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 |
|---|---|---|---|---|
561,921 | <p>So far I have,
$$
\lim_{x\to 1} \frac{\frac{x}{\sqrt{x^2+1}} - \frac{1}{\sqrt{1^2+1}}}{x-1}=\lim_{x\to 1} \frac{\frac{x}{\sqrt{x^2+1}} - \frac{1}{\sqrt{2}}}{x-1}
$$</p>
<p>I have no idea how to keep going with this, every way I try I get stuck and can't do anything with it. </p>
| Zackkenyon | 79,927 | <p>Set $f(x)=\frac{x}{\sqrt{x^2+1}}$, $f'(x) = \frac{\sqrt{x^2+1}-\frac{x^2}{\sqrt{x^2+1}}}{x^2+1}$ by the quotient rule (I hope, I'm terrible at math).</p>
<p>The answer to your question is $f'(1)$ which shouldn't be hard to calculate.</p>
|
145,612 | <p>Why are isosceles triangles called that — or called anything? Why is their class given a name? Why did they find their way into the <em>Elements</em> and every single elementary geometry text and course ever since? Did no one ever ask himself, "What use is this, or why is it interesting?"?</p>
<p>Here are som... | Community | -1 | <p>If you're ever doing a geometrical construction involving two radii of a circle with their endpoints joined by a line, you'll probably need some facts about isosceles triangles.</p>
|
3,051,643 | <blockquote>
<p>Find the volume of intersection of the cylinder<br>
{<span class="math-container">$ x^2 +
y^2 \leq 1 $</span>} , {<span class="math-container">$ x^2 + z^2 \leq 1$</span>}, {<span class="math-container">$ y^2 + z^2 \leq 1$</span>}.</p>
</blockquote>
<p>i am having tough time finding the volume how... | David G. Stork | 210,401 | <p><a href="https://i.stack.imgur.com/rpOTQ.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/rpOTQ.png" alt="enter image description here"></a></p>
<p><span class="math-container">$$V = 16 \int\limits_0^{\pi/4} \int\limits_0^1 s \sqrt{1 - s^2 \cos^2 \theta}\ ds\ d\theta = 8(2 - \sqrt{2}) $$</span></p... |
1,969,625 | <p>Which of the following statements is/are true?</p>
<p>$A.$ $\sin(\cos x)=x$ has exactly one root in $[0,\pi/2]$</p>
<p>$B.$ $\cos(\sin x)=x$ has exactly one root in $[0,\pi/2]$</p>
<p>$C.$ Both $A$ and $B$ are true.</p>
<p>$D.$ Both $A$ and $B$ are false.</p>
<p>I tried as $\cos(\sin x):[0,\pi/2]\rightarrow [0,... | Nitin Uniyal | 246,221 | <p>To seek solution of sin(cos$x$)$=x$ in $[0,π/2]$, take $f(x)=sin(cosx)$ ans notice that its graph originates at the point $(0,sin$1$)$ and ends at $(π/2,0)$ on X-axis. Moreover, $f(x)$ is monotonically decreasing in $[0,π/2]$ as $f'(x)<0$ therein, hence it will intersect the oblique line $y=x$ exactly once in $[0... |
1,466,618 | <p>Find the equation of the locus of the intersection of the lines below<br>
$y=mx+\sqrt{m^2+2}$<br>
$y=-\frac{ 1 }{ m }x+\sqrt{\frac{ 1 }{ m^2 +2}}$</p>
<hr>
<p>By <a href="https://www.desmos.com/calculator/rpif4yiotx" rel="nofollow">graphing</a>, I have got an ellipse as locus : $x^2+\dfrac{y^2}{2}=1$.<br>
The give... | Konstantinos Michailidis | 50,350 | <p>Solving the system you get</p>
<p>$x = - \frac{m}{{\sqrt {{m^2} + 2} }},y = \frac{2}{{\sqrt {{m^2} + 2} }}$</p>
<p>Now squaring and adding you get</p>
<p>${x^2} + {y^2} = \frac{{{m^2}}}{{{m^2} + 2}} + \frac{4}{{{m^2} + 2}} = {\left( {\sqrt {\frac{{{m^2} + 4}}{{{m^2} + 2}}} } \right)^2}$</p>
<p>which is a circle... |
1,466,618 | <p>Find the equation of the locus of the intersection of the lines below<br>
$y=mx+\sqrt{m^2+2}$<br>
$y=-\frac{ 1 }{ m }x+\sqrt{\frac{ 1 }{ m^2 +2}}$</p>
<hr>
<p>By <a href="https://www.desmos.com/calculator/rpif4yiotx" rel="nofollow">graphing</a>, I have got an ellipse as locus : $x^2+\dfrac{y^2}{2}=1$.<br>
The give... | Harish Chandra Rajpoot | 210,295 | <p>Notice, </p>
<p>Solving the given equations of the straight lines: $y=mx+\sqrt{m^2+2}$ & $y=-\frac{1}{m}x+\sqrt{\frac{1}{m^2+2}}$,</p>
<p>as follows<br>
$$mx+\sqrt{m^2+2}=-\frac{1}{m}x+\sqrt{\frac{1}{m^2+2}}$$
$$mx+\frac{x}{m}=\frac{1}{\sqrt{m^2+2}}-\sqrt{m^2+2}$$
$$\frac{m^2+1}{m}x=\frac{1-m^2-2}{\sqrt{m^2+2}... |
4,147,856 | <p>If <span class="math-container">$(a_n)_{n\ge 0}$</span> is a sequence of complex numbers and <span class="math-container">$\lim_{n\to \infty} a_n=L\neq 0$</span> then
<span class="math-container">$$\lim_{x\to 1^-} (1-x)\sum_{n\ge 0} a_n x^n=L$$</span></p>
<p>EDIT: The initial question had a typo from the place I'd s... | David C. Ullrich | 248,223 | <blockquote>
<blockquote>
<p>If <span class="math-container">$\lim_na_n=L$</span> then <span class="math-container">$\lim_{x\to1^-}(1-x)\sum_{n=0}^\infty a_nx^n=L$</span>.</p>
</blockquote>
</blockquote>
<p>Hint: The formula for the sum of a geometric series shows that <span class="math-container">$$(1-x)\sum a_nx^n=L+... |
390,532 | <p>I'm trying to solve (for $x$) some problems such as $\arctan(0)=x$, $\arcsin(-\frac{\sqrt{3}}{{2}})=x$, etc.</p>
<p>What is the best way to go about this? So far, I have been trying to solve the problems intuitively (e.g. I ask myself <em>what value of sine will give me $-\frac{\sqrt{3}}{{2}}$?</em>), maybe drawing... | Maesumi | 29,038 | <p>A combination of unit circle and "famous triangles" is perhaps the best strategy. Have drawing of the three famous triangles, $30,60,90$, with sides $1,2,\sqrt 3$, and $45,45,90$, with sides $1,1,\sqrt 2$ and $0,90,90$ with sides $0,1,1$. These will give you the reference angle you are looking for. Then for findin... |
2,668,275 | <p>I have two independent, exponential RVs $X$ and $Y$ that both have the same parameter. I am trying to find the distribution of $Y$ given that $X>Y$. So far, I have: </p>
<p>$$P(Y|X>Y) = P(Y=y|X>Y) = P(Y=y, X>Y)/P(X>Y)$$</p>
<p>and I don't know how to proceed at this point. I understand how to get th... | Daniel Schepler | 337,888 | <p>As a variation on the topologist's sine curve example, I will construct a "topologist's helix":</p>
<p>$$X \subseteq \mathbb{R}^3, X := \left\{ \left(\cos t, \sin t, \frac{1}{t} \right) \mid t > 0 \right\} \cup \{ (\cos t, \sin t, 0) \mid t \in [0, 2 \pi) \}.$$</p>
<p>The picture is of a helix winding around th... |
1,601,575 | <p>$1/\sin50^\circ + √3/\cos50^\circ=4$</p>
<p>I have tried it as:
LHS
$(\cos50+√3 \sin50)/\sin50\cos50$</p>
<p>$(2\cos50+2√3 \sin50)/2\sin50\cos50$</p>
<p>$(2\cos50+2√3 \sin50)/(\sin100)$</p>
<p>Now, whats next??</p>
| Ian Miller | 278,461 | <p>Continuing...
