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 |
|---|---|---|---|---|
3,600,139 | <blockquote>
<p>Solve <span class="math-container">$y''-y'-y=\cos x$</span>.</p>
</blockquote>
<p>After first solving the homogeneous equation we know that the solution to it is <span class="math-container">$$y(x)=a\sin(x)+b\cos(x).$$</span></p>
| user577215664 | 475,762 | <p><span class="math-container">$$y''-y'-y=\cos x$$</span>
The characteristic polynomial is:
<span class="math-container">$$\implies r^2-r-1=0$$</span>
<span class="math-container">$$r=\dfrac {1\pm \sqrt 5}2$$</span>
Therefore the solution to the homogeneous equation is:
<span class="math-container">$$y=e^{x/2}(c_1e^{\... |
503,808 | <p>Let $\mathbb{K} = \mathbb{Q}[\sqrt[3]{5}] \ $, and let $\mathbb{L}$ be the normal closure of $\mathbb{K}$. </p>
<p>Let $\mathbb{O}_{\mathbb{K}} \ $ be the integral closure of $\mathbb{Z}$ in $K$ and $\mathbb{O}_{\mathbb{L}} \ $ be the integral closure of $\mathbb{Z}$ in $\mathbb{L}$. I want to find the factorizatio... | paul garrett | 12,291 | <p>For a prime $p$ in $\mathbb Z$ unramified in an extension $K=\mathbb Q(\beta)$, if we are merely <em>close</em> to the true ring of algebraic integers, meaning that $p$ does not divide the discriminate of $\mathbb Z[\beta]$, then the <em>localization</em> at $p$ of the true ring of integers is the same as the locali... |
1,960,793 | <p>Considering the (not most common) definition:</p>
<blockquote>
<p>A set is <em>infinite</em> if it is equipotent to a proper subset of itself. A set is <em>finite</em> if it is not infinite.</p>
</blockquote>
<p>How can I prove that a set $A$ is finite iif it is equipotent to $J_n=\{1,…,n\}$ for some $n{\in}\mat... | G. Snapsmath | 376,415 | <p>Suppose $A$ is not equipotent with any of the $J_n$. Then we see that we can produce a surjection from $A$ to $\mathbb{N}$. If not then there is some maximum $n\in \mathbb{N}$ for $f(A)$. From there you can whittle it down to a bijection to a $J_n$. Then $A$ you can conclude by contradiction that $A$ is equipotent t... |
2,471,217 | <p>Contradiction or contra-positive? Or is direct easier? </p>
<p>∀n∈N, (n>3∧ n is prime)→∃q∈N,(n=6q+1∨n=6q+5)</p>
| DeepSea | 101,504 | <p>Consider $ n\mod 6$. Thus any $n > 3$ can be written as $n = 6q + k$ with $0 \le k < 6$. For if $k = 0,2,3,4$, $n$ is divisible by $2$ or $3$ or $6$ so not a prime. Thus for $n$ to be prime, $k = 1$ or $5$.</p>
|
3,745,452 | <p>I'm evaluating the area of a parametric function and I'm stuck at this point.
