qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
2,993,551 | <p>I'm studying for a first year Discrete Mathematics course, I found this question on a previous paper and am lost on how to solve:</p>
<blockquote>
<p>Let <span class="math-container">$n$</span> be a fixed arbitrary integer, prove that there are infinitely
many integers <span class="math-container">$m$</span> s.... | Jimmy R. | 128,037 | <p><strong>Hint:</strong><span class="math-container">$$\det{\pmatrix{1&1&1\\1 & 2 & 3\\ 2& -1&1}}=5\neq0$$</span></p>
|
1,305,481 | <p>Let $C$ denote the circle $|z|=1$ oriented counterclockwise. Show that</p>
<p>i)$\int_Cz^ne^{\frac{1}{z}}dz=\frac{2\pi i}{(n+1)!}$ for $n=0,1,2$</p>
<p>ii)$\int_C e^{z+\frac{1}{z}}dz=2\pi i\sum_{n=0}^\infty\frac{1}{n!(n+1)!}$</p>
<p>I'm stuck in this exercise, because</p>
<p>$$\int_Cz^ne^{\frac{1}{z}}dz=2\pi i *... | kobe | 190,421 | <p>You made an error in the last line, where you wrote</p>
<p>$$z^n \sum_{n = 0}^\infty \frac{z^{-n}}{n!} = \sum_{n = 0}^\infty \frac{1}{n!}.$$</p>
<p>This equation does not even make sense since the summation index $n$ cannot appear outside the summation. What you should have is</p>
<p>$$z^n e^{1/z} = z^n \sum_{k =... |
1,428,905 | <p>I have two functions:</p>
<p>$n!$</p>
<p>$2^{n^{2}}$</p>
<p>What is the difference between the growth of these two? My thought is that $2^{n^2}$ grows much faster than $n!$. </p>
| Quang Hoang | 91,708 | <p>We know $2^{n}$ grows much faster than $n$, so
$$2^{n^2}>2^{1+2+\cdots+n}$$
grows much faster than
$$n!=1\cdot 2\cdots n.$$</p>
|
1,035,877 | <p>I'd really appreciate if someone could help me so I could get going on these problems, but this is confusing me... and it's been holding me up for the last couple hours. </p>
<p>How can I find the volume of the solid when revolving the region bounded by $y=1-\frac{1}{2}x$, $y=0$, and $x=0$ about the line $ x=-1$? H... | Idris Addou | 192,045 | <p>The shape is a triangle, vetrex are the points (0,0), (0,1) and (2,0). You can use tube methode: $V=\int_{0}^{2}2\pi x(1-\frac{1}{2}x)dx=\frac{4}{3}\pi $ units.</p>
|
2,406,587 | <p>Isn't the concept of homomorphism and isomorphism in abstract algebra analogous to functions and invertible functions in set theory respectively? That's one way to quickly grasp the concept into the mind?</p>
| mathreadler | 213,607 | <p>An eigenvector to a matrix is a vector that when multiplied with the matrix maps onto itself up to scaling with a scalar of the underlying field.</p>
<p>Eigen is the german word for own. It basically means something that maps back to itself.</p>
<p>What you need to learn to understand this language in the context ... |
10,949 | <p>Is it known whether every finite abelian group is isomorphic to the ideal class group of the ring of integers in some number field? If so, is it still true if we consider only imaginary quadratic fields?</p>
| ABC | 3,728 | <p>The smallest abelian group which is not the class group of an imaginary quadratic field is $(\mathbf{Z}/3 \mathbf{Z})^3$. There are six other groups of order
$< 100$ which do not occur in this way, of orders
$32$, $27$, $64$, $64$, $81$, and $81$ respectively.
The groups $(\mathbf{Z}/3 \mathbf{Z})^2$ and $(\mathb... |
1,673,771 | <p>I was wondering if this proof is valid. </p>
<p>I use $[x]$ to denote the floor of $x$.</p>
<p><strong>Problem</strong> </p>
<p>Prove that</p>
<p>$$[mx] = \sum_{k=0}^{m-1} \, \bigg[x+\frac{k}{m} \bigg]$$</p>
<p>where $m \in \mathbb{N}$ and $x \in \mathbb{R}$.</p>
<p><strong>Proof</strong></p>
<p>Let $m \in \m... | Brian M. Scott | 12,042 | <p>There are a couple of small bugs, but the argument is basically correct. When you partition $[0,1)$, you want either $\bigcup_{k=0}^{m-1}\left[\frac{k}m,\frac{k+1}m\right)$ or $\bigcup_{k=1}^m\left[\frac{k-1}m,\frac{k}m\right)$. In the next sentence you should not be considering all $m$ values of $p$: you already ha... |
2,147,571 | <p>Both $A$ and $B$ are a random number from the $\left [ 0;1 \right ]$ interval.</p>
<p>I don't know how to calculate it, so i've made an estimation with excel and 1 million test, and i've got $0.214633$. But i would need the exact number.</p>
| heropup | 118,193 | <p>This is <a href="http://www.math.hawaii.edu/~dale/putnam/1993.pdf" rel="nofollow noreferrer">Problem 1993-B3 from the 54th Putnam exam.</a></p>
<p>The solution becomes obvious if we look at a graph of $(B,A)$ in the Cartesian unit square $[0,1]^2$. Then the value $A/B$ is the slope of the line segment joining $(B,... |
3,020,988 | <p>Here's my attempt at an integral I found on this site.
<span class="math-container">$$\int_0^{2\pi}e^{\cos2x}\cos(\sin2x)\ \mathrm{d}x=2\pi$$</span>
<strong>I'm not asking for a proof, I just want to know where I messed up</strong></p>
<p>Recall that, for all <span class="math-container">$x$</span>,
<span class="ma... | Pseudo Professor | 559,658 | <p>If you want a full solution of the integral using complex analysis, going along the lines of what Seewoo Lee recommend, you can solve the integral as follows:</p>
<p>First consider the integral <span class="math-container">$$\int_{C} \frac{e^z}{z}dz$$</span></p>
<p>where <span class="math-container">$C$</span> is th... |
1,269,447 | <p>I consider the space $C^1[a, b]$ of (complex) functions that are at least once differentiable on $[a, b]$. I want to show that</p>
<p>$$||f||_{C^1} := ||f||_\infty + ||f'||_\infty$$</p>
<p>defines a norm on $C^1[a, b]$.</p>
<p>Now it's easy to see that $||f||_{C^1}$ is non-negative, and that it's zero iff f = 0, ... | Tito Piezas III | 4,781 | <p>(<em>More a comment</em>.) If we allow the <em>non-principal square root</em>,
$$\begin{cases}
2x^2+y^2=1,\\
x^2 + y \sqrt{1-x^2}=1\color{red}{\pm} (1-y)\sqrt{x}
\end{cases}$$
the $+$ case has one real solution, but the $-$ case has <strong><em>three</em></strong> real solutions: $(x,y)=(0,1)$ and two which surprisi... |
322,134 | <p>$$2e^{-x}+e^{5x}$$</p>
<p>Here is what I have tried: $$2e^{-x}+e^{5x}$$
$$\frac{2}{e^x}+e^{5x}$$
$$\left(\frac{2}{e^x}\right)'+(e^{5x})'$$</p>
<p>$$\left(\frac{2}{e^x}\right)' = \frac{-2e^x}{e^{2x}}$$
$$(e^{5x})'=5xe^{5x}$$</p>
<p>So the answer I got was $$\frac{-2e^x}{e^{2x}}+5xe^{5x}$$</p>
<p>I checked my answ... | amWhy | 9,003 | <p>Not correct: $(e^{5x})'\neq 5xe^{5x}$</p>
<p>$$(e^{5x})' = 5e^{5x}$$</p>
<p>$$(ae^{bx})' = abe^{bx}$$</p>
<p>It's because of the chain rule, and because $\frac d{dx}(e^x) = e^x$.</p>
|
3,218,525 | <p>Let <span class="math-container">$f:[0,1] \to [0, \infty)$</span> is a non-negative continuous function so that <span class="math-container">$f(0)=0$</span> and for all <span class="math-container">$x \in [0,1]$</span> we have <span class="math-container">$$f(x) \leq \int_{0}^{x} f(y)^2 dy$$</span><br>
Now consider ... | User8128 | 307,205 | <p>Clearly <span class="math-container">$A$</span> is non-empty since <span class="math-container">$f(0) = 0$</span> and thus by continuity there is <span class="math-container">$\delta > 0$</span> so that <span class="math-container">$f(y) \le 1/2$</span> for all <span class="math-container">$y \in [0,\delta]$</spa... |
142,993 | <p>I'm challenging myself to figure out the mathematical expression of the number of possible combinations for certain parameters, and frankly I have no idea how.</p>
<p>The rules are these:</p>
<p>Take numbers 1...n. Given m places, and with <em>no repeated digits</em>, how many combinations of those numbers can be ... | Will Orrick | 3,736 | <p>Let's first choose the three columns. There are four cases,</p>
<ol>
<li>all columns even</li>
<li>two columns even, one odd</li>
<li>one column even, two odd</li>
<li>all columns odd</li>
</ol>
<p>It either of cases (1) and (4), we have four rows we could choose for the lowest numbered column, three for the next... |
160,801 | <p>Here is a vector </p>
<p>$$\begin{pmatrix}i\\7i\\-2\end{pmatrix}$$</p>
<p>Here is a matrix</p>
<p>$$\begin{pmatrix}2& i&0\\-i&1&1\\0 &1&0\end{pmatrix}$$</p>
<p>Is there a simple way to determine whether the vector is an eigenvector of this matrix?</p>
<p>Here is some code for your conven... | Neil_P | 67,649 | <p>Carl Woll's answer seems to be broken in the newest Mathematica. Here is a slight modification that makes it work</p>
<pre><code>EigenvectorQ[matrix_, vector_] := MatrixRank[Join[matrix.vector, vector, 2]] == 1
</code></pre>
<p><code>MatrixRank</code> gives the number of linearly dependent columns.
