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 |
|---|---|---|---|---|
69,542 | <p>My first question here would fall into the 'ask Johnson' category if there was one (no pressure Bill). I'm interested in constructing a uniformly convex Banach space with conditional structure without using interpolation. The constructions of Ferenczi and Maurey-Rosenthal both use interpolation. </p>
<p>Using ex... | Adi Tcaciuc | 7,872 | <p>I think James also showed that if $X$ does not contain almost isometric copies of $\ell_1^2$ (he called such a space uniformly non-square) then $X$ <strong>is</strong> superreflexive. This is no longer true for $n>2$, as James later constructed a non-reflexive, uniformly non-octahedral (no almost isometric copies... |
139,232 | <p>Let $O$ be an operad in $\mathtt{SETS}$. Assume that $O(0)$ is empty and $O(1)$ only consists of the identity. Assume for simplicity that $O$ is monochromatic, i.e. we have no labels on the in/outputs. Assume also for simplicity that the operad is plain, i.e. neither symmetric nor braided. So the operads in question... | James Griffin | 110 | <p>This needs some checking but for the free non-symmetric operad generated by a single operation in arity 2, the category (PRO) I think you get is the free monoidal category generated by a single object $X$ and a single morphism $\alpha$ from $X^2$ to $X$.</p>
<p>But the nerve of this category is a classifying space ... |
2,002,601 | <p>Given n+1 data pairs $(x_0,y_0)...(x_n,y_n)$ for j=0,1,2...,n we have
$p_j=\prod_{i\neq j}(x_j-x_i)$ and $\psi(x)=\prod_{i=0}^n(x-x_i)$.</p>
<p>I am having trouble determining what $\psi(x_j)$ is and what $\psi'(x_j)$ would be. </p>
<p>I feel like $\psi(x_j)= 0$ because it would contain the $x_j-x_j$ term... But ... | Svinto | 263,547 | <p><strong>Hint:</strong> Which function does $f_n$ converge pointwise to? Is it continuous?</p>
|
4,481,695 | <p>I tried substituting <span class="math-container">$x+3$</span> to see if I could simplify in any way, but couldn't think of anything. Also tried using <span class="math-container">$\ln$</span> and <span class="math-container">$\exp$</span>, but in the end just got to <span class="math-container">$\ln(0)$</span>. Can... | Átila Correia | 953,679 | <p><strong>HINT</strong></p>
<p>Are you acquainted to the derivative definition?</p>
<p><span class="math-container">\begin{align*}
\lim_{x\to-3}\frac{4^{\frac{x+3}{5}} - 1}{x + 3} = \lim_{x\to-3}\frac{4^{\frac{x+3}{5}} - 4^{\frac{-3 + 3}{5}}}{x - (-3)}
\end{align*}</span></p>
<p>Can you take it from here?</p>
|
2,480,528 | <blockquote>
<p>Find a formula for $\prod_{i=1}^{2n-1} \left(1-\frac{(-1)^i}{i}\right)$ then prove it. </p>
</blockquote>
<p>I assumed that $\prod_{i=1}^{2n-1} \left(1-\frac{(-1)^i}{i}\right)=\frac{2n}{2n-1}$ after doing a few cases from above then I tried to prove it with induction would this be a fair approach or ... | Robert Z | 299,698 | <p>Something went wrong in your evaluation.
After the substitution $z=a+ib$, you should have
$$(1+a)^2+b^2=|1+a+ib|^2=|1-b-ia|^2=(1-b)^2+(-a)^2$$
that is, after a few simplifications, $a=-b$.</p>
|
4,066,512 | <p>Find <span class="math-container">$\sum_{j=0}^{n}\sum_{i=j}^{n} {n \choose i}{i \choose j}$</span>.</p>
<p>I don't know how to double summations like this very well. Can someone expand this to show how the <span class="math-container">$i=j$</span> thing works?</p>
<p>I tried the following:
<span class="math-containe... | robjohn | 13,854 | <p><span class="math-container">$$
\begin{align}
\sum\limits_{i=j}^n\binom{n-j}{i-j}
&=\sum\limits_{i=0}^{n-j}\binom{n-j}{i}\tag1\\[3pt]
&=2^{n-j}\tag2
\end{align}
$$</span>
Explanation:<br />
<span class="math-container">$(1)$</span>: subsitute <span class="math-container">$i\mapsto i+j$</span><br />
<span cla... |
4,066,512 | <p>Find <span class="math-container">$\sum_{j=0}^{n}\sum_{i=j}^{n} {n \choose i}{i \choose j}$</span>.</p>
<p>I don't know how to double summations like this very well. Can someone expand this to show how the <span class="math-container">$i=j$</span> thing works?</p>
<p>I tried the following:
<span class="math-containe... | crankk | 202,579 | <p>Change the order of summation (overall there are only finitely many summands). For that note that</p>
<p><span class="math-container">\begin{align*}
M:&=\{ (i,j)\in\mathbb{N}^2~:~j=0,...,n~~i=j,...,n\}\\
&= \{(i,j)\in \mathbb{N}^2~:~i=0,...,n~~j=0,...,i\}.
\end{align*}</span></p>
<p>Therefore we can change ... |
3,382,241 | <p>I am trying to find the smallest <span class="math-container">$n \in \mathbb{N}\setminus \{ 0 \}$</span>, such that <span class="math-container">$n = 2 x^2 = 3y^3 = 5 z^5$</span>, for <span class="math-container">$x,y,z \in \mathbb{Z}$</span>. Is there a way to prove this by the Chinese Remainder Theorem?</p>
| Sam | 616,072 | <p>First of all the number of ways in which you can fill 5 identical boxes with 25 identical balls when none of them are empty will be <span class="math-container">$25-5+5-1\choose 5-1$</span> or <span class="math-container">${24\choose 4} = 10626$</span>.</p>
<p>If you want to use inclusion-exclusion principle, Total... |
2,800,015 | <p>Prove $p(x)=\frac{6}{(\pi x)^2}$ for $x=1,2,...$where $p$ is a probability function. and $E[X]$ doesn't exists.</p>
<p><b> My work </b></p>
<p>I know $\sum _{n=1}^\infty\frac{1}{n^2}=\frac{\pi^2}{6}$</p>
<p>Moreover,</p>
<p>$p(1)=\frac{6}{\pi^2}$<br>
$p(2)=\frac{6}{\pi^24}$<br>
$p(3)=\frac{6}{\pi^29}$<br>
$p(4)... | LeoCenturion | 565,209 | <p>$\mathbb{E[x]} = \sum_{i=1}^\infty x\frac{6}{\pi^2 x^2} = \frac{6}{\pi^2} \sum_{i=1}^\infty \frac{1}{x} $ which is the harmonic series and is divergent, thus the random variable X with probability function as described above doesn't have a finite Expected value </p>
|
306,588 | <p>I'll first explain what Mobius inversion says, and then state what I am fairly sure the equivariant version is. I can write out a proof, but I also can't believe this hasn't been done already; this is a request for references to where it has already been done.</p>
<p><b>Ordinary Mobius Inversion</b> Let $P$ be a fi... | Cihan | 21,848 | <p>This is more of a long comment. I am not sure I understand your construction, but the sort of alternating sum you take has a preimage in the Burnside ring, and is often called the "Lefschetz invariant" by finite group theorists. One of the Representation and Cohomology books by Benson has a section about this. Also... |
3,597,172 | <h2>The problem</h2>
<p>Let <span class="math-container">$f: \mathbb{R} \to \mathbb{R}$</span></p>
<p>Determine <span class="math-container">$f(x)$</span> knowing that </p>
<p><span class="math-container">$ 3f(x) + 2 = 2f(\left \lfloor{x}\right \rfloor) + 2f(\{x\}) + 5x $</span>, where <span class="math-container">$... | Ilmari Karonen | 9,602 | <p>Just in case, check the conventions your textbook is using, and specifically whether matrices conventionally act from left (on column vectors) or from right (on row vectors).</p>
<p>In particular, for your matrix <span class="math-container">$$A = \begin{bmatrix}1 & -2 & 3 & 0 \\ 0 & 0 & 0 &... |
271,255 | <p>I'm reading <em>Mathematica Programming</em> by Leonid Shifrin. And it said</p>
<blockquote>
<p><code>ClearAll</code> serves to clear all definitions (including attributes) for a given symbol (or symbols), and not to clear definitions of all global symbols in the system (it is a common mistake to mix these two thing... | MarcoB | 27,951 | <p>Leonid, I think, was cautioning against using <code>ClearAll[]</code> alone in hopes that this would clear all definitions and provide a "clean slate". The closest equivalent to that might be <code>ClearAll["Global`*"]</code>, although that still only clears user-generated definitions and not tho... |
812,939 | <p>How to find the sequences of $\sin n$ (n=natural number) sub-sequence limits? I know that it is a $[-1;1]$, but how to proof?</p>
<p><strong>Edit:</strong>
Is it true, that sin(n), with all natural numbers have different value? How to proof? If that is true, it is posible found bijection between N->[-1;1] ... | André Nicolas | 6,312 | <p><strong>Outline:</strong> Consider the points $P(n)=(\cos n,\sin n)$ as $n$ ranges over the positive integers. It is enough to show that this set of points is dense in the unit circle. </p>
<p>Since $\pi$ is irrational, we have $P(m)\ne P(n)$ if $m\ne n$. It follows by the Pigeonhole Principle that given any $\epsi... |
812,939 | <p>How to find the sequences of $\sin n$ (n=natural number) sub-sequence limits? I know that it is a $[-1;1]$, but how to proof?</p>
<p><strong>Edit:</strong>
Is it true, that sin(n), with all natural numbers have different value? How to proof? If that is true, it is posible found bijection between N->[-1;1] ... | Math.StackExchange | 86,086 | <p>It's easy to prove that $\forall n,m\in\mathbb{N}, \sin n \neq \sin m$ if $n \neq m $. </p>
<p>If $\sin{n}=\sin{m}$, then $n=m+2\pi k (k\in\mathbb{Z})$ by periodicity. So, $\pi=\dfrac{n-m}{2k}\in\mathbb{Q}$ But since the left side is clearly irrational number, this contradiction implies the above fact. </p>
|
2,270,861 | <p>What follows is part of Exercise 1.34 from Pillay's <em>Introduction to Stability Theory</em>. Suppose the following:</p>
<ol>
<li>$M \prec N$.</li>
<li>$N$ is $|M|^+$-saturated.</li>
<li>$p \in S_1(M)$, $q \in S_1(N)$.</li>
<li>$q \supset p$ is a coheir of $p$.</li>
</ol>
<p>Construct a sequence $(a_i \mid i &l... | Chappers | 221,811 | <p>You'll kick yourself: the integrand of $F'(y)+2yF(y)$ is the derivative with respect to $x$ of
$$ e^{-x^2}\sin{2xy}, $$
which vanishes at the endpoints. Hence the integral is zero.</p>
|
3,673,950 | <p>I realized that matrix transformation must be a linear transformation, but linear is not necessary matrix.
