uses Taylor series.</p>
<p>Built a round $x=0$, $$\sec(a+x)=\sec (a)+x \tan (a) \sec (a)+O\left(x^2\right)$$ $$(x+a)\sec(a+x)=a \sec (a)+x (\sec (a)+a \tan (a) \sec (a))+O\left(x^2\right)$$</p>
|
4,198,805 | <p>A <a href="https://en.wikipedia.org/wiki/Sober_space" rel="nofollow noreferrer">sober space</a> is a topological space such that every irreducible closed subset is the closure of exactly one point. Looking for examples I convinced myself that the following is true.</p>
<blockquote>
<p>Every finite <span class="math... | diracdeltafunk | 19,006 | <p>This is true. It suffices to show that every finite irreducible space has a generic point, since <span class="math-container">$T_0$</span> implies that generic points are unique. So, let <span class="math-container">$X$</span> be a finite irreducible space. Then <span class="math-container">$X$</span> is the union o... |
751,053 | <p>T : Rn → Rm is a linear transformation where n,m>= 2</p>
<p>Let V be a subspace of Rn
and let W ={T(v ) | v ∈ V}
. Prove completely that W
is a subspace of Rm. </p>
<p>For this question how do I show that the subspace is non empty, holds under scaler addition and multiplication! I have never proved subspaces with... | Community | -1 | <ul>
<li>We have $T(0_{\Bbb R^n})=0_{\Bbb R^m}\in W$ since $T$ is linear so $W\ne\emptyset$.</li>
<li>Let $y_1,y_2\in W$ and $\alpha\in\Bbb F$ then there's $x_1,x_2\in V$ s.t.
$$T(x_1)=y_1\qquad T(x_2)=y_2$$
but since $V$ is a linear subspace then $\alpha x_1+x_2\in V$ and by linearity of $T$ we have
$$T(\alpha x_1+ x_... |
878,373 | <p>Often I have heard about the link between Algebra (in particular Representations of Groups and Algebras) and some "indefinite" field of Physics.</p>
<p>I have a good preparation in Algebra and Representation Theory (in particular about Representations of Lie Algebras), and I'm fascinated with Physics. My idea is t... | user44670 | 44,670 | <p>I'm sorry, I was very stupid. We can just apply Ito's formula and get:
\begin{align*}
& \int_a^b f(t)dW_t=f(b)W_b-f(a)W_a-\int_a^b W_t f'(t)dt.
\end{align*}
This yields that indeed we can find the desired pathwise upper bound of $\int_a^b f(t)dW_t$ in terms of $||f||$, $||f'||$, $\sup_{t \in [a,b]}|W(t)|$.</p>
|
2,010,069 | <p>I am looking on the solution to this problem presented in the book <em>"Fifty Challenging Problems in Probability with Solutions"</em> by Mosteller (p.18-19).</p>
<blockquote>
<p>On average, how many times must a die be thrown until one gets a 6?</p>
</blockquote>
<p>There are many ways to solve this problem as... | Bobbie D | 317,218 | <p>Using the property of the transpose $\langle A^Tw,v\rangle = \langle w, Av\rangle$, I get:</p>
<p>$$\pmatrix{a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23}}\pmatrix{x \\ y \\ z} = \pmatrix{\langle A^T\pmatrix{1 \\ 0}, \pmatrix{x \\ y \\ z}\rangle \\ \langle A^T\pmatrix{0 \\ 1}, \pmatrix{x \\ y... |
2,010,069 | <p>I am looking on the solution to this problem presented in the book <em>"Fifty Challenging Problems in Probability with Solutions"</em> by Mosteller (p.18-19).</p>
<blockquote>
<p>On average, how many times must a die be thrown until one gets a 6?</p>
</blockquote>
<p>There are many ways to solve this problem as... | Jair Taylor | 28,545 | <p>It is easiest to see this in one dimension first.</p>
<p>Our goal is to show that any linear transformation $T: \mathbb{R}^n \rightarrow \mathbb{R}$ can be represented in the form $Tu = \beta^Tu$ for some $n$-dimensional vector $\beta$. Say that $u \in \mathbb{R}^n$; let $e_1, \ldots, e_n$ be the standard basis ve... |
119,481 | <p>I need to prove the following trigonometric identity:
$$ \frac{\sin^2(\frac{5\pi}{6} - \alpha )}{\cos^2(\alpha - 4\pi)} - \cot^2(\alpha - 11\pi)\sin^2(-\alpha - \frac{13\pi}{2}) =\sin^2(\alpha)$$</p>
<p>I cannot express $\sin(\frac{5\pi}{6}-\alpha)$ as a function of $\alpha$. Could it be a textbook error?</p>
| Pedro | 23,350 | <p>Some important translations: </p>
<p>$$\tag 1\sin(x\pm 2 \pi) = \sin x $$