<span class="math-container">$$S=2\pi \int_{0}^{2\pi} x\cos(x)\sqrt{x^2+1} \mathrm{d}x.$$</span>
Is there anyway to work this problem further without a calculator?</p>
| Claude Leibovici | 82,404 | <p>There is probably no antiderivative. So, basically, you are asking to perform a numerical integration without calculator.</p>
<p>The only thing I am able to think about is to write <span class="math-container">$\sqrt{1+x^2} \sim x$</span> and then
<span class="math-container">$$S=2\pi \int_{0}^{2\pi} x\cos(x)\sqrt{x... |
3,762,306 | <p>I have the equation <span class="math-container">$$(e^x-1)-k\arctan(x) = 0$$</span> where <span class="math-container">$0<k \leq \frac 2\pi$</span> and I was wondering how I would go about starting to determine the amount of real roots of this equation. So far I have just manipulated the equation to get different... | Robert Israel | 8,508 | <p>Of course <span class="math-container">$x=0$</span> is always a solution. Otherwise, write the equation as <span class="math-container">$$k = \frac{e^x-1}{\arctan(x)}$$</span>
Call the right side <span class="math-container">$f(x)$</span>. The singularity at <span class="math-container">$x=0$</span> is removable, ... |
7,354 | <p>some of my students refer to there being an invisible $-1$ in front of the expression $-(x + 4)$ or in the exponent of $x$. While it is not phrased mathematically, I am ok with them saying this because it reminds them to distribute fully before simplifying etc. It got me thinking though is there a reason to not te... | Frank Newman | 5,104 | <p>The invisible $1$ makes sense when students are just getting started in using certain formulas.</p>
<p>In the equation $y = x + 4$, if students are new to the idea of $y = mx + b$, then it can help to ask the class, "if you had to put some number to the left of the $x$, what would it be?" Someone will probably say... |
1,615,201 | <p>I just solved a long problem for my physics w/calculus homework that required a simplification using a quadratic formula. The "textbook" (flipItPhysics) came up with a different simplification than mine but it turns out they are equivalent. I can't, for the life of me, figure out how to simplify mine into theirs.</p... | Brian M. Scott | 12,042 | <p>It’s a little easier to go the other way: assuming that $q>0$, we have</p>
<p>$$q\sqrt{\frac{Q}q}=\sqrt{q^2}\cdot\sqrt{\frac{Q}q}=\sqrt{q^2\cdot\frac{Q}q}=\sqrt{qQ}\;.$$</p>
<p>If $q<0$, $q=-\sqrt{q^2}$, so you get $-\sqrt{qQ}$; since you have a $\pm$ sign, it doesn’t matter: as long as $q\ne 0$, you get bot... |
1,815,918 | <p>Let $G$ be a group and $|G|=p^n$ for some prime $p$. If $f:G\to H$ is a surjective homomorphism, how do I know $H=f(G)$ also has cardinality a power of $p$?</p>
| Tsemo Aristide | 280,301 | <p>Hint: $f(x)={1\over2}((f(x)+f(-x)))+{1\over 2}(f(x)-f(-x)))$.</p>
<p>$g(x)={1\over2}((f(x)+f(-x)))$</p>
<p>$h(x)={1\over2}((f(x)-f(-x)))$</p>
|
24,412 | <p>Trying to plot the following two functions to show points of intersection.</p>
<pre><code>2 x + y - 1 == 0,
x - y + 2 == 0
ContourPlot[{2 x + y - 1 == 0, x - y + 2 == 0}, {x, -5, 5}, {y, -5, 5}]
</code></pre>
<p>The above shows the plots, but I find it difficult to see the point of intersection. I suspect that th... | ubpdqn | 1,997 | <p>You could put these linear equations in matrix form:</p>
<pre><code>i = LinearSolve[a = {{2, 1}, {1, -1}}, b = {1, -2}]
ContourPlot[Evaluate@Thread[a.{x, y} == b], {x, -3, 3}, {y, -3, 3},
Epilog -> {Red, PointSize[0.02], Point[i]},
PlotLegends -> "Expressions"]
</code></pre>
<p><a href="https://i.stack.i... |
333,265 | <p>Can $|-x^2 | < 1 $ imply that $-1<x<1$?
My steps are as follows?