$$\frac{2\cos50+2\sqrt{2}\sin50}{\sin100}=4\times\frac{\frac12\cos50+\frac{\sqrt{3}}{2}\sin50}{\sin80}$$
$$=4\times\frac{\sin30\cos50+\cos30\sin50}{\sin80}$$
$$=4\times\frac{\sin80}{\sin80}$$
$$=4$$</p>
|
255,483 | <p>How to transform this infinite sum</p>
<p>$$1+\sum_{i\geq1}\frac{x^i}{(1-x)(1-x^2)\cdots(1-x^i)}$$</p>
<p>to an infinite product</p>
<p>$$\prod_{i\geq1}\frac{1}{1-x^i}$$</p>
| mjqxxxx | 5,546 | <p>Let
$$a_{n}=\frac{x^{n}}{(1-x)(1-x^2)\cdots(1-x^{n})}$$
and
$S_{n}=\sum_{i=0}^{n}a_i$. We want to show that
$$
S_{n}=\frac{1}{(1-x)(1-x^2)\cdots(1-x^{n})}.$$
Clearly this is true for $n=1$, since $S_0=a_0=1$; and assuming it's true for $n=k$, we have
$$
\begin{eqnarray}
S_{k+1}&=&S_{k}+a_{k+1} \\ &=&... |
255,483 | <p>How to transform this infinite sum</p>
<p>$$1+\sum_{i\geq1}\frac{x^i}{(1-x)(1-x^2)\cdots(1-x^i)}$$</p>
<p>to an infinite product</p>
<p>$$\prod_{i\geq1}\frac{1}{1-x^i}$$</p>
| robjohn | 13,854 | <p>Start by taking the difference
$$
\begin{align}
\prod_{i=1}^k\frac1{1-x^i}-\prod_{i=1}^{k-1}\frac1{1-x^i}
&=\left(\frac1{1-x^k}-1\right)\prod_{i=1}^{k-1}\frac1{1-x^i}\\
&=\frac{x^k}{1-x^k}\prod_{i=1}^{k-1}\frac1{1-x^i}\\
&=\frac{x^k}{\prod_{i=1}^k(1-x^i)}\tag{1}
\end{align}
$$
Summing both sides of $(1)$... |
4,588,868 | <p>As the title suggests, this is a college entrance exam practice problem from Japan, it is as follows:</p>
<blockquote>
<p>Given a scalene triangle <span class="math-container">$\triangle ABC$</span>, prove that it is a right triangle if <span class="math-container">$\sin(A)\cos(A)=\sin(B)\cos(B)$</span></p>
</blockq... | ACB | 947,379 | <p>@OscarLanzi's reasoning is correct and easy. Alternatively you can do:</p>
<p><span class="math-container">$$\sin A\cos A=\sin B\cos B$$</span>
<span class="math-container">$$\sin2A=\sin2B$$</span>
<span class="math-container">$$\sin2A-\sin2B=0$$</span>
<span class="math-container">$$2\cos(\underbrace{A+B}_{\pi-C})\... |
29,177 | <p>I had a quick look around here and google before, and didn't find any answer to this particular question, and it's beginning to really irritate me now, so I'm asking here:</p>
<p>How is one supposed to write l (little L), 1 (one) and | (pipe) so that they don't all look the same? One of my teachers draws them all a... | Arturo Magidin | 742 | <p>Sometimes one has to modify one's handwriting. When I started taking math courses in college, I realized that writing "$\mathsf{x}$" and "$\mathsf{y}$" both as two strokes just led to frequent confusions. I purposely started writing $\mathcal{x}$ instead of $\mathsf{x}$, and $\mathcal{y}$ instead of $\mathsf{y}$; I... |
2,852,020 | <p>Five identical(indistinguishable) balls are to be randomly distributed into $4$ distinct boxes. What is the probability that exactly one box remains empty?</p>
<p>we can choose the empty box in $4$ ways.
We should fit the $5$ balls now into $3$ distinct boxes in $^7C_5$ ways.
The correct answer must be: $\frac37$ ... | Key Flex | 568,718 | <p>Let $A$ be the empty box, and $B$ be the box containing $2$ balls and $C$ is $n-2$ boxes containing $1$ ball. So, there are $2\dbinom{n}{2}$ ways of arranging the letter sequence $ABC.....C$. Then the size of the sample is $\dbinom{n+n-1}{n}$ because it is equivalent to the number of ways of placing $n$ $0$'s and $n... |
2,852,020 | <p>Five identical(indistinguishable) balls are to be randomly distributed into $4$ distinct boxes. What is the probability that exactly one box remains empty?</p>
<p>we can choose the empty box in $4$ ways.
We should fit the $5$ balls now into $3$ distinct boxes in $^7C_5$ ways.
The correct answer must be: $\frac37$ ... | awkward | 76,172 | <p>Here is a solution by way of an extension of the Principle of Inclusion / Exclusion (PIE).</p>
<p>Say that an arrangement of balls in the boxes has "Property $i$" if box $i$ is empty, for $i=1,2,3,4$. Let $S_j$ be the total of the probabilities of all the arrangements with $j$ of the properties, for $j=1,2,3,4$. ... |
997,634 | <p>Evaluate
$$\int_0^R\int_0^\sqrt{R^2-x^2} e^{-(x^2+y^2)} \,dy\,dx$$ </p>
<p>using polar coordinates.</p>
<p>My answer is $-\frac{1}{2}R(e^{-R^2+x^2}-1)$ but I want to confirm if that's correct</p>
<p>And also, when I change from $dy\,dx$ to $dr \,d\theta$ ...how do I know if it should be $dr\,d\theta$ or $d\theta... | Dylan | 135,643 | <p>We want to integrate
$$I = \iint_D e^{-(x^2+y^2)} \, dA $$</p>
<p>with $D$ being the region bounded by $0 \le y \le \sqrt{R^2-x^2}$ and $ 0 \le x \le R$. This is a quarter circle of radius $R$ in the first quadrant. In polar coordinates this is equivalent to $0 \le r \le R $ and $0 \le \theta \le \pi/2$. Using a ch... |
44,226 | <p>Let $D$ denote the unit complex 1-dimensional disc, together with the hyperbolic metric $h_D=\frac{4dz\wedge d\bar{z}}{(1-|z|^2)^2}$of curvature $-1$. By Nash's embedding theorem, we can always embed the disc $D$ real-analytically and isometrically into real Euclidean space ${\mathbb{R}}^n$ for some large $n$. (I th... | David Roberts | 4,177 | <p>There is a body of work on embedding Riemann surfaces into $\mathbb{C}^n$ for small $n$ (it's possible for large $n$). If I recall correctly, Riemann surfaces can be holomorphically embedded in $\mathbb{C}^3$ easily, but the sharp lower bound that is known for complex manifolds of complex dimension $n > 1$ (whic... |
107,915 | <p>I randomly place $k$ rooks on an (arbitrarily sized) $N$ by $M$ chessboard. Until only one rook remains, for each of $P$ time intervals we move the pieces as follows:</p>
<p>(1) We choose one of the $k$ rooks on the board with uniform probability. </p>
<p>(2) We choose a direction for the rook, $(N, W, E, S)$, w... | Aaron Golden | 16,518 | <p>Joseph O'Rourke beat me to it, but here is <em>yet another</em> set of data. This is starting from an 8x8 board completely filled with rooks. The x-axis is the number of moves needed to eliminate all but one rook. The y-axis is the number of times this number of moves was sufficient, divided by the number of tria... |
107,915 | <p>I randomly place $k$ rooks on an (arbitrarily sized) $N$ by $M$ chessboard. Until only one rook remains, for each of $P$ time intervals we move the pieces as follows:</p>
<p>(1) We choose one of the $k$ rooks on the board with uniform probability. </p>
<p>(2) We choose a direction for the rook, $(N, W, E, S)$, w... | Barry Cipra | 15,837 | <p>It occurred to me it might be of interest to see what happens if you start with a board completely clogged with rooks*, so I decided to pluck the lowest-hanging nontrivial fruit and examine the $2\times2$ case, which features 5 distinct states: the starting state $S$ with 4 rooks, a trio state $T$, a pair of double... |
2,318,669 | <p>So I have this differential equation</p>
<p>$-0.4 \cdot 9.81+\frac{1}{100}v^2=0.4 v'$</p>
<p>I was able to solve it which gives me </p>
<p>$\ln(\frac{v+20}{v-20})=t+c$</p>
<p>My problem is I can't isolate $v$ after that i get it in this form and also when I try to find the constant $c$ knowing that $v(0) = 0$ I ... | Jan Eerland | 226,665 | <p>Well, in general:</p>
<p>$$\text{a}+\frac{\text{v}\left(t\right)^2}{\text{b}}=\text{c}\cdot\text{v}'\left(t\right)\space\Longleftrightarrow\space\int\frac{\text{c}\cdot\text{v}'\left(t\right)}{\text{a}+\frac{\text{v}\left(t\right)^2}{\text{b}}}\space\text{d}t=\int1\space\text{d}t\tag1$$</p>
<p>So, for the integral... |
2,141,406 | <p>I want to show that the function $f : \mathbb{R}\times \mathbb{R} \to \mathbb{R}$ defined by </p>
<p>$ f(x,y)=
\begin{cases}
\frac{xy}{x^{2}+y^{2}},& \text{if } (x,y)\neq (0,0)\\ 0, & \text{otherwise}
\end{cases}
$</p>
<p>is continuous on $\mathbb{R} \times \mathbb{R} \setminus{(0,0)}$.... | Adren | 405,819 | <p>The maps $(-\frac\pi2,\frac\pi2)\to\mathbb{R},x\mapsto\tan(x)$ and $\mathbb{R}\to(-\frac\pi2,\frac\pi2),x\mapsto\arctan(x)$ are bijections and each one is the inverse of the other one.</p>
<p>Hence we have :</p>
<p>$$\forall x\in\mathbb{R},\,\tan(\arctan(x))=x$$</p>
<p>and</p>
<p>$$\forall x\in(-\frac\pi2,\frac\... |
3,496,594 | <p>I have to factorize the polynomial <span class="math-container">$P(X)=X^5-1$</span> into irreducible factors in <span class="math-container">$\mathbb{C}$</span> and in <span class="math-container">$\mathbb{R}$</span>, this factorisation happens with the <span class="math-container">$5$</span>th roots of the unity. <... | Oscar Lanzi | 248,217 | <p>There are various ways to render separate values of <span class="math-container">$\cos(2\pi/5)$</span> and <span class="math-container">$\cos(4\pi/5)$</span> given the equation</p>
<p><span class="math-container">$1+2\cos(2\pi/5)+2\cos(4\pi/5)=0$</span></p>
<p>I demonstrate one approach. Separate out the factor o... |
4,508,558 | <p>I'm trying to make sure that I have correctly proved Munkres' Lemma 2.1, which is left to the reader. The lemma states:</p>
<blockquote>
<p>Let <span class="math-container">$f: A \to B$</span>. If there are functions <span class="math-container">$g: B \to A$</span> and <span class="math-container">$h: B \to A$</span... | José Carlos Santos | 446,262 | <p>Your answer is correct and, actually, we don't have <span class="math-container">$$(1\ \ 2\ \ 3)\circ(1\ \ 2\ \ 4\ \ 3)\circ(1\ \ 4\ \ 3\ \ 2)=(1\ \ 3)\circ(2\ \ 4).\tag1\label{1}$$</span> For instance, the LHS of \eqref{1} maps <span class="math-container">$1$</span> into <span class="math-container">$1$</span>, si... |
654,239 | <p>Let $\phi_{\alpha}(z)=\frac{z-\alpha}{1-\bar{\alpha}z}$ for $0<|\alpha|<1$</p>
<p>Find all the line $L$ in the complex plane such that $\phi_{\alpha} (L)=L$</p>
<p>Can you help me?</p>
| Robert Israel | 8,508 | <p>Hint: Möbius transformations take lines and circles to lines and circles. The lines are distinguished from circles in that they go through $\infty$ (in the extended complex plane, i.e. the Riemann sphere). What does this transformation do to $\infty$, and what does it map to $\infty$?</p>
|
2,163,948 | <p><strong>Question:</strong></p>
<blockquote>
<p>Does there exist a Riemannian manifold, with a point $p \in M$, and <strong>infinitely many</strong> points $q \in M$ such that there is <strong>more than one</strong> minimizing geodesic from $p$ to $q$?</p>
</blockquote>
<p><strong>Edit:</strong></p>
<p>As demons... | Jack Lee | 1,421 | <p>Take $M$ to be the following cylinder in $\mathbb R^3$:
$$
M = \{(x,y,z): x^2 + y^2=1\}.
