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 |
|---|---|---|---|---|
4,242,765 | <p>Let <span class="math-container">$X_k = 1$</span> with probability <span class="math-container">$0.5$</span> and <span class="math-container">$X_k = -1$</span> with probability 0.5, and let <span class="math-container">$X_k$</span> be independent random variables <span class="math-container">$k = (1,2,...,n)$</span>... | Oliver Díaz | 121,671 | <p>Recall that
<span class="math-container">$$\cos z=\sum^\infty_{k=0}\frac{(-1)^kz^{2k}}{(2k)!}=1-\frac{z^2}{2}(1+g(z))$$</span>
where <span class="math-container">$\lim_{z\rightarrow0}g(z)=0$</span>.</p>
<p>Also, <a href="https://math.stackexchange.com/a/4128931/121671">recall</a> that for any sequence of complex num... |
2,825,789 | <p>I struggle to understand the following theorem (not the proof, I can't even validate it to be true). Note: I don't have a math background.</p>
<blockquote>
<p>If S is not the empty set, then (f : T → V) is injective if and only if Hom(S, f) is injective.</p>
<p>Hom(S, f) : Hom(S, T) → Hom(T, V)</p>
</blockquote>
<p>... | Fred | 380,717 | <p>$f(x)=1/x$ is continuous on $[1, \infty)$, but $\int_1^{\infty} f(x) dx = \infty$.</p>
|
2,649,283 | <p>There are these two questions that my professor posted, and they absolutely stumped me:</p>
<p>$ \vdash (\exists x. \bot) \implies P $
and
$(\exists x. \top) \vdash (\forall x. \bot ) \implies P$.</p>
<p>What do I even do with the $(\exists x. \bot)$ part? It got me stuck for quite some time. Any help will be ap... | Peter Smith | 35,151 | <p>In a syntax that allows vacuous quantification (bad and un-Fregean, but now sadly usually permitted), $\exists x\,\phi$ is equivalent to plain $\phi$, where $\phi$ is closed, without free variables, and so $\exists x\,\bot$ is equivalent to plain $\bot$. And you know, presumably, about the ex falso principle $\vdash... |
1,819,841 | <p>Is it possible to find out what the "limit" is as the number of iterations of compositions a certain function with itself, like a trig function for example, tends to infinity? </p>
<ol>
<li><p>$\sin(\sin(\sin(\dots(\sin(x))\dots)))$ </p></li>
<li><p>$\cos(\cos(\cos(\dots(\cos(x))\dots)))$</p></li>
</ol>
<p>In the ... | E. Joseph | 288,138 | <p>Yes it is possible.</p>
<p>Just consider the sequence defined by induction:</p>
<p>$$\begin{cases} u_0=x \\ u_{n+1}=f(u_n). \end{cases}$$</p>
<p>where $f$ is the function you consider (here $\sin$ and $\cos$).</p>
<p>Then look at the limit</p>
<p>$$\lim_{n\to\infty} u_n$$</p>
<p>with the usuals theorems.</p>
|
1,819,841 | <p>Is it possible to find out what the "limit" is as the number of iterations of compositions a certain function with itself, like a trig function for example, tends to infinity? </p>
<ol>
<li><p>$\sin(\sin(\sin(\dots(\sin(x))\dots)))$ </p></li>
<li><p>$\cos(\cos(\cos(\dots(\cos(x))\dots)))$</p></li>
</ol>
<p>In the ... | Community | -1 | <p>Observe that $\sin(x) \in [-1,1]$ and for $x \in [-1,1]$ we have $\lvert \sin(x) \lvert \leq \lvert x \rvert$. This shows that it converges.</p>
<p>In fact, you could also just use Banachs Fixed Point theorem. And you'll see that the sequence will converge to the fixedpoints of $\sin$ and $\cos$ (in the interval $[... |
1,925,867 | <p>I can't find any. For saying $H$ is a subgroup of $G$ we have notation but it seems none exists for subrings.</p>
| D_S | 28,556 | <p>There doesn't seem to be any standard notation for "is a subring of." If $S, R$ are rings, and one writes $S \subseteq R$ (literally, $S$ is a subset of $R$), then it is tacitly assumed that the operations on $S$ making it into a ring are restricted from those of $R$. It follows from here that $0_S = 0_R$, but if ... |
3,792,135 | <p><strong>Question:</strong> Sum of the series <span class="math-container">$1-3x^2+5x^4 - ... + (-1)^{n-1} (2n-1)x^{2n-2} = \sum\limits_{n=0}^{\infty} (-1)^{n-1} (2n-1)x^{2n-2}$</span></p>
<p>My first idea is to integrate to get <span class="math-container">$\int f(x) dx = x -x^3 + x^5 - ... + (-1)^{ n-1}x^{2n-1} = \... | Community | -1 | <p><span class="math-container">$1-3x^2+5x^4-...=\sum_{n=1}^{\infty} (-1)^{n-1}(2n-1)x^{2n-2}=\big( \sum_{n=1}^{\infty} (-1)^{n-1}x^{2n-1} \big)'=\big(\frac{-1}{x}\sum_{n=1}^{\infty}(-x^2)^{n} \big)'=\big(\frac{-1}{x}\times (\frac{1}{1+x^2}-1) \big)'=\big(\frac{x}{1+x^2} \big)'=\frac{1-x^2}{(1+x^2)^2}$</span></p>
<p>w... |
3,792,135 | <p><strong>Question:</strong> Sum of the series <span class="math-container">$1-3x^2+5x^4 - ... + (-1)^{n-1} (2n-1)x^{2n-2} = \sum\limits_{n=0}^{\infty} (-1)^{n-1} (2n-1)x^{2n-2}$</span></p>
<p>My first idea is to integrate to get <span class="math-container">$\int f(x) dx = x -x^3 + x^5 - ... + (-1)^{ n-1}x^{2n-1} = \... | ratatuy | 812,151 | <p><span class="math-container">$x-x^3+x^5-x^7+\dots=\frac{x}{1+x^2}=\int f(x)\mathrm dx$</span>,
<span class="math-container">$f(x)=\left(\frac{x}{x^2+1}\right)'=\frac{1-x^2}{(1+x^2)^2}$</span></p>
|
2,062,706 | <p>I have the following function:</p>
<p>\begin{equation}
f(q,p) = q \sqrt{p} + (1-q) \sqrt{1 - p}
\end{equation}</p>
<p>Here, $q \in [0,1]$ and $p \in [0,1]$.</p>
<p>Now, given some value $q \in [0,1]$ what value should I select for $p$ in order to maximize $f(q,p)$? That is, I need to define some function $g(q)$ s... | Ahmed S. Attaalla | 229,023 | <p>Hint:</p>
<p>Now we take some constant $q$ so our function becomes a function with only $p$ varying:</p>
<p>$$\begin{equation}
f(p) = q \sqrt{p} + (1-q) \sqrt{1 - p}
\end{equation}$$</p>
<p>With the constraint $0 \leq p \leq 1$. To maximize one must consider critical points in our interval, and the endpoints. Kee... |
2,789,002 | <p>How can I calculate the height of the tree? I am with geometric proportionality.</p>
<p><img src="https://i.stack.imgur.com/m4zMD.png"></p>
| Rhys Hughes | 487,658 | <p>Welcome to Math.SE. It is always better if you show us the work you've done on a problem, and where you got stuck. I'll answer your question but please bear this in mind for next time. </p>
<p>The length from the man to the base of the small tree is $16m$. The length from the man to the big tree is $26m(10m+16m)$
T... |
2,048,054 | <p>I need to find signed distance from the point to the intersection of 2 hyperplanes. I was quite sure that this is something that every mathematician do twice a week :) But not found any good solution or explanation for same problem.</p>
<p>In my case the hyperplanes is defined as $y = w'*x + x_0$, but it is ok to d... | Ben Grossmann | 81,360 | <p>The answer is no. For example, $\{1/n:n\in \Bbb N\} \cup \{0\}$ is disconnected but compact.</p>
|
213,338 | <p>Suppose that $f$ is analytic in the unit disc D = {$z \in \mathbb{C}$ : |$z$| < 1} and $|$f($z$)$| \le 1/(1-|$z$|)$ for all $z\in D$.</p>
<p>Let $f($z$)= \sum _{n=0}^{\infty } a_nz^n$ be the power series expansion of f about $0$.</p>
<p>Prove that $$|a_n| \le (n+1)(1+1/n)^{n} < e(n+1)$$</p>
| Asaf Karagila | 622 | <p>We can rewrite $x^x=e^{x\ln x}$, now using continuity of exponentiation we know that $$\lim_{x\to 0}e^{x\ln x}=e^{\lim_{x\to 0} x\ln x}$$</p>
<p>Calculating $\lim\limits_{x\to0} x\ln x$ is simpler, and it is indeed $0$ (you can use L'Hospital to prove this limit), now we have: $$\lim_{x\to 0}x^x=\lim_{x\to0} e^{x\l... |
3,320,193 | <blockquote>
<p>If given <span class="math-container">$P(B\mid A) =4/5$</span>, <span class="math-container">$P(B\mid A^\complement)= 2/5$</span> and <span class="math-container">$P(B)= 1/2$</span>, what is the probability of <span class="math-container">$A$</span>?</p>
</blockquote>
<p>I know I need to apply Bayes ... | Community | -1 | <p>You know that <span class="math-container">$$P(B)=P(B|A)P(A)+P(B|A^C)P(A^C)$$</span> and <span class="math-container">$P(A^C)=1-P(A)$</span>. From there, it's just plugging in and solving for <span class="math-container">$P(A)$</span>.</p>
|
2,895,284 | <blockquote>
<p>Find $\frac{d}{dx}\frac{x^3}{{(x-1)}^2}$</p>
</blockquote>
<p>I start by finding the derivative of the denominator, since I have to use the chain rule. </p>
<p>Thus, I make $u=x-1$ and $g=u^{-2}$. I find that $u'=1$ and $g'=-2u^{-3}$. I then multiply the two together and substitute $u$ in to get:</p... | Bernard | 202,857 | <p>You're mixing the product rule and the quotient rule. You can apply each of them, but not simultaneously.</p>
<ul>
<li>by the <em>product rule</em>: remember $\dfrac{\mathrm d}{\mathrm dx}\Bigl(\dfrac1{x^n}\Bigr)=-\dfrac n{x^{n+1}}$, so
\begin{align}
\dfrac{\mathrm d}{\mathrm dx}\biggl(\dfrac{x^3}{(1-x)^2}\biggr)&a... |
2,103,436 | <p>Suppose we have the vector space $V$ and the non-empty subspace $W$. I know there is a theorem that states that if $\bar{v}_1$ and $\bar{v}_2$ are vectors in a subspace $W$ then the vector $(\bar{v}_1 + \bar{v}_2)$ will also be in the subspace $W$. However is the converse true? Would having the vector $(\bar{v}_1 + ... | Mark Fischler | 150,362 | <p>Of course not. Let $P_W(\vec{v})$ be the projection of $\vec{v}$ onto subspace $W$.</p>
<p>Then as long as $$\vec{v}_1- P_W(\vec{v}_1)=- (\vec{v}_2-P_W(\vec{v}_2))$$ their sum will be in subspace $W$.</p>
|
2,316,042 | <p><strong>Problem:</strong> Consider the set of all those vectors in $\mathbb{C}^3$ each of whose coordinates is either $0$ or $1$; how many different bases does this set contain?</p>
<p>In general, if $B$ is the set of all bases vectors then,
$$B=\{(x_1,x_2,x_3),(y_1,y_2,y_3),(z_1,z_2,z_3)\}.$$</p>
<p>There are $8... | Max Herrmann | 172,142 | <p>The set contains 29 different bases out of 35 possible permutations (excluding the zero vector).