Can someone give me an example of a linear transformation that is not matrix transformation?</p>
| user786879 | 786,879 | <p>I'm going to assume that, by "matrix transformation", you mean a linear transformation of the form
<span class="math-container">$$T : \Bbb{R}^n \to \Bbb{R}^m : v \mapsto Av$$</span>
where <span class="math-container">$A$</span> is an <span class="math-container">$m \times n$</span> real matrix. We could also replace... |
3,673,950 | <p>I realized that matrix transformation must be a linear transformation, but linear is not necessary matrix.
Can someone give me an example of a linear transformation that is not matrix transformation?</p>
| Disintegrating By Parts | 112,478 | <p>Let <span class="math-container">$X=C[0,1]$</span> be the linear space of continuous complex functions on <span class="math-container">$[0,1]$</span>, and let <span class="math-container">$(Tf)(x)=xf(x)$</span>. <span class="math-container">$T$</span> is a linear transformation on <span class="math-container">$C[0,1... |
2,956,050 | <p>I think the title is pretty clear about the problem. Should I try to find the joint probability of <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> and decide if <span class="math-container">$\, f_{X,Y}(x,y) = f_X(x) \cdot f_Y(y)$</span>? If so, how am I going to find a joint distr... | Leucippus | 148,155 | <p>For <span class="math-container">$a+e^a\ln x = x+e^a\ln a = a+e^x\ln a$</span> there are some oddities. For instance take the first and third segments, ie <span class="math-container">$a + e^{a} \, \ln x = a + e^{x} \, \ln a$</span> then <span class="math-container">$x=a$</span>. For the second and third segments, <... |
2,956,050 | <p>I think the title is pretty clear about the problem. Should I try to find the joint probability of <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> and decide if <span class="math-container">$\, f_{X,Y}(x,y) = f_X(x) \cdot f_Y(y)$</span>? If so, how am I going to find a joint distr... | Claude Leibovici | 82,404 | <p><span class="math-container">$x=a$</span> is clearly a solution.</p>
<p>To find <span class="math-container">$a$</span>, Consider <span class="math-container">$$a+e^a\ln x = x+e^a\ln a $$</span> and solve for <span class="math-container">$x$</span>. The result is given in terms of Lambert function
<span class="math-... |
187,197 | <p>I have a logic expression:
<code>f0[a0_, a1_, a2_, a3_] := a0 And Not a1 And Not a2 And a3 Or Not a0 And a2 And a3 Or Not a0 And a1 And a3</code>, I know I should use <code>BooleanTable</code>, but it cannot generate a table like below.</p>
<p>How to generate a truth table in mathematica like below?</p>
<p><a hre... | Community | -1 | <p><code>f = a0 && ! a1 && ! a2 && a3</code></p>
<p><code>TableForm[BooleanTable[{a0, a1, a2, a3, f}, {a0, a1, a2, a3}],
TableHeadings -> {None, {a0, a1, a2, a3, f}}]</code></p>
<p>Apparently there are resources like <a href="http://mathworld.wolfram.com/TruthTable.html" rel="nofollow nor... |
2,013,115 | <p>I tried to do this problem in the following way:</p>
<p>As, $x^2+1 + \langle 3 , x^2+1 \rangle= 0 + \langle 3 , x^2+1 \rangle \implies x^2+1 \equiv 0 \implies x^2 \equiv -1.$</p>
<p>Also, $3+ \langle 3 , x^2+1 \rangle=0 +\langle 3 , x^2+1 \rangle \implies 3 \equiv 0$.</p>
<p>Now, any element of $\mathbb{Z}[x]/\l... | Stahl | 62,500 | <p>Looks right, except for one nit-picky thing: the $a$ and $b$ in your $ax + b$ aren't elements of $\Bbb Z/3\Bbb Z$, they're elements of $\{0,1,2\}\subseteq\Bbb Z$ (you could also take $a,b\in S$ for any fixed system $S$ of representatives of $\Bbb Z/3\Bbb Z$ in $\Bbb Z$).</p>
|
3,965,668 | <p>I started off my proof by of course stating that the different right triangles I would be comparing should have the same area A. I was able to show that what the question is asking is true visually and computationally using the Pythagorean Theorem, and even using the triangle inequality, but I don't really know how ... | mathcounterexamples.net | 187,663 | <p><strong>Hint</strong></p>
<p>The answer is positive and follows from:</p>
<ul>
<li>The fact that a branch of the logarithm is uniquely defined on an open disk <span class="math-container">$D(a,r)$</span> such that <span class="math-container">$0 \notin D(a,r)$</span> by its value <span class="math-container">$f(a)$<... |
4,539,637 | <blockquote>
<p>If the digits <span class="math-container">$7,7,3,2$</span>, and 1 are randomly arranged from left to right, what is the probability both of the 7 digits are to the left of the 1 digit?</p>
</blockquote>
<p>The answer is <span class="math-container">$1/3$</span> because <span class="math-container">$1 7... | user170231 | 170,231 | <p>I have not been able to simplify the last result, but I did manage to figure out the path from double sum to hypergeometric function.</p>
<p>It turns out that either alternative to the "third option" mentioned in OP are more useful. We have for instance</p>
<p><span class="math-container">$$\frac{\Gamma(m)... |
3,136,568 | <p><a href="https://i.stack.imgur.com/STONY.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/STONY.jpg" alt="enter image description here"></a></p>
<p>I seem to be struggling with this particular question. It is my understanding that in this situation, where du does not equal dx, that you must manipu... | user170231 | 170,231 | <p>With <span class="math-container">$u=5t$</span>, the differential is <span class="math-container">$\mathrm du=5\,\mathrm dt$</span>, or <span class="math-container">$\mathrm dt=\frac{\mathrm du}5$</span>. The "manipulation" you mention is really just a matter of keeping track of this factor of <span class="math-cont... |
2,820,779 | <p>So i have this integral </p>
<p>$$\int_{\sqrt[3]{4}}^{\sqrt[3]{3+e}}x^2 \ln(x^3-3)\,dx.$$</p>
<p>I was thinking of using u subsitution to make everything easier. </p>
<p>I made $u = x^3-3$ and $du = 3x^2dx$.</p>
<p>So I would then re-write my integral as </p>
<p>$$1/3\int_{\sqrt[3]{4}}^{\sqrt[3]{3+e}} \ln(x^3-3... | JohnKnoxV | 431,468 | <p>After your substitution, your integral should be
$ 1/3 \int_1^e \ln(u) du$.</p>
|
117,619 | <p>I need to evaluate the following real convergent improper integral using residue theory (vital that i use residue theory so other methods are not needed here)
I also need to use the following contour (specifically a keyhole contour to exclude the branch cut):</p>
<p><a href="https://i.stack.imgur.com/4wwwj.png" rel... | Batominovski | 72,152 | <p>Since a solution involving contour integration has been given, I am providing an alternative method without contour integration. Let $u:=\sqrt{x}$. Then, the integral $I:=\displaystyle\int_0^\infty\,\frac{\sqrt{x}}{x^3+1}\,\text{d}x$ equals
$$I=2\,\int_0^\infty\,\frac{u^2}{u^6+1}\,\text{d}u=\int_{-\infty}^{+\infty... |
2,491,394 | <p>So, my problem is with Axiom 5 of the proof, where Gödel considers necessary existence as a property. However, by his own definition, a 'property' applies to objects including those whose necessary existence has not even been proven, as can be inferred from Theorem 1. This, to me, seems like the perfect example of q... | Christopher Rose | 296,508 | <p>I havent read too far into it, but i think these authors embedded the higher order modal argument into higher order logic, and proved their embedding is consistent:
<a href="https://www.google.com/url?sa=t&source=web&rct=j&url=http://page.mi.fu-berlin.de/cbenzmueller/papers/C40.pdf&ved=0ahUKEwiciojM1... |
1,456,444 | <p>How can I go about solving this Pigeonhole Principle problem? </p>
<p>So I think the possible numbers would be: $[3+12], [4+11], [5+10], [6+9], [7+8]$</p>
<p>I am trying to put this in words...</p>
| Shailesh kumar | 31,530 | <p>You can divide your set into three group </p>
<ol>
<li>selected </li>
<li>Can't selected </li>
<li>total- {selected + can't selected}</li>
</ol>
<p>As your selected set increase , you can observe that your second group is also increasing and both are equal. In the end , your total group become zero. </p>
<p>So ,... |
849,093 | <p>After being introduced to the non-elementary function through an attempt to evaluate $\int x \tan (x)$, an interesting question occurred to me:
Can the non-elementary functions be decomposed to elementary ones? For instance, the logarithm, an elementary, can be decomposed into multiplication (e.g. $\ln x=y$ is the ... | David K | 139,123 | <p>It seems you mean a very general kind of "decomposition" in which you are allowed to rewrite the functions in terms of some procedure involving some sequence of "elementary" operations.</p>
<p>It seems to me that any "decomposition" in any reasonable sense would have to be capable of being described using a finite ... |
819,830 | <p>Is the idea of a proof by contradiction to prove that the desired conclusion is both true and false or can it be any derived statement that is true and false (not necessarily relating to the conclusion)? Or can it simply be an absurdity that you know is false but through your derivation comes out true?</p>
| DonAntonio | 31,254 | <p>You want the area between $\;y=\sqrt x\;,\;\;y=2\;$ and $\;x=0\;$ revolved around the $\;x$-axis, thus you get</p>
<p>$$\pi\int\limits_0^4\left(2^2-(\sqrt x)^2\right)dx=\pi\left(16-\frac124^2\right)=8\pi$$</p>
|
611,788 | <p>I'm here to ask you guys if my logic is correct.
