qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
1,045,920 | <p>Let T: $M_{2x2} \rightarrow M_{2x2}$ be defined by</p>
<p>T(A) = $\begin{bmatrix}
1 & 3 \\[0.3em]
-1 & 1 \\[0.3em]
\end{bmatrix} A $</p>
<p>Is T a linear transformation?</p>
<p>I want to show that the operations for linear transformations hold. However, I am not sure if I am... | Ben Grossmann | 81,360 | <p>It should be given in your text at some point that matrix multiplication satisfies the following properties (among some other important ones): for compatible matrices $A,B,C$ and scalar $\alpha$:</p>
<ul>
<li>$A(B + C) = AB + AC$</li>
<li>$A(\alpha B) = (\alpha A)B = \alpha(AB)$</li>
</ul>
<p>That should be enough... |
1,045,920 | <p>Let T: $M_{2x2} \rightarrow M_{2x2}$ be defined by</p>
<p>T(A) = $\begin{bmatrix}
1 & 3 \\[0.3em]
-1 & 1 \\[0.3em]
\end{bmatrix} A $</p>
<p>Is T a linear transformation?</p>
<p>I want to show that the operations for linear transformations hold. However, I am not sure if I am... | Milo Brandt | 174,927 | <p>It looks like you've already proved everything you desire to; you've got it in the wrong order though; you ought to write out $T(A+B)$ as the sum of two matrices (as you do on the rightmost expression) and <em>then</em> note that this equals $T(A)+T(B)$ (as you do in the middle expression). (Your proof that $T$ pres... |
3,075,987 | <p>So I am currently studying Calculus and Linear Algebra and I came across the same concepts that is being applied in a lot of the proofs that I read for Calculus and Linear algebra but not capable of fully understanding it.</p>
<p><strong>Claim:</strong> There exists a <span class="math-container">$\lambda \in \math... | Bijayan Ray | 627,419 | <p>one hint to understand is</p>
<p>in a finite dimensional space zero vector is the only vector orthogonal to all vectors </p>
<p>so from the last equation of your proof take the things in one side and use the above statement</p>
|
1,582,742 | <blockquote>
<p>I have this so easy question, yet it is giving me a headache</p>
<p>Consider the following two subsets of <span class="math-container">$R^3$</span></p>
<p><span class="math-container">$S_2 = [(1,2,3),(4,5,6),(7,8,9),(10,11,12)]$</span></p>
<p><span class="math-container">$S_3 = [(1,0,-1),(1,0,1)]$</span... | Javier | 241,291 | <p>You have correctly deduce that $S_2$ consists on dependent vectors. That means that one of those vectors (let's say $v_4$ can be expressed as a lineal combination of $v_1,v_2,v_3$. In that case the space that $\left[ v_1,v_2,v_3,v_4 \right]$ spans is the same that the space that $\left[ v_1,v_2,v_3 \right]$. You can... |
1,582,742 | <blockquote>
<p>I have this so easy question, yet it is giving me a headache</p>
<p>Consider the following two subsets of <span class="math-container">$R^3$</span></p>
<p><span class="math-container">$S_2 = [(1,2,3),(4,5,6),(7,8,9),(10,11,12)]$</span></p>
<p><span class="math-container">$S_3 = [(1,0,-1),(1,0,1)]$</span... | David G. Stork | 210,401 | <p>All your vectors in $S_2$ are over the form $(1,2,3) + i\ (3,3,3)$ for $i = 1,2,3$. Thus $(1,2,3)$ and the single vector $(3,3,3)$ can generate all your set, and thus form all linear combinations of your set. Hence the intrinsic dimensionality is $2$ (not $3$). </p>
|
1,619,208 | <blockquote>
<p>A person walks downhill at 10 km/h, uphill at 6 km/h and on the plane at 7.5 km/h. If the person takes 3 hours to go from a place A to another place B, and 1 hour on the way back, the distance between A and B is
$\begin{array}1 1. 15 km && 2. 23.5 km \\ 3. 16 km && 4. \text{Given dat... | sinbadh | 277,566 | <p>Let us suppose that you know $r$ and want to know $k$ such that $\binom{k}{r}=c$, with $c\in \mathbb{N}$. Let us name $(*)$ this equation.</p>
<p>For $r=0$, $\binom{k}{r}=1$, and you can easly find the solution of $(*)$ if there exists.</p>
<p>If $r=1,2,3,4$ you can solve $(*)$ relatively easy, cause $\binom{r}{2}... |
2,856,568 | <p>In the example below I get the first part. May I know why they're dividing by 2 for the second part? I'm asking because I feel the answer for second part should also be $\binom{10}{5}$. </p>
<p>In the first part simply by choosing 5 people, we're already splitting the squad into two teams. So I don't really see a d... | Community | -1 | <p>This is from Sheldon Ross's book a First Course in Probability, I remember that example confusing me. Sometimes it helped me to think of it this way - assume that the first 5 picked will be on the red jersey team and the remaining will be on the yellow jersey team for option 1. Then a team of A,B,C,D,E in red jersey... |
3,403,464 | <p>Suppose <span class="math-container">$f\in L^2(X\times Y)$</span> where <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> are finite measure spaces and we have a product measure <span class="math-container">$\mu$</span>.</p>
<p>Suppose further that <span class="math-container">$g\i... | Aloizio Macedo | 59,234 | <p>This is mostly a pointwise statement. That <span class="math-container">$\omega$</span> is a volume form means that <span class="math-container">$\omega_p(X_1,\cdots,X_n) \neq 0$</span> if and only if <span class="math-container">$X_1,\cdots,X_n$</span> is a basis of <span class="math-container">$T_pM$</span>. (<spa... |
1,905,397 | <p>I want to know if the popular Sudoku puzzle is a Cayley table for a group. </p>
<p>Methods I've looked at: Someone I've spoken to told me they're not because counting the number of puzzle solutions against the number of tables with certain permutations of elements, rows and columns, the solutions are bigger than th... | flawr | 109,451 | <p>Another approach: We are considering a group of order $9 = 3^2$ so a prime squared. <a href="https://math.stackexchange.com/questions/64371/showing-non-cyclic-group-with-p2-elements-is-abelian">It is well known</a> that groups of order $p^2$ must be abelian. (abelian = commutative) Now a soduku can never represent a... |
3,620,409 | <blockquote>
<p>If a matrix and its transpose both have the same eigenvectors, is it necessarily symmetric?</p>
</blockquote>
<p>It's clear to see how if <span class="math-container">$A$</span> = <span class="math-container">$A^T$</span>, they would have the same eigenvectors, but is it the only way? And how would y... | Batominovski | 72,152 | <blockquote>
<p><strong>Proposition.</strong> Let <span class="math-container">$A\in\text{Mat}_{n\times n}(\mathbb{R})$</span> be such that <span class="math-container">$A$</span> is diagonalizable over <span class="math-container">$\mathbb{R}$</span>. Then, <span class="math-container">$A$</span> and <span class="... |
194,569 | <blockquote>
<p>How can we find an integer $n$ such that $\mathbb{Z}[\frac{1}{20},\frac{1}{32}]=\mathbb{Z}[\frac{1}{n}]$?</p>
</blockquote>
<p>Thanks in advance!</p>
| i. m. soloveichik | 32,940 | <p>Let $n=\operatorname{lcm}(32,20)=160$ then $R=Z[1/32,1/20]\subset Z[1/n]$ since 1/32=5/160, 1/20=8/160.
On the other hand $1/160=4\cdot 1/32\cdot 1/20\in R$. So $n=160$ is a valid answer</p>
|
3,141,004 | <p>I am just a high school student, so please excuse me if this question sound silly to you.</p>
<p>How do I graph this : <span class="math-container">$\dfrac{9}{x} - \dfrac{4}{y} = 8$</span></p>
<p>I could do trial and error, but is there a more systematic way that I can use to graph this equation.</p>
| José Carlos Santos | 446,262 | <p>Yes, there is. Note that<span class="math-container">\begin{align}\frac9x-\frac4y=8&\iff\frac4y=\frac9x-8\\&\iff y=\frac4{\frac9x-8}=\frac{4x}{9-8x}=-\frac12+\frac9{18-16x}.\end{align}</span>So, graph the function <span class="math-container">$x\mapsto-\dfrac12+\dfrac9{18-16x}$</span>.</p>
|
1,161,089 | <p>Sorry if this may seem trivial - I just started studying Group Theory. This is the problem:</p>
<blockquote>
<p>Prove that <span class="math-container">$(g,h) \rightarrow hg$</span> does not define a group action with <span class="math-container">$g$</span> acting on <span class="math-container">$h$</span>. Prove in... | Uncountable | 215,630 | <p>You throw the coin $x$ times. Now the probability that the first $y$ are heads and the second $x-y$ are tails is:
$$\frac{1}{2^{y}}\frac{1}{2^{x-y}}=\frac{1}{2^x}$$
But we can choose the $y$ heads out of the $x$ throws in ${x\choose y}$ ways. So:
$$P(y\text{ heads in } x\text{ coin tosses})={x\choose y}\frac{1}{2^x}... |
2,304,168 | <p>Let's say we have a random point inside a circle . My intuition tells me that that If I draw vectors from all the circumference points to that point and find their sum the result will be zero.</p>
<p>Am I correct ? How can we prove this ?</p>
<p>Edit: I am asking about the sum of the 'vectors'</p>
| William Elliot | 426,203 | <p>bd A = cl A - int A.<br>
int A subset A.<br>
Thus<br>
cl A - A subset cl A - int A = bd A.</p>
|
3,306,110 | <p>the expression <span class="math-container">$x(x-9)$</span> is resolved in the manner
<span class="math-container">$$x(x-9)=x^2-9x$$</span>
but why not like this?