Here are the six elements of the set which do not form a basis:</p>
<p><span class="math-container">$$
\begin{eqnarray}
(0, 0, 1), (0, 1, 0), (0, 1, 1) \\
(0, 0, 1), (1, 0, 0), (1, 0, 1) \\
(0, 0, 1), (1, 1, 0), (1, 1, ... |
205,479 | <p>There are $K$ items indexed $X_1, X_2, \ldots, X_K$ in the pool. Person A first randomly take $K_A$ out of these $K$ items and put them back to the pool. Person B then randomly take $K_B$ out of these $K$ items. What is the expectation of items that was picked by B but not taken by A before?</p>
<p>Assuming $K_A \g... | André Nicolas | 6,312 | <p>I would prefer to index the objects by the <em>numbers</em> $1,2,\dots,K$. </p>
<p>Define the indicator random variable $W_i$ by $W_i=1$ if item $i$ was taken by $B$ but not taken by $A$ before, and by $W_i=0$ otherwise. Let $p=\Pr(W_i=1)$. Then
$$p=\frac{K_B}{K}\left(1-\frac{K_A}{K}\right).$$
This is because the e... |
2,155,071 | <p>You Throw Two Dice. One of the possible many outcomes that may occur is that you get a six on each die (this outcome is called a double six). How many times must you throw the two dice in order for the probability of getting a double six (on one of your throws) to be at least .50?</p>
<p>I completed this problem ... | PattuX | 390,101 | <p>Ask yourself the following:</p>
<p>$p(double \space six)=$</p>
<p>That means $p(no \space double \space six)=$</p>
<p>Chaining this: $p(no \space double \space six \space in \space n \space tries)=$</p>
<p>Again negating: $p(at \space least \space one \space double \space six \space in \space n \space tries)=$</... |
2,155,071 | <p>You Throw Two Dice. One of the possible many outcomes that may occur is that you get a six on each die (this outcome is called a double six). How many times must you throw the two dice in order for the probability of getting a double six (on one of your throws) to be at least .50?</p>
<p>I completed this problem ... | Doug M | 317,162 | <p>The probability of getting a double 6 on any throw is $\frac {1}{36}$</p>
<p>Your probability of getting not getting any 1 double 6s on N throws is $(1-\frac 1{36})^N$</p>
<p>Solve for N such that
$(1-\frac 1{36})^N < 0.5$</p>
<p>$N \log (\frac {35}{36}) < \log {\frac 12}\\
N > \frac {\log 2}{\log 36-\l... |
244,433 | <p>I have a list:</p>
<pre><code>data = {{2.*10^-9, 0.0025}, {4.*10^-9, 0.0025}, {6.*10^-9, 0.0025}, {8.*10^-9, 0.0025}, {1.*10^-8, 0.0025}, {7.*10^-9, 0.0023}, {3.*10^-9, 0.0025},...}
</code></pre>
<p>And I wanted to remove every third pair and get</p>
<pre><code> newdata = {{2.*10^-9, 0.0025}, {4.*10^-9, 0.0025}, {8.... | bill s | 1,783 | <p>Another way to construct the needed indices:</p>
<pre><code>data[[Union[Range[1, Length[data], 3], Range[2, Length[data], 3]]]]
{{2.*10^-9, 0.0025}, {4.*10^-9, 0.0025}, {8.*10^-9, 0.0025}, {1.*10^-8, 0.0025}, {3.*10^-9, 0.0025}}
</code></pre>
<p>Similarly:</p>
<pre><code>data[[Complement[Range[Length[data]], Range[... |
1,400,352 | <p>Confusion with the eccentricity of ellipse. On <a href="https://en.wikipedia.org/wiki/Ellipse#Directrix" rel="nofollow noreferrer">wikipedia</a> I got the following in the directrix section of ellipse.</p>
<blockquote>
<p>Each focus F of the ellipse is associated with a line parallel to the minor axis called a di... | Prakhar Londhe | 261,867 | <p>The ellipse can also be defined as the locus of points whose distance from the focus is proportional to the horizontal distance from a vertical line known as the conic section directrix, where the ratio is $<1$. Letting $e$ be the ratio and $d$ the distance from the center at which the directrix lies, then in ord... |
4,459,439 | <p>Suppose <span class="math-container">$G$</span> is an abelian finite group, and the number of order-2 elements in <span class="math-container">$G$</span> is denoted by <span class="math-container">$N$</span>.</p>
<p>I have found that <span class="math-container">$N= 2^n-1$</span> for some <span class="math-container... | Maxime Cazaux | 666,222 | <p>One can be very explicit using the following fact : a finite abelian group is isomorphic to <span class="math-container">$\bigoplus_{i=1}^n \frac{\mathbb{Z}}{n_i \mathbb{Z}}$</span>, so you only have to count the number of <span class="math-container">$n_i$</span>'s which are even.
