qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
1,913,320 | <blockquote>
<p>Let <span class="math-container">$A=(a_{ij})_{n\times n}$</span> and <span class="math-container">$A=(a_{ij})_{n\times n}$</span> be two upper triangular matrices, i.e. <span class="math-container">$a_{ij}=b_{ij}=0$</span> whenever <span class="math-container">$i>j$</span>.</p>
<p><span class="math-c... | Sri-Amirthan Theivendran | 302,692 | <p>Typically one proves that every natural number $n>1$ is divisible by a prime by strong induction. The idea of the proof is to construct a number which is not divisible by any prime supposing that we had only finitely many primes $p_1, \dotsc, p_n$. This number is $p_1p_2\cdots p_n+1$ which leads to a contradictio... |
1,005,154 | <p>I don't know how to advance in the following <em><strong>problem</strong></em>:</p>
<p>Let $X$, $Y$ and $Z$ independent random variables equally distributed with uniform distribution over $[0,1]$.</p>
<ul>
<li>Find the joint pdf of $W:=XY$ and $V:=Z^2$.</li>
</ul>
<hr>
<p><em><strong>I tried to</strong></em> ans... | Did | 6,179 | <blockquote>
<p>I don't know what limits to use.</p>
</blockquote>
<p>Note that $x=w/u$, $y=u$, $z=\sqrt{v}$ with $0\leqslant x,y,z\leqslant1$ hence the domain of integration is $$0\leqslant w/u,u,\sqrt{v}\leqslant1,$$ or, equivalently, $$0\leqslant w\leqslant u\leqslant1,\qquad0\leqslant v\leqslant1.$$</p>
<blockq... |
2,445,655 | <p>Challenge: A Good Deal</p>
<p>You are currently learning some important aspects of collusion and cartels. This challenge puts you in the position of a bad guy, namely a price-fixing sales manager. Suppose that you find yourself in a so-called “smoke-filled room” to fix prices for the upcoming year with the sales ma... | José Carlos Santos | 446,262 | <p>\begin{align}\lim_{x\to-\infty}2x+\sqrt{4x^2+x}&=\lim_{x\to-\infty}\frac{\left(2x+\sqrt{4x^2+x}\right)\left(2x-\sqrt{4x^2+x}\right)}{2x-\sqrt{4x^2+x}}\\&=\lim_{x\to-\infty}\frac{-x}{2x-\sqrt{4x^2+x}}\\&=-\lim_{x\to-\infty}\frac1{2+\sqrt{4+\frac1x}}\text{ (because $x<0$)}\\&=-\frac14.\end{align}</p... |
2,445,655 | <p>Challenge: A Good Deal</p>
<p>You are currently learning some important aspects of collusion and cartels. This challenge puts you in the position of a bad guy, namely a price-fixing sales manager. Suppose that you find yourself in a so-called “smoke-filled room” to fix prices for the upcoming year with the sales ma... | cip999 | 483,763 | <p>$$\begin{align*} \lim_{x \rightarrow -\infty} 2x + \sqrt{4x^2 + x} & = \lim_{x \rightarrow \infty} \sqrt{4x^2 - x} - 2x = \\ & = \lim_{x \rightarrow \infty} \frac{(\sqrt{4x^2 - x} - 2x)(\sqrt{4x^2 - x} + 2x)}{\sqrt{4x^2 - x} + 2x} = \\ & = \lim_{x \rightarrow \infty} -\frac{x}{\sqrt{4x^2 - x} + 2x} = \bo... |
2,194,190 | <p>I need to check the irreducibility of $p(x) \in F[x]$, where $F$ is a finite field.
I have read and checked on several exercises on the internet. Their solutions are as follows:</p>
<p>For instance, let $p(x)$ an arbitrary polynomial in $\mathbb{Z}_5[x]$. </p>
<p>If $p(x)$ has no zeros in $\mathbb{Z}_5$, then they... | Yunus Syed | 422,304 | <p>In $Z_5[x]$ the polynomial $p(x) = x^4 + 1$. This can be checked to have no zeroes in your field.</p>
<p>It seems the definition you are using is incorrect. Reducible means factorable into polynomials of lesser degree.</p>
|
2,903,163 | <p>I don't really know whether to put this in Physics forums since it is relating to Mechanics, or Math since the question is actually about the math being done. Don't criticize me over it.</p>
<p>So for the question: I was doing some review problems on Lagrange's equations, KE+PE, and I found <a href="http://wwwf.impe... | Quiver | 564,698 | <p>I think this question belongs to PSE! But whatever, here's your answer: you have to remember that $\dot x$ is a complete derivative of $x$ with respect to time. Going to a new representation of $x$ in a new system, like in your case $x(r,\theta,\phi)$ for the spherical coordinates, where, and this is important, all ... |
2,903,163 | <p>I don't really know whether to put this in Physics forums since it is relating to Mechanics, or Math since the question is actually about the math being done. Don't criticize me over it.</p>
<p>So for the question: I was doing some review problems on Lagrange's equations, KE+PE, and I found <a href="http://wwwf.impe... | Ahmed S. Attaalla | 229,023 | <p>This is a (relatively tedious) application of the chain and product rule.</p>
<hr>
<p>$$z=r \cos \theta$$</p>
<p>$$\frac{dz}{dt}=\frac{d}{dt} \left( r \cos \theta \right)$$</p>
<p>Applying the product rule,</p>
<p>$$=\frac{dr}{dt} \cos \theta+ \frac{d \cos \theta}{dt} r$$</p>
<p>Applying the chain rule,</p>
<... |
1,501,595 | <blockquote>
<p>Let $A$ be an integral domain. Show that $\dim(A)=0 \iff A$ is a field.</p>
</blockquote>
<p>The backward implication is trivial.</p>
<p>For the forward implication, if we can show that $1 \in <a>$, where $a(\neq 0) \in A$. Then, we are done. However, I don't know how to show it. </p>
<p>Any ... | rschwieb | 29,335 | <p>Since maximal ideals are prime, "zero-dimensional" becomes the same thing as "prime ideals are maximal ideals."</p>
<p>In a domain $\{0\}$ is prime, and in a zero-dimensional domain it is also maximal.</p>
<p>So, there are no other ideals besides $\{0\}$ and $A$. Do you know what this implies about $A$? (There are... |
184,719 | <p>Let $\Sigma_g$ be a Riemann surface of genus $g\geq 2$ and $G=\pi_1(\Sigma_g)$.
Let $\pi\colon \mathbb{H}\to \Sigma_g$ be the universal covering map. What kind of surface is $\mathbb{H}/[G,G]$? </p>
<p>Moreover, what is $[G,G]$; e.g. if $g=2$?</p>
| Misha | 21,684 | <p>Let me start by interpreting the question "What kind of surface is $S$?" in the case of a general connected oriented topological surface (without boundary). (I am considering only oriented surfaces just for simplicity of discussion.)
If $S$ had finite complexity, i.e., would be homeomorphic to the interior of a com... |
2,624,837 | <p>What is the integral of $$\int a^{x-1}dx?$$</p>
<p>is it $$\frac{a^{x-1}}{\log(a)} + c?$$</p>
<p>How can we derive the proper integral? Also can you please tell me the definite integral with limits, say b to c?</p>
| Dr. Sonnhard Graubner | 175,066 | <p>it is $$\frac{1}{a}\int a^xdx=\frac{1}{a}\frac{a^x}{\ln(a)}+C$$</p>
|
2,479,918 | <p>Every vector space $V$ could be embedded into $V^{\ast}$ (see <a href="https://en.wikipedia.org/wiki/Dual_basis" rel="noreferrer">here</a>) after choosing a basis, for a given vector $v \in V$ denote this embedding by $v^{\ast}\in V^{\ast}$. Now for given vector spaces $V_1, \ldots, V_k$ over some field $F$, let $V ... | Pedro | 23,350 | <p>I haven't read all the details of your construction, but if you look at Halmos' classic text on linear algebra, <em>Finite dimensional vector spaces</em>, you will see he defines $V\otimes W$ as the linear dual of bilinear forms $V\times W\to k$, where $k$ is the base field, which seems to be what you suggest. This... |
74,271 | <p>Hello, all!</p>
<p>I have a big sum of log-normal (with location parameter $\mu$ and scale parameter $\sigma$) random variables $X_i$ $\sum_{i=1}^N X_i$ with $N \gg 1$.