$$\tag {1'}\cos(x\pm 2 \pi) = \cos x $$
$$\tag 2\cot(x\pm \pi)= \cot x$$
$$\tag {2'}\tan(x\pm \pi)= \tan x$$</p>
<p>$$\tag 3 \sin \left(\frac \pi 2 -x \right)=\cos x$$
$$\tag 4 \cos \left(\frac \pi 2 -x \right)=\sin x$$
$$\tag 5 \sin(\pi-x)=... |
1,878,734 | <p>Is it true that if an isomorphism $f$ maps a cyclic group $G$ to group $H$ that $H$ must also be cyclic? It seems intuitive but until I can actually prove it I'm always a bit dubious to believe it. </p>
| florence | 343,842 | <p>Suppose $G$ is cyclic and $\phi: G\to H$ is surjective. Let $G$ be generated by $a\in G$, i.e. $G = \langle a\rangle$. Let $b\in H$. Then there exists some $x\in G$ so that $b = \phi(x)$, as $\phi$ is surjective. Further, since $G$ is cyclic, we have $x = a^n$ for some $n$. Then $b = \phi(x) = \phi(a^n) =\phi(a)^n$.... |
2,793,983 | <p>For example I find myself wanting to write $x$ is an element of the integers from $1$ to $50$,</p>
<p>Is this the quickest way? </p>
<p>$x\in \left[ 1,50\right] \cap \mathbb{N} $</p>
<p>Also is this standard on here? $\mathbb{N} = \{0, 1, 2,\dotsc \}$,
$\mathbb{ℤ}_+ = \{1, 2, \dotsc \}$.</p>
| user21820 | 21,820 | <p>As others have said, you should always define non-standard notation, but here is one that you can consider (and is actually valid syntax in some programming languages):</p>
<blockquote>
<p>$[a\,..b]$ represents the integers from $a$ to $b$ inclusive.</p>
</blockquote>
<p>This is also compatible with the conventi... |
1,985,552 | <p>Where $p_n \rightarrow p$. I'm trying to prove that for $E=\{ p_n : n \in \mathbb{N}$ and $lim_{n\rightarrow \infty} p_n =p \}$, then $Cl(E)=E \cup \{p \}$ and $Cl(E)$ is compact. </p>
<p>Also, I'm currently using the definition of limit points as p is a limit point if $\forall r>0, (E \cap N_r(p)) \backslash \{... | Jack D'Aurizio | 44,121 | <p>The answer is given by the coefficient of $x^{30}$ in $A(x)\cdot B(x)\cdot C(x)\cdot D(x)$ where
$$A(x)=(x^1+x^2+x^2+x^4+x^5),\quad B(x)=(x^4+x^5+x^6+x^7+x^8+x^9)\\C(x)=(x^6+x^7+x^8+x^9+x^{10}+x^{11}),\quad D(x)=(x^{10}+x^{11}+x^{12}+x^{13}+x^{14}+x^{15})$$
That is also the coefficient of $x^9$ in $E(x)\cdot F(x)\cd... |
1,985,552 | <p>Where $p_n \rightarrow p$. I'm trying to prove that for $E=\{ p_n : n \in \mathbb{N}$ and $lim_{n\rightarrow \infty} p_n =p \}$, then $Cl(E)=E \cup \{p \}$ and $Cl(E)$ is compact. </p>
<p>Also, I'm currently using the definition of limit points as p is a limit point if $\forall r>0, (E \cap N_r(p)) \backslash \{... | Felix Marin | 85,343 | <p>$\newcommand{\bbx}[1]{\,\bbox[8px,border:1px groove navy]{{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\newcommand{\ic... |
4,483,507 | <p>How can I change <span class="math-container">$\dfrac{-(3-\sqrt{3})}{(3+\sqrt{3})}$</span> to <span class="math-container">$\dfrac{1-\sqrt{3}}{1+\sqrt{3}}$</span>?</p>
<p>Background:</p>
<p>I tried solving <span class="math-container">$\tan(345°)$</span> with the trigonometric angle <em><strong>sum/difference</stron... | Suzu Hirose | 190,784 | <p>The logic of the proof is correct, but it reads like word salad. For example,</p>
<blockquote>
<p>Let <span class="math-container">$d$</span> be the standard Euclidean metric.</p>
</blockquote>
<p>You've put that in twice, and after you've already used <span class="math-container">$d$</span>. You're considering a me... |
1,627,619 | <p>Could anyone please check my solution to the following problem?</p>
<blockquote>
<p><strong>Problem:</strong> Let $f(x, y) = (x^2 + y^2)e^{-(x^2 + y^2)}$. Find global extrema of $f$ on $M = {\mathbf R}^2$.</p>
</blockquote>
<p><strong>Proposed solution:</strong> Taking partial derivatives of $f$, we conclude tha... | Travis Willse | 155,629 | <p>This looks correct to me, but one can treat this a little more efficiently: If $f$ achieves a global extremum at $(x, y)$, then the map $g: [0, \infty) \to \Bbb R$ defined by $$g(r) := r^2 e^{-r^2}$$ achieves a global extremum at $\sqrt{x^2 + y^2}$ and vice versa. Since $g$ is differentiable, to determine the latter... |