$$
Then let $p$ be the point $(1,0,0)\in M$. If $q$ is any point of the form $(-1,0,z)$, then there are two minimizing geodesics from $p$ to $q$. </p>
|
3,665,879 | <p>We all are familiar with the sum and difference formulas for <span class="math-container">$\sin$</span> and <span class="math-container">$\cos$</span>, but is there an analogue for the sum and difference formulas for secant and cosecant? That is, </p>
<p><span class="math-container">$$\csc (A\pm B) = ?$$</span> an... | Kavi Rama Murthy | 142,385 | <p>If it is not bounded there exist <span class="math-container">$x_n,y_n$</span> such that <span class="math-container">$d(x_n,y_n) \to \infty$</span>. There exist subsequences <span class="math-container">$x_{n_k}$</span> and <span class="math-container">$y_{n_k}$</span> converging to some points <span class="math-co... |
73,550 | <p>I am learning a bit about <a href="http://en.wikipedia.org/wiki/PGF/TikZ" rel="nofollow noreferrer">TikZ</a> and found a nice feature in its graphics, that I am having hard time duplicating with <code>Graphics3D</code>. It is making a <code>Cylinder</code>, where the bottom will have part of its edge, that is behind... | DavidC | 173 | <p>Not just what you requested but perhaps a bit closer to the tikz rendering.</p>
<pre><code>g1 = Graphics3D[{Opacity[.8], EdgeForm[{Thick}], Glow[Pink], Black,
Cylinder[]}, Boxed -> False];
g2 = Graphics3D[{Opacity[.8], EdgeForm[{Thick}], Glow[Pink], Black,
Cylinder[]}];
GraphicsGrid[{{g1, g2}}]
</code... |
3,389,422 | <p>How to transform this limit <span class="math-container">$\lim_{h \to 0}\left(1+h\right)^{\frac{x}{h}}=e^x$</span> into <span class="math-container">$\lim_{h \to 0}\left(1+xh\right)^{\frac{1}{h}}=e^x$</span>?</p>
<p>This seems like I can't simplify this limit because I end up with zero divisors.</p>
<p>Can anyone ... | trancelocation | 467,003 | <p>Just set <span class="math-container">$t=xh \Rightarrow \frac{1}{h} = \frac{x}{t}$</span>.</p>
<p>Hence,
<span class="math-container">$$\lim_{h\to 0} (1+xh)^{\frac{1}{h}}= \lim_{t\to 0}\left((1+t)^{\frac{1}{t}}\right)^x = e^x$$</span></p>
<p><strong>Additional info after comment:</strong></p>
<p>The other way rou... |
3,389,422 | <p>How to transform this limit <span class="math-container">$\lim_{h \to 0}\left(1+h\right)^{\frac{x}{h}}=e^x$</span> into <span class="math-container">$\lim_{h \to 0}\left(1+xh\right)^{\frac{1}{h}}=e^x$</span>?</p>
<p>This seems like I can't simplify this limit because I end up with zero divisors.</p>
<p>Can anyone ... | José Carlos Santos | 446,262 | <p>Since <span class="math-container">$\lim_{h\to0}(1+h)^{\frac1h}=e$</span>, you have<span class="math-container">$$\lim_{h\to0}(1+xh)^{\frac1{xh}}=e$$</span>too and therefore<span class="math-container">\begin{align}\lim_{h\to0}(1+xh)^{\frac1h}&=\lim_{h\to0}\left((1+xh)^{\frac1{xh}}\right)^x\\&=\left(\lim_{h\... |
48,746 | <p>Let's assume that I have some particular signal on the finite time interval which is described by function <span class="math-container">$f(t)$</span>. It could be, for instance, a rectangular pulse with amplitude <span class="math-container">$a$</span> and period T; Gauss function with <span class="math-container">$... | xslittlegrass | 1,364 | <p>The the roughness on the surface is due to the noise in the original data, so that the first derivative is not smooth.</p>
<p>This shows the first derivative of the x and y components using original data, we can see it's very noisy</p>
<pre><code>tvals = parametrizeCurve[points];
m = 3;
knots = Join[ConstantArra... |
2,058,939 | <p>Let $f:\mathbb{R^3}$ $\rightarrow$ $\mathbb{R}$, defined as: </p>
<p>$$f(x,y,z)=\begin{cases} \left(x^2+y^2+z^2\right)^p \exp\left(\frac{1}{\sqrt{x^2+y^2+z^2}}\right)& ,\,\text{if }\quad(x,y,z) \ne (0,0,0)\quad \\
0 &,\,\text{o.w}
\end{cases}$$</p>
<p>Where $\,p\in \mathbb{R}$. Is this function is contin... | hamam_Abdallah | 369,188 | <p>Using the well-known limit</p>
<p>$$\forall p\in \mathbb R\;\;\lim_{X\to +\infty}\frac{e^X}{X^p}=+\infty$$ </p>
<p>we get</p>
<p>$$\lim_{x\to 0}f(x,0,0)=+\infty$$</p>
<p>thus, the function is not continuous at $0$ but it is at $\mathbb R\setminus \{0\}$.</p>
|
4,351,497 | <p>Northcott Multilinear Algebra poses a problem. Consider R-modules <span class="math-container">$M_1, \ldots, M_p$</span>, <span class="math-container">$M$</span> and <span class="math-container">$N$</span>. Consider multilinear mapping</p>
<p><span class="math-container">$$
\psi: M_1 \times \ldots \times M_p \righta... | Svyatoslav | 869,237 | <p>We can also try to dig a bit deeper. Knowing that <a href="https://www.google.com/search?q=erf%20laplace%20transform&rlz=1C1GCEU_ruRU866RU868&oq=erf%20laplace%20transform&aqs=chrome..69i57.11914j0j15&sourceid=chrome&ie=UTF-8" rel="nofollow noreferrer">Laplace Transform</a> of <span class="math-co... |
619,040 | <p>An exponential object $B^{A}$ is defined to be the representing object of the functor $$\mathcal{C}\left(- \times A,B\right): \mathcal{C} \rightarrow Set$$
or equivalently, as the terminal object of $\left(-\times A \downarrow B\right)$. The dual concept is of the co-exponential object which is the initial object o... | Eric Wofsey | 86,856 | <p>If there is "co-exponential" <span class="math-container">$f:B\to C\times A$</span> as you suggest, let <span class="math-container">$f_1:B\to C$</span> and <span class="math-container">$f_2:B\to A$</span> be the compositions of <span class="math-container">$f$</span> with the projections. The universal property of... |
801,464 | <p>Let $2^n-1$ be a prime number. If $1<i<n$, I need to prove that $2^n-1$ does not divide $1+2^{{2i}}$. Any comment would be appreciated. </p>
| Barry Cipra | 86,747 | <p>You can prove more generally that if $n$ is any integer greater than $2$, then $2^n-1$ does not divide $2^m+1$ for any $m$. </p>
<p>If $n\gt2$, it's clearly the case that $2^n-1$ does not divide $2^m+1$ for $m\lt n$, since $2^n-1\gt2^m+1$. </p>
<p>Now consider the <em>smallest</em> $m$ such that $2^n-1$ divides ... |
987,054 | <p>Prove that the sequence
$$b_n=\left(1+\frac{1}{n}\right)^{n+1}$$
Is decreasing.</p>
<p>I have calculated $b_n/b_{n-1}$ but it is obtain:
$$\left(1-\frac{1}{n^2}\right)^n \left(1+\frac{1}{n}\right)^n$$
But I can't go on.</p>
<p>Any suggestions please?</p>
| Martín-Blas Pérez Pinilla | 98,199 | <p>Hint: using the tools of differential calculus, study the function
$$x\longmapsto \left(1+\frac{1}{x}\right)^{x+1}$$
for $x>0$.</p>
|
987,054 | <p>Prove that the sequence
$$b_n=\left(1+\frac{1}{n}\right)^{n+1}$$
Is decreasing.</p>
<p>I have calculated $b_n/b_{n-1}$ but it is obtain:
$$\left(1-\frac{1}{n^2}\right)^n \left(1+\frac{1}{n}\right)^n$$
But I can't go on.</p>
<p>Any suggestions please?</p>
| DeepSea | 101,504 | <p>Consider $f(x) = (x+1)\ln (x+1) - (x+1)\ln x$ on $x \geq 1$, we have:</p>
<p>$f'(x) = \ln\left(1 + \dfrac{1}{x}\right) - \dfrac{1}{x} < 0$ because it is true that $e^r \geq 1 + r$, and apply this for $r = \dfrac{1}{x} > 0$. From this the answer follows.</p>
|
2,356,813 | <p>Let $f:[0,\infty)\to\mathbb R$ be a function in $C^2$ such that
$\lim_{x\to\infty} (f(x)+f'(x)+f''(x)) = a.$
Prove that $\lim_{x\to\infty} f(x)=a$</p>
| stewbasic | 197,161 | <p>Let $h(x)=f(x)+f'(x)+f''(x)-a$, so $h(x)\to0$ as $x\to\infty$. Let $\omega=e^{2\pi i/3}$ and
$$
f_1(x)=\frac1{\omega-\bar\omega}\left(e^{\omega x}\int_0^xe^{-\omega t}h(t)\;dt-e^{\bar\omega x}\int_0^xe^{-\bar\omega t}h(t)\;dt\right).