<span class="math-container">$$x(x-9)=x^2-9$$</span>
where does the <span class="math-container">$9x$</span> come from?</p>
| Ethan Bolker | 72,858 | <p>Remember the distributive law
<span class="math-container">$$
a(b+c) = ab + ac .
$$</span></p>
<p>In your expression, <span class="math-container">$a=x$</span>, <span class="math-container">$b=x$</span> and <span class="math-container">$c = -9$</span>.</p>
<p>Then <span class="math-container">$x$</span> times <spa... |
4,508,532 | <blockquote>
<p>Solve the equation <span class="math-container">$a \circ x \circ b = c$</span> in the group <span class="math-container">$S_4$</span> when <span class="math-container">$$a=\begin{pmatrix}1&2&3&4\\ 2&3&1&4\end{pmatrix}, b= \begin{pmatrix}1&2&3&4\\ 4&1&2&3\e... | José Carlos Santos | 446,262 | <p>Your answer is correct and, actually, we don't have <span class="math-container">$$(1\ \ 2\ \ 3)\circ(1\ \ 2\ \ 4\ \ 3)\circ(1\ \ 4\ \ 3\ \ 2)=(1\ \ 3)\circ(2\ \ 4).\tag1\label{1}$$</span> For instance, the LHS of \eqref{1} maps <span class="math-container">$1$</span> into <span class="math-container">$1$</span>, si... |
1,904,786 | <p>How can I get a good approximation of the sum</p>
<p>$$\sum_{s=1}^\infty x^s\ln(s)$$</p>
<p>by hand ?</p>
<p>If I only consider the part of the function, where it is strictly decreasing, I can bound the sum by integrals. But how can I approximate the sum upto this point ?</p>
| Simply Beautiful Art | 272,831 | <p>$|x|<1$</p>
<p>$$f(x):=\sum_{s=2}^\infty x^s\ln(s)$$</p>
<p>$$f(x)+f(-x)=2x^{-2}\sum_{s=2}^\infty x^{2s}\ln(2s)$$</p>
<p>$$=2x^{-2}\sum_{s=2}^\infty x^{2s}\left(\ln(2)+\ln(s)\right)$$</p>
<p>$$f(x)+f(-x)=2\ln(2)\frac{x^2}{1-x^2}+2\frac{f(x^2)}{x^2}$$</p>
<p>So as long as we can approximate it for $x>0$, t... |
4,097,772 | <p>Consider <span class="math-container">$\mathbb{D}_{R} = \{z \in \mathbb{C}: |z| < R\}$</span>. For given <span class="math-container">$f \in \mathscr{O}(\mathbb{D}_{R})$</span> - space of holomorphic functions on the open disk, denote by <span class="math-container">$c_n$</span> the <span class="math-container">$... | Community | -1 | <p>It suffices to prove that, for every <span class="math-container">$R>r>\delta>0$</span>, if <span class="math-container">$||f_\lambda-f||_{r, \infty}\to 0$</span> then <span class="math-container">$f_\lambda$</span> converges uniformly to <span class="math-container">$f$</span> on <span class="math-contain... |
1,038,219 | <ol>
<li>(n) people put their name in a hat. </li>
<li>Each person picks a name out of the hat to buy a gift for. </li>
<li>If a person picks out themselves they put the name back into the hat.</li>
<li>If the last person can only pick themselves then the loop is invalid and either<br>
. start again<br>
. or step bac... | Ross Millikan | 1,827 | <p>You are asking for the chance of a single cycle given that you have a <a href="http://en.wikipedia.org/wiki/Derangement" rel="nofollow">derangement</a>. For $n$ people, the number of derangements is the closest integer to $\frac {n!}e$ To have a cycle, person $1$ has $n-1$ choices, then that person has $n-2$ choic... |
251,607 | <p>The simplest Laguerre polynomials are
$$
L_k(x)=(\frac{d}{dx}-1)^k\left(\frac{x^k}{k!}\right).
$$
I would like to find a simple reference for proving or disproving the following assertions.</p>
<p>(1) All the $k$ zeroes of $L_k$ are simple and located on the positive half-line.</p>
<p>(2) The largest zero of $L_k... | john mangual | 1,358 | <p>hmm derivatives interlace zeros... however $ a^+=1- d/dx$ behaves nicely on polynomials. It should preserve positivity of the roots and shift the leading zero by $+n$ with a small correction term where $n = \deg p$. here we have $x=0$ as a root with multiplicity $k$ and they should hop forward a bit</p>
|
1,177,111 | <p>Let $(a_n)$ be defined by $a_1=1$, $a_{n+1}=a_n+{1\over a_n}$. Prove there exists $\lim_\limits{n\to \infty}{a_n}$ and find $\lim_\limits{n\to \infty}{a_n\over n}$. </p>
<p>I tried many things but they wouldn't work. I would appreciate your help.</p>
| Clement C. | 75,808 | <ul>
<li>By induction, prove $a_n \geq 1$ for all $n$.</li>
<li>Use this to show $(a_n)_n)$ is strictly increasing.</li>
<li>By monotonicity, it must converge in $\mathbb{R}\cup\{+\infty\}$.</li>
<li>By continuity of the function $f\colon x > 0 \mapsto x+\frac{1}{x}$, if the limit $\ell$ exists it must satisfy $\ell... |
3,724,594 | <p>Let <span class="math-container">$g: \mathbb{R}^2 \to \mathbb{R}$</span>, <span class="math-container">$g(x,y) = |xy|$</span>. Find all the <span class="math-container">$(x,y) \in \mathbb{R}^2$</span> where <span class="math-container">$g$</span> is differentiable.</p>
<p>I tried to compute the partial derivatives:<... | Allawonder | 145,126 | <blockquote>
<p>So what if I always choose <span class="math-container">$\epsilon=\infty$</span>? Then it is guaranteed that the distance between <span class="math-container">$f(x)$</span> and <span class="math-container">$L$</span> is less than <span class="math-container">$\epsilon$</span>, and, as a bonus, <em><span... |
2,229,473 | <p>I am trying to solve the following question:</p>
<blockquote>
<p>If $K$ is a field and $R=K[x]/(x^n)$.and $r=a_0+a_1x +\dots+a_{n-1}x^{n-1}$ is an element in $R$ with $a_0\neq 0$, prove that $r$ is a unit and find its inverse.</p>
</blockquote>
<p>To prove it is a unit, I know that $\gcd(r,x^n)=1$, so there exis... | egreg | 62,967 | <p>I'd prefer to write $\xi$ for the image of $x$ in $K[x]/(x^n)$, just to be sure we're working in the right context.</p>
<p>What you want to prove is that $r=a_0+a_1\xi+\dots+a_{n-1}\xi^{n-1}$ is invertible in $R$.</p>
<p>Hints:</p>
<ol>
<li>$a\xi^k$ is nilpotent, for every $a\in K$ and every positive integer $k$;... |
2,229,473 | <p>I am trying to solve the following question:</p>
<blockquote>
<p>If $K$ is a field and $R=K[x]/(x^n)$.and $r=a_0+a_1x +\dots+a_{n-1}x^{n-1}$ is an element in $R$ with $a_0\neq 0$, prove that $r$ is a unit and find its inverse.</p>
</blockquote>
<p>To prove it is a unit, I know that $\gcd(r,x^n)=1$, so there exis... | ancient mathematician | 414,424 | <p>Of course @egreg has given the best solution (and I gave it +1) by putting it all in a general context, but I can never resist the "idiot" answer to this question. It is clear from the Binomial Theorem that the inverse of $a_0+a_1x +\dots+a_{n-1}x^{n-1}+(x^n)$ is
$$
\sum_{k=0}^\infty (-1)^k a_0^{-k-1}(a_1x +\dots+a_... |
10,352 | <p>I teach high school geometry and see many of my students fall in to the trap of "it looks like it, so it's true" -- a <a href="https://en.wikipedia.org/wiki/Van_Hiele_model" rel="noreferrer">Van Hiele Level 0 to 1</a> thought process. For instance, when talking about parallel line angle relationships, once they real... | Jasper | 1,147 | <p>This model is really describing three different concepts.</p>
<p>1-2: What does it look like? This is "basic" shape recognition.</p>
<p>2-3: Practical geometry. How many sides does a shape have? What kinds of angles does the shape have? Can shapes be put next to each other without any gaps?</p>
<p>4-5: Pro... |
629,756 | <p>Consider the eigenvalue problem
\begin{equation}
\left\{ \begin{array}{l}
\Phi \in C^{2}(\mathbb{R}) \ \text{and bounded }\\
-\Phi^{''}(x)=\lambda\Phi(x), \ x\in \mathbb{R}.
