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 |
|---|---|---|---|---|
885,450 | <blockquote>
<p>After covering a distance of 30Km with a uniform speed, there got some
defect in train engine and therefore its speed is reduced to 4/5 of
its original speed. Consequently, the train reaches its destination 45
minutes late. If it had happened after covering 18Km of distance, the
train would h... | MCT | 92,774 | <p>Wolfram can actually find it: <a href="http://www.wolframalpha.com/input/?i=lim%28n+to+infinity%29+prod%28k+%3D+0+to+%282%5E%282%5En+-+1%29+%2B+1%29%29+%281+%2B+1%2F%282%5E2%5En+%2B+2k%29%29" rel="nofollow">http://www.wolframalpha.com/input/?i=lim%28n+to+infinity%29+prod%28k+%3D+0+to+%282%5E%282%5En+-+1%29+%2B+1%29%... |
105,040 | <p>This question in stackExchange remained unanswered. </p>
<p>Let $\mathbb F$ be a finite field. Denote by $M_n(\mathbb F)$ the set of matrices of order $n$ over $\mathbb F$ . For a matrix $A∈M_n(\mathbb F)$ what is the cardinality of $C_{M_n(\mathbb F)} (A)$ , the centralizer of $A$ in $M_n(\mathbb F)$? There a... | Alexander Chervov | 10,446 | <p>This is just elementary comment, but may be too long for comment, probably you know it, but just for completeness.</p>
<p>I happened to ask almost (but not exactly) the same question some days ago: </p>
<p><a href="https://mathoverflow.net/questions/104439/conjugcy-classes-in-glf-2-glf-q">Conjugcy classes in GL(F_... |
3,846,717 | <p>Denote <span class="math-container">$\mathbb{F}=\mathbb{C}$</span> or <span class="math-container">$\mathbb{R}$</span>.</p>
<p><strong>Theorem (Cauchy - Schwarz Inequality).</strong> <em>If <span class="math-container">$\langle\cdot,\cdot\rangle$</span> is a semi-inner product on a vector space <span class="math-con... | user | 505,767 | <p>The statements are related to the quadratic equation</p>
<p><span class="math-container">$$ax^2+bx+c=0 \implies x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$$</span></p>
<p>since when <span class="math-container">$b^2-4ac>0$</span> then <span class="math-container">$\sqrt{b^2-4ac}$</span> exists and we always have two disti... |
300,745 | <p>If a function is uniformly continuous in $(a,b)$ can I say that its image is bounded?</p>
<p>($a$ and $b$ being finite numbers).</p>
<p>I tried proving and disproving it. Couldn't find an example for a non-bounded image. </p>
<p>Is there any basic proof or counter example for any of the cases?</p>
<p>Thanks a mi... | Ittay Weiss | 30,953 | <p>Hint: Prove first that a uniformly continuous function on an open interval can be extended to a continuous function on the closure of the interval. </p>
|
300,745 | <p>If a function is uniformly continuous in $(a,b)$ can I say that its image is bounded?</p>
<p>($a$ and $b$ being finite numbers).</p>
<p>I tried proving and disproving it. Couldn't find an example for a non-bounded image. </p>
<p>Is there any basic proof or counter example for any of the cases?</p>
<p>Thanks a mi... | David Mitra | 18,986 | <p>If, for instance, $\limsup\limits_{x\rightarrow a^+} f(x)=\infty$, then given any $\delta>0$, one could find $x$ and $y$ with $a<x<\delta$ and $a<y<\delta$ such that $|f(x)-f(y)|>1$. Could $f$ then be uniformly continuous on $(a,b)$?</p>
<p>Note if $f$ is continuous on $(a,b)$ and unbounded, the... |
741,436 | <p>I get stuck at the following question:</p>
<p>Consider the matrix<br>
$$A=\begin{bmatrix}
0 & 2 & 0 \\
1 & 1 & -1 \\
-1 & 1 & 1\\
\end{bmatrix}$$</p>
<p>Find $A^{1000}$ by using the Cayley-Hamilton theorem.</p>
<p>I find the characteristic polynomial by $P(A) = -A^{3} + 2A^2 = 0$ (by Cayle... | 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{\down}{\... |
566,993 | <p>Suppose $f(z)=1/(1+z^2)$ and we want to find the power series in $a=1$. I think we have to write $1/(1+z^2)=1/(1+(z-1)+1)^2=1/(1+(1+(z-1)^2+2(z-1)))$, but I'm stuck here.</p>
| Giorgio Mossa | 11,888 | <p><strong>Edit:</strong> Andreas Caranti have pointed out, this proof work iff the all the elements of the group have finite order. So it doesn't answer completely but
is more a partial result.</p>
<p>Let $x,a \in G$ then by hypothesis we have that there's $b \in G$ such that $ab=x$ then
$$a^{-1}x^2a=a^{-1}(ab)^2a=a... |
566,993 | <p>Suppose $f(z)=1/(1+z^2)$ and we want to find the power series in $a=1$. I think we have to write $1/(1+z^2)=1/(1+(z-1)+1)^2=1/(1+(1+(z-1)^2+2(z-1)))$, but I'm stuck here.</p>
| dan_fulea | 550,003 | <p>The following solution is the same as in the <a href="https://math.stackexchange.com/questions/4270671/prove-that-a-certain-group-is-abelian/4272460#4272460">duplicate</a>, just made shorter. As in the answer of <a href="https://math.stackexchange.com/users/62565/boris-novikov">Boris Novikov</a>, for every <span cla... |
131,179 | <p>Consider the assumptions</p>
<pre><code>$Assumptions = {Element[a,Reals], Element[z,Complexes]}
</code></pre>
<p>I'm looking for a test, to be applied on <code>a</code> and <code>z</code>, that gives <code>True</code> if the argument is a complex number such as <code>a</code> and <code>False</code> if it's real su... | ubpdqn | 1,997 | <p>There are a number of ways, e.g.:</p>
<pre><code>f[x_] := With[{ri = ReIm /@ N[x]},
ri /. {{_, 0} :> False, {_, _} :> True}]
</code></pre>
<p>Testing:</p>
<pre><code>r = RandomReal[1, 10];
c = Complex @@@ RandomReal[1, {10, 2}];
test = Join[r, c][[RandomSample[Range@20]]];
Thread[{test, f[test]}] // Grid... |
3,006,022 | <p>I have a radioactive decay system to solve for <span class="math-container">$x(t)$</span> and <span class="math-container">$y(t)$</span> (no need for <span class="math-container">$z(t)$</span>):
<span class="math-container">$$\begin{cases}x'=-\lambda x\\
y'=\lambda x-\mu y\\
z'=\mu y\end{cases}$$</span>
with the ini... | Robert Z | 299,698 | <p>The differential equation
<span class="math-container">$$y'+\mu y=\lambda e^{-\lambda t}$$</span>
is <a href="https://en.wikipedia.org/wiki/Linear_differential_equation#Non_homogeneous_equation_with_constant_coefficients" rel="nofollow noreferrer">linear non-homogeneous with constant coefficients</a>. The general so... |
3,012,416 | <p>I know the answer is obvious: In <span class="math-container">$\mathbb{Z}$</span> the only solutions of <span class="math-container">$xy=-1$</span> are <span class="math-container">$x=-y=1$</span> and <span class="math-container">$x=-y=-1$</span>.