60,152 | <p>What is the motivation behind topology?</p>
<p>For instance, in real analysis, we are interested in rigorously studying about limits so that we can use them appropriately. Similarly, in number theory, we are interested in patterns and structure possessed by algebraic integers and algebraic prime numbers.</p>
<p>So... | Qiaochu Yuan | 232 | <p><a href="https://math.stackexchange.com/questions/31859/what-concept-does-an-open-set-axiomatise">This math.SE question</a> may be relevant, but not pedagogically optimal. </p>
<p>Pedagogically I think the simplest answer is to axiomatize topological spaces via the <a href="http://en.wikipedia.org/wiki/Kuratowski_c... |
457,557 | <p>Use a triple integral to find the volume of the solid: The solid enclosed by the cylinder $$x^2+y^2=9$$ and the planes $$y+z=5$$ and $$z=1$$<br>
This is how I started solving the problem, but the way I was solving it lead me to 0, which is incorrect. $$\int_{-3}^3\int_{-\sqrt{9-y^2}}^{\sqrt{9-y^2}}\int_{1}^{5-y}dzdx... | apnorton | 23,353 | <p>Ok. So you have the triple integral:
$$\begin{align}
\int_{-3}^3\int_{-\sqrt{9-y^2}}^{\sqrt{9-y^2}}\int_1^{5-y} \;dz\;dx\;dy
&= \int_{-3}^3\int_{-\sqrt{9-y^2}}^{\sqrt{9-y^2}}4-y\;dx\;dy \\
&=\int_{-3}^34x-xy\Bigg|_{-\sqrt{9-y^2}}^{\sqrt{9-y^2}}\;dy \\
&=\int_{-3}^38\sqrt{9-y^2}-2y\sqrt{9-y^2}\;dy \\
&am... |
1,917,790 | <p>Can anyone help me to solve this? </p>
<blockquote>
<p>Determine the value or values of $k$ such that $x + y + k = 0$ is tangent to the circle $x^2+y^2+6x+2y+6=0$.</p>
</blockquote>
<p>I don't know how to calculate the tangent.</p>
| Michael Hoppe | 93,935 | <p>The tangent's gradient is $-1$. From $$(x+3)^2+(y+1)^2=4$$ we know that the normal's gradient is $(y+1)/(x+3)$, so we must have $(y+1)/(x+3)=1$, that is $y+1=x+3$, hence $2(x+3)^2=4$. We conclude $x=-3\pm\sqrt2$ and $y=x+2=-1\pm\sqrt2$.</p>
<p>So the points where the tangents touch the circle are $(-3\pm\sqrt2,-1... |
1,740,458 | <p>I saw somewhere on Math Stack that there was a way of finding integrals in the form $$\int \frac{dx}{a+b \cos x}$$ <strong>without</strong> using Weierstrass substitution, which is the usual technique. </p>
<p>When $a,b=1$ we can just multiply the numerator and denominator by $1-\cos x$ and that solves the problem ... | user5713492 | 316,404 | <p>Kepler found the substitution when he was trying to solve the equation
$$\ell=mr^2\frac{d\nu}{dt}=\text{constant}$$
where $\ell$ is the orbital angular momentum, $m$ is the mass of the orbiting body, the true anomaly $\nu$ is the angle in the orbit past periapsis, $t$ is the time, and $r$ is the distance to the attr... |
484,367 | <p>I've been trying to find a tight upper bound for the series</p>
<p>$$S (x) = e^{-x} \sum_{k=0}^{\infty} \frac{x^k}{k!} \sqrt{k+1}$$</p>
<p>So far, I've managed to get a reasonable bound for small values of $x$ by using the inequality $\sqrt{k+1} \leq \sqrt{\frac{k^{2}}{4} + k + 1} = \frac{k}{2} + 1 ~\forall~k \geq... | Raymond Manzoni | 21,783 | <p>An asymptotic expansion for your series $\;\displaystyle S (x) := e^{-x} \sum_{k=0}^{\infty} \sqrt{k+1}\frac{x^k}{k!} \;$
seems to be, as $\,x\to +\infty$ :
$$S(x)\sim\sqrt{x}\left(1+\frac3{8\;x}+\frac 1{128\;x^2}+\frac 9{1024\;x^3}+O\left(\frac 1{x^4}\right)\right)$$
I have no proof for that sorry... (the ideas us... |
3,046,979 | <p>Recently I've been trying to tackle the integral <span class="math-container">$\int_0^{\frac{\pi}{2}} \ln(\sin(\theta))d\theta$</span> using the Beta function
<span class="math-container">$$\frac{B(\frac{x}{2},\frac{1}{2})}{2}=\int_0^{\frac{\pi}{2}} \sin^{x-1}(\theta)d\theta=\frac{\sqrt{\pi}}{2}\left(\Gamma\left(\fr... | Jack D'Aurizio | 44,121 | <p>Cleaner approach: for any <span class="math-container">$\alpha\geq 0$</span>,</p>
<p><span class="math-container">$$ \int_{0}^{\pi/2}\left(\sin\theta\right)^{\alpha}\,d\theta = \int_{0}^{1}\frac{u^\alpha}{\sqrt{1-u^2}}\,du = \frac{1}{2}\int_{0}^{1}v^{\frac{\alpha-1}{2}}(1-v)^{-1/2}\,dv = \frac{\Gamma\left(\frac{\al... |
