qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
2,183,809 | <p>The problem:</p>
<blockquote>
<p>If $p$ is prime and $4p^4+1$ is also prime, what can the value of $p$ be?</p>
</blockquote>
<p>I am sure this is a pretty simple question, but I just can't tackle it. I don't even known how I should begin...</p>
| Joffan | 206,402 | <p>If $p\ne 5$, then $p^4\equiv 1 \bmod 5$ by Fermat's Little Theorem. Thus $4p^4+1\equiv 0 \bmod 5$, that is $4p^4{+}1$ is divisible by $5$ and since $4p^4>2^4>5$ this is not prime.</p>
<p>If $p=5$ , $4p^4+1=2501$ is not prime either. So there is no such $p$.</p>
|
2,185,072 | <p>Let A be the matrix: $$\begin{pmatrix} 1&2&3&2&1&0\\2&4&5&3&3&1\\1&2&2&1&2&1 \end{pmatrix}$$.</p>
<p>Show that {$\bigl( \begin{smallmatrix} 1 \\ 4\\3\end{smallmatrix} \bigr)$, $\bigl( \begin{smallmatrix} 3\\4\\1 \end{smallmatrix} \bigr)$} is a basis for th... | puhsu | 417,064 | <p>You can just show, that rank of matrix $A$ is equal to $2$ (you actually did it by reducing $A$). So if rank is $2$ the dimension of a column space is also two. It means that there are two elements in the basis of $A$ column space. And you only need to notice that vectors $(1, 4, 3)^T$ and $(3, 4, 3)^T$ are linearly... |
2,185,072 | <p>Let A be the matrix: $$\begin{pmatrix} 1&2&3&2&1&0\\2&4&5&3&3&1\\1&2&2&1&2&1 \end{pmatrix}$$.</p>
<p>Show that {$\bigl( \begin{smallmatrix} 1 \\ 4\\3\end{smallmatrix} \bigr)$, $\bigl( \begin{smallmatrix} 3\\4\\1 \end{smallmatrix} \bigr)$} is a basis for th... | dantopa | 206,581 | <p>Building on the insights of @puhsu, you know you have 2 vectors that span the image, columns 1 and 3. Can you construct the target vectors in this basis?
$$
\left[ \begin{array}{r}
1 \\ 4 \\3
\end{array} \right]
=
\alpha
\left[ \begin{array}{r}
1 \\ 2 \\ 1
\end{array} \right]
+
\beta
\left[ \begin{array}{r}... |
2,328,567 | <p>I'm trying to search for any kind of development in the mathematics (science, astronomy, even astrology or other kind of early studies that envolve any kind of math) expecially in early england and in the carolingian empire. </p>
<p>The problem I have is that it seams that math died in there: every work seems to be... | Nigel Overmars | 96,700 | <p>Continuing where you stopped: Since $f$ is bijective, we have that $f(f^{-1}(U)) = U$ and since $U$ is open, we have that $f(V)$ is open (in $Y$)</p>
<p>To prove that $f^{-1}$ is continuous, we should prove that for all open sets $U \in X$, $(f^{-1})^{-1}(U)$ is open in $Y$, since $(f^{-1})^{-1}(U) = f(U)$. And si... |
1,469,859 | <p>If I have a positive <code>x</code>, are there more integers below <code>x</code> or above <code>x</code>?</p>
<p>I was discussing this with some friends and we came up with two opposing ideas:</p>
<ol>
<li>No, since you can always count one more in either direction.</li>
<li>Yes, since the infinite amount of numb... | cr001 | 254,175 | <p>Actually there are an equal number of integers:
Denote the positive integer $n$. Then the two sets you are talking about are $S_1=\{x\mid x\in \mathbb{Z} \land x>n \}$ and $S_2=\{x\mid x\in \mathbb{Z} \land x<n \}$</p>
<p>Then consider the following function $f:S_1\rightarrow S_2, f(x) =2n-x$. This is a bijec... |
3,840,643 | <p>Assume that given three predicates are presented below:</p>
<p><span class="math-container">$H(x)$</span>: <span class="math-container">$x$</span> is a horse</p>
<p><span class="math-container">$A(x)$</span>: <span class="math-container">$x$</span> is an animal</p>
<p><span class="math-container">$T(x,y)$</span>: <s... | Taroccoesbrocco | 288,417 | <p>As correctly suggested in lemontree's <a href="https://math.stackexchange.com/a/3840664/288417">answer</a>, "Horses' tails are tails of animals" can be formalized as <span class="math-container">$\forall x \forall y \big((H(y) \land T(x,y)) \to A(y) \big)$</span> or more precisely, <span class="math-contai... |
275,308 | <p>Problems with calculating </p>
<p>$$\lim_{x\rightarrow0}\frac{\ln(\cos(2x))}{x\sin x}$$</p>
<p>$$\lim_{x\rightarrow0}\frac{\ln(\cos(2x))}{x\sin x}=\lim_{x\rightarrow0}\frac{\ln(2\cos^{2}(x)-1)}{(2\cos^{2}(x)-1)}\cdot \left(\frac{\sin x}{x}\right)^{-1}\cdot\frac{(2\cos^{2}(x)-1)}{x^{2}}=0$$</p>
<p>Correct answer i... | lab bhattacharjee | 33,337 | <p>$\lim_{x\to0}\cos 2x=1$ not $0$ unlike $\sin x$ so we can not write $\lim_{x\rightarrow0}\frac{\ln(2\cos^{2}(x)-1)}{(2\cos^{2}(x)-1)}$ to utilize $\lim_{y\to 0}\frac{\ln(1+y)}y=1$</p>
<p>Instead, we can try in the following way: $$\frac{\ln(1-2\sin^2x)}{x\sin x}=(-2)\frac{\ln(1-2\sin^2x)}{(-2\sin^2x)}\frac{\sin x}x... |
2,135,918 | <p>Does L'Hopitals Rule hold for second derivative, third derivative, etc...? Assuming the function is differential up to the $k$th derivative, is the following true?</p>
<p>$$\lim_{x\to c} \frac{f(x)}{g(x)} = \lim_{x\to c} \frac{f'(x)}{g'(x)} = \lim_{x\to c} \frac{f''(x)}{g''(x)} = \cdots = \lim_{x\to c} \frac{f^{(k)... | Simply Beautiful Art | 272,831 | <p>Well, let's see. Under the specified conditions <strong><em>and</em></strong> $f(c)=g(c)=0$ (or both go to $\pm\infty$), then</p>
<p>$$\lim_{x\to c}\frac{f(x)}{g(x)}=\lim_{x\to c}\frac{f'(x)}{g'(x)}$$</p>
<p>Now, if $f'(x)$ and $g'(x)$ still hold all those conditions and $f'(c)=g'(c)=0$ (or both go to $\pm\infty$... |
2,704,955 | <p>In my test on complex analysis I encountered following problem:</p>
<blockquote>
<p>Find $\oint\limits_{|z-\frac{1}{3}|=3} z \text{Im}(z)\text{d}z$</p>
</blockquote>
<p>So first I observed that function $z\text{Im}(z)$ is not holomorphic at least on real axis. Therefore we have to intgrate using parametrization.... | W99 | 546,048 | <p>You can also use Stokes'theorem. Since $z\mathrm{Im}(z)=\frac{1}{2i}(z^2-z\overline{z})$ and $dx\wedge dy=\frac{1}{2i}d\overline{z}\wedge dz$ we have
$$
\oint_{|z-\frac{1}{3}|=3}\frac{z^2-z\overline{z}}{2i}dz=\iint_{|z-\frac{1}{3}|\le3}-z\frac{d\overline{z}\wedge dz}{2i}=\iint_{(x-\frac{1}{3})^2+y^2\le 9}(-x-iy)dx\w... |
1,071,321 | <p>The problem is quite simple to formulate. If you have a large group of people (n > 365), and their birthdays are uniformly distributed over the year (365 days), what's the probability that every day of the year is someone's birthday?</p>
<p>I am thinking that the problem should be equivalent to finding the number o... | Robert Israel | 8,508 | <p>Use the <a href="http://en.wikipedia.org/wiki/Inclusion%E2%80%93exclusion_principle" rel="nofollow">inclusion-exclusion principle</a>. For a given set of $m$ days, the probability that nobody has a birthday on those days is $(1 - m/365)^n$. </p>
<p>EDIT: There are $365 \choose m$ such sets. So the probability th... |
4,496,913 | <blockquote>
<p>If <span class="math-container">$x=\frac12(\sqrt[3]{2009}-\frac{1}{\sqrt[3]{2009}})$</span>, what is the value of <span class="math-container">$(x+\sqrt{1+x^2})^3$</span>?</p>
</blockquote>
<p>I solved this problem as follow,</p>
<p>Assuming <span class="math-container">$\sqrt[3]{2009}=\alpha$</span> , ... | Ninad Munshi | 698,724 | <p>Here's a rather elegant approach that involves pattern matching. Let <span class="math-container">$x=\sinh t$</span>. Then</p>
<p><span class="math-container">$$(x+\sqrt{1+x^2})^3 = (\sinh t + \cosh t)^3 = e^{3t}$$</span></p>
<p>And from comparing definitions</p>
<p><span class="math-container">$$\sinh t = \frac{1}{... |
1,593,679 | <p>While proving that $$\int^{\infty}_0 \frac{\sin x}xdx$$
I saw the Laplace Transform proof. <br>
It used that $$\cal L\left\{\frac{\sin t}{t}\right\}=\int^\infty_0 \cal L\left\{\sin(t)\right\}d\sigma$$
So for understanding it, I tried:
$$\cal L\left\{\frac{\sin t}{t}\right\}=\int^\infty_0e^{-st}\frac{\sin t}{t}dt=\i... | Albert | 85,959 | <p>You can write,</p>
<p>$$
\int_{0}^{\infty}\frac{\sin t}{t}dt=L\bigg\{\frac{\sin t}{t}\bigg\}_{s=0}=\Bigg(\int_{s}^{\infty}L\big\{\sin t\big\}du\Bigg)_{s=0}\\
=\Bigg(\int_{s}^{\infty}\frac{1}{1+u^{2}}du\Bigg)_{s=0}=\Bigg(\tan^{-1}(u)\Bigg|^{\infty}_{s}\Bigg)_{s=0}=\frac{\pi}{2}.