<code>Join[l1, ... |
353,480 | <p>Is $f(x)=\ln(x)$ uniformly continuous on $(1,+\infty)$? If so, how to show it?</p>
<p>I know how to show that it is not uniformly continuous on $(0,1)$, by taking $x=\frac{1}{\exp(n)}$ and $y = \frac{1}{\exp(n+1)}$.</p>
<p>Also, on which interval does $\ln(x)$ satisfy the Lipschitz condition?</p>
| Community | -1 | <p><strong>HINT</strong> Every differentiable function that has bounded derivative on a set $X$ is uniformly continuous on $X$.</p>
|
2,128,380 | <p>Okay, I must admit that I am lost on how to do this. I have looked up videos and tutorials about this, and they helped a little. The main thing is that my professor asked for us to solve this without using the "determinant method." I have just started linear algebra, so I am still trying to understand determinants a... | Anna SdTC | 410,766 | <p>By defiition, the cross product of $A$ and $B$ is a vector $(u,v,w)\in\mathbb{R}^3$ that is perpendicular to both of them.</p>
<p>So, both doth products should be zero:
$$(A_x,A_y,A_z)\cdot(u,v,w)=A_xu+A_yv+A_zw=0,$$
$$(B_x,B_y,B_z)\cdot(u,v,w)=B_xu+B_yv+B_zw=0.$$</p>
<p>From here, you can find expressions of two ... |
365,986 | <p>If $A$ is an $n \times n$ matrix with $\DeclareMathOperator{\rank}{rank}$ $\rank(A) < n$, then I need to show that $\det(A) = 0$.</p>
<p>Now I understand why this is - if $\rank(A) < n$ then when converted to reduced row echelon form, there will be a row/column of zeroes, thus $\det(A) = 0$</p>
<p>However, I... | not all wrong | 37,268 | <p>$f(x+cy,y,z,\cdots) = f(x,y,z,\cdots) + cf(y,y,z,\cdots)=f(x,y,z,\cdots)$ using multilinearity and the alternating property respectively.</p>
<p>Hence you can add columns together without changing the determinant. But the rank is $<n$ if and only if some linear combination of the columns is trivial, in which cas... |
3,227,788 | <p>Let <span class="math-container">$f: D(0,1)\to \mathbb C$</span> be a holomorphic function. How to show that there exists a sequence <span class="math-container">$\{z_n\}$</span> in <span class="math-container">$D(0,1)$</span> such that <span class="math-container">$|z_n| \to 1$</span> and <span class="math-containe... | Conrad | 298,272 | <p>The way to do it is to first take out the (finitely many) zeros with a finite Blaschke product <span class="math-container">$B$</span> - which has absolute value <span class="math-container">$1$</span> on the circle- so get <span class="math-container">$g = \frac{f}{B}$</span> analytic, no zeros, <span class="math-c... |
1,564,729 | <p>Find:
$$
L = \lim_{x\to0}\frac{\sin\left(1-\frac{\sin(x)}{x}\right)}{x^2}
$$</p>
<p>My approach:</p>
<p>Because of the fact that the above limit is evaluated as $\frac{0}{0}$, we might want to try the De L' Hospital rule, but that would lead to a more complex limit which is also of the form $\frac{0}{0}$. </p>
<p... | Olivier Oloa | 118,798 | <p><em>In a slightly different way</em>, using the Taylor expansion, as $x \to 0$,
$$
\sin x=x-\frac{x^3}6+O(x^5)
$$ gives
$$
1-\frac{\sin x}x=\frac{x^2}6+O(x^4)
$$ then
$$
\sin \left( 1-\frac{\sin x}x\right)=\frac{x^2}6+O(x^4)
$$ and</p>
<blockquote>
<p>$$
\frac{\sin \left( 1-\frac{\sin x}x\right)}{x^2}=\frac16+O(x... |
148,185 | <p>Let $ X = \mathbb R^3 \setminus A$, where $A$ is a circle. I'd like to calculate $\pi_1(X)$, using van Kampen. I don't know how to approach this at all - I can't see an open/NDR pair $C,D$ such that $X = C \cup D$ and $C \cap D$ is path connected on which to use van Kampen. </p>
<p>Any help would be appreciated. Th... | Community | -1 | <p>You can add a point to get $S^3-A$, without changing the fundamental group (this follows from van Kampen's theorem). Now $S^3$ minus <em>any point</em> is homeomorphic to $\mathbb{R}^3$, so choose this <em>any point</em> to lie on $A$! This gives a new space, still with the same fundamental group, but now you've g... |
3,760,253 | <blockquote>
<p>The diagram shows the line <span class="math-container">$y=\frac{3x}{5\pi}$</span> and the curve <span class="math-container">$y=\sin$</span>
<span class="math-container">$x$</span> for <span class="math-container">$0\le x\le \pi$</span>.</p>
<p>Find (as an exact value) the enclosed area shown shaded in... | univer | 809,451 | <p>L<span class="math-container">$= 1−x+x^2−⋯$</span><br>
<span class="math-container">$=(1+x^2+x^4+⋯) − (x+x^3+x^5⋯)$</span><br>
<span class="math-container">$=(1+x^2+x^4+⋯) − x(1+x^2+x^4⋯)$</span><br>
<span class="math-container">$=(1+x^2+x^4+⋯)(1−x)$</span><br>
<span class="math-container">$=\frac{1}{(1-x^2)}(1-x)$<... |
2,604,093 | <p>I would like to study the convergence of the series:</p>
<p>$$\sum_{n=1}^\infty \frac{\log n}{n^2}$$</p>
<p>I could compare the generic element $\frac{\log n}{n^2}$ with $\frac{1}{n^2}$ and say that
$$\frac{1}{n^2}<\frac{\log n}{n^2}$$ and $\frac{1}{n^2}$ converges but nothing more about.</p>
| BenB | 336,000 | <p>Here is one way to show it directly. Note that $$\lim_{n \rightarrow \infty} \frac{\sqrt{n}}{\log(n)} = \infty$$
(if you need convinced, just apply L'Hospital's Rule).</p>
<p>Thus $\exists N \in \mathbb{N}, \text{ such that }\sqrt{n} > \log(n) \quad \forall n > N. \quad$ So we write:
$$
\begin{align*}
\sum_{... |
3,208,822 | <p>I'm looking at a STEP question and I'm a little confused by the logic of the method, and i'm really hoping someone could clarify what is going on for me. I have a good knowledge (At least I thought), as some STEP II and III questions are accessible but this one , I just can't wrap my head around - there must be a ga... | José Carlos Santos | 446,262 | <p>For each <span class="math-container">$x\in[0,1]$</span>, <span class="math-container">$\bigl\lfloor nf(x)\bigr\rfloor\leqslant nf(x)<\bigl\lfloor nf(x)\bigr\rfloor+1$</span> and therefore<span class="math-container">$$\frac{\bigl\lfloor nf(x)\bigr\rfloor}n\leqslant f(x)<\frac{\bigl\lfloor nf(x)\bigr\rfloor}n+... |
886,243 | <p>Evaluate</p>
<p><img src="https://latex.codecogs.com/gif.latex?%0A%24%245050%20%5Cfrac%20%7B%5Cleft(%20%5Csum%20_%7Br%3D0%7D%5E%7B100%7D%20%5Cfrac%20%7B%7B100%5Cchoose%20r%7D%7D%7B50r%2B1%7D%5Ccdot%20(-1)%5Er%5Cright)%20-%201%7D%7B%5Cleft(%20%5Csum%20_%7Br%3D0%7D%5E%7B101%7D%5Cfrac%7B%7B101%5Cchoose%20r%7D%7D%7B50r... | SuperAbound | 140,590 | <p>This expression can be simplified using <a href="http://en.wikipedia.org/wiki/Beta_function" rel="nofollow">Beta</a> and <a href="http://en.wikipedia.org/wiki/Gamma_function" rel="nofollow">Gamma</a> functions. <br/><br/>
The sum in the numerator is
\begin{align}
\sum^{100}_{r=0}\binom{100}{r}\frac{(-1)^r}{50r+1}
&a... |
552,474 | <p>If there are,
Are there unity <strong>(but not division)</strong> rings of this kind?