My problem is that I want to formally prove it and I don't know how t... | Michael Burr | 86,421 | <p>Suppose that <span class="math-container">$xy=-1$</span>, then <span class="math-container">$-xy=1$</span>, so <span class="math-container">$x$</span> and <span class="math-container">$y$</span> are units (by definition). Since the only units in <span class="math-container">$\mathbb{Z}$</span> are <span class="math... |
419,176 | <p>Let $f:\left\{x\in\mathbb{R}^n\vert\parallel x\parallel<1\right\}\rightarrow\mathbb{R}$ be an one-to-one bounded continuous function.<br>
I want to construct such $f$ which is not uniformly continuous.<br><br>
In this case, I thought I can construct $f$ with a restriction $n=2$.<br>
But I'm confused because $f$ i... | Vishal Gupta | 60,810 | <p>At least for $n =1$, any such function will have a limit on the end points $-1,1$ in this case and hence will have an extension on the interval $[-1,1]$ and hence will be uniformly continuous. So, you cannot construct a function you want in the case $n=1$.</p>
<p>For higher dimensions, the main thing is to examine ... |
2,871,892 | <p><a href="https://i.stack.imgur.com/XLen7.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/XLen7.png" alt="Q1"></a></p>
<p>Solution is 4. </p>
<p>Original matrix is simply [v1;v2;v3;v4]. It forms an identity matrix. Hence the only alteration of the determinant comes from row 1 operation where v1 i... | hmakholm left over Monica | 14,366 | <p>If you take the equation
$$ \tag{*} f(x) = x^n $$
and "set $x=0$", what you get is <em>not</em> $f(x)=0$, but
$$ f(0) = 0^n $$
which does not define a function -- it only says what the function value must be at $0$ (and doesn't even do that until you decide what $n$ is).</p>
<p>As @lulu pointed out in comments, you... |
204,106 | <p>I would like to know what the definition of a short proof is.</p>
<p>In Lance Fortnow’s article “<a href="http://cacm.acm.org/magazines/2009/9/38904-the-status-of-the-p-versus-np-problem/fulltext" rel="nofollow">The Status of the P Versus NP Problem</a>”, Communications of the ACM, Vol. 52 No. 9, he says,</p>
<blo... | zeb | 2,363 | <p>Allow me to muddy the waters a little bit. Before we start making statements about the lengths of proofs, we should first formally define what a proof <em>is</em>. For that, we want the concept of a <em>proof system</em>.</p>
<h2>What is a proof system?</h2>
<p>A proof system is a Turing machine which runs in poly... |
4,463,105 | <p>Write down the radius of convergence of the power series, obtained by the Taylor expansion of the analytic functions about the stated point, in</p>
<p><span class="math-container">$f(z) = \frac{(z+20)(z+21)}{(z-20i)^{21} (z^2 +z+1)}$</span> about <span class="math-container">$z = 0$</span>.</p>
<p>My attempt: Since ... | José Carlos Santos | 446,262 | <p>The radius of convergence is the distance from <span class="math-container">$0$</span> to the nearest singularity. And the nearest singularities are <span class="math-container">$-\frac12\pm\frac{\sqrt3}2i$</span> (the roots of <span class="math-container">$z^2+z+1$</span>), whose distance to <span class="math-conta... |
667,371 | <p>I try to solve this equation:
$$\sqrt{x+2}+\sqrt{x-3}=\sqrt{3x+4}$$</p>
<p>So what i did was:</p>
<p>$$x+2+2*\sqrt{x+2}*\sqrt{x-3}+x-3=3x+4$$</p>
<p>$$2*\sqrt{x+2}*\sqrt{x-3}=x+5$$</p>
<p>$$4*{(x+2)}*(x-3)=x^2+25+10x$$</p>
<p>$$4x^2-4x-24=x^2+25+10x$$</p>
<p>$$3x^2-14x-49$$</p>
<p>But this seems to be wrong! ... | gt6989b | 16,192 | <p>first step on the right hand side should be $3x+5$ not $3x+4$</p>
<p>Then proceeding the same way:
$$
\begin{split}
4(x^2-x-6) &= x^2 + 12x + 36 \\
3x^2 - 16x - 60 &= 0
\end{split}
$$
Since the discriminant of that equation is $16^2 - 4 \cdot 3 \cdot 60 < 0$ there are no solutions.</p>
<p><strong>UPDATE... |
2,076 | <p>I'm attempting for the first time to create a map within <em>Mathematica</em>. In particular, I would like to take an output of points and plot them according to their lat/long values over a geographic map. I have a series of latitude/longitude values like so:</p>
<pre><code> {{32.6123, -117.041}, {40.6973, -111.9}... | kglr | 125 | <p>data:</p>
<pre><code>latlong = {{32.6123, -117.041}, {40.6973, -111.9}, {34.0276, -118.046},
{40.8231, -111.986}, {34.0446, -117.94}, {33.7389, -118.024},
{34.122, -118.088}, {37.3881, -122.252}, {44.9325, -122.966},
{32.6029, -117.154}, {44.7165, -123.062}, {37.8475, -122.47},
{32.6833, -117.098}, {44.4881, -12... |
2,076 | <p>I'm attempting for the first time to create a map within <em>Mathematica</em>. In particular, I would like to take an output of points and plot them according to their lat/long values over a geographic map. I have a series of latitude/longitude values like so:</p>
<pre><code> {{32.6123, -117.041}, {40.6973, -111.9}... | Artes | 184 | <p>Given latitude/longitude values: </p>
<pre><code>list = {{32.6123, -117.041}, {40.6973, -111.9}, {34.0276, -118.046}, \
{40.8231, -111.986}, {34.0446, -117.94}, {33.7389, -118.024}, \
{34.122, -118.088}, {37.3881, -122.252}, {44.9325, -122.966}, \
{32.6029, -117.154}, {44.7165, -123.062}, {3... |
2,076 | <p>I'm attempting for the first time to create a map within <em>Mathematica</em>. In particular, I would like to take an output of points and plot them according to their lat/long values over a geographic map. I have a series of latitude/longitude values like so:</p>
<pre><code> {{32.6123, -117.041}, {40.6973, -111.9}... | Jason B. | 9,490 | <p>Just in case anyone comes across this post in a search, here is a one-liner here for version 10 and after. If <code>latlong</code> is the list from the OP, then you can do this</p>
<pre><code>GeoGraphics[{Red, PointSize[.02], Point@GeoPosition@latlong}]
</code></pre>
<p><img src="https://i.stack.imgur.com/tgU6N.p... |
2,355,579 | <blockquote>
<p><strong>Problem:</strong> James has a pile of n stones for some positive integer n ≥ 2. At each step, he
chooses one pile of stones and splits it into two smaller piles and writes the product
of the new pile sizes on the board. He repeats this process until every pile is exactly
one stone.</p>
... | emma | 406,460 | <p>I think the first step would be to try to figure out what the formula for the sum would be. If James has $n$ stones, one way he could work through the process is to remove one stone from the pile each time (putting it into its own 1-stone pile). So at first he would have two piles, one with $n-1$ stones and the othe... |
3,182,587 | <p>I think it is simpler if I just focus on the derivation of <span class="math-container">$\cos(z)=\frac{e^{iz}+e^{-iz}}{2}$</span>.</p>
<p>I start from the formulae for the complex exponential, assuming <span class="math-container">$z=x+iy$</span>: <span class="math-container">$e^{iz}=e^{-y}*(\cos(x)+i\sin(x))$</spa... | Kandinskij | 657,309 | <p>Consider <span class="math-container">$e^{ix}$</span> and <span class="math-container">$e^{-ix}$</span> remember that sine is an odd function and cosine is an even function:</p>
<p><span class="math-container">$$e^{ix}=\cos(x)+i\sin(x)$$</span>
<span class="math-container">$$e^{-ix}=\cos(-x)+i\sin(-x)=\cos(x)-i\sin... |
93,499 | <p>Let $P$ be a normal sylow $p$-subgroup of a finite group $G$.</p>
<p>Since $P$ is normal it is the unique sylow $p$-subgroup.</p>
<p>I would like to say if $\phi$ is an automorphism then $\phi(P)$ is also a sylow $p$-subgroup. Then uniqueness would finish the proof. But is that true?</p>
<p>Does an automorphism o... | Community | -1 | <p>Let $\phi:G \to G$ be the automorphism. We'll prove $\phi(P)$ is a $p$-Sylow Subgroup of $G$.</p>
<p><em>$\phi$ takes identity to itself</em>:</p>
<p>For any $e_G \in G$, the identity in G, $\phi(e_G.e_G)=\phi(e_G)\phi(e_G)$
which proves the result.(from cancellation law in a group)</p>
<p><em>$\phi$ takes invers... |
1,346,039 | <p>Let <span class="math-container">$F$</span> be a field of characteristic prime to <span class="math-container">$n$</span>, and let <span class="math-container">$F^a$</span> be an algebraic closure of <span class="math-container">$F$</span>. Let <span class="math-container">$\zeta$</span> be a primitive <span class="... | mich95 | 229,072 | <p>I am really not sure of this, but here's what I could do!
Let $\tau$ be in the Galois group. $\tau$ is uniquely determined by what it does to $\zeta$. And $\tau_{k}(\zeta)=\zeta^{k}$. Now if $m_{1}$ and $m_{2}$ lie in the same coset of $S$, then $m_{1}m_{2}^{-1} \in S$, So $m_{1}m_{2}^{-1}=t$ for some $t$ in $S$. Bu... |
2,806,432 | <p>Let $(\mathbb{R}^N,\tau)$ a topological space, where $\tau$ is the usual topology.
Let $A\subset\mathbb{R}^N$ a compact. If $(A_n)_n$ is a family of open such that
\begin{equation}
\bigcup_nA_n\supset A,
\end{equation}
then, from compact definition
\begin{equation}
\bigcup_{i=1}^{k}A_i\supset A
\end{equation}
Now, ... | Riccardo Sven Risuleo | 419,249 | <p>The first thing to notice is that $B$ throws 50 coins and makes 225 points.