3,179,505 | <p>Help me please , I am not able to solve this problem.I have tried in many ways to figure out such as Ration test , Integral test , Comparison test , Limit Comparison Test , Root Test but i can't find the way out . This is my first question and i'm not good at English. If there is something wrong or you are not comfo... | 5xum | 112,884 | <p>Taking the discrete metric on <span class="math-container">$X_1\times X_2$</span> will probably be enough to find a counterexample...</p>
|
4,257,962 | <p>By definition - A real number is algebraic if it is a root of a non-zero polynomial equation with rational coefficients. What does non-zero polynomial equation mean?</p>
<p>Well, an equation f(x) = x -5, becomes zero when x = 5, so this is a zero polynomial equation. Is the definition saying that the equation should... | Tito Eliatron | 84,972 | <p>Let <span class="math-container">$P_0(x):=0$</span> be the ZERO-POLYNOMIAL.</p>
<p>If we not avoid this pathological case, then every real number would be algebraic, since every real number is a "solution" of the equation <span class="math-container">$P_0(x)=0$</span> (even <span class="math-container">$\p... |
3,970,959 | <blockquote>
<p><span class="math-container">$S = \frac{1}{1001} + \frac{1}{1002}+ \frac{1}{1003}+ \dots+\frac{1}{3001}$</span>.</p>
</blockquote>
<blockquote>
<p>Prove that <span class="math-container">$\dfrac{29}{27}<S<\dfrac{7}{6}$</span>.<br></p>
</blockquote>
<p>My Attempt:<br>
<span class="math-container... | TonyK | 1,508 | <h2>Upper bound <span class="math-container">$\dfrac{7}{6}$</span></h2>
<p>Your question describes a method of breaking the range up into sub-ranges, and then estimating the sum over a sub-range by noting that each term is <span class="math-container">$\le$</span> the first term. But we can do much better if we estimat... |
3,970,959 | <blockquote>
<p><span class="math-container">$S = \frac{1}{1001} + \frac{1}{1002}+ \frac{1}{1003}+ \dots+\frac{1}{3001}$</span>.</p>
</blockquote>
<blockquote>
<p>Prove that <span class="math-container">$\dfrac{29}{27}<S<\dfrac{7}{6}$</span>.<br></p>
</blockquote>
<p>My Attempt:<br>
<span class="math-container... | Anindya Prithvi | 811,225 | <p><span class="math-container">$$\sum_{n=1}^{2001}\frac{1}{1000\left(1+\frac{n}{1000}\right)} =\int_{0}^{2}\frac{1}{\left(1+x\right)}dx \approx \ln3$$</span></p>
<p><span class="math-container">$$\frac{29}{27} < \ln3 <\frac{7}{6}$$</span></p>
<p>Used:</p>
<ol>
<li>Summation to integration when step size is small... |
3,080,230 | <p>On <a href="https://en.wikipedia.org/wiki/Net_(mathematics)#Properties" rel="nofollow noreferrer">Wikipedia</a> it states that a space <span class="math-container">$X$</span> is compact if and only if every net has a convergent subnet. It then states that a net in the product topology has a limit if and only if each... | SmileyCraft | 439,467 | <p>The problem in the reasoning in the OP is that a net can admit no subnet that is a sequence. This is because a subnet needs to be final. For example if the index set of the subnet is <span class="math-container">$\omega_1$</span>, the first uncountable ordinal, then there exists no final function <span class="math-c... |
56,847 | <p>What are the angles formed at the center of a tetrahedron if you draw lines to the vertices?</p>
<p>I'm trying to make these:</p>
<p><img src="https://i.stack.imgur.com/FRUi8.jpg" alt="caltrop"> </p>
<p>I need to know what angles to bend the metal.</p>
| Mark Bennet | 2,906 | <p>(Assuming the tetrahedron is supposed to be regular) Take the tetrahedron with vertices $(1,1,1), (1,-1,-1), (-1,1,-1), (-1,-1,1)$, which has centre at the origin, and use the dot product formula:</p>
<p>$a\cdot b = |a| |b|\cos\theta$ </p>
<p>which gives $\cos\theta=-\frac13$</p>
|