$$| -x^2| < 1 $$
$$-1<(-x^2)< 1 $$
$$-1<(-x^2)< 1 $$
$$-1<x^2< 1 $$
$$\sqrt{-1}<x< \sqrt 1 $$</p>
<p>I'm actually looking for the radius of convergence for the power series of $\frac{1}{1-x^2}$:</p>
<p>$$\fra... | Dan Rust | 29,059 | <p>Be careful with the last line. $\sqrt{-1}$ is not a real number so it doesn't make much sense to say $x$ is greater than it. You can, however, deduce from $x^2\geq 0$ for all real numbers, that $0\leq x^2<1$ which implies that $-1<x<1$ (you can see this from the graph of $y=x^2$. For a more rigorous proof, ... |
1,318,884 | <blockquote>
<p>Let S be a piecewise smooth oriented surface in <span class="math-container">$\mathbb{R}^3$</span> with positive oriented piecewise smooth boundary curve <span class="math-container">$\Gamma:=\partial S$</span> and <span class="math-container">$\Gamma : X=\gamma(t), t\in [a,b]$</span> a rectifiable para... | Atvin | 215,617 | <p>Hint: For any $a,b$ real numbers: $\min(a,b)+\max(a,b)=a+b$.</p>
<p>Now, if we have $a=a_1^{p_1} a_2^{p_2}\ldots$ and similarly with $b$, if you use the equation I just mentioned for all $p_i$, you will get, that $\gcd(a,b)\cdot\operatorname{lcm}(a,b)=ab$.</p>
|
970,882 | <p>This is homework so no answers please</p>
<p>We have Multiplication map $F:O(n)\times O(n)\to O(n)$ defined as $F(A,B)=AB$ $F:O(n)\times O(n)\to O(n)$, where $O(n)=\{A\in M(n\times n):AA^{t}=id\}$.
The smooth structure of O(n) comes from ImpFT: consider smooth map $f:M(n\times n)\to Sym(n\times n)$ defined as $f(A)... | Daniel Valenzuela | 156,302 | <p>$GL_n$ is a Lie group, in particular multiplication is smooth. $O(n)$ is just a sub manifold which is closed under multiplication, hence $F$ is smooth.</p>
<p>So to prove your homework it would be easier to prove that multiplaction on $GL_n$ is smooth, or even easier (almost trivial) that multiplication on $M(n\ti... |
100,433 | <p>I am trying to export some largish integer arrays to HDF5 and know that every entry in them would fit into an unsigned 8 bit integer array. As the default "DataFormat" for export that Mathematica is using is a 32bit integer array the resulting files are unnecessarily large. Does anyone know the correct syntax to exp... | Albert Retey | 169 | <p>The following seems to work, but I'd also be interested in other, possibly simpler or clearer versions...</p>
<pre><code>Export["int8.h5",
{
"Datasets" -> {"/data" -> RandomInteger[{2^8 - 1}, {100, 100}]},
"DataFormat" -> {"UnsignedInteger8"}
},
"Rules"
]
</code></pre>
<p>Two notes: </p>
<o... |
3,398,164 | <p>I am working through the math behind homographies, but my math skills are a bit rusty.</p>
<p>A homography can be calculated with 8 corresponding points (4-4) because the homography matrix has 8 degrees of freedom. This is because, <a href="https://docs.opencv.org/master/d9/dab/tutorial_homography.html#lecture_16" ... | amd | 265,466 | <p>Suppose that <span class="math-container">$\mathbf y=H\mathbf x$</span>. Then by elementary properties of matrix multiplication, <span class="math-container">$(\lambda H)\mathbf x = \lambda(H\mathbf x)=\lambda\mathbf y$</span>, so when <span class="math-container">$\lambda\ne0$</span>, <span class="math-container">$... |
2,961,592 | <p>We know 4 is a real number but how can we prove that it is a complex number? How can be describe it in the a+ib form??</p>
| Michael Rozenberg | 190,319 | <p>Because <span class="math-container">$$\sqrt{y}-\sqrt{x}=\frac{y-x}{\sqrt{y}+\sqrt{x}}>0,$$</span>
<span class="math-container">$$y-\sqrt{xy}=\sqrt{y}(\sqrt{y}-\sqrt{x})>0$$</span> and
<span class="math-container">$$\sqrt{xy}-x=\sqrt{x}(\sqrt{y}-\sqrt{x})>0.$$</span></p>