$$</p>
|
3,743,282 | <p>One definition of complex sympletic group I have encountered is (sourced from <a href="https://en.wikipedia.org/wiki/Symplectic_group" rel="nofollow noreferrer">Wikipedia</a>):
<span class="math-container">$$Sp(2n,F)=\{M\in M_{2n\times 2n}(F):M^{\mathrm {T} }\Omega M=\Omega \}$$</span></p>
<p>What is the motivation ... | paul garrett | 12,291 | <p>Such issues become clearer when we do not commit to a basis of a vector space, and do not depend on matrices to express either linear maps or quadratic forms or...</p>
<p>So, given a finite-dimensional <span class="math-container">$k$</span>-vectorspace <span class="math-container">$V$</span>, with <span class="math... |
445,127 | <p>I need to prove this limit:</p>
<blockquote>
<p>Given $f:(-1,1) \to \mathbb{R}\,$ and $\,f(x)>0,\,$ if $\,\lim_{x\to 0} \left(f(x) + \dfrac{1}{f(x)}\right) = 2,\,$ then $\,\lim_{x\to 0} f(x) = 1$.</p>
</blockquote>
| Christian Bueno | 86,451 | <p>Let $z=\lim_{x\to 0} f(x)$, then the first limit can be written as, $z+\dfrac{1}{z}=2$. Equivalently, we can write, $z^2 -2z +1=0$ or $(z-1)^2=0$.</p>
<p>Thus $z=1$ and so $\lim_{x\to 0} f(x)=1$.</p>
|
2,159,083 | <p>I am currently translating a research paper from French (which I do not speak well). I have made good progress with copious use of google translate & switching between French & English versions of articles on wikipedia, coupled with knowledge in the given field. However, I am stuck on the following ($M$ is a... | Community | -1 | <p>The main idea connecting logic to set theory is the idea that <em>propositions are subsets</em>.</p>
<p>On the one hand, if you are interpreting propositional logic in some (set-theoretic) domain $D$ of discourse, then each proposition gets interpreted as a subset of $D$, and connectives are operations on subsets.<... |
891,137 | <p>Count all $n$-length strings of digits $0, 1,\dots, m$ that have an equal number of $0$'s and $1$'s. Is there a closed form expression?</p>
| DSinghvi | 148,018 | <p>Hint : go by calculus (increasing or decreasing)</p>
<p>See the domain ant that's all</p>
|
3,369,438 | <p><span class="math-container">$\begin{array}{|l} \forall xP(x) \vee \forall x \neg P(x) \quad premise
\\ \exists xQ(x) \rightarrow \neg P(x) \quad premise \\ \forall xQ(x) \quad premise
\\\hline \begin{array}{|l} \forall xP(x) \quad assumption
\\\hline \vdots \quad
\\ \forall x\neg P(x) \quad \end{array}
\\\begin{a... | lemontree | 344,246 | <p>What you did so far is correct, and you only have to fill in the "..."s in your subproofs.</p>
<p>The second subproof is easy: You already got your conclusion <span class="math-container">$\forall x \neg P(x)$</span> as an assumption -- just reiterate (R) that line and you're done.</p>
<p>As for the first subproof... |
35,964 | <p>This is kind of an odd question, but can somebody please tell me that I am crazy with the following question, I did the math, and what I am told to prove is simply wrong:</p>
<p>Question:
Show that a ball dropped from height of <em>h</em> feet and bounces in such a way that each bounce is $\frac34$ of the height of... | Brandon Carter | 1,016 | <p>Hint: You are forgetting to count the distance traveled on the way up from the bounce. If the initial drop is from height $h$, the ball bounces up $\frac{3h}{4}$ and then drops from the same distance, and so on.</p>
|
192,784 | <p>First let me try to describe in more details below the approach of
"reordering" digits of Pi, which is used in OEIS A096566</p>
<p><a href="https://oeis.org/A096566" rel="nofollow noreferrer">https://oeis.org/A096566</a></p>
<p>and what I have done analyzing it so far.</p>
<p>I am looking at first 620 "reordered"... | Community | -1 | <blockquote>
<p>I also received suggestion that in order to test whether this discussed above feature is only characteristic to (some initial digits of) Pi, the same reordering should be applied to the some significant number of randomly generated very long strings of decimal digits - to see if there the same pattern... |
40,500 | <blockquote>
<p>What are the most
fundamental/useful/interesting ways in
which the concepts of Brownian motion,
martingales and markov chains are
related?</p>
</blockquote>
<p>I'm a graduate student doing a crash course in probability and stochastic analysis. At the moment, the world of probability is a conf... | Simon Lyons | 9,564 | <p>If X is a continuous martingale of finite variation such that $X_0 = 0$, then $P(X_t = 0 \ \ \forall t) = 1$. </p>
|
59,486 | <p>Chern-Weil theory tells us that the integral Chern classes of a flat bundle over a compact manifold (i.e. a bundle admitting a flat connection) are all torsion. Given a compact manifold $M$ whose integral cohomology contains torsion, one can then ask which (even-dimensional) torsion classes appear as the Chern clas... | Tom Goodwillie | 6,666 | <p>Small correction: For a non-compact manifold $M$, the group $H_{2k-1}(M)$ might not be finitely generated. In this case Chern-Weil does not imply that the $k$th Chern class of a flat bundle on $M$ has finite order. Rather, it just implies that it belongs to the subgroup $Ext(H_{2k-1}(M),\mathbb Z)\subset H^{2k}(M)$.... |
2,372,743 | <p>Show that $$\lim_{x\to\infty} \int_{1}^x e^{-y^2} dy$$
exist and is in $[e^{-4},1]$</p>
<p>I cannot find a good minoration, I must show it whitout using the double integration, I totally don't know how to solve it</p>
| Brevan Ellefsen | 269,764 | <p><strong>Existence:</strong> Note that $e^{-x^2} \le e^{-x}$ for all $x \in [1,\infty)$ </p>
<hr>
<p><strong>Bounds:</strong>
$$\begin{align}
\int_1^x e^{-t^2} dt &=\int_1^3 e^{-t^2} dt +\int_3^\infty e^{-t^2} dt\\
&<\int_1^3 e^{-t^2} dt +\int_3^\infty e^{-t} dt
\end{align}$$
Estimate the finite integr... |
4,133,760 | <p>Dr Strang in his book linear algebra and it's applications, pg 108 says ,when talking about the left inverse of a matrix( <span class="math-container">$m$</span> by <span class="math-container">$n$</span>)</p>
<blockquote>
<p><strong>UNIQUENESS:</strong> For a full column rank <span class="math-container">$r=n . A x... | Still Learning | 362,881 | <p>As an add on to user169852’s proof, I would note that Strang’s argument that <span class="math-container">$ A^T A $</span> is invertible if A has full column rank is fairly simple.</p>
<p>He shows that for any matrix A, <span class="math-container">$ A^T A $</span> has the same nullspace as A:</p>
<p>(1) Clearly the... |
1,221,586 | <p>How would one compute?