If $n_H$ and $n_T$ are the numbers of heads and tails, 50 throws make it so that
$$ n_H + n_T = 50.$$
Given that a heads gives 1 point and a tails 8 points, we can set up the equation
$$ 1 \cdot n_H + 8 \cdot n_T = 225.$$</p>
<p>Solving the... |
1,369,409 | <p>I have a bit of an advanced combination problem that has left me stumped for a few days. Essentially my question is if you have n sets of items, and you can select a different number of items from each set, how do you compute the combinations without first creating new sets.</p>
<p>An example in pictures:
I have th... | Graham Kemp | 135,106 | <p>Assuming "compute" means "count".</p>
<blockquote>
<p>And now I would like to compute all of the combinations with the criteria that 2 elements be selected from set A (3 elements), 1 element is selected from set B (3 elements), and 2 elements are selected from set C (4 elements). </p>
</blockquote>
<p>$${^{3}{\r... |
1,100,812 | <p>Here is the statement:
($\bf{Tonelli}$) If $f\in L^+(X,Y)$, then $\displaystyle g:x\mapsto\int_Yf_xd\nu$ is $\mathcal{M}$-measurable,\
$\displaystyle h:y\mapsto \int_Xf^yd\mu$ is $\mathcal{N}$-measurable (so $g\in L^+(X)$ and $h\in L^+(Y)$). And $$\displaystyle \int_{X\times Y}fd\mu \times\nu=\int_Xgd\mu=\int_Yhd\... | pre-kidney | 34,662 | <p>Here is where the a.e. is coming from. Suppose we have a measurable function $g$, and we know that
$$
\int |g|\ d\mu<\infty.
$$
Then it is intuitively clear that $|g|<\infty$, because integrating $\infty$s would give you $\infty$. However, we have to hedge our statement because integration w.r.t. a measure "fo... |
2,589,019 | <p>I am aware that there is a result saying that if $L_1, L_2$ are two algebraically closed fields of the same characteristic, then either $L_1$ is isomorphic to the subfield of $L_2$, or $L_2$ is isomorphic to a subfield of $L_1$.</p>
<p>I am unsure however of how to prove this result. I am quite surprised that this ... | Eric Wofsey | 86,856 | <p>Let $k$ be the prime field of the common characteristic of $L_1$ and $L_2$, let $B_1$ be a transcendence basis for $L_1$ over $k$, and let $B_2$ be a transcendence basis for $L_2$ over $k$. Then either $|B_1|\leq |B_2|$ or $|B_2|\leq |B_1|$; suppose WLOG that $|B_1|\leq |B_2|$. Choose an injection $i:B_1\to B_2$; ... |
381,566 | <p>I know practically nothing about fractional calculus so I apologize in advance if the following is a silly question. I already tried on math.stackexchange.</p>
<p>I just wanted to ask if there is a notion of fractional derivative that is linear and satisfy the following property <span class="math-container">$D^u((f)... | Terry Tao | 766 | <p>There are basically no interesting solutions to this equation beyond first and zeroth order operators, even if one only imposes the stated constraint for <span class="math-container">$n=2$</span>.</p>
<p>First, we can <a href="https://en.wikipedia.org/wiki/Polarization_identity" rel="nofollow noreferrer">depolarise<... |
1,700,590 | <p>Let $A=\{a_1\ldots,a_m \}$ be a set of linearly independent vectors. Suppose that each $a_j$ $(j=1,\ldots,m)$ can be written as a linear combination of vectors in the set $B=\{b_1,\ldots,b_n\}$.</p>
<p>Then how to show that $m \le n$?</p>
<p>I have tried as follows:</p>
<p>Since $a_j \in A$, we have $a_j \in \o... | mordecai iwazuki | 167,818 | <p>You are almost there. Consider for the sake of contradiction that $m>n$ but you still have $A\subseteq$span$(B)$. Then, you have a set of $m>n$ vectors in a space spanned by $n$ vectors so the set $\{a_j\}$ must be linearly dependent, a contradiction to our assumptions so it must be that $m\leq n$.</p>
|
200,278 | <p>Say I have two TimeSeries:</p>
<pre><code>x = TimeSeries[{2, 4, 1, 10}, {{1, 2, 4, 5}}]
y = TimeSeries[{6, 2, 6, 3, 9}, {{1, 2, 3, 4, 5}}]
</code></pre>
<p>x has a value at times: 1,2,4,5</p>
<p>y has a value at times: 1,2,3,4,5</p>
<p>I would like to build a list of pairs {<span class="math-container">$x_i$</sp... | Sjoerd Smit | 43,522 | <p><code>TimeSeriesThread</code> is probably the tool for the job when you specify the right options:</p>
<pre><code>x = TimeSeries[{2, 4, 1, 10}, {{1, 2, 4, 5}}];
y = TimeSeries[{6, 2, 6, 3, 9}, {{1, 2, 3, 4, 5}}];
DeleteMissing[
TimeSeriesThread[Identity, {x, y}, ResamplingMethod -> None]["Values"],
1, 1
]
</co... |
1,942,641 | <p>Recall that a function $f: A \rightarrow B$ is continuous means that for any open set $\beta \in B$, $f^{-1}(\beta)$ is open, where $f^{-1}$ is the preimage set.</p>
<p>Say that $f$ is not continuous. Then, that means that there exists some open set $\beta$ in $B$ such that $f^{-1}(\beta)$ is...not open?</p>
<p>Wh... | Thomas Andrews | 7,933 | <p>It's really as simple as noting that $a+b\geq 2\sqrt{ab}$, and similarly for $a+c$ and $b+c$, then multiplying it out.</p>
|
1,942,641 | <p>Recall that a function $f: A \rightarrow B$ is continuous means that for any open set $\beta \in B$, $f^{-1}(\beta)$ is open, where $f^{-1}$ is the preimage set.</p>
<p>Say that $f$ is not continuous. Then, that means that there exists some open set $\beta$ in $B$ such that $f^{-1}(\beta)$ is...not open?</p>
<p>Wh... | ADAM | 312,147 | <p>Using AM-GM, we get that
$$\frac{a+b}{2} \geq \sqrt{ab} $$
$$\Rightarrow a+b \geq 2 \sqrt{ab}$$</p>
<p>Similar manipulations show that<br>
$$ b+c \geq 2 \sqrt{bc}$$
$$ a+c \geq 2 \sqrt{ac}$$</p>
<p>Multiplying all these inequalities together ( this is allowed here, since all numbers involved are positive), we get ... |
9,484 | <p>Let <span class="math-container">$F(k,n)$</span> be the number of permutations of an n-element set that fix exactly <span class="math-container">$k$</span> elements.</p>
<p>We know:</p>
<ol>
<li><p><span class="math-container">$F(n,n) = 1$</span></p>
</li>
<li><p><span class="math-container">$F(n-1,n) = 0$</span></p... | Hans-Peter Stricker | 2,672 | <p>see <a href="https://oeis.org/A000166" rel="nofollow noreferrer">Subfactorial or rencontres numbers, or derangements: number of permutations of n elements with no fixed points</a></p>
|
9,484 | <p>Let <span class="math-container">$F(k,n)$</span> be the number of permutations of an n-element set that fix exactly <span class="math-container">$k$</span> elements.</p>
<p>We know:</p>
<ol>
<li><p><span class="math-container">$F(n,n) = 1$</span></p>
</li>
<li><p><span class="math-container">$F(n-1,n) = 0$</span></p... | Sam OT | 59,264 | <p>Various links on this answer have expired, so I thought I would add an answer.</p>
<p>One can use inclusion--exclusion. First, note (as in @ReidBarton's answer) that
<span class="math-container">$$ F(k,n) = \binom kn F(0,n-k). $$</span>
So it is sufficient to only study permutations with no fixed points.