I have to calculate limit of this:
$$\lim_{n\rightarrow\infty}\sqrt[n]{\sum_{k=1}^n (k^{999} + \frac{1}{\sqrt k})}$$
At first point. I see it's some limit of $$\lim_{n\rightarrow\infty}\sqrt[n]{1^{999} + \frac{1}{\sqrt 1} + 2^{999} + \frac{1}{\sqrt 2} \dots + n^{99... | Igor Rivin | 109,865 | <p>Yes. Take any orthonormal basis of $\mathbb{R}^4,$ call it $v_1, v_2, v_3, v_4$ Then the circles $\cos t v_1 + \sin t v_2$ will be disjoint from $\cos s v_3 + \sin s v_4.$</p>
|
73,424 | <p>I'm working on a PhD project that involves parameter estimation for diffusion processes. I'm based in a machine learning research group, and the emphasis here is strongly on "practical" research. </p>
<p>I've developed some theory, and now I'm starting to look for real-world problems to apply it to. To this end, I'... | ShawnD | 17,219 | <p>You might look at the introduction of Bernt Oskendal's <em>Stochastic Differential Equations</em>. He gives seven motivating problems (at least he does in the edition I have) for studying stochastic calculus. One big area that you have not mentioned is the applications of stochastic calculus to finance.</p>
|
73,424 | <p>I'm working on a PhD project that involves parameter estimation for diffusion processes. I'm based in a machine learning research group, and the emphasis here is strongly on "practical" research. </p>
<p>I've developed some theory, and now I'm starting to look for real-world problems to apply it to. To this end, I'... | Tim van Beek | 1,478 | <p>Please also have a look at the page about <a href="http://www.azimuthproject.org/azimuth/show/Stochastic+resonance" rel="nofollow">stochastic resonance</a> on Azimuth (there are links to review papers in the reference section of that page).</p>
<p>While stochastic resonance has been "invented" to explain the glacia... |
2,174,912 | <p>I' just so stumped right now. I want to get $x^{n}$ to equal $x^{2n+1}$. Right now I have that:
$$(\sqrt{x})^{2n} = x^n$$
But I don't know what to do to x to get:
$$x^n = \{\text{something done to $x$}\}^{2n+1}$$</p>
| Community | -1 | <p>Recall the following exponent law:</p>
<p>$$x^{ab} = (x^a)^b$$</p>
<p>In your case, you have this:</p>
<p>$$x^n = (x^a)^{2n+1}$$</p>
<p>Note that this is just a more mathematical way of stating exactly what you have in your question:</p>
<blockquote>
<p>But I don't know what to do to x to get:
$$x^n = \{\te... |
2,197,967 | <p>Can someone explain how is the RHS concluded? I did with sample numbers and it is all correct. but I can't figure out how C(12,6) comes to play.
$$
\binom{12}{0} + \binom{12}{1} + \binom{12}{2} + \binom{12}{3} + \binom{12}{4} + \binom{12}{5} = (2^{12} - \binom{12}{6}) / 2
$$</p>
| PSPACEhard | 140,280 | <p><strong>Hint:</strong> We have
$$
\sum_{i=0}^n \binom{n}{i} = 2^n
$$
and
$$
\binom{n}{i} = \binom{n}{n-i}
$$</p>
|
119,876 | <pre><code>Module[{x},
f@x_ = x;
p@x_ := x;
{x, x_, x_ -> x, x_ :> x}
]
?f
?p
</code></pre>
<p>gives</p>
<pre><code>{x$17312, x$17312_, x_ -> x, x_ :> x}
f[x_]=x
p[x_]:=x
</code></pre>
<p>but I'd like to get</p>
<pre><code>{x$17312, x$17312_, x$17312_ -> x$17312, x$17312_ :> x$17312}
f[x$17312... | Mr.Wizard | 121 | <p>This issue has been discussed before in</p>
<ul>
<li><a href="https://mathematica.stackexchange.com/q/72758/121">I define a variable as local to a module BUT then the module uses its global value! Why?</a> </li>
</ul>
<p>Regarding your motivation a solution of mine, which you linked to yourself, is shown in</p>
... |
402,802 | <p>I have read that $$y=\lvert\sin x\rvert+ \lvert\cos x\rvert $$ is periodic with fundamental period $\frac{\pi}{2}$.</p>
<p>But <a href="http://www.wolframalpha.com/input/?i=y%3D%7Csinx%7C%2B%7Ccosx%7C" rel="nofollow">Wolfram</a> says it is periodic with period $\pi$.</p>
<p>Please tell what is correct.</p>
| robjohn | 13,854 | <p><strong>Hint 1:</strong> $|\sin(x)|$ and $|\cos(x)|$ have period $\pi$</p>
<p><strong>Hint 2:</strong> $(|\sin(x)|+|\cos(x)|)^2=1+|2\sin(x)\cos(x)|=1+|\sin(2x)|$</p>
|
3,244,073 | <p>Let <span class="math-container">$A = \{1, 3, 5, 9, 11, 13\}$</span> and let <span class="math-container">$\odot$</span> define the binary operation of multiplication modulo <span class="math-container">$14$</span>.</p>
<p>Prove that <span class="math-container">$(A, \odot)$</span> is a group. </p>
<p>While comple... | dan_fulea | 550,003 | <p>The function to be integrated (and any related function that will ultimately solve the problem) hates and wants to avoid the points <span class="math-container">$0,1$</span>, so the pink contour would be a first guess to apply complex analysis (<em>C.A.</em> for short below):</p>
<p><a href="https://i.stack.imgur.c... |
1,908,844 | <p>The following example is taken from the book "Introduction to Probability Models" of Sheldon M. Ross (Chapter 5, example 5.4).</p>
<blockquote>
<p>The dollar amount of damage involved in an automobile accident is an
exponential random variable with mean 1000. Of this, the insurance
company only pays that amou... | epi163sqrt | 132,007 | <blockquote>
<p>This identity is known as <em><a href="https://en.wikipedia.org/wiki/Vandermonde%27s_identity" rel="nofollow">Vandermonde's identity</a></em>.</p>
</blockquote>
<p>In order to show the relationship with binomials $(1+x)^n$ it is convenient to introduce the <em>coefficient of</em> o... |
1,574,003 | <p>I know that if A and C are finite sets then |AxC|=|A||C|. This makes the problem quite simple but the sets may not be finite. </p>
<p>I am guessing that the concept of cardinally of infinite sets and ℵ <sub>0</sub> are part of the solution but those are concepts that my class did not go into much and I do not... | Clive Newstead | 19,542 | <p><strong>Hint:</strong> Let $f : A \to B$ and $g : C \to D$ be bijections. Find a bijection $A \times C \to B \times D$ in terms of $f$ and $g$.</p>
|
1,466,198 | <p>I was solving some mathematical questions and have come across the situation, where I need to divide 3900/139. Here is my question, </p>
<p>a. Can I assume 139 to 140 for the ease of division?</p>