Are there non-unity rings of this kind?</p>
<p>Sorry, I forgot writting the non division condition.</p>
| anon | 11,763 | <p>Yes. In fact not only can we force every nonzero element to not be a zero divisor, we can force every nonzero element to be <em>invertible</em>. You get a division ring (also called skew field).</p>
<p>Perhaps one of the most famous historical examples: the quaternions. They are defined by</p>
<p>$$\Bbb H=\{a+bi+b... |
103,545 | <p>It is well-known that, given a normalized eigenform $f=\sum a_n q^n$, its coefficients $a_n$ generate a number field $K_f$. </p>
<p>In their 1995 <a href="http://www.math.mcgill.ca/darmon/pub/Articles/Expository/05.DDT/paper.pdf">paper</a> "Fermat's Last Theorem", Darmon, Diamond, and Taylor remark that, at the tim... | Rob Harron | 1,021 | <p>For (1), see Ribet's wonderful article <em>Galois representations attached to eigenforms with Nebentypus</em> (<a href="http://dx.doi.org/10.1007/BFb0063943" rel="nofollow">http://dx.doi.org/10.1007/BFb0063943</a>). It's proposition 3.2.</p>
|
3,482,376 | <blockquote>
<p>Suppose <span class="math-container">$f$</span> is differentiable on <span class="math-container">$[0,\infty)$</span> and <span class="math-container">$\displaystyle \lim_{x \to \infty} \frac{f(x)}{x} = 0$</span>. Show that <span class="math-container">$\displaystyle \liminf_{x \to \infty}|f'(x)| = 0$... | Franklin Pezzuti Dyer | 438,055 | <p>It seems easier to prove this using the contrapositive - here‘s a sketch. </p>
<p>Suppose that <span class="math-container">$\liminf |f‘(x)|=k > 0$</span>, so that the infimum is not equal to zero. It must then be the case that either <span class="math-container">$f’(x)>k$</span> or <span class="math-containe... |
3,482,376 | <blockquote>
<p>Suppose <span class="math-container">$f$</span> is differentiable on <span class="math-container">$[0,\infty)$</span> and <span class="math-container">$\displaystyle \lim_{x \to \infty} \frac{f(x)}{x} = 0$</span>. Show that <span class="math-container">$\displaystyle \liminf_{x \to \infty}|f'(x)| = 0$... | user284331 | 284,331 | <p>Let <span class="math-container">$\epsilon>0$</span> and choose some <span class="math-container">$M>0$</span> such that <span class="math-container">$\left|\dfrac{f(x)}{x}\right|<\epsilon$</span> for all <span class="math-container">$x\geq M$</span>.</p>
<p>Now we choose by Mean Value Theorem some <span c... |
1,630,733 | <p>I seem to be having a lot of difficulty with proofs and wondered if someone can walk me through this. The question out of my textbook states:</p>
<blockquote>
<p>Use a direct proof to show that if two integers have the same parity, then their sum is even.</p>
</blockquote>
<p>A very similar example from my notes... | Alex Provost | 59,556 | <p>Yes, you can use the definitions directly. If $a,b$ are even then like you say we have $a = 2m$ and $b = 2n$, so $a+b = 2m + 2n = 2(m+n)$, which is even.</p>
<p>Similarly, if $a,b$ are odd then we have $a = 2m + 1$ and $b = 2n + 1$, and so $a+b = (2m +1) + (2n + 1) = 2(m + n + 1)$, which is also even.</p>
|
1,473,418 | <p>I have difficulties with this question : </p>
<p>Given the ODE named (1) : $$x'=y+\sin (x^2y)$$ $$y'=x+\sin(xy^2)$$</p>
<p>and the :</p>
<p><strong>Definition.</strong> A <em>Petal</em> is a solution $(x(t),y(t))$ that verifies $\displaystyle \lim _{t \to \pm \infty} (x(t),y(t)) =(0,0)$.</p>
<p>How can I show th... | JMP | 210,189 | <p>Your approach is fine except for you are missing the repeated digit. Here is a lazy solution:</p>
<p>Imagine we only have $0,1,2,3,4,5,6$, then your sums become:</p>
<ul>
<li>$0: 6\times5\times4=120$</li>
<li>$5: 5\times5\times4=100$</li>
</ul>
<p>Therefore there are at least $220$ and at most $390$ solutions, th... |
3,678,033 | <p>let Y be an ordered set in the order topology.let X be a topological space and let <span class="math-container">$f,g:{X\to Y}$</span> be continuous function.</p>
<p>show that the set <span class="math-container">$A=\{x\in X\mid f(x)\le g(x)\}$</span> is closed in X.
I used the complement A and Hasdorf, but I didn't... | Didier | 788,724 | <p>If <span class="math-container">$f$</span> and <span class="math-container">$g$</span> are continuous, what can you say about <span class="math-container">$f-g : X \to Y$</span> ? Thus, can you express <span class="math-container">$A$</span> as the reverse image of <span class="math-container">$f-g$</span> of a suit... |
2,983,877 | <p>I once saw a function for generating successively more-precise square root approximations, <span class="math-container">$f(x) = \frac{1}{2} ({x + \frac{S}{x}})$</span> where S is the square for which we are trying to calculate <span class="math-container">$\sqrt S$</span>. And the function works really well, generat... | Ken Draco | 431,886 | <p><a href="https://math.stackexchange.com/questions/2225186/approximation-for-value-of-2x-without-using-calculator/2225416#2225416">Please see the link description at the very bottom</a></p>
<p>Let's first derive a formula for calculating square roots manually without calculus (and later for more general cases of var... |
2,078,796 | <p>In a contest problem book, I found a reference to Newton's little formula that may be used to find the <em>nth</em> term of a numeric sequence. Specifically, it is a formula that is based on the differences between consecutive terms that is computed at each level until the differences match. </p>
<p>An example appl... | Ahmed S. Attaalla | 229,023 | <p>Suppose that we have a sequence:</p>
<p>$$a_0,a_1,...a_k$$</p>
<p>And we want to find the function of $n$ that defines $a_n$.</p>
<p>To do this we start by letting $a_{n+1}-a_n=\Delta a_n$ and we call this operation on $a_n$ the forward difference. Then given $\Delta a_n$ we can find $a_n$. Sum both sides of the ... |
481,173 | <p>The most common way to find inverse matrix is $M^{-1}=\frac1{\det(M)}\mathrm{adj}(M)$. However it is very trouble to find when the matrix is large.</p>
<p>I found a very interesting way to get inverse matrix and I want to know why it can be done like this. For example if you want to find the inverse of $$M=\begin{b... | bubba | 31,744 | <p>The method you describe is called the Gauss-Jordan method. There is a nice description and an informal discussion of why it works <a href="http://www.mathsisfun.com/algebra/matrix-inverse-row-operations-gauss-jordan.html" rel="nofollow">here</a>.</p>
<p>It's much more efficient and stable than the method based on d... |
1,943,478 | <p>So the prompt is merely an existence proof--just find a $u$ and $v$ that work. Well, I'm unfortunately a little stuck on getting started.</p>
<p>I know that $Q \in SO_4(\mathbb R) \implies QQ^T = I \text{ and } \det(Q) = 1$.</p>
<p>I tried to solve $Qx = uxv$ for $u,v$ but I was not able to do so successfully. Thi... | Emilio Novati | 187,568 | <p>It is a consequence of the fact that any 4D rotation can be canonically decomposed into a left-isoclinic and a right-isoclinic rotation.
As you can see in the <a href="https://en.wikipedia.org/wiki/Rotations_in_4-dimensional_Euclidean_space#Isoclinic_decomposition" rel="nofollow">reference</a> this means that any $A... |
488,258 | <p>What are the last two digits of $11^{25}$ to be solved by binomial theorem like $(1+10)^{25}$?
If there is any other way to solve this it would help if that is shown too.</p>
| Gerry Myerson | 8,269 | <p>Here's another way. $11\times11=121$ ends in 21. $11\times21=231$ ends in 31. $11\times31=341$ ends in 41. See the pattern for the last two digits of powers of 11? Now prove that the pattern continues to hold, and then see what that tells you about $11^{25}$. </p>
|
935,331 | <p>Previously, to integrate functions like $x(x^2+1)^7$ I used integration by parts. Today we were introduced to a new formula in class: $$\int f'(x)f(x)^n dx = \frac{1}{n+1} {f(x)}^{n+1} +c$$
I was wondering how and why this works. Any help would be appreciated. </p>
| Community | -1 | <p>By the chain rule we have</p>
<p>$$(g\circ f)'=(g'\circ f)\times f'$$
Now what we get if we take $g(x)=x^n$?</p>
|
640,554 | <p>For the system
$$
\left\{
\begin{array}{rcrcrcr}
x &+ &3y &- &z &= &-4 \\
4x &- &y &+ &2z &= &3 \\
2x &- &y &- &3z &= &1
\end{array}
\right.