$$
It can be verified that
$$
f_1(x)+f_1'(x)+f_1''(x)=h(x).
$$
Thus setting $f_... |
2,158,636 | <p>I am trying to find the value of $$\sum_{k=1}^{\infty}{\frac{1}{(k+2)(k+3)}}$$</p>
<p>I do not believe it is geometric, it cannot be divided into two fractions that both converge, but it definitely does converge, and, according to WolframAlpha, to $\frac{1}{3}$. Any way I can easily show this with no more than basi... | rubik | 2,582 | <p>By partial fractions decomposition, we have
$$\frac{1}{(k + 2)(k + 3)} = \frac1{k + 2} - \frac1{k + 3}$$
Now, write down the partial sum of the series:
$$S_n = \left(\frac{1}{3}-\color{#4488dd}{\frac{1}{4}}\right)+\left(\color{#4488dd}{\frac{1}{4}}-\color{#ff4444}{\frac{1}{5}}\right)+\bigg(\color{#ff4444}{\frac{1}{5... |
3,849,851 | <p>During my research work I found a non-linear differential equation <span class="math-container">$y'''+y^2y'=0$</span>. Now I am stuck here. Please help me solve this.</p>
| Physor | 772,645 | <p>For <span class="math-container">$y' = 0$</span> it is easy to find the solution. Otherwise we have
<span class="math-container">\begin{align}
&y'''+y^2y'=0 \\
\int dx \implies& y''+\frac{y^3}{3}=C \\
\text{($y'$ is a nonzero function), }\ \cdot y' \implies &y'y''+\frac{y^3y'}{3}=Cy' \\
\int dx \implies ... |
1,860,615 | <p>I have a simple quadratic (with $x^2$) equation, x can Be complex too:</p>
<p>$$x^2+x+1=0$$</p>
<p>But it could be any equation, the equation above is just an example. I need to compute $x_1^{10}+x_2^{10}$, but it could have another exponents (ex: $x_1^{50}+x_2^{50}$).</p>
<p>I need to know, on a general case, ho... | Bernard | 202,857 | <p><em>Newton-Girard's relations</em> are very general, so we can redo the computations in the case of two roots.</p>
<p>Set $x+y= s; \enspace xy=p$. We want to compute <em>recursively</em> the sums of powers
$$P_n=x^n+y^n$$
as polynomials in $s$ and $p$. </p>
<p><em>Initialisation:</em></p>
<p>$$P_0=2,\quad P_1=s=... |
3,270,944 | <p>Let <span class="math-container">$A$</span> be a bounded linear operator on a separable Hilbert space <span class="math-container">${\cal H}$</span>, and suppose that <span class="math-container">$A$</span> is distinct from its adjoint <span class="math-container">$A^*$</span>. </p>
<p><strong>Question:</strong> Ca... | azif00 | 680,927 | <p>We have
<span class="math-container">\begin{align} f(x+h)=&2(x+h)^2+1 \\ =&2(x^2+2xh+h^2)+1 \\ =& 2x^2+4xh+2h^2+1 \end{align}</span>
Then
<span class="math-container">$$\frac{f(x+h)-f(x)}{h}=\frac{(2x^2+4xh+2h^2+1)-(2x^2+1)}{h}=\frac{h(4x+2h)}{h}=4x+2h$$</span>
if we consider <span class="math-container"... |
70,176 | <p>So I can do something like this which I like:</p>
<pre><code>Manipulate[i, {i, {1,2,3,4}}]
</code></pre>
<p>It lets me pick which specific values I want to allow to be chosen for my function. But that list appears to be very limiting.</p>
<p>Lets say I have a list and each element contains a list of two elements ... | WReach | 142 | <p>The problem is caused by ambiguity in the control-inferencing logic used by <code>Manipulate</code> and <code>Control</code> in the absence of an explicit control type specification. A <code>Manipulate</code> value with a list of pairs is a valid specification for a <code>Slider</code>, <code>SetterBar</code>, <cod... |
4,113,376 | <p>Given following predicates:</p>
<p><span class="math-container">$$
F_1 = (\forall x)(F(x) \leftrightarrow G(x)) \text{ and } F_2 = (\forall x)F(x) \leftrightarrow (\forall x)G(x)
$$</span></p>
<p>I think that they are not equivalent, but if it possible to prove that?</p>
| Bram28 | 256,001 | <p>Consider a domain with exactly two objects, one of which has property <span class="math-container">$F$</span> but not property <span class="math-container">$G$</span>, while the other one has property <span class="math-container">$G$</span>, but not property <span class="math-container">$F$</span>.</p>
<p>Then <span... |
751,138 | <p>Show that if $3\mid(a^2+1)$ then $3$ does not divide $(a+1)$. </p>
<p>using proof of contradiction </p>
<p>can someone prove this using contradiction method please</p>
| Community | -1 | <p>For the original question, if $3 \nmid a$, then $3 \nmid a^2$ since $3$ is prime (consider the factorization of $a$ and $a^2$), but this doesn't imply anything about whether $3$ divides $a^2 + 1$.</p>
<p>Here's a different approach: Suppose $3$ is a divisor of both $a^2 + 1$ and $a + 1$. Can you justify why $$3 \mi... |
751,138 | <p>Show that if $3\mid(a^2+1)$ then $3$ does not divide $(a+1)$. </p>
<p>using proof of contradiction </p>
<p>can someone prove this using contradiction method please</p>
| Bill Dubuque | 242 | <p>$3\mid \color{blue}{a^2\!+\!1},\,\color{#0a0}{a\!+\!1}\,\Rightarrow\,3\mid\color{blue}{a^2\!+\!1}-\overbrace{(\color{#0a0}{a\!+\!1})(a\!-\!1)}^{\large a^2\,-\,1} = \color{#c00}2,\,$ a contradiction, completing your proof.</p>
<p><strong>Or</strong> modly: $\ 3\mid a\!+\!1\,\Rightarrow\, {\rm mod}\ 3\!:\ a\equiv -1\... |
293,921 | <p>The problem I am working on is:</p>
<p>An ATM personal identification number (PIN) consists of four digits, each a 0, 1, 2, . . . 8, or 9, in succession.</p>
<p>a.How many different possible PINs are there if there are no restrictions on the choice of digits?</p>
<p>b.According to a representative at the author’s... | Muhamed Huseinbašić | 227,964 | <p>Regarding part [c], I do not think that there is something wrong with the book. As <strong>@Mack</strong> said in one of his comments, thief has to guess only two second numbers. Since the given restrictions actually do not apply on this order of numbers, he has to guess among 100 (10*10) possible combinations of th... |
386,799 | <blockquote>
<p>P1086: For a closed surface, the positive orientation is the one for which the normal vectors point outward from the surface, and inward-pointing normals give the negative orientation.</p>
<p>P1087: If <span class="math-container">$S$</span> is a smooth orientable surface given in parametric form by a v... | Triatticus | 75,861 | <p>Well there is a definite answer for each example here, but as Ted mentions its definitely up to the situation in general. All these problems likely fall under the blanket statement that the positive normal vector is the outward pointing one to the surface ( a usual choice). I find the best way to determine direction... |
2,519,623 | <p>How do I calculate the side B of the triangle if I know the following:</p>
<p>Side $A = 15 \rm {cm}
;\beta = 12^{\circ}
;\gamma= 90^{\circ}
;\alpha = 78^{\circ}
$</p>
<p>Thank you.</p>
| Cm7F7Bb | 23,249 | <p>Let us simplify the game a little bit. Suppose that balls $1,\ldots,16$ are red and $17,\ldots,20$ are green. We win if we draw a green ball. The distribution of green balls among the three drawn balls is <a href="https://en.wikipedia.org/wiki/Hypergeometric_distribution" rel="nofollow noreferrer">hypergeometric</a>... |
2,519,623 | <p>How do I calculate the side B of the triangle if I know the following:</p>
<p>Side $A = 15 \rm {cm}
;\beta = 12^{\circ}
;\gamma= 90^{\circ}
;\alpha = 78^{\circ}
$</p>
<p>Thank you.</p>
| NewBee | 473,224 | <p>First, find out probability of not winning then 1-that.</p>
<p>Total no of outcomes(A) : 20C1 x 19C1 x 18C1</p>
<p>Favourable outcomes for not winning(B) : 16C1 x 15C1 x 14C1 (Selecting balls numbered <17)</p>
<p>So the probability of not winning(C) : B/A = 28/57</p>
<p>Thus, required probability : 1 - C = 1 ... |
4,114,034 | <p>In Linear Algebra Done Right by Axler, there are two sentences he uses to describe the uniqueness of Linear Maps (3.5) which I cannot reconcile. Namely, whether the uniqueness of Linear Maps is determined by the choice of 1) <em>basis</em> or 2) <em>subspace</em>. These two seem like very different statements to me ... | Gary Moon | 477,460 | <p>Define <span class="math-container">$\Phi(x) = \phi(x) - x$</span>. We then want to show that <span class="math-container">$\Phi$</span> has two zeros on <span class="math-container">$[-1,1]$</span>. That <span class="math-container">$\Phi(-1) > 0$</span>, <span class="math-container">$\Phi(\frac{1}{3}) < 0$</... |