|
2,459,263 | <p>If $p$ and $q$ are two points on a $n$-dimensional manifold $M$, then their tangent spaces are two $n$-dimensional vector spaces. So algebraically they have the same structure, but they are not the same because $p$ and $q$ are different points. I can understand they are not the same by visualization. How is it state... | Amitai Yuval | 166,201 | <p>The algebraic statement is very simple: </p>
<p>Even though the spaces $T_pM$ and $T_qM$ are isomorphic, there are many different isomorphisms between these two spaces, none of which is "special" in any way or can be distinguished from the others. This is why one cannot state that the two spaces are equal to one an... |
1,900,365 | <p>Apparently this is a true statement, but I cannot figure out how to prove this. I have tried setting </p>
<p>$$(m)(m + 1)(m + 2)(m + 3) = (m + 4)^2 - 1 $$</p>
<p>but to no avail. Could someone point me in the right direction? </p>
| David Quinn | 187,299 | <p>You can take the product of the four consecutive integers as $$(m-1)m(m+1)(m+2)$$</p>
<p>Adding $1$ and simplifying, we get $$m^4+2m^3-m^2-2m+1$$
$$=m^4+m^2+1+2m^3-2m^2-2m$$
$$=(m^2+m-1)^2$$</p>
|
1,900,365 | <p>Apparently this is a true statement, but I cannot figure out how to prove this. I have tried setting </p>
<p>$$(m)(m + 1)(m + 2)(m + 3) = (m + 4)^2 - 1 $$</p>
<p>but to no avail. Could someone point me in the right direction? </p>
| gnasher729 | 137,175 | <p>You can solve this without much thinking. </p>
<p>By expanding the product, we find $m(m+1)(m+2)(m+3) = m^4 + 6m^3 + 11m^2 + 6m$. That's a bit bigger than $(m^2)^2 = m^4$. $(m^2+c)^2 = (m^4 + 2m^2c + c^2)$ is still too small because there is no term $m^3$. $(m^2+cm)^2 = m^4 + 2cm^3+c^2m^2$ looks better, especially ... |
2,542,242 | <p>When I tried solving for the inverse of $f(x)=\frac{1-x}{-x}$, I got this:</p>
<p>$f^{-1}(x)=\frac{1}{1-x}$</p>
<p>I know that the way to check my answer would be to take the inverse of the inverse I just found, but this is what I get:</p>
<p>$f^{-1}(f^{-1}(x))=1-\frac{1}{x}=\frac{x-1}{-x}$</p>
<p>The last part,... | spaceisdarkgreen | 397,125 | <p>You use the <a href="https://en.wikipedia.org/wiki/Leibniz_integral_rule" rel="nofollow noreferrer">Leibniz formula</a> $$ \frac{d}{dx}\int_{a(x)}^{b(x)} f(x,t)\;dt = f(x,b(x))b'(x) - f(x,a(x))a'(x) + \int_{a(x)}^{b(x)}\frac{\partial}{\partial x}f(x,t)\;dt$$ and get $$ \frac{d}{dx}\int_0^xf(x-t)\;dt = f(0)+\int_0^xf... |
3,779,996 | <p>Let <span class="math-container">$(X_1,...,X_n)$</span> be a random sample with PDF <span class="math-container">$f(x;\theta) = \frac{x}{\theta}\exp(-x^2/(2\theta)), \theta > 0$</span></p>
<p>I want to show that the likelihood ratio test of <span class="math-container">$H_0 : \theta \le \theta_0$</span> against <... | Michael Hardy | 11,667 | <p><span class="math-container">\begin{align}
L(\theta) & = \frac{t}{\theta^n}\exp\left(\frac{-s}{2\theta} \right) \\[8pt]
\ell(\theta) = \log L(\theta) & = -n\log\theta - \frac s {2\theta} + (\text{something not depending on } \theta) \\[8pt]
\ell\,'(\theta) & = \frac{-n}\theta + \frac s {\theta^2} = \frac... |
354,512 | <p>Find the minimal polynomial of the matrix M:
\begin{pmatrix}
0 & 0 & 0 & \dots & 0 & a_{1}\\
1 & 0 & 0 & \dots & 0 & a_{2}\\
0 & 1 & 0 & \dots & 0 & a_{3} \\
\dots & \dots \\
0 & 0 & 0 & \dots & 1 & a_{n}
\end{pmatrix}</p>
<p>Let's ... | Marc van Leeuwen | 18,880 | <p>Your computation shows that $M^n(e_1)-a_nM^{n-1}(e_1)-\cdots-a_2M(e_1)-a_{1}e_1=0$, and since $e_1,M(e_1),\ldots,M^{n-1}(e_1)$ are linearly independent, no nonzero polynomial of lower degree than$~n$ has this property of the polynomial $P=x^n-a_nx^{n-1}-\cdots-a_2x-a_1$. Then the minimal polynomial $\mu$ of$~M$, whi... |