$$
what is the condition to determine if there is no solution or unique solution or infinite solut... | Ben | 116,271 | <p>The important concept here is linear dependence versus linear independence. As shown in the examples posted by others, linear dependence occurs when one equation in the system of equations can be shown to be a multiple of another. This is ultimately what Gaussian elimination or computing the determinant reveals. In ... |
129,890 | <p>The code</p>
<pre><code>x0 = 0.25; T = 20; u1 = -0.03; u2 = 0.07; u3 = -0.04;
a = 1/100; t0 = 5; omega = 2;
a = 0.01; dis[x_] := a/(Pi (x^2 + a^2))
P[t_] := If[t <= t0, Sin[omega t], 0]
u[t_] := u1 HeavisideTheta[t - 0.8] +
u2 HeavisideTheta[t - 1.64] + u3 HeavisideTheta[t - 3.33]
pde = a D[w[x, t],... | xzczd | 1,871 | <p><code>eerr</code> is a warning, not an error, it just suggests the possibility of trouble and doesn't always mean the output you obtained is wrong. Indeed, the solution given by <code>NDSolve</code> with the default setting seems to be erroneous, but according to my test, with a spatial grid <strong>dense enough</st... |
332,583 | <p>I'm a high school student, and I have to write a 4000-word research paper on mathematics (as part of the IB Diploma Programme). Among my potential topics were cellular automata and the Boolean satisfiability problem, but then I thought that maybe there was a connection between the two. Variables in Boolean expressio... | Robert Israel | 8,508 | <p>If you could find an efficient way to solve SAT, you'd become very rich and famous. That's not likely to happen when you're still in high school. What you might be able to do, though, is get your cellular automaton to go through all possible values of the variables, and check the value of the Boolean expression fo... |
332,583 | <p>I'm a high school student, and I have to write a 4000-word research paper on mathematics (as part of the IB Diploma Programme). Among my potential topics were cellular automata and the Boolean satisfiability problem, but then I thought that maybe there was a connection between the two. Variables in Boolean expressio... | Sonia | 8,415 | <p>The simple answer is Yes. Can you do it in a simple way? probably not. Cook showed in 2004 that you can indeed compute anything that a turing-machine could on elemetry celular automata using Rule 110 and careful selection of start conditions. There is probably no simple rule or start conditions to do this though. </... |
119,506 | <p>Let $\kappa$ be a singular cardinal, and let $\langle \kappa_i \mid i<\mathrm{cf}(\kappa) \rangle$ be an increasing sequence of regular cardinals cofinal in $\kappa$. Recall that a scale on $\Pi_{i<\mathrm{cf}(\kappa)} \kappa_i$ is a sequence $\langle f_\alpha \mid \alpha < \kappa^+ \rangle$ such that:</p>
... | Chris Lambie-Hanson | 26,002 | <p>I have a negative answer assuming some mild cardinal arithmetic assumptions. Namely, if $(\kappa_i)^i < \kappa$ for every $i<\mathrm{cf}(\kappa)$, then there can be no scale with the desired property. This is true, for example, whenever $\mathrm{cf}(\kappa) = \omega$ or $\kappa$ is strong limit. We also make t... |
987,620 | <p>$P$ and $Q$ are two distinct prime numbers. How can I prove that $\sqrt{PQ}$ is an irrational number?</p>
| Yiorgos S. Smyrlis | 57,021 | <p>If
<span class="math-container">$$
\sqrt{pq}=\frac{m}{n}, \quad (m,n)=1,
$$</span>
then
<span class="math-container">$$
n^2pq=m^2, \tag{$\star$}
$$</span>
which means that <span class="math-container">$p\mid m^2$</span> and hence <span class="math-container">$p\mid m$</span>. Thus <span class="math-container">$m=pm_... |
3,856,370 | <p>this is the result in the book(Discrete mathematics and its applications) I was reading.</p>
<ol>
<li><span class="math-container">$n^d\in O(b^n)$</span></li>
</ol>
<p>where <span class="math-container">$b>1$</span> and <span class="math-container">$d$</span> is positive</p>
<p>and</p>
<ol start="2">
<li><span cl... | J.G. | 56,861 | <p>By a <a href="https://en.wikipedia.org/wiki/Schwinger_parametrization" rel="nofollow noreferrer">Schwinger parametrization</a>, this integral is<span class="math-container">$$\begin{align}\frac{1}{4i}\int_0^\infty dx\int_0^\infty dy(e^{ix}-e^{-ix}-2ix)y^2e^{-xy}&=\frac{1}{4i}\int_0^\infty dy\left(\frac{1}{y-i}-\... |
1,422,990 | <p>How to show that $(2^n-1)^{1/n}$ is irrational for all integer $n\ge 2$?</p>
<p>If $(2^n-1)^{1/n}=q\in\Bbb Q$ then $q^n=2^n-1$ which doesn't seem right, but I don't get how to prove it.</p>
| Hagen von Eitzen | 39,174 | <p>If the $n$th root of an integer is rational, then it is in fact an integer (any prime occuring in the dneominator of $\frac xy$ occurs also in the denominator of $\frac{x^n}{y^n}$). As $1^n<2^n-1<2^n$, this is not possible.</p>
|
1,892 | <p>Although whether $$ P = NP $$ is important from theoretical computer science point of view, but I fail to see any practical implication of it.</p>
<p>Suppose that we can prove all questions that can be verified in polynomial time have polynomial time solutions, it won't help us in finding the actual solutions. Conv... | Community | -1 | <p>There is an interesting heuristic to suggest that P is actually not NP. It is that, roughly, the task of finding out a proof of a statement is an NP task, but that of verifying it is a P task. From our actual experience that verifying a proof is far easier than finding one, we can intuitively expect P != NP to hold ... |
2,995,408 | <blockquote>
<p><span class="math-container">$$
\lim_{x\to 2^-}\frac{x(x-2)}{|(x+1)(x-2)|}=
\lim_{x\to 2^-}\left(\frac{x}{|x+1|}\cdot \frac{x-2}{|x-2|}\right)
$$</span></p>
</blockquote>
<p>So as the title says, is it okay to separate function under absolute value like this (i.e In form of Products) as shown in the... | Yuval Gat | 450,141 | <p>Yes, <span class="math-container">$|ab|=|a||b|$</span> holds for all <span class="math-container">$a, b\in\mathbb{R}$</span>.</p>
|
2,838,312 | <p>What is the Fourier transform of $\mathrm{e}^{ik|x|}$? Here, $k > 0$ is real.</p>
<p>I use the definition $$ F(\omega) = \int_{-\infty}^\infty \mathrm{e}^{-i\omega x} f(x) \mathrm{d}x.$$</p>
<p>Thanks!</p>
| mathworker21 | 366,088 | <p>If $|z| = r$, then, using that $|a_i| \le r-1$ for each $i$, $$|a_{n-1}z^{n-1}+\dots+a_1z+a_0| \le (r-1)r^{n-1}+(r-1)r^{n-2}+\dots+(r-1)r+(r-1) = r^n-1 < |z^n|$$ So by Rouché, we see that the number of zeroes of $p$ in $\Delta_r(0)$ is the same as the number of zeroes as $z^n$ in $\Delta_r(0)$, namely $n$.</p>
|
2,178,714 | <p>Let $ f: (-1,1) \rightarrow \mathbb{R}$ be a bounded and continuous function . Prove that the function $ g(x)=(x^{2}-1)f(x) $ is uniformly continuous on $ (-1,1)$ . $$ $$ My little approach is, Since $f$ is bounded on $(-1,1)$ , there is positive $M \in \mathbb{R}$ such that </p>
<p>$$\forall x \in (-1,1)\,|f(x)|... | edm | 356,114 | <p>Define a new function on a larger domain $G:[-1,1]\to \Bbb R$ by $$G(x):=\begin{cases}
g(x) &\text{if $x\in(-1,1)$}\\
0 &\text{if $x=1$ or $-1$}
\end{cases}.$$
You can check that $G$ is continuous (specifically, you just need to check this at $-1$ and $1$). This follows from boundedness of $f$ and you apply ... |
3,400,123 | <blockquote>
<p>Proof that <span class="math-container">$$\sum_{l=1}^{\infty} \frac{\sin((2l-1)x)}{2l-1}
=\frac{\pi}{4}$$</span> when <span class="math-container">$0<x<\pi$</span></p>
</blockquote>
<p>The chapter we are working on is about Fourier series, so I guess I'd need to use that some how.</p>
<p>My id... | Jack D'Aurizio | 44,121 | <p>For short:
<span class="math-container">$$\sum_{k\geq 0}\frac{\sin((2k+1)x)}{2k+1}=\text{Im}\!\sum_{k\geq 0}\frac{(e^{ix})^{2k+1}}{2k+1}=\text{Im}\,\text{arctanh}(e^{ix})=\frac{1}{2}\text{Im}\log\left(\frac{1+e^{ix}}{1-e^{ix}}\right)=\frac{1}{2}\text{Arg}\left(i\cot\frac{x}{2}\right)=\frac{\pi}{4}. $$</span></p>
<p... |
615,093 | <p>How to prove the following sequence converges to $0.5$ ?
$$a_n=\int_0^1{nx^{n-1}\over 1+x}dx$$
What I have tried:
I calculated the integral $$a_n=1-n\left(-1\right)^n\left[\ln2-\sum_{i=1}^n {\left(-1\right)^{i+1}\over i}\right]$$
I also noticed ${1\over2}<a_n<1$ $\forall n \in \mathbb{N}$.</p>
<p>Then I wrote... | Felix Marin | 85,343 | <p>$\newcommand{\+}{^{\dagger}}%
\newcommand{\angles}[1]{\left\langle #1 \right\rangle}%
\newcommand{\braces}[1]{\left\lbrace #1 \right\rbrace}%
\newcommand{\bracks}[1]{\left\lbrack #1 \right\rbrack}%
\newcommand{\ceil}[1]{\,\left\lceil #1 \right\rceil\,}%
\newcommand{\dd}{{\rm d}}%
\newcommand{\ds}[1]{\displayst... |
615,093 | <p>How to prove the following sequence converges to $0.5$ ?