How could I estimate convergence rate to a gaussian distribution relative to $\mu$ and $\sigma$?</p>
<p>Thank you.</p>
| Brendan McKay | 9,025 | <p>If you search at Google Scholar for "sum of lognormal" or "sum of log-normal" (using the quotation marks), you will find several papers devoted to this question.</p>
|
393,378 | <p>Let <span class="math-container">$K=\mathbb{Q}(x)$</span> be the rational functions in one variable <span class="math-container">$x$</span> and let the automorphisms <span class="math-container">$\phi,\psi$</span> of <span class="math-container">$K$</span> be defined as <span class="math-container">$\phi(x)=-\frac{1... | R.P. | 17,907 | <p>The answer as to the surjectivity of <span class="math-container">$\alpha$</span> is <strong>no</strong>. As in algebraic number theory, the simplest way to prove that an element is not a norm is by local considerations. Let us consider
<span class="math-container">$$
y=\frac{(x^3-3x-1)(x^3+3x^2-1)}{x^2(x+1)^2},
$$<... |
760,926 | <p>Show that $\binom{n}{0} - \binom{n}{1} + \binom{n}{2} - ...+(-1)^k * \binom{n}{k} = (-1)^k * \binom{n-1}{k}$.</p>
<p>I know this has to do with permutations and combination problems, but I'm not sure how would I start with this problem. </p>
| Igor | 66,242 | <p>I assume that $n \geq 1$ and $K \geq 0$. We need to prove that
$\sum_{k=0}^{K}(-1)^k\binom{n}{k}=(-1)^K\cdot\binom{n-1}{K}$</p>
<p>I like proofs by induction. So we fix some $n \geq 1$ and our base case is $K = 0$. In this case we have (recall that for any $~n~$ $\binom{n}{0} = 1$):</p>
<p>$LHS: ~~ \sum_{k=0}^{0}(... |
670,522 | <p>In my very young mathematical career, I have worked a lot with modular forms. Recently, I worked as a teaching assistant in a course about geometry. At the end of the course, we dealt with hyperbolic geometry. It seems as if there is some relation between hyperbolic geometry and modular forms, for example, why is it... | dmk | 88,878 | <p>The OP, I assume, knows this, but in the interest of making this question more searchable, this is Exercise 3.3 in Axler's <em>Linear Algebra Done Right</em> (as well as the reason I was looking up questions like this).</p>
<p>I can only imagine what grading twenty or thirty proofs for this (and for a few other res... |
2,210,893 | <p>A lot of times when proving for example inequalities like $$x \leq y$$
for real numbers $x,y$ the argument looks like
$$x \leq y + \varepsilon$$
for all $\varepsilon > 0$, hence $x \leq y$. </p>
<p>Now this is obviously very intuitive, but is there a "proof" that this conclusion is correct? And is it always suf... | Olivier | 111,247 | <p>Suppose $x < y + \varepsilon$ for all $\varepsilon >0$, but that $x>y$. Then, taking $\varepsilon = x-y >0$, you obtain $x<x$, a contradiction.</p>
|
3,853,723 | <p>While brushing up on my old discrete mathematics skills I stumbled upon this problem that I can't solve.</p>
<p>In <span class="math-container">$\mathbb{R^2}$</span> the middle point of two coordinates is <span class="math-container">$(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2})$</span>. Show that given five points in <sp... | David Lui | 445,002 | <p>As <span class="math-container">$k$</span>-modules, <span class="math-container">$M_n(k)$</span> is isomorphic to <span class="math-container">$k^{(n^2)}$</span>, and similarly, <span class="math-container">$M_n(B) \sim B^{(n^2)}$</span>. Therefore, since tensor product distributes over direct sum, <span class="math... |
291,957 | <p>Does there exist a simple expression for integrals of the form,</p>
<p>$I = \int_{-\infty}^0 H_n(u) H_m(u)\, \mathrm{e}^{-u^2}\,du$,</p>
<p>where $m$ and $n$ are nonnegative integers and $H_n$ is the $n$'th (physicists') Hermite polynomial?</p>
<p>When $n+m$ is even, the symmetry of the integrand and the orthogon... | Tom Davis | 864,833 | <p>Although this is a very old question, I think it has another answer that may be useful to people.</p>
<p>I prefer the probabilists definition
<span class="math-container">$$
He_{\alpha}(x) = 2^{-\frac{\alpha}{2}}H_\alpha\left(\frac{x}{\sqrt{2}}\right)
$$</span>
so the integral becomes
<span class="math-container">$$... |
2,359,700 | <p>Given the vector space, $ C(-\infty,\infty)$ as the set of all continuous functions that are always continuous, is the set of all exponential functions, $U=\{a^x\mid a \ge 1 \}$, a subspace of the given vector space?</p>
<p>As far as I'm aware, proving a subspace of a given vector space only requires you to prove c... | Robert Z | 299,698 | <p>Hint. Try by evaluating the limit along the parabola $y=mx^2$ with $m\in\mathbb{R}$. What do you obtain? The limit depends on $m$?</p>
|
42,301 | <p>everyone! I am sorry, but I am an abcolute novice of Mathematica (to be more precise this is my first day of using it) and even after surfing the web and all documents I am not able to solve the following system: </p>
<pre><code>Solve[{y*(((y*x)/(beta*b))^(1/(beta - 1)) - v) - c*alpha ==
0, ((x/alpha))*(((y*x)... | user1066 | 106 | <p>Not very efficient, I suspect, but two other (related) possibilities:</p>
<pre><code>#[[Position[Ordering@Ordering@#[[All, 3]], 1, 1, 1][[1, 1]]]] &@mya
</code></pre>
<p>=> </p>
<blockquote>
<p>{0, 2, 5}</p>
</blockquote>
<pre><code>Pick[#, Ordering@Ordering@#[[All, 3]], 1] &@mya
</code></pre>
<p>=> <... |
18,686 | <p>Suppose you have an arbitrary triangle with vertices $A$, $B$, and $C$. <a href="http://www.cs.princeton.edu/~funk/tog02.pdf">This paper (section 4.2)</a> says that you can generate a random point, $P$, uniformly from within triangle $ABC$ by the following convex combination of the vertices:</p>
<p>$P = (1 - \sqrt{... | Ross Millikan | 1,827 | <p>I would argue that if it is true for any triangle, it is true for all of them, as we can find an affine transformation between them. So I would pick my favorite triangle, which is $A=(0,0), B=(1,0), C=(0,1)$. Then the point is $(\sqrt{r_1}(1-r_2),r_2\sqrt{r_1})$ and we need to prove it is always within the triangl... |
18,686 | <p>Suppose you have an arbitrary triangle with vertices $A$, $B$, and $C$. <a href="http://www.cs.princeton.edu/~funk/tog02.pdf">This paper (section 4.2)</a> says that you can generate a random point, $P$, uniformly from within triangle $ABC$ by the following convex combination of the vertices:</p>
<p>$P = (1 - \sqrt{... | Samrat Mukhopadhyay | 83,973 | <p>I'm starting with the argument provided by @Ross Millikan. Let $A=(0,0),\ B=(1,0),\ C=(0,1)$. Then the point chosen according to the given equation is $P=(X,Y)=(\sqrt{r_1}(1-r_2),r_2\sqrt{r_1})$. Now clearly, $0\leq X,Y \leq 1$ and $X+Y\leq \sqrt{r_1}\leq 1$. Now the problem is to show that $\mathbb{P}(X\leq x, Y\le... |