2,549,891 | <p>Suppose that $n$ different letters are sent to $n$ different addresses on the same street, one to each address. A drunk mailman randomly delivers the letters to the $n$ addresses on the street, one to each address. What is the expected number of letters that were received at correct addresses? Find the probability t... | Robert Israel | 8,508 | <p>The number of correctly delivered letters is $X = \sum_{i=1}^n X_i$, where $X_i = 1$ if letter number $i$ is delivered correctly, $0$ otherwise. Then
$\mathbb E[X_i] = \mathbb P(X_i = 1) = 1/n$, and $\mathbb E[X] = \sum_{i=1}^n \mathbb E[X_i] = 1$.</p>
|
3,521,382 | <p>Let <span class="math-container">$X$</span> be a metric space with inner product space. Suppose that there is a sequence, <span class="math-container">$ \lbrace x_{n} \rbrace $</span>, in <span class="math-container">$X$</span> such <span class="math-container">$\lim_{n \to \infty}\|x_{n}\|=\|x\|$</span> and such <... | Idris Addou | 192,045 | <p>choose a=x_n and b=x, and remember that y is any point in X so you can choose it to be x. So...</p>
|
4,409,290 | <p>Can I have a matrix <span class="math-container">$Q$</span> which is orthogonal because each of the column vectors dot products with each other is 0? Or must only satisfy <span class="math-container">$QQ^T=I$</span>. For example consider the following matrix <span class="math-container">$Q$</span>:</p>
<p><span cla... | Ian | 83,396 | <p>The columns are mutually orthogonal but it is not an orthogonal matrix because the columns are not normalized. I agree the terminology is weird.</p>
|
266,124 | <p>A palindrome is a number or word that is the same when read forward
and backward, for example, “176671” and “civic.” Can the number obtained by writing
the numbers from 1 to n in order (n > 1) be a palindrome?</p>
| Hagen von Eitzen | 39,174 | <p>No. With $n=1$, we do have a palindrome of course.
But for $n>1$ we can clearly exclude the case $n\le 10$.
In fact, we need $n\equiv 1\pmod {10}$, as the palindromic string ms
must end in "$\ldots 1$".
Let $k\ge1$ with $10^k<n<10^{k+1}$. Then there is exactly one position in the assumed palindromic string... |
499,044 | <p>I "know" that $\mathbb{C} \otimes_\mathbb{R} \mathbb{C} \cong \mathbb{C} \oplus \mathbb{C}$ as rings, but I don't really know it, what I mean with this is that I don't know any explicit isomorphism $f: \mathbb{C} \otimes_\mathbb{R} \mathbb{C} \rightarrow \mathbb{C} \oplus \mathbb{C}$. I suspect that such an isomorph... | anon | 11,763 | <p>This isomorphism would be of $\Bbb R$-algebras. Clearly both ${\Bbb C}\otimes_{\Bbb R}\Bbb C$ and $\Bbb C\oplus\Bbb C$ are four-dimensional as $\Bbb R$-vector spaces. If ${\Bbb C}\otimes_{\Bbb R}{\Bbb C}\cong{\Bbb C}\oplus{\Bbb C}$ then there must be <strong>central orthogonal idempotents</strong> in the algebra ${\... |
2,658,563 | <p><strong>(Brazil National Olympiad)</strong></p>
<p><em>Let $n$ be a positive integer. In how many ways can we distribute $n+1$ toys to $n$ kids, such that each kid gets at least one toy?</em></p>
<p><strong><em>My approach</em></strong>:</p>
<p>For each child we can assign a number $k$ to it, representing the to... | Anton Grudkin | 346,332 | <p>One may distribute toys in 3 steps: </p>
<ol>
<li>[$n$ ways] Choose one kid of $n$; </li>
<li>[$\binom{n+1}{2}$ ways] Give him two of $n+1$ toys; </li>
<li>[$(n-1)!$ ways] Distribute remaining $n-1$ toys to $n-1$ kids (one toy to one kid). .</li>
</ol>
<p>Multiplying counts of ways to perform each step we get $\fr... |
123,018 | <p>I'm doing some homework for a computer science class. It's been so long since I've done math, I have a question that assumes math knowledge that confuses me.</p>
<p>Given:
<em>Whether a diophantine polynomial in a single variable has integer roots.</em></p>
<p>With the given question I need to determine if that qu... | André Nicolas | 6,312 | <p>There are various definitions of Diophantine equation, not all equivalent. But one standard definition goes as follows. Let $P(x_1,x_2,\dots,x_k)$ be a polynomial with <em>integer</em> coefficients. A <em>solution</em> of the Diophantine equation $P(x_1,x_2,\dots,x_k)=0$ is a $k$-tuple $(x_1,x_2,\dots,x_k)$ of <em>i... |
4,029,619 | <p><strong>I worked out a solution but don't know if its the right one. Is this the right way to approach the problem? Any help would be appreciated.</strong></p>
<p>First, the number of non-negative integer solutions for <span class="math-container">$W+X+Y+Z = 15$</span> can be calculated using stars and bars:</p>
<p>... | Benjamin Wang | 463,578 | <p>The “<span class="math-container">$\le6”$</span> constraints are annoying to work with. To fix it,we can substitute <span class="math-container">$$A=6-W, \quad B=6-X, \quad C=6-Y, \quad D = 6 - Z.$$</span></p>
<p>Now, the constraints become <span class="math-container">$A+B+C+D\le 4\times 6 - 15=9$</span>, and <span... |
4,029,619 | <p><strong>I worked out a solution but don't know if its the right one. Is this the right way to approach the problem? Any help would be appreciated.</strong></p>
<p>First, the number of non-negative integer solutions for <span class="math-container">$W+X+Y+Z = 15$</span> can be calculated using stars and bars:</p>
<p>... | Math Lover | 801,574 | <p>There is a mistake in your <em>last</em> term. The answer should be <span class="math-container">$180$</span> and not <span class="math-container">$172$</span>. Also you can simplify the working.</p>
<p>There are multiple ways to tackle the problem -</p>
<ul>
<li>Solve as is using P.I.E what you did</li>
<li>Simplif... |
3,325,250 | <p>Here is the proof that every Hilbert space is refexive:</p>
<p>Let <span class="math-container">$\varphi\in\mathcal{H^{**}}$</span> be arbitrary. By Riesz, there is a unique <span class="math-container">$f_\varphi\in\mathcal{H^*}$</span> with </p>
<p><span class="math-container">$\varphi(f)=\langle\,f,f_\varphi\ra... | Chris Eagle | 693,182 | <p>The point is that <span class="math-container">$\varphi \in \mathcal{H}^{**}$</span> was arbitrary, and your proof shows that it agrees with another element of <span class="math-container">$\mathcal{H}^{**}$</span> which has a particular form, thus showing that every element of <span class="math-container">$\mathcal... |
3,368,655 | <p>I came across a problem that asked if it is posible for a function to be Riemann integrable function in <span class="math-container">$[0,+\infty)$</span> but also <span class="math-container">$|f(x)|\geq 1$</span> for all <span class="math-container">$x\geq 0$</span>. </p>
<p>At first I thought it was imposible, bu... | zhw. | 228,045 | <p>Let <span class="math-container">$a_0=0,$</span> <span class="math-container">$a_n = \sum_{k=1}^{n}1/k, n\ge 1.$</span> Then <span class="math-container">$0=a_0<a_1<a_2 < \cdots $</span> and <span class="math-container">$a_n\to \infty.$</span> Define</p>
<p><span class="math-container">$$f=\sum_{n=1}^{\inf... |
1,101,371 | <p>Any book that I find on abstract algebra is somehow advanced and not OK for self-learning. I am high-school student with high-school math knowledge. Please someone tell me a book can be fine on abstract algebra? Thanks a lot. </p>
| Ethan Bolker | 72,858 | <p>When I was in high school (60 years ago) I stumbled on W. W. Sawyer's <em>A Concrete Approach to Abstract Algebra</em>. Google found it free at <a href="https://archive.org/stream/AConcreteApproachToAbstractAlgebra/Sawyer-AConcreteApproachToAbstractAlgebra#page/n5/mode/2up" rel="nofollow">https://archive.org/stream... |
3,882,261 | <p>I have the following question. It's basically my first day doing complex numbers, so I am absolutely lost here. <a href="https://i.stack.imgur.com/JTebK.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/JTebK.png" alt="enter image description here" /></a>
I have read that the modulus-arg form is
<sp... | Narasimham | 95,860 | <p>There is no unique parametrization. Difference of squares being unity is the common criterion. Some are simpler than the others. One possibility is</p>
<p><span class="math-container">$$ x/a=\frac12 (t+1/t), y/b =\frac12(t-1/t)$$</span></p>
<p>which is constructed on Pythagorean triplets. I.e.,take</p>
<p><span cla... |