1,357,405 | <p>Under what conditions will the cubic equation
$ax^3 + bx^2 + cx + d$ where $a,b,c,d \in \mathbb R$ yield roots which have negative real parts? (All roots must have negative real parts)</p>
<p><strong><em>Motivation</strong>:
I am studying a dynamical system i.e, Chua circuit, in $ \mathbb R^3 $and wish to analyze i... | i. m. soloveichik | 32,940 | <p>You can separate $x$ into the real and imaginary parts, $x=u+Iv$, to get two equations that must be satisfied. For one of them we can solve for $v$ and substitute to get the equation $$u^3+(b/a)u^2+((2ac+2b^2)/8a^2)u+(1/8)(bc-da)/a^2$$ for $u$. So you want this to have three negative roots $u$. To answer that you ca... |
1,357,405 | <p>Under what conditions will the cubic equation
$ax^3 + bx^2 + cx + d$ where $a,b,c,d \in \mathbb R$ yield roots which have negative real parts? (All roots must have negative real parts)</p>
<p><strong><em>Motivation</strong>:
I am studying a dynamical system i.e, Chua circuit, in $ \mathbb R^3 $and wish to analyze i... | Piquito | 219,998 | <p>There is always a real root and you have three possibilities: </p>
<p>Let $F(x)= ax^3+bx^2+cx+d$ ; the discriminant $\Delta$ is defined by
$$\Delta=(bc)^2+18abcd-4ac^3-4db^3-27(ad)^2$$
(1) $\Delta>0$: $F(x)=a(x-x_1)(x-x_2)(x-x_3)$; the three roots, $x_1, x_2, x_3,$ are real.</p>
<p>(2) $\Delta=0$: $F(x)=a(x-x... |
4,114,962 | <blockquote>
<p>Prove that the product of four consecutive natural numbers are not the square of an integer</p>
</blockquote>
<p>Would appreciate any thoughts and feedback on my suggested proof, which is as follows:</p>
<p>Let <span class="math-container">$f(n) = n(n+1)(n+2)(n+3) $</span>.
Multiplying out the expressio... | Community | -1 | <p>Your solution is perfect and detailed.</p>
<p>Another alternative shorter approach based on observation is if we can consider <span class="math-container">$n(n+2)=k$</span> and then redefine the function as:
<span class="math-container">$$\begin{align*}f(k) &= k(k+2)\\ f(k) &= (k+1)^2-1\end{align*}$$</span><... |
958,267 | <p>When I was doing problems in my textbook, I came across this problem:</p>
<blockquote>
<p>The velocity of a heavy meteorite entering the earth's atmosphere is inversely proportional to $\sqrt s$ when it is s kilometers from the earth's center. Show that the meteorite's acceleration is inversely proportional to $... | Eweler | 179,156 | <p>Because $s$ is strictly a function of time, you can't differentiate the velocity with respect to time without using the chain rule. Perhaps putting this in more familiar notation, $v = v(s(t)) $</p>
<p>Just using the chain rule, $$ \frac{dv}{dt} = \frac{dv}{ds}\frac{ds}{dt} = v\frac{b}{s^{3/2}} = \frac{k'}{s^2}$$</... |
3,490,454 | <p>Definition from Munkres, the textbook we are following: If <span class="math-container">$Y$</span> is a compact Hausdorff space and <span class="math-container">$X$</span> is a proper subspace of <span class="math-container">$Y$</span> whose closure equals <span class="math-container">$Y$</span>, then <span class="m... | Henno Brandsma | 4,280 | <p>Look at <span class="math-container">$$X=\{(x,y)\in \Bbb R^2\mid (x+1)^2 + y^2= 1\} \cup \{(x,y)\in \Bbb R^2\mid (x-1)^2 + y^2= 1\}$$</span></p>
<p>which is the subspace of the plane of two circles intersecting exactly at <span class="math-container">$(0,0)$</span>.</p>
<p><span class="math-container">$X$</span> i... |
4,003,019 | <p>So I've just started learning topology and a lot of the definitions confuse me.