\end{array} \right.
\end{equation}</p>
<p>and assume that the solution is given by $\Phi(x)=Ae^{i\sqrt{\lambda}x}+Be^{-i\sqrt{\lambda}x}$.<... | Community | -1 | <p>If it is not true that $\lambda \geq 0$, then $\operatorname{Im} \sqrt{\lambda} \neq 0$, i.e. $\sqrt{\lambda} = a + ib$ with $b \neq 0$. Thus
$$C\exp(i \sqrt{\lambda} x) = C \exp(iax)\exp(-bx)$$
is unbounded as $x \to \infty$ or as $x \to -\infty$ depending on the sign of $b$. Thus if a solution of the form you pro... |
3,490,416 | <p>Problem:</p>
<blockquote>
<p>From any fixed point in the axis of a general conic, a line is drawn perpendicular to the tangent at <span class="math-container">$P$</span> (on the conic) meeting <span class="math-container">$SP$</span> in <span class="math-container">$R$</span> where <span class="math-container">$S... | Intelligenti pauca | 255,730 | <p>I'll show the proof for an ellipse, the other cases are analogous.</p>
<p>In an ellipse, the bisector <span class="math-container">$PH$</span> of <span class="math-container">$\angle SPS'$</span> (where <span class="math-container">$S'$</span> is the second focus) is perpendicular to the tangent at <span class="mat... |
857,207 | <p>I need help to confirm my answer for the following question</p>
<p>"There is an alphabet of size 40 and this alphabet is used for forming messages in a communication system. If 10 of these alphabets can be used only as the first or last symbols in the message and the rest 30 can be used anywhere, how many messages ... | johannesvalks | 155,865 | <p>Only 10 for the first and the last and 30 for the innerpart thus</p>
<p>$$
10 + 10 \times 10 + 10 \times 30 \times 10 + 10 \times 30^2 \times 10 + 10 \times 30^3 \times 10 + \cdots + 10 \times 30^8 \times 10
$$</p>
<p>Thus</p>
<p>$$
10 + 100 \sum_{k=0}^{8} 30^k = 10 + 100 \frac{30^9 - 30}{30-1}.
$$</p>
<hr>
<p>... |
190,914 | <p>In <a href="https://math.stackexchange.com/questions/38350/n-lines-cannot-divide-a-plane-region-into-x-regions-finding-x-for-n">this</a> post it is mentioned that $n$ straight lines can divide the plane into a maximum number of $(n^{2}+n+2)/2$ different regions. </p>
<p>What happens if we use circles instead of lin... | hmakholm left over Monica | 14,366 | <p>A qualitative approach:</p>
<p>Consider a group of $n$ <em>lines</em> that divide the plane into maximally many regions. The intersections between the lines all fall within a bounded area of the plane. Now replaces each of the lines with a huge circle, so large that within the bounded area-of-intersections it devia... |
3,990,110 | <p>I was seeking the other day for intuition about the Hölder inequality and more precisely the geometric intuition behind the relation <span class="math-container">$\frac{1}{p} + \frac{1}{q} = 1$</span>.</p>
<p>I found <a href="http://www.tricki.org/article/Geometric_view_of_H%C3%B6lders_inequality" rel="nofollow nore... | epi163sqrt | 132,007 | <p>At first we look at the geometrical situation of the convex hull of three points in general position in <span class="math-container">$\mathbb{R^2}$</span>.</p>
<p><strong>Convex hull of three points with <span class="math-container">$(a,b)$</span> inside:</strong></p>
<p>In OPs referenced article we have a triangle ... |
3,990,110 | <p>I was seeking the other day for intuition about the Hölder inequality and more precisely the geometric intuition behind the relation <span class="math-container">$\frac{1}{p} + \frac{1}{q} = 1$</span>.</p>
<p>I found <a href="http://www.tricki.org/article/Geometric_view_of_H%C3%B6lders_inequality" rel="nofollow nore... | mlg4080 | 102,585 | <p>Focusing more on intuition, and landing on the geometry how integrals <em>scale</em>, let us check out a simple case.</p>
<p>Consider <span class="math-container">$\int_{0}^{r} 1 dx = r$</span>. Applying Holder's inequality we would get on the opposite side <span class="math-container">$\left( \int_{0}^{r} 1 dx \rig... |
3,672,378 | <p>Consider the following Primal LP </p>
<p><span class="math-container">$$
\min_{x_1,x_2,x_3} x_1+x_2+x_3 $$</span>
subject to constraints,
<span class="math-container">$$
x_1+x_2 \ge a \\
x_1+x_3 \ge b \\
x_2+x_3 \ge c\\
x_1 \ge d \\
x_2 \ge e \\
x_3 \ge f
$$</span></p>
<p>Here <span class="math-container">$$a,b,c... | Jean Marie | 305,862 | <p>Add the first 3 inequalities, then add the last 3 ones : you get</p>
<p><span class="math-container">$$x_1+x_2+x_3 \ge M \ \ \ \ \text{with} \ \ \ \ M:=\max((a+b+c)/2,d+e+f)$$</span></p>
<p>Therefore, taking all <span class="math-container">$x_k$</span> equal to <span class="math-container">$M$</span> gives a feas... |
1,083,301 | <p>I'm reading Knuth/Graham/Patashnik's Concrete Mathematics:</p>
<blockquote>
<p><img src="https://i.imgur.com/YpjYJwd.png" alt="enter image description here"></p>
</blockquote>
<p>I don't understand how he goes from $(r-k){r \choose r-k}$ to $r{r-1 \choose r-k-1}$ using $(5.6)$. The mentioned property has a $k$ i... | Red Banana | 25,805 | <ul>
<li>Using the symmetry:</li>
</ul>
<p>$\displaystyle(r-k){r\choose k}=(r-k){r\choose r-k}$</p>
<ul>
<li>Now using $(5.6)$:</li>
</ul>
<p>$\begin{eqnarray*}
{(r-k){r\choose r-k}}&=&{(r-k) \frac{r}{r-k}{n-1 \choose n-k-1}} \\
{}&&{}\\
{}&&{}\\
{}&=&{\frac{(r-k)r}{(r-k)} {n... |
3,408,320 | <p>Let <span class="math-container">$T : R^{3} \to R^{3}$</span> be the linear transformation defined by <span class="math-container">$T(x,y,z) = (x+3y+2z,3x+4y+z,2x+y-z)$</span>. Find the rank of <span class="math-container">$T^{2}$</span> and <span class="math-container">$T^{3}$</span>.</p>
<p>I formed the matrix of... | Graham Kemp | 135,106 | <p>Because the order of the groups is not important you must count <em>fewer</em> of the arrangements.</p>
|
1,783,841 | <p>If a function $f(x)$ has a derivative $f'(x)$ then where $f'(x_0) = 0$ there is an extreme point at $x=x_0$.