$$ \oint_{|z|=1} \frac{dz}{\sin \frac{1}{z}} $$</p>
<p>Can one "generalize" the contour theorem and take the infinite series of the residues at each singularity? </p>
| GEdgar | 442 | <p>Another method for this case: Think of the outside of the unit disk as your domain. Go around the contour in the opposite direction. Outside the unit disk, the function
$$
\frac{1}{\sin\frac{1}{z}}
$$
is meromorphic. It has only one singularity in that region: a pole with residue $-1/6$ at $\infty$. So the value... |
1,916,297 | <p>Let $\{F_n\}_{n=1}^\infty $ be the Fibonacci sequence (defined by
$ F_n=F_{n-1}+F_{n-2}$ with $F_1=F_2=1$).</p>
<p>Is it true that for every integer number $N$ there exist positive integers
$n,m,k,l$ such that $N=F_n+F_m-F_k-F_l$ ?</p>
| user133281 | 133,281 | <p><strong>Short answer:</strong> No, because the Fibonacci sequence grows exponentially.</p>
<p><strong>Longer answer:</strong> First suppose $\{n,m\} \cap \{k,\ell\} \neq \emptyset$, say $n=l$. Then $N = F_m - F_k$. This implies that $N = F_m - F_k \geq F_m - F_{m-1} = F_{m-2}$. It follows that $m = O(\log(N))$ beca... |
628,236 | <p>I am trying to give a name to this axiom in a definition: </p>
<p>$(X \bullet R) \sqcup (Y \bullet S) \equiv (X \sqcup Y) \bullet (R \sqcup S)$</p>
<p>(for all $X, Y, R, S$) where $\sqcup$ is the join of a lattice and $\bullet$ is some binary operation. It feels related to monotonicity/distributivity but I don't k... | Community | -1 | <p>Alternatively, it is a quadratic polynomial in $Z$. Complete the square in $Z$. You will something like $(Z-A)^2+B$. For it to be reducible $B$ must be a square (over the complex numbers). But $B$ is again a quadratic polynomial in the remaining variables. If it is a square, it is reducible. Repeat the same work as ... |
628,236 | <p>I am trying to give a name to this axiom in a definition: </p>
<p>$(X \bullet R) \sqcup (Y \bullet S) \equiv (X \sqcup Y) \bullet (R \sqcup S)$</p>
<p>(for all $X, Y, R, S$) where $\sqcup$ is the join of a lattice and $\bullet$ is some binary operation. It feels related to monotonicity/distributivity but I don't k... | Igor Rivin | 109,865 | <p>Yet another way: substitute some value for $Y,$ and check that the resulting polynomial is irreducible. My favorite value is $Y=2.$</p>
|
810,190 | <blockquote>
<p>Page 29 of Source 1: Denote the complex conjugate by * :
$\mathbf{u \cdot v} = \sum_{1 \le i \le n} u_i^*v_i = (\mathbf{v \cdot u})^*$</p>
<p><a href="http://www.math.sunysb.edu/~eitan/la13.pdf" rel="nofollow">Page 1 of Source 2:</a> $\mathbf{u \cdot v} = \mathbf{u}^T\mathbf{ \bar{v} }$.</p>
... | Branimir Ćaćić | 49,610 | <p>So, as you observed, there are two conventions in the literature for Hermitian inner products:</p>
<ol>
<li>anti-linearity in the first argument, linearity in the second (Sources 1 and 3);</li>
<li>linearity in the first argument, anti-linearity in the second (Source 2).</li>
</ol>
<p>Convention (1) is standard in... |
133,936 | <p>I am trying to understand a part of the following theorem:</p>
<blockquote>
<p><strong>Theorem.</strong> Assume that $f:[a,b]\to\mathbb{R}$ is bounded, and let $c\in(a,b)$. Then, $f$ is integrable on $[a,b]$ if and only if $f$ is integrable on $[a,c]$ and $[c,b]$. In this case, we have
$$\int_a^bf=\int_a^cf+\in... | Jonathan Prieto-Cubides | 29,241 | <p>i look your question, and i'm thinking the following.
By Definition, with the partition $P$ you have:
$ U(f,P) - L(f,P) <\epsilon$ so the difference will be near zero, because we say in terms inf and sup for sums Riemman, and $U(f,P)\geq L(f,P)$.</p>
<p>So, If $P_1=P\cap [a,c]$ and $P_2 = P \cap [c,b]$ should b... |
1,318,934 | <blockquote>
<p>Let $q$ be an odd prime power. Consider the map $f:\Bbb F_{q^3} \rightarrow \Bbb F_{q^3}$, defined by
$$f(x)=\alpha x^q+\alpha^q x$$
for some fixed $\alpha \in \Bbb F_{q^3} \setminus \{ 0 \}$.
Show that $f$ is a bijection.</p>
</blockquote>
<p>Hint: If $\beta \in \ker(f) \setminus \{ 0 \}$ c... | Adam Hughes | 58,831 | <p>Consider</p>
<p>$$g(x) =f(x)-f(a)=\alpha(x-a)^q+\alpha^q(x-a)$$</p>
<p>since $q\equiv 0$ in our field and the binomial theorem holds. Now if $f$ has a double value, say $f(a)$ which is taken on twice, then $g$ has a two zeros. However, if $b\ne a$ is such a pair of zeros, we have</p>
<p>$$\alpha(b-a)^q+\alpha^q(b... |
239,544 | <p>I'm working with the code below, which works well for my purposes; however, instead of the canonical ordering seen in the output, I want the output produced in the order I give in the Cases line of code. Any help is most appreciated.</p>
<pre><code>ClearAll;
m = {{N11, "x"}, {N12, "y"}, {N19, &q... | thorimur | 63,584 | <p>Welcome to MMA SE! The assumption <code>Element[t, Reals]</code> is only used inside <code>Refine</code>, and so in a sense is not "known to" any evaluations outside of that expression. <code>Refine</code> doesn't change the nature of <code>A</code>; it merely spits out an expression produced from its argu... |
239,544 | <p>I'm working with the code below, which works well for my purposes; however, instead of the canonical ordering seen in the output, I want the output produced in the order I give in the Cases line of code. Any help is most appreciated.</p>
<pre><code>ClearAll;
m = {{N11, "x"}, {N12, "y"}, {N19, &q... | E. Chan-López | 53,427 | <p>Using <strong>ComplexExpand</strong> command:</p>
<pre><code>A = 5 t^2 + 3 t + 4;
ComplexExpand[Conjugate[A]]
</code></pre>
<p>For expressions within radicals, we must use the <strong>Refine</strong> command for the <strong>ComplexExpand</strong> command to work properly. Here is an example:</p>
<p>Without Refine co... |
195,006 | <p>I am not very familiar with mathematical proofs, or the notation involved, so if it is possible to explain in 8th grade English (or thereabouts), I would really appreciate it.</p>
<p>Since I may even be using incorrect terminology, I'll try to explain what the terms I'm using mean in my mind. Please correct my term... | Tarnation | 37,670 | <p>The short answer is, yes. </p>
<p>Every infinite set has a countably infinite subset. In lay terms, 'countably infinite' means that we can index each element of the set with an integer, such as $a_5$. Say your original set, call it $S$,has a total ordering on it (I am assuming this is what you mean); this says we h... |
3,201,996 | <p>I have an 8-digit number and you have an 8-digit number - I want to see if our numbers are the same without either of us passing the other our actual number. Hashing the numbers is the obvious solution. However, if you send me your hashed number and I do not have it - it is very easy to hash all the permutations of ... | Ross Millikan | 1,827 | <p>There are only <span class="math-container">$10^8$</span> eight digit numbers, even if you allow leading zeros. No deterministic operation can increase the entropy of this. Your example of squaring is a good one. It makes the number larger, but does not increase the number of results. You need more possible code... |
1,225,122 | <p>So I am currently studying a course in commutative algebra and the main object that we are looking at are ideals generated by polynomials in n variables. But the one thing I don't understand when working with these ideals is when we reduce the generating set to something much simpler. For e.g.</p>
<p>Consider the I... | Bernard | 202,857 | <p>$I=\langle(x-1)(x-3),(x-1)(x+2)\rangle=(x-1)\langle x-3,x+2\rangle =(x-1)A$, since $x-3$ and $x+2$ are coprime.</p>
<p>You can't apply the same reasoning to $\langle x^3-x^2+x\rangle$ and $\langle x\rangle$.</p>
|
3,283,600 | <p>"Find all the complex roots of the following polynomials</p>
<p>A) <span class="math-container">$S(x)=135x^4 -324x^3 +234x^2 -68x+7$</span>, knowing that all its real roots belong to the interval <span class="math-container">$(0.25;1.75)$</span></p>
<p>B)<span class="math-container">$M(x)=(x^3 -1+i)(5x^3 +27x^2 -2... | Michael Rozenberg | 190,319 | <p>Since <span class="math-container">$\frac{1}{3}$</span> is a root of <span class="math-container">$S$</span>, we obtain:
<span class="math-container">$$S=135x^4-324x^3+234x^2-68x+7=$$</span>