254,030 | <p>I am following a course in basic algebra, and we have covered rings & groups in class, but I am having trouble visualising them. Are there applications of group &/or ring theory that can be more easily visualized than the abstract object? For instance, are there objects, or properties of objects, that behave... | still_learning | 42,808 | <p>Dihedral groups arise frequently in art and nature. Many of the
decorative designs used on floor coverings, pottery, and buildings have
one of the dihedral groups as a group of symmetry. Corporation logos
are rich sources of dihedral symmetry. Chrysler’s logo has D5 as a
symmetry group, and that of Mercedes-Benz has... |
258,704 | <p>How can I solve a system of linear congruences as such?</p>
<p><span class="math-container">$$\begin{align*}
3x+2y+28z &= 9 \pmod {29} \\
5x+27y+z &= 9 \pmod {29} \\
2x+y+z &= 6 \pmod {29}
\end{align*}$$</span></p>
<p>I tried it this way as a system of equations, but no luck:</p>
<pre><code>eqn1 = Full... | cvgmt | 72,111 | <pre><code>Solve[{3*x + 2*y + 28*z == 9 + (29*i), 5*x + 27*y + z == 9 + (29*j),
2*x + y + z == 6 + (29*k)}, {x, y, z}, {i, j, k}, Integers,
GeneratedParameters -> c]
</code></pre>
<blockquote>
<p><code>{{x -> ConditionalExpression[ 24 + 29 c[1], (c[1] | c[2] | c[3]) ∈ Integers], y -> ConditionalExpressio... |
3,534,377 | <p>I've followed this tutorial (<a href="http://web.eecs.utk.edu/~jplank/plank/papers/CS-96-332.pdf" rel="nofollow noreferrer">http://web.eecs.utk.edu/~jplank/plank/papers/CS-96-332.pdf</a>) which introduces Reed-Solomon coding and therefore covers finite fields. My problem is with one of the examples of how the logari... | Bram28 | 256,001 | <p>As already pointed out in the comments, <span class="math-container">$A$</span> and <span class="math-container">$B$</span> are sets, not statements.</p>
<p>Still, there are obvious connections between sets and logical statements. For example, the set <span class="math-container">$A \cap B$</span> is the set of all... |
3,534,377 | <p>I've followed this tutorial (<a href="http://web.eecs.utk.edu/~jplank/plank/papers/CS-96-332.pdf" rel="nofollow noreferrer">http://web.eecs.utk.edu/~jplank/plank/papers/CS-96-332.pdf</a>) which introduces Reed-Solomon coding and therefore covers finite fields. My problem is with one of the examples of how the logari... | R. Burton | 614,269 | <p>For any set <span class="math-container">$S$</span>, there is a predicate <span class="math-container">$P_S$</span> such that <span class="math-container">$x\in S\iff P_S(x)$</span> (in other words, <span class="math-container">$S=\{x:P_S(x)\}$</span>). For instance, if we are working in the domain of integers we ca... |
959,219 | <p>let $a,b,c>0$, and such
$$a^2+b^2+c^2<2ab+2bc+2ca$$</p>
<p>show that
$$a^4+b^4+c^4+6(a^2b^2+b^2c^2+a^2c^2)+4abc(a+b+c)<4(ab+bc+ac)(a^2+b^2+c^2)$$</p>
<p>I know this indentity:
$$a^2+b^2+c^2-2(ab+bc+ac)
=-(\sqrt{a}+\sqrt{b}+\sqrt{c})(-\sqrt{a}+\sqrt{b}+\sqrt{c})(\sqrt{a}-\sqrt{b}+\sqrt{c})(\sqrt{a}+\sqrt{b... | James Harrison | 135,585 | <p>$$
\because a,b,c > 0\\
(a+b+c)^4 = a^4 + b^4 + c^4 + 4(a^3 b + a^3 c + b^3 a + b^3 c + c^3 a + c^3 b) + 6(a^2 b^2 + a^2 c^2 + b^2 c^2) + 12abc(a+b+c) > a^4 + b^4 + c^4 + 6(a^2 b^2 + b^2 c^2 + a^2 c^2 ) + 4abc(a+b+c)$$</p>
<p>From your starting condition we get:</p>
<p>$$
a^2 + b^2 + c^2 - 2(ab+ac+bc) <0 ... |
3,236,067 | <p>I am having some trouble understanding where some linear boundary conditions are derived from </p>
<p>The following is an extract from my lecture notes on boundary value problems for second-order Linear ODE's</p>
<blockquote>
<p>In this section we are going to consider the different situation when some condition... | Community | -1 | <p>There is no derivation. Those are the three types of boundary conditions generally seen.</p>
<p><span class="math-container">$$ y(x_{1}) = b_{1}, y(x_{2}) = b_{2} $$</span></p>
<p>is known as a Dirichlet Boundary Condition</p>
<p><span class="math-container">$$ y^{'}(x_{1}) = b_{1}, y^{'}(x_{2}) = b_{2} $$</span>... |
3,017,928 | <p>Doing some self study from the text <em>Basic Mathematics</em> by Serge Lang
I ran into an exercise question which I can't seem to wrap my head around.