If we call this number <span class... |
145,303 | <p>Another question about the convergence notes by Dr. Pete Clark:</p>
<p><a href="http://alpha.math.uga.edu/%7Epete/convergence.pdf" rel="nofollow noreferrer">http://alpha.math.uga.edu/~pete/convergence.pdf</a></p>
<p>(I'm almost at the filters chapter! Getting very excited now!)</p>
<p>On page 15, Proposition 4.6 st... | Pete L. Clark | 299 | <p>The missing implication "separable metrizable implies second countable" is rather easy to prove -- as Carl's answer shows -- but the proof should still appear in the notes. </p>
<p>I have uploaded a new version taking this into account. Thanks for bringing this to my attention.</p>
|
2,853,401 | <p>Assume $E\neq \emptyset $, $E \neq \mathbb{R}^n $. Then prove $E$ has at least one boundary point. (i.e $\partial E \neq \emptyset $).</p>
<p>================= </p>
<p>Here is what I tried.<br>
Consider $P_0=(x_1,x_2,\dots,x_n)\in E,P_1=(y_1,y_2,\dots,y_n)\notin E $.<br>
Denote $P_t=(ty_1+(1-t)x_1,ty_2+(1-t)x_2,\... | William Elliot | 426,203 | <p>Let S be a connected space.<br>
Assume A subset S and empty $\partial$A.<br>
Since $\partial$A = $\overline A$ $\cap$ $\overline{S-A},$
S = (S-A)$^o$ $\cup$ A$^o.$ </p>
<p>Thus either (S-A)$^o$ or A$^o$ is empty.<br>
If A$^o$ is empty, (S-A)$^o$ = S. So S = S - A and A is empty.<br>
If (S-A)$^o$ is empty, S = A$^... |
3,583,778 | <p>This problem came to me when I was solving another binomial coefficient summation problem in this site.</p>
<p>I want to prove that <span class="math-container">$\sum_{i=0}^{p}{\sum_{j=0}^{q+i}{\sum_{k=0}^{r+j}{\binom{p}{i}\binom{q+i}{j}\binom{r+j}{k}}}}=4^{p}3^{q}2^{r}$</span></p>
| Z Ahmed | 671,540 | <p><span class="math-container">$$S=\sum_{i=0}^{p} \sum_{j=0}^{q+i} \sum_{k=0}^{r+j} {p \choose i} {q+i \choose j} {r+j \choose k}$$</span></p>
<p>First do the <span class="math-container">$k$</span> sum, then
<span class="math-container">$$S=\sum_{i=0}^{p} \sum_{j=0}^{q+i} {p \choose i} {q+i \choose j} 2^{r+j}$$</sp... |
1,123,777 | <p><strong><span class="math-container">$U$</span> here represents the upper Riemann Integral.</strong></p>
<p><img src="https://i.stack.imgur.com/GbNm2.jpg" alt="enter image description here" /></p>
<p><img src="https://i.stack.imgur.com/KtRI4.jpg" alt="enter image description here" /></p>
<p><img src="https://i.stack... | Paul Vithayathil | 211,283 | <p>I noticed that you had for n = 2 the solutions as 1+1, 2+0 but 2+0 cannot be a solution based on the problem constraints.</p>
<p>Edit: My bad, as I was working through 2+0 did not make sense but 2 is clearly a solution. But I was able to prove this by thinking for each n, how many ways can you sum to n with k numbe... |
2,064,984 | <p>I am learning about the student t-test. </p>
<p>I am struggling, however, to be given a reasonable explanation why the standard deviation of the standard normal distribution curve is 1. </p>
<p>It says "The Standard Normal Variable is denoted Z and has mean 0 and S.D 1..."</p>
<p>"... this is written as Z ~ N(0,1... | Michael Hardy | 11,667 | <p>It's hard to be sure what your question means. Suppose $X$ is normally distributed and has expected value $\mu$ and standard deviation $\sigma$. Let $Z = \dfrac {X-\mu} \sigma,$ so that $X = \mu + \sigma Z.$ Then $Z$ has expected value $0$ and standard deviation $1$ and is normally distributed. The one member of thi... |
4,112,958 | <p>This is a Number Theory problem about the extended Euclidean Algorithm I found:</p>
<p>Use the extended Euclidean Algorithm to find all numbers smaller than <span class="math-container">$2040$</span> so that <span class="math-container">$51 | 71n-24$</span>.</p>
<p>As the eEA always involves two variables so that <s... | Joffan | 206,402 | <p>This is asking you to find solutions to <span class="math-container">$71n-24\equiv 0 \bmod 51$</span>, which equivalently is <span class="math-container">$20n\equiv 24\bmod 51$</span>.</p>
<p>To do this you can proceed by finding the modular multiplicative inverse of <span class="math-container">$20 \bmod 51$</span>... |
4,345,671 | <p>I have a series of cubic polynomials that are being used to create a trajectory. Where some constraints can be applied to each polynomial, such that these 4 parameters are satisfied.
-Initial Position
-final Position
-Initial Velocity
-final Velocity</p>
<p>The polynomials are pieced together such that the ends of o... | Karl | 279,914 | <p>The problem is that your pieces are not triangles and don't "approach triangles in the limit" in the necessary way. For example, the proportion of shaded area above height <span class="math-container">$\frac h2$</span> in your figure doesn't approach <span class="math-container">$\frac14$</span> of the tot... |
764,905 | <p>Calculate $$\int_{D}(x-2y)^2\sin(x+2y)\,dx\,dy$$ where $D$ is a triangle with vertices in $(0,0), (2\pi,0),(0,\pi)$.</p>
<p>I've tried using the substitution $g(u,v)=(2\pi u, \pi v)$ to make it a BIT simpler but honestly, it doesn't help much.</p>
<p>What are the patterns I need to look for in these problems so I ... | Disintegrating By Parts | 112,478 | <p>Your integral can be evaluated as an iterated integral and bunch of integration-by-parts:
$$
\int_{0}^{\pi}\int_{0}^{2\pi-2y}(x-2y)^{2}\sin(x+2y)dxdy \\
= \int_{0}^{\pi}\int_{2y}^{2\pi}(x-4y)^{2}\sin(x)\,dx dy \\
= \int_{0}^{\pi}\left(\left.-(x-4y)^{2}\cos(x)\right|_{x=2y}^{x=2\pi}+\int_{2y}^{2\pi}2(x-4y)\c... |
7,647 | <p>Given a polyhedron consists of a list of vertices (<code>v</code>), a list of edges (<code>e</code>), and a list of surfaces connecting those edges (<code>s</code>), how to break the polyhedron into a list of tetrahedron?</p>
<p>I have a convex polyhedron.</p>
| Joseph Malkevitch | 1,618 | <p>There are polyhedra which are homeomorphic to a sphere with the property that every edge which is not already an edge of the polyhedron lies completely in the exterior of the polyhedron. Sometimes these polyhedra are called Lennes Polyhedra. These polyhedra can not be subdivided into tetrahedra using existing vertic... |
4,059,420 | <p>If f is holomorphic at every point on the open disc <span class="math-container">$$D=\{z:|z|\lt1\}$$</span>
I want to show that f is constant</p>
| qwfwq | 899,490 | <p><span class="math-container">$\infty$</span> can't be the biggest number, because it isn't a number. It's just a symbol used in limits. Ok, you can consider compactifications of real or complex numbers, but that's certainly not what you had in mind, because both are very far from "counting".</p>
|
4,059,420 | <p>If f is holomorphic at every point on the open disc <span class="math-container">$$D=\{z:|z|\lt1\}$$</span>
I want to show that f is constant</p>
| Oscar Lanzi | 248,217 | <p>Set ordinalities:</p>
<p><span class="math-container">$0,1,2,...,\omega,\omega+1,\omega+2,...,\omega^n,...,\omega^{\omega},...,\omega^{\omega^{\omega^{...}}}=\epsilon_0,...$</span></p>
<p>One might say it's hugely, vastly, mind-bogglingly big.</p>
|
399,804 | <p>The Question was:</p>
<blockquote>
<p>Express $2\cos{X} = \sin{X}$ in terms of $\sin{X}$ only.</p>
</blockquote>
<p>I have had dealings with similar problems but for some reason, due to I believe a minor oversight, I am terribly vexed.</p>
| Kris Williams | 38,143 | <p>Given the equation $$2 \cos(x) = \sin(x)$$ and the instruction to write solely in terms of $\sin(x)$, I would begin by looking for an identity that involves $\cos(x)$, the term we want to transform and $\sin(x)$ the term we want to write everything in. This leaves us with the identity $$\cos(x)^2 + \sin(x)^2 =1.$$ W... |
399,804 | <p>The Question was:</p>
<blockquote>
<p>Express $2\cos{X} = \sin{X}$ in terms of $\sin{X}$ only.</p>
</blockquote>
<p>I have had dealings with similar problems but for some reason, due to I believe a minor oversight, I am terribly vexed.</p>
| Kushashwa Ravi Shrimali | 42,058 | <p>As already mentioned by the users , you must use the identity : </p>
<p>$$\large \sin^2 x + \cos^2 x = 1 ........... \boxed{1} $$ </p>
<p>Here is , how to start solving the problem : </p>
<p>$$\large \text{Given : } \quad 2\cos x = \sin x ......... \boxed{2}$$
Now, simplifying equation 1 further :</p>
<p>$$\larg... |
399,804 | <p>The Question was:</p>
<blockquote>
<p>Express $2\cos{X} = \sin{X}$ in terms of $\sin{X}$ only.</p>
</blockquote>
<p>I have had dealings with similar problems but for some reason, due to I believe a minor oversight, I am terribly vexed.</p>
| robjohn | 13,854 | <p>Of course, since $\sin^2(x)+\cos^2(x)=1$, if $2\cos(x)=\sin(x)$, squaring and adding $4\sin^2(x)$ to both sides yields
$$
4=5\sin^2(x)\tag{1}
$$
Of course, $(1)$ also has solutions where $2\cos(x)=-\sin(x)$. Knowledge of
$\sin(x)$ fully determines $|\cos(x)|$, but says nothing about the sign of $\cos(x)$. For this ... |
506,720 | <p>Hi how can I find the dimension of a vector space? For example :
$V = \mathbb{C} , F = \mathbb{Q}$
what is the dimension of $V$ over $\mathbb{Q}$?</p>
| Mhenni Benghorbal | 35,472 | <p><strong>Hint:</strong> Check this basis for $\mathbb{C}$ </p>
<blockquote>
<p>$$ \left\{ 1, i\right\}. $$</p>
</blockquote>
<p>Now, you need to know what the definition of dimension is? </p>
|
4,183,263 | <p>If Tychonoff's theorem is true, why closed ball in <span class="math-container">$\mathbb{R}^n$</span> is not compact?</p>
<p>The theorem says that if <span class="math-container">$X_i$</span> is compact, for every <span class="math-container">$i\in I$</span>, so <span class="math-container">$\prod_{i\in I}X_i$</span... | Hoang Nguyen | 934,337 | <p><span class="math-container">$\prod_{n \in \mathbb{N}}[-1,1]$</span> is indeed compact under product topology. I believe you misunderstand the structure of a closed ball, that is a closed ball <span class="math-container">$B_{d}(x, \epsilon)$</span> is defined to be the set <span class="math-container">$\{y: d(x,y) ... |
1,599,886 | <p>What is the proper way of proving : the density operator $\hat{\rho}$ of a pure state has exactly one non-zero eigenvalue and it is unity, i.e,</p>
<p>the density matrix takes the form (after diagonalizing):
\begin{equation}
\hat{\rho}=
{\begin{bmatrix}
1 & 0 & \cdots & 0 \\
0 & 0 & \cdots &... | Martin Argerami | 22,857 | <p>Since $\langle \psi|\psi\rangle=1$, we have
$$
(|\psi\rangle\langle\psi|)^2=|\psi\rangle\langle\psi|\psi\rangle\langle\psi|=|\psi\rangle\langle\psi|
$$
Then $|\psi\rangle\langle\psi|$ is a projection, and its eigenvalues are $0$ and $1$. Since we know that the trace of $|\psi\rangle\langle\psi|$ is one we conclude t... |
752,045 | <p>Write the area $D$ as the union of regions. Then, calculate $$\int\int_Rxy\textrm{d}A.$$</p>
<p>First of all I do not get a lot of parameters because they are not defined explicitly (like what is $A$? what is $R$?).</p>
<p>Here is what I did for the first question:</p>
<p>The area $D$ can be written as:</p>
<p>$... | mookid | 131,738 | <p>You should write
$$
D =\{(x,y): -1<x<1, -1<y<1+x^2
\}
-
\{ (x,y): 0<x<1, -\sqrt{x}<y<\sqrt{x}
\}
$$</p>
|
752,045 | <p>Write the area $D$ as the union of regions. Then, calculate $$\int\int_Rxy\textrm{d}A.$$</p>
<p>First of all I do not get a lot of parameters because they are not defined explicitly (like what is $A$? what is $R$?).</p>
<p>Here is what I did for the first question:</p>
<p>The area $D$ can be written as:</p>
<p>$... | Ellya | 135,305 | <p>It will take a while but to write as unions you need to break things down I.e. for the top left section you can write as $\{-1\leq x\leq 1, 0\leq y\leq 1\}\cup\{-1\leq x\leq 1,1\leq y\leq 1+x^2\}$. And continue this for the other areas.</p>
<p>Btw $R$ refers to the whole region you have written as unions, and A jus... |
1,634,325 | <blockquote>
<p><strong>Problem</strong>:
Is there sequence that sublimit are $\mathbb{N}$? If it's eqsitist prove this.</p>
</blockquote>
<p>I try to solve this problem by guessing what type of sequence need to be. <br>For example:
$a_n=(-1)^n$ has two sublimit $\{1,-1\}$.