My main problem is that some of the definitions are quite inconsistent to me for example in topology without tears the author says that a topology <span class="math-container">$\tau$</span> is a set of subsets of <span class="math-contai... | Arthur | 15,500 | <p>"A set of sets" is a somewhat abstract notion. It is, formally, what a topology is, but people also need to know how to think about them. Having an informal "hierarchy" of container names gives newcomers some mental intuitive help (or at least, that's the idea).</p>
<p>Thus we get "a family ... |
110,867 | <p>I want to prove that the function $f(x) := x^3$ for all real $x$ defines a homeomorphism from $\mathbb{R}$ to $\mathbb{R}$. But I am finding it difficult to prove that the inverse map is continuous!</p>
| Alex Becker | 8,173 | <p>Suppose $x^{1/3}$ is not continuous at some $x_0\in\mathbb R$, so for some $\epsilon>0$ we have a sequence $(x_n)$ such that $\lim\limits_{n\to\infty} x_n = x_0$ yet $|x_n^{1/3}-x_0^{1/3}|>\epsilon$ for all $n$. We must have either infinitely many $n$ such that $x_n^{1/3}-x_0^{1/3}$ or infinitely many $n$ such... |
867,207 | <p>The Efron-Stein inequality sais that if $X_1,\ldots,X_n$ are independent random variables on say $R^n$, and $f:R^n \rightarrow R$ s.t. $Z:=f(X_1,\ldots,X_n)$ has finite variance, then</p>
<p>$$\operatorname{Var}(X)\le \sum_{i=1}^n E[(Z-E^{(i)}[Z]]$$</p>
<p>where $E^{(i)}$ denotes conditional expectation taken w.r.... | hexaflexagonal | 162,141 | <p>A relation on $A$ is simply a subset of the Cartesian product $A \times A$. For $A=\{a,b,c,d\}$, both $(b,c)$ and $(b,d)$ are contained in $A \times A$; therefore, $\{(b,c),(b,d)\}$ is a relation on $A$. </p>
|
3,143,532 | <blockquote>
<p>Find <span class="math-container">$\det A$</span> and <span class="math-container">$\text{Tr} A$</span> for the matrix <span class="math-container">$A\in M_n(\mathbb{Q})$</span> such that <span class="math-container">$\sqrt[n]{p}$</span> is an eigenvalue of <span class="math-container">$A$</span>, whe... | FDP | 186,817 | <p>For <span class="math-container">$\theta \in [0;\pi]$</span>,
<span class="math-container">\begin{align}
J(\theta)&=\int_0^\infty \ln\left(1-\frac{2\cos(2\theta)}{x^2}+\frac{1}{x^4}\right) \,dx
\end{align}</span>
Perform the change of variable <span class="math-container">$y=\dfrac{1}{x}$</span>,</p>
<p><span c... |
1,552,775 | <p>In how many ways 5 blue pens and 6 black pens can be distributed to 6 children?</p>
<hr>
<p>To do that I used:
$\text{Coefficient of } x^6 \text{ in } ((1+x+x^2+x^3+x^4+x^5)(1+x+x^2+x^3+x^4+x^5+x^6))$
and got answer = 6</p>
<p>but options given are:</p>
<p>a) 97020</p>
<p>b) 116424</p>
<p>c) 8008</p>
<p>d) 67... | Brian M. Scott | 12,042 | <p>It doesn’t. The coefficient of $x^6$ in your polynomial is the number of ways to pick $6$ of the $11$ pens. For instance, the term $x^2\cdot x^4$ (before you collect terms) corresponds to choosing $2$ blue and $4$ black pens.</p>
<p>HINT: This problem is quite different. Let’s forget about the black pens for a mome... |
3,676,719 | <p>I've recently started to learn about differential equations and I am having a hard time solving any of them.</p>
<p>I feel like I'm missing some steps.</p>
<p>Therefore, could I please know how to solve:</p>
<p><span class="math-container">$(ye^x +y)dx+ye^{(x+y)}dy=0$</span></p>