The question is:</p>
<p>Express the following expressions in the form <span class="math-container">$2^m3^na^rb^s$</span> ,where <span class="math-container">$m,n,r,... | Xavier Stanton | 620,797 | <p>I also have another way to do it. If you distribute the <span class="math-container">$8 a^2 b^3$</span> with the <span class="math-container">$27 a^4$</span>, you will get <span class="math-container">$216 a^6 b^3$</span>. Then, distribute that value with <span class="math-container">$2^5 a b$</span> and you will ge... |
615,375 | <p>I would appreciate if somebody could help me with the following problem</p>
<p>Q: Let $f:[0,1]\longrightarrow \Bbb R$ be a continuously function such that
$$m\leq f(x)\leq M, m+M=1 (m:\text{minimum of} f(x), M:\text{maximum of} f(x) )$$
Prove that for every $x \in[0,1]$, there exists $c\in[0,1]$ such that $f(c)=1-... | Mathronaut | 53,265 | <p>Note that, $m= 1-M \le 1-f(x) \le 1-m =M$, Now apply Intermediate Value theorem to get $c\in [0,1]$ st $f(c)=1-f(x)$</p>
|
61,047 | <p>I can add the value of a slider to the right of it using the Appearance-->Labelled option, but what if I want to add text after the automatic label. How can I do that?</p>
<p>Normally I want to do this to show the units of the value. For example, if the slider label is "4.7", I might want it to read "4.7 meters".</... | Nasser | 70 | <p>There are many ways to do this. The most basic is to use <code>Control</code>, added few versions earlier just for this purpose. Here is an example. <code>Control</code> can be inserted inside <code>Row</code> or <code>Column</code> or <code>Grid</code> for example</p>
<pre><code>Manipulate[x,
Row[{Control[{{x, 1,... |
1,363,213 | <p>I am given a chessboard of size $8*8$. In this chessboard there are two holes at positions $(X1,Y1)$ and $(X2,Y2)$. Now I need to find the maximum number of rooks that can be placed on this chessboard such that no rook threatens another. </p>
<p>Also no two rooks can threaten each other if there is hole between the... | hmakholm left over Monica | 14,366 | <p>For most positions of the holes there seems to be room for 10 rooks.</p>
<p>If the holes are in the same row (not on the edge, not touching each other, and with at least two rows above and below it), then place 7 rooks as</p>
<pre><code> R
R
R o R o R
R
R
</code></pre>
<p>There are then 3 rows and ... |
1,363,213 | <p>I am given a chessboard of size $8*8$. In this chessboard there are two holes at positions $(X1,Y1)$ and $(X2,Y2)$. Now I need to find the maximum number of rooks that can be placed on this chessboard such that no rook threatens another. </p>
<p>Also no two rooks can threaten each other if there is hole between the... | David K | 139,123 | <p>We can get an algorithm to determine the maximum number of rooks
by considering attacks along ranks (parallel to one axis of the board)
separately from attacks along files (parallel to the other axis).</p>
<p>So first consider only attacks along ranks.
A rank with a single hole in it can still hold only one rook if... |
773,324 | <p>I know Dijkstra's algorithm to find the shortest way between 2 nodes, but is there a way to find the shortest path between 3 nodes among $n$ nodes? Here are the details:</p>
<p>I have $n$ nodes, some of which are connected directly and some of which are connected indirectly, and I need to find the shortest path bet... | ml0105 | 135,298 | <p>I'd say Phicar's solution of Floyd-Warshall's all-pairs, shortest paths algorithm is your best choice. Of course, this problem is NP-Hard. It's the optimal Hamiltonian Path problem, which is equivalent to the Traveling Salesman Problem.</p>
<p>The Floyd-Warshall algorithm can be executed in polynomial time. However... |
2,304,448 | <blockquote>
<p>Does any group homomorphism $\Bbb Z \to \Bbb Z/n$ have kernel isomorphic to $\Bbb Z$?</p>
</blockquote>
<p>Here $n$ is any natural number.</p>
| MichaelGaudreau | 570,438 | <p>I like Tomek's answer a lot. But here is another answer that is essentially the same but is phrased slightly differently. </p>
<p>First observe that the proof of the usual open mapping theorem still works if instead of <span class="math-container">$Y$</span> Banach and <span class="math-container">$T$</span> surjec... |
1,712,256 | <p>I'm given some equations.</p>
<p>The first one, $x^3+2x^2-8x+1$ wants me to find the tangent line at $x=2$.</p>