<br>
$a_n=n
\times\sin(\frac{\pi}{2})$ h... | Community | -1 | <p>There is such a sequence. For example make it by counting. Fix a natural $n$, first count to $n$, then start again and count to $n+1$, then to $n+2$ etc. </p>
|
887,200 | <p>So I have the permutations:
$$\pi=\left( \begin{array}{ccc}
1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\
2 & 3 & 7 & 1 & 6 & 5 & 4 & 9 & 8
\end{array} \right)$$
$$\sigma=\left( \begin{array}{ccc}
1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9... | amWhy | 9,003 | <p>You've done just fine - EXCEPT: the first is $\sigma \pi$ and the second is $\pi\sigma$.</p>
<p>$$\sigma \pi = \sigma\circ \pi = \sigma(\pi(i))\quad i = 1, 2, \ldots, 9\, $$ So we can only find $\sigma(\pi(i))$ after finding $\pi(i)$.</p>
|
179,230 | <p>I draw <a href="http://reference.wolfram.com/language/ref/Cos.html" rel="nofollow noreferrer"><code>Cos</code></a> function using the code line : </p>
<pre><code>GraphicsColumn[
{
Plot[Cos[0.0625*Pi x], {x, 0, 40*Pi}, Axes -> False],
Plot[Cos[0.0625*Pi x], {x, 0, 40*Pi}, Axes -> False],
Plot[Cos... | user7739 | 7,739 | <p>I know this post could be shorter, but I wanted to outline a methodology, as in teach a man to fish..., so please bear with me. The first point to make is that the geometric scale of x vs y is easiest to manage from within the Graphics[] environment. Each Plot[] command creates its own coordinate context and so you ... |
2,883,023 | <p>Find the number of zeros of $f(z)=z^6-5z^4+3z^2-1$ in $|z|\leq1$. </p>
<p>My attempts have not gotten far. </p>
<p>I know we can examine the related equation $f(w)=w^3-5w^2+3w-1$ in $|w|\leq1$, letting $w=z^2$.</p>
<p>It is clear that $f(w)=0$ for $|w|=1$ if and only if $w=-1$. </p>
<p>My main problem is that th... | Doug M | 317,162 | <p>Rouche's theorem.</p>
<p>if $|g(z)| > |h(z)|$ for all $z$ along some closed contour. $g(z) + h(z)$ has as many zeros inside the contour as $g(z)$ has inside the contour.</p>
<p>But what about when $|g(z)| = |h(z)|$? Then there is the possibility that some zeros lie on the contour.</p>
<p>In this case
let $g(... |
96,080 | <p>The empty clause is a clause containing no literals and by definition is false.</p>
<p>c = {} = F</p>
<p>What then is the empty set, and why does it evaluate to true?</p>
<p>Thanks!</p>
| Isnax | 554,236 | <p>You can get an intuition why this is so by observing that:</p>
<ol>
<li>A disjunction is true iff there exists a member which is true. In an empty disjunction (empty clause) there is no such member, so it is always false.</li>
<li>A conjunction is true iff no member which is false exists. An empty conjunction (empt... |
974,656 | <p><img src="https://i.stack.imgur.com/LyqzL.jpg" alt="enter image description here"></p>
<p>One way to solve this and my book has done it is by : </p>
<p><img src="https://i.stack.imgur.com/2wYSn.jpg" alt="enter image description here"></p>
<hr>
<p>This is a well known way, but I have a different method, and it se... | Petite Etincelle | 100,564 | <p>Suppose $a_0 < 0$ and $a_n >0$</p>
<p>Firstly we have $P(0) = a_0 < 0$.</p>
<p>$P$ is of even degree, which means $\lim_{x \to +\infty} P(x) = \lim_{x\to -\infty}P(x) = +\infty$, since $a_n > 0$</p>
<p>Apply IVT on $[0, +\infty)$ and $(-\infty, 0]$, we find two roots for $P(x)$.</p>
<p>If $a_0 > 0... |
28,532 | <p><code>MapIndexed</code> is a very handy built-in function. Suppose that I have the following list, called <code>list</code>:</p>
<pre><code>list = {10, 20, 30, 40};
</code></pre>
<p>I can use <code>MapIndexed</code> to map an arbitrary function <code>f</code> across <code>list</code>:</p>
<pre><code>{f[10, {1}],... | Michael E2 | 4,999 | <p>Another way, using <code>MapIndexed</code>'s functionality, like <a href="https://mathematica.stackexchange.com/users/5">rm-rf</a>'s:</p>
<pre><code>mapAtIndexed[f_, expr_, pos_, levelspec_: 1, opts : OptionsPattern[MapIndexed]] :=
Module[{f0},
f0[x_, p : Alternatives @@ pos] := f[x, p];
f0[x_, _] := x;
MapI... |
113,797 | <p>I'm trying to extract every 21st character from this text, s (given below), to create new strings of all 1st characters, 2nd characters, etc.</p>
<p>I have already separated the long string into substrings of 21 characters each using</p>
<pre><code> splitstring[String : str_, n_] :=
StringJoin @@@ Partitio... | Edmund | 19,542 | <p>The last substring does not contain 21 characters but you can use the padded form of <code>Partition</code> to get sublist of equal length with padding. Then it is just a <code>Transpose</code> and <code>StringJoin</code> to the result. I use <code>"&"</code> for padding which you can substitute for any charact... |
172,894 | <p>Suppose $A$ is an integral domain with integral closure $\overline{A}$ (inside its fraction field), $\mathfrak{q}$ is a prime ideal of $A$, and $\mathfrak{P}_1,\ldots,\mathfrak{P}_k$ are the prime ideals of $\overline{A}$ lying over $\mathfrak{q}$. Show that $\overline{A_\mathfrak{q}} = \bigcap\overline{A}_\mathfrak... | Cantlog | 37,134 | <p>First notice that without extra hypothesis, the integral closure needs not be finite over $A$, and there might be infinitely many prime ideals lying over a given one in $A$. Neverthless, the equality holds. </p>
<p>It is easy to see that the LHS is contained in the RHS because the latter is integrally closed and co... |
464,426 | <p>Find the limit of $$\lim_{x\to 1}\frac{x^{1/5}-1}{x^{1/3}-1}$$</p>
<p>How should I approach it? I tried to use L'Hopital's Rule but it's just keep giving me 0/0.</p>
| André Nicolas | 6,312 | <p>L'Hospital's Rule works just fine, and in one step. Remember that you are taking the limit as $x$ approaches $1$.</p>
|
464,426 | <p>Find the limit of $$\lim_{x\to 1}\frac{x^{1/5}-1}{x^{1/3}-1}$$</p>
<p>How should I approach it? I tried to use L'Hopital's Rule but it's just keep giving me 0/0.</p>
| Michael Hardy | 11,667 | <p>If $u^{15}= x$ then $x^{1/5}=(u^{15})^{1/5}=u^3$, and similarly $x^{1/3}=u^5$. Therefore
$$
\frac{x^{1/5}-1}{x^{1/3}-1} = \frac{u^3-1}{u^5-1}= \frac{(u-1)(u^2+u+1)}{(u-1)(u^4+u^3+u^2+u+1)}.