$$a_n=\int_0^1{nx^{n-1}\over 1+x}dx$$
What I have tried:
I calculated the integral $$a_n=1-n\left(-1\right)^n\left[\ln2-\sum_{i=1}^n {\left(-1\right)^{i+1}\over i}\right]$$
I also noticed ${1\over2}<a_n<1$ $\forall n \in \mathbb{N}$.</p>
<p>Then I wrote... | Claude Leibovici | 82,404 | <p>There is another way of adressing the problem (but it could be totally out of scope depending of what you are supposed to use for this). </p>
<p>It can be established that<br>
a(n) = (n / 2) (-PolyGamma[0, n/2] + PolyGamma[0, (1 + n)/2])<br>
Now, a Taylor development of a(n) built around infinity gives an approxim... |
308,565 | <p>Suppose I have a one parameter flat family of complex surfaces (regular, of general type) whose general fibre is smooth. Is it possible for the central fibre to have singularities which are not canonical? If so, how bad can they be? </p>
| Miguel González | 11,528 | <p>The central fiber can even be everywhere non-reduced. This can happen when you take the central fiber of the global image of a global map whose general fiber map is the canonical embedding of a surface of general type with very ample canonical map but its central fiber map is the canonical map of a surface of gener... |
80,432 | <p>I have a question that I've been wondering about the past day or so while trying to relate measure theory back to some general topology.</p>
<p>Is it ever possible for some family of (open) sets in $\mathbb{R}^2$ to be both a base for the usual topology on $\mathbb{R}^2$, as well as a semiring?</p>
<p>To avoid con... | Henno Brandsma | 4,280 | <p>In a non-trivial $T_1$ connected, locally connected space this can never happen, I think: as soon as we can find non-empty proper connected open subsets $U \subset V$ in a base/semiring, where the inclusion is proper, then $V \setminus U$ cannot be open (so in particular cannot be a disjoint union of base elements) ... |
80,432 | <p>I have a question that I've been wondering about the past day or so while trying to relate measure theory back to some general topology.</p>
<p>Is it ever possible for some family of (open) sets in $\mathbb{R}^2$ to be both a base for the usual topology on $\mathbb{R}^2$, as well as a semiring?</p>
<p>To avoid con... | Arturo Magidin | 742 | <p>I believe here's a proof for $\mathbb{R}^2$, but I'm not positive how general this can be; as Henno Brandsma mentions, connectivity seems to be key.</p>
<p>Assume you have a base for $\mathbb{R}^2$ that is closed under finite intersections. Let $A$ be a nonempty basic open set, and let $B$ be a connected component ... |
153,426 | <p>Let $r=a/b$ be a rational number in lowest terms, larger than $1$,
and not an integer (so $b > 1$).</p>
<blockquote>
<p><strong>Q</strong>. Does the sequence
$$ \lfloor r \rfloor, \lfloor r^2 \rfloor, \lfloor r^3 \rfloor,
\ldots, \lfloor r^n \rfloor, \ldots$$
always contain an infinite number of primes?<... | Gerry Myerson | 3,684 | <p>The question is much too hard. Forman and Shapiro proved that $[r^n]$ is <em>composite</em> infinitely often for $r=3/2$ and for $r=4/3$. Dubickas and his students have found a few more results along these lines. </p>
<p>EDIT: Here is a <a href="http://vddb.laba.lt/fedora/get/LT-eLABa-0001%3aE.02~2012~D_20121017_11... |
153,426 | <p>Let $r=a/b$ be a rational number in lowest terms, larger than $1$,
and not an integer (so $b > 1$).</p>
<blockquote>
<p><strong>Q</strong>. Does the sequence
$$ \lfloor r \rfloor, \lfloor r^2 \rfloor, \lfloor r^3 \rfloor,
\ldots, \lfloor r^n \rfloor, \ldots$$
always contain an infinite number of primes?<... | Aaron Meyerowitz | 8,008 | <p>I recently asked a <a href="https://mathoverflow.net/questions/153721">related question</a> (inspired by this one!) . I accepted an answer but also provided <a href="https://mathoverflow.net/questions/153721/floors-of-powers-of-reals-how-much-do-the-first-few-determine-the-next/153923#153923">an answer</a> of my own... |
1,856,530 | <blockquote>
<p>Prove that the product of five consecutive positive integers cannot be the square of an integer.</p>
</blockquote>
<p>I don't understand the book's argument below for why $24r-1$ and $24r+5$ can't be one of the five consecutive numbers. Are they saying that since $24-1$ and $24+5$ aren't perfect squa... | TonyK | 1,508 | <p>$24r-1$ and $24r+5$ are also divisible neither by $2$ nor by $3$. So they must also be coprime to the remaining four numbers, and thus must be squares.</p>
<p>But this is impossible, because we already know that $24r+1$ is a square, and two non-zero squares can't differ by $2$ or $4$.</p>
<p>For the second part: $... |
3,523,205 | <p>The given series of function is as follow</p>
<blockquote>
<p><span class="math-container">$$\sum_{n=1}^\infty x^{n-1}(1-x)^{2}$$</span>
prove that given series is uniformaly convergent on <span class="math-container">$[0,1]$</span></p>
</blockquote>
<p><strong>The solution i tried</strong>-The given series fo... | astro | 587,159 | <p>The answer above is correct. Other approach would be to show that, since <span class="math-container">$f$</span> is continuous and <span class="math-container">$f(1)=0$</span> then given <span class="math-container">$\varepsilon >0$</span> there exists <span class="math-container">$\delta >0$</span> such that ... |
732,121 | <p>I'm having trouble seeing why the bounds of integration used to calculate the marginal density of $X$ aren't $0 < y < \infty$.</p>
<p>Here's the problem:</p>
<p>$f(x,y) = \frac{1}{8}(y^2 + x^2)e^{-y}$ where $-y \leq x \leq y$, $0 < y < \infty$ </p>
<p>Find the marginal densities of $X$ and $Y$.</p>
... | Brian Tung | 224,454 | <p>The constraints are $-y \leq x \leq y$ and $0 < y < \infty$. That first constraint is equivalent to</p>
<p>$$
|x| \leq y
$$</p>
<p>which, when combined with the second constraint, naturally yields</p>
<p>$$
|x| \leq y < \infty
$$</p>
<p>Whence the integration limits.</p>
|
1,480,720 | <p>How many times do you have to flip a coin such that the probability of getting $2$ heads in a row is at least $1/2$?</p>
<p>I tried using a Negative Binomial:
$P(X=2)=$$(n-1)\choose(r-1)$$p^r\times(1-p)^{n-r} \geq 1/2$ where $r = 2$ and $p = 1/4$. However, I don't get a value of $n$ that makes sense.</p>
<p>Than... | Birdman2246 | 271,207 | <p>NOTE: This proof is <strong>incorrect</strong>. Please see the other proofs.</p>
<p>I'm no statistician, but I'll do my best to help you:</p>
<p>Consider some number of coin tosses $n$. With every $n$ coin tosses, there are $n-1$ opportunities to get heads two times in a row. Since with every coin toss there is a ... |
1,449,450 | <p>Is there a way to prove which one of these is bigger? $e^{(a+b)}$ or $e^a + e^b$?</p>
<p>Thanks</p>
| lab bhattacharjee | 33,337 | <p>$$xy>x+y\iff x(y-1)>y$$</p>
<p>If $x,y>0,$</p>
<p>If $y-1>0, x(y-1)>y\iff x>\dfrac y{y-1}$</p>
<p>Else $x(y-1)>y\iff x<\dfrac y{y-1}$</p>
|
744,034 | <p>How do I show that for all integers $n$, $n^3+(n+1)^3+(n+2)^3$ is a multiple of $9$?