And where $f''(x_0)=0$ there is an inflection point at $x=x_0$.</p>
<p>I am asking are there any significance or applications where $f(x_0)=f'(x_0)$ at some point $x=x_0$?</p>
<p>I am referring to a case li... | J.G | 293,121 | <p>If I'm interpreting this correctly, I believe you are asking if there is anything special about this happening locally, i.e at a certain point $x_0$, does the fact that $f(x_0)=f'(x_0)$ mean anything? I would say no; note that by just translating the function appropriately, you can make this happen at any point. To ... |
1,254,981 | <p>The undergrad course is called intro the applied math, and it covers: "The unit introduces some of the principal mathematical techniques such as difference equations, differential equations and partial differential equations."</p>
<p>We are not given a textbook and I really want to study from one, the closer it mat... | Skipe | 234,395 | <p>You could try these books:
- An Introduction to Ordinary Differential Equations by James C Robinson
- Differential Equations by Richard Bronson and Gabriel Costa</p>
<p>These are the books I used to use and they are pretty good, and your course, from your description, looks a lot like my first year differential equ... |
1,254,981 | <p>The undergrad course is called intro the applied math, and it covers: "The unit introduces some of the principal mathematical techniques such as difference equations, differential equations and partial differential equations."</p>
<p>We are not given a textbook and I really want to study from one, the closer it mat... | Hans Lundmark | 1,242 | <p>It's hard to tell exactly how well it fits your course, but a really great book in general is
<a href="http://rads.stackoverflow.com/amzn/click/0738204536" rel="nofollow">Nonlinear Dynamics And Chaos</a> by Steven Strogatz. (I wholeheartedly agree with all the rave reviews on Amazon. Besides, <a href="http://www.wir... |
375,348 | <p>Prove that $\mathbb Z_{m}\times\mathbb Z_{n} \cong \mathbb Z_{mn}$ implies $\gcd(m,n)=1$.</p>
<p>This is the converse of the Chinese remainder theorem in abstract algebra. Any help would be appreciated.</p>
<p>Thanks!</p>
| André Nicolas | 6,312 | <p><strong>Hint:</strong> Every element of $\mathbb{Z}_m\times \mathbb{Z}_n$ has order $\le$ the lcm of $m$ and $n$.</p>
|
3,808,774 | <p>I have this limit I've been trying to solve, and so far I can only solve it using the property of geometric series.</p>
<p>So we get
<span class="math-container">$$\lim_{n \to \infty}\frac{1}{3^2}+\frac{1}{3^3}+...+\frac{1}{3^n}=\sum_{n=2}^{\infty}\left(\frac{1}{3}\right)^n=\frac{1}{1-\frac{1}{3}}=\frac{3}{2}$$</spa... | fleablood | 280,126 | <blockquote>
<p><span class="math-container">$\lim_{n \to \infty}\frac{1}{3^2}+\frac{1}{3^3}+...+\frac{1}{3^n}=\sum_{n=2}^{\infty}\left(\frac{1}{3}\right)^n=\frac{1}{1-\frac{1}{3}}=\frac{3}{2}$</span></p>
</blockquote>
<p>This is not correct as <span class="math-container">$\sum_{n=\color{red}0}^{\infty} (\frac 13)^n =... |
197,470 | <p>I'm new in Mathematica software, and I need help with a simple exercise. </p>
<p>Given the list</p>
<pre><code>list = {x, 3.2, y, 1, 2.679, 3, z, 4, -7.9};
</code></pre>
<p>I need to make some rules to generate a new list with only three elements.</p>
<ul>
<li>first element is the product of its symbols </li>
<l... | kglr | 125 | <p>You can use <a href="https://reference.wolfram.com/language/ref/GroupBy.html" rel="noreferrer"><code>GroupBy</code></a> in several ways:</p>
<pre><code>GroupBy[list, Head, Apply[Times]] /@ {Symbol, Integer, Real}
</code></pre>
<blockquote>
<p>{x y z, 12, -67.7251} </p>
</blockquote>
<p>or using a combination of... |
2,330,368 | <p>I have calculate the sum of the following power series: $$\sum_{n=0}^{\infty} (-1)^n(n+1)^2x^n$$</p>
<p>I've determined the convergence radius $R = 1$convergence interval $I_c = (-1,1)$ and the convergence set $A = (-1,1)$ for this power series. Furthermore, I've tired to bring it to a from which can be easily comp... | Claude Leibovici | 82,404 | <p>First, let $x=-y$ which makes $$S=\sum_{n=0}^{\infty} (-1)^n(n+1)^2x^n=\sum_{n=0}^{\infty} (n+1)^2y^n=\sum_{n=0}^{\infty} (n^2+2n+1)\,y^n$$ Now, use $n^2=n(n-1)+n$ which makes $$S=\sum_{n=0}^{\infty} (n(n-1)+3n+1)\,y^n$$ that is to say $$S=\sum_{n=0}^{\infty} n(n-1)\,y^n+3\sum_{n=0}^{\infty} n\,y^n+\sum_{n=0}^{\inft... |
505,804 | <p>Let $X$ be a metric space and let $G$ be a group of homeomorphisms $X \to X$ acting on $X$. We say $G$'s action is properly discontinuous in case for every $x \in X$ and compact $K \subseteq X$, there are at most finitely many $g \in G$ such that $g(x) \in K$. Equivalently (and this is not hard to show), $G \cdot x$... | lessthanepsilon | 129,626 | <p>see the errata posted here: <a href="http://www.personal.psu.edu/sxk37/errata.pdf" rel="noreferrer">http://www.personal.psu.edu/sxk37/errata.pdf</a></p>
<p>where katok writes:</p>
<p>p.27 l.9: Replace “metric” by “locally compact metric”</p>
<p>p.27 l.10: Replace “homeomorphisms” by “isometries”.</p>
<p>p.27 l.-... |
1,642,247 | <p>How to prove that any regular map $\phi : \Bbb P^1 \to \Bbb A^n(\Bbb C)$ maps $\Bbb P^1$ to point. Now $\phi=(F_1/G_1,...,F_n/G_n)$ where $F_i/G_i$ is a regular function.
Now how do I conclude? </p>
| Marra | 27,071 | <p>If $\phi$ is regular, by continuity its image is a limited set in $\mathbb{A}^n(\mathbb{C})$. Using any chart of $\mathbb{P}^1$, you can obtain a holomorphic mapping from $\mathbb{C}^1$ to $\mathbb{A}^n(\mathbb{C})$ which is globally holomorphic and limited. Therefore it is constant, that is, its image is only one p... |
1,642,247 | <p>How to prove that any regular map $\phi : \Bbb P^1 \to \Bbb A^n(\Bbb C)$ maps $\Bbb P^1$ to point. Now $\phi=(F_1/G_1,...,F_n/G_n)$ where $F_i/G_i$ is a regular function.
Now how do I conclude? </p>
| basket | 294,706 | <p>Let $k$ be algebraically closed. A morphism $f: \mathbb{P}^r \to \mathbb{A}^n$ induces a $k$-homomorphism $\varphi: \mathcal{O}(\mathbb{A}^n) = k[x_1,\dots,x_n] \to \mathcal{O}(\mathbb{P}^r) = k$. As a $k$-homomorphism, $\varphi$ is surjective and so it must be the quotient map by a maximal ideal $\mathfrak{m}_P$. T... |
1,769,297 | <p>I have been reading <em>Representations and Characters of Groups</em> by Gordon James and Martin Liebeck. I encountered the following construction of an $\mathbb{R}G$-module from a $\mathbb{C}G$-module.</p>
<p><a href="https://i.stack.imgur.com/gXuGt.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.co... | guest | 187,930 | <p>First of all, to make things more comfortable for you I will do all calculations on $\mathbb{C}G$-modules using the following observation: if $V_0$ is an $\mathbb{R}G$-module of real dimension $n$ with character $\chi$, the complexification $\mathbb{C}\otimes_\mathbb{R} V_0=V$ is a $\mathbb{C}G$-module ($G$ acting o... |
1,938,341 | <p>Prove, that $\forall n \in \mathbb{N}$ the following identity holds:
$$a^n - b^n = (a-b)\sum_{k=0}^{n-1} a^k b^{n-k-1}.$$</p>
<hr>
<p>For $n = 1$ we get $a^1 - b^1 = a - b$ on LHS, and on RHS we get $(a-b)\sum_{k=0}^{0}a^k b^{n-k-1} = a-b.$ Both sides are equal, so identity hold for $n = 1$.</p>
<hr>
<p>Assume t... | B. Goddard | 362,009 | <p>You can re-index the first sum in your last expression. If you decrease $k$ by one in the summand, you can pay for it by increasing the limits. So you'd get
$$\sum_{k=1}^{m+1} a^kb^{m-k} - \sum_{k=0}^{m} a^kb^{m-k}.$$</p>
<p>Now if you separate the last term of the first sum and the first term of the
second sum y... |
2,014,156 | <p><a href="http://www.wolframalpha.com/input/?i=lim%20(sum%201%2F(a%2Bn)%5E2,%20n%20%3D%201%20to%20infinity),%20a%20to%20infinity" rel="nofollow noreferrer">WolframAlpha</a> says that the sequence
$$
S_n = \sum\limits_{i=1}^{\infty}\frac{1}{(n+i)^2}
$$
tends to $0$. The proof given by the site uses notations unfamilia... | marty cohen | 13,079 | <p>The proof can be rewritten
to make it simpler and clearer.</p>
<p>$\begin{array}\\
\sum\limits_{i=1}^N\frac{1}{(n+i)^2}