$$
Do the obvious cancelation. Then ask: as $x\to1$, then $u\to\text{what}$? Then finding the limit is easy.</p>
<p>You cou... |
51,555 | <p>Consider the set $(s_1, ..., s_N) \in S$, where all $s_i$ are positive integers selected from some interval $[M, L]$ and the sum of any $k$ integers in $S$ is required to be unique and to have a distance of at least $d$ from all other possible sums of $k$ integers in the set. For example, if $k = 2$, the sum of any... | Gerry Myerson | 8,269 | <p>Here's a start. The number of choices of $k$ elements is $N\choose k$ (or a little more, if repeats are allowed), and each sum is between $kM$ and $kL$, so you must have $${N\choose k}\lt kL-kM+2$$ which gives you an upper bound on $N$. It doesn't tell you whether you can achieve that bound, nor an efficient way to ... |
51,555 | <p>Consider the set $(s_1, ..., s_N) \in S$, where all $s_i$ are positive integers selected from some interval $[M, L]$ and the sum of any $k$ integers in $S$ is required to be unique and to have a distance of at least $d$ from all other possible sums of $k$ integers in the set. For example, if $k = 2$, the sum of any... | Julián Aguirre | 4,791 | <p>This is an algorithm that I am convinced is optimal, although I have not tried to prove it. First of all, we may assume that $L=0$ (consider ${s_1-L,\dots,s_N-L}$). For the algorithm I will describe, we may also asume that $d=1$.</p>
<p>The key is to observe that if $\{s_1,\dots,s_N\}$ is such that all sums of subs... |
3,999,996 | <p>I know how you can show this geometrically, but is there any way to prove this algebraically?</p>
| paulinho | 474,578 | <p>Since <span class="math-container">$\arcsin x$</span> is bounded above by <span class="math-container">$\pi / 2$</span>, by monotonicity of the integral we have
<span class="math-container">$$\int_0^1 x \arcsin x dx \leq \frac \pi 2 \int_0^1 x dx = \frac \pi 4$$</span></p>
|
1,501,876 | <blockquote>
<p>I want to prove $A_n$ has no subgroups of index 2. </p>
</blockquote>
<p>I know that if there exists such a subgroup $H$ then $\vert H \vert = \frac{n!}{4}$ and that $\vert \frac{A_n}{H} \vert = 2$ but am stuck there. I have tried using the proof that $A_4$ has no subgroup of order 6 to get some idea... | MooS | 211,913 | <p>This elementary argument works for all $n$ at once and does not need simplicity of the alternating group: Let $N$ be of index $2$ in $A_n$ and let $x$ be a $3$-cycle. We have $xN=x^4N=(xN)^4=N$, where the latter holds since the factor group is of order $2$. We derive that $N$ contains all $3$-cycles. But it is a wel... |
2,869,898 | <p>I want to prove that <span class="math-container">$$
f(x,y)=
\begin{cases} \frac{xy^2}{x^2+y^2} &\text{ if }(x,y)\neq (0,0)\\
0 &\text{ if }(x,y)=(0,0)
\end{cases}
$$</span>
is not differentiable at <span class="math-container">$(0,0)$</span>.</p>
<p>I thought that I can prove that it is not continuous arou... | Fred | 380,717 | <p>Another approach: assume that $f$ is differentiable at $(0,0)$. For each $v=(u,w) \in \mathbb R^2$ with $u^2+w^2=1$ we then have</p>
<p>$$(*) \quad \frac{\partial f}{\partial v}(0,0)=v \cdot gradf(0,0).$$</p>
<p>Since </p>
<p>$$ (**) \quad\frac{\partial f}{\partial v}(0,0)= \lim_{t \to 0}\frac{f(tv)-f(0,0)}{t}=\f... |
2,869,898 | <p>I want to prove that <span class="math-container">$$
f(x,y)=
\begin{cases} \frac{xy^2}{x^2+y^2} &\text{ if }(x,y)\neq (0,0)\\
0 &\text{ if }(x,y)=(0,0)
\end{cases}
$$</span>
is not differentiable at <span class="math-container">$(0,0)$</span>.</p>
<p>I thought that I can prove that it is not continuous arou... | Rene Schipperus | 149,912 | <p>Look at the directional derivative:</p>
<p>$$d_{\mathbf{v}}f=\lim\limits_{t\to 0}\frac{f(\mathbf{x}+t\mathbf{v})-f(\mathbf{x})}{t}$$</p>
<p>in your case you have for $\mathbf{x}=(0,0)$</p>
<p>$$d_{(a,b)}f=\lim\limits_{t\to 0}\frac{\frac{t^3ab^2}{t^2(a^2+b^2)}}{t}=\frac{ab^2}{a^2+b^2}$$ Thus the directional deriv... |
86,422 | <p>In his notes in Algebraic Number theory, J S Milne gives the following as an example of an unramified Abelian extension :</p>
<p>$ K = \mathbb Q (\sqrt{-5})$ having a quadratic extension $L = \mathbb Q (\sqrt{-1}, \sqrt{-5})$.
Then, $L/K$ has discriminant a unit, so it ramifies. </p>
<p>My question is, considerin... | David Loeffler | 2,481 | <p>You are slipping up because $i$ does not generate the ring of integers of $L$ as an $\mathcal{O}_K$-algebra: we have $\mathcal{O}_K = \mathbb{Z}[\sqrt{-5}]$, but $\mathcal{O}_L = \mathbb{Z}\left[i, \frac{1 + \sqrt{5}}{2}\right] \ne \mathbb{Z}[i, \sqrt{-5}]$. Hence the discriminant of $L/K$ is not the same as the dis... |
86,422 | <p>In his notes in Algebraic Number theory, J S Milne gives the following as an example of an unramified Abelian extension :</p>
<p>$ K = \mathbb Q (\sqrt{-5})$ having a quadratic extension $L = \mathbb Q (\sqrt{-1}, \sqrt{-5})$.