Do I use induction for showing this? If not what do I use and how? And is this question asking me to prove it or show it? How do I show it? </p>
| user140943 | 140,943 | <p>Well $$n^3+(n+1)^3+(n+2)^3=n^3+(n^3+3n^2+3n+1)+(n^3+6n^2+12n+8)=3n^3+9n^2+15n+9=3(n^3+3n^2+5n+3)=3(n+1)(n^2+2n+3)$$ Suppose $n=3k$ or $n=3k+1$. Then $$n^2+2n+3=3(3k^2+2k+1)$$ or $$n^2+2n+3=9k^2+6k+1+6k+2+3=9k^2+12k+6=3(3k^2+4k+2)$$ Can you take it from here?</p>
|
428,415 | <p>I tried using integration by parts twice, the same way we do for $\int \sin {(\sqrt{x})}$
but in the second integral, I'm not getting an expression that is equal to $\int x\sin {(\sqrt{x})}$.</p>
<p>I let $\sqrt x = t$ thus,
$$\int t^2 \cdot \sin({t})\cdot 2t dt = 2\int t^3\sin(t)dt = 2[(-\cos(t)\cdot t^3 + \int... | lab bhattacharjee | 33,337 | <p>Integrating by parts, we get </p>
<p>if $n\ne-1,$</p>
<p>$$\int x^n\cos\sqrt xdx= \frac{x^{n+1}\cos\sqrt x}{n+1}+\frac1{2(n+1)}\int x^{n+\frac12}\sin\sqrt x dx$$</p>
<p>$$\int x^n\sin\sqrt xdx= \frac{x^{n+1}\sin\sqrt x}{n+1}-\frac1{2(n+1)}\int x^{n+\frac12}\cos\sqrt x dx$$</p>
<p>Putting $n=\frac12$ in the first... |
1,275,848 | <p>Given two numbers $x$ and $y$, how to check whether $x$ is divisible by <strong>all</strong> prime factors of $y$ or not?, is there a way to do this without factoring $y$?.</p>
| NovaDenizen | 109,816 | <p>There's a pretty simple algorithm based on gcd.</p>
<ol>
<li>$c := \gcd(x,y)$. </li>
<li>$z := y / c$.</li>
<li>If $z = 1$, terminate. All prime factors in y were also in $x$.</li>
<li>$c := \gcd(x,z)$.</li>
<li>If $c = 1$, terminate. $z > 1$ and $z$ divides $y$, but no prime factor of $z$ divides $x$.</li>
<l... |
262,745 | <p>I need to find the normal vector of the form Ax+By+C=0 of the plane that includes the point (6.82,1,5.56) and the line (7.82,6.82,6.56) +t(6,12,-6), with A=1.</p>
<p>Of course, this is easy to do by hand, using the cross product of two lines and the point. There's supposed to be an automated way of doing it, though,... | Rudy Potter | 57,047 | <p>We have a point</p>
<pre><code>pt1 = {6.82, 1, 5.56};
</code></pre>
<p>And we have a line</p>
<pre><code>ln = {7.82, 6.82, 6.56} + t {6, 12, -6};
</code></pre>
<p>We can get two more points from the line</p>
<pre><code>pt2 = ln /. t -> 0;
pt3 = ln /. t -> 1;
</code></pre>
<p>Then we can borrow the example in t... |
262,745 | <p>I need to find the normal vector of the form Ax+By+C=0 of the plane that includes the point (6.82,1,5.56) and the line (7.82,6.82,6.56) +t(6,12,-6), with A=1.</p>
<p>Of course, this is easy to do by hand, using the cross product of two lines and the point. There's supposed to be an automated way of doing it, though,... | Roman | 26,598 | <p>Based on <a href="https://mathematica.stackexchange.com/a/239071/26598">this answer</a>, we find the equation of a plane through three points with</p>
<pre><code>p1 = {6.82, 1, 5.56};
p2 = {7.82, 6.82, 6.56};
d2 = {6, 12, -6};
v = First@NullSpace[Append[#, 1] & /@ {p1, p2, p2 + d2}]
(* {-0.106949, 0.0273527,... |
4,243,030 | <p>I tried to evaluate the integral <span class="math-container">$$ \oint_c\dfrac{dz}{\sin^2 z}$$</span> where <span class="math-container">$c$</span> is a circle <span class="math-container">$|z|=1/2$</span>. The only pole within <span class="math-container">$c$</span> is <span class="math-container">$z=0$</span> and ... | Mr.Gandalf Sauron | 683,801 | <p>You can also find the residue at <span class="math-container">$z=0$</span> by directly evaluating the coefficient of <span class="math-container">$\frac{1}{z}$</span> in :-</p>
<p><span class="math-container">$$\frac{1}{\sin^{2}(z)}=(z-\frac{z^{3}}{3!}+\frac{z^{5}}{5!}-...)^{-2}=z^{-2}(1-\frac{z^{2}}{3!}+\frac{z^{4}... |
802,877 | <blockquote>
<p>Find $\displaystyle\lim_{n\to\infty} n(e^{\frac 1 n}-1)$ </p>
</blockquote>
<p>This should be solved without LHR. I tried to substitute $n=1/k$ but still get indeterminant form like $\displaystyle\lim_{k\to 0} \frac {e^k-1} k$. Is there a way to solve it without LHR nor Taylor or integrals ?</p>
<p>... | Paramanand Singh | 72,031 | <p>Again this turns out to be a very nice question. Without making any assumptions on $e$ or $e^{x}$ it is possible to show that the limit $\lim_{n \to \infty}n(e^{1/n} - 1)$ exists. To be more general we can show that for any real number $x > 0$ the limit $$f(x) = \lim_{n \to \infty}n(x^{1/n} - 1)$$ exists. We need... |
74,805 | <p>Could anyone give me some explanation how can I compute sum of multibranched functions? For example, if I have $z=re^{i\varphi}$ then $$\sqrt{z}=\sqrt{r}(\cos(\varphi/2+k\pi)+i\sin (\varphi/2+k\pi))$$ for some integer $k$. Therefore, is $$\sqrt{z}+\sqrt{z}=2\sqrt{r}(\cos(\varphi/2+k\pi)+i\sin (\varphi/2+k\pi))$$ for... | Did | 6,179 | <p>The first condition, that $f'(0)/f'0)=g'(0)/g(0)$, is equivalent to $A=-z/w$ where $z$ and $w$ are the complex number $z=B-C-\mathrm i(B+C)$ and $w=B-C+\mathrm i(B+C)$. If $B$ and $C$ are real numbers, $w=\bar z$ hence $|A|=1$. (The only case when $w=0$ is $B=C=0$, and then $g(0)=0$ hence $g'(0)/g(0)... |
74,805 | <p>Could anyone give me some explanation how can I compute sum of multibranched functions? For example, if I have $z=re^{i\varphi}$ then $$\sqrt{z}=\sqrt{r}(\cos(\varphi/2+k\pi)+i\sin (\varphi/2+k\pi))$$ for some integer $k$. Therefore, is $$\sqrt{z}+\sqrt{z}=2\sqrt{r}(\cos(\varphi/2+k\pi)+i\sin (\varphi/2+k\pi))$$ for... | Peđa | 15,660 | <p>From the first condition we can conclude :</p>
<p>$(A=-1 \lor k=0 \lor B=C) \land (A=1 \lor k=0 \lor B=-C)$</p>
<p>From the second condition we have that:</p>
<p>$(k=0 \lor C=Be^{2ka}) \land (k=0 \lor C=-Be^{2ka})$</p>
<p>so... </p>
<p>$a)$if $k=0 \Rightarrow A$ is undetermined</p>
<p>$b)$if $B=\pm C=0 \Righta... |
3,479,940 | <p>Say we're given a set of <span class="math-container">$d$</span> vectors <span class="math-container">$S=\{\mathbf{v}_1,\dots,\mathbf{v}_d\}$</span> in <span class="math-container">$\mathbb{R}^n$</span>, with <span class="math-container">$d\leq n$</span> (obviously). We want to test in an efficient way if S is linea... | tch | 352,534 | <p>The standard numerical approaches would be computing a (rank revealing) QR decomposition or SVD. Both <a href="https://docs.scipy.org/doc/numpy-1.10.4/reference/generated/numpy.linalg.matrix_rank.html" rel="nofollow noreferrer">Numpy</a> and <a href="https://www.mathworks.com/help/matlab/ref/rank.html" rel="nofollow... |
58,870 | <p>I am teaching a introductory course on differentiable manifolds next term. The course is aimed at fourth year US undergraduate students and first year US graduate students who have done basic coursework in
point-set topology and multivariable calculus, but may not know the definition of differentiable manifold. I ... | AFK | 1,985 | <p>Thierry Aubin's book "A course in differential geometry" is really good for an introductory course. It covers the basic definitions of manifolds and vector bundles, orientability and integration (Stokes formula) and then focuses on Riemannian geometry defining the Levi-Civita connection, curvature tensor etc... <... |
294,519 | <p>The problem I am working on is:</p>
<p>Translate these statements into English, where C(x) is “x is a comedian” and F(x) is “x is funny” and the domain consists of all people.</p>
<p>a) $∀x(C(x)→F(x))$ </p>
<p>b)$∀x(C(x)∧F(x))$</p>
<p>c) $∃x(C(x)→F(x))$ </p>
<p>d)$∃x(C(x)∧F(x))$</p>
<h2>-----------------------... | Peter Smith | 35,151 | <p>The homework question said "Translate these statements into English". And that means normal English, such as a native English speaker might use. Now, is</p>
<blockquote>
<p>For every person, if they are a comedian, then they are funny.</p>
</blockquote>
<p>normal English in that sense? Would <em>you</em> ever sa... |
947,358 | <p>Okay $g(x)= \sqrt{x^2-9}$</p>
<p>thus, $x^2 -9 \ge 0$</p>
<p>equals $x \ge +3$ and $x \ge -3$</p>
<p>thus the domains should be $[3,+\infty) \cup [-3,\infty)$ how come the answer key in my book is stating $(−\infty, −3] \cup[3,\infty)$. </p>