This is kn... |
156,479 | <p>Let $S$ be a compact oriented surface of genus at least $2$ (possibly with boundary). Let $X$ be a connected component of the space of embeddings of $S^1$ into $S$.</p>
<p>Question : what is the fundamental group of $X$? My guess is that the answer is $\mathbb{Z}$ with generator the loop of embeddings obtained by... | Sam Nead | 1,650 | <p>Edit - </p>
<p>As pointed out by Igor below, my "proof" of the first bullet point is incomplete. I'll leave the rest of the post here: perhaps some kind soul will fix the gap. </p>
<p>Original - </p>
<p>Let's restrict attention to the case where all embeddings are smooth and where $S$ has negative Euler characte... |
216,532 | <p>How do I find the limit of something like</p>
<p>$$ \lim_{x\to \infty} \frac{2\cdot3^{5x}+5}{3^{5x}+2^{5x}} $$</p>
<p>?</p>
| Hagen von Eitzen | 39,174 | <p><strong>Hint:</strong> $2\cdot 3^{5x}+5=3^{5x}\cdot (2+\frac5{3^{5x}})$ and $3^{5x}+ 2^{5x}=3^{5x}\cdot(1+(\frac23)^{5x})$.</p>
|
216,532 | <p>How do I find the limit of something like</p>
<p>$$ \lim_{x\to \infty} \frac{2\cdot3^{5x}+5}{3^{5x}+2^{5x}} $$</p>
<p>?</p>
| Cameron Buie | 28,900 | <p>More generally, if you're trying to determine limiting behavior of a function of form <span class="math-container">$$\frac{f(x)}{g(x)}$$</span> as <span class="math-container">$x\to\infty$</span>, and the limit is of form "<span class="math-container">$\pm\frac\infty\infty$</span>", then you can look for a... |
2,782,658 | <p>Here's what I have to show:</p>
<blockquote>
<p>If $x\le y+z$ for every $z>0$ then $x\le y$.</p>
</blockquote>
<p>I tried proof by contradiction but it does not seem to work. Suppose $y<x$. Then $x-y>0$. Let $z_{0}=x-y$. Then $x\le y+z_{0} \Rightarrow x\le x$ which is true. Had $x$ been strictly less th... | fleablood | 280,126 | <p>"Had x been strictly less than y+z, it would have worked."</p>
<p>So instead of making $z = x-y$ make $0 < z < x-y$.</p>
<p>Then $x \le y + z < y + x-y = x$.</p>
|
2,782,658 | <p>Here's what I have to show:</p>
<blockquote>
<p>If $x\le y+z$ for every $z>0$ then $x\le y$.</p>
</blockquote>
<p>I tried proof by contradiction but it does not seem to work. Suppose $y<x$. Then $x-y>0$. Let $z_{0}=x-y$. Then $x\le y+z_{0} \Rightarrow x\le x$ which is true. Had $x$ been strictly less th... | robjohn | 13,854 | <p>You are almost there. Assume that $x\gt y$, then set $z=\frac{x-y}2\gt0$. That would mean that $x=y+2z\gt y+z$ and $z\gt0$, which is a contradiction.</p>
|
3,405,622 | <p>Let <span class="math-container">$\int_{0}^{1}fg \text{ }d\mathbb{P}=0$</span>, for all <span class="math-container">$f$</span> <span class="math-container">$\in$</span> <span class="math-container">$L^{\infty}([0,1],\mathbb{P})$</span> and <span class="math-container">$g$</span> be a fixed function in <span class="... | Marios Gretsas | 359,315 | <p>Yes since the dual of <span class="math-container">$L^1$</span> is <span class="math-container">$L^{\infty}$</span> </p>
<p><span class="math-container">$T_g(f):=\int_0^1fg$</span> is a bounded linear functional <span class="math-container">$T:L^{\infty} \to \Bbb{C}$</span></p>
<p>and <span class="math-container"... |
2,857,769 | <blockquote>
<p>Find $t$ such that $$\lim_{n\to\infty} \frac {\left(\sum_{r=1}^n r^4\right)\cdot\left(\sum_{r=1}^n r^5\right)}{\left(\sum_{r=1}^n r^t\right)\cdot\left(\sum_{r=1}^n r^{9-t}\right)}=\frac 45.$$</p>
</blockquote>
<p>At first sight this question scared the hell out of me. I tried using the general known... | Robert Z | 299,698 | <p>Hint. Note that for $a>0$
$$F(a):=\lim_{n\to\infty} \frac{1}{n^{a+1}}\left(\sum_{r=1}^n r^a\right)=
\lim_{n\to\infty} \frac{1}{n}\left(\sum_{r=1}^n \left(\frac{r}{n}\right)^a\right)\to \int_0^1x^a dx=\frac{1}{a+1}.$$
Then, as $n\to\infty$,
$$\frac {\left(\sum_{r=1}^n r^4\right)\cdot\left(\sum_{r=1}^n r^5\right)}{... |
2,857,769 | <blockquote>
<p>Find $t$ such that $$\lim_{n\to\infty} \frac {\left(\sum_{r=1}^n r^4\right)\cdot\left(\sum_{r=1}^n r^5\right)}{\left(\sum_{r=1}^n r^t\right)\cdot\left(\sum_{r=1}^n r^{9-t}\right)}=\frac 45.$$</p>
</blockquote>
<p>At first sight this question scared the hell out of me. I tried using the general known... | Paramanand Singh | 72,031 | <p>The key here is Cesaro-Stolz. Using Cesaro-Stolz we have $$f(t) =\lim_{n\to\infty} \frac{1}{n^{t+1}}\sum_{r=1}^{n}r^t=\lim_{n\to \infty} \frac{n^t} {n^{t+1}-(n-1)^{t+1}}=\frac{1}{t+1}$$ Dividing the numerator and denominator of the given expression by $n^{11}$ we can see that the desired limit is $$\frac{f(4)f(5)}{f... |
1,613,171 | <p>On page $61$ of the book <a href="http://solmu.math.helsinki.fi/2010/algebra.pdf" rel="nofollow">Algebra</a> by Tauno Metsänkylä, Marjatta Näätänen, it states</p>
<blockquote>
<p>$\langle \emptyset \rangle =\{1\},\langle 1 \rangle =\{1\}. H\leq G \implies \langle H \rangle =H$</p>
</blockquote>
<p>where $H \leq ... | 5xum | 112,884 | <p>$\emptyset$ is not a subgroup of $G$, because it is not a group. Because $\langle X\rangle$ is <strong>by definition</strong></p>
<blockquote>
<p>The smallest subgroup of $H$ that includes $X$</p>
</blockquote>
<p>then if $X=\emptyset$, it is equal to $\{1\}$</p>
|
4,086,995 | <p>Let <span class="math-container">$ABCD$</span> be a rectangle.</p>
<p>Given:</p>
<p><span class="math-container">$A(2;1)$</span></p>
<p><span class="math-container">$C(5;7)$</span></p>
<p><span class="math-container">$\overline{BC}=2\overline{AB}$</span>.</p>
<p>I tried to solve it, but after using the Pythagoras th... | Arturo Magidin | 742 | <p>Jose Carlos Santos has explained why the automorphism group of <span class="math-container">$\mathbb{Z}$</span> is very constrained, while that of <span class="math-container">$\mathbb{Z}_n$</span> is not as restricted, leading to more possibilities.</p>
<p>Paraphrasing US Supreme Court justices, I write separately ... |
4,086,995 | <p>Let <span class="math-container">$ABCD$</span> be a rectangle.</p>
<p>Given:</p>
<p><span class="math-container">$A(2;1)$</span></p>
<p><span class="math-container">$C(5;7)$</span></p>
<p><span class="math-container">$\overline{BC}=2\overline{AB}$</span>.</p>
<p>I tried to solve it, but after using the Pythagoras th... | user909807 | 909,807 | <p>All these groups are 1-generated. Thus, the image of a generator must be a generator. (Isomorphisms preserve such things.) But
<span class="math-container">$\Bbb Z$</span> has only two generators, whereas <span class="math-container">$\Bbb Z_n$</span> can have many. In fact, the number of automorphisms of <span cl... |
1,844,374 | <p>Why does the "$\times$" used in arithmetic change to a "$\cdot$" as we progress through education? The symbol seems to only be ambiguous because of the variable $x$; however, we wouldn't have chosen the variable $x$ unless we were already removing $\times$ as the symbol for multiplication. So why do we? I am very cu... | Allie | 315,111 | <p>This is primarily done to emphasize different multiplication operations in terms of vector and multidimensional calculus. In particular, this is to emphasize that the dot product $\cdot$ is mechanically different from the cross product $\times$, although in operations on objects of one dimension, they are virtually ... |
1,844,374 | <p>Why does the "$\times$" used in arithmetic change to a "$\cdot$" as we progress through education? The symbol seems to only be ambiguous because of the variable $x$; however, we wouldn't have chosen the variable $x$ unless we were already removing $\times$ as the symbol for multiplication. So why do we? I am very cu... | DpS | 350,927 | <p>It's because . stands for any binary operation which might look like 'multiplication' in some particular setting, or might be a substitute for multiplication. Hence, it is more general in nature.</p>
|
1,844,374 | <p>Why does the "$\times$" used in arithmetic change to a "$\cdot$" as we progress through education? The symbol seems to only be ambiguous because of the variable $x$; however, we wouldn't have chosen the variable $x$ unless we were already removing $\times$ as the symbol for multiplication. So why do we? I am very cu... | Robert Soupe | 149,436 | <p>The lowercase letter $x$ and the multiplication cross $\times$ (<code>\times</code> in TeX, <code>&times;</code> in HTML) are very different symbols. One can be used to represent a variable, as you have already halfway surmised, but it shouldn't be used to denote any kind of multiplication, whereas the other can... |
1,844,374 | <p>Why does the "$\times$" used in arithmetic change to a "$\cdot$" as we progress through education? The symbol seems to only be ambiguous because of the variable $x$; however, we wouldn't have chosen the variable $x$ unless we were already removing $\times$ as the symbol for multiplication. So why do we? I am very cu... | Community | -1 | <p>$\times$ is clearly visible.</p>
<p>$\cdot$ is more discrete and is often left implicit.</p>
<p>With maturity, we become able to supply the operator where required.</p>
|
250,687 | <p>I'm doing a sanity check of the following equation:
<span class="math-container">$$\sum_{j=2}^\infty \frac{(-x)^j}{j!}\zeta(j) \approx x(\log x + 2 \gamma -1)$$</span></p>
<p>Naive comparison of the two shows a bad match but I suspect one of the graphs is incorrect.</p>
<ol>
<li>Why isn't there a warning?</li>
<li>H... | Mariusz Iwaniuk | 26,828 | <p>Another way using:</p>
<p><span class="math-container">$$\sum _{n=1}^{\infty } \left(\exp \left(-\frac{x}{n}\right)-1+\frac{x}{n}\right)=\int_0^{\infty } \left(\frac{x}{-1+e^z}-\frac{\sqrt{x}
J_1\left(2 \sqrt{x} \sqrt{z}\right)}{\left(-1+e^z\right) \sqrt{z}}\right) \, dz$$</span></p>
<pre><code> f[x_] := NIntegra... |
3,124,412 | <p>I'm having a hard time coming up with a formal proof by cases method for this set of premises and conclusion. Note that <span class="math-container">$\neg$</span> refers to negation and <span class="math-container">$\wedge$</span> denotes AND. | Denotes subproof</p>
<ol>
<li><p><span class="math-container">$\neg(A... | Lutz Lehmann | 115,115 | <p>Insert the Taylor expansion
<span class="math-container">$$
f(x_k)=f(ξ+e_k)=0+f'(ξ)e_k+\frac12f''(ξ)e_k^2+...