<span class="math-container">$$=135x^4-45x^3-279x^3+93x^2+141x^2-47x-21x+7=$$</span>
<span class="math-container">$$=(3x-1)(45... |
588,488 | <p>I know that for the harmonic series
$\lim_{n \to \infty} \frac1n = 0$ and
$\sum_{n=1}^{\infty} \frac1n = \infty$.</p>
<p>I was just wondering, is there a sequence ($a_n =\dots$) that converges "faster" (I am not entirely sure what's the exact definition here, but I think you know what I mean...) than $\frac1n$ to ... | Eric Thoma | 35,667 | <p>If we use the notion of a partial sum:</p>
<p>$$S_n = \sum_{k=1}^n a_k$$</p>
<p>you are asking for a series in which the partial sums diverge more slowly than $\sum_{k=1}^n 1/k$ as $n\to\infty$. For this to happen, we just need to find $a_k$ such that $a_k < 1/k$ for all $k>c$ where $c$ is some value.</p>
<... |
249,332 | <p>Consider the following list</p>
<pre><code>list={{{0,0,0},0},{{0,0,1},a},{{0,0,-1},-a},{{1,0,1},b},{{1,0,0},-b},{{1,1,1},a+b},{{1,1,1},a-b},{{-1,0,-1},{-a-b}},{{-1,0,-1},{-a+b}}};
</code></pre>
<p>how can this list be sorted such that a>b>0, therefore the expected result would be</p>
<pre><code>list={{{1,1,1},... | Lukas Lang | 36,508 | <p>After fixing your list (the last two items had a list wrapped around the second element), the following works:</p>
<pre><code>list = {{{0, 0, 0}, 0}, {{0, 0, 1}, a}, {{0, 0, -1}, -a}, {{1, 0, 1}, b}, {{1, 0, 0}, -b}, {{1, 1, 1}, a + b}, {{1, 1, 1}, a - b}, {{-1, 0, -1}, -a - b}, {{-1, 0, -1}, -a + b}};
Sort[list, F... |
312,649 | <p>Hello how to show the following:</p>
<p>Let $(X,\tau)$ be a topological space then a single point is compact but not
necessarily closed.</p>
<p>Thank you!</p>
| Brian M. Scott | 12,042 | <p>HINT: Let $X$ be the <a href="http://en.wikipedia.org/wiki/Sierpi%C5%84ski_space" rel="nofollow">Sierpiński space</a>: $X=\{0,1\}$, and the open sets are $\varnothing,\{1\}$, and $X$. Is $\{1\}$ a compact set in $X$? Is it a closed set?</p>
|
312,649 | <p>Hello how to show the following:</p>
<p>Let $(X,\tau)$ be a topological space then a single point is compact but not
necessarily closed.</p>
<p>Thank you!</p>
| Community | -1 | <p>Take $\Bbb{R}$ with the trivial topology. The set $\{0\}$ is compact but is not closed.</p>
|
757,917 | <p>According to <a href="http://www.wolframalpha.com/input/?i=sqrt%285%2bsqrt%2824%29%29-sqrt%282%29%20=%20sqrt%283%29" rel="nofollow">wolfram alpha</a> this is true: $\sqrt{5+\sqrt{24}} = \sqrt{3}+\sqrt{2}$</p>
<p>But how do you show this? I know of no rules that works with addition inside square roots.</p>
<p>I not... | lab bhattacharjee | 33,337 | <p>$$5+\sqrt{24}=(\sqrt3)^2+(\sqrt2)^2+2\cdot\sqrt2\cdot\sqrt3=(\sqrt3+\sqrt2)^2$$</p>
|
3,684,331 | <p>I am working on a problem from my Qual</p>
<p>"Let <span class="math-container">$T:V\to V$</span> be a bounded linear map where <span class="math-container">$V$</span> is a Banach space. Assume for each <span class="math-container">$v\in V$</span>, there exists <span class="math-container">$n$</span> s.t. <span cla... | Conrad | 298,272 | <p>Maybe an easier way to do it, though conceptually similar with @Integrand</p>
<p>By multiplying out and simplifying the terms, while noting that <span class="math-container">$x(x+\pi)^2-x^3=\pi x(x+2\pi)=\pi(x+\pi)^2-\pi^3$</span>, the inequality is equivalent to:</p>
<p><span class="math-container">$(\pi-\sin x)(... |
268,635 | <p>Consider the triangle formed by randomly distributing three points on a circle. What is the probability of the center of the circle be contained within the triangle?</p>
| Christian Blatter | 1,303 | <p>Let $A$ be the first point chosen, $M$ the center of the circle, and $AA'$ the diameter through $A$. The probability that the other two points $B$ and $C$ lie to either side of $AA'$ is ${1\over2}$. Given that this is the case, the central angles $X:=\angle(AMB)$ and $Y:=\angle(AMC)$ are independently and uniforml... |
268,635 | <p>Consider the triangle formed by randomly distributing three points on a circle. What is the probability of the center of the circle be contained within the triangle?</p>
| Rayhunter | 76,236 | <p>Take any triangle that contains the centre. Connect each vertex with the centre and continue the lines until they cross the circle. Now form the three 'outer' triangles, as shown in Figure 1. </p>
<p>These sets of 4 triangles are uniquely determined by any one of them, meaning that you can reconstruct all of them i... |
1,058,515 | <p>determine whether $v_1=(1,1,2)$ , $v_2=(1,0,1)$ and $v_3=(2,2,3)$ span $\mathbb{R}^3$</p>
<p>I think that it must prove that it linear Independence </p>
<p>let $α_1v_1+α_2v_2+α_3v_3=(0,0,0)$</p>
<p>so what next , is this a correct way or I'm completely wrong </p>
<p>please guide me to understand this</p>
<p>t... | ml0105 | 135,298 | <p>You have three vectors and $dim(\mathbb{R}^{3}) = 3$. So all you have to test for is linear independence. You can do this by constructing a matrix $A$ as JohnD suggested. Then just take $det(A)$. The vectors form a basis iff $det(A) \neq 0$.</p>
|
1,058,515 | <p>determine whether $v_1=(1,1,2)$ , $v_2=(1,0,1)$ and $v_3=(2,2,3)$ span $\mathbb{R}^3$</p>
<p>I think that it must prove that it linear Independence </p>
<p>let $α_1v_1+α_2v_2+α_3v_3=(0,0,0)$</p>
<p>so what next , is this a correct way or I'm completely wrong </p>
<p>please guide me to understand this</p>
<p>t... | Community | -1 | <p>There is a theorem in linear algebra which says that any set of $n$ linearly independent vectors in an $n$-dimensional space is a basis for that space. $\Bbb R^3$ is $3$-dimensional (I assume you've proven this at some point this semester?) and you've got $3$ vectors. Thus those vectors form a basis for $\Bbb R^3$... |
3,080,664 | <p>Hi I've almost completed a maths question I am stuck on I just can't seem to get to the final result. The question is:</p>
<p>Find the Maclaurin series of <span class="math-container">$g(x) = \cos(\ln(x+1))$</span> up to order 3.</p>
<p>I have used the formulas which I won't type out as I'm not great with Mathjax ... | José Carlos Santos | 446,262 | <p>Since<span class="math-container">$$\cos x=1-\frac{x^2}2+\cdots,$$</span>you have<span class="math-container">\begin{align}\cos\bigl(\log(x+1)\bigr)&=1-\frac12\left(x-\frac{x^2}2+\frac{x^3}3-\cdots\right)^2+\cdots\\&=1-\frac{x^2}2+\frac{x^3}2+\cdots\end{align}</span>Note that this argument works because the ... |
155,373 | <p>Consider $X,Y \subseteq \mathbb{N}$. </p>
<p>We say that $X \equiv Y$ iff there exists a bijection between $X$ and $Y$.</p>
<p>We say that $X \equiv_c Y$ iff there exist a bijective computable function between $X$ and $Y$. </p>
<p>Can you show me some examples in which the two concepts disagree?</p>
| Kaveh | 468 | <p>The structure with the computable equivalence (which is defined as $A\leq_T B$ and $B \leq_T A$) is called <a href="http://en.wikipedia.org/wiki/Turing_degree" rel="nofollow">Turing degrees</a> and has a very rich structure (unlike the usual bijection).</p>
|
1,220,002 | <p>I am performing the ratio test on series and I come across many situations where I have any of those 3 in the numerator and denominator and I was wondering what I could cancel as n approaches infinity.</p>
| user157227 | 157,227 | <p>If I am interpreting your question correctly, here is how you can relate $n^n$ and $(n+1)^n$.</p>
<p>$$\lim_{n\to\infty} \frac{(n+1)^n}{n^n} = \lim_{n\to\infty} \left(\frac{n+1}{n}\right)^n = \lim_{n\to\infty} \left(1 + \frac{1}{n}\right)^n = e.$$</p>
<p>More generally,</p>
<p>$$\lim_{n\to\infty} \frac{(n+k)^n}{n... |
1,220,002 | <p>I am performing the ratio test on series and I come across many situations where I have any of those 3 in the numerator and denominator and I was wondering what I could cancel as n approaches infinity.</p>
| Alessandro17 | 224,351 | <p>You should know the fundamental limit
$$ \lim_{n\to +\infty} \frac{(n+1)^n}{n^n} = \lim_{n\to +\infty} \Bigl(\frac{n+1}{n}\Bigr)^n = \lim_{n\to +\infty} \Bigl(1 + \frac{1}{n}\Bigr)^n = e $$
so when $n$ approaches $+\infty$ the expressions $n^n$ and $(n+1)^n$ differ only by a constant.