Then, $L/K$ has discriminant a unit, so it ramifies. </p>
<p>My question is, considerin... | Franz Lemmermeyer | 3,503 | <p>A quick way of seeing what's going on is using the fact that
$L = K(i) = K(\sqrt{5})$; the fact that the different above (say of $L/{\mathbb Q}(i)$ divides the different below (e.g. of ${\mathbb Q}(\sqrt{5})/{\mathbb Q}$) shows (take norms) that the discriminant of $L/K$ divides both $-4$ and $5$, hence is trivial. ... |
4,139,141 | <p>If the conditions of the theorem are met for some ordinary differential equation, then we are guaranteed that a solution exists. However, I don't fully understand what it means for a solution to exist. If we can show that a solution exists, does that mean that it can be found explicitly using known methods? Or, are ... | Christopher K | 101,768 | <p>Here's some guidance to each of your questions. Hopefully this helps!</p>
<ol>
<li>For all <span class="math-container">$a \in A$</span>, there exist disjoint open sets <span class="math-container">$V_{a} \ni a$</span> and <span class="math-container">$U_{a} \ni x$</span>. Since <span class="math-container">$A = \cu... |
772,315 | <p>Statement: $\limsup\limits_{n\to\infty} c_n a_n = c \limsup\limits_{n\to\infty} a_n$</p>
<p>Please help find a counterexample to this statement if $c<0$.</p>
<p>Edit: also suppose $c_n \to c$ and $\limsup a_n$ is finite</p>
| Community | -1 | <p>Let $c_n = -1 \to -1 = c $, and $a_n = (-1)^n$. then</p>
<p>$$ limsup (c_na_n) = 1$$</p>
<p>$$ -1 ( \limsup(a_n) ) = -1 $$</p>
|
2,173,918 | <p>Let $f(z)=\sum\limits_{k=1}^\infty\frac{z^k}{1-z^k}$. I want to show that this series represents a holomorphic function in the unit disk. I'm, however, quite confused. For example, is $f(z)$ even a power series? It doesn't look as such. Here's what I have so far come up with.</p>
<blockquote>
<p>Proof:</p>
</bloc... | reuns | 276,986 | <blockquote>
<p>Theorem : if $\sum_{n=0}^\infty a_n z^n$ converges absolutely for $|z| < R$ then it is analytic/holomorphic for $|z| < R$.</p>
</blockquote>
<p>For $|z| < 1$ : $$\sum_{k=1}^\infty\frac{z^k}{1-z^k}=\sum_{k=1}^\infty\sum_{n=1}^\infty z^{kn} = \sum_{m=1}^\infty z^{m} \tau(m)$$ where $\tau(m)=\... |
4,041,758 | <p>If you have a line passing through the middle of a circle, does it create a right angle at the intersection of the line and curve?</p>
<p>More generally, is it valid to define an angle created between a line and a curve? Is the tangent to the curve at the point of intersection a valid interpretation (I.e a semi circ... | dbmag9 | 73,894 | <p>As you identify in your question, the real point of contention is the definition of <em>angle</em>. As the other answers have indicated, if your definition includes angles at the intersection of two curves then a semicircle certainly has two right angles.</p>
<p>However, as the controversy on Twitter (a few days ago... |
964,989 | <p>Question: Find the order of $(1/2)^{1/2}$, $(1/e)^{1/e}$, $(1/3)^{1/4}$ without using calculator.</p>
<p>Extra constraint: You only have about 150 seconds to do it, failing to do so will eh... make you run out of time on the exam which affects the chance of admitting into graduate school!</p>
<p>Back to the questi... | Community | -1 | <p>$$\left(\frac13\right)^{\frac14}>\left(\frac14\right)^{\frac14}=\left(\frac12\right)^{\frac12}.$$</p>
<p>Knowing there is a minimum at $x=e$,</p>
<p>$$\left(\frac1e\right)^\frac1e<\left(\frac12\right)^{\frac12}<\left(\frac13\right)^{\frac14}.$$</p>
<p>Hint:</p>
<p>Take $\log(x^x)=x\log x$, derivative $\... |
4,492,104 | <p>I'm looking for closed-form expressions for the integral <span class="math-container">$$I:=\int_0^\infty \ln \left(1-\frac{\sin x}{x} \right) dx .$$</span>
Some related integrals that I've found <a href="https://math.stackexchange.com/questions/2354232/is-it-possible-to-evaluate-int-0-infty-log1-coshx-fracx2ex-dx?rq... | metamorphy | 543,769 | <p>More precise computation of the given integral: <span class="math-container">$$I=-5.031379591902842520548271636746403412607399342991051\cdots$$</span></p>
<p>This is computed using <a href="https://pari.math.u-bordeaux.fr/" rel="nofollow noreferrer">PARI/GP</a>. The integrand is oscillating, so <code>intnum</code> c... |
4,492,104 | <p>I'm looking for closed-form expressions for the integral <span class="math-container">$$I:=\int_0^\infty \ln \left(1-\frac{\sin x}{x} \right) dx .$$</span>
Some related integrals that I've found <a href="https://math.stackexchange.com/questions/2354232/is-it-possible-to-evaluate-int-0-infty-log1-coshx-fracx2ex-dx?rq... | Quanto | 686,284 | <p>Expand as follows
<span class="math-container">$$\int_0^\infty \ln \left(1-\frac{\sin x}{x} \right) dx
=-\sum_{n\ge1}\frac1n \int_0^\infty \left( \frac{\sin x}{x} \right)^n dx $$</span>
where <a href="https://math.stackexchange.com/q/307510/686284"><span class="math-container">$\int_0^\infty \left( \frac{\sin x}{x... |
4,492,104 | <p>I'm looking for closed-form expressions for the integral <span class="math-container">$$I:=\int_0^\infty \ln \left(1-\frac{\sin x}{x} \right) dx .$$</span>
Some related integrals that I've found <a href="https://math.stackexchange.com/questions/2354232/is-it-possible-to-evaluate-int-0-infty-log1-coshx-fracx2ex-dx?rq... | Tyma Gaidash | 905,886 | <p>The final sum looks like a <a href="https://www.wolframalpha.com/input?i=Kapteyn+Series" rel="nofollow noreferrer">Kapteyn Series</a>. Maybe it can be simplified or is the solution to a transcendental equation. From @metamorphy’s answer with <a href="https://www.wolframalpha.com/input?i=besselj" rel="nofollow norefe... |
1,383,380 | <p>On page 12 of Stein, Shakarchi textbook 'Complex analysis', the authors state that the <em>Cauchy-Riemann equations link complex and real analysis</em>. I have completed courses on real and complex analysis, but I feel that this is somewhat of an over-statement. But perhaps it is just me which doesnt have a good eno... | David C. Ullrich | 248,223 | <p>Why is there not a fixed point $f(x)=x$? Nobody said there wasn't.</p>
<p>To show there does exist an $x$ with $f(x)=2x$, let $g(x)=f(x)-2x$. Then $g(0)=f(0)\ge 0$, while $g(1)=f(1)-2\le0$. So the intermediate value theorem shows there exists $x$ with $g(x)=0$.</p>
<p>(Or, if you have a theorem saying any map from... |
464,489 | <p>We are given $H = \{(1),(13),(24),(12)(34),(13)(24),(14)(23),(1234),(1432)\}$ is a subgroup of $S_4$. Also assume $K = \{(1),(13)(24)\}$ is a normal subgroup of $H$. Show $H/K$ isomorphic to $Z_2\oplus Z_2$. </p>
<p>This is just a practice question (not assignment). So I have tried finding $H/K$ explicitly.</p>
<p... | user77404 | 77,404 | <p>After Tobias comment. I realized the order of (1234)K is actually 2 and not 4. Since
(1234)K(1234)K = (1234)(1234)K = (13)(24)K = K. We know there are only 2 groups of order 4. That is $Z_4$ and $Z_2 \bigoplus Z_2$. Since we see $H/K$ does not have any elements of order 4 it is not cyclic and cannot be isomorphic t... |
1,017,965 | <p>Just as the title, my question is what is the matrix representation of Radon transform (Radon projection matrix)? I want to have an exact matrix for the Radon transformation. </p>
<p>(I want to implement some electron tomography algorithms by myself so I need to use the matrix representation of the Radon transforma... | AnonSubmitter85 | 33,383 | <p>A couple of years ago I was looking for the same answer. You can see that thread <a href="https://stackoverflow.com/questions/12166562/radon-transform-matrix-representation">here</a>. However, I don't think my solution is the only way to do it nor necessarily the best way.</p>
<p>We know it's a linear transform, so... |
3,725,007 | <p>Consider the rings, <span class="math-container">$Z_2[x]/(1 + x^2)$</span> and <span class="math-container">$ Z_2[x]/(1 + x + x^2)$</span>, despite having different polynomial as divisor, I have been told that -</p>
<p><span class="math-container">$$Z_2[x]/(1 + x^2) = \{0, 1, x, 1 + x\}$$</span></p>
<p>and</p>
<p><s... | Preston Lui | 556,396 | <p>The way I will approach it is to prove that <span class="math-container">$\lim_{x \to \infty}G(x)\le G(\pi)$</span>, where <span class="math-container">$G(x) is \int_{0}^{x}\frac{sin(y)}y dy $</span>. The reason being if <span class="math-container">$f(y)=\frac{sin(y)}y$</span>, then <span class="math-container">$\i... |
891,370 | <p>I got the function $8.513 \times 1.00531^{\Large t} = 10$. The task is to solve $t$. The correct answer is $t = 31$. How do I get there ?.</p>
| beep-boop | 127,192 | <p>Hint: $$8.513 \cdot 1.00531^t=10 \iff 1.00531^t=\frac{10}{8.513} \iff \ln[1.00531^t]=\ln\left[\frac{10}{8.513}\right] $$
$$\iff t\ln[1.00531]=\ln\left[\frac{10}{8.513}\right] \iff t= \quad?$$</p>
|
891,370 | <p>I got the function $8.513 \times 1.00531^{\Large t} = 10$. The task is to solve $t$. The correct answer is $t = 31$. How do I get there ?.</p>
| amWhy | 9,003 | <ul>
<li><p>Isolate $ 1.00531^{\large t}$ on the left side of the equation.</p></li>
<li><p>Take the $\ln$ of each side of that equation.</p></li>
<li><p>And use the fact that $\ln a^b = b \ln a$.</p></li>
<li><p>Solve for $t$ as you would any first degree polynomial.</p></li>
</ul>
|
16,795 | <p>Consider a finite simple graph $G$ with $n$ vertices, presented in two different but equivalent ways:</p>
<ol>
<li>as a logical formula $\Phi= \bigwedge_{i,j\in[n]} \neg_{ij}\ Rx_ix_j$ with $\neg_{ij} = \neg$ or $ \neg\neg$ </li>
<li>as an (unordered) set $\Gamma = \lbrace [n],R \subseteq [n]^2\rbrace$ </li>
</ol>
... | Tony Huynh | 2,233 | <p>The <em>set</em> { $[n], R, \neg R$ } does not actually specify a graph, since we cannot distinguish between edges and non-edges. The <em>triple</em> ($[n], R, \neg R$ ) does specify a graph since by convention we can say that the second coordinate specifies the edges and the third specifies the non-edges. Thus, (... |
382,526 | <p>I can't calculate the Integral:</p>
<p>$$
\int_{0}^{1}\frac{\sqrt{x}}{\sqrt{1-x^{6}}}dx
$$</p>
<p>any help would be great!</p>
<p>p.s I know it converges, I want to calculate it.</p>
| Shuhao Cao | 7,200 | <p>Doing the substitution first $x^3 = \cos\theta$ , then $\theta/2 = t$, would lead us to a simpler integral:
$$
\int^{\frac{\pi}{2}}_0 \frac{1}{3\sqrt{\cos\theta}} d\theta = \int^{\frac{\pi}{4}}_0 \frac{2}{3\sqrt{1 - 2\sin^2t}} dt
$$
and this is <a href="http://mathworld.wolfram.com/EllipticIntegraloftheFirstKind.htm... |
958,099 | <p>I have the following question:</p>
<blockquote>
<p>For real $x$, $f(x) = \frac{x^2-k}{x-2}$ can take any real value. Find the range of values $k$ can take.</p>
</blockquote>
<p>Here is how I commenced:</p>
<p>$$
y(x-2) = x^2-k \\
-x^2 + xy - 2y + k = 0\\
$$</p>
<p>So we have $a=-1$, $b=y$, $c=(-2y+k)$. In orde... | Yulia V | 86,800 | <p>$$y^2 -8y + 4k = (y-4)^2+4(k-4)$$</p>
<p>For $y=4$, we need $k \geq 4$ to ensure that the discriminant is positive.</p>
|
3,040,110 | <p>What is the Range of <span class="math-container">$5|\sin x|+12|\cos x|$</span> ?</p>
<p>I entered the value in desmos.com and getting the range as <span class="math-container">$[5,13]$</span>.</p>
<p>Using <span class="math-container">$\sqrt{5^2+12^2} =13$</span>, i am able to get maximum value but not able to fi... | Makina | 603,955 | <p>Another possible approach.</p>
<p>For the first quadrant: <span class="math-container">$5\sin(x) + 12\cos(x) = 13\sin(x + \arccos(\frac{5}{13}))$</span>. Can follow from there for the rest of the quadrants. Can also try alternative forms for arguments in order to adapt to the values of <span class="math-container">... |
2,279,120 | <p>How do we prove that for any complex number $z$ the minimum value of $|z|+|z-1|$ is $1$ ?