| Dr. Sonnhard Graubner | 175,066 | <p>it must be $(x-3)(x+3)\geq 0$ and this means $x\geq 3$ or $x\le -3$.</p>
|
947,358 | <p>Okay $g(x)= \sqrt{x^2-9}$</p>
<p>thus, $x^2 -9 \ge 0$</p>
<p>equals $x \ge +3$ and $x \ge -3$</p>
<p>thus the domains should be $[3,+\infty) \cup [-3,\infty)$ how come the answer key in my book is stating $(−\infty, −3] \cup[3,\infty)$. </p>
| kingW3 | 130,953 | <p>So we have that $$(x-3)(x+3)\geq0$$ Now this happens when either both $x-3$ and $x+3$ are positive or both are negative.Now solving $$x-3\geq0\land x+3\geq 0\implies x\geq 3\land x\geq -3\\x-3\leq0\land x+3\leq0\implies x\leq3\land x\leq-3$$
Now the first solution is $x\geq3$ and the second is $x\leq-3$</p>
|
3,034,441 | <p>How can I prove that <span class="math-container">$\lim_{n\to\infty} \frac{2^n}{n!}=0$</span>?</p>
| Jack D'Aurizio | 44,121 | <p>In other terms we want to evaluate</p>
<p><span class="math-container">$$ \sum_{n\geq 1}(-1)^{n+1}\left(H_{n(n+1)/2}-H_{n(n-1)/2}\right)=\int_{0}^{1}\sum_{n\geq 1}(-1)^{n+1}\frac{x^{n(n+1)/2}-x^{n(n-1)/2}}{x-1}\,dx$$</span>
where the theory of modular forms ensures
<span class="math-container">$$ \sum_{n\geq 0} x^{... |
3,034,441 | <p>How can I prove that <span class="math-container">$\lim_{n\to\infty} \frac{2^n}{n!}=0$</span>?</p>
| robjohn | 13,854 | <p><strong>Approximating the Sum</strong></p>
<p>Applying the Euler-Maclaurin Sum Formula to <span class="math-container">$\frac1n$</span>, we get
<span class="math-container">$$
\begin{align}
H_n
&=\log(n)+\gamma+\frac1{2n}-\frac1{12n^2}+\frac1{120n^4}-\frac1{252n^6}+\frac1{240n^8}-\frac1{132n^{10}}\\
&\phanto... |
216,021 | <p>Suppose $A$ and $B$ are $n \times n$ matrices. Assume $AB=I$. Prove that $A$ and $B$ are invertible and that $B=A^{-1}$.</p>
<p>Please let me know whether my proof is correct and if there are any improvements to be made.</p>
<p>Assume $AB=I$. Then $(AB)A=IA=A$. So, $A(BA)=AI=A$. Then $BA=I$. Therefore $AB=BA=I... | littleO | 40,119 | <p>Suppose $AB = I$. ($A$ and $B$ are $n \times n$ matrices.)</p>
<p>First note that $R(A) = \mathbb{R}^n$. (If $y \in \mathbb{R}^n$, then
$y = Ax$, where $x = By$.)</p>
<p>It follows that $N(A) = \{0\}$.</p>
<p>We wish to show that $BAx = x$ for all $x \in \mathbb{R}^n$.
So let $x \in \mathbb{R}^n$, and let $z = ... |
2,917,439 | <p>Suppose we flip two fair coins and roll one fair six-sided die.</p>
<p>What is the conditional probability that the number of heads equals the number
showing on the die, conditional on knowing that the die showed 1?</p>
<p>Let's define the following:</p>
<p>$A=\{\text{#H = # on die}\}$</p>
<p>$B=\{\text{# on die... | stagerf | 492,266 | <p>Let $\xi_1$ be the number of heads from two coin tosses and $\xi_2$ be value of a die after a roll.</p>
<p>$\xi_1$ is binomially distributed with $n=2$ and $p=\frac{1}{2}$, so $p_{\xi_1}(x)=_2C_x \cdot \left ( \frac{1}{2} \right )^2$. $\xi_2$ has a discrete uniform distribution with $p_{\xi_2}(x)=\frac{1}{6}$.</p>
... |
3,366,569 | <p>I am trying to solve the following problem;</p>
<p>Write all elements of the following set: <span class="math-container">$ A=\left \{ x\in\mathbb{R}; \sqrt{8-t+\sqrt{2-t}}\in\mathbb{R}, t\in\mathbb{R} \right \}$</span> .</p>
<p>My assumption is that the solution is <span class="math-container">$\mathbb{R}$</span> ... | Kavi Rama Murthy | 142,385 | <p>Hint: try <span class="math-container">$y=\sum\limits_{k=0}^\infty a_nx^{n}$</span>. The coefficient of <span class="math-container">$x^{n}$</span> on the left side becomes <span class="math-container">$(n^{2}-1)a_n$</span>. Equate this to the coefficient of <span class="math-container">$x^{n}$</span> on the right s... |
1,596 | <p><a href="https://en.wikipedia.org/wiki/Lenna" rel="nofollow noreferrer">Lenna</a> is commonly used as an example placeholder image. I also recently used it in an <a href="https://mathematica.stackexchange.com/questions/87693/how-to-put-an-imported-image-in-a-disk/87699#87699">answer on the site</a>. However, as the ... | Community | -1 | <blockquote>
<p>If a professor makes a sexist joke, a female student might well find it so disturbing that she is unable to listen to the rest of the lecture [<a href="http://www.spertus.com/ellen/Gender/pap/node10.html#SECTION00410000000000000000" rel="nofollow">2</a>]. Suggestive pictures used in lectures on image ... |
2,170,278 | <p>The answer is $\binom {13}1 \binom42 \binom{12}3 \binom 41^3$</p>
<p>I want to break the last term and see what happens. [Struggling with the concept so trying to work with it as much as possible].</p>
<p>$\binom 41^3$ means that $\heartsuit \diamondsuit \spadesuit$ is different from $\diamondsuit \heartsuit \spad... | David K | 139,123 | <p>The difference between the two principles is that the chain of implications <em>terminates</em> in downward induction, whereas in infinite descent it does not terminate.</p>
<p>Examine the two following statements carefully to see how they are different:</p>
<blockquote>
<p>A) From <span class="math-container">$k\in... |
262,319 | <pre><code>ContourPlot[
EuclideanDistance[{-5, 0}, {x, y}]*
EuclideanDistance[{5, 0}, {x, y}], {x, -15, 15}, {y, -11, 11},
Contours -> Range[5, 150, 20], Frame -> False,
ContourLabels -> (Text[Style[#3, Directive[Blue, 15]], {#1, #2}] &),
AspectRatio -> Automatic,
ColorFunction -> (If[# &l... | kglr | 125 | <pre><code>cp = ContourPlot[EuclideanDistance[{-5, 0}, {x, y}] EuclideanDistance[{5, 0}, {x, y}],
{x, -15, 15}, {y, -11, 11},
Contours -> Range[5, 150, 20], Frame -> False,
ContourLabels -> (Text[Style[#3, Directive[Blue, 15]], {#1, #2}] &),
AspectRatio -> Automatic,
ColorFunction -> (I... |
262,319 | <pre><code>ContourPlot[
EuclideanDistance[{-5, 0}, {x, y}]*
EuclideanDistance[{5, 0}, {x, y}], {x, -15, 15}, {y, -11, 11},
Contours -> Range[5, 150, 20], Frame -> False,
ContourLabels -> (Text[Style[#3, Directive[Blue, 15]], {#1, #2}] &),
AspectRatio -> Automatic,
ColorFunction -> (If[# &l... | cvgmt | 72,111 | <p>Follow the approach by @Bob Hanlon.</p>
<p>I think the problem come from the Complex Function <code>EuclideanDistance</code> since it contain <code>Sqrt</code> and <code>Abs</code>.</p>
<pre><code>EuclideanDistance[{a, b}, {x, y}]
(* Sqrt[Abs[a - x]^2 + Abs[b - y]^2] *)
</code></pre>
<p>So sometimes I avoid to use ... |
178,028 | <p>I am given $G = \{x + y \sqrt7 \mid x^2 - 7y^2 = 1; x,y \in \mathbb Q\}$ and the task is to determine the nature of $(G, \cdot)$, where $\cdot$ is multiplication. I'm having trouble finding the inverse element (I have found the neutral and proven the associative rule.</p>
| Bill Dubuque | 242 | <p><strong>Hint</strong> $\ \alpha\, \bar\alpha = 1\:\Rightarrow\: \bar\alpha = 1/\alpha$</p>
|
21,201 | <p>Next Monday, I'll have an interview at Siemens for an internship where I have to know about fluid dynamics/computational fluid dynamics. I'm not a physicist, so does somebody have a suggestion for a good book where I can read about some basics? Thank you very much.</p>
| Ben Webster | 66 | <p>There are many group actions on sets which are linearly equivalent but not equivalent as actions. In fact, every group other than the cyclic group has one. This follows from some easy linear algebra:</p>
<ul>
<li>the number of irreducible reps over <span class="math-container">$\mathbb{Q}$</span> is the number of ... |
1,006,354 | <ul>
<li>A multiple choice exam has 175 questions. </li>
<li>Each question has 4 possible answers. </li>
<li>Only 1 answer out of the 4 possible answers is correct. </li>
<li>The pass rate for the exam is 70% (123 questions must be answered correctly). </li>
<li>We know for a fact that 100 questions were answered corre... | Joshua Mundinger | 106,317 | <p>The question is essentially what the probability of answering more than 22 questions out of 75 correctly. </p>