<p>If so, how will I know what percentage of error I am introducing? How can I ensure that I am adding very less value... | fleablood | 280,126 | <p>N/139 = real answer</p>
<p>N/140 = your answer</p>
<p>your answer/real answer = (N/140)/(N/139) = 139/140. </p>
<p>Your answer will be 1/140 too small. </p>
<p>====</p>
<p>in general if you replace p with (p + n) your result will by factor of n/(p+n)</p>
<p>Replace 487 with 500 and your be off by a factor of 1... |
2,111,402 | <p>Simple exercise 6.2 in Hammack's Book of Proof. "Use proof by contradiction to prove"</p>
<p>"Suppose $n$ is an integer. If $n^2$ is odd, then $n$ is odd"</p>
<p>So my approach was:</p>
<p>Suppose instead, IF $n^2$ is odd THEN $n$ is even</p>
<p>Alternatively, then you have the contrapositive, IF $n$ is not even... | NeedForHelp | 392,893 | <p>To prove
$$
n^2\text{ is odd}\implies n\text{ is odd}\tag{1}
$$
by contradiction, you need to prove that
$$
n^2\text{ is odd}\wedge n\text{ is even}\tag{2}
$$
is false. That is, you need to suppose that $n^2$ is odd <strong>and</strong> that $n$ is even and obtain a contradiction from those two statements.</p>
<p>T... |
3,637,283 | <p>How would I find the fourth roots of <span class="math-container">$-81i$</span> in the complex numbers? </p>
<p>Here is what I currently have: </p>
<p><span class="math-container">$w = -81i$</span> </p>
<p><span class="math-container">$r = 9$</span> </p>
<p><span class="math-container">$\theta = \arctan (-81)$</... | vonbrand | 43,946 | <p>Use Euler's formula: If the complex number is <span class="math-container">$z = \rho e^{i \theta} = \rho (\cos \theta + i \sin \theta)$</span> (polar coordinates; <span class="math-container">$\rho, \theta$</span> are reals), then:</p>
<p><span class="math-container">$\begin{align*}
z^\alpha
&= \rho^\alph... |
3,408,082 | <blockquote>
<p><span class="math-container">$\textbf{Definition}$</span>: We say <span class="math-container">$f:\mathbb{R} \to \mathbb{R}$</span> is <em>intersecting</em> if for every nonempty <span class="math-container">$A \subset \mathbb{R}$</span>, <span class="math-container">$f[A] \cap A \neq \varnothing$</span... | antkam | 546,005 | <p>A bunch of partial results... (<strong>updated</strong> 10/25 10:35 EDT with a stronger result for finite case)</p>
<p>Define the <em>deviant set</em> as <span class="math-container">$D(f) = \{x \in \mathbb{R}: f(x) \neq x\}$</span>. Obviously, <em>deviation</em> <span class="math-container">$= | D(f)|$</span>.</p>... |
1,982,102 | <p>If I wanted to figure out for example, how many tutorial exercises I completed today.</p>
<p>And the first question I do is <strong>question $45$</strong>, </p>
<p>And the last question I do is <strong>question $55$</strong></p>
<p>If I do $55-45$ I get $10$.</p>
<p>But I have actually done $11$ questions:<br>
$... | Rob Arthan | 23,171 | <p>Suggestion: if you'd done questions $1$ to $N$, you'd have done $N$ questions. So if you start at question $44$ and finish at question $55$, subtract $43$ from both $44$ and $55$ to reduce to the easy case where the question numbers begin with $1$.</p>
|
66,199 | <p>Say I have the following lists of rules:</p>
<pre><code>case1 = {a -> 1, b -> 3, c -> 4, e -> 5}
case2 = {c -> 3, a -> 1, w -> 2}
case3 = {x -> 5, y -> 2, z -> 0, c -> 2}
</code></pre>
<p>How do I write a function <code>myfun[]</code>, to select the value of "c" in each case?</p>
... | Dr. belisarius | 193 | <pre><code>myfun[x_] := c /. x
myfun /@ {case1, case2, case3}
(* {4, 3, 2} *)
</code></pre>
<p>But please note that if you inadvertently assign a value to the symbol <code>c</code>, it goes astray and can't be repaired by tricks done only on <code>myfun[]</code> since it "corrupts" your cases lists.<br>
Considering th... |
66,199 | <p>Say I have the following lists of rules:</p>
<pre><code>case1 = {a -> 1, b -> 3, c -> 4, e -> 5}
case2 = {c -> 3, a -> 1, w -> 2}
case3 = {x -> 5, y -> 2, z -> 0, c -> 2}
</code></pre>
<p>How do I write a function <code>myfun[]</code>, to select the value of "c" in each case?</p>
... | Bob Hanlon | 9,362 | <p>An alternative to @belisarius' use of formal symbols to overcome the problem caused by c having been assigned a value</p>
<pre><code>case1 = {a -> 1, b -> 3, c -> 4, e -> 5};
case2 = {c -> 3, a -> 1, w -> 2};
case3 = {x -> 5, y -> 2, z -> 0, c -> 2};
myfun[x_] := Cases[x, (c -> ... |
1,439,004 | <p>I am trying to come up with a counting argument for: $\sum_{k=1}^{n}q^{k-1} = \frac{q^n-1}{q-1}$. I am trying to base it off of counting the left side as the sum of the (k-1) length words from an alphabet of size q for $k=1$ to $k=n-1$, but I can't seem to come up with a fitting argument to count the right side of ... | Graham Kemp | 135,106 | <p>The proof is not a combinatorial argument. It's an algebraic argument.</p>
<p>$$\require{cancel}\begin{align}
(q-1)\sum_{k=1}^n q^{n-1} & = (q-1)(q^{n-1}+q^{n-2}+\ldots + q^1+q^0)
\\[1ex] & = (q^n+\cancel{q^{n-1}+\ldots + q^2 + q^1}) -( \cancel{q^{n-1}+ \ldots+q^2+q^1}+q^0)
\\[2ex] & = q^n -1
\en... |
1,038,579 | <p>The question is from Joseph .J Rotman's book - Introduction to the Theory of Groups and it goes like this:
<br/> $A,B,C$ are subgroups of $G$, so $A\leq B$, prove that if $(AC=BC\ \text{and}\ A\cap C=B\cap C)$. (we do not assume that either AB or AC is a subgroup) than A=B.
<br/><br/>
I need you guys to tell me if ... | user 59363 | 192,084 | <p>One can also do shorter: take any $b\in B$. Since $B\subseteq BC=AC$ there exist $a\in A,c\in C$ such that $b=ac$. Then $a^{-1}b=c$; looking at the left hand side, this is in $B$ (since $A\subseteq B$), looking at the right hand side, this is in $C$, altogether it is in $B\cap C$ and hence in $A\cap C$. In particula... |
3,905,197 | <p>Stirling's Formula states that <span class="math-container">$\Gamma(z+1) \sim \sqrt{2 \pi z} (\frac{z}{\mathbb{e}})^{z}$</span> as <span class="math-container">$z \rightarrow \infty$</span>. I need to prove the following identity using Stirling's formula:</p>
<p><span class="math-container">$$ (2n)! \sim \frac{2^{2n... | robjohn | 13,854 | <p>This can be done via Integration by Parts.