$$
|z|+|z-1|=|z|+|-(z-1)|\geq|z-(z-1)|=|z-z+1|=|1|=1\\\implies|z|+|z-1|\geq1
$$</p>
<p>But, when I do as follows
$$
|z|+|z-1|\geq|z+z-1|=|2z-1|\geq2|z|-|1|\geq-|1|=-1
$$
Since LHS can never be less than 0, $|z|+|z-1|\geq0$</p>... | Vim | 191,404 | <p>By the triangle inequality:
$$|z|+|z-1|\ge |z-(z-1)|=1$$
And $1$ is attained when $z=0$, say.</p>
|
3,846,339 | <p>Suppose I have the inequality <span class="math-container">$(\frac{A}{B})^X < (\frac{C}{D})\cdot(\frac{E}{F})^Y$</span> and I want X by itself.</p>
<p>Can I do this <span class="math-container">$X\cdot \log(\frac{A}{B}) < \log(\frac{C}{D})\cdot(Y\cdot \log(\frac{E}{F}))$</span>?
Am I breaking any rules on the ... | KingLogic | 503,528 | <p>One of the logarithm rules is <span class="math-container">$\log (ab)=\log a + \log b$</span>. Therefore, on the right, it should be <span class="math-container">$\log (\frac{C}{D}) + Y \log (\frac{E}{F})$</span>.</p>
|
275,785 | <p>Let $a_{1}, a_{2}, \ldots, a_{n}$, $n \geq 3$. Prove that at least one of the number $(a_{1}+a_{2}\ldots +a_{n})^{2}-(n^2-n+2)a_{i}a_{j}$ is greater or equal with $0$ for $1 \leq i < j \leq n$.</p>
<p>I don't know at least how to catch this problem .
Thanks :)</p>
| pre-kidney | 34,662 | <p>Suppose for contradiction that the condition didn't hold. This gives you a set of ${n \choose 2}$ sharp inequalities in the $(i,j)$ which you can multiply together, to yield a sharp inequality of homogeneous polynomials of degree $n(n-1)$. However, the reverse inequality can be proven via "bashing" techniques, so it... |
1,393,154 | <p><span class="math-container">$4n$</span> to the power of <span class="math-container">$3$</span> over <span class="math-container">$2 = 8$</span> to the power of negative <span class="math-container">$1$</span> over <span class="math-container">$3$</span></p>
<p>Written Differently for Clarity:</p>
<p><span class="m... | haqnatural | 247,767 | <p>$${ \left( 4n \right) }^{ \frac { 3 }{ 2 } }={ \left( 8 \right) }^{ -\frac { 1 }{ 3 } }\\ \left( { \left( 4n \right) }^{ \frac { 3 }{ 2 } } \right) ^{ 2/3 }=\left( { \left( { 2 }^{ 3 } \right) }^{ -\frac { 1 }{ 3 } } \right) ^{ 2/3 }\\ 4n=2^{ -\frac { 2 }{ 3 } }\\ n=\frac { 2^{ -\frac { 2 }{ 3 } } }{ 4 } =... |
2,348,909 | <p>Example 7.3 of Baby Rudin states that the sum
\begin{align}
\sum_{n=0}^{\infty}\frac{x^2}{(1 + x^2)^n}
\end{align}
is, for $x \neq 0$, a convergent geometric series with sum $1 + x^2$. This confuses me. As far as I know, a geometric series is a series of the form
\begin{align}
\sum_{n=0}^{\infty}x^n,
\end{align}
a... | farruhota | 425,072 | <p>You should write the first few terms to understand its behavior:
$$\sum_{n=0}^{\infty}\frac{x^2}{(1 + x^2)^n}=x^2+\frac{x^2}{1+x^2}+\frac{x^2}{(1+x^2)^2}+\frac{x^2}{(1+x^2)^3}+\cdots=$$
$$x^2\left(1+\frac{1}{1+x^2}+\frac{1}{(1+x^2)^2}+\frac{1}{(1+x^2)^3}+\cdots\right)\stackrel{x\ne 0}=$$
$$x^2\cdot \frac{1}{1-\frac{... |
2,843,560 | <p>If $\sin x +\sin 2x + \sin 3x = \sin y\:$ and $\:\cos x + \cos 2x + \cos 3x =\cos y$, then $x$ is equal to</p>
<p>(a) $y$</p>
<p>(b) $y/2$</p>
<p>(c) $2y$</p>
<p>(d) $y/6$</p>
<p>I expanded the first equation to reach $2\sin x(2+\cos x-2\sin x)= \sin y$, but I doubt it leads me anywhere. A little hint would be... | lab bhattacharjee | 33,337 | <p><strong>Generalization</strong> :</p>
<p>Using <a href="https://math.stackexchange.com/questions/17966/how-can-we-sum-up-sin-and-cos-series-when-the-angles-are-in-arithmetic-pro">How can we sum up $\sin$ and $\cos$ series when the angles are in arithmetic progression?</a>,</p>
<p>$$f(n)=\dfrac{\sum_{k=0}^{n-1}\si... |
81,267 | <p>I have the following problem: I have a (a lot)*3 table, meaning that I have 3 columns, say X, Y and Z, with real values. In this table some of the rows have the same (X,Y) values, but with different value of Z. For instance</p>
<pre><code>{{12.123, 4.123, 513.423}, {12.123, 4.123, 33.43}}
</code></pre>
<p>have th... | Bob Hanlon | 9,362 | <pre><code>data = Table[{x, y, RandomReal[]},
{x, 3}, {y, 3}, {3}] // Flatten[#, 2] &
</code></pre>
<blockquote>
<p>{{1, 1, 0.987008}, {1, 1, 0.682772}, {1, 1, 0.923863}, {1, 2,
0.464991}, {1, 2, 0.963954}, {1, 2, 0.829995}, {1, 3, 0.773942}, {1, 3, 0.550081}, {1, 3,
0.0821332}, {2, 1, 0.466804}, {2... |
353 | <p>One of the challenges of undergraduate teaching is logical implication. The case by case definition, in particular, is quite disturbing for most students, that have trouble accepting "false implies false" and "false implies true" as true sentences. </p>
<blockquote>
<p><strong>Question:</strong> What are good poi... | Tim Seguine | 208 | <p>One of the things that really helped me (not to learn it, but to appropriately apply it), back when I originally learned this myself was the equivalence between $p\implies q$ and $\neg p \vee q$. I constantly reminded myself of it when manipulating logic statements. Make sure the students are aware of this equivalen... |
246,808 | <p>Trying to solve the following PDE with BC <code>T==1</code> on a spherical cap of a unit sphere and <code>T==0</code> at infinity (approximated as <code>r==(x^2 + y^2 + z^2)^0.5==40^0.5</code>) and the flux over the remaining surfaces taken to be zero (only half domains has been specified due to symmetry reasons):</... | user21 | 18,437 | <p>Here are a couple of different ideas for generating the meshes. This works in 12.3</p>
<pre><code>Needs["NDSolve`FEM`"]
Pe = 15;
p = 2/10;
reg = RegionDifference[Cuboid[{-2.5, 0, -3.8}, {7.5, 4, -0.8}],
Ball[]];
bmesh = ToBoundaryMesh[reg, AccuracyGoal -> 3.5];
</code></pre>
<p>The key here is the <... |
468,487 | <p>I have calculated the likelihood of an event to be $1$ in $1.07 \times 10^{2867}$.</p>
<p>I'm looking for a way to describe to a layperson how unlikely this event is to occur, but the number is so mind boggling large I can't find a way to put it into words.</p>
<p>Any suggestions would be appreciated</p>
| user5402 | 81,595 | <p>There's about $n=10^{80}$ atoms in the universe and about $p=4\times 10^{27}$ atoms in the human body. If you choose arbitrarily $p$ atoms in the universe, what is the probability that these atoms are exactly the atoms you are composed of at the instant you finish this sentence?</p>
<p>It's $\binom{n}{p}^{-1}$</p>
|
3,283,606 | <p>Good Evening,</p>
<p>I know this is a basic question, but I haven't been able to find a clear explanation for how to simplify the follow equation:
<span class="math-container">$$n\log_2n=10^6$$</span>
Solving this equation is part of the solution for Problem 1-1 from the Intro. to Algorithms book by CLRS:
<a href="... | Ross Millikan | 1,827 | <p>You are doing one dimensional <a href="https://en.wikipedia.org/wiki/Root-finding_algorithm" rel="nofollow noreferrer">root finding</a> on the function <span class="math-container">$f(n)=10^6-n \log n$</span>. This is a large subject, a chapter in every numerical analysis book. The simplest algorithm to describe i... |