$$</span>
However, apart from not knowing what <span class="math-container">$p_2$</span> is relative to <span class="math-container">$p_1$</span>, it also does not make systematic use of <span class="math-co... |
3,124,412 | <p>I'm having a hard time coming up with a formal proof by cases method for this set of premises and conclusion. Note that <span class="math-container">$\neg$</span> refers to negation and <span class="math-container">$\wedge$</span> denotes AND. | Denotes subproof</p>
<ol>
<li><p><span class="math-container">$\neg(A... | Carl Christian | 307,944 | <p>Let <span class="math-container">$\xi$</span> denote the root and let <span class="math-container">$x_0 \not = x_1$</span> denote the initial approximation. We assume that <span class="math-container">$f(x_1) \not = f(x_2)$</span> such that the secant step
<span class="math-container">$$x_2 = x_1 - \frac{x_1 -x_0}{f... |
70,976 | <blockquote>
<p>I'm considering the ring $\mathbb{Z}[\sqrt{-n}]$, where $n\ge 3$ and square free. I want to see why it's not a UFD.</p>
</blockquote>
<p>I defined a norm for the ring by $|a+b\sqrt{-n}|=a^2+nb^2$. Using this I was able to show that $2$, $\sqrt{-n}$ and $1+\sqrt{-n}$ are all irreducible. Is there some... | Chris Eagle | 5,203 | <p>If $n$ is even, then $2$ divides $\sqrt{-n}^2=-n$ but does not divide $\sqrt{-n}$, so $2$ is a nonprime irreducible. In a UFD, all irreducibles are prime, so this shows $\mathbb{Z}[\sqrt{-n}]$ is not a UFD.</p>
<p>Similarly, if $n$ is odd, then $2$ divides $(1+\sqrt{-n})(1-\sqrt{-n})=1+n$ without dividing either of... |
3,580,068 | <p>Let <span class="math-container">$f:[0,1] \rightarrow \mathbb{R}$</span> continuous such that <span class="math-container">$\int_0^1 xf(x)\,dx=\frac{\pi}{4}$</span>. Prove that there is <span class="math-container">$c\in (0,1)$</span> such that <span class="math-container">$c^3f(c)+cf(c)-1=0$</span>.</p>
<p>Here is... | LHF | 744,207 | <p>Since <span class="math-container">$$\displaystyle\frac{\pi}{4}=\int_0^1\frac{1}{1+x^2}\, dx$$</span> </p>
<p>we can write the condition as:</p>
<p><span class="math-container">$$\int_0^1\left(xf(x)-\frac{1}{1+x^2}\right)\,dx=0$$</span></p>
<p>and from the mean value theorem, there exists some <span class="math-c... |
3,580,068 | <p>Let <span class="math-container">$f:[0,1] \rightarrow \mathbb{R}$</span> continuous such that <span class="math-container">$\int_0^1 xf(x)\,dx=\frac{\pi}{4}$</span>. Prove that there is <span class="math-container">$c\in (0,1)$</span> such that <span class="math-container">$c^3f(c)+cf(c)-1=0$</span>.</p>
<p>Here is... | Milo Brandt | 174,927 | <p>You can get this somewhat easily by examining cases: since <span class="math-container">$f$</span> is continuous, so is the function taking <span class="math-container">$c$</span> to <span class="math-container">$c^3f(c)+cf(c)-1$</span>, meaning that if this quantity is never zero, it must either always be negative ... |
260,516 | <p>I was inspired by <a href="https://math.stackexchange.com/questions/2062960/there-exist-infinite-many-n-in-mathbbn-such-that-s-n-s-n-frac1n2?noredirect=1#comment4336226_2062960">this</a> topic on Math.SE.<br>
Suppose that $H_n = \sum\limits_{k=1}^n \frac{1}{k}$ - $n$th harmonic number. Then</p>
<h2>Conjecture</h2>
... | Gerhard Paseman | 3,402 | <p>I am letting $A = H_n - \lfloor H_n \rfloor$ and computing it in a dumb way, and then computing $1/nA$ when $A \lt 1/n $. For $n=83$ I get $1/nA \approx 5.825$, which is the largest such value I find so far, and which means 83 is one of three values of $n$ which are allowed when $\epsilon$ is 0.3. A smarter way t... |
3,308,291 | <p>I have an array of numbers (a column in excel). I calculated the half of the set's total and now I need the minimum number of set's values that the sum of them would be greater or equal to the half of the total. </p>
<p>Example:</p>
<pre><code>The set: 5, 5, 3, 3, 2, 1, 1, 1, 1
Half of the total is: 11
The least a... | Sayan Goswami | 595,808 | <p>1+1=2 and 4+4=3 is not in your set. So it is not a semigroup.
Ok, now you can restrict your operation as, you will not alow a*a, i.e, if you say that the operation between same element is not allowed, i.e., 1+1 and 4+4 is not defined then {0,1,4} is closed. But it is not so much interesting.</p>
|
2,582,046 | <p>My professor showed the following false proof, which showed that complex numbers do not exist. We were told to find the point where an incorrect step was taken, but I could not find it. Here is the proof: (Complex numbers are of the form <span class="math-container">$\rho e^{i\theta}$</span>, so the proof begins the... | ArsenBerk | 505,611 | <p>Because for complex numbers, $e^{zc}\ne(e^z)^c$ for $z,c \in \mathbb{C}$.</p>
|
1,641,255 | <p>I am having difficulty find the centroid of the region that is bound by the surfaces $x^2+y^2+z^2-2az=0$ and $3x^2+3y^2-z^2=0$ (lying above $xy$ plane, consider the inner region). I know the first surface is a sphere, while the second is an infinite cone.</p>
<p>I just dont know how to even approach it as I dont kn... | David G. Stork | 210,401 | <p>For the first equation, complete the square in $z$... that is, convert the equation to the form $x^2 + y^2 + (z - q)^2 = r^2$. That will give you a sphere centered at $\{0,0,q\}$ of radius $r$. You must express $q$ and $r$ in terms of your $a$.</p>
<p>For the second equation, note that the region is rotationally ... |
1,778,440 | <p>I'm new to combinatorics, Although I understood most of the concepts this one baffles me.</p>
<blockquote>
<p>How many words exist that have exactly $5$ distinct consonants and $2$ identical vowels?</p>
</blockquote>
<p>The Answer is $$\binom{21}{5} \binom{5}{1} \frac{7!}{2!}$$</p>
<p>My doubt is:
Why do we ... | lomot | 338,275 | <p>Since we have 2 <strong>identical</strong> vowels, we should to choose only one "kind" of vowels to put it in two places.</p>
|
1,778,440 | <p>I'm new to combinatorics, Although I understood most of the concepts this one baffles me.</p>
<blockquote>
<p>How many words exist that have exactly $5$ distinct consonants and $2$ identical vowels?</p>
</blockquote>
<p>The Answer is $$\binom{21}{5} \binom{5}{1} \frac{7!}{2!}$$</p>
<p>My doubt is:
Why do we ... | gt6989b | 16,192 | <p>You are selecting 1 vowel (since your problem requires 2 <em>identical</em> ones) and then choosing where to place consonants and vowels in $7!/2!$ ways</p>
|
3,309,511 | <p>Prove that there exists infinitely many pairs of positive real numbers <span class="math-container">$x$</span> and <span class="math-container">$y$</span> such that <span class="math-container">$x\neq y$</span> but <span class="math-container">$ x^x=y^y$</span>.</p>
<p>For example <span class="math-container">$\tfr... | Especially Lime | 341,019 | <p>In fact is is true for any <span class="math-container">$a<b$</span> (in this case, <span class="math-container">$a=1/4, b=1/2$</span>) and any function <span class="math-container">$f$</span> satisfying</p>
<ul>
<li><span class="math-container">$f(a)=f(b)$</span> and</li>
<li><span class="math-container">$f$</... |
94,525 | <p>I am trying to solve the equation
$$z^n = 1.$$</p>