1,988,419 | <p>Any hint for proving this? If $Y$ is a subspace of $X$, what I am able to find is a closed subset $V$ in $Y$, hence $\mbox{cl}_Y(V)$ is compact, whose closure is contained in a neighborhood of a point $x$, by regularity of $Y$. Restricted to $X$, this $V_x=V \cap X$ is closed. I dont see any way to prove that $V_x$ ... | Taumatawhakatangihangakoauauot | 144,375 | <p>The result you wish to prove is not true. To give a counterexample I will use a characterization of locally compact subspaces of locally compact Hausdorff spaces found in <a href="https://math.stackexchange.com/a/644089">this answer</a>:</p>
<blockquote>
<p>If a subspace $Y$ of a locally compact Hausdorff space $... |
4,174 | <p>I'm developing a course that focuses on the transistion from arithmetic to algebraic thinking, particularly in grades 5-8. We will do this through focus on the common core. I'm also putting together a collection of suggested readings from the math education literature. I would be interested to hear your suggestio... | Joseph Malkevitch | 1,865 | <p>Although these two NCTM books cover K-12 (there are items specifically directed at middle school level) they have ideas related to how to develop algebraic thinking: The Ideas of Algebra, K12, 1988 Yearbook, A. Coxford and A. Shulte, editors, and Developing Mathematical Reasoning in Grades K-12, 1999 Yearbook, Lee S... |
2,764,818 | <blockquote>
<p>Let $f(x)=ax^3+bx^2+cx+d$, be a polynomial function, find relation between $a,b,c,d$ such that it's roots are in an arithmetic/geometric progression. (separate relations)</p>
</blockquote>
<p>So for the arithmetic progression I took let $\alpha = x_2$ and $r$ be the ratio of the arithmetic progressio... | Somos | 438,089 | <p>The condition you want is <span class="math-container">$\;D:=(x_1^2-x_2x_3)(x_2^2-x_1x_3)(x_3^2-x_1x_2)=0.$</span> This can be written as <span class="math-container">$\;D=e_1^3e_3-e_2^3\;$</span> where <span class="math-container">$\;e_1,e_2,e_3\;$</span> are the elementary symmetric functions of the roots. This si... |
587,217 | <p>I know that the units of 2 by 2 matrices with integer entries must have a determinant of 1 or -1, and I have proved that if the determinant is zero then the matrix is not a unit, however I am wondering how you would go about proving that matrices with determinants other than 1 and -1 are not units?</p>
| user1337 | 62,839 | <p>Given a matrix of integers $$\begin{pmatrix}a & b \\ c & d \end{pmatrix} $$ it's inverse is $$ \frac{1}{ad-bc} \begin{pmatrix}d & -b \\ -c & a \end{pmatrix} .$$ This will be a matrix of integers if and only if $\frac{1}{ad-bc} \in \mathbb Z$.</p>
|
2,643,099 | <p>Can someone help me explain why it is true that</p>
<p>$$\sin(\pi/2-\theta)=\sqrt{1-\sin^2\theta}$$</p>
<p>When answering please explain the different relation which is used</p>
<p>Thanks</p>
| user | 505,767 | <p>Simply note that</p>
<ul>
<li>$1-\sin^2\theta=\cos^2 \theta$</li>
<li>$\sin \left(\frac{\pi}2-\theta\right)=\cos \theta$</li>
</ul>
<p>thus, since RHS is non negative </p>
<p>$$\sin \left(\frac{\pi}2-\theta\right)=\sqrt{1-\sin^2\theta}$$</p>
<p>is true if and only if $\sin \left(\frac{\pi}2-\theta\right)=\cos \t... |
2,643,099 | <p>Can someone help me explain why it is true that</p>
<p>$$\sin(\pi/2-\theta)=\sqrt{1-\sin^2\theta}$$</p>
<p>When answering please explain the different relation which is used</p>
<p>Thanks</p>
| Guy Fsone | 385,707 | <p>$$ \sin \left(\frac{\pi}2-\theta\right) =\cos\theta \not = |\cos\theta| =\sqrt{\cos^2\theta}=\sqrt{1-\sin^2\theta}$$</p>
|
3,161,371 | <p>For <span class="math-container">$p, q$</span> prime, if <span class="math-container">$q$</span> divides an integer <span class="math-container">$n$</span> but <span class="math-container">$p$</span> does not, show that <span class="math-container">$\text{gcd}(n, pq) = q$</span></p>
<p>This statement sort of remind... | Bill Dubuque | 242 | <p>More generally for any <span class="math-container">$\,p,q\in\Bbb Z\!:\,$</span> <span class="math-container">$\, \color{#c00}{(p,n)}=1\,\Rightarrow\, (pq,n) = (q,n),\,$</span> because</p>
<p><span class="math-container">$$ (pq,n) = (pq,nq,n)=(\color{#c00}{(p,n)}q,n) = (q,n)$$</span></p>
<p>This is indeed one form... |
173,466 | <p>For a given matrix <code>M[n]</code> of size $ n\times n $ I want to define the following list of matrix-expressions:</p>
<pre><code>n=1
{Tr[M[1]]}
n=2
{Tr[M[2]]^2,Tr[M[2].M[2]]}
n=3
{Tr[M[3]]^3,Tr[M[3]]Tr[M[3].M[3]],Tr[M[3].M[3].M[3]]}
</code></pre>
<p>How could I generalize this relation for arbitrary $ n $? I... | Αλέξανδρος Ζεγγ | 12,924 | <p>How about this:</p>
<pre><code>listVonMat[M_] := Module[{n = Length[M], inter1, inter2},
inter1 = TakeDrop[Table[M, n], #] & /@ Range[n - 1, 0, -1];
inter2 = {Times @@ Tr /@ #1, Tr @ (Dot @@ #2)} & @@@ inter1;
Times @@@ inter2
]
</code></pre>
<p>Maybe numerical results ar... |
230,204 | <p>Let $X$ be a compact, oriented Riemann manifold. Let $\pi_{P}: P \rightarrow X$ be a principal $G$-bundle over $X$, for a compact Lie group $G$. Let $(M, \omega)$ be a symplectic manifold endowed with a symplectic action of $G$. Denote by $\mathcal{N}:=C^{\infty}(P,M)^{U(1)}$ the space of smooth $G$-equivariant maps... | Peter Michor | 26,935 | <p>Yes, I think $\Omega$ is closed. I will add a proof later.
Yes, as a mapping $T\mathcal N \to T^*\mathcal N$ it is injective, but it can never be surjective, since $T_n\mathcal N$ is a Frechet space, whereas its dual $T^*_u\mathcal N$ is a DF-space (generalized functions of distributions) which can never be isomorph... |
4,133,782 | <p>I am having trouble finding a formula that connects the two and can produce an answer. Anyone know how this is done? I tried y=mx+b, m=3, and b=5-a. But I don't know what to do next or did I even start right.</p>
| Community | -1 | <p>If <span class="math-container">$A$</span> is <span class="math-container">$m \times n$</span>, then the following are equivalent:</p>
<ol>
<li><span class="math-container">$A$</span> has full column rank <span class="math-container">$n$</span></li>
<li>The columns of <span class="math-container">$A$</span> are line... |
4,401,460 | <p>I'm wondering what other tools there are aside from radicals can be used to extend fields in the context of solving polynomials. Since <span class="math-container">$S_5$</span> isn't solvable, constructing a field with a Galois group of <span class="math-container">$S_5$</span> with respect to <span class="math-cont... | Young Toom | 1,035,157 | <p>My idea is Markov Chain.
Three states, representing <span class="math-container">$0,1,2$</span>. Start at <span class="math-container">$0$</span>, every turn uniformly go to one of the three states.We want to calculate <span class="math-container">$E_0(\sigma_0)$</span>.</p>
<p>Denote <span class="math-container">$E... |
918,689 | <p>of 5 be selected that contain /at least/ 1 of the broken bulbs?</p>
<p>So far, I have tried only 1 method, as it's the only one I've been taught, but I don't know if I am doing it right.