<span class="math-container">$$
\begin{align}
&\int_0^\infty\frac1{x^{2n+3}}\left(\sin(x)-\sum_{k=0}^n\frac{(-1)^kx^{2k+1}}{(2k+1)!}\right)\mathrm{d}x\\
&=\frac{-1}{(2n+1)(2n+2)}\int_0^\infty\frac1{x^{2n+1}}\left(\sin(x)-\sum_{k=0}^{n-1}\frac{(-1)^kx^{2k+1}}{(2k+1)!... |
2,245,408 | <blockquote>
<p>How is the following result of a parabola with focus <span class="math-container">$F(0,0)$</span> and directrix <span class="math-container">$y=-p$</span>, for <span class="math-container">$p \gt 0$</span> reached? It is said to be <span class="math-container">$$r(\theta)=\frac{p}{1-\sin \theta} $$</spa... | Cye Waldman | 424,641 | <p>The equation for a parabola in the complex plane is</p>
<p>$$z=\frac{1}{2}p(u+i)^2\\
y=pu\\
x=\frac{1}{2}p(u^2-1)
$$</p>
<p>I think you would have to say</p>
<p>$$r=|z|\\
\theta=\arg(z)$$</p>
<p>to get the true polar form.</p>
<p>Ref: Zwikker, C. (1968). <em>The Advanced Geometry of Plane Curves and Their Appli... |
4,251,233 | <p>Find</p>
<p><span class="math-container">$\int\frac{x+1}{x^2+x+1}dx$</span></p>
<p><span class="math-container">$\int \frac{x+1dx}{x^2+x+1}=\int \frac{x+1}{(x+\frac{1}{2})^2+\frac{3}{4}}dx$</span></p>
<p>From here I don't know what to do.Write <span class="math-container">$(x+1)$</span> = <span class="math-container... | Z Ahmed | 671,540 | <p>Hint: Write <span class="math-container">$(x+1)=(2x+1)/2+1/2$</span></p>
|
4,251,233 | <p>Find</p>
<p><span class="math-container">$\int\frac{x+1}{x^2+x+1}dx$</span></p>
<p><span class="math-container">$\int \frac{x+1dx}{x^2+x+1}=\int \frac{x+1}{(x+\frac{1}{2})^2+\frac{3}{4}}dx$</span></p>
<p>From here I don't know what to do.Write <span class="math-container">$(x+1)$</span> = <span class="math-container... | Siong Thye Goh | 306,553 | <p><span class="math-container">\begin{align}
\int \frac{x+1}{x^2+x+1}\, dx &= \int \frac{x+\frac12 + \frac12}{x^2+x+1} \, dx \\
&= \int \frac{x+\frac12}{x^2+x+1} \, dx + \frac12 \int \frac1{x^2+x+1}\, dx \\
&=\frac12\ln |x^2+x+1| + \frac12 \int\frac1{(x+\frac12)^2 + \frac34} \, dx
\end{align}</span></p>
<p... |
3,410,150 | <p>If we try solving it by finding <span class="math-container">$f''(x)$</span> then it is very long and difficult to do, so my teacher suggested a way of doing it, he said find nature of all the roots of <span class="math-container">$f(x) =f'(x)$</span>, and on finding nature of the roots we got them to be real(but no... | N. S. | 9,176 | <p>Recall the Leibnitz differentiation formula</p>
<p><span class="math-container">$$(fg)^{(n)}=\sum_{k=0}^n f^{(k)} g^{(n-k)}$$</span></p>
<p>Then
<span class="math-container">$$\left( (x-a)^3(x-b)^3 \right)'=3(x-a)^2(x-b)^3+3(x-a)^3(x-b)^2=3(x-a)^2(x-b)^3[x-a+x-b]$$</span>
<span class="math-container">$$\left( (x-... |
262,425 | <p>I'm trying to integrate a function that involves a <em>finite</em> sum:</p>
<p><span class="math-container">$$\int_{-\infty}^{\infty}\sum_{j=1}^n (e^{-b t^2}r_j) \,dt$$</span></p>
<p>I think it should be possible to take the exponent <em>outside</em> the sum:</p>
<p><span class="math-container">$$\int_{-\infty}^{\in... | Michael E2 | 4,999 | <p>Using <code>linearExpand</code> from <a href="https://mathematica.stackexchange.com/questions/64422/how-to-do-algebra-on-unsolved-integrals/64447#64447">How to do algebra on unevaluated integrals?</a> :</p>
<pre><code>Clear[linearExpand];
linearExpand[e_, x_, head_] :=
e //. {op : head[arg_Plus, __] :> Distrib... |
1,005,186 | <p>the difference between a positive integer, n, and its cube is 4896. Compute n.
Please give solution and detailed explanation! Thank you ver much!
I tried and got 17, but what i did is to try numbers one by one, so i would really appreciate if anyone can tell me the right and systematic way to tackle this question??<... | Did | 6,179 | <blockquote>
<p>I don't know what limits to use.</p>
</blockquote>
<p>Note that $x=w/u$, $y=u$, $z=\sqrt{v}$ with $0\leqslant x,y,z\leqslant1$ hence the domain of integration is $$0\leqslant w/u,u,\sqrt{v}\leqslant1,$$ or, equivalently, $$0\leqslant w\leqslant u\leqslant1,\qquad0\leqslant v\leqslant1.$$</p>
<blockq... |
1,005,186 | <p>the difference between a positive integer, n, and its cube is 4896. Compute n.
Please give solution and detailed explanation! Thank you ver much!
I tried and got 17, but what i did is to try numbers one by one, so i would really appreciate if anyone can tell me the right and systematic way to tackle this question??<... | Vladimir Vargas | 187,578 | <p>Notice that:</p>
<p>$$f_{WVU}(w,v,u)=|\boldsymbol{J(h)}|f_{XYZ}(h(x,t,z))=|\boldsymbol{J(h)}|f_X\left(\dfrac{w}{u}\right)\chi_{[0,1]}(w)f_Y(u)\chi_{[w,1]}(u)f_Z(\sqrt{v})\chi_{[0,1]}(v).$$</p>
|
1,172,893 | <p>My textbook says I should solve the following integral by first making a substitution, and then using integration by parts:</p>
<p>$$\int cos\sqrt x \ dx$$</p>
<p>The problem is, after staring at it for a while I'm still not sure what substitution I should make, and hence I'm stuck at the first step. I thought abo... | abel | 9,252 | <p>make a subs $u = \sqrt x, x = u^2, dx = 2u du$ now the integral $\int \cos \sqrt x \, dx$ is transformed into $$2\int u \cos u \, du = 2 \int u d (\sin u) =2\left( u\sin u - \int \sin u \, du\right) = 2\left( u\sin u +\cos u +C\right)$$</p>
|
713,098 | <p>The answer to my question might be obvious to you, but I have difficulty with it. </p>
<p>Which equations are correct:</p>
<p>$\sqrt{9} = 3$</p>
<p>$\sqrt{9} = \pm3$</p>
<p>$\sqrt{x^2} = |x|$</p>
<p>$\sqrt{x^2} = \pm x$</p>
<p>I'm confused. When it's right to take an absolute value? When do we have only one va... | Klaas van Aarsen | 134,550 | <p>In the real numbers, $\sqrt x$ is <em>defined</em> to be positive.</p>
<p>In the complex numbers, $\sqrt z$ is a <em>multivalued function</em> that indeed yields 2 values. In that case we have a <em>principal value</em> of $\sqrt 9$ that is $3$.</p>
|
2,715,374 | <p>We know that \begin{equation*}
a_0+\cfrac{1}{a_1+\cfrac{1}{a_2+\cfrac{1}{a_3+\cfrac{1}{\ddots+\cfrac{1}{a_n}}}}}=[a_0,a_1, \cdots, a_n]
\end{equation*}</p>
<p>If $\frac{p_n}{q_n}=[a_0,a_1, \cdots, a_n]$.</p>
<blockquote>
<p>How to prove that $$
\begin{pmatrix}
p_n & p_{n-1} \\
q_n & q_{n-1} \... | user | 505,767 | <p>When we multiply for $(x-n)$ we need to set $x\neq n$ that is precisely the solution we obtain.</p>
|
367,204 | <p>I'm trying to prove that $\mathbb Z_p^*$ ($p$ prime) is a group using the Fermat's little theorem to show that every element is invertible.</p>
<p>Thus using the Fermat's little theorem, for each $a\in Z_p^*$, we have $a^{p-1}\equiv1$ (mod p). The problem is to prove that p-1 is the least positive integer which $a^... | egreg | 62,967 | <p>You can't show that $p-1$ is the least positive integer $r$ such that $a^r\equiv 1\pmod{p}$, because in general it isn't: for instance, the least integer for $a=1$ is $1$.</p>
<p>But all you need is to find an element which acts as an inverse:</p>
<p>$$a\cdot a^{p-2} \equiv 1 \pmod{p}$$</p>
<p>so that, for any $\... |
3,516,241 | <p>Consider the equation:</p>
<p><span class="math-container">$$ x ^ 4 - (2m - 1) x^ 2 + 4m -5 = 0 $$</span></p>
<p>with <span class="math-container">$m \in \mathbb{R}$</span>. I have to find the values of <span class="math-container">$m$</span> such that the given equation has all of its roots real.</p>
<p>This is ... | P. Lawrence | 545,558 | <p>Put m=2. Then u is not real, so x is not real. Instead, if <span class="math-container">$ m \ge 5/4 $</span> write out the two quadratic factors and use the condition for them(it's the same condition for each factor) to have real roots.After simpplification, you finally get <span class="math-container">$$ (2m-7)(2m-... |
3,451,301 | <p>The following classical generalization</p>
<blockquote>
<p><span class="math-container">$$\sum_{n=1}^\infty\frac{(-1)^{n}H_n}{n^{2a}}=-\left(a+\frac 12\right)\eta(2a+1)+\frac12\zeta(2a+1)+\sum_{j=1}^{a-1}\eta(2j)\zeta(2a+1-2j)$$</span>
where <span class="math-container">$\eta(a)=\sum_{n=1}^\infty\frac{(-1)^{n-1}}{n^... | Ali Shadhar | 432,085 | <p>In the question body in Eq <span class="math-container">$(3)$</span>, we reached</p>
<p><span class="math-container">$$\sum_{n=1}^\infty\frac{(-1)^nH_n}{n^{2a}}=-\left(a+\frac12\right)\eta(2n+1)+\frac1{2(2a-1)!}\int_0^\infty\frac{\ln^{2a-1}(x)\ln(1+x)}{x(1+x)}dx\tag{1}$$</span></p>
<hr />
<p>From following the same ... |
2,189,445 | <p>I try to solve this:
$$
\frac{\partial^{2} I}{\partial b \partial a} = I.