On the other hand we have
$$ \l... |
1,906,332 | <p>I know every complex differentiable function is continuous. I would like to know if the converse is true. If not, could someone give me some counterexamples?</p>
<p><strong>Remark:</strong> I know this is not true for the real functions (e.g. $f(x)=|x|$ is a continuous function in $\mathbb R$ which is not different... | Surb | 154,545 | <p>No, for example, any non constant function s.t. for all $z$ you have $f(z)\in \mathbb R$. As you mentioned, $$f(z)=|z|$$
is a good counter-example. But a beautiful property, is that any complex differentiable function are infinitely differentiable. </p>
|
1,906,332 | <p>I know every complex differentiable function is continuous. I would like to know if the converse is true. If not, could someone give me some counterexamples?</p>
<p><strong>Remark:</strong> I know this is not true for the real functions (e.g. $f(x)=|x|$ is a continuous function in $\mathbb R$ which is not different... | operatorerror | 210,391 | <p>A classic example is $f(z)=\bar{z}$. It is continuous everywhere but there is no open disk in which it is holomorphic, and therefore it is nowhere holomorphic.</p>
|
11,724 | <p>What is the compelling need for introducing a theory of $p$-adic integration?</p>
<p>Do the existing theories of $p$-adic integration use some kind of analogues of Lebesgue measures? That is, do we put a Lebesgue measure on $p$-adic spaces, and just integrate real or complex valued functions on $p$-adic spaces, or ... | Community | -1 | <p>I must say I do not know anything much about it; but the following is the general idea. </p>
<p>Note that in the case of cohomology theories for smooth projective varieties over $\mathbb C$, there are various canonical isomorphisms between de Rham cohomology, Betti cohomology, etale cohomology, etc.. For instance t... |
164,321 | <p>I'm searching for a suitable (hopefully simple enough) solution to the following form of integral:</p>
<p>$$\int_0^\infty \mathrm{d}x~x^n J_\nu(a x) J_\nu(b x) K_\mu(c x) $$</p>
<p>Where $n$, $\nu$, and $\mu$ are all integers, and $a$, $b$, and $c$ are all real and positive.</p>
<p>If not generally, a specific ca... | Zurab Silagadze | 32,389 | <p>The integral should be calculable by the Mellin-transform technique. See the calculation of the similar (but different) integral in <a href="http://www.opticsinfobase.org/josaa/abstract.cfm?uri=josaa-7-7-1218" rel="nofollow">http://www.opticsinfobase.org/josaa/abstract.cfm?uri=josaa-7-7-1218</a> (Analysis of propaga... |
4,330,852 | <blockquote>
<p>Let <span class="math-container">$A_1 ,\ldots ,A_m$</span> be subsets of an <span class="math-container">$n$</span>-element set such that <span class="math-container">$|A_i \cap A_j |\leq t$</span> for all <span class="math-container">$i\neq j$</span>. Prove that <span class="math-container">$$\sum_{i=... | nonuser | 463,553 | <p>Use of <a href="https://en.wikipedia.org/wiki/Boole%27s_inequality#Bonferroni_inequalities" rel="nofollow noreferrer">Bonferroni inequalitie</a>: <span class="math-container">$$n\geq \Big|\bigcup _{i=1}^n A_i\Big| \geq \sum _{i=1}^m|A_i|-\sum _{1\leq i<j\leq m} |A_i\cap A_j| \geq \sum _{i=1}^m|A_i| - t\cdot {m\ch... |
4,618,975 | <p>I've read the definition of the ring homomorphism:<br />
<strong>Definition</strong>. Let <span class="math-container">$R$</span> and <span class="math-container">$S$</span> be rings. A ring homomorphism is a function <span class="math-container">$f : R → S$</span> such that:<br />
(a) For all <span class="math-cont... | Math.mx | 15,654 | <p><span class="math-container">$f\colon R\longrightarrow R\otimes_\mathbb{Z} S$</span> where <span class="math-container">$S$</span> is another unital ring
given by <span class="math-container">$f(r)=r\otimes 1$</span></p>
<p><span class="math-container">$f\colon R\longrightarrow M_n(R)$</span> given by <span class="m... |
1,856,409 | <p>In business with 80 workers, 7 of them are angry. If the business leader visits and picks 12 randomly, what is the probability of picking 12 where exactly 1 is angry? </p>
<p>(7/80)<em>(73/79)</em>(72/78)<em>(71/77)</em>(70/76)<em>(69/75)</em>(68/74)<em>(67/73)</em>(66/72)<em>(65/71)</em>(64/70)*(63/69)*12=0.413458... | Arthur | 15,500 | <p>There is one easier way to do it, which even lets you directly use the result from the first part as part of your calculations: "at least $2$" and "at most $1$" are complementary events, which means that the probability of one of them happening is equal to one minus the probability of the other happening, or written... |
1,519,251 | <p>This question was in school maths challenge.
I dont know how to approach this one.. any help would be appreciated.</p>
| JMoravitz | 179,297 | <p>First, note that $n!$ will end in a number of zeroes equal to $\sum\limits_{i=1}^\infty \lfloor\frac{n}{5^i}\rfloor$</p>
<p>Alternatively, this could be written as $\sum\limits_{i=1}^n a_i$ where $a_i$ is the largest integer value such that $5^{a_i}\mid i$.</p>
<p>This can be seen by the fact that in the expansion... |
275,527 | <pre><code>Sqrt[Matrix[( {
{0, 1},
{-1, 0}
} )]] /. f_[Matrix[x__]] :> Matrix[MatrixFunction[f, x]]
</code></pre>
<p><code>Matrix</code> is an undefined symbol but I want to define some substitutions with it.</p>
| mikado | 36,788 | <p>If you look at the <code>FullForm</code> of <code>Sqrt</code>, you see that it is silently converted to a <code>Power</code>, a function taking two arguments, not one. Your pattern only matches functions of a single argument.</p>
<pre><code>FullForm[Sqrt[x]]
(* Power[x, Rational[1, 2]]*)
</code></pre>
|
602,248 | <p>This is the system of equations:
$$\sqrt { x } +y=7$$
$$\sqrt { y } +x=11$$</p>
<p>Its pretty visible that the solution is $(x,y)=(9,4)$</p>
<p>For this, I put $x={ p }^{ 2 }$ and $y={ q }^{ 2 }$.
Then I subtracted one equation from the another such that I got $4$ on RHS and factorized LHS to get two factors in te... | Igor Rivin | 109,865 | <p>Solve the first equation for $x,$ the second for $y,$ to get</p>
<p>$$ x = (y-7)^2.$$</p>
<p>$$ y = (x-11)^2.$$</p>
<p>Substitute the value for $x$ into the second equation, to get:</p>
<p>$$y = ((y-7)^2 -11)^2 = (y^2-14 y + 38)^2.$$ This is a quartic equation, but dividing through by $y-4,$ you get an irreducib... |
602,248 | <p>This is the system of equations:
$$\sqrt { x } +y=7$$
$$\sqrt { y } +x=11$$</p>
<p>Its pretty visible that the solution is $(x,y)=(9,4)$</p>
<p>For this, I put $x={ p }^{ 2 }$ and $y={ q }^{ 2 }$.
Then I subtracted one equation from the another such that I got $4$ on RHS and factorized LHS to get two factors in te... | Max | 2,633 | <p>Second equation implies $y$ is a perfect square, first one implies it's at most $7$. There are only 3 options - $y=0$, $y=1$, $y=4$. The first two don't work (they lead to $x=49$ and $x=36$ respectively), so $y=4$, $x=9$ is the only integral solution.</p>
|
1,384,203 | <p>The additive character of a finite field $\mathbb F_q$ is obtained by using the trace function to base field $F_p$.</p>
<p>Can we write some of them into a middle field. Using trace function from $\mathbb F_{q^n}$ to $\mathbb F_q$? If yes, how?</p>
| xxxxxxxxx | 252,194 | <p>Not for an additive character; since a finite field has characteristic $p$, an additive map from $\mathbb{F}_{q^{n}} \to \mathbb{C}$ must send every element to a $p$th root of unity. If you wrote into a middle field $\mathbb{F}_{q}$, how would you use these elements to obtain an additive map? I may be misunderstand... |
210,071 | <p>I have some lines of code that generate numerical solutions to equations. Then I want to combine two of these in a piecewise function. The way I did it is the following -lf1[r] and lf4[r] are the aforementioned numerical solutions</p>
<pre><code>test[r_] :=
Piecewise[{{lf1[r], 0.688199 <= r <= 10}, {lf4[r],... | Carl Woll | 45,431 | <p>You can use a single plot if you separate the <a href="http://reference.wolfram.com/language/ref/Piecewise" rel="noreferrer"><code>Piecewise</code></a> conditions into separate <a href="http://reference.wolfram.com/language/ref/ConditionalExpression" rel="noreferrer"><code>ConditionalExpression</code></a> objects.</... |
1,804,360 | <p>The following is a classically valid deduction for any propositions <span class="math-container">$A,B,C$</span>.