I tried doing C(100,1)/C(100,5) but it just doesn't seem right. Is it? If it isn't, what am I doing wrong?</p>
| thanasissdr | 124,031 | <p>Another way is the following one:</p>
<p>$\bullet$ Let's suppose that you want to have <strong>exactly 1</strong> non - working bulb.</p>
<p>Then, you need 4 working bulbs out of the 99 and 1 non-working bulb out of 2. </p>
<p>Then, the ways you can choose the 4 working bulbs are $\binom{98}{4}$ and the ways you ... |
1,386,343 | <p>Let $P$ be an idempotent $n \times n$ matrix ($P^2 = P$). What is $(I + P)^{-1}$? I've been thinking about this problem for a while, but can't find an answer. I tried a few examples, but I'm not sure what the general pattern is.</p>
| sebigu | 32,185 | <p>If $0 \neq 2$ in the field and $P^2=P$, then the minimal Polynomial of $P$ divides $f := x^2-x$, which means it is $f$, $x$, or $x-1$. If it is $x$, $P=0$, and if it is $x-1$, $P=1$. Those cases are clear.</p>
<p>So suppose it is $x^2-x$. Then $I+P$ has minimal polynomial $(x-1)(x-2)=x^2-3x+2$. This means that $I$ ... |
267,236 | <blockquote>
<p><strong>Possible Duplicate:</strong><br />
<a href="https://math.stackexchange.com/questions/30156/why-is-this-entangled-circle-not-a-retract-of-the-solid-torus">Why is this entangled circle not a retract of the solid torus?</a></p>
</blockquote>
<p>I am stuck with exercise 16 (c), pag.39 of Hatcher's ... | Chris Gerig | 22,295 | <p><em>Hint:</em> View the subspace "$A$" as a path in the space $A$, and then as a path in the space $S^1\times D^2$. Then see what that means about the desired retraction and inclusion maps.</p>
|
2,349,124 | <p>I keep on hitting a road block in trying to solve this, especially when trying to prove it going from the right hand side to the left hand side. </p>
| Patrick Stevens | 259,262 | <p>Suppose $X=\emptyset$; then the RHS is $\emptyset \cup Y$.</p>
<p>Suppose $X \not = \emptyset$; then say $x \in X$. Then two mutually exclusive cases:</p>
<ul>
<li>$x \in Y^c$. Then we have $x \in (X \cap Y^c) \cup (X^c \cap Y)$, but $x \in Y^c$ also implies $x \not \in Y$ so $Y$ can't be equal to $(X \cap Y^c) \c... |
112,503 | <p>I am working the a subject guide on involving $L$-Systems and have the alphabet $A = \{a, b, c\}$. The initiator is the string $a$ and the rules of substitution $a \to ba$, $b \to ccb$, $c \to a$. </p>
<p>The study guide gives the first five generations as:</p>
<p>$$[a] \to [ba] \to [ccba] \to [acba] \to [aaba] \t... | William Vickery | 334,699 | <p>Lindenmayer systems make all substitutions at once; this is one of their defining features. This is mentioned in the book The Algorithmic Beauty of Plants on page 3 of chapter 1, "In Chomsky grammars productions are applied sequentially, whereas in L-systems they are applied in parallel and simultaneously replace al... |
2,491,577 | <p>What is the meaning of phrase,<strong>"Compactness and Connectedness are intrinsic properties of a topological space"</strong>?</p>
| Alex Mathers | 227,652 | <p>This means that these properties are preserved by homeomorphism (the natural notion of equivalence for topological spaces).</p>
<p>To be more precise, you could make the following definition:</p>
<blockquote>
<p><strong>Definition</strong>: Let $P$ be a property of a topological space $X$. We say that $P$ is a <... |
2,543,169 | <p>The question is pretty self explanatory, but I’ve encountered situations where, for the length of some vector $\vec{a}$, to denote the length (or magnitude, which ever you prefer) as either $\| \vec{a}\|= \sqrt{a_1^2+a_2^2+\ldots+a_n^2}$ or $|\vec{a}|= \sqrt{a_1^2+a_2^2+\ldots+a_n^2}$ and I was wondering which notat... | mlk | 155,406 | <p>I would say, it really depends on the context. Both are widely accepted and understood. However, the nice thing is, that the two notations allow you to distinguish between two notions of "length". So in one extreme, in an introductory course on vectors I would write
$$\|a\| = \sqrt{|a_1|^2+ |a_2|^2+...+|a_n|^2}$$
to... |
2,835,172 | <p>What is the probability of a number (picked at random) from set $A= \{1,2...,6\},$ being larger than a number (picked at random) from set $B= \{1,2...,10\}.$ What would the probability be if sets $A$ and $B$ were generalized: Set $A=\{n,n+1,...,n+x\},$ and set $B = \{m,m+1,...,m+x\}?$ This is well beyond my realm of... | BruceET | 221,800 | <p><strong>Extended Comment:</strong> As indicated in the Comment by @HagenvonEitzen, one way to work the initial problem (on the probability D6 shows a larger value than D10) is to enumerate
cases. In particular, you might make a $10 \times 6$ array of possible pairs
of outcomes and highlight the pairs that satisfy yo... |
2,835,172 | <p>What is the probability of a number (picked at random) from set $A= \{1,2...,6\},$ being larger than a number (picked at random) from set $B= \{1,2...,10\}.$ What would the probability be if sets $A$ and $B$ were generalized: Set $A=\{n,n+1,...,n+x\},$ and set $B = \{m,m+1,...,m+x\}?$ This is well beyond my realm of... | Graham Kemp | 135,106 | <p>Use the Law of Total Probability: for example $d6, d10$ the results of independen six and ten sided dice.</p>
<p>$$\begin{align}\mathsf P(d6>d10) &= \mathsf P(d10>6)\mathsf P(d6>d10\mid d10>6)+\mathsf P(d10\leq 6)\mathsf P(d6>d10\mid d10\leq 6) \\ &=\tfrac 4{10}\cdot 1+\tfrac 6{10}\cdot\maths... |
2,380,456 | <p>Given this graph:</p>
<p><a href="https://i.stack.imgur.com/5hDje.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/5hDje.png" alt="enter image description here"></a></p>
<p>We can assume there is a linearity in the semi-curves shown above (<em>note that <code>air</code> line is not my concern her... | k.stm | 42,242 | <p>Not sure if there’s a neat direct way. But I learnt that the trick is to add another more conceptual equivalent statement.</p>
<blockquote>
<p>Let $K / F$ be a finite extension of fields and let $L$ be an algebraically closed field containing $K$. So we assume $F ⊆ K ⊆ L$. The following are equivalent:</p>
<... |
2,380,456 | <p>Given this graph:</p>
<p><a href="https://i.stack.imgur.com/5hDje.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/5hDje.png" alt="enter image description here"></a></p>
<p>We can assume there is a linearity in the semi-curves shown above (<em>note that <code>air</code> line is not my concern her... | Bach | 497,335 | <p>I have found a direct way proving this:</p>
<p>Suppose <span class="math-container">$K$</span> is the splitting field of <span class="math-container">$g(x)\in F[x]$</span> and <span class="math-container">$p(x)\in F[x]$</span> is irreducible over <span class="math-container">$F$</span>. Moreover, <span class="math-c... |
2,127,494 | <p>Given two $3$D vectors $\mathbf{u}$ and $\mathbf{v}$ their cross-product $\mathbf{u} \times \mathbf{v}$ can be defined by the property that, for any vector $\mathbf{x}$ one has $\langle \mathbf{x} ; \mathbf{u} \times \mathbf{v} \rangle = {\rm det}(\mathbf{x}, \mathbf{u},\mathbf{v})$.
From this a number of properties... | Jaroslaw Matlak | 389,592 | <p><strong>Hint</strong></p>
<p>$$|\mathbf{u}\times \mathbf{v}| = |\mathbf{u}|\cdot|\mathbf{v}|\cdot|\sin \alpha|$$
$$\langle \mathbf{u}, \mathbf{v}\rangle = |\mathbf{u}|\cdot|\mathbf{v}|\cdot|\cos \alpha|$$
where $\alpha$ is an angle between vectors $\mathbf{u}$ and $\mathbf{v}$</p>
|
2,127,494 | <p>Given two $3$D vectors $\mathbf{u}$ and $\mathbf{v}$ their cross-product $\mathbf{u} \times \mathbf{v}$ can be defined by the property that, for any vector $\mathbf{x}$ one has $\langle \mathbf{x} ; \mathbf{u} \times \mathbf{v} \rangle = {\rm det}(\mathbf{x}, \mathbf{u},\mathbf{v})$.
From this a number of properties... | DC75 | 376,199 | <p>I went through some books and found something I am happier with. It comes from <em>Euclidean and Non-Euclidean Geometry: An Analytic Approach</em>, by Ryan (p.85, or something like this)</p>
<p>Essentially it goes as follows : $\mathbf{n} = \mathbf{u} \times \mathbf{v}$ is defined by the property that for every $\m... |
4,118 | <p>I've recently dipped my toes into the world of number theory; and I've bought a book that to me is quite unconventional: R. P. Burn, <em>A Pathway into Number Theory</em>. I've yet to put the book through its paces, but it seems agreeable enough to me. The book is unique in that it poses a sequence of questions to y... | bzm3r | 2,013 | <p>I am going to copy and paste my answer from another question on this site, because I think one would be hard pressed to beat it in terms of the number of suggestions it covers, and the general quality with which it presents these suggestions:</p>
<blockquote>
<p>You might be interested in the expansive answers th... |
398,388 | <p>The classification of finite simple groups has been called one of the great intellectual achievements of humanity, but I don't even know one single application of it. Even worse, I know a lot of applications of simple <em>modules</em> over some ring/algebra <span class="math-container">$A$</span>, but I can barely k... | JoshuaZ | 127,690 | <p>The best case bound for <a href="https://en.wikipedia.org/wiki/Jordan%E2%80%93Schur_theorem" rel="nofollow noreferrer">the Jordan-Schur theorem</a> uses heavily the classification, and that theorem shows up in a lot of different contexts.</p>
|
3,439,626 | <p>I need to proof the following statement:</p>
<p>Let <span class="math-container">$a, b, n \in \Bbb{Z}$</span> with <span class="math-container">$ n≥ 2, gcd(a,n)=1$</span>. Proof that if <span class="math-container">$s_{1},s_{2}$</span> are solutions to <span class="math-container">$ax\equiv b \pmod{n}$</span>, the... | Rushabh Mehta | 537,349 | <p>Note that if <span class="math-container">$s_1,s_2$</span> are solutions, <span class="math-container">$as_1\equiv as_2\equiv b\pmod n$</span>, so <span class="math-container">$as_1\equiv as_2\pmod n$</span>. </p>