&= \sum\limits_{i = n+1}^{n+N}\frac{1}{i^2}\\
&< \sum\limits_{i = n+1}^{n+N}\frac{1}{i(i-1)}
\qquad\text{since } i(i-1) < i^2\\
&= \sum\limits_{i = n+1}^{n+N}\left( \frac1{i-1}-... |
2,108,245 | <p>While studying Fourier analysis last semester, I saw an interesting identity:</p>
<p>$$\sum_{n=1}^{\infty}\frac{1}{n^2-\alpha^2}=\frac{1}{2\alpha^2}-\frac{\pi}{2\alpha\tan\pi\alpha}$$
whenever $\alpha \in \mathbb{C}\setminus \mathbb{Z}$, which I learned two proofs using Fourier series and residue calculus.</p>
<p>... | Renascence_5. | 286,825 | <p>For the partial fraction decomposition of the cotangent
$$\pi \cot \pi z = \frac{1}{z} + \sum_{n\ge 1}\left( \frac{1}{z-n} + \frac{1}{z+n}\right) = \lim_{k\to\infty} \sum_{n=-k}^k \frac{1}{z-n}$$
hence
\begin{align*}
\sum_{n=1}^\infty \frac{1}{n^2-\alpha ^2}
&= \lim_{k\to\infty}\sum_{n=1}^k \frac{1}{n^2-\alpha^2... |
2,500,723 | <blockquote>
<p>A price $p$ (in dollars) and demand $x$ for a product are related by $$\left(2x^2\right)-2xp+50p^2 = 20600$$ If the price is increasing at a rate of $2$ dollars per month when the price is $20$ dollars, find the rate of change of the demand.</p>
</blockquote>
<p>I was a little confused on how to proc... | John Alexiou | 3,301 | <p>Blindly follow the rule that $\boxed{{\rm d}f = \frac{\partial f}{\partial x} {\rm d}x + \frac{\partial f}{\partial p} {\rm d}p}$.</p>
<p>In this case:</p>
<p>$$ 2x^2-2xp+50p^2 = 20600 $$</p>
<p>$$ {\rm d} \left( 2x^2-2xp+50p^2 = 20600 \right) $$</p>
<p>$$ \frac{\partial}{\partial x} \left( 2x^2-2xp+50p^2 = 20... |
3,735,388 | <blockquote>
<p>They ask me to solve
<span class="math-container">$$y' +2y + \int_{0}^{x} y(t)dt = f(x)$$</span> with <span class="math-container">$y(0)=0$</span> and
<span class="math-container">$$
f(x) =
\begin{cases}
0, & x < 5 \\
2, & x \geq 5
\end{cases}
$$</span></p>
</blockquote>
<p>I don't know how ... | Sameer Baheti | 567,070 | <blockquote>
<p>Solve the following differential equation:
<span class="math-container">$$y^\prime +2y + \int_{0}^{x} y(t)dt = f(x)\tag{1}$$</span>
where <span class="math-container">$y(0)=0$</span>, and
<span class="math-container">$$f(x)= \begin{cases} 0 & x < 5 \\ 2 & x \geq 5 \end{cases}$$</span></p>
</b... |
1,029,515 | <p>Show that $\gcd(x,x+2)$ is $1$ if $x$ is odd and $2$ if $x$ is even.</p>
<p>I am looking for a much simpler proof beside the one which I have posted.</p>
| lab bhattacharjee | 33,337 | <p>If integer $d$ divides $x,x+2;d$ must divide $(x+2-x)=2$</p>
<p>Observe that $x,x+2$ have same parity as $x+2-x=2$ which is even </p>
<p>Now if $x$ is even $\iff x+2$ is even, then clearly $2$ divides both, hence $(2a,2a+2)=2$</p>
<p>If $x$ is odd, $2$ can not divide $x\implies (2b+1,2b-1)=1$</p>
|
1,627,830 | <p>I have to calcuate this limit: $$\lim _{x\to 0}\left(\frac{1-\frac{\left(1+x\right)^{\frac{1}{x}}}{e}}{x}\right)$$</p>
<p>I have no idea how to start, maybe taylor series?</p>
<p>Thanks.</p>
| Emilio Novati | 187,568 | <p>Hint:</p>
<p>Substitute $y=\frac{1}{x}$, so your limit becomes
$$
\lim_{y \to \infty}y\left(1-\frac{\left(1+\frac{1}{y} \right)^y}{e} \right)
$$
now use the series expansion at $\infty$
$$
\left(1+\frac{1}{y}\right)^y=e-\frac{e}{2y} + O \left(\frac{1}{y^2}\right)
$$
and you have the result .</p>
|
60,486 | <p>I have noted several instances <a href="https://mathematica.stackexchange.com/a/51377/7167">here</a> and <a href="https://mathematica.stackexchange.com/a/55412/7167">here</a> where <code>ConvexHullMesh</code>, introduced in v10 has been used to greatly simplify some geometry problems. Determining the volume of a re... | RunnyKine | 5,709 | <p>For fun, here is an approach that uses the graphics generated from the mesh. Lets generate some points:</p>
<pre><code>SeedRandom[0]
pts = RandomReal[5, {50, 3}];
chull = ConvexHullMesh @ pts;
</code></pre>
<p>Create a graphics object from it and discretize it:</p>
<pre><code>gr = Graphics3D[GraphicsComplex[MeshC... |
404,372 | <p>There are $7$ empty seats on a bus and four people get on. How many different ways can they be seated?</p>
<p>Would it be ${}^7C_{4}$ or ${}^7P_{4}$?</p>
| Michiel Van Couwenberghe | 79,783 | <p>I think the answer is $$ 4! C_7^4 = 7 * 6 * 5 *4 $$ Where the C is a notation for the combination of 4 out of 7. Explanation: first you pick 4 out of 7 seats which will be taken and then you determine the person for each seat.</p>
|
127,645 | <p>In Lay's Linear Algebra and Its Applications textbook, he defines the matrix
$$A=\begin{bmatrix}
4 & 11 & 14\\8 & 7 & -2
\end{bmatrix}$$
and claims that the transformation $T(x)=A x$ maps the sphere to an ellipse.</p>
<p><a href="https://i.stack.imgur.com/grGkv.png" rel="nofollow noreferrer"><img sr... | corey979 | 22,013 | <p>Generating equidistributed points on a sphere <a href="https://mathematica.stackexchange.com/a/119341/22013">according to this post</a>:</p>
<pre><code>pts = With[{points = 200, samples = 40000, iterations = 20},
Nest[With[{randoms = Join[#, RandomPoint[Sphere[], samples]]},
Normalize @ Mean @ randoms[[#... |
127,645 | <p>In Lay's Linear Algebra and Its Applications textbook, he defines the matrix
$$A=\begin{bmatrix}
4 & 11 & 14\\8 & 7 & -2
\end{bmatrix}$$
and claims that the transformation $T(x)=A x$ maps the sphere to an ellipse.</p>
<p><a href="https://i.stack.imgur.com/grGkv.png" rel="nofollow noreferrer"><img sr... | yarchik | 9,469 | <pre><code>x = Cos[ϕ] Sin[θ];
y = Sin[ϕ] Sin[θ];
z = Cos[θ];
A = {{4, 11, 14}, {8, 7, -2}};
ParametricPlot[
A.{x, y, z}, {θ, 0, Pi}, {ϕ, 0, 2 Pi},
MeshStyle -> {Red, Blue}, ColorFunction -> "MintColors"]
</code></pre>
<p><a href="https://i.stack.imgur.com/JCOSp.png" rel="noreferrer"><img src="https://i.stack.... |
2,055,695 | <p>Use mathematical induction to prove each of the following statements.
let <span class="math-container">$$g(n) = 1^3 + 2^3 + 3^3 + ... + n^3$$</span></p>
<p>Show that the function <span class="math-container">$$g(n)= \frac{n^2(n+1)^2}{4}$$</span> for all n in <strong>N</strong></p>
<p>so the base case is just g(1) ... | K Split X | 381,431 | <p>Base Case:</p>
<p>Show that $g(n)$ holds for $n=1$.</p>
<p>(You've done this already).</p>
<hr>
<p>Inductive Hypothesis:</p>
<p>Assume that $g(n)$ holds for arbitrary $k \in \mathbb{N}$, where $k \geq 1$ (since you already proved base case).</p>
<p>Thus we have that:</p>
<p>$$g(k) = \frac{k^2(k+1)^2}{4}$$</p>... |
2,055,695 | <p>Use mathematical induction to prove each of the following statements.
let <span class="math-container">$$g(n) = 1^3 + 2^3 + 3^3 + ... + n^3$$</span></p>
<p>Show that the function <span class="math-container">$$g(n)= \frac{n^2(n+1)^2}{4}$$</span> for all n in <strong>N</strong></p>
<p>so the base case is just g(1) ... | prokaryoticeukaryote | 217,977 | <p>Assuming you are asking what the inductive step is, you simply need to assume that $g(n) =\frac{n^2(n+1)^2}{4}$ and then use it to evaluate g(n+1) like so:
\begin{equation}
g(n+1) = g(n) + (n+1)^3
\end{equation}
\begin{equation}
g(n+1) = \frac{n^2(n+1)^2}{4} + (n+1)^3
\end{equation}
\begin{equation}
g(n+1) = \frac{4... |
31,113 | <p>Zagier has a very short proof (<a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1041893" rel="noreferrer">MR1041893</a>, <a href="https://www.jstor.org/stable/2323918" rel="noreferrer">JSTOR</a>) for the fact that every prime number $p$ of the form $4k+1$ is the sum of two squares.
The proof defines an inv... | yoyo | 129,839 | <p>[This is more of a comment to Franz Lemmermeyer's answer.]</p>
<p>Let <span class="math-container">$F(x,y)=ax^2+bxy+cy^2=[a,b,c]$</span> be an integral (<span class="math-container">$a,b,c\in\mathbb{Z}$</span>) binary quadratic form of discriminant <span class="math-container">$\Delta=b^2-4ac$</span>, and let <span ... |
31,113 | <p>Zagier has a very short proof (<a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1041893" rel="noreferrer">MR1041893</a>, <a href="https://www.jstor.org/stable/2323918" rel="noreferrer">JSTOR</a>) for the fact that every prime number $p$ of the form $4k+1$ is the sum of two squares.