<span class="math-container">$\def\imp{\rightarrow}$</span></p>
<blockquote>
<p><span class="math-container">$A \imp B \lor C \vdash ( A \imp B ) \lor ( A \imp C )$</span>.</p>
</blockquote>
<p>But I'... | Paul | 939,509 | <p>That is <strong>G</strong>(Gödel–Dummett logic: <a href="https://en.wikipedia.org/wiki/Intermediate_logic#Properties_and_examples" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Intermediate_logic#Properties_and_examples</a>).
<span class="math-container">$$
\forall A\forall B\forall C((A\to B\lor C)\to (A\... |
4,590,713 | <p>If I know the McLaurin series of
<span class="math-container">$$f(x)=\sum_{n=0}^\infty a_n x^n,$$</span>
can I say something about the McLaurin series of
<span class="math-container">$$e^{f(x)}=\sum_{n=0}^\infty b_n x^n?$$</span>
In other words, what is the relation between <span class="math-container">$a_n$</span> ... | Gary | 83,800 | <p>A complete asymptotic expansion may be obtained as follows. First
<span class="math-container">$$
\sum\limits_{k = 1}^n {\sqrt {k\log k} } \le \sqrt {n\log n} + \sum\limits_{k = 1}^{n - 1} {\int_k^{k + 1} {\sqrt {t\log t} \,{\rm d}t} } = \sqrt {n\log n} + \int_1^n {\sqrt {t\log t} \,{\rm d}t}
$$</span>
and
<spa... |
3,817,760 | <p>In geometry class, it is usually first shown that the medians of a triangle intersect at a single point. Then is is explained that this point is called the centroid and that it is the balance point and center of mass of the triangle. Why is that the case?</p>
<p>This is the best explanation I could think of. I ho... | Intelligenti pauca | 255,730 | <p>I think your approach is very simple and effective: you can substitute every chord of a triangle parallel to a fixed side with a mass at its midpoint, proportional to the length of the chord: the centre of mass of the triangle is the same as the centre of mass of those masses. But those masses are aligned along a me... |
1,190,798 | <p>I'm having difficulty understanding when to use $\cos$ and $\sin$ to find $x$ and $y$ components of a vector.
Do we always use $\cos$ for $x$-component or what?</p>
| dk2ax | 223,875 | <p>One thing I've found useful is having a mental picture of the <span class="math-container">$\sin$</span> and <span class="math-container">$\cos$</span> functions. If you draw your vector and split it up into components like so:</p>
<p><img src="https://i.stack.imgur.com/SJOhg.gif" alt="enter image description here" ... |
38,594 | <p>One of the definitions of a Galois extension is that $E/K$ is Galois iff $E$ is the splitting field of some <strong>separable</strong> polynomial $f(x) \in K[x]$, yes?</p>
<p>I want to understand why the following is true:</p>
<blockquote>
<p>Let $f(x) \in \mathbb{Q}[x]$ be a polynomial and let $F$ be a splittin... | Qiaochu Yuan | 232 | <p>We don't. What we do know is that every irreducible polynomial over a field of characteristic zero is separable, and $f$ is the product of its irreducible factors, so its splitting field is a compositum of separable extensions, hence separable. (In other words, the splitting field of $f$ is the splitting field of a ... |
3,963,517 | <blockquote>
<p>Write to power series of <span class="math-container">$$f(x)=\frac{\sin(3x)}{2x}, \quad x\neq 0.$$</span></p>
</blockquote>
<p>I try to get a series for <span class="math-container">$\sin(3x)$</span> with <span class="math-container">$x=0$</span> and multiplying the series with <span class="math-contain... | Clerni | 866,477 | <p>Using the Taylor series for the sine:
<span class="math-container">$$\sin{3x}=\sum_{n=0}^{\infty} (-1)^{n}\frac{(3x)^{2n+1}}{(2n+1)!}$$</span></p>
<p>We can write the function <span class="math-container">$f(x)$</span> as:</p>
<p><span class="math-container">$$f(x)=\frac{\sin{3x}}{2x}=\frac{1}{2x}\sum_{n=0}^{\infty}... |
1,563,004 | <p>Assume we that we calculate the expected value of some measurements $x=\dfrac {x_1 + x_2 + x_3 + x_4} 4$. what if we dont include $x_3$ and $x_4$, but instead we use $x_2$ as $x_3$ and $x_4$. Then We get the following expression $v=\dfrac {x_1 + x_2 + x_2 + x_2} 4$.</p>
<p>How do I know if $v$ is a unbiased estimat... | Prabhdeep Singh | 294,228 | <p>$$\dfrac {\Bbb d} {\Bbb dx} [10x \dfrac {\ln x} {\ln 10}] =\dfrac {\frac {\Bbb d} {\Bbb dx} (10x \ln x)} {\ln 10} = \dfrac {\frac {\Bbb d} {\Bbb dx} [10^x] \ln x + 10^x \frac {\Bbb d} {\Bbb dx} \ln x} {\ln 10} = \dfrac {\ln 10 10^x \ln x + \frac 1 x 10^x} {\ln 10} = \dfrac {\ln 10 10^x \ln x + \frac {10^x} x} {\ln 1... |
858,250 | <p>I would like to evaluate the following alternating sum of products of binomial coefficients:
$$\sum_{k=0}^{m} (-1)^k \binom m k \binom n k .$$
I had the idea to use Pascal recursion to re-express $\binom n k$ so that we always have $m$ as the upper index and I have been able to come up with nice expressions for
$$ ... | Mhenni Benghorbal | 35,472 | <p>Here is a nice closed form in terms of <a href="http://en.wikipedia.org/wiki/Jacobi_polynomials" rel="nofollow">Jacobi polynomials</a> </p>
<blockquote>
<p>$$ \sum_{k=0}^{m} (-1)^k \binom m k \binom n k = P_{m}^{(0,-n-m-1)}(3) . $$</p>
</blockquote>
|
440,452 | <blockquote>
<p>Let $b,c \in \mathbb{Z} $ and let $n \in \mathbb{N} $, $n \ge 2. $ Let $f(x) = x^{n} -bx+c$. Prove that $$\hbox{disc} (f(x)) = n^{n }c^{ n-1}-(n-1)^{n-1 }b^{n }.$$</p>
</blockquote>
<p>Here $\hbox{disc} (f(x)) = \prod_{i} f'(\alpha_{i} )$ where $\alpha_{1}, \dots, \alpha_{n}$ are the roots of $f(x)$.... | Jyrki Lahtonen | 11,619 | <p>Two solutions. The first is based on resultants. As I wasn't 100% confident in my handling of the leading coefficients, I calculated the discriminant also using the definition. I am keeping the first (worse) solution for "educational purposes".</p>
<hr>
<p>A promising way is to use the description of the discrimin... |
3,830,971 | <p>The function <span class="math-container">$f(x)=\cot^{-1} x$</span> is well known to be neither even nor odd because <span class="math-container">$\cot^{-1}(-x)=\pi-\cot^{-1} x$</span>. it's domain is <span class="math-container">$(-\infty, \infty)$</span> and range is <span class="math-container">$(0, \pi)$</span>.... | Z Ahmed | 671,540 | <p>I think the Range of <span class="math-container">$f(x)=\cot^{-1} x$</span> as <span class="math-container">$(0,\pi)$</span> and hence the mixed parity of this function gets preference as then <span class="math-container">$f(x)$</span> is continuous in the its domain <span class="math-container">$(-\infty, \infty)$... |
1,430,369 | <p>I know how to prove this by induction but the text I'm following shows another way to prove it and I guess this way is used again in the future. I'm confused by it.</p>
<p>So the expression for first n numbers is:
$$\frac{n(n+1)}{2}$$</p>
<p>And this second proof starts out like this. It says since:</p>
<p>$$(n+1... | Mark Bennet | 2,906 | <p>The proof is a trick, of course. Working in the opposite direction, the idea is to write $$2\sum_{r=1}^nr=\sum 2r=\sum \left ((2r+1)-1\right)=\sum (2r+1)-\sum 1=\sum (2r+1)-n$$</p>
<p>This seems elaborate, but the point is to write as much of the sum as possible in a form which cancels. We use $2r+1$ for this becau... |
3,625,739 | <p>My question is from Humphreys <em>Introduction to Lie Algebras and Representation Theory</em>. There is a lemma in section 6.3 which says, if <span class="math-container">$\phi: L \to \mathfrak{gl}(V)$</span> is a representation of a semisimple Lie algebra <span class="math-container">$L$</span> then <span class="ma... | José Carlos Santos | 446,262 | <p>Because a linear map <span class="math-container">$\psi$</span> from a <span class="math-container">$1$</span>-dimensional vector space <span class="math-container">$V$</span> into itself can only be of the form <span class="math-container">$v\mapsto\lambda v$</span> and then <span class="math-container">$\lambda=\o... |
50,479 | <p>Can the number of solutions $xy(x-y-1)=n$ for $x,y,n \in Z$ be unbounded as n varies?</p>