144,364 | <p>I have been in a debate over <a href="http://9gag.com/" rel="noreferrer">9gag</a> with this new comic: <a href="http://9gag.com/gag/4145133" rel="noreferrer">"The Origins"</a></p>
<p><img src="https://i.stack.imgur.com/BOEnP.jpg" alt=""-1 doesn't have a square root?" "Here come imaginary numbers&... | Asaf Karagila | 622 | <p>The question whether something has a root or not must include a setting. Definitions do not appear magically, they require some preexisting framework.</p>
<p>In the setting of the real numbers negative numbers do not have a square root. In the setting of the complex numbers negative numbers <em>do</em> have a squar... |
144,364 | <p>I have been in a debate over <a href="http://9gag.com/" rel="noreferrer">9gag</a> with this new comic: <a href="http://9gag.com/gag/4145133" rel="noreferrer">"The Origins"</a></p>
<p><img src="https://i.stack.imgur.com/BOEnP.jpg" alt=""-1 doesn't have a square root?" "Here come imaginary numbers&... | Logan M | 8,473 | <p>I think Asaf's answer, while correct, misses some of the point. It's fairly clear from context that the OP wants to know whether writing $i=\sqrt{-1}$ makes sense in the complex numbers. This is essentially a matter of convention. You can define it that way, but any way you define the function $\sqrt{z}$ it won't ha... |
962,287 | <p>I am trying to isolate x in the equation $$(x-20)^{2} = -(y-40)^{2} - 525.$$ How can I do it?</p>
| MPW | 113,214 | <p>If you are looking for real solutions, there can be none. The left side is nonnegative, and the right side is strictly negative.</p>
<p>Otherwise, for complex solutions, you can take square roots of both sides and add $20$. Don't forget that you get two equations (with $\pm$) when you take square roots:</p>
<p>$$x... |
3,773,695 | <p>I have been trying to get some upper bound on the coefficient of <span class="math-container">$x^k$</span> in the polynomial
<span class="math-container">$$(1-x^2)^n (1-x)^{-m}, \text{ $m \le n$}.$$</span></p>
<p>A straightforward calculation shows that for even <span class="math-container">$k$</span>, the coefficie... | Moko19 | 618,171 | <p>There are four operations we can perform to convert two similar 2D graphs: translation, reflection, rotation, and uniform scaling</p>
<p>It is obvious that the first three of these cannot change the existence of local extreme points.</p>
<p>If we uniformly scale an equation by a factor of <span class="math-container... |
3,277,555 | <p>For a math class I was given the assignment to make a game of chance, for my game the person must roll 4 dice and get a 6, a 5, and a 4 in a row in 3 rolls or less to qualify. the remaining dice must be over 3 for you to win. my question though is how can I find out the probability of rolling the 6,5, and 4 in a sin... | Graham Kemp | 135,106 | <p>You want the probability for obtaining 4,5,6, and one other number, or of obtaining 4,5, and 6, where one from these is rolled twice, from a roll of four independent
dice.</p>
<ul>
<li>The first event is a selection of one from three numbers (1,2,3) as the fourth number, and an arrangement of the four distinct num... |
1,110,543 | <p>I am dealing with galois theory at the moment and I came across with an example in the lecture and I got a question:</p>
<p>Let $K=\mathbb Q$ and $L=\mathbb Q(\sqrt{2},\sqrt{3})\subset \mathbb C$. Lets consider the field-extension $L/K$</p>
<p>Because of $[L:K]=[L:K(\sqrt{2})]\cdot [K(\sqrt{2}):K]=4$</p>
<p><stro... | Ennar | 122,131 | <p>Actually, you cannot argue that $[L/K(\sqrt 2)] = 2$ without knowing the degree of minimal polynomial of $\sqrt 3$ over $K(\sqrt 2)$. What you do know is that the degree is at most $2$, since it is the root of $x^2-3$. Thus, you can conclude $[L/K]\leq 4$ and that it is divisible by $2$, because $[K(\sqrt 2)/K] = 2$... |
3,504,422 | <blockquote>
<p>Find: <span class="math-container">$$\displaystyle\lim_{x\to
\infty}\left(\frac{\ln(x^2+3x+4)}{\ln(x^2+2x+3)}\right)^{x\ln x}$$</span></p>
</blockquote>
<p>My attempt:</p>
<p><span class="math-container">$\displaystyle\lim_{x\to\infty}\left(\frac{\ln(x^2+3x+4)}{\ln(x^2+2x+3)}\right)^{x\ln x}=\lim_{... | Paramanand Singh | 72,031 | <p>Whenever you see an expression of type <span class="math-container">$\{u(x) \} ^{v(x)} $</span> under limit it is best to take logarithms. This makes your expressions simpler and easier to type and write thereby allowing you to focus on the problem more efficiently.</p>
<p>Thus if <span class="math-container">$L$</... |
1,891,496 | <p>For example, can we say: $\infty=\lim\limits_{n\rightarrow\infty} n < \aleph_0$?</p>
<p>These are two different types of structures. The limit being like the length, extension, or just generic magnitude and the other being cardinality of a set. Can we compare magnitude to cardinality?</p>
<p>Intuitively, we can... | Asaf Karagila | 622 | <p>The reason the answer is negative is that
$$\huge\underline{\underline{\color{red}{\textbf{Cardinals are not real numbers.}}}}$$</p>
<p>What do I mean by that? For finite cardinals we can nicely match the natural numbers with the ordinals, the finite cardinals, the iterated sums of the unity of the real numbers, or... |
1,891,496 | <p>For example, can we say: $\infty=\lim\limits_{n\rightarrow\infty} n < \aleph_0$?</p>
<p>These are two different types of structures. The limit being like the length, extension, or just generic magnitude and the other being cardinality of a set. Can we compare magnitude to cardinality?</p>
<p>Intuitively, we can... | Community | -1 | <p>There is an ordered set of extended natural numbers $\mathbb{N} \cup \{ \infty \}$.</p>
<p>The ordered class of cardinal numbers has an initial segment $\mathbb{N} \cup \{ \aleph_0 \}$.</p>
<p>These two ordered sets happen to be isomorphic. This fact is pretty much the <em>entirety</em> of the relationship between... |
335,929 | <p>Let $A=\{1, 2,\dots,n\}$
What is the maximum possible number of subsets of $A$ with the property that any two of them have exactly one element in common ?</p>
<p>I strongly suspect the answer is $n$, but can't prove it.</p>
| Jonathan Rich | 66,596 | <p><strong>Hint:</strong> For a set of length $i > 2$, no other set can have any subset of that set $i \ge 2$ as a subset.</p>
|
148,032 | <p>What is the larger of the two numbers?</p>
<p>$$\sqrt{2}^{\sqrt{3}} \mbox{ or } \sqrt{3}^{\sqrt{2}}\, \, \; ?$$
I solved this, and I think that is an interesting elementary problem. I want different points of view and solutions. Thanks!</p>
| chharvey | 15,501 | <p>$$\sqrt{2}^{\sqrt{3}} \approx 1.414^{1.732} \approx 1.822$$
$$\sqrt{3}^{\sqrt{2}} \approx 1.732^{1.414} \approx 2.174$$
$$\text{The rest is clear.}$$</p>
|
1,524,615 | <p>I'm trying to solve some task and I'm stuck. I suppose that I will be able to solve my problem, if I'll find elementary way to calculate $\lim_{x \to \infty}\sqrt[x-1]{\frac{x^x}{x!}}$ for $x \in \mathbb{N}_+$.<br>
My effort: I had prove, that $x! \geq (\frac{x+1}{e})^x$, so (cause $x^x>x!$):</p>
<p>$$
\left(\fr... | Tacet | 186,012 | <p>Elementary solution to this problem:<br>
Fact:
$$\lim_{n\to\infty}\frac{a_{n+1}}{a_n} = g \in \mathbb{R} \Longrightarrow \lim_{n\to\infty}\sqrt[n]{a_n} = g$$
So, we can take $a_n = \frac{n^n}{n!}$, then $\frac{a_{n+1}}{a_n} = \frac{(n+1)^{n+1}}{n^n}\cdot\frac{n!}{(n+1)!}=(1+\frac{1}{n})^n (n+1) \cdot \frac{1}{(n+1)}... |
1,987,480 | <p>This question has been bugging me for a while now and I want to know where I'm going wrong. </p>
<blockquote>
<p>There are $20$ tickets in a raffle with one prize. What should each ticket cost if the prize is \$80 and the expected gain to the organizer is \$30?</p>
</blockquote>
<p>Now I can get the right answer... | msm | 350,875 | <p>Your mistake (in your own calculation) is that you assume all tickets are sold and the winner is the last person who buys the last ticket. This is not what always happens. <em>Any</em> of the tickets can win (including the first one!) and it is likely that some of them are not sold at all.</p>
|
16,584 | <p>In the definition of vertex algebra, we call the vertex operator state-field correspondence, does that mean that it is an injective map??