<p>Taking $\log$ on both sides I get $n\log(z) = \log(1) = 0$.</p>
<p>$\implies$ $n = 0$ or $\log(z) = 0$</p>
<p>$\implies$ $n = 0$ or $z = 1$.</p>
<p>But I clearly missed out $(-1)^{\text{even numbers}}$ which is equal to $1$.</p>
<p>How do I solve this equati... | David Mitra | 18,986 | <p>Working in the reals, take the $n$th root of both sides. </p>
<p>If $n$ is even then you'd write $(z^n)^{1/n}=|z|$.</p>
<p>For example $z^2=1 \iff (z^2)^{1/2}=1^{1/2} \iff |z|=1$.</p>
<p>For odd powers, you could say, for example:
$z^3=1 \iff (z^3)^{1/3}=1^{1/3} \iff z =1$.</p>
|
94,525 | <p>I am trying to solve the equation
$$z^n = 1.$$</p>
<p>Taking $\log$ on both sides I get $n\log(z) = \log(1) = 0$.</p>
<p>$\implies$ $n = 0$ or $\log(z) = 0$</p>
<p>$\implies$ $n = 0$ or $z = 1$.</p>
<p>But I clearly missed out $(-1)^{\text{even numbers}}$ which is equal to $1$.</p>
<p>How do I solve this equati... | Robert Israel | 8,508 | <p>If you want to do this properly you need complex numbers. The logarithm is a multivalued function; $z^n = 1$ is equivalent not to $n \log z = \log 1$ but to
$n \log z = 2 \pi i m$ where $m$ is an arbitrary integer. If $n \ne 0$ this says
$\log z = (2 \pi i m)/n$ and $z = e^{2 \pi i m/n}$. In particular with $n =... |
1,114 | <p>Or more specifically, why do people get so excited about them? And what's your favorite easy example of one, which illustrates why I should care (and is not a group)?</p>
| Charles Siegel | 622 | <p>As above, a groupoid is an object where every morphism is an isomorphism, and generalizes groups. As for why to get excited about them, they're useful in classification of things. Like, say you want to understand vector bundles on a space. One method of doing so is constructing the "stack" of vector bundles on th... |
1,114 | <p>Or more specifically, why do people get so excited about them? And what's your favorite easy example of one, which illustrates why I should care (and is not a group)?</p>
| David Zureick-Brown | 2 | <p>A set is an example of a groupoid, and I care about groupoids as a generalization of <strong>sets</strong> as opposed to groups. My most fundamental tool is Yoneda's lemma, which says that one can think of a category C as being embedded in the category C-hat of presheaves (C-hat := Hom(C,Sets)); this is a really use... |
1,114 | <p>Or more specifically, why do people get so excited about them? And what's your favorite easy example of one, which illustrates why I should care (and is not a group)?</p>
| Qiaochu Yuan | 290 | <p>Personally, the reason I'm interested in groupoids is something called <a href="http://math.ucr.edu/home/baez/counting/" rel="noreferrer">groupoid cardinality</a> and some other related ideas (the link contains a lot of other links). A motivating idea here is that certain sets X of algebraic objects have the proper... |
1,114 | <p>Or more specifically, why do people get so excited about them? And what's your favorite easy example of one, which illustrates why I should care (and is not a group)?</p>
| Kevin H. Lin | 83 | <p>Let me expand a bit on what Dave said.</p>
<p>The Yoneda lemma tells us that given an object $X$ of a category $\mathcal C$, the (covariant, contravariant, whatever) functor $h_X : \mathcal C \to \mathsf{Set}$, which sends an object $Y$ to the set $\mathsf{Hom}(Y,X)$, can be thought of as the "same" as the object $... |
1,114 | <p>Or more specifically, why do people get so excited about them? And what's your favorite easy example of one, which illustrates why I should care (and is not a group)?</p>
| Ari | 673 | <p>While the categorical definition of groupoid is the most concise, you can also think of a groupoid as being like a group, except where multiplication is only partially defined, rather than being defined for any pair of elements. Here are a few of my favorite examples:</p>
<ul>
<li><p>Given a vector bundle E, the g... |
744,952 | <p>Is it true that a map between ${\bf T1}$ topological spaces $f:X \to Y$ is surjective iff the induced geometric morphism $f:Sh(Y) \to Sh(X)$ is a surjection (i.e. its inverse image part $f^*$ is faithful)?</p>
<p>In "Sheaves in Geometry and Logic" a proof is given, but the the "if" part leaves me a bit unsatisfied ... | E.P. | 30,935 | <p>It is obvious that this cannot hold unless $f$ is continuous at $0$, because otherwise the value $f(0)$ becomes decoupled to the continuum of values that determine the integral, and the result is meaningless. Therefore, whatever route you take must be based upon this continuity condition.</p>
<p>So, how do you do t... |
2,930,292 | <p>I'm currently learning the unit circle definition of trigonometry. I have seen a graphical representation of all the trig functions at <a href="https://www.khanacademy.org/math/trigonometry/unit-circle-trig-func/unit-circle-definition-of-trig-functions/a/trig-unit-circle-review" rel="nofollow noreferrer">khan academ... | Kurt Schwanda | 592,591 | <p>At least in the first quadrant of the unit circle, the tangent of the angle is equal to the length of the tangent segment connecting the point on the circle to the x-axis.</p>
<p>So in your image, imagine a radius being drawn from the origin to the green point on the circle's circumference (let's call it <span clas... |
2,845,085 | <p>Find $f(5)$, if the graph of the quadratic function $f(x)=ax^2+bx+c$ intersects the ordinate axis at point $(0;3)$ and its vertex is at point $(2;0)$</p>
<p>So I used the vertex form, $y=(x-2)^2+3$, got the quadratic equation and then put $5$ instead of $x$ to get the answer, but it's wrong. I think I shouldn't hav... | Rory Daulton | 161,807 | <p>You are right that one solution is to use the vertex form</p>
<p>$$y=a(x-h)^2+k$$</p>
<p>but you have $k$ wrong. If the vertex is at $(2,0)$ then $h=2$ (which you have) and $k=0$ (which you got wrong). So the equation is now</p>
<p>$$y=a(x-2)^2+0$$
or
$$y=a(x-2)^2$$</p>
<p>Now use the fact that $(0,3)$ is on the... |
514,922 | <p>I need to prove the following affirmation: If $ \lim x_{2n} = a $ and $ \lim x_{2n-1} = a $, prove that $\lim x_n = a $ (in $ \mathbb{R} $ )</p>
<p>It is a simple proof but I am having problems how to write it. I'm not sure it is the right way to write, for example, that the limit of $(x_{2n})$ converges to a:</p>
... | kedrigern | 97,299 | <p>$$\forall\varepsilon>0 \; \exists n_1\in\mathbb{N} \; \forall k>n_1 |a_{2k} - a|<\varepsilon
\\ \forall\varepsilon>0 \; \exists n_2\in\mathbb{N} \; \forall k>n_2 |a_{2k+1} - a|<\varepsilon$$
Therefore if we set $n'=\max\{n_1,n_2\}$ we get
$$\forall\varepsilon>0 \; \exists n'\in\mathbb{N} \; \for... |
2,918,091 | <p>Suppose I want to find the locus of the point $z$ satisfying $|z+1| = |z-1|$</p>
<p>Let $z = x+iy$</p>
<p>$\Rightarrow \sqrt{(x+1)^2 + y^2} = \sqrt{(x-1)^2 + y^2}$ <br/>
$\Rightarrow (x+1)^2 = (x-1)^2$ <br/>
$\Rightarrow x+1 = x-1$ <br/>
$\Rightarrow 1= -1$ <br/>
$\Rightarrow$ Loucus does not exist</p>
<p>Is my a... | cansomeonehelpmeout | 413,677 | <p>Instead of taking the square-root of both sides, try to expand to get $$(x-1)^2=(x+1)^2\\x^2-2x+1=x^2+2x+1\\-2x=2x\\x=0$$</p>
|
1,928,149 | <p>I have the following general question about geodesics.