<p>Hence, <span class="math-container">$n\mid a(s_2-s_1)$</span>. But, since <span class="math-contain... |
513,779 | <p>If $a,b\in\mathbb{N}$ are odd</p>
<p>then demonstrate:
$$ {\sqrt{a^2 + b^2}} \not\in \mathbb{Q}$$ </p>
<p>I try to guess that $$ {\sqrt{a^2 + b^2}} \in\mathbb{Q}.$$ Then i write $$ {\sqrt{a^2 + b^2}= m/n}.$$ After that: $$ {n\sqrt{a^2 + b^2}= m}$$ , I raised at squared and i have like $$ n^2(a^2+ b^2)=m^2 $$ and ... | imranfat | 64,546 | <p>You may remember that Pythagorean triplets of a right triangle are of the form, a²-b², 2ab and a²+b², the latter one being the hypotenuse. Well, the second terms says it all...</p>
|
4,190,492 | <p>I offer a proposition with both a proof and a counterexample. Thus, either the proof is incorrect, or the counterexample is not actually a counterexample, or both. Which is it?</p>
<p><strong>Proposition.</strong> Given a function <span class="math-container">$h(x)$</span> which is twice differentiable, strictly con... | Theo Bendit | 248,286 | <p>In your proof, you write</p>
<blockquote>
<p>For the statement to hold, we need <span class="math-container">$g'(h(x)) \leq 0 $</span> and <span class="math-container">$g''(h(x)) \leq 0$</span>. Thus <span class="math-container">$g'$</span> must be weakly decreasing (and concave), a contradiction.</p>
</blockquote>
... |
4,190,492 | <p>I offer a proposition with both a proof and a counterexample. Thus, either the proof is incorrect, or the counterexample is not actually a counterexample, or both. Which is it?</p>
<p><strong>Proposition.</strong> Given a function <span class="math-container">$h(x)$</span> which is twice differentiable, strictly con... | Lee White | 468,437 | <p>For the second derivative of <span class="math-container">$f\left(x\right)$</span>, <span class="math-container">$g^\prime \left(h\left(x\right)\right) \leq 0 $</span> and <span class="math-container">$g^\prime \left(h\left(x\right)\right) \leq 0$</span> is sufficient condition of <span class="math-container">$f\lef... |
202,040 | <p>I'd like to get separate plots for the functions in a list, and I'm trying the following, which doesn't work. What is the correct way to do that?</p>
<pre><code>Table[ContourPlot3D[f, {x, -2, 2}, {y, -2, 2}, {z, -2, 2}], {f, {x + y + z + x y z == 0, x + y + z^2 + x y z^2 == 0, x + y^2 + z + x y^2 z == 0}}]
</code><... | Rohit Namjoshi | 58,370 | <p>Using <code>Table</code></p>
<pre><code>Table[ContourPlot3D[
Evaluate@f, {x, -2, 2}, {y, -2, 2}, {z, -2, 2}], {f, {x + y + z + x y z == 0,
x + y + z^2 + x y z^2 == 0, x + y^2 + z + x y^2 z == 0}}]
</code></pre>
<p><a href="https://i.stack.imgur.com/Be7F0.png" rel="nofollow noreferrer"><img src="https://i.sta... |
1,416,998 | <p>In the definition of martingales, one finds in Stroock and Varadhan (Multidimensional Diffusion processes - page 20) the strange request that it be right-continuous process.</p>
<p><a href="https://i.stack.imgur.com/0Nni7.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/0Nni7.png" alt="enter image... | 5xum | 112,884 | <p>There is some slight confusion about what you really proved, but it's minor (even though, rigorously speaking, the proof is all wrong).</p>
<p>You can actually remove the first sentence "Then $ax+by=d$".</p>
<p>The argument would flow better like so:</p>
<ul>
<li>We know: by Bezout's theorem, if there exists a pa... |
69,272 | <p>By the way, does anyone know how to prove in an elementary way (i.e. expanding) that $\prod_1^n (1+a_i r)$ tends to $e^r=\sum \frac{r^k}{k!}$ as you let $\max|a_i|\to 0$ with $0\leq a_i \leq 1$ and $\sum a_i = 1$? An easy solution goes by writing the product with the exponential function so that you get the exponent... | Phil Isett | 7,193 | <p>Thanks, Anthony, for finding this solution. I was completely at a loss for how to handle all the indices. If you don't mind, I would like to write down one version of the argument that you've given in full detail.</p>
<p>Claim: Under the hypotheses of the question $1 = k! \sum_{i_1 < \ldots < i_k} a_{i_1}... |
138,800 | <h1>Background</h1>
<p>I have a block of code, reproduced at the bottom of this post, consisting of combined \$PreRead and \$PrePrint statements, that automatically formats outputs as 'input = output', and also allows easy inline combination of math and text (allowing the text to be placed either before, after, or bot... | userrandrand | 86,543 | <p>I had a similar issue. I wanted to prevent the function in PrePrint from applying when I asked it not to. I did not use any systematic conditions where the function would be automatically suppressed (like if the expression to be evaluated is a plot or an integer).</p>
<p>My solution was to use an inert global variab... |
2,091,766 | <p>Suppose $h:R \longrightarrow R$ is differentiable everywhere and $h'$ is continuous on $[0,1]$, $h(0) = -2$ and $h(1) = 1$. Show that:
<p> $|h(x)|\leq max(|h'(t)| , t\in[0,1])$ for all $x\in[0,1]$</p>
<p>I attempted the problem the following way:
Since $h(x)$ is differentiable everywhere then it is also continuous ... | Fred | 380,717 | <p>If $a$ and $b$ are rational, then all your sets $S$ from above contain no irrational number !</p>
<p>Your turn !</p>
|
2,018,239 | <p>I have to show, using induction, that $2^{4^n}+5$ is divisible by $21$. It is supposed to be a standard exercise, but no matter what I try, I get to a point where I have to use two more inductions.</p>
<p>For example, here is one of the things I tried:</p>
<p>Assuming that $21 |2^{4^k}+5$, we have to show that $21... | Sarvesh Ravichandran Iyer | 316,409 | <p>You may have to use a few tricks here. The base case is clear.</p>
<p>Note that $2^{(4^n)} - 2^{(4^{n-1})} = 2^{(4^{n-1})}(2^{(3 \cdot 4^{n-1})} -1)$</p>
<p>So we have to prove that $2^{3 \cdot 4^{n-1}} - 1$ is a multiple of $21$. To do this, note that for $n \geq 2$,we can use modular arithmetic: since $3 \cdo... |
1,933,744 | <p>I simulated the following situation on my pc. Two persons A and B are initially at opposite ends of a sphere of radius r. Both being drunk, can take exactly a step of 1 unit(you can define the unit, i kept it at 1m) either along a latitude at their current location, or a longitude. A and B are said to meet, if the a... | Daniel Robert-Nicoud | 60,713 | <p>For now, I will give a reformulation of this problem in terms that should make it easier to attack with analytic methods (at least to get results on the asymptotic behavior) and greatly simplify simulations. In a second time, I will maybe also attempt to solve it, but I don't guarantee any kind of success.</p>
<p>M... |
2,464,756 | <p>When I was trying to prove a relation from solid state physics, I reached this mathematical problem. In the equation</p>
<p>$$\sum_{i=1}^Nm_ix_i=n$$</p>
<p>$m_i$ and $n$ are known integers, $N=3$, and $x_i$ are unknown integers. Also we know that the greatest common factor of $\left\{m_i\right\}$ is 1. I don't nee... | B. Mehta | 418,148 | <p>If the greatest common divisor of $m_1, \dots, m_N$ divides $n$, then this has a solution, by <a href="https://en.wikipedia.org/wiki/B%C3%A9zout%27s_identity#For_three_or_more_integers" rel="noreferrer">Bézout's identity</a>. If not, there is no solution, since the gcd will divide the left for any choice of $x_i$, ... |
3,794,158 | <p>I am trying to prove that: Let <span class="math-container">$(M,d)$</span> an metric space and <span class="math-container">$(x_n)$</span>,<span class="math-container">$(y_n)$</span> sequences in <span class="math-container">$M$</span> such that <span class="math-container">$d(x_n,y_n) \leq \frac{1}{n}$</span> <spa... | Ben Grossmann | 81,360 | <p><strong>Hint:</strong> Our goal is the following: given an <span class="math-container">$\epsilon > 0$</span>, prove that there exists an <span class="math-container">$N$</span> such that <span class="math-container">$y_n \in B_{\epsilon}(L)$</span> whenever <span class="math-container">$n \geq N$</span>.</p>
<p>... |
933,604 | <p>Hi can anyone solve these two questions using logs and indices</p>
<p>a.
$$4^{2x}-2^{x+1}=48$$</p>
<p>b.
$$6^{2x+1}-17*{6^x}+12=0$$</p>
<p>Thanks.</p>
| lab bhattacharjee | 33,337 | <p>I believe the last question to be $6^{2x+1}-17(6^x)+12=0$</p>
<p>$$\iff6(6^x)^2-17(6^x)+12=0$$</p>
<p>$$6^x=\frac{17\pm\sqrt{17^2-4\cdot6\cdot12}}{2\cdot6}=\frac{17\pm1}{12}=\frac32,\frac43$$</p>
|
2,327,273 | <p>If a tree has 5 vertices of degree 2, 3 vertices of degree 3, 4 vertices of degree 4, then how many leaves are there in that tree? </p>
<p>I know the tree has at least 12 vertices and so it must have at least 11 edges. Also the number of leaves must be odd but I could not proceed further. </p>
| Prajwal Kansakar | 49,781 | <p>If $k$ is the number of leaves then the total number of vertices in the tree is $12+k$ with $11+k$ edges and the sum of the degrees is $\sum\deg(v)=(5\times 2)+(3\times 3)+(4\times 4)+(k\times 1)=35+k$. Now by handshaking lemma,
$$35+k=2(11+k)\Rightarrow k=13.$$</p>
|
2,979,226 | <p>Consider you are given following </p>
<blockquote>
<p><span class="math-container">$$\biggr (x-\dfrac{2}{x^2}\biggr )^6$$</span></p>
</blockquote>
<p>I'm trying to evaluate the constant term. What I've done so far is given below</p>
<p><span class="math-container">$$\sum^{6}_{n = 0} \binom{6}{r}x^{6-r}\times (... | lab bhattacharjee | 33,337 | <p>It should be <span class="math-container">$$\binom6rx^{6-r}(-2x^{-2})^r=?$$</span></p>
|
4,045,238 | <p>I was working on the problems in Mathematical Methods for Physics and Engineering by Riley,Hobson & Bence.