$$
I guessed $ I = C e^{a+b} $, but it's not the general solution. So, how to find the last one?</p>
| Peter Smith | 35,151 | <p>(A) The "Barber paradox" is not really a paradox, properly so called. </p>
<p>What we have here is a perfectly good proof by reductio that there can't exist someone in the village who shaves all and only those in the village who don't shave themselves. For suppose there is such a person, $B$. Then, by hypothesis, f... |
314,238 | <p>Let $R$ be a ring and $\mathfrak{m},\mathfrak{m'}$ two ideals of $R$.</p>
<p>Suppose that $\frac{R}{\mathfrak{m}}$ and $\frac{R}{\mathfrak{m'}}$ are isomorphic. Can i san say that $\mathfrak{m}$ and $\mathfrak{m'}$ are isomorphic too?</p>
| Zev Chonoles | 264 | <p>No; for example, let $R=\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/4\mathbb{Z}$, let $$\mathfrak{m}=\mathbb{Z}/2\mathbb{Z}\times2\mathbb{Z}/4\mathbb{Z}=\{(\overline{0},\overline{0}),(\overline{1},\overline{0}),(\overline{0},\overline{2}),(\overline{1},\overline{2})\}$$
and
$$\mathfrak{m}'=0\times\mathbb{Z}/4\mathbb{Z}=\... |
314,238 | <p>Let $R$ be a ring and $\mathfrak{m},\mathfrak{m'}$ two ideals of $R$.</p>
<p>Suppose that $\frac{R}{\mathfrak{m}}$ and $\frac{R}{\mathfrak{m'}}$ are isomorphic. Can i san say that $\mathfrak{m}$ and $\mathfrak{m'}$ are isomorphic too?</p>
| Math Gems | 75,092 | <p>Let $\rm\:R = \Bbb Q[x_1,x_2,x_3,\ldots].\:$ Then $\rm\: R\,\cong\, R/(x_1\!)\,\cong\, R/(x_1,x_2)\cong R/(x_1,x_2,x_3)\,\cong\, \cdots$</p>
|
3,121,361 | <p>Given <span class="math-container">$G$</span> has elements in the interval <span class="math-container">$(-c, c)$</span>. Group operation is defined as:
<span class="math-container">$$x\cdot y = \frac{x + y}{1 + \frac{xy}{c^2}}$$</span></p>
<p>How to prove closure property to prove that G is a group?</p>
| Peter Szilas | 408,605 | <p>Hint:</p>
<p>1)<span class="math-container">$\binom{n}{k}\frac{1}{n^k} \le \frac {1}{k!}, k \in
\mathbb{N}$</span>.</p>
<p>2)<span class="math-container">$(1+ \frac{1}{n})^n =$</span></p>
<p><span class="math-container">$\sum_{k=0}^{n} \binom{n}{k}(\frac{1}{n})^k \le \sum_{k=0}^{n}\frac{1}{k!}$</span></p>
<p>3)... |
413,719 | <p>Would I be correct in saying that they correspond to all points in $\mathbb{R}^3$? Or a line in $\mathbb{R}^3$?</p>
| Community | -1 | <p>On their own they're simply vectors in $\mathbb{R}^3$. However, if you were to take all linear combinations of these vectors</p>
<p>$$c_1(1,0,0) + c_2(0,1,0) + c_3(0,0,1) \text { where } c_1,c_2,c_3 \in \mathbb{R}$$</p>
<p>Then this would give you the entire space of $\mathbb{R}^3$. In more technical language, $\b... |
1,071,564 | <p>Let's a call a directed simple graph $G$ on $n$ labelled vertices <strong>good</strong> if every vertex has outdegree 1 and, when considered as if it were undirected, it is connected. How many good graphs of size $n$ are there?</p>
<p>Here's my work so far. Let's call this number $T(n)$. Clearly, $T(2) = 1$: there'... | Asinomás | 33,907 | <p>Let $f(n)$ be what you want. Denote a graph acceptable if it's connected components (when viewed as an undirected graph) Are all good.(This is the same as saying it is of regular out-degree 1)</p>
<p>Then the number of acceptable graphs is $(n-1)^n$.</p>
<p>But we can also count the number of acceptable graphs by ... |
1,071,564 | <p>Let's a call a directed simple graph $G$ on $n$ labelled vertices <strong>good</strong> if every vertex has outdegree 1 and, when considered as if it were undirected, it is connected. How many good graphs of size $n$ are there?</p>
<p>Here's my work so far. Let's call this number $T(n)$. Clearly, $T(2) = 1$: there'... | Marko Riedel | 44,883 | <p>I would like to contribute some ideas even though I don't have as much
time as I'd like at the moment. If I understand this problem correctly
then the class of graphs under consideration call it $\mathcal{Q}$ is
in a set-of relationship with the class of endofunctions call it
$\mathcal{E}$ with the latter bei... |
1,832,320 | <p>I know there are n linearly independent and n + 1 affinely independent vectors in $\mathbb{R}^n$. But how many convexly independent there are?</p>
<p>I think there are infinity number of them because if I have a convex polytope I can always add another point that is "outside" of said polytope. </p>
<p>But I'm not ... | Tsemo Aristide | 280,301 | <p>Hint: On finite dimensional spaces, two metrics are equivalent.</p>
|
2,611,656 | <p>Suppose $x = 1/t$. So now $x$ is a function of $t$, i.e., $x(t)$.</p>
<p>So $$\frac{dx(t)}{dt} = -t^{-2} \Rightarrow dx(t) = -t^{-2}dt$$</p>
<p>This problem is from the textbook: <code>advanced mathematical methods for scientists and engineers</code></p>
<p><a href="https://i.stack.imgur.com/SqKnQ.png" rel="nofol... | Peter Szilas | 408,605 | <p>Correct me if wrong :</p>
<p>$x=x(t)$, differentiable on an interval $I$, and $x'(t)\not=0,$</p>
<p>then the inverse function $t=t(x)$ exists, and is differentiable $x=x(t).$</p>
<p>Let $F(t(x))$ arbitrary, differentiable, then:</p>
<p>$\dfrac{d}{dx} F(t(x)) = \dfrac{d}{dt} F(t)×\dfrac{d}{dx}t(x),$ I.e.,</p>
<p... |
2,979,315 | <p>Let <span class="math-container">$X$</span> be a continuous random variable with uniform distribution between <span class="math-container">$0$</span> and <span class="math-container">$1$</span>. Compute the distribution of <span class="math-container">$Y = \sin(2\pi X)$</span>.</p>
<p><span class="math-container">$... | José Carlos Santos | 446,262 | <p>Let <span class="math-container">$R$</span> be the set of all reflections. Fix <span class="math-container">$r_0\in R$</span>. For each <span class="math-container">$r\in R$</span>, <span class="math-container">$r\circ{r_0}^{-1}$</span> is an isometry of the cube. But, since <span class="math-container">$r_0$</span>... |
2,261,500 | <p>I try to prove that statement using only Bachet-Bézout theorem (I know that it's not the best technique). So I get $k$ useful equations with $n_1$ then $(k-1)$ useful equations with $n_2$ ... then $1$ useful equation with $n_{k-1}$. I multiply all these equations to obtain $1$ for one side. For the other side I'm lo... | Thomas Andrews | 7,933 | <p>You are making this way too complicated.</p>
<p>There exists integers $x_1,x_2$ such that $n_1x_1+n_2x_2=1$. Letting $x_3=x_4=\cdots=x_k=0$. Then $\sum_{i=1}^{k} n_ix_i=1$ so the $x_i$ must be relatively prime.</p>
<p>Basically, $\gcd(x_1,x_2,\dots,x_k)\mid\gcd(x_1,x_2)$.</p>
|
1,637,748 | <p>I was given the following thing to prove:</p>
<p>$$\lim_{n \to \infty} {d(n) \over n} = 0$$
where $d(n)$ is the number of divisors of n.</p>
<p>I'm so sure how to approach this question. One way I thought of is to use the UFT to turn the expression to:</p>
<p>$$\lim_{n \to \infty} {\prod (x_i + 1) \over \prod p_i... | W-t-P | 181,098 | <p>It is actually true that $\lim_{n\to\infty} \frac{d(n)}{n^c}=0$ for any $c>0$, but if you only want it for $c=1$, the simplest proof would be the following. The divisors $\delta\mid n$ can be organized into pairs $(\delta,n/\delta)$, and in every pair the smallest divisor is $\min\{\delta,n/\delta\}\le\sqrt n$. I... |
114,371 | <p>A sector $P_1OP_2$ of an ellipse is given by angles $\theta_1$ and $\theta_2$. </p>
<p><img src="https://i.stack.imgur.com/mdmq2.png" alt="A sector of an ellipse"></p>
<p>Could you please explain me how to find the area of a sector of an ellipse?</p>
| Community | -1 | <p>Scale the entire figure along the $y$ direction by a factor of $a/b$. The ellipse becomes a circle of radius $a$, and the two angles become $\tan^{-1}(\frac ab\tan\theta_1)$ and $\tan^{-1}(\frac ab\tan \theta_2)$. The area of the original elliptical sector is $b/a$ times the area of the circular sector between these... |