I know the following equation for a geodesic $\sigma$ on a manifold $M\subset R^n$ of dimension $m$, written in local coordinates:
$${\sigma^k}^{''} (t) + \Gamma_{i,j}^k {\sigma^{i}}'{\sigma^{j}}'=0,$$</p>
<p>for $i,j,k=1, \dots, m$.</p>
<p>Now, if I have a cu... | janmarqz | 74,166 | <p>With a parametrization $\Phi$ from an open set $U$ of $\Bbb R^m$ and a curve $\sigma$ in $U$ you will get $\gamma=\Phi\circ\sigma$. For that $\sigma$ you ought to get your $\sigma^k$ in the geodesic equations.</p>
|
1,168,968 | <p>So I'm doing some cryptography assignment and I'm dealing with a modular arithmetic in hexadecimal. Basically I have the values for $n$ and the remainder $x$, but I need to find the original number $m$, e.g.</p>
<p>$$m \mod 0x6e678181e5be3ef34ca7 = 0x3a22341b02ad1d53117b.$$</p>
<p>I just need a formula to calculat... | user26486 | 107,671 | <p><strong>Hint:</strong></p>
<p>$$\frac{5+\sqrt{x^2+5}}{x-6}=\frac{\frac{5}{x}+\sqrt{\frac{x^2}{x^2}+\frac{5}{x^2}}}{\frac{x}{x}-\frac{6}{x}}=\frac{\frac{5}{x}+\sqrt{1+\frac{5}{x^2}}}{1-\frac{6}{x}}$$</p>
<p>$$\lim_{x\to\infty}\frac{\frac{5}{x}+\sqrt{1+\frac{5}{x^2}}}{1-\frac{6}{x}}=\frac{0+\sqrt{1+0}}{1-0}=1$$</p>
|
2,743,288 | <p>I need help with this exponential equation: $5^{x+2}\ 2^{4-x} = 1000 $</p>
<p>We know that $ 1000 = 10^3$, so:</p>
<p>$$\ln(5^{x+2}\cdot2^{4-x}) = \ln10^3 \implies\ln(5^{x+2}) + \ln(2^{4-x}) = \ln10^3$$</p>
<p>In the next step I use that: $\ln(a^x) = x\ln(a)$</p>
<p>$$(x+2)\ln 5 + (4-x)\ln 2 = 3\ln 10$$</p>
<p... | Michael Hoppe | 93,935 | <p>$1000=5^{x+2}2^{4-x}=5^32^35^{x-1}2^{1-x}\iff 1=(5/2)^{x-1}\iff x=1$</p>
|
2,743,288 | <p>I need help with this exponential equation: $5^{x+2}\ 2^{4-x} = 1000 $</p>
<p>We know that $ 1000 = 10^3$, so:</p>
<p>$$\ln(5^{x+2}\cdot2^{4-x}) = \ln10^3 \implies\ln(5^{x+2}) + \ln(2^{4-x}) = \ln10^3$$</p>
<p>In the next step I use that: $\ln(a^x) = x\ln(a)$</p>
<p>$$(x+2)\ln 5 + (4-x)\ln 2 = 3\ln 10$$</p>
<p... | danimal | 202,026 | <p>Remember that $\ln 5$, $\ln2$ etc are just (real) numbers, so they can be used like real numbers, so:
$$(x+2)\ln(5) + (4-x)\ln(2) = 3\ln(10)$$
expanding the brackets and factoring the $x$ s:
$$x(\ln5 -\ln2) = 3\ln10-2\ln5-4\ln2$$
and combining the logs
$$x\ln(5/2) = \ln1000-\ln25-\ln16 = \ln\left({1000\over 16\times... |
1,836,190 | <p>I've been working on a problem and got to a point where I need the closed form of </p>
<blockquote>
<p>$$\sum_{k=1}^nk\binom{m+k}{m+1}.$$</p>
</blockquote>
<p>I wasn't making any headway so I figured I would see what Wolfram Alpha could do. It gave me this: </p>
<p>$$\sum_{k=1}^nk\binom{m+k}{m+1} = \frac{n((m+2... | Leucippus | 148,155 | <p>The series can also be seen as the following.
\begin{align}
\sum_{k=0}^{n} k \, \binom{m+k}{m+1} &= \frac{1}{m+1} \, \sum_{k=0}^{n} k \, \frac{(m+1)_{k}}{k!} \\
&= \frac{1}{m+1} \, \left[ \sum_{k=0}^{n-2} \frac{(m+1)_{k+2}}{k!} + \sum_{k=0}^{n-1} \frac{(m+1)_{k+1}}{k!} \right] \\
&= (m+2) \, \frac{(n-1) ... |
512,768 | <p>I am trying to intuitively understand why the solution to the following problem is $-2$. $$\lim_{x\to\infty}\sqrt{x^2-4x}-x$$
$$\lim_{x\to\infty}(\sqrt{x^2-4x}-x)\frac{\sqrt{x^2-4x}+x}{\sqrt{x^2-4x}+x}$$
$$\lim_{x\to\infty}\frac{x^2-4x-x^2}{\sqrt{x^2-4x}+x}$$
$$\lim_{x\to\infty}\frac{-4x}{\sqrt{x^2-4x}+x}$$
$$\lim_{... | Paramanand Singh | 72,031 | <p>Neglecting lower order terms while dealing with certain limits is a sort of "thumb rule" which applies in certain circumstances and its proper justification in those circumstances is based on the standard rules of limits. You are trying to assume that this thumb rule is more fundamental than the rules of limits and ... |
57,232 | <p>Given a Heegaard splitting of genus $n$, and two distinct orientation preserving homeomorphisms, elements of the mapping class group of the genus $n$ torus, is there a method which shows whether or not these homeomorphisms, when used to identify the boundaries of the pair of handlebodies, will produce the same $3$-m... | John Sidles | 11,394 | <p>The high level of abstraction that number theorists sustain is a continual source of amazement to me ... doesn't anyone want to see what $\hat\mu(k,n)$ concretely <i>looks</i> like? </p>
<p>So let's do it! Mainly for fun (and as a gesture of respect for Gil), here is a density plot of $n^{1/2} \hat\mu(k,n)$ for all... |
4,411,247 | <blockquote>
<p>If <span class="math-container">$G$</span> is finite group, how to prove that <span class="math-container">$f(g)=ag$</span>, <span class="math-container">$a \in G$</span>, is a bijection for all <span class="math-container">$g \in G$</span>? Here <span class="math-container">$ag$</span> is <span class="... | Berci | 41,488 | <p>The inverse of <span class="math-container">$g\mapsto ag$</span> is simply <span class="math-container">$h\mapsto a^{-1}h$</span>.</p>
|
2,713,311 | <p>$ \lim_{x \to \infty} [\frac{x^2+1}{x+1}-ax-b]=0 \ $ then show that $ \ a=1, \ b=-1 \ $</p>
<p><strong>Answer:</strong></p>
<p>$ \lim_{x \to \infty} [\frac{x^2+1}{x+1}-ax-b]=0 \\ \Rightarrow \lim_{x \to \infty} [\frac{x^2+1-ax^2-ax-bx-b}{x+1}]=0 \\ \Rightarrow \lim_{x \to \infty} \frac{2x-2ax-a-b}{1}=0 \\ \Righta... | user | 505,767 | <p>Note that</p>
<p>$$\frac{x^2+1}{x+1}-ax-b=\frac{x^2+1-ax^2-bx-ax-b}{x+1}=\frac{x^2(1-a)-x(a+b)-b+1}{x+1}$$</p>
<p>and in order to have limi zero we need</p>
<ul>
<li>$(1-a)=0 \implies a=1$</li>
<li>$(a+b)=0\implies b=-1$</li>
</ul>
<p>indeed</p>
<p>$$\frac{x^2(1-1)-x(1-1)-(-1)+1}{x+1}=\frac{2}{x+1}\to 0$$</p>
|
2,401,281 | <blockquote>
<p>Show that for $\{a,b,c\}\subset\Bbb Z$ if $a+b+c=0$ then $2(a^4 + b^4+ c^4)$ is a perfect square. </p>
</blockquote>
<p>This question is from a math olympiad contest. </p>
<p>I started developing the expression $(a^2+b^2+c^2)^2=a^4+b^4+c^4+2a^2b^2+2a^2c^2+2b^2c^2$ but was not able to find any useful... | Dr. Sonnhard Graubner | 175,066 | <p>plugging $$c=-a-b$$ in the term $$2(a^4+b^4+c^4)$$ we get $$4\, \left( {a}^{2}+ab+{b}^{2} \right) ^{2}$$ and this is a perfect square.</p>
|
2,401,281 | <blockquote>
<p>Show that for $\{a,b,c\}\subset\Bbb Z$ if $a+b+c=0$ then $2(a^4 + b^4+ c^4)$ is a perfect square. </p>
</blockquote>
<p>This question is from a math olympiad contest. </p>
<p>I started developing the expression $(a^2+b^2+c^2)^2=a^4+b^4+c^4+2a^2b^2+2a^2c^2+2b^2c^2$ but was not able to find any useful... | lab bhattacharjee | 33,337 | <p>$$a^2+b^2+c^2=(a+b+c)^2-2(ab+bc+ca)=?$$</p>
<p>Now $$(a^2)^2+(b^2)^2+(c^2)^2=(a^2+b^2+c^2)^2-2(a^2b^2+b^2c^2+c^2a^2)$$</p>
<p>$$a^2b^2+b^2c^2+c^2a^2=(ab+bc+ca)^2-abc(a+b+c)=?$$</p>
|
1,714 | <p>I know of two good mathematics videos available online, namely:</p>
<ol>
<li>Sphere inside out (<a href="https://www.youtube.com/watch?v=BVVfs4zKrgk" rel="nofollow noreferrer">part I</a> and <a href="https://www.youtube.com/watch?v=x7d13SgqUXg" rel="nofollow noreferrer">part II</a>)</li>
<li><a href="https://www.yo... | javier | 914 | <p>The "Touching Soap Films" series by Springer. about minimal surfaces. Some excerpts of the video are available here:
<a href="http://page.mi.fu-berlin.de/polthier/video/Touching/Scenes.html" rel="nofollow">http://page.mi.fu-berlin.de/polthier/video/Touching/Scenes.html</a></p>
|
1,714 | <p>I know of two good mathematics videos available online, namely:</p>
<ol>
<li>Sphere inside out (<a href="https://www.youtube.com/watch?v=BVVfs4zKrgk" rel="nofollow noreferrer">part I</a> and <a href="https://www.youtube.com/watch?v=x7d13SgqUXg" rel="nofollow noreferrer">part II</a>)</li>
<li><a href="https://www.yo... | Charles Siegel | 622 | <p>Not lecture videos or anything, but the stuff from <a href="https://www.youtube.com/results?search_query=oliver+labs&search_type=&aq=f" rel="nofollow noreferrer">Oliver Labs</a> is very good for just illustrating geometric stuff, like blowups and dual curves.</p>
|
1,714 | <p>I know of two good mathematics videos available online, namely:</p>