The proof defines an inv... | David Leep | 476,997 | <p>A recent article gives another very elegant proof of the two-square theorem in the Zagier style, but is easier. See</p>
<ul>
<li>Stan Dolan, <em>A very simple proof of the two-squares theorem</em>,
The Mathematical Gazette, Volume 105, Issue 564 (2021) p. 511. doi:<a href="https://doi.org/10.1017/mag.2021.120" rel=... |
2,907,487 | <p>Trying to better understand how an "interpretation" works in logic: <a href="https://en.wikipedia.org/wiki/Interpretation_(logic)" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Interpretation_(logic)</a></p>
<p>For the most part it seems to be the idea that we have some syntactical symbol and we can assig... | Mauro ALLEGRANZA | 108,274 | <p>We may follow Wiki's entry carefully [<em>emphasis added</em>]:</p>
<blockquote>
<p>An interpretation is an assignment of <em>meaning</em> to the symbols of a formal language. Many formal languages used in mathematics, logic, and theoretical computer science are defined in solely syntactic terms, and as such do not ... |
4,295,884 | <p>The challenge is to solve this equation <span class="math-container">$2^{x}+7^{y}=9^{z}$</span> in positive integers. The obvious solution is <span class="math-container">$x=y=z=1$</span>. Using brute force, I found <span class="math-container">$3$</span> possible solutions:
<span class="math-container">\begin{eqnar... | Mike | 544,150 | <p>This is a PARTIAL answer. What remains to show is that there are not an infinite number of solutions for <span class="math-container">$x=1$</span>. Assume <span class="math-container">$x\ge 3$</span>. Then as in Conner's answer,
<span class="math-container">$$2^x = (3^z-7^v)(3^z+7^v).$$</span></p>
<p>This gives for ... |
275,558 | <p>I recently had a homework problem asking to prove that the direct product of rings (or rings with identity) are still rings (with identity), and it seemed really silly to go through all the steps in order to conclude what was extremely trivial. </p>
<p>My question is, are there instances when this is not the case? ... | Dietrich Burde | 83,966 | <p>Some properties <em>by definition</em> do not necessarily "maintain the original algebraic structure" for the direct product, such as, say, being "irreducible".
Perhaps the question gets more focus if we stay with groups (or rings, modules and algebras). <br>
The direct product of simple groups need not be simple, ... |
935,358 | <p>What is an example of a six-element group that is not abelian?</p>
<p>I can't think of any. It is very possible that I am overthinking this. Thank you for any help.</p>
| N. F. Taussig | 173,070 | <p>The symmetric group, $S_3$, on the vertices of an equilateral triangle is non-abelian. It is generated by a 120 degree counterclockwise rotation and a reflection over the perpendicular bisector of one of the sides. As you can check, it is a non-abelian group with six elements since the reflection and rotation do n... |
155,143 | <p>Let $X$ be a Banach space an let $(e_n)_{n=1}^{\infty}$ be a Schauder basis of $X$. If we denote the sequence of the natural projections associated with $(e_n)_{n=1}^{\infty}$ by $(S_n)_{n=1}^{\infty}$ then by Uniform Boundedness principle we can observe that $$\sup_n||S_n||< \infty. $$</p>
<p>And the number $K=... | alpha | 45,681 | <p>I know one situation where the basis constant plays a role. This is the Paley-Wiener theorem on the perturbation of bases (nota bene, not the more famous one on the characterisation of the Fourier transform of $L^2$ functions). It is a quantitative version of the intuitively obvious result that the isomorphic cha... |
2,998,479 | <p>Solving the integration problem:</p>
<p><span class="math-container">$$\int \frac{4^x-5^x}{7^x} \, dx$$</span></p>
<p>Not sure how to solve this problem, I don't see a way to incorporate integration by parts into this problem or even some kind of substitution.</p>
<p>Seems that taking the natural log of each elem... | Tianlalu | 394,456 | <p><strong>HINT:</strong>
<span class="math-container">$$\frac{4^x-5^x}{7^x}=\left(\frac47\right)^x-\left(\frac57\right)^x,$$</span>
and you have the formula (by putting <span class="math-container">$a=4/7$</span> and <span class="math-container">$5/7$</span>)
<span class="math-container">$$\int a^x~\mathrm dx=?$$</sp... |
834,451 | <p>Simple question from a calc 3 beginner. Visually I cannot imagine the span of two vectors, what does this necessarily mean? For example my text mentions if two vectors are parallel their span is a line, otherwise a plane. Can anyone elaborate?</p>
| Hayden | 27,496 | <p>Assuming it makes sense that the span of a single vector is a line, we can imagine the two vectors in 3-space. Because the span of each vector lies within the space of each of them, we can draw the two lines that are in the direction of these two vectors:</p>
<ul>
<li><p>if the two lines are equal, then this is al... |
834,451 | <p>Simple question from a calc 3 beginner. Visually I cannot imagine the span of two vectors, what does this necessarily mean? For example my text mentions if two vectors are parallel their span is a line, otherwise a plane. Can anyone elaborate?</p>
| Tobias Kildetoft | 2,538 | <p>Here is a visual way of thinking about this (at least in small enough dimension that you can visualize it).</p>
<p>Say you have some collection of vectors and you place yourself at the origin. Each vector determines a direction (the span does not care if you replace a vector by some non-zero scalar multiple), and t... |
2,410,118 | <p>Edit: Excuse me, the numerator of the given solution should've been a rising factorial. That said, I still don't understand where it comes from?</p>
<p>The question is as follows :</p>
<blockquote>
<p>n friends visit k exhibitions, each person visits only 1 exhibition. Find the number of possibilities if</p>
<p>b) O... | Hagen von Eitzen | 39,174 | <p>We can try something of the form
$$a_n=\alpha n+\beta (-1)^nn+\gamma.$$
From $$\begin{align}\alpha(2n+1)+2\gamma&=a_n+a_{n+1}\\&=(k+1, 2k+1,3k+1,4k+1, 5k+1,\ldots )\\&=nk+1,\end{align}$$
we find $\alpha=\frac k2$, $\gamma=\frac12-\frac k4$. From this we infer $\beta=\frac k4-\frac12$, and - hooray! - tha... |
2,322,938 | <p>The powerset of the empty set is {$\emptyset$}. Attempt:</p>
<p>The power set of the powerset of the empty set is { { {$\emptyset$} } }? </p>
<p>Next I want to find the powerset of the power set of the powerset of the empty set P(P(P($\emptyset$))). </p>
| Akiva Weinberger | 166,353 | <p>Here's a visualization of $\emptyset$ (called $V_0$ here), $\mathcal P(\emptyset)$ (called $V_1$ here), $\mathcal P(\mathcal P(\emptyset))$ (called $V_2$ here), etc. An empty box represents the empty set. A box containing only an empty box represents the set containing only the empty set, and so forth.</p>
<p><a hr... |
1,702,139 | <p>I am reading through the <a href="http://www.computervisionmodels.com/" rel="nofollow noreferrer">Computer Vision: Models, Learning, and Inference</a> book to get an understanding of computer vision. The author describes the high-level steps taken to arrive at one of the derivations, but I'm stuck in working through... | Robert Israel | 8,508 | <p>Hint: what kind of singularity can such a map have at $0$? </p>
|
1,702,139 | <p>I am reading through the <a href="http://www.computervisionmodels.com/" rel="nofollow noreferrer">Computer Vision: Models, Learning, and Inference</a> book to get an understanding of computer vision. The author describes the high-level steps taken to arrive at one of the derivations, but I'm stuck in working through... | Yiorgos S. Smyrlis | 57,021 | <p><strong>Answer.</strong> $\,f(z)=az\,$ or $\,f(z)=a/z$, for some $a\ne 0$.</p>
<p>Hint. At $z=0$ the function $f$ has an isolated singularity. It can only be a removable one or a simple pole. If it is removable, $f$ should vanish there.</p>
|
973,771 | <p>I am trying to find </p>
<p>$$ \lim_{x\to 0} \frac{e^{\alpha x} -e^{\beta x}}{\sin(\alpha x) + \sin(\beta x) }$$</p>
<p>but dont have a clue where to start, could someone give me a hint please ?</p>
| user184394 | 184,394 | <p>Using Wolfram Alpha:</p>
<p>lim (e^(alpha x)-e^(beta x))/(sin(alpha x)+sin(\beta x)) as x->0</p>
<p>Answer:</p>
<p>(alpha-beta)/(alpha+beta)</p>
<p><a href="http://www.wolframalpha.com/input/?i=lim+%28e%5E%28alpha+x%29-e%5E%28beta+x%29%29%2F%28sin%28alpha+x%29%2Bsin%28%5Cbeta+x%29%29+as+x-%3E0" rel="nofollow">ht... |
2,046,729 | <p>In my book there is a solved question . <a href="https://i.stack.imgur.com/O4LpL.gif" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/O4LpL.gif" alt="enter image description here" /></a></p>
<p>In that question I could not understand from where does <span class="math-container">$(3\lambda + 2)$</span> ... | Jorge Medina | 279,252 | <p>Your objective here is to figure out where the terms of both sequences are the same. In general you do this by expressing one variable (n) in terms of the other (m) and then expressing both in terms of a third variable (λ). This way you can see which terms in Tn are the same as which terms in Tm, just by plugging in... |
219,793 | <p>I have the following function </p>
<p><code>f[z_] := Piecewise[{{Cos[z] - 17 Sin[z]/z, z > 0}, {Cosh[z] - 17 Sinh[z]/z, z <= 0}}]</code></p>
<p>I am interested in the regions where <span class="math-container">$\vert f(z)\vert \leq 1$</span>, I tried to shade that region using <code>Filling</code>, but my re... | kglr | 125 | <p>Assuming (1) possible codes are triples with distinct elements and (2) "in the wrong position" means <em>not necessarily in the correct position</em>:</p>
<pre><code>ClearAll[containsOneCP, containsOne, containsTwo, containsNone]
containsOneCP[lst_] := Module[{abc = MapThread[Equal, {#, lst}]},
BooleanCoun... |
3,322,760 | <p>I am trying to solve the non linear ODE</p>
<p><span class="math-container">$$\displaystyle \frac{d^2y} {dx^2}=a^2(y+y^3)$$</span></p>
<p>With the boundary conditions that it vanish at <span class="math-container">$\pm\infty$</span>.