251,028 | <p>I want to make <span class="math-container">$S[\{,\cdots, \}]$</span> as follows</p>
<p>First input of <span class="math-container">$S$</span> is given list <span class="math-container">$\{1,2,3,\cdots, n\}$</span> and it produces <span class="math-container">$s_{123\cdots n}$</span></p>
<p>Further, if the ordering ... | Andrzej | 79,522 | <p>If you want an input to be a <code>List</code> then e.g.:</p>
<pre><code>S[list_] := Subscript[S, StringDelete[ToString[Sort[list]], {",", " ","{", "}"}]]
</code></pre>
<p>If a <code>Sequence</code> then e.g.:</p>
<pre><code>R[seq__] := Subscript[R, StringDelete[ToString[{seq}... |
210,849 | <p>Let $p$ be an odd prime and $a$ be an integer with $\gcd(a, p) = 1$. Show that $x^2 - a \equiv 0 \mod p$ has either $0$ or $2$ solutions modulo $p$</p>
<p>I am clueless with this one. Hints please.</p>
| N. S. | 9,176 | <p>If $x^2-a=0 \mod p$ has some solution $b$, it means that $b^2=a$, and hence you original question becomes </p>
<p>$$x^2-b^2 \equiv 0 \mod p$$</p>
<p>Can you prove now that this equation has exactly two solutions?</p>
|
210,849 | <p>Let $p$ be an odd prime and $a$ be an integer with $\gcd(a, p) = 1$. Show that $x^2 - a \equiv 0 \mod p$ has either $0$ or $2$ solutions modulo $p$</p>
<p>I am clueless with this one. Hints please.</p>
| DonAntonio | 31,254 | <p>For $\,p=2\,$ the claim fails, as $\,x^2-1=x^2+1=(x+1)^2=0\pmod 2\,$ has one unique solution.</p>
<p>If $\,p\,$ is odd and $\,x^2-a=0\pmod p\,$ has a solution $\,b\,$, then</p>
<p>$$x^2-a=x^2-b^2=(x-b)(x+b)=0\pmod p\Longleftrightarrow$$</p>
<p>$$ x=b\pmod p\,\,or\,\,x=-b\pmod p$$</p>
<p>And now you only have to ... |
2,780,043 | <p>In the Navy we had bunk beds with a locker and lock both built in. You could gain access with a combination lock on the side that you could reprogram the code for. The lock was nothing more than several push buttons. Ive wondered for a long time now how many possible combinations there were. I believe the lock h... | Ray Chou | 560,580 | <p>There are $\dbinom{n}{k}$ ways to choose k objects from a set of $n$ buttons, and there are $S(k,j)$ (Stirling numbers of the second kind) ways to partition these $k$ labeled objects into $j$ different (unlabeled) sets, and $j!$ ways to order these sets. In total, that makes</p>
<p>$$\sum_{k=0}^{n}\left[\dbinom{n}{... |
623,703 | <blockquote>
<p>Find the exact value of $\tan\left ( \sin^{-1} \left ( \dfrac{\sqrt{2}}{2} \right )\right )$ without using a calculator. </p>
</blockquote>
<p>I started by finding $\sin^{-1} \left ( \dfrac{\sqrt{2}}{2} \right )=\dfrac{\pi}{4}$</p>
<p>So, $\tan\left ( \sin^{-1} \left ( \dfrac{\sqrt{2}}{2} \right )\r... | Archis Welankar | 275,884 | <p>We have to find the exact value of <span class="math-container">$\;\tan\left(\sin^{-1}\dfrac{1}{\sqrt{2}}\right)$</span>.</p>
<p>Since <span class="math-container">$\;\sin(45)=\dfrac{1}{\sqrt{2}}\;,\;x=45\;$</span> so <span class="math-container">$\;\tan(45)=1\;,\;$</span> hence we are done. Now <span class="math-c... |
10,666 | <p>My question is about <a href="http://en.wikipedia.org/wiki/Non-standard_analysis">nonstandard analysis</a>, and the diverse possibilities for the choice of the nonstandard model R*. Although one hears talk of <em>the</em> nonstandard reals R*, there are of course many non-isomorphic possibilities for R*. My question... | Joel David Hamkins | 1,946 | <p>Let me offer one counterpoint to John's excellent answer.</p>
<p>Under the Continuum Hypothesis, the ultrapower version of R* will be saturated in any countable language. That is, it will realize all finitely realizable countable types with countably many parameters. Thus, by the usual back-and-forth construction, ... |
467,574 | <p>Using permutation or otherwise, prove that $\displaystyle \frac{(n^2)!}{(n!)^n}$ is an integer,where $n$ is a positive integer.</p>
<p>I have no idea how to prove this..!!I am not able to even start this Can u give some hints or the solution.!cheers.!!</p>
| Davood | 477,916 | <p><strong>Remark</strong> : For every (prime) number $p$:
$$
n
\cdot
\lfloor \dfrac{n}{p^{\alpha}} \rfloor
\leq
\lfloor \dfrac{n^2}{p^{\alpha}} \rfloor
\
.
$$
<strong>Proof</strong> :
Let's denote by $m$ and $p$ the integral part and fractional part of
$\dfrac{n}{p^{\alpha}}$ </p>
<p>$$
m=\lfloor \dfrac{n... |
18,895 | <p>I know that the answer is $C(8,2)$, but I don't get, why.
Can anyone, please, explain it?</p>
| ThomasMcLeod | 4,735 | <p>$C[8, 2] = \dfrac{8!}{2!(8-2)!} = (8!/6!)/2! = (8*7)/2 = 28$. Think of it this way. The $8$ is the choice of the first bit, and the $7$ is the choice of the second bit (it's only $7$ because there are only $7$ bits available after the first bit is decided). The $2$ represents the number of permutations of the chosen... |
894,909 | <p>Given </p>
<p>$(x+3)(y−4)=0 $</p>
<p>Quantity $A = xy $</p>
<p>Quantity $B = -12 $</p>
<p>A Quantity $A$ is greater.<br>
B Quantity $B$ is greater.<br>
C The two quantities are equal.<br>
D The relationship cannot be determined from the information given. </p>
<p>How is the answer D and not C ? </p>
<p>Pl... | Vikram | 11,309 | <p>$(x+3)(y-4)=0 ...(I)\Rightarrow$ either $x=-3$ <strong>OR</strong> $y=4$ <strong>OR</strong> $ x=-3$ <strong>AND</strong> $y=4$.</p>
<p>If $x=-3$, then $y$ can assume any value and when $y=4$, $x$ can assume any value to satisfy eqn (I)</p>
<p>In both the cases $A=xy$ can be anything, so we can not be sure about ... |
1,098,438 | <p>This problem is really bothering me for some time, I appreciate if you have some idea and insight.</p>
<blockquote>
<p>Prove that</p>
<p>$$2^{2^n}+5^{2^n}+7^{2^n}$$</p>
<p>is divisible by $39$ for all natural numbers $n$.</p>
</blockquote>
<p>There was a suggestion that this should be done by mathemati... | wendy.krieger | 78,024 | <p>You first find $2^2$, $5^2$ and $7^2$, $\mod 39$, to be $4$, $25$ and $10$ respectively. These add to $39$.</p>
<p>Their squares are $16$, $1$, and $22$, which also add to $39$. This is power $2^2$.</p>
<p>The squares of these are $22$, $1$, $16$, which is the same as before, Thus if it is true for $2^n$, it's ... |
1,098,438 | <p>This problem is really bothering me for some time, I appreciate if you have some idea and insight.</p>
<blockquote>
<p>Prove that</p>
<p>$$2^{2^n}+5^{2^n}+7^{2^n}$$</p>
<p>is divisible by $39$ for all natural numbers $n$.</p>
</blockquote>
<p>There was a suggestion that this should be done by mathemati... | N. S. | 9,176 | <p>Here is an alternate approach. </p>
<p>As $3$ and $13$ are prime, it follows that for any $a$ not divisible by $3$ and $13$ we have
$$a^2 \equiv 1 \pmod{3}$$
$$a^{12} \equiv 1 \pmod{13} $$</p>
<p>From the first one you get immediately that $a^{2^n} \equiv 1 \pmod{3}$ for $a \in \{ 2,5,7 \}$.</p>
<p>The second one... |
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