<p>The chance that we will answer all 75 incorrectly is simply $(3/4)^{75}$. The probability that we will answer one particular question correctly and all the rest incorrectly is $(1/4)(3/4)^{74}$. Thus, t... |
402,214 | <p>I recently obtained "What is Mathematics?" by Richard Courant and I am having trouble understanding what is happening with the Prime Number Unique Factor Composition Proof (found on Page 23).</p>
<p>The first part:</p>
<blockquote>
<p><img src="https://i.stack.imgur.com/h5rCh.png" alt="enter image description he... | Key Ideas | 78,535 | <p>It appears that the troublesome aspect of the proof is the negative contradictory form of induction that is employed, i.e. the contrapositive of complete induction (Fermat's infinite descent). This is easily eliminated by rewriting the proof to use <a href="https://en.wikipedia.org/wiki/Mathematical_induction#Comple... |
3,245,223 | <p>so I am supposed to solve a proof which seems fairly easy, but the negative exponents in <span class="math-container">$$\sum_{k=0}^n \binom nk\ (\frac{(-1)^k}{k+1})= \frac{1}{n+1}$$</span> are making this question very difficult for me. I have tried using binomial theorem on the right side with <span class="math-con... | Z Ahmed | 671,540 | <p>The binomial theorem gives:
<span class="math-container">$$(1-x)^n= \sum_{k=0}^{n} (-1)^k x^k$$</span>
Integrate w.r.t. x both sides from <span class="math-container">$x=0$</span> to <span class="math-container">$x=1$</span>, to get
<span class="math-container">$$\left .\sum_{k=0}^{n} (-1)^k {n \choose k}\frac{x^{k... |
1,177,493 | <p>If $p$ is a prime and $p \equiv 1 \bmod 4$, how many ways are there to write $p$ as a sum of two squares? Is there an explicit formulation for this?</p>
<p>There's a theorem that says that $p = 1 \bmod 4$ if and only if $p$ is a sum of two squares so this number must be at least 1. There's also the Sum of Two Squar... | Gerry Myerson | 8,269 | <p>If $n$ has two distinct expressions as a sum of two squares, $n=a^2+b^2=c^2+d^2$, then $n$ divides $(ac+bd)(ac-bd)$. But then you can show $n$ doesn't divide either $ac+bd$ or $ac-bd$ (it's a bit tricky), from which it follows that $n$ isn't prime. </p>
<p>Then the contrapositive is that if $n$ is prime it doesn't ... |
1,575,671 | <p>The whole question is that <br>
If $f(x) = -2cos^2x$, then what is $d^6y \over dx^6$ for x = $\pi/4$?</p>
<p>The key here is what does $d^6y \over dx^6$ mean?</p>
<p>I know that $d^6y \over d^6x$ means 6th derivative of y with respect to x, but I've never seen it before.</p>
| Kaster | 49,333 | <p>If you recall the definition of the derivative, then you can write
\begin{align}
y'(x) &= \lim_{h \to 0} \frac {y(x+h) - y(h)}h \\
y''(x) &= \lim_{h \to 0} \frac {y'(x+h) - y'(x)}h = \lim_{h \to 0} \frac{\frac {y(x+2h) - y(x+h)}h - \frac {y(x+h) - y(x)}h}h = \lim_{h \to 0} \frac {y(x+2h) - 2y(x+h) + y(x)}{h^... |
1,738,153 | <p>I know the definition is given as follows:</p>
<p>A map $p: G \rightarrow GL(V)$ such that $p(g_1g_2)=p(g_1)p(g_2)$ but I still do not really understand what this means</p>
<p>Can someone help me gain some intuition for this - perhaps a basic example?</p>
<p>Thanks</p>
| Peter Franek | 62,009 | <p>The intuition is that $G$ is the group of symmetry of something. For example, if you study classical physics, it may be the group of Euclidean transformations. $V$ is a space whose elements are objects (velocity, force, electron in quantum mechanics,...) as seen by an observer. The action of the group corresponds to... |
151,937 | <p>In <code>FindGraphCommunities</code>, how can one find the vertices associated with the edges that are found to connect one or more communities?</p>
| David G. Stork | 9,735 | <p>Start with a community graph:</p>
<pre><code>CommunityGraphPlot[g = RandomGraph[{20, 50}]]
</code></pre>
<p>Find the list of vertexes in each community:</p>
<pre><code>mycommunitylists = FindGraphCommunities[g]
</code></pre>
<p>(*</p>
<p>{{3, 5, 9, 11, 12, 13, 14, 16, 18}, {1, 2, 7, 10, 15, 19, 20}, {4, 6,
8... |
2,877,576 | <p>Is it true that in equation $Ax=b$,
$A$ is a square matrix of $n\times n$, is having rank $n$, then augmented matrix $[A|b]$ will always have rank $n$?</p>
<p>$b$ is a column vector with non-zero values.
$x$ is a column vector of $n$ variables.</p>
<p>If not then please provide an example.</p>
| Phil H | 554,494 | <p><a href="https://i.stack.imgur.com/QJ9We.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/QJ9We.jpg" alt="enter image description here"></a></p>
<p>We can treat this by maximizing the cross sectional area A.</p>
<p>$A = (100 - x^2)^{.5} (20 + x)$ where $x$ is the appex height of the roof</p>
<p>... |
1,538,496 | <p>I came across this riddle during a job interview and thought it was worth sharing with the community as I thought it was clever:</p>
<blockquote>
<p>Suppose you are sitting at a perfectly round table with an adversary about to play a game. Next to each of you is an infinitely large bag of pennies. The goal of the... | zahbaz | 176,922 | <p>JMoravitz has a great post. I'm happy to have arrived at the same answer, and would like to share a thought that aided my process. Consider an extreme case to find out whether to go first or second:</p>
<blockquote class="spoiler">
<p> The problem specifies pennies and a circular table. If there is a winning stra... |
1,961,727 | <p>As far as I understood <a href="https://en.wikipedia.org/wiki/Gram%E2%80%93Schmidt_process">Gram–Schmidt orthogonalization</a> starts with a set of linearly independent vectors and produces a set of mutually orthonormal vectors that spans the same space that starting vectors did.</p>
<p>I have no problem understand... | Sean Lake | 153,694 | <p>You can also get into function spaces where it's not clear what the basis you can just grab from is. The Legendre polynomials can be constructed by starting with the functions $1$ and $x$ on the interval $x \in [-1,1]$, and using Gram-Schmidt orthogonalization to construct the higher order ones. The second order pol... |
3,124,158 | <p>So what I want to prove is
<span class="math-container">$$ |xy+xz+yz- 2(x+y+z) + 3| \leq |x^2+y^2+z^2-2(x+y+z)+3| $$</span>
for <span class="math-container">$x,y,z\in \mathbb{R}$</span>, and I'm aware that the RHS is just <span class="math-container">$|(x-1)^2+(y-1)^2+(z-1)^2|$</span>.</p>
<p>Now I'm able to prove ... | Michael Rozenberg | 190,319 | <p>Since <span class="math-container">$$x^2+y^2+z^2-2(x+y+z)+3=\sum_{cyc}(x-1)^2\geq0,$$</span> we need to prove that <span class="math-container">$$ x^2+y^2+z^2-2(x+y+z)+3 \geq xy+xz+yz- 2(x+y+z) + 3\geq$$</span>
<span class="math-container">$$\geq-(x^2+y^2+z^2-2(x+y+z)+3)$$</span> </p>
<p>The left inequality it's... |
1,913,689 | <blockquote>
<p>Let $f: X \rightarrow Y$ be a function. $A \subset X$ and $B \subset Y$.
Prove $A \subset f^{-1}(f(A))$.</p>
</blockquote>
<p>Here is my approach. </p>
<p>Let $x \in A$. Then there exists some $y \in f(A)$ such that $y = f(x)$. By the definition of inverse function, $f^{-1}(f(x)) = \{ x \in X$ suc... | Michael Hardy | 11,667 | <p>"there exists some $y\in f(A)$ such that $y=f(x)$" is a cumbersome way of expressing it. I'd just say "let $y = f(x)$." Also, I would avoid using the same letter, $x$, in two different senses, especially in view of the fact that not every point whose image under $f$ is equal to $f(x)$ needs to be the same as $x.$<... |
1,017,707 | <p>Are there any proofs of this equality online? I'm just looking for something very simply that I can self-verify. My textbook uses the result without a proof, and I want to see what a proof would look like here.</p>
| mookid | 131,738 | <p>\begin{align}
e^{inx} &= (\cos x + i\sin x)^n
\\&= \sum_{k=0}^n \binom nk\cos^{n-k}(x) i^k \sin^k (x)
\\&= \sum_{k=0}^{\lfloor n/2\rfloor} \binom n{2k}\cos^{n-2k}(x) i^{2k} \sin^{2k} (x) +
\sum_{k=0}^{\lfloor (n-1)/2\rfloor} \binom n{2k+1}\cos^{n-2k-1}(x)
i^{2k+1} \sin^{2k+1} (x)
\end{align}</p>
<... |
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