<ol>
<li>Sphere inside out (<a href="https://www.youtube.com/watch?v=BVVfs4zKrgk" rel="nofollow noreferrer">part I</a> and <a href="https://www.youtube.com/watch?v=x7d13SgqUXg" rel="nofollow noreferrer">part II</a>)</li>
<li><a href="https://www.yo... | bhwang | 239 | <p>An excellent (and very lively) overview of basic one-variable calculus: Calculus I in 20 minutes: <a href="http://www.youtube.com/watch?v=EX_is9LzFSY" rel="nofollow">Part I</a>, <a href="http://www.youtube.com/watch?v=Q9OkFTDG4fY" rel="nofollow">Part II</a>.</p>
|
1,714 | <p>I know of two good mathematics videos available online, namely:</p>
<ol>
<li>Sphere inside out (<a href="https://www.youtube.com/watch?v=BVVfs4zKrgk" rel="nofollow noreferrer">part I</a> and <a href="https://www.youtube.com/watch?v=x7d13SgqUXg" rel="nofollow noreferrer">part II</a>)</li>
<li><a href="https://www.yo... | Andy Putman | 317 | <p>Most of the talks at MSRI are videotaped and placed on the web here:</p>
<p><a href="http://www.msri.org/communications/vmath/index_html">http://www.msri.org/communications/vmath/index_html</a></p>
|
1,714 | <p>I know of two good mathematics videos available online, namely:</p>
<ol>
<li>Sphere inside out (<a href="https://www.youtube.com/watch?v=BVVfs4zKrgk" rel="nofollow noreferrer">part I</a> and <a href="https://www.youtube.com/watch?v=x7d13SgqUXg" rel="nofollow noreferrer">part II</a>)</li>
<li><a href="https://www.yo... | Ryan Budney | 1,465 | <p>This isn't purely a math video, it's an interview with Peter Woit and it is something of a summary of the main issues discussed on his blog and in his book. He talks about math vs. physics culture, especially the string theory community.</p>
<p>edit: the link appears to have changed.</p>
<p><a href="https://bigthin... |
1,714 | <p>I know of two good mathematics videos available online, namely:</p>
<ol>
<li>Sphere inside out (<a href="https://www.youtube.com/watch?v=BVVfs4zKrgk" rel="nofollow noreferrer">part I</a> and <a href="https://www.youtube.com/watch?v=x7d13SgqUXg" rel="nofollow noreferrer">part II</a>)</li>
<li><a href="https://www.yo... | Daniel Pape | 11,176 | <p>Some talks on history by some leading mathematicians (mostly in French):</p>
<p><a href="http://www.archivesaudiovisuelles.fr/FR/_LibraryThemas.asp?thema=541" rel="nofollow">http://www.archivesaudiovisuelles.fr/FR/_LibraryThemas.asp?thema=541</a></p>
|
1,714 | <p>I know of two good mathematics videos available online, namely:</p>
<ol>
<li>Sphere inside out (<a href="https://www.youtube.com/watch?v=BVVfs4zKrgk" rel="nofollow noreferrer">part I</a> and <a href="https://www.youtube.com/watch?v=x7d13SgqUXg" rel="nofollow noreferrer">part II</a>)</li>
<li><a href="https://www.yo... | Jesus Martinez Garcia | 1,887 | <p>David Cox's lectures in toric varieties at <a href="http://www.msri.org/web/msri/scientific/workshops/summer-graduate-workshops/show/-/event/Wm463" rel="nofollow">MSRI</a></p>
<p>Something really good to end the evening with :)</p>
|
1,714 | <p>I know of two good mathematics videos available online, namely:</p>
<ol>
<li>Sphere inside out (<a href="https://www.youtube.com/watch?v=BVVfs4zKrgk" rel="nofollow noreferrer">part I</a> and <a href="https://www.youtube.com/watch?v=x7d13SgqUXg" rel="nofollow noreferrer">part II</a>)</li>
<li><a href="https://www.yo... | jfm314 | 25,910 | <p>I have compiled a list (1500+) of math videos at <a href="http://pinterest.com/mathematicsprof/">http://pinterest.com/mathematicsprof/</a> . If anyone is aware of others, please send them to me. </p>
|
1,714 | <p>I know of two good mathematics videos available online, namely:</p>
<ol>
<li>Sphere inside out (<a href="https://www.youtube.com/watch?v=BVVfs4zKrgk" rel="nofollow noreferrer">part I</a> and <a href="https://www.youtube.com/watch?v=x7d13SgqUXg" rel="nofollow noreferrer">part II</a>)</li>
<li><a href="https://www.yo... | user48365 | 48,365 | <p>So much maths video in <a href="http://nptel.ac.in/" rel="nofollow">http://nptel.ac.in/</a> National Programme on Technology Enhanced Learning</p>
|
480,828 | <p>We can suppose that we will create a new number system with essentially two imaginaries that do not interact. (Besides this, all quantities are taken to be integers) For example, we have an $i_1$ and an $i_2$. Then we could say</p>
<p>$$(a+b i_1)(c+d i_1) = ac + (ad + bc)i_1-bd$$</p>
<p>and, similarly for $i_2$... | hmakholm left over Monica | 14,366 | <p><strong>BEWARE. This answer is as wrong as it is possible for it to be without actually containing any technically false statements. See my other answer.</strong> I'm leaving this up because the references to ring theory and quaternions are probably still helpful.</p>
<hr>
<p>If $i_1 i_2=0$, then $i_1$ and $i_2$ a... |
480,828 | <p>We can suppose that we will create a new number system with essentially two imaginaries that do not interact. (Besides this, all quantities are taken to be integers) For example, we have an $i_1$ and an $i_2$. Then we could say</p>
<p>$$(a+b i_1)(c+d i_1) = ac + (ad + bc)i_1-bd$$</p>
<p>and, similarly for $i_2$... | hmakholm left over Monica | 14,366 | <p>Um, scratch the conclusions of my earlier answer. You get into more trouble than mere zero divisors:</p>
<p>$$1 = (-1)\cdot (-1) = i_1^2\cdot i_2^2 = (i_1 i_2)^2 = 0^2 = 0 $$</p>
<p>so everything collapses!</p>
<p>My observation that what you get is a quotient ring was technically right, but I failed to notice th... |
1,452,425 | <p>From what I have been told, everything in mathematics has a definition and everything is based on the rules of logic. For example, whether or not <a href="https://math.stackexchange.com/a/11155/171192">$0^0$ is $1$ is a simple matter of definition</a>.</p>
<p><strong>My question is what the definition of a set is?<... | bof | 111,012 | <p>The definition of <em>set</em> depends on what kind of set theory you're using. Here are two examples.</p>
<p>In the kind of set theory described in the appendix of <a href="https://en.wikipedia.org/wiki/John_L._Kelley" rel="nofollow">John L. Kelley</a>'s <em>General Topology</em> (available at the <a href="https:/... |
1,452,425 | <p>From what I have been told, everything in mathematics has a definition and everything is based on the rules of logic. For example, whether or not <a href="https://math.stackexchange.com/a/11155/171192">$0^0$ is $1$ is a simple matter of definition</a>.</p>
<p><strong>My question is what the definition of a set is?<... | Vlad | 229,317 | <p>I disagree with people who say that <em>set</em> is undefined.
I am also not very fond of definitions which describe <em>set</em> as an <em>"object"</em>, a <em>"list"</em>, or a <em>"collection"</em> of something.
Each of above approaches raise somewhat difficult questions, with <em>"collection"</em> being the leas... |
1,893,280 | <p>How to show $\frac{c}{n} \leq \log(1+\frac{c}{n-c})$ for any positive constant $c$ such that $0 < c < n$?</p>
<p>I'm considering the Taylor expansion, but it does not work...</p>
| kobe | 190,421 | <p>By the mean value theorem, $e^{c/n} = 1 + e^{\alpha}(c/n)$ for some $\alpha\in (0, c/n)$. Since $e^{\alpha} \le e^{c/n}$, then $e^{c/n} \le 1 + e^{c/n}(c/n)$. So $(1 - c/n)e^{c/n} \le 1$, or </p>
<p>$$e^{c/n} \le \frac{1}{1 - c/n} = \frac{n}{n-c} = 1 + \frac{c}{n-c}$$</p>
<p>Now take logarithms.</p>
|
971,160 | <p>So, this is actually 2 questions in 1. I apologize if that is bad practice, but I didn't want to write 2 questions when they're a word different. So, I have</p>
<ol>
<li>Prove or disprove that if $a|(sb+tc), \forall s,t \in\mathbb{Z}$, then $a|b$, and $a|c$.</li>
</ol>
<p>and then,</p>
<ol start="2">
<li>Prove or... | N. S. | 9,176 | <ol>
<li><p>What happens when one of $s,t$ is one and the other one is zero?</p></li>
<li><p>$2014| 2014 \cdot 1 + 2014 \cdot 3$ but $2014$ doesn't divide $1$ or $3$...</p></li>
</ol>
|
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