I am thinking that it might be better to deal with it an integral equation</p>
<... | Kavi Rama Murthy | 142,385 | <p>In (A) we cannot write a rational number <span class="math-container">$x$</span> as <span class="math-container">$y+z$</span> with <span class="math-container">$x$</span> rational and <span class="math-container">$y$</span> irrational because <span class="math-container">$z=x-y$</span> would then be rational. In fac... |
57,261 | <p>In Wellin's <a href="http://rads.stackoverflow.com/amzn/click/1107009464" rel="nofollow noreferrer">Programming with Mathematica</a> book, here's one of his implementations of the Newton method, where the iteration runs until the error tolerance is reached.</p>
<p><img src="https://i.stack.imgur.com/EnGoj.png" alt=... | eldo | 14,254 | <p>"Is this a better way?" </p>
<p>Yes, but I think there are alternatives:</p>
<p><a href="http://reference.wolfram.com/language/ref/Nest.html" rel="nofollow noreferrer">Nest</a></p>
<pre><code>newton1[fun_, xi_, n_] := With[{f = fun/D[fun, x]}, Nest[# - f /. x -> # &, 2., n]]
newton1[x^3 - 2, 2., 10]
</cod... |
43,043 | <p>I'm working through Carl Bender's <a href="http://youtu.be/LYNOGk3ZjFM">Mathematical Physics lectures</a> on YouTube (which are great fun), and I'd like <em>Mathematica</em>'s help solving terms in the perturbation series. It would be convenient if expressions like</p>
<pre><code>SeriesCoefficient[Sum[a[n] b^n, {n,... | Michael E2 | 4,999 | <p>How funny - I was just doing this the other day (different equation, with a series in <code>x^(1/2)</code>). Here is what I came up with (actually, I'm following Isaac Newton's method, I believe):</p>
<pre><code>Clear[sc];
(* series coefficient of x in f[b, x] == 0 *)
sc[f_, n_] := Nest[
Append[#, a /. First @ S... |
2,164,338 | <p>I am stuck up with this simple problem related to averages. Please help me out in explaining the complete solution to this problem.</p>
<blockquote>
<p>Consider a class of <span class="math-container">$40$</span> students whose average weight is <span class="math-container">$40\ \mathrm{kg}$</span>. <span class="... | just_curious | 420,614 | <p>Hint: Find the new total weight (which you claim you can find), divide by $40+m$, find the $m$ maximizing the expression!</p>
|
2,164,338 | <p>I am stuck up with this simple problem related to averages. Please help me out in explaining the complete solution to this problem.</p>
<blockquote>
<p>Consider a class of <span class="math-container">$40$</span> students whose average weight is <span class="math-container">$40\ \mathrm{kg}$</span>. <span class="... | Juned Hashmi | 489,322 | <p>My simple approach to the question:</p>
<p>Every new joinee should contribute to maximize average weight.</p>
<p>Therefore contribution if $1$ student joins $= 1 \times 9 = 9$ kg</p>
<p>And this will be maximum when both the variable is equal i.e., $5 \times 5 = 25$</p>
<p>Therefore maximum average weight $= 40 ... |
871,221 | <blockquote>
<p>Let $F$ be a field. Show that the field of quotients of $F$ is ring isomorphic to $F$.</p>
</blockquote>
<p><strong>Attempt:</strong> Let $F'$ be the field of Quotients of the field $F$. </p>
<p>Let $\Phi:F \rightarrow F'$ such that $\Phi(x)=x/1$ </p>
<p>$x/1$ refers to the equivalence class contai... | symplectomorphic | 23,611 | <p>You're confusing two things: the set of solutions to the normal equations, and the set of vectors over which you are searching for the least squared error. The second set is just the set of all vectors $x$ in whatever vector space you're working on. The squared error varies as you choose different vectors from the s... |
415,743 | <p><strong>Given Problem</strong> is to solve this separable differential equation:</p>
<p>$$y^{\prime}=\frac{y}{4x-x^2}.$$</p>
<p><strong>My approach</strong>: was to build the integral of y':</p>
<p>$$\int y^{\prime} = \int \frac{y}{4x-x^2}dy = \frac{y^2}{2(4x-x^2)}.$$</p>
<p>But now i am stuck in differential eq... | Alex Wertheim | 73,817 | <p>You seem to be slightly confused - where did the $dy$ come from? Why did you make that choice? You should apply separation of variables to solve this problem. </p>
<p>That is, given </p>
<p>$$ y' = \frac{y}{4x-x^{2}}$$</p>
<p>Write this as: </p>
<p>$$ \frac{dy}{dx} = \frac{y}{4x-x^{2}}$$</p>
<p>Separating the v... |
3,051,452 | <p>How can I prove that <span class="math-container">$\mathbb Q \times \mathbb Q$</span> is connected? I was trying to prove that it isn't because <span class="math-container">$\mathbb Q$</span> is not connected but I can't find two open <span class="math-container">$A,B$</span> to prove it.</p>
| José Carlos Santos | 446,262 | <p>You know that <span class="math-container">$\mathbb Q$</span> is not connected. That means that there is a set <span class="math-container">$A\subset\mathbb Q$</span> such that <span class="math-container">$A$</span> is open and closed in <span class="math-container">$\mathbb Q$</span> and that furthermore <span cla... |
30,339 | <p>I need to construct a vector similar to:</p>
<pre><code>v[x_]:={0, 0, x, 0, 0, 2*x, 0, 0, 3*x, ....., 0, 0, n*x};
</code></pre>
<p>where <code>n=10^9</code>. How can I make such a vector wisely?</p>
| Pankaj Sejwal | 1,561 | <p>Just one more way,but is not packed array,</p>
<pre><code>Table[{0, 0, i x} /. List -> Sequence, {i, 1, 4}]
</code></pre>
<blockquote>
<p>{0, 0, x, 0, 0, 2 x, 0, 0, 3 x, 0, 0, 4 x}</p>
</blockquote>
|
1,763,377 | <p>Let $T:V \to V$ be a linear operator such that $T^5 \neq 0,$ but $T^6 = 0.$ Suppose $V$ is isomorphic to $\mathbb{R}^6.$ Prove that there does not exist an $S:V \to V$ such that $S^2 = T.$ Does the answer change if $V$ is isomorphic to $\mathbb{R}^{12}?$</p>
<p>My thoughts for the first part: Suppose such an $S$ ... | Bernard | 202,857 | <p><strong>Hint:</strong></p>
<p>This is a consequence of <em>Hamilton-Cayley's theorem</em>.</p>
|
240,894 | <p>I am importing an excel sheet into Mathematica. The result is a rectangular array. The array contains sequences of commas. Whenever a cell is empty in the Excel sheet consecutive commas appear in the imported array. How to get rid of them? So I am looking for a function <span class="math-container">$f$</span> such t... | A little mouse on the pampas | 42,417 | <pre><code>DeleteCases[{1, 2, 3, 2, 1, , , , 6, 5, 4, 3, , , , 8}, Null]
</code></pre>
|
368,745 | <p>Here $\mu$ is a probability measure. Another similar question is: is it a $\sigma$-compact space? Thank you in advance!</p>
| naaah | 73,716 | <p>This very rarely happens.</p>
<p>A real or complex topological vector space is locally compact if and only it is finite-dimensional. A Banach space is $\sigma$-compact if and only if it is finite-dimensional. For $L^\infty(\mu)$ of a probability measure this happens if and only if $\mu$ has finitely many atoms and ... |
1,463,152 | <p>Let $$f(x)= \begin{cases}
e^{-1/x} & x>0 \\
0 & x\le 0
\end{cases}
$$
I want to show that $f^{(n)}(0)=0$ for every $n\in\mathbb{N}$.</p>
<p>My idea is to do this by induction on $n$. I can compute the first derivative by taking the limit of $$\frac{f(x)-f(0)}{x-0}=\frac{e^{-1/x}}{x}$$ as ... | vonbrand | 43,946 | <p>Use generating functions. To simplify later work, use the recurrence backwards to get <span class="math-container">$a_0 = -5/9$</span>. Define <span class="math-container">$A(z) = \sum_{n \ge 0} a_n z^n$</span>, shift the recurrence by 2, multiply by <span class="math-container">$z^n$</span> and sum over <span class... |
1,358,140 | <p>I haven't been able to solve this limit, so help would be appreciated.
L'Hopital is not allowed.