<p>x,y are integral points on an Elliptic Curve and are easy to find using enumeration of divisors of n (assuming n can be factored).</p>
<p>If yes, will large number of solutions give moderate rank EC?</p>
<p>If one drops $... | Chris Wuthrich | 5,015 | <p>With the transformation <span class="math-container">$X = -n/x$</span> and <span class="math-container">$Y= ny/x$</span>, the curve becomes isomorphic to the Weierstrass model
<span class="math-container">$$ E_n\colon \ \ Y^2 - X\ Y - n\ Y = X^3.$$</span>
The points in question are exactly the integral points in <s... |
59,949 | <p>How to track <code>Initialization</code> step by step? Any tricky solution is ok. One condition, <code>x</code> should be <code>DynamicModule</code> variable.</p>
<pre><code>DynamicModule[{x},
Dynamic[x],
Initialization :> (
x = 1;
Pause@1;
x = 2;
Pause@1;
x = 3;
)]... | WReach | 142 | <p>By default, <code>DynamicModule</code> uses <code>SynchronousInitialization -> True</code>. This causes the initialization to be performed on the preemptive link, disabling any updates to the front-end. In particular, print statements, cell creation and dynamic box updates will all be deferred until the initial... |
851,647 | <p>I have
$$
X_i, i=1,...,n_1,
$$
$$
Y_j, j=1,...,n_2,
$$
$$
Z_k, k=1,...,n_3,
$$
whereat $X_i, Y_j$ and $Z_k$ are all independent. Does then follow that
$$
\bar{X}=\frac{1}{n_1}\sum_{i=1}^{n_1}X_i, \bar{Y}=\frac{1}{n_2}\sum_{=1}^{n_2}Y_j, \bar{Z}=\frac{1}{n_3}\sum_{k=1}^{n_3}Z_k
$$
independent, too?</p>
| Vishwa Iyer | 71,281 | <p>$$V = ((F + \frac{PG}{B}) * A^M) + \frac{HP}{B}$$
Solving it for $P$ means isolating $P$. So we want $P$ by itself. First we multiply both sides by $B$. $$BV = A^MBF+A^MPG+HP$$ $$BV = A^MBF+P(A^MG+H)$$ $$BV - A^MBF = P(A^MG+H)
$$ $$P = \frac{BV-A^MBF}{A^MG+H}$$</p>
|
160,392 | <p>Did I solve the following quadratic equation correctly.</p>
<p>$$W(W+2)-7=2W(W-3)$$</p>
<p>I got.</p>
<p>$$W^2-8W+7$$
Then for my solution I got.</p>
<p>$$(W-1)(W-7)$$</p>
| Gigili | 181,853 | <p>$$W(W+2)-7=2W(W-3)$$</p>
<p>$$\Downarrow$$</p>
<p>$$W^2+2W-7=2W^2-6W$$</p>
<p>$$\Downarrow$$</p>
<p>$$W^2-8W+7=0$$</p>
<p>$$\Downarrow$$</p>
<p>$$(W-1)(W-7)=0$$</p>
<p>You're right, well done. The solutions are $W=7$ and $W=1$.</p>
<p><strong>EDIT</strong>: It should be written as I showed above, you've omit... |
4,023,366 | <p>So the question is pretty much described in the title already. I have to show the following result. I have tried it but am failing to do so. Anyone who can please help me in understanding its proof. I have attached the picture of formulas that may prove to be helpful.</p>
<p><span class="math-container">$\exists x.P... | Simone | 586,627 | <p>Hint: the proof has 3 instances of an elimination rule.</p>
<p>Consider that the premise has as the principal operator a Vel, whereas on the right we have an existential quantifier.</p>
<p>Try to work backward to see what you need: clearly the last rule will be an application of <span class="math-container">$\exists... |
1,660,361 | <blockquote>
<p>$$x^4+x^2-2$$</p>
</blockquote>
<p>How do I write in a that can be used for partial fraction? taking out $x$ does not help, what is the process? </p>
<p>using Wolfram I got to $(x^2+2)(x-1)(x+1)$</p>
| John | 314,739 | <p>Find the greatest common factor $k$ amongst the exponents of the nonzero powers of $x$ and perform a substitution of $y$ for $x^k$. Often, this substitution is the first step in seeing directly how to quickly factor a relatively simple polynomial into its irreducible parts, for, the polynomial in $y$, from the subst... |
9,880 | <p>The ban on edits that change only a few characters can be quite annoying in some cases. I don't have a problem with trying to keep trivial spelling corrections and such out of the peer review queue, but sometimes a small edit is semantically significant. For example, it makes perfect sense to edit an otherwise good ... | PA6OTA | 127,690 | <p>I've run into the same problem. This is particularly bad for math, as one character can really dramatically change the essence of a question. </p>
<p>Of course, there are workarounds suggested, like adding a comment describing the nature of your edit. Maybe that's the way to go.</p>
|
3,837,856 | <p>In <a href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.29.6143&rep=rep1&type=pdf" rel="nofollow noreferrer">Fast Exact Multiplication by the Hessian</a> equation 1,</p>
<p><span class="math-container">$O(\left\Vert\Delta w\right\Vert^2)$</span> gets taken from RHS to LHS and <span class="math-c... | Alex Ortiz | 305,215 | <p>When we write <span class="math-container">$x = O(z)$</span>, where <span class="math-container">$x$</span> is some expression, and <span class="math-container">$z$</span> is some other quantity, then we mean that <span class="math-container">$x$</span> is a quantity such that <span class="math-container">$|x| \le C... |
412,944 | <p>Is it mathematically acceptable to use <a href="https://math.stackexchange.com/questions/403346/prove-if-n2-is-even-then-n-is-even">Prove if $n^2$ is even, then $n$ is even.</a> to conclude since 2 is even then $\sqrt 2$ is even? Further more using that result to also conclude that $\sqrt [n]{2}$ is even for all n?<... | john | 79,781 | <p>Note that that theorem begins with "Suppose $n$ is an integer, and that $n^2$ is even." So it does not hold when considering $\sqrt{2}$</p>
|
2,086,598 | <p>Let's start with a well known limit:
$$\lim_{n \to \infty} \left(1 + {1\over n}\right)^n=e$$</p>
<p>As $n$ approaches infinity, the expression evaluates Euler's number $e\approx 2.7182$. Why that number? What properties does the limit have that leads it to 2.71828$\dots$ and why is this number important? Where can ... | Community | -1 | <p>I know only one amazing property. </p>
<hr>
<p>For any number $a$, $(a^x)^\prime = ba^x$,where $\displaystyle b = \lim_{\Delta x \to 0} {a^{\Delta x} - 1\over \Delta x} = st\left({a^{\Delta x} - 1\over \Delta x}\right)$</p>
<p>eg :- </p>
<blockquote>
<p>$$(10^x)^\prime = b(10^x)$$
Where $$b = \lim_{\Delta... |
2,086,598 | <p>Let's start with a well known limit:
$$\lim_{n \to \infty} \left(1 + {1\over n}\right)^n=e$$</p>
<p>As $n$ approaches infinity, the expression evaluates Euler's number $e\approx 2.7182$. Why that number? What properties does the limit have that leads it to 2.71828$\dots$ and why is this number important? Where can ... | Ng Chung Tak | 299,599 | <p>Firstly,</p>
<p>\begin{align*}
e^{x} &= 1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\ldots \\
(e^{x})' &= e^{x}
\end{align*}</p>
<p>Secondly,</p>
<p>\begin{align*}
y &= e^{x} \\
\frac{dy}{dx} &= y\\
x &= \ln y \\
\frac{dx}{dy} &= \frac{1}{y} \\
\frac{d}{dy} \ln y &= \frac{1}{y} \\
... |
416,514 | <p>I really think I have no talents in topology. This is a part of a problem from <em>Topology</em> by Munkres:</p>
<blockquote>
<p>Show that if $A$ is compact, $d(x,A)= d(x,a)$ for some $a \in A$. </p>
</blockquote>
<p>I always have the feeling that it is easy to understand the problem emotionally but hard to expr... | DonAntonio | 31,254 | <p>Hints:</p>
<p>(1) By definition of distance between points and set:</p>
<p>$$D:=d(x,A):=\inf_{a\in A}d(x,A)$$</p>
<p>(2) By definition of infimum:</p>
<p>$$\forall\,n\in\Bbb N\;\exists\,a_n\in A\;\;s.t.\;\;D\le d(x,a_n)\le D+\frac1n$$</p>
<p>(3) The sequence $\,\{a_n\}\,$ has a subsequence </p>
<p>$$\,\{a_{n_k... |
1,136,458 | <p>Suppose $G$ is an abelian group. Then the subset $H= \big\{ x\in G\ | \ x^3=1_G \big\}$ is a subgroup of $G$.</p>
<p>I was able to show the statement is correct but the weird thing is that I didn't use the fact $G$ is abelian. <strong>Is it possible that the fact $G$ is abelian is redundant?</strong></p>
<p>Here ... | Geoff Robinson | 13,147 | <p>In fact, the sequence $( 1 + \frac{1}{n})^{n}$ is an increasing sequence, which is bounded above by $e$, and $e$ is in fact the least upper bound for the sequence, hence is the limit.