<p>The second, (x^1.5) - (x^1/2) wants me to find the tangent line at $x=4$.</p>
<p>How would I go about solving this algebraically? I have to be able to prove the answers are $y=12x-23$ and $y=2.75x-5$... | Brian Tung | 224,454 | <p>If you have a function $f(x)$, then at a given point $x_0$, the tangent line has slope $f'(x_0)$ and goes through the point $(x_0, f(x_0))$. The general equation of a line through a point $(x_0, y_0)$ and slope $m$ is</p>
<p>$$
y-y_0 = m(x-x_0)
$$</p>
<p>So in the first problem, $x_0 = 2$ and $f(x) = x^3+2x^2-8x+... |
512,118 | <p>Suppose $X$ is a metric space, $z$ is in $X$ and $(x_n)$ is a sequence in $X$. </p>
<p>Then what does it mean to say that, $z$ is in the "<em>closure of every tail of $(x_n)$</em>."</p>
<p>What does "<em>closure</em>" of every tail, mean ?</p>
| Prahlad Vaidyanathan | 89,789 | <p>It means that, for any $n \in \mathbb{N}$, consider the set
$$
S_n = \{x_n, x_{n+1}, x_{n+2}, \ldots\}
$$
Then, $z\in \overline{S_n}$ for all $n$ (here, $S_n$ is a tail of the sequence)</p>
|
2,003 | <p>I use some custom shortcut keys in <code>KeyEventTranslations.tr</code>. One is for the <code>Delete All Output</code> function: </p>
<pre><code>Item[KeyEvent["w", Modifiers -> {Control}],
FrontEnd`FrontEndExecute[FrontEnd`FrontEndToken["DeleteGeneratedCells"]]]
</code></pre>
<p>or simply:</p>
<pre><code>... | Albert Retey | 169 | <p>This just adds another hack for the <code>Quit</code> without confirm. It's not especially nice and I also haven't tested it in <code>KeyEventTranslations.tr</code> but it works from a Button with <code>Evaluator -> None</code> in versions 6,7 and 8 on Windows:</p>
<pre><code>FrontEnd`FrontEndExecute[{
FrontEn... |
449,413 | <p>I'm trying to construct a norm on the space $\mathcal{D}(\Omega) := \{ f \in C^\infty(\Omega) | f $ is compactly supported on $ \Omega \}$ where $\Omega$ is an open subset of $\mathbb{R}$. I want this norm to include, somehow, the $L^\infty$-norms of <strong>all</strong> the derivatives of the smooth function to whi... | youler | 25,895 | <p>I'm not an expert in functional analysis, but I think this answers your question.</p>
<p>The topology of $\mathcal{D}(\Omega)$ is not induced by a norm. There is a discussion of this in chapter 1 of Rudin's Functional Analysis book. Briefly, a topological vector space whose topology is induced by a norm is locally ... |
439,745 | <blockquote>
<p>Prove:$|x-1|+|x-2|+|x-3|+\cdots+|x-n|\geq n-1$</p>
</blockquote>
<p>example1: $|x-1|+|x-2|\geq 1$</p>
<p>my solution：(substitution)</p>
<p>$x-1=t,x-2=t-1,|t|+|t-1|\geq 1,|t-1|\geq 1-|t|,$</p>
<p>square,</p>
<p>$t^2-2t+1\geq 1-2|t|+t^2,\text{Since} -t\leq -|t|,$</p>
<p>so proved.</p>
<p><em>ques... | Christian Blatter | 1,303 | <p><strong>(Edited)</strong></p>
<p>Given a real-valued data set ${\bf y}=(y_k)_{1\leq k\leq n}$ the function
$$f(x):=\sum_{k=1}^n |x-y_k|$$
assumes its minimum at the <em>median</em> $\mu$ of ${\bf y}$. When $y_1\leq y_2\leq \ldots\leq y_n$ and $n=2m+1$ then $\mu=y_{m+1}$ (the "middle value"), and if $n=2m$ then $f$ ... |
1,250,132 | <p>Below is part of a solution to a critical points question. I'm just not sure how the equation on the left becomes the equation on the right. Could someone please show me the steps in-between? Thanks.</p>
<blockquote>
<p>$$\frac{-1}{x^2}+2x=0 \implies 2x^3-1=0$$</p>
</blockquote>
| Nikita | 509,540 | <p>For $x\neq 0$, we have $0= - \frac{1}{x^2} + 2x =- \frac{1}{x^2} + \frac{x^2}{x^2} \times 2x = - \frac{1}{x^2} + \frac{2x^3}{x^2} = \frac{2x^3 -1}{x^2}$.<p>
So we have $\frac{2x^3 -1}{x^2} = 0$ which will give us $2x^3 -1 = 0.$<p>
And we are done.</p>
|
1,569,728 | <p>Find one $z\in \mathbb{C}$ in the inequality $|z-25i|\le 15$ that has the largest argument ($\arg (z)$)</p>
<p>The inequality is equivalent to $x^2+(y-25)^2\le 15^2$ that represents the set of points in the circle of radius $15$ and center coordinate $C(0,25)$.</p>
<p>In this set, how to find one complex number wh... | robjohn | 13,854 | <p><strong>Geometric Approach</strong></p>
<p>The image below shows how to compute the point in the given region with the greatest argument.</p>
<p><img src="https://i.stack.imgur.com/5nKt0.png" alt="enter image description here"></p>
<hr>
<p><strong>Calculus Approach</strong></p>
<p>The argument is an increasing ... |