Thanks in advance.</p>
| lab bhattacharjee | 33,337 | <p>HINT:</p>
<p>Set $x-1=h$</p>
<p>$$\lim_{h\to0}\dfrac{(h+1)e^{(1+h)^2}-e}h=\lim_{h\to0}e^{(1+h)^2}+e\cdot\lim_{h\to0}\dfrac{e^{2h+h^2}-1}h$$</p>
<p>Now $$\lim_{h\to0}\dfrac{e^{2h+h^2}-1}{2h+h^2}\cdot\lim_{h\to0}\dfrac{2h+h^2}h$$</p>
|
1,358,140 | <p>I haven't been able to solve this limit, so help would be appreciated.
L'Hopital is not allowed.
Thanks in advance.</p>
| Mark Viola | 218,419 | <p>The best way to proceed has already been presented. I thought that it might be instructive to see another approach. To that end, we write</p>
<p>$$e^{x^2}=e+e(x^2-1)+O(x^2-1)^2$$</p>
<p>so that</p>
<p>$$\begin{align}
\frac{xe^{x^2}-e}{x-1}&=\frac{e(x-1)+e(x^2-1)^2+O(x^2-1)^2}{x-1}\\\\
&=3e+O(x-1)\\\\
&a... |
1,358,140 | <p>I haven't been able to solve this limit, so help would be appreciated.
L'Hopital is not allowed.
Thanks in advance.</p>
| Paramanand Singh | 72,031 | <p>\begin{align}
L &= \lim_{x \to 1}\frac{xe^{x^{2}} - e}{x - 1}\notag\\
&= \lim_{x \to 1}\frac{xe^{x^{2}} - e}{x^{2} - 1}\cdot\frac{x^{2} - 1}{x - 1}\notag\\
&= \lim_{x \to 1}\frac{xe^{x^{2}} - e}{x^{2} - 1}\cdot(x + 1)\notag\\
&= 2\lim_{x \to 1}\frac{xe^{x^{2}} - xe + xe - e}{x^{2} - 1}\notag\\
&=... |
167,463 | <p>How many permutations of the letters of the word "DEFEATED" are there in which the E's are separated from each other?</p>
<p>I relied on the fact that: (The number of permutations of the letters in this word with NO restriction) - (The number of permutations of the letters in this word with the 3 E's are s... | Old John | 32,441 | <p>My preferred method for this goes as follows:</p>
<p>First work out the number of permutations of the letters which are not E: this is just $5!/2$ (didvide by 2 for the number of ways of rearranging the identical Ds).</p>
<p>Now imagine the these letters written something like this: .D.F.A.T.D.</p>
<p>If the Es a... |
1,531,488 | <p>I hope to know the singularity and pole of $\frac{z^2 e^z}{1+e^{2z}}$.</p>
<p>I try $\frac{z^2 e^z}{1+e^{2z}} = \frac{z^2}{e^{-z}+e^{z}}$ and observe that the denominator seems like cosine function. So I think the singularites are i(2n+1)$\pi$/2. But when I try to evaluate the poles, I fail and different values of ... | zahbaz | 176,922 | <p>Since $\frac{z^2 e^z}{1+e^{2z}} = \frac{z^2}{2\cosh z}$, let's start with</p>
<p>$$f(z)=\frac{z^2}{2\cos iz}$$</p>
<p>Now both $z^2$ and $\cos iz$ are entire. The function can be singular only where $z_{0_n} = iz = \frac{\pi +2\pi n}{2}$ and since the zeroes of the denominators are first order, we have</p>
<p>$$\... |
285,965 | <p>I was told to assume $f(x)$ is a polynomial with degree $d\geq 1$ with integer coefficients and positive leading coefficient.</p>
<p>(i) I need to show that there are infinitely many $x$ such that $f(x)$ isn't prime.</p>
<p>(ii) I also need to show that if $f(x_0) = m$, where $m>0$, then $f(x) \equiv 0 \pmod m$... | Math Gems | 75,092 | <p><strong>Hint</strong> $\ $ For some $\,x_0\,$ we have $\,f(x_0)= m\ne \pm1\: $ (see the note below for one proof). </p>
<p>It follows that $\,m\mid f(mn+x_0)\,$ since $\,{\rm mod\ } m\!:\ f(mn+x_0)\equiv f(x_0)\equiv m \equiv 0.\:$ </p>
<p><strong>Note</strong> $\ $ If $\,f(x) = \pm 1\,$ for all $\,x\in \Bbb Z\,$... |
2,210,560 | <p>As I understand it, <strong>partial orders</strong> are binary relations that are:</p>
<ul>
<li>Reflexive</li>
<li>Anti-symmetric</li>
<li>Transitive</li>
</ul>
<p>An example would be $\subseteq$ for sets</p>
<p>And if we add totality to this, we get a total (or linear) order, so a <strong>total order</strong> is... | Fabio Somenzi | 123,852 | <p>Transitivity is the fundamental property of all relations that we call "something something" order. Of course, an equivalence relation is also transitive, and in fact is also a <em>preorder</em>. </p>
<p>So, maybe, one can start from transitive relations, split them according to whether they are reflexive, irrefl... |
2,210,560 | <p>As I understand it, <strong>partial orders</strong> are binary relations that are:</p>
<ul>
<li>Reflexive</li>
<li>Anti-symmetric</li>
<li>Transitive</li>
</ul>
<p>An example would be $\subseteq$ for sets</p>
<p>And if we add totality to this, we get a total (or linear) order, so a <strong>total order</strong> is... | egreg | 62,967 | <p>A <em>strict partial order</em> is a relation that's irreflexive and transitive (asymmetric is a consequence). This is the most common definition.</p>
<p>Actually, this notion is completely equivalent to the notion of <em>partial order</em> (a reflexive, antisymmetric and transitive relation).</p>
<p>Indeed, if $X... |
1,385,197 | <p>For $n\in\mathbb N$,
$$a_{n+1}=2a_n+\frac{1}{a_n},\quad a_1=1.
$$
Can any one give an explicit formula for all $a_n$? If such an explicit general formula doesn't exist, please explain it. I've tried to figure out the $n$-iterated function $f^{(n)}$ where $f(x)=2x+1/x$ or even $f(\tan(t))$. But in either cases, I fai... | vonbrand | 43,946 | <p>Take the recurrence, and square:</p>
<p>$$
a^2_{n + 1} = 4 a^2_n + 4 + \frac{1}{a^2_n}
$$</p>
<p>We know that $a_n \to \infty$, so as a first approximation you have for $b_n = a^2_n$:</p>
<p>$$
b_{n + 1} = 4 b_n + 4
$$</p>
<p>For $b_0 = 1$ this one has solution:</p>
<p>$$
b_n = (n + 4) \cdot 4^{n - 1}
$$</p>
<... |
3,454,810 | <blockquote>
<p>Study the following sequence of numbers:
<span class="math-container">$$\forall n\in\mathbb{N}, u_n=\sum_{k=n}^{2n}\frac{k}{\sqrt{n^2+k^2}}$$</span></p>
</blockquote>
<p>I tried to calculate <span class="math-container">$u_{n+1}-u_n$</span>, but I couldn't simplify the expression.</p>
<p>Plotting ... | Khosrotash | 104,171 | <p><span class="math-container">$$\forall n\in\mathbb{N}, u_n=\sum_{k=n}^{2n}\frac{k}{\sqrt{n^2+k^2}}\\ u_{n+1}=\sum_{k=n+1}^{2n+2}\frac{k}{\sqrt{(n+1)^2+k^2}}\\u_{n+1}-u_{n}=\underbrace{\sum_{k=n+1}^{2n+2}\frac{k}{\sqrt{(n+1)^2+k^2}}}_{n+2 \space terms}-\underbrace{\sum_{k=n}^{2n}\frac{k}{\sqrt{n^2+k^2}}}_{n+1 \space ... |
121,852 | <p>I am creating a Java game with collisions. I found myself stuck on the following problem.</p>
<p>I have got two known lines: $y$ and $i.$ $i$ is the inbound direction and $o$ the outbound direction, therefore both angles are the same. From $y$ and $i,$ I have calculated $\alpha.$ I also know the coordinates of the ... | Américo Tavares | 752 | <p>This is an approach using standard methods from analytic geometry and trigonometry.</p>
<p><img src="https://i.stack.imgur.com/xFi62.jpg" alt="enter image description here"></p>
<p>For convenience instead of $(x,y)$ I call the coordinates of the point of intersection $(h,k)$. Since the line $o$ passes through $(h,... |
79,991 | <p>I'm attempting to MapThread a function of two lists that requires the index of the list values. For example,</p>
<pre><code>MapThread[#1*i+#2*j&,{{a,b,c},{e,f,g}}]
</code></pre>
<p>Where #1 represents the value from list 1, #2 the value from list 2, i the index of list 1, and j the index of list 2. The expecte... | kglr | 125 | <p>Update:</p>
<pre><code>ClearAll[imtF]
imtF[foo_] := Module[{i = 1}, foo[#, i++] & /@ Transpose@#] &
</code></pre>
<p>Examples:</p>
<pre><code>imtF[#2 (Plus @@ #) &][{{a, b, c}, {e, f, g}}]
(* {a + e, 2 (b + f), 3 (c + g)} *)
xx = {{a, b, c}, {e, f, g}, {x, y, z}};
imtF[#2 (Plus @@ #) &][xx]
(* {a... |
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