Are there some physical interpretations about state-field correspondence ? Or why we need state-field correspondence in physical viewpoint??
Does it have some relations to highe... | Pavel Etingof | 3,696 | <p>Yes, the state-field map $v\mapsto Y(v,z)$ is an injective map, since by the axioms of VOA, $Y(v,z)1|_{z=0}=v$. </p>
<p>The state-field correspondence appears in 2-dimensional field theory because such a field theory attaches an amplitude to a "pair of pants" (a 2-sphere with 3 holes). Namely, if you regard two of ... |
2,156,109 | <p>How do I find the limit:</p>
<p>$$\lim_{x\to\infty}x\sin(\tan\frac1x)$$</p>
| S.C.B. | 310,930 | <p>Set $x=\frac{1}{t}$. Note that $$\lim_{x \to \infty} x \sin \left(\tan \frac{1}{x}\right)=\lim_{t \to 0^{+}} \frac{\sin (\tan t)}{t}$$
Now, note that $$\lim_{t \to 0^{+}} \frac{\sin (\tan t)}{t}=\lim_{t \to 0^{+}} \frac{\sin (\tan t)}{\tan t} \times \frac{\tan t}{t}=\lim_{t \to 0^{+}} \frac{\sin (\tan t)}{\tan t} \t... |
2,156,109 | <p>How do I find the limit:</p>
<p>$$\lim_{x\to\infty}x\sin(\tan\frac1x)$$</p>
| Tom | 386,571 | <p>Rewrite the limit as </p>
<p>\begin{align}
L:=\lim\limits_{x\rightarrow\infty}x\sin\left(\tan\frac{1}{x}\right)=\lim\limits_{x\rightarrow\infty}\frac{\sin\left(\tan\frac{1}{x}\right)}{\frac{1}{x}}
\end{align}</p>
<p>Since both the numerator and the denominator tend to $0$ in the limit, this is an indeterminate for... |
4,321,122 | <p>In <a href="https://terrytao.wordpress.com/tag/selberg-sieve/" rel="nofollow noreferrer">Tao's blog</a>, one of <a href="https://en.wikipedia.org/wiki/Landau%27s_problems" rel="nofollow noreferrer">Landau's problems</a> is interpreted in the setting of sieve theory. More precisely, the twin prime conjecture leads to... | G. Fougeron | 188,609 | <p>We have :</p>
<p><span class="math-container">$(1+z)(1+w)-1 = z+w+zw$</span></p>
<p>Thus :</p>
<p><span class="math-container">$|(1+z)(1+w)-1| = |z+w+zw| \le |z| + |w| + |zw| = (1+|z|)(1+|w|) -1$</span></p>
<p>QED</p>
|
4,321,122 | <p>In <a href="https://terrytao.wordpress.com/tag/selberg-sieve/" rel="nofollow noreferrer">Tao's blog</a>, one of <a href="https://en.wikipedia.org/wiki/Landau%27s_problems" rel="nofollow noreferrer">Landau's problems</a> is interpreted in the setting of sieve theory. More precisely, the twin prime conjecture leads to... | dxiv | 291,201 | <p>To answer the <code>solution-verification</code> part of the question, this step is wrong.</p>
<blockquote>
<p>one obtains</p>
<p><span class="math-container">\begin{align}
(1+z)|(1+w)| \leq |(1+z)||(1+w)|
\end{align}</span></p>
<p>Since <span class="math-container">$(1+w) \leq |(1+w)|$</span>, it follows that</p>
<... |
1,794,221 | <p>I am asked to show that the tangent space of $M$={ $(x,y,z)\in \mathbb{R}^3 : x^{2}+y^{2}=z^{2}$} at the point p=(0,0,0) is equal to $M$ itself.</p>
<p>I have that $f(x,y,z)=x^{2}+y^{2}-z^{2}$ but as i calculate $<gradf_p,v>$ i get zero for any vector.Where am i making a disastrous error?</p>
| Siddharth Bhat | 261,373 | <p>Think of it in terms of what the kernel represents. In a sense, the kernel of a homomorphism $\phi: G \to H$ represents the "degree of failure" of injectivity of the map.</p>
<p>If the kernel is larger than trivial, then this means that multiple elements in $G$ get compressed to one element in $H$. For this to not ... |
747,789 | <p>I've been reading some basic classical algebraic geometry, and some authors choose to define the more general algebraic sets as the locus of points in affine/projective space satisfying a finite collection of polynomials $f_1, \dots, f_m$ in $n$ variables without any more restrictions. Then they define an algebraic ... | Georges Elencwajg | 3,217 | <p>a) A useful trick for showing irreducibility of an algebraic set $X$ is to exhibit an open dense subset $X_0\subset X$ which is known to be irreducible.<br>
In particular the closure $X\subset \mathbb P^n$ of <em>any</em> algebraic irreducible subset $X_0\subset \mathbb A^n$ is irreducible. </p>
<p>For exampl... |
1,811,612 | <p>We have $5$ normal dice. What is the chance to get five $6$'s if you can roll the dice that do not show a 6 one more time (if you do get a die with a $6$, you can leave it and roll the others one more time. Example: first roll $6$ $5$ $1$ $2$ $3$, we will roll $4$ dice and hope for four $6$s or if we get $6$ $6$ $2$... | angryavian | 43,949 | <p>Just FYI, <em>dice</em> is the plural form of <em>die</em>, e.g. "roll one die" or "roll two dice."</p>
<p>Consider doing the game with a single die. The probability of rolling a six on the first time is $1/6$, and the probability of failing the first time but succeeding the second time is $(5/6) \cdot (1/6)$. So t... |
221,729 | <p>Till now, I have proved followings;</p>
<p>Suppose $X,Y$ are metric spaces and $E$ is dense in $X$ and $f:E\rightarrow Y$ is uniformly continuous. Then,</p>
<ol>
<li><p>$Y=\mathbb{R}^k \Rightarrow \exists$ a continuous extension.</p></li>
<li><p>$Y$ is compact $\Rightarrow \exists$ a continuous extension.</p></li>... | apnorton | 23,353 | <p>I found this question (and the first answer) helpful:
<a href="https://math.stackexchange.com/questions/23938/big-o-notation-and-asymptotics?rq=1">Big-O Notation and Asymptotics</a></p>
<p>For example, <span class="math-container">$f(n)$</span> is <span class="math-container">$O(g(n))$</span>. Then, <span class="m... |
4,120,827 | <p>Let's assume <span class="math-container">$P_1=(x_1, y_1)$</span> and <span class="math-container">$P_2=(x_2, y_2)$</span> and <span class="math-container">$P_3=(x_3, y_3)$</span>.</p>
<p>How to find the closest distance between <span class="math-container">$P_3$</span> and the line segment between <span class="math... | Vishu | 751,311 | <p>The line segment can be parameterized as <span class="math-container">$$(x,y) = (tx_2+(1-t)x_3,ty_2+(1-t)y_3) $$</span> for <span class="math-container">$0\le t\le 1$</span>. The distance (squared) from <span class="math-container">$P_1$</span> to any point <span class="math-container">$(x,y)$</span> on this line is... |
4,120,827 | <p>Let's assume <span class="math-container">$P_1=(x_1, y_1)$</span> and <span class="math-container">$P_2=(x_2, y_2)$</span> and <span class="math-container">$P_3=(x_3, y_3)$</span>.</p>
<p>How to find the closest distance between <span class="math-container">$P_3$</span> and the line segment between <span class="math... | user | 293,846 | <p>Let <span class="math-container">$d_{ij}$</span> and <span class="math-container">$d_{i(jk)}$</span> denote the distance between points <span class="math-container">$i,j$</span> and the distance between the point <span class="math-container">$i$</span> and the line <span class="math-container">$(jk)$</span> respecti... |
546,701 | <p>Find the number of positive integers $$n <9,999,999 $$ for which the sum of the digits in n equals 42.</p>
<p>Can anyone give me any hints on how to solve this?</p>