In Problem 2.34 (d) I'm supposed to find this integral: <span class="math-container">$$J=\int\frac{dx}{x(x^n+a^n)}.$$</span>
I used partial fractions and arrived at the form
<span class="math-container">$$J... | Ishraaq Parvez | 736,904 | <p><span class="math-container">\begin{gather*}
Let\ I=\int \frac{dx}{x\left( x^{n} +a^{n}\right)} =\int \frac{dx}{x^{n+1}} \cdotp \frac{1}{1+\frac{a^{n}}{x^{n}}}\\
Let\ 1+\frac{a^{n}}{x^{n}} =t\\
\frac{-n\cdotp a^{n}}{x^{n+1}} dx=dt\\
\frac{dx}{x^{n+1}} =\frac{-dt}{n\cdotp a^{n}}\\
I=\int \frac{-dt}{n\cdotp a^{n}} \cd... |
4,045,238 | <p>I was working on the problems in Mathematical Methods for Physics and Engineering by Riley,Hobson & Bence.
In Problem 2.34 (d) I'm supposed to find this integral: <span class="math-container">$$J=\int\frac{dx}{x(x^n+a^n)}.$$</span>
I used partial fractions and arrived at the form
<span class="math-container">$$J... | imranfat | 64,546 | <p>Alternatively, perform a u-sub. Let <span class="math-container">$x=1/t$</span>. What will happen is that
(after little algebraic simplification) you get a monomial numerator which is one degree lower than a binomial denominator consisting of an <span class="math-container">$x$</span> term and a constant. Integratio... |
4,573,566 | <p>So I have to find the bifurcation points of the system: <span class="math-container">$\dot{x}=(ax-x^3+x^5)(x-a+2)$</span>, where <span class="math-container">$a\in\mathbb{R}$</span> is a parameter.</p>
<p>Attempt:<br />
I know that a bifurcation point is the point, where there is a change in stability or number of f... | MathWonk | 301,562 | <p>One extra tip that augments the prior answer. The easy way to locate the fixed points is to first use the equation <span class="math-container">$\dot x=0$</span> by setting each factor equal to zero, and then (the sneaky part) plot the solution set for each factor by expressing <span class="math-container">$a$</spa... |
4,573,566 | <p>So I have to find the bifurcation points of the system: <span class="math-container">$\dot{x}=(ax-x^3+x^5)(x-a+2)$</span>, where <span class="math-container">$a\in\mathbb{R}$</span> is a parameter.</p>
<p>Attempt:<br />
I know that a bifurcation point is the point, where there is a change in stability or number of f... | boojum | 882,145 | <p>Another way in which the problem can be addressed without referring to graphs is to consider that the function in the differential equation <span class="math-container">$ \ \dot{x} \ = \ f(x) \ $</span> is a sixth-degree polynomial for which the partial factorization is <span class="math-container">$ \ f(x) \ = \ x... |
3,362,000 | <p>From listing the first few terms, I suspect that the sequence is increasing, so I wanted to use mathematical induction to verify my suspicion.</p>
<p>I have assumed that <span class="math-container">$a_k<a_{k+1}$</span>, I don't see how I can obtain <span class="math-container">$a_{k+1}<a_{k+2}$</span> becaus... | J. W. Tanner | 615,567 | <p>The sequence is increasing. </p>
<p>Since <span class="math-container">$a_{n-1}>0, $</span> it follows that <span class="math-container">$\dfrac1{a_{n-1}}>0$</span>, and therefore that <span class="math-container">$a_n=a_{n-1}+\dfrac1{a_{n-1}}>a_{n-1}$</span>.</p>
|
166,013 | <p>Ordinary (connective) complex $K$-theory is the algebraic $K$ theory of the topological ring $\mathbb{C}$ with analytic topology. One can also study the $K$ theory of $\mathbb{C}$ with discrete topology. Weibel, in his $K$-theory book, computes the torsion in its coefficient ring. I would like to know the torsion-fr... | Dan Ramras | 4,042 | <p>According to Corollary 22.4 here:
<a href="http://www.math.uiuc.edu/~dan/Papers/KTheoryOfFields.pdf" rel="nofollow">http://www.math.uiuc.edu/~dan/Papers/KTheoryOfFields.pdf</a>
the K-theory of an algebraically closed field is divisible. According to the "structure theorem for divisible abelian groups", discussed h... |
3,953,681 | <p>I have a basic question but I have failed in solving it. I have the equation of a cylinder which is <span class="math-container">$y^2 + z^2 = r^2$</span> (centered in the x-axis). The parametric equation (dependent on <span class="math-container">$L$</span> and <span class="math-container">$s$</span>) is <span class... | lab bhattacharjee | 33,337 | <p>Hint:</p>
<p><span class="math-container">$$\int\dfrac{dx}{x\sqrt{a-bx^2}}=\int\dfrac{bx\ dx}{bx^2\sqrt{a-bx^2}}$$</span></p>
<p>Let <span class="math-container">$\sqrt{a-bx^2}=u\implies du=\dfrac{{-bx}}{\sqrt{a-bx^2}}$</span> and <span class="math-container">$bx^2=a-u^2$</span></p>
<p>More generally for <span clas... |
1,574,663 | <p>I'm a first time Calc I student with a professor who loves using $e^x$ and logarithims in questions. So, loosely I know L'Hopital's rule states that when you have a limit that is indeterminate, you can differentiate the function to then solve the problem. But what do you do when no matter how much you differentiate,... | SchrodingersCat | 278,967 | <p><strong>HINT:</strong> </p>
<p>As for your problem, divide both numerator and denominator by $e^x$. You'll get your limit as $1$.</p>
<p>In mathematics, logic, representation and arrangement play an extremely vital role. So always check that you have arranged your expression properly. Else repeated applications of... |
1,574,663 | <p>I'm a first time Calc I student with a professor who loves using $e^x$ and logarithims in questions. So, loosely I know L'Hopital's rule states that when you have a limit that is indeterminate, you can differentiate the function to then solve the problem. But what do you do when no matter how much you differentiate,... | abel | 9,252 | <p>why would you want to use l'hopitals on this? for $x$ large positive $e^{-x} = \frac 1{e^x}$ which is small. therefore $$\frac{e^x + e^{-x}}{e^x -e^{-x}} = \frac{e^x}{e^x}+\cdots \to 1 \text{ as } x \to \infty.$$</p>
|
3,734,216 | <p>Say <span class="math-container">$A$</span> and <span class="math-container">$B$</span> are operators on Hilbert spaces <span class="math-container">$H_A,H_B$</span> respectively. If the Hilbert spaces are finite dimensional, then I know the tensor <span class="math-container">$A\otimes B$</span> can be represented ... | tomasz | 30,222 | <p><span class="math-container">$\DeclareMathOperator{\End}{End}$</span>
Yes, the Kronecker product formula works, and not just for Hilbert spaces.</p>
<p>More precisely, in the non-topological case, if <span class="math-container">$V,W$</span> are <span class="math-container">$k$</span>-vector spaces with (algebraic) ... |
2,010,693 | <p>How can I prove that $x_{n+1}=c+\sqrt{x_n}$, $x_1=a>0$ and $c>0$ converges?
I know that the limit (if it exists) is $L={{2c+1+\sqrt{4c+1}}\over 2}$.
I have already prove that if $x_1<L$ then $x_n<L$ so its bounded from above but how can I prove that if $x_1<L$ then the sequence is increasing?