The answer is that there is a fault in the computer's accuracy, there is no counterexample to the theory. If the computer contradicts... |
3,225,151 | <p><span class="math-container">$ZFC+V=L$</span> implies that <span class="math-container">$P(\mathbb{N})$</span> is a subset of <span class="math-container">$L_{\omega_1}$</span>. But I’m wondering what layer of the constructible Universe contains a smaller set.</p>
<p>My question is, what is the smallest ordinal <s... | Not Mike | 480,631 | <p>What you want is the least <span class="math-container">$\delta \in \mathsf{On}$</span> such that <span class="math-container">$L_\delta \vDash "\mathsf{ZFC}^{-}+V=HC"$</span> (where <span class="math-container">$V=HC$</span> is the assertion that every set is hereditarily countable.) Since the theory "<span class="... |
747,997 | <p>in <a href="https://math.stackexchange.com/questions/239521/why-no-trace-operator-in-l2">Why no trace operator in $L^2$?</a>
it is mentioned, that there exists a linear continuous trace operator from $L^2(\Omega)$ to $H^\frac12(\partial\Omega)$* for sufficiently smooth boundary. Can you give me any reference for thi... | Jan | 59,638 | <p>I recommend you have a look into Girault&Raviart's book on <em>Finite element methods for Navier-Stokes equations</em>. Chapter 1 provides a survey on basic concepts on Sobolev spaces including also the pretty involved definition of the trace spaces. The result you are out for is given in Thm. 1.5. </p>
<p>Pers... |
747,997 | <p>in <a href="https://math.stackexchange.com/questions/239521/why-no-trace-operator-in-l2">Why no trace operator in $L^2$?</a>
it is mentioned, that there exists a linear continuous trace operator from $L^2(\Omega)$ to $H^\frac12(\partial\Omega)$* for sufficiently smooth boundary. Can you give me any reference for thi... | bastienchaudet | 390,718 | <p>In Girault&Raviart's book, Thm 1.5 does not apply in this case. Indeed, the continuity of the trace operator from $W^{s,p}(\Omega)$ to $W^{s-\frac{1}{p},p}(\Omega)$ holds only if $s-\frac{1}{p}>0$ (this is implied by the assumptions).</p>
|
1,638,490 | <p>Yes, I know the title is bizarre. I was urinating and forgot to lift the seat up. That made me wonder: assuming I maintain my current position, is it possible for the toilet seat (assume it is a closed, but otherwise freely deformable curve) to be moved/deformed such that stream does not pass through the hole anymor... | Milo Brandt | 174,927 | <p>Purely topologically, the situation here is the same as having a ring (a toilet seat) surrounding a ray (with you at the end of the ray, urine in the middle, and the toilet anchored "off at infinity"). So, one can simply move the ring off the ray, since the ray has an end. Bringing this into real world terms, this l... |
2,967,626 | <p>I have a graph that is 5280 units wide, and 5281 is the length of the arc.</p>
<p><img src="https://i.stack.imgur.com/4rFK4.png"></p>
<p>Knowing the width of this arc, and how long the arc length is, how would I calculate exactly how high the highest point of the arc is from the line at the bottom?</p>
| saulspatz | 235,128 | <p>The usual way of getting rid of square roots is to square both sides. Of course, this increases the degrees, but at least it gets rid of the root. Rewrite the last inequality as <span class="math-container">$$r-{b^2r\over z^2}< b\sqrt{1-{b^2\over r^2}}\tag{1}$$</span> Now the right-hand side is nonnegative, so... |
55,071 | <p><strong>Bug introduced in 10.0.0 and persisting through 10.3.0 or later</strong></p>
<hr>
<p>I've upgraded my home installation of <em>Mathematica</em> from version 9 to 10 today on a Windows 8.1 machine, and I'm getting a weird font issue - the fonts are not anti-aliased, and look unbalanced and weird. Just look:... | Community | -1 | <p>As a follow-up to Tetsuo Ichii's <a href="https://mathematica.stackexchange.com/a/55356/31159">answer</a>, the thinnings observed in version 10 are also present in 11.0.1:</p>
<p><a href="https://i.stack.imgur.com/BSF4X.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/BSF4X.png" alt="enter image d... |
31,948 | <p>A Finsler manifold is defined as a differentiable manifold with a metric defined on so that any well-defined curves of finite arc length is given by a generalized arc length integral of an asymmetric norm over each tangent space defined at a point. This generalizes the Riemannian manifold structure since the norm is... | Deane Yang | 613 | <p>No, a Finsler metric is in general not semiriemannian. As you and José indicate, a semiriemannian metric is always given by a nondegenerate quadratic form on tangent vectors at each point in the manifold. In other words, if you fix a basis of the tangent space for a given point, then the norm of that vector is given... |
183,233 | <p>I am having some problems joining sigma algebras. So I have:
<span class="math-container">$$\textit{K} =\left \{ A\cap B: A\in \sigma (D),B\in \sigma (E) \right \}$$</span>
and I need to show
<span class="math-container">$$\sigma(K)= \sigma (D,E)$$</span></p>
<h3>What I've done so far</h3>
<p>I intend to do this by ... | tomasz | 30,222 | <p>Notice that $K$ contains $A$ and $B$. The definition of $\sigma(A,B)$ would be $\sigma(A\cup B)$, I think. Not that it makes for much of a difference, but still...</p>
<p>I think you're overthinking it. I believe that the first part is actually <em>harder</em> (if only a little bit).</p>
<p>Also, you might want to... |
183,233 | <p>I am having some problems joining sigma algebras. So I have:
<span class="math-container">$$\textit{K} =\left \{ A\cap B: A\in \sigma (D),B\in \sigma (E) \right \}$$</span>
and I need to show
<span class="math-container">$$\sigma(K)= \sigma (D,E)$$</span></p>
<h3>What I've done so far</h3>
<p>I intend to do this by ... | Stéphane Laurent | 38,217 | <p>Your definition of $K$ has no sense: you should say for instance $K=\sigma(X\cap Y, X \in\sigma(A), Y\in\sigma(B))$ (because $A \in \sigma(A)$ is puzzling).</p>
<p>The first inclusion $\sigma(K) \subset \sigma(A,B)$ is easy: it follows from the inclusions $\sigma(A) \subset \sigma(A,B)$ and $\sigma(B) \subset \sig... |
651,707 | <p>Let $A=(a_{ij})\in \mathbb{M}_n(\mathbb{R})$ be defined by</p>
<p>$$
a_{ij} =
\begin{cases}
i, & \text{if } i+j=n+1 \\
0, & \text{ otherwise}
\end{cases}
$$
Compute $\det (A)$</p>
<hr>
<blockquote>
<p>After calculation I get that it may be $(-1)^{n-1}n!$. Am I right?</p>
</blockquote>
| dato datuashvili | 3,196 | <p>for example let us take $3X3$ matrix,then it would be following matrix</p>
<p>a=[0 0 1;0 2 0;3 0 0]</p>
<p>a =</p>
<pre><code> 0 0 1
0 2 0
3 0 0
det(a)
ans =
-6
</code></pre>
<p>in your case if we compute $(-1)^{n-1}*n!=(-1)^2*n!=6$</p>
<p>maybe it is $(-1)^{n}*n!$</p>
|
3,792,242 | <p>A large part of the set theory is devoted to infinities of various kinds, and this has been built on Cantor's groundbreaking work on uncountable sets. However, even Cantor's proof is based on the assumption that certain sets exist, namely that the power set of a countably infinite set exists. Sure, after assuming th... | Doctor Who | 812,056 | <p>Firstly, let's discuss the situation in classical logic (ordinary logic where <span class="math-container">$P \lor \neg P$</span> is a tautology). Suppose the existence of a set <span class="math-container">$S$</span>, and suppose that the set <span class="math-container">$\{f : S \to 2\}$</span> exists, where <span... |
3,341,471 | <p>Let <span class="math-container">$M$</span> be a multiplication operator on <span class="math-container">$L^2(\mathbb{R})$</span> defined by <span class="math-container">$$Mf(x) = m(x) f(x)$$</span> where <span class="math-container">$m(x)$</span> is continuous and bounded. Prove that <span class="math-container">$M... | Kavi Rama Murthy | 142,385 | <p><span class="math-container">$X$</span> is closed ans bounded, hence compact. If <span class="math-container">$\lambda \notin X$</span> the there exists <span class="math-container">$\epsilon >0$</span> such that <span class="math-container">$|\lambda -y| >\epsilon$</span> for all <span class="math-container">... |
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