440,615 | <p>Let $R$ be the region in the first quadrant bounded above by the circle $(x-1)^2 + y^2 = 1$ and below by the line $y = x$ . Sketch the region $R$ and evaluate the double integral $\iint 2y \;\mathrm dA$ . </p>
| André Nicolas | 6,312 | <p>We will do it using rectangular coordinates, though polar is tempting in any problem that involves circles.</p>
<p>The sketch is an essential part of the work. (I do not think I could do the calculation of the integral without a picture.)</p>
<p>We have a circle centre $(1,0)$ and radius $1$, and the familiar lin... |
898,495 | <p>A standard pack of 52 cards with 4 suits (each having 13 denominations) is well shuffled and dealt out to 4 players (N, S, E and W).</p>
<p>They each receive 13 cards.</p>
<p>If N and S have exactly 10 cards of a specified suit between them. </p>
<p>What is the probability that the 3 remaining cards of the suit a... | Philwy | 810,389 | <p><a href="https://i.stack.imgur.com/FlXWg.png" rel="nofollow noreferrer">Click to see my answer.</a></p>
<p>See the picture. Thank you.</p>
|
1,868,440 | <p>In a game , there are <code>N</code> numbers and <code>2</code> player(<code>A</code> and <code>B</code>) . If <code>A</code> and <code>B</code> pick a number and replace it with one of it's divisors other than itself alternatively, how would I conclude who would make the last move? (Notice that eventually when the ... | PM 2Ring | 207,316 | <p>As I mentioned in the comments, this is essentially a version of the ancient game known as Nim. This game has been thoroughly analysed and the <a href="https://en.wikipedia.org/wiki/Nim" rel="nofollow">Wikipedia article</a> gives a good description of the winning strategy for Nim, so that will not be repeated here. ... |
19,962 | <p><a href="http://en.wikipedia.org/wiki/Covariance_matrix" rel="nofollow">http://en.wikipedia.org/wiki/Covariance_matrix</a></p>
<pre><code>Cov(Xi,Xj) = E((Xi-Mi)(Xj-Mj))
</code></pre>
<p>Is the above equivalent to:</p>
<pre><code>(Xi-Mi)(Xj-Mj)
</code></pre>
<p>I don't understand why the expectancy of (Xi-Mi)(Xj-... | Henry | 6,460 | <p>There is a difference between a random variable and its expectation. Take a standard fair six sided die. It can take any of the values {1,2,3,4,5,6}, but its expectation is 3.5. </p>
<p>Similarly $(X_i - \mu_i)(X_j - \mu_j)$ is a random variable with several possible values while $\mathrm{E}[(X_i - \mu_i)(X_j - ... |
4,425,234 | <p>Let <span class="math-container">$f:[1,\infty)\rightarrow [1,\infty)$</span> be a function such that for every <span class="math-container">$x\in [1,\infty)$</span>, <span class="math-container">$f(f(x))=2x^{2}-3x+2$</span>. I am required to show that <span class="math-container">$f$</span> is bijective and also to ... | Ross Millikan | 1,827 | <p><span class="math-container">$f(t_n,y_n)$</span> is the (approximation of) the derivative of <span class="math-container">$y$</span> at <span class="math-container">$t_n$</span>. <span class="math-container">$h$</span> is the time step, so the change in <span class="math-container">$y$</span> over the time step is ... |
1,213,613 | <p>I am stuck trying to solve this equation. </p>
<p>B<sup>2</sup>(B + 11.97) = 238.67</p>
<p>This is for my math class, we solved this equation and got to that final form and I know that one solution is 3.88 but I don't know how to get it mathematically. I tried using Horner's Method but I don't know how to make it ... | Claude Leibovici | 82,404 | <p>You have $$B^2(B + 11.97) = 238.67$$ Rewrite is as $$B^2\Big(B+\frac{1197}{100}\Big)=\frac{23867}{100}$$ Now define $B=\frac{x}{100}$ so the equation becomes $$\frac{x^2}{10000}\Big(\frac{x}{100}+\frac{1197}{100}\Big)=\frac{23867}{100}$$ Multiple by $100$ inside the parentheses and the rhs to get $$\frac{x^2}{10000}... |
1,213,613 | <p>I am stuck trying to solve this equation. </p>
<p>B<sup>2</sup>(B + 11.97) = 238.67</p>
<p>This is for my math class, we solved this equation and got to that final form and I know that one solution is 3.88 but I don't know how to get it mathematically. I tried using Horner's Method but I don't know how to make it ... | Mark Bennet | 2,906 | <p>Treat it as $B^2(A+11.97)=238.67$ with $A\approx B$ to get a quick estimate, which can be improved with standard methods.</p>
<p>Start by approximating $B^2(A+12)=240$ with $A=0$ so that $B^2\approx 20$ and $B\approx 4$. So set $A=4$ and $B^2\approx 15$ so that $B\approx 3.7$</p>
<p>That's close with no calculator... |
2,414,472 | <blockquote>
<p>Let $(a_n)_{n\geq2}$ be a sequence defined as
$$
a_2=1,\qquad a_{n+1}=\frac{n^2-1}{n^2}a_n.
$$
Show that
$$
a_n=\frac{n}{2(n-1)},\quad\forall n\geq2
$$
and determine $\lim_{n\rightarrow+\infty}a_n$.</p>
</blockquote>
<p>I cannot show that $a_n$ is $\frac{1}{2}\frac{n}{n-1}$. Some helps? </p>
... | Franklin Pezzuti Dyer | 438,055 | <p>Try induction.</p>
<p>First of all, notice that if $n=2$, then
$$\frac{1}{2}\frac{n}{n-1}=1=a_1$$
Which proves that the explicit formula holds for $a_1$. Then suppose that for some $k$, the formula holds. Then
$$a_k=\frac{1}{2}\frac{k}{k-1}$$
and so, using the recursive definition,
$$a_{k+1}=\frac{k^2-1}{k^2}a_k$$
... |
2,569,267 | <p><a href="https://gowers.wordpress.com/2011/10/16/permutations/" rel="nofollow noreferrer">This</a> article claims:</p>
<blockquote>
<p>we simply replace the number 1 by 2, the number 2 by 4, and the number 4 by 1</p>
<p>....I start with the numbers arranged as follows: 1 2 3 4 5 6. After doing the permutation (124) ... | Dietrich Burde | 83,966 | <p>The text says "Let me illustrate this with $n=6$ and those two permutations. I start with the numbers arranged as follows: 1 2 3 4 5 6. After doing the permutation $(124)$ the numbers are arranged as 2 4 3 1 5 6." So indeed $1$ goes to $2$, and $2$ goes to $4$, and $4$ goes to $1$, if you read this top-down:
$$
\beg... |
2,878,448 | <p>What could be a possible approach to find the proof of:</p>
<blockquote>
<p>$\binom{2k+1}{k}$ is odd when $k=2^m-1$, otherwise $\binom{2k+1}{k}$ is even.</p>
</blockquote>
<p>I have seen some similar problems in <a href="https://math.stackexchange.com/questions/317163/prove-if-n-2k-1-then-binomni-is-odd-for-0-le... | P. Grabowski | 549,313 | <p>You could just use Legendre's formula or have some fun with the binomial expansion of $\left ( 1 + x \right)^{2k+1}$ modulo 2.</p>
<p>[<a href="https://en.wikipedia.org/wiki/Legendre%27s_formula" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Legendre%27s_formula</a> ][1]</p>
|
3,461,762 | <blockquote>
<p>Is it true that, for any Pythagorean triple <span class="math-container">$4ab > c^2$</span>?</p>
</blockquote>
<p>So this came up in a proof I was working on and it seems experimentally correct from what I've tried and I would imagine the proof is similar to proving,</p>
<p><span class="math-containe... | fleablood | 280,126 | <p>If <span class="math-container">$a^2 + b^2 = c^2$</span> then </p>
<p><span class="math-container">$c^2 < 4ab$</span> is the same thing as claiming <span class="math-container">$a^2 + b^2 < 4ab$</span></p>
<p>This is true if and only if <span class="math-container">$a^2 - 2ab + b^2 < 2ab$</span> which is ... |
1,206,460 | <p>This is the question :
Prove that the set of all the words in the English language is countble (the set's cardinality is אo)
A word is defined as a finite sequence of letters in the English language.</p>
<p>I'm not really sure how to start this. I know that a finite union of countble sets is countble and i think th... | peter.petrov | 116,591 | <p>The set $S_n$ of the English words with length $n$ is finite (this is almost obvious). So it's also countable. Why is it finite? The set $A_n$ of all sequences with length $n$ made up of latin characters is finite as it contains $26^n$ elements. Only some of these sequences are meaningful/actual English words. So $S... |