1,569,728 | <p>Find one $z\in \mathbb{C}$ in the inequality $|z-25i|\le 15$ that has the largest argument ($\arg (z)$)</p>
<p>The inequality is equivalent to $x^2+(y-25)^2\le 15^2$ that represents the set of points in the circle of radius $15$ and center coordinate $C(0,25)$.</p>
<p>In this set, how to find one complex number wh... | the_candyman | 51,370 | <p>Having posed that $z = x+iy$, then:
$$|z-25i|\le 15 \Rightarrow x^2 +(y-25)^2 \le 15^2.$$</p>
<p>Recall that:</p>
<p>$$\arg(z) =
\begin{cases}
f(y) & x > 0 \\
f(y) + \pi & x<0 \wedge y \ge 0 \\
f(y) - \pi & x<0 \wedge y < 0\\
\frac{\pi}{2} & x = 0 \wedge y > 0 \\
-\frac{\pi}{2} &... |
4,090,259 | <p>I began watching Gilbert Strang's lectures on Linear Algebra and soon realized that I lacked an intuitive understanding of matrices, especially as to why certain operations (e.g. matrix multiplication) are defined the way they are. Someone suggested to me 3Blue1Brown's video series (<a href="https://youtube.com/play... | user408858 | 408,858 | <p>First of all, simply by using matrix multiplication, you can rewrite <span class="math-container">$$Ax$$</span></p>
<p>into some equations</p>
<p><span class="math-container">$$a_{11}x_1+\cdots+a_{1n}x_n$$</span></p>
<p><span class="math-container">$$\cdots$$</span></p>
<p><span class="math-container">$$a_{m1}x_1+\c... |
1,221,914 | <blockquote>
<p>Let <span class="math-container">$P(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0 ,\ n\ge 1$</span> have <span class="math-container">$n$</span> roots <span class="math-container">$x_1,x_2,\ldots,x_n \le -1$</span> and <span class="math-container">$a_0^2+a_1a_n=a_n^2+a_0a_{n-1}$</span>.
Find all such <... | Mark Fischler | 150,362 | <p>Unless I misunderstand the problem,
$P(x) = x^2+2x+1$ meets the condition: Both roots are $-1$ and
$$a_0^2+a_1a_2 = a_2^2 + a_0a_1=3$$
Was the problem to find <em>all</em> such $P(x)$?</p>
|
1,036,636 | <p>The following statement makes sense intuitively, but is there a way to prove it mathematically? (This is something we make use of in applied optimization in calculus.)</p>
<blockquote>
<p>If $f$ is continuous on an interval $I$ and $x_0$ is the <strong>only</strong> relative (local) extremum, then $x_0$ is actua... | David Holden | 79,543 | <p>suppose I is closed. if $x_0$ is a local maximum, then if it is not an absolute maximum $\exists x_1. f(x_1) \gt f(x_0)$. if $x_0 \ne \sup\{x | x \in I\}$ we may assume w.l.o.g $x_1 \gt x_0$ </p>
<p>since $I'=[x_0,x_1]$ is compact $f$ attains a minimum value on $I'$, say at $x'$, contradicting the assumption that... |
14,735 | <p>This question is somewhat related to <a href="https://mathematica.stackexchange.com/questions/4576/changing-the-order-of-elements-in-a-chart-legend">this</a> one.</p>
<p>Let's say this is the <code>BarChart</code> i want to make:</p>
<pre><code>BarChart[Range[5], ChartStyle -> "Rainbow", BarOrigin -> Left,
... | Artes | 184 | <p>The appropriate function for symbolic representation of complex functions and numbers is <code>ComplexExpand</code>, e.g. </p>
<pre><code>ComplexExpand @ Table[(-1)^(k/3), {k, 3}]
</code></pre>
<blockquote>
<pre><code>{1/2 + (I Sqrt[3])/2, -(1/2) + (I Sqrt[3])/2, -1}
</code></pre>
</blockquote>
<p>For this specif... |
36,735 | <p>In Peter J. Cameron's book "Permutation Groups" I found the following quote</p>
<blockquote>
<p>It is a slogan of modern enumeration theory that the ability to count a set is closely related to the ability to pick a random element from that set (with all elements equally likely).</p>
</blockquote>
<p>Indeed, one... | Aaron Meyerowitz | 8,008 | <p>I suspect not. You did not ask about approximate counting or approximately uniform sampling (and of course that isn't what Cameron means). There are many situations where the desired set to enumerate sits in a larger set where we can easily and uniformly find a random element (for example: permutations of 1..n avoid... |
194,664 | <p>How to generate a list of fixpoint free permutations of n elements in mathematica?</p>
| Ulrich Neumann | 53,677 | <p>The problem is symmetric in a , {x,a} and {x,-a} solve the equation.
Try </p>
<pre><code>pic= ContourPlot[(Sin[a ]/a ) - x == 0, {x, 0, 1}, {a, 0, Pi},FrameLabel -> {x, a}]
</code></pre>
<p><a href="https://i.stack.imgur.com/pyJex.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/pyJex.png" al... |
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