I would... | Mark Viola | 218,419 | <p>Note that we have</p>
<p>$$x_{n+1}-x_n=\sqrt{x_n}-\sqrt{x_{n-1}}=\frac{x_n-x_{n-1}}{\sqrt{x_n}+\sqrt{x_{n-1}}}$$</p>
<p>Hence, if $c+\sqrt{x_1}>x_1$, then we see from $(1)$ by inductive reasoning that the sequence is always increasing.</p>
<p>Similarly, if $c+\sqrt{x_1}<x_1$, then the sequence is decreasing... |
869,337 | <p>"Abstract index" and "coordinate free notations" are often submitted as alternatives to Einstein Summation notation. Could you illustrate their use using an example?</p>
<p>Here's a sum written in Einstein's notation:</p>
<p>$a_{ij}b_{kj} = a_{i}b_{k}$</p>
<p>How would you rewrite it in a modern way? </p>
| Gro-Tsen | 84,253 | <p>Even assuming $R = k$ is a field (which I will do throughout), there is no really satisfactory simple condition, but there is a name associated to the phenomenon: Tschirnhaus transformations. And we can say a few things about them.</p>
<p>Specifically, if $k$ is a field and $f,g$ are two monic polynomials of the s... |
181,499 | <p>In many of the classes that I teach, I require students to learn the basics of Mathematica which we use throughout the semester to do computations and to submit homeworks (in notebook form). Some students really like this and some... not so much. </p>
<p>Since I teach in an engineering department, almost everyone a... | Αλέξανδρος Ζεγγ | 12,924 | <p>When a language, e.g., Python, not emphasizing but has to talk about "functional programming", usually it speaks about three functions: <code>map</code>, <code>filter</code> and <code>reduce</code>. I always think comparison a good approach to learn things, so below I share the comparison I made before.</p>
<p><a h... |
1,602,271 | <p>Someone can help me to solve this differential equation with method of undetermined coefficient.
$$ y''-2y'+y=x\sin x$$
Thanks</p>
| ultrainstinct | 177,777 | <p>So first you need to get the solution to the homogeneous ODE, and the characteristic equation is $$r^2-2r+1=0=(r-1)^2,$$ so you have two repeated real roots, you know what to do with that from there. Now onto the particular solution. Since we have a $\sin x$ multiplied by a polynomial, we have $$y_p = (Ax+B)\sin x +... |
213,872 | <p>I'm learning probability theory and I see the half-open intervals $(a,b]$ appear many times. One of theorems about Borel $\sigma$-algebra is that</p>
<blockquote>
<p>The Borel $\sigma$-algebra of ${\mathbb R}$ is generated by inervals of the form $(-\infty,a]$, where $a\in{\mathbb Q}$. </p>
</blockquote>
<p>Also... | George Frank | 30,674 | <p>I think it's because the distribution function in the discrete case is the sum of probabilities from minus infinity up to and including x; but minus infinity is not a number so that end of the interval is open, i.e., has no end point.</p>
|
3,041,656 | <p>I need some help in a proof:
Prove that for any integer <span class="math-container">$n>6$</span> can be written as a sum of two co-prime integers <span class="math-container">$a,b$</span> s.t. <span class="math-container">$\gcd(a,b)=1$</span>.</p>
<p>I tried to go around with "Dirichlet's theorem on arithmetic ... | user760041 | 760,041 | <p>We know that <span class="math-container">$n>6$</span> and we need to prove that any integer <span class="math-container">$n>6$</span> can be written as the form of two co-primes. So it has a very simple proof.
We know that any integer <span class="math-container">$n$</span> and <span class="math-container">$n... |
642,631 | <p>What is $[\mathbb{Q}(i,\sqrt{2},\sqrt{3}):\mathbb{Q}]$?</p>
<p>On the one hand, we have $[\mathbb{Q}(i,\sqrt{2},\sqrt{3}):\mathbb{Q}(i,\sqrt{2})]\cdot[\mathbb{Q}(i,\sqrt{2}):\mathbb{Q}(i)]\cdot[\mathbb{Q}(i):\mathbb{Q}]=2^3=8.$</p>
<p>On the other hand, the minimum polynomial in $\mathbb{Q}[x]$ containing $i,\sqrt... | Hagen von Eitzen | 39,174 | <p>The polynomial you give is not irreducible and is not the minimal polynomial of a single $\alpha$ with $\mathbb Q(\alpha)=\mathbb Q(i,\sqrt 2,\sqrt 3)$. Out of the blue I suggest that $\alpha=i+\sqrt 2+\sqrt 3$ has the desired property - try to find its minimal polynomial.</p>
|
203,456 | <p>Please help me proof $\log_b a\cdot\log_c b\cdot\log_a c=1$, where $a,b,c$ positive number different for 1.</p>
| Aang | 33,989 | <p>Let $\log_b a=x\implies b^x=a,$</p>
<p>$ \log_c b=y\implies c^y=b$ and</p>
<p>$\log_a c=z\implies a^z=c$</p>
<p>Now, $a^z=c\implies (b^x)^z=c\implies ((c^y)^z)^x=c\implies c^{xyz}=c\implies xyz=1$ assuming $c\neq 0,1$</p>
<p>Thus, $xyz=1\implies \log_b a\cdot\log_c b\cdot \log_a c=1$</p>
|
4,164,960 | <p>I'm trying to prove <span class="math-container">$$P \implies Q \vdash \neg Q \implies\lnot P$$</span>
with natural deduction but I'm kind of stuck. I tried going from the conclusion side, which lead me to this:
<span class="math-container">$$ \frac{\frac{[P] \quad
\bot}{[\neg Q] \qquad
\neg P \qquad(\neg E)}}{\qqua... | Rob Arthan | 23,171 | <p>Unfortunately, (1) I don't know of a good way of drawing proof trees in MathJax and (2) there is no universally agreed set of natural deduction rules. Here is a proof presented as a sequence of labelled steps using rules similar to those presented <a href="https://en.wikipedia.org/wiki/Natural_deduction#Introduction... |
4,164,960 | <p>I'm trying to prove <span class="math-container">$$P \implies Q \vdash \neg Q \implies\lnot P$$</span>
with natural deduction but I'm kind of stuck. I tried going from the conclusion side, which lead me to this:
<span class="math-container">$$ \frac{\frac{[P] \quad
\bot}{[\neg Q] \qquad
\neg P \qquad(\neg E)}}{\qqua... | Dan Christensen | 3,515 | <p>(Posted after previous answer was accepted 3 hours ago)</p>
<p>Same proof, but this may be a little more readable format (screenshot from my proof checker):</p>
<p><a href="https://i.stack.imgur.com/ezosz.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/ezosz.png" alt="enter image description here"... |
9,629 | <p>are people facing problem of not loading latex symbols in MSE? I have high speed internet connection but I am facing this problem from yesterday,any suggestion?It says "math processing error" if my connection is low speed but this is not the case, I am just watching all latex symbols instead of compiled complete pi... | 75064 | 75,064 | <p>The timing suggests that the problem was related to the release of <a href="http://www.mathjax.org/mathjax-v2-2-now-available/" rel="nofollow">MathJax 2.2</a>: </p>
<blockquote>
<p>During the time that the files are making their way out to the CDN’s servers, there may be a mixture of files in a browser cache, and... |
237,708 | <p>Does the series </p>
<p>$$\sum_{n=1}^{\infty}\log n - (\log n)^{n/(n+1)}$$</p>
<p>converge?</p>
| N. S. | 9,176 | <p>By AM-GM:</p>
<p>$$\sqrt[n+1]{ \log(n)^n} \leq \frac{n\log(n)+1}{n+1}$$</p>
<p>Thus</p>
<p>$$ \log n- (\log n)^{\frac{n}{n+1}} \geq \log(n)-\frac{n\log(n)+1}{n+1}=\frac{\log(n)-1}{n+1}$$</p>
<p>Now, Limit Comparison test tells you that since $\sum \frac{\log n}{n}$ is divergent, this series is also divergent.</p... |
14,552 | <p>What are good examples of proofs by induction that are relatively low on algebra? Examples might include simple results about graphs.</p>
<p>My aim is to help students get a sense of the logical form of an induction proof (in particular proving a statement of the form 'if $P(k)$ then $P(k+1)$'), independent of the ... | Brendan W. Sullivan | 80 | <p>How about the <strong>Tower of Hanoi</strong> puzzle and finding the optimal number of moves? </p>
<p>This link describes the recursive solution procedure and a proof of optimality using induction.</p>
<p><a href="https://proofwiki.org/wiki/Tower_of_Hanoi" rel="nofollow noreferrer">https://proofwiki.org/wiki/Tower... |
14,552 | <p>What are good examples of proofs by induction that are relatively low on algebra? Examples might include simple results about graphs.</p>