1,206,460 | <p>This is the question :
Prove that the set of all the words in the English language is countble (the set's cardinality is אo)
A word is defined as a finite sequence of letters in the English language.</p>
<p>I'm not really sure how to start this. I know that a finite union of countble sets is countble and i think th... | barak manos | 131,263 | <p>There are $26$ letters in the English language.</p>
<p>Consider each letter as one of the digits on base $27$:</p>
<ul>
<li>$A=1$</li>
<li>$B=2$</li>
<li>$C=3$</li>
<li>$\dots$</li>
<li>$Z=26$</li>
</ul>
<p>Then map each word to the corresponding integer on base $27$, for example:</p>
<p>$\text{BAGDAD}=217414_{2... |
1,448,476 | <p>Let's assume that we are given $f_{X}(x)=0.5e^{-|x|}$, with x being in the set of all real numbers and Y=$|X|^{1/3}$. If I'm asked to find the pdf of Y, do I just follow the formula and do the following?</p>
<p>$f_{Y}(Y)$=$f_{x}(g^{-1}(y))$|$g^{-1}$'(y) to get something like:
$0.5e^{-|y^{1/3}|} |y^{-2/3}/3|$</p>
... | Leo | 244,657 | <p>I let:</p>
<p>$$ y=f(x)=x^2\cos(1/x) $$</p>
<p>then let,</p>
<p>$$ u=x^2 $$ so $$ du/dx=2x $$ and $$ v=\cos(1/x) $$</p>
<p>so using the chain rule $$dv/dx=-\sin(1/x)*-1/x^2$$ $$dv/dx=1/x^2.\sin(1/x)$$</p>
<p>Then applying the product rule: $dy/dx=v*du/dx+u*dv/dx$,</p>
<p>$$f'(x)=2x\cos(1/x)+x^2.1/x^2\sin(1/x)=... |
3,328,822 | <blockquote>
<p>How do I evaluate <span class="math-container">$$\displaystyle\int^{\infty}_0 \exp\left[-\left(4x+\dfrac{9}{x}\right)\right] \sqrt{x}\;dx?$$</span> </p>
</blockquote>
<p>To my knowledge the following integral should be related to the Gamma function.</p>
<p>I have tried using the substitution <span c... | dan_fulea | 550,003 | <p>(I started an answer involving only "plain computations", but was not quick enough, maybe it is time now to complete and submit, rather then removed the typed formulas and quit the post. I am posting an alternative solution in the hope it looks simpler from some point of view, although there is a lot to be typed.)</... |
1,450,497 | <p>Consider the class of topological spaces $\langle X,\mathcal T\rangle$ such that the following are equivalent for $A\subseteq X$:</p>
<ul>
<li>$A$ is a $G_\delta$ set with respect to $\mathcal T$</li>
<li>$A\in\mathcal T$ or $X\smallsetminus A\in\mathcal T$</li>
</ul>
<p>Open sets, of course, are always $G_\delta$... | marty cohen | 13,079 | <p>A really trivial proof
for positive $x$ and $y$.</p>
<p>If $x|y$,
then
$y = ax$
where $a \ge 1$.</p>
<p>If $y|x$,
then
$x = by$
where $b \ge 1$.</p>
<p>Therefore
$x = by
=bax
$
so
$ba = 1$.
Since $a\ge 1$ and
$b \ge 1$,
we must have
$a = b = 1$
so $x = y
$.</p>
|
2,101,756 | <p>From the power series definition of the polylogarithm and from the integral representation of the Gamma function it is easy to show that:
\begin{equation}
Li_{s}(z) := \sum\limits_{k=1}^\infty k^{-s} z^k = \frac{z}{\Gamma(s)} \int\limits_0^\infty \frac{\theta^{s-1}}{e^\theta-z} d \theta
\end{equation}
The identity ... | Tushant Mittal | 272,305 | <p>Just take $\log_3 (y) = t $ and $\log_3 (x) = s $.
The equations become
$$ s + t =5$$
$$ st = 6 $$
Thus, s,t are the roots of the equation $a^2-5a+6 =0$ whose obvious roots are 2,3.</p>
<p>Therefore, $(y=8,x=27)$ or $(x=8,y=27)$ are the solutions. </p>
|
3,115,347 | <p>Let <span class="math-container">$f:(0,\infty) \to \mathbb R$</span> be a differentiable function and <span class="math-container">$F$</span> on of its primitives. Prove that if <span class="math-container">$f$</span> is bounded and <span class="math-container">$\lim_{x \to \infty}F(x)=0$</span>, then <span class="m... | GNUSupporter 8964民主女神 地下教會 | 290,189 | <p>Let <span class="math-container">$D_i$</span> be the outcome of the <span class="math-container">$i$</span>-th die. Denote <span class="math-container">$D = (D_1,\dots,D_4)$</span>.</p>
<blockquote>
<p>So the denominator/sample space should be 6464 right?</p>
</blockquote>
<ul>
<li>Sample space <span class="mat... |
3,941,106 | <p>Let <span class="math-container">$K\subseteq\mathbb R$</span> be compact and <span class="math-container">$h:K\to\mathbb R$</span> be continuous and <span class="math-container">$\varepsilon>0$</span>. By the Stone-Weierstrass theorem, there is a polynomial <span class="math-container">$p:K\to\mathbb R$</span> wi... | Kavi Rama Murthy | 142,385 | <p><span class="math-container">$\|p\|\leq \epsilon +\|h\|$</span>. Let <span class="math-container">$q=\frac {\|h\|} {\epsilon+\|h\|} p$</span>. The <span class="math-container">$q$</span> is a polynomial and <span class="math-container">$\|q\| \leq \|h\|$</span>. Now <span class="math-container">$\|q-h\| \leq \|q-p\|... |
4,004 | <p>This is related to <a href="https://math.stackexchange.com/q/133615/26306">this post</a>, please read the comments.</p>
<p>What is the usual way of dealing with that kind of problems on math.SE?
(By "that kind of problems" I mean someone posting tasks from an ongoing contest.)</p>
<p>I mean I did email the contest... | Michael Joyce | 17,673 | <p>In my opinion, one thing that should be done is to make an effort to change the culture from one of posting complete solutions to problems to one of posting hints and general strategies that explain the methodology behind the problem. This does not directly address contest cheating, as it is still possible to post ... |
4,004 | <p>This is related to <a href="https://math.stackexchange.com/q/133615/26306">this post</a>, please read the comments.</p>
<p>What is the usual way of dealing with that kind of problems on math.SE?
(By "that kind of problems" I mean someone posting tasks from an ongoing contest.)</p>
<p>I mean I did email the contest... | Community | -1 | <p>Sequence of same or almost same math questions (possibly from online math contest). I have flagged the moderators for attention. Unfortunately, I do not know what online math contest it is to email the coordinators.</p>
<p><a href="https://math.stackexchange.com/questions/222524/whats-the-probability-that-x-y-is-le... |
1,522,216 | <p>I want to show that following:
$$\left(\frac{n^2-1}{n^2}\right)^n\sqrt{\frac{n+1}{n-1}}\leq 1; ~~n\geq 2$$ and $n$ is an integer. </p>
<p>After some simplifications, I got left hand-side as
$$LHS:\left(1-\frac{1}{n}\right)^{n-\frac{1}{2}} \left(1+\frac{1}{n}\right)^{n+\frac{1}{2}}$$
It is clear that the 1st term is... | Archis Welankar | 275,884 | <p>Ths collinearity is X-Z-Y now XY is 10 and XZ is 3 so YZ has to be 7.So proxuct has to be 7. And Z-X-Y so 13 so product =13.7=91</p>
|
1,657,557 | <p>For example, how would I enter y^(IV) - 16y = 0? </p>
<p>typing out fourth derivative, and putting four ' marks does not seem to work. </p>
| Adriano | 76,987 | <p>Typing <a href="http://www.wolframalpha.com/input/?t=crmtb01&f=ob&i=y%27%27%27%27%20-%2016y%20%3D%200" rel="nofollow"><code>y'''' - 16y = 0</code></a> or <a href="http://www.wolframalpha.com/input/?t=crmtb01&i=(d%5E4%2Fdx%5E4%20y)%20-%2016y%20%3D%200" rel="nofollow"><code>(d^4/dx^4 y) - 16y = 0</code></a... |
3,354,566 | <p>I see integrals defined as anti-derivatives but for some reason I haven't come across the reverse. Both seem equally implied by the fundamental theorem of calculus.</p>
<p>This emerged as a sticking point in <a href="https://math.stackexchange.com/questions/3354502/are-integrals-thought-of-as-antiderivatives-to-avo... | AccidentalFourierTransform | 289,977 | <h2>Weak derivatives.</h2>
<p>This is essentially the way one defines a <a href="https://en.wikipedia.org/wiki/Weak_derivative" rel="noreferrer">weak derivative</a>. If a function is not differentiable in the traditional sense, but it is integrable, then one may define a weaker notion of derivative through duality: th... |
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