| totaam | 104,482 | <p>I know this is not the answer you are looking for, but I thought it was an amusing one-liner brute force solution (takes less than 1 minute to run) - and may allow you to verify your own result:</p>
<pre><code>python -c "print len([i for i in xrange(10000000) \
if sum([int(x) for x in str(i)])==42])... |
546,701 | <p>Find the number of positive integers $$n <9,999,999 $$ for which the sum of the digits in n equals 42.</p>
<p>Can anyone give me any hints on how to solve this?</p>
| wolfies | 74,360 | <p>The question is just a variation on this one:</p>
<p><a href="https://math.stackexchange.com/questions/372624/probability-of-random-integers-digits-summing-to-12/">Probability of random integer's digits summing to 12</a></p>
<p>which provides several neat theoretical solutions. </p>
<p>For computational evalu... |
546,701 | <p>Find the number of positive integers $$n <9,999,999 $$ for which the sum of the digits in n equals 42.</p>
<p>Can anyone give me any hints on how to solve this?</p>
| yo' | 43,247 | <p>A solution through generating functions:</p>
<p>Imagine a language of words $\newcommand\L{\mathcal{L}}\L=(a^{\{0,9\}}b)^7$, where $a^{\{0,9\}}$ means "any number of $a$'s betwees $0$ and $9$". A word in this language corresponds to $7$ digits in $0,\dots,9$. The sum of the digits is exactly the number of $a$'s in ... |
546,701 | <p>Find the number of positive integers $$n <9,999,999 $$ for which the sum of the digits in n equals 42.</p>
<p>Can anyone give me any hints on how to solve this?</p>
| robjohn | 13,854 | <p>Consider the product $(1+x+x^2+x^3+\dots+x^9)^7$. Each choice of a term in each factor corresponds to the choice of a digit in a $7$ digit number. The coefficient of $x^{42}$ corresponds to the number of ways for the sum of those digits to be $42$.</p>
<p>Thus, the answer is the coefficient of $x^{42}$ in
$$
\left(... |
2,721,992 | <p>I would like to see the fact that the components of a vector transform differently (controvariant transformation) than the unit bases vectors (covariant transformation) for the specific case of cartesian to polar coordinate transformation. </p>
<p>The polar unit vectors $\hat{r}$ and $\hat{\theta}$ can be expressed... | DosGatos | 548,738 | <p>As @TedShifrin pointed out, the correct transformation for the dual is <span class="math-container">$(A^{-1})^T$</span>.</p>
<p>For instance, in special relativity contravariant vectors transform as</p>
<p><span class="math-container">$ V^{\mu '} = \Lambda ^{\mu '} _{\,\nu} V^{\nu}$</span>,</p>
<p>where <span cla... |
3,276,332 | <p>I'm studying for my exam in discrete mathematics and found the following problem on last years exam:</p>
<p>Find a closed formula without using induction for <span class="math-container">$\sum_{k=0}^n k^3$</span>.</p>
<p>I tried it by finding the Generating Function first:</p>
<p><span class="math-container">$F(x... | J. W. Tanner | 615,567 | <p>If you'll grant me that <span class="math-container">$f(n)=\sum_{k=0}^n k^3=an^4+bn^3+cn^2+dn+e,$</span></p>
<p>then we have this system to solve:</p>
<p><span class="math-container">$$ f(0)=0=e\\
f(1)=1=a+b+c+d+e\\
f(2)=9=16a+8b+4c+2d+e\\ f(3)=36=81a+27b+9c+3d+e \\ f(4)=100=256a+64b+16c+4d+e.
$$</span></p>
<p>Fr... |
3,276,332 | <p>I'm studying for my exam in discrete mathematics and found the following problem on last years exam:</p>
<p>Find a closed formula without using induction for <span class="math-container">$\sum_{k=0}^n k^3$</span>.</p>
<p>I tried it by finding the Generating Function first:</p>
<p><span class="math-container">$F(x... | amd | 265,466 | <p>To use a generating function to solve this, it helps to know that the ordinary generating function for the sequence <span class="math-container">$\{n^m\}_{n=0}^\infty$</span> is <span class="math-container">$\left(x\frac d{dx}\right)^m\frac1{1-x}$</span>: differentiation multiplies the coefficient of <span class="ma... |
163,296 | <p>For positive real numbers $x_1,x_2,\ldots,x_n$ and any $1\leq r\leq n$ let $A_r$ and $G_r$ be , respectively, the arithmetic mean and geometric mean of $x_1,x_2,\ldots,x_r$.</p>
<p>Is it true that the arithmetic mean of $G_1,G_2,\ldots,G_n$ is never greater then the geometric mean of $A_1,A_2,\ldots,A_n$ ?</p>
<p>... | Community | -1 | <p>Here's a proof for $n=2$. Apply the Cauchy-Schwarz inequality to the vectors $(\sqrt{a},\sqrt{b})$ and $(1/2,1/2)$ to get $${\sqrt{a}+\sqrt{b}\over 2}\leq\sqrt{a+b\over 2}.$$ Multiply by $\sqrt{a}$ to obtain $${a+\sqrt{ab}\over 2}\leq \sqrt{a\left({a+b\over 2}\right)}.$$</p>
|
2,867,907 | <p>Is the function </p>
<p>$$ f\left(\frac{x}{\epsilon}\right)=\exp\left(\frac{2\pi ix}{\epsilon}\right)=g(x) $$</p>
<p>equivalent to zero ? in the limit $ \epsilon \to 0 $ ?</p>
<p>If I take the derivative $$\frac{ g(x+\epsilon)-g(x)}{\epsilon} $$ is $0$ because the function $g(x)$ has a period 'epsilon'</p>
<p>Al... | Kavi Rama Murthy | 142,385 | <p>$|g(x)|=1$ for all real numbers $x$ and all $\epsilon >0$. How can $g$ be equivalent to $0$?</p>
|
3,251,233 | <blockquote>
<p>Calculate <span class="math-container">$\int_3^4 \sqrt {x^2-3x+2}\, dx$</span> using Euler's substitution</p>
</blockquote>
<p><strong>My try:</strong>
<br><span class="math-container">$$\sqrt {x^2-3x+2}=x+t$$</span>
<span class="math-container">$$x=\frac{2-t^2}{2t+3}$$</span>
<span class="math-conta... | eranreches | 208,983 | <p><strong>Hint:</strong> Make another substitution <span class="math-container">$z=2t+3$</span>. However ugly it turns out to be, you only need to calculate integrals of the form <span class="math-container">$x^{\alpha}$</span> for <span class="math-container">$\alpha$</span> integer.</p>
|
3,251,233 | <blockquote>
<p>Calculate <span class="math-container">$\int_3^4 \sqrt {x^2-3x+2}\, dx$</span> using Euler's substitution</p>
</blockquote>
<p><strong>My try:</strong>
<br><span class="math-container">$$\sqrt {x^2-3x+2}=x+t$$</span>
<span class="math-container">$$x=\frac{2-t^2}{2t+3}$$</span>
<span class="math-conta... | logo | 587,007 | <p>Using <a href="https://en.wikipedia.org/wiki/Euler_substitution" rel="nofollow noreferrer">the third substitution of Euler</a></p>
<p><span class="math-container">$$\sqrt{{{x}^{2}}-3x+2}=\sqrt{\left( x-1 \right)\left( x-2 \right)}=\left( x-1 \right)t$$</span>
<span class="math-container">$$x=\frac{2-{{t}^{2}}}{1-{{... |
2,897,340 | <p>my attempt for
(i)</p>
<p>$\left. \begin{array} { l } { \cot ( \theta ) = - \frac { 1 \cdot \sqrt { 3 } } { \sqrt { 3 } \sqrt { 3 } } } \\ { 1 \cdot \sqrt { 3 } = \sqrt { 3 } } \end{array} \right.$</p>
<p>$\cot ( \theta ) = - \frac { \sqrt { 3 } } { 3 }$</p>
<p>(ii)</p>
<p>$\left. \begin{array} { l } { \text { ... | Mostafa Ayaz | 518,023 | <p>We know that $$\cot\left(-\dfrac{\pi}{3}\right)=-\dfrac{\sqrt 3}{3}$$ and $$\cot( \theta+\pi)=\cot \theta$$similarly $$\cos\left(\dfrac{\pi}{3}\right)=\dfrac{1}{2}$$and $$\cos^2(\theta+\pi)=\cos^2\theta$$therefore $$(i)\qquad x=k\pi-\dfrac{\pi}{3}\\(ii)\qquad x=k\pi\pm\dfrac{\pi}{3}$$</p>
|
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