<p>My aim is to help students get a sense of the logical form of an induction proof (in particular proving a statement of the form 'if $P(k)$ then $P(k+1)$'), independent of the ... | Steven Gubkin | 117 | <p>I am going to try the following activity as a first introduction to Mathematical Induction on Monday next week. I will let you know how it goes.</p>
<p>The implication <span class="math-container">$P(k) \implies P(k+1)$</span> let's you "hop around" the natural numbers, deciding the proof of new statement... |
611,529 | <p>$$i^3=iii=\sqrt{-1}\sqrt{-1}\sqrt{-1}=\sqrt{(-1)(-1)(-1)}=\sqrt{-1}=i
$$</p>
<p>Please take a look at the equation above. What am I doing wrong to understand $i^3 = i$, not $-i$?</p>
| robjohn | 13,854 | <p>Since $i^2=-1$ by definition, $i^3=i^2\cdot i=-i$.</p>
<p>$\sqrt{a}\sqrt{b}=\sqrt{ab}$ is only guaranteed for positive real $a$ and $b$.</p>
|
611,529 | <p>$$i^3=iii=\sqrt{-1}\sqrt{-1}\sqrt{-1}=\sqrt{(-1)(-1)(-1)}=\sqrt{-1}=i
$$</p>
<p>Please take a look at the equation above. What am I doing wrong to understand $i^3 = i$, not $-i$?</p>
| haughtonomous | 116,620 | <p>Someone posted earlier that it is i x i x i which is -1 x i, ie -i (but the post seems to have been deleted!</p>
<p>But that's all there is to it. It doesn't matter what i represents, the algebra will be consistent. But to prove it, substituting the value of i, ie (√-1), you get</p>
<p>√-1 x √-1 x √-1
= -1 x √-1
... |
3,242,921 | <p>Prove that the equation<span class="math-container">$$x^4+(a-2)x^3+(a^2-2a+4)x^2-x+1=0$$</span>
does not admit <span class="math-container">$$x=-2$$</span> as a triple root.</p>
| Servaes | 30,382 | <p>A set of vectors does <em>not</em> span the whole space if and only if it is contained in a hyperplane. So there exist sets of <span class="math-container">$2^{n-1}$</span> vectors that do not span the whole space, and every set of <span class="math-container">$2^{n-1}+1$</span> does span the whole space.</p>
<p>Th... |
2,128,991 | <p>What kind of rule or formula this kind of equations uses?</p>
<p>For example we have:</p>
<p>$$a=e^{x}$$</p>
<p>How come it is equal to:</p>
<p>$$\ln a =x$$</p>
<p>Tried to find some kind of rule for that about how it works, but didn't found anything.</p>
| S.C.B. | 310,930 | <p>The rule we have is the definition of <a href="https://en.wikipedia.org/wiki/Logarithm">logarithim</a>.The logarithim, by definition is the inverse of the exponential function. </p>
<p>Note that by the definition of logarithim $$a=b^{x}$$ becomes $$\log_{b} a=x$$
Yours is just a case when $b=e$. Note $\log_{e} x =\... |
2,128,991 | <p>What kind of rule or formula this kind of equations uses?</p>
<p>For example we have:</p>
<p>$$a=e^{x}$$</p>
<p>How come it is equal to:</p>
<p>$$\ln a =x$$</p>
<p>Tried to find some kind of rule for that about how it works, but didn't found anything.</p>
| amWhy | 9,003 | <p>Note $\log_a x$ is the inverse of the function $a^x$. When we speak of the natural log of $x$, it is written $\ln(x)$, which is simply shorthand for $\log_e(x)$.</p>
<p>$$a = e^x$$</p>
<p>Since $\log x$, or in this case the natural log $\log_e x = \ln x$ is a strictly increasing function, we can take the $\ln$ of... |
3,715,484 | <p>As the title saying , the question is how to find the radius <span class="math-container">$R$</span> of convergence of <span class="math-container">$\sum_{n=1}^{\infty}\frac{\sin n}{n} x^n$</span>. My method is as the following:</p>
<p>When <span class="math-container">$x=1$</span>, it is well known that the ser... | zkutch | 775,801 | <p>Let's consider
<span class="math-container">$$f(x,y)= \begin{cases}
\frac{x^2y}{x^4+y^2}, & x^2+y^2 \ne 0 \\
0, & x=y=0
\end{cases}$$</span>
This function have derivative on any line which contain <span class="math-container">$(0,0)$</span>, because it's <span class="math-container">$0$</span> on both axis a... |
2,103,706 | <p>I tried to prove this by induction.</p>
<p>Base case $n=1$, $5$ vertices. I just drew a pentagon, which has $5$ vertices of degree $2$</p>
<p>Then I assume for $n=k$,$4k+1$ vertices, there is at least one vertex with degree $2n$. The number of edges for this graph is $\dfrac{(4k+1)(4k)}{4}=(4k+1)(k)$</p>
<p>Then ... | Joffan | 206,402 | <p>The <a href="https://en.wikipedia.org/wiki/Complement_graph" rel="nofollow noreferrer">complementary graph</a> to $G$ adds all edges that are missing from the complete graph $K_{4n+1}$, and removes all existing edges. In $K_{4n+1}$ every vertex has $4n$ edges, so the complementing process changes each vertex degree... |
4,203,079 | <p>I’m trying to grasp the idea behind quotient spaces and reading <a href="https://en.m.wikipedia.org/wiki/Quotient_space_(topology)" rel="nofollow noreferrer">this</a> wikipedia article. In the section ”Examples” they have the unit square <span class="math-container">$S^2$</span> homeomorphism example, which I tought... | José Carlos Santos | 446,262 | <p>The equivalence relation <span class="math-container">$\sim$</span> is this one:</p>
<ul>
<li>if <span class="math-container">$p\in I^2\setminus\partial I^2$</span>, then <span class="math-container">$p\sim q$</span> if and only if <span class="math-container">$q=p$</span>;</li>
<li>if <span class="math-container">$... |
1,528,507 | <p>So I started reading Conjecture and Proof by Miklos Laczkovich and one of the first proofs he provides is that of the irrationality of the square root of two. I am aware there are alternative proofs (one of which is geometric and another that uses the fundamental theorem of arithmetic) but I have a few questions abo... | Henno Brandsma | 4,280 | <p>The proof assumes that you know what fractions are and how to compute with them, and also that they have more than one representation: if $\frac{p}{q}$ represents a fraction, then $\frac{2p}{2q}$ represents the <em>same</em> fraction. We use this fact to add fractions ($\frac{1}{2} + \frac{1}{4} = \frac{2}{4} + \fra... |
1,645,130 | <p>Is there any known explicit bijection between these two sets? </p>
<p>I know it can be proved that such bijection exists using two injections and Schröder–Bernstein theorem, but I wanted to know whether some explicit bijection is known. I failed to find any except ones constructed awkwardly from the Schröder–Bernst... | Hagen von Eitzen | 39,174 | <p>First note that the set $\mathcal P_{\text{fin}}(\Bbb N)$ of <em>finite</em> subsets of $\Bbb N_0$ is in bijection with $\Bbb N_0$:
$$ \begin{align}\alpha\colon \mathcal P_{\text{fin}}(\Bbb N)&\to \Bbb N\\A&\mapsto \sum_{k\in A}2^k\end{align}$$</p>
<p>Every real number $a\in[0,1)$ as a binary expansion $a=... |
142,819 | <p>I am currently studying Serge Lang's book "Algebra", on page 25 it is proved that if $G$ is a cyclic group of order $n$, and if $d$ is a divisor of $n$, then there exists a unique subgroup $H$ of $G$ of order $d$.</p>
<p>I have trouble seeing why the proof (as explained below) settles the uniqueness part.</p>
<p>T... | Manos | 11,921 | <p>Let $H, H'$ be two subgroups of $G$ of order $d$. Let $m \mathbb{Z} = f^{-1}(H)$ and $m' \mathbb{Z} = f^{-1}(H')$. Then $H = f(m \mathbb{Z})$ and $H' = f(m' \mathbb{Z})$. But $G/H, G/H'$ have the same order. Also by the canonical isomorphism given at the bottom of p. 17, $G/H$ is isomorphic to $\mathbb{Z} / m \mathb... |
452,653 | <p>If $f:X\rightarrow Y$ is initial in category <strong>Top</strong> then
it is easy to proof that </p>
<blockquote>
<p>(!) the topology on $X$ is the set of preimages of open sets in $Y$. </p>
</blockquote>
<p>Just construct topology $Z$ having
the same underlying subset as $X$ and let the set of these preimages
s... | jimjim | 3,936 | <p>Edited : thanks to Did</p>
<p>There is no need for any test, the terms of series tend to $0$ monotonically alternating sign, that is sufficient that series converges. (There was nothing regrading the absolute convergence in the question, so why bother with it?)</p>
<p>More over the limit of the series L is :</p>
... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.