<p>So far, I've looked around the... | Noob mathematician | 779,382 | <p>Your statement is right but the following is not true for any <span class="math-container">$n≥2$</span> : "for a matrix A of size <span class="math-container">$n$</span>, if <span class="math-container">$trace(A)=0$</span> then <span class="math-container">$A^2=0$</span>.</p>
<p>If <span class="math-container">$A=0... |
439,750 | <p>I have to check if $\int_{0}^\infty \mathrm 1/(x\ln(x)^2)\,\mathrm dx $ is convergent or divergent.</p>
<p>My approach was to integrate the function , hence : $\int_{0}^\infty \mathrm 1/(x\ln(x)^2)\,\mathrm dx=-\lim_{x \to \infty} 1/\ln(x)+ \lim_{x \to 0} 1/\ln(x)=0 $</p>
<p>Still my book says that it is diverg... | Mikasa | 8,581 | <p>Look at the following improper integral:
$$
\int_1^2f(x)dx
$$
Certainly, $\lim_{x\to 1^+}(x-1)f(x)=+\infty$ so the Comparison test admits the series is divergent.</p>
|
285,975 | <p>I'm not sure I understand what to do with what's given to me to solve this. I know it has to do with the relationship between velocity, acceleration and time.</p>
<blockquote>
<p>At a distance of <span class="math-container">$45m$</span> from a traffic light, a car traveling <span class="math-container">$15 m/sec$</... | dinoboy | 43,912 | <p>OK, so we have the useful lemma that for integers $a,b$ we have $(a-b)|(f(a) - f(b))$. This follows essentially from the fact that $(a-b)|(a^n - b^n)$ for any $n$. This is why $f(x) \equiv 0 \pmod{m}$ whenever $x \equiv x_0 \pmod{m}$. </p>
<p>Now, how does this help with part i)? Suppose the conclusion is false, s... |
285,975 | <p>I'm not sure I understand what to do with what's given to me to solve this. I know it has to do with the relationship between velocity, acceleration and time.</p>
<blockquote>
<p>At a distance of <span class="math-container">$45m$</span> from a traffic light, a car traveling <span class="math-container">$15 m/sec$</... | Hagen von Eitzen | 39,174 | <p>For (ii) note that $(x_0+km)^n=x_0^n+{n\choose 1}x_0^{n-1}m+{n\choose 2}x_0^{n-2}m^2+\ldots +m^n = x_0^n+m\cdot(\ldots)$, therefore in general $f(x_0+km)\equiv f(x_0)\pmod m$.</p>
<p>There are at most finitely many $x$ with $f(x)=0$, so we can pick $x_0$ such that $m:=f(x_0)\ne 0$.
Among the infinitely many $x=x_0+... |
345,652 | <p>I try to understand why by definition </p>
<ol>
<li>$[c_0,c_1,\ldots,c_n]=[c_0,[c_1,\ldots,c_n]]$ and also </li>
<li>$[c_0,c_1,\ldots,c_n]=[c_0,c_1,\ldots,c_{n-2},[c_{n-1},c_n]]$ .</li>
</ol>
<p>Those are continued fractions, and $1$ and $2$ are notes I have in the lecture summery.</p>
<p>But we can add bracke... | Yoni Rozenshein | 36,650 | <p>Usually one wouldn't write such a thing as $[c_0; [c_1, \ldots, c_n]]$, because elements appearing in a continued fraction representation are supposed to be positive integers (except the one before the semicolon, if there is one, which may be a negative integer).</p>
<p>But it doesn't contradict anything to write t... |
3,163,525 | <p>How would I expand the following function as a power series, around <span class="math-container">$\eta=0$</span>?</p>
<p><span class="math-container">$$g_0(1,\eta)=\frac{\left(\frac{PV}{NkT}\right)_0-1}{4\eta}$$</span></p>
<p>Note that:</p>
<p><span class="math-container">$$\left(\frac{PV}{NkT}\right)_0=1+\frac{3... | Community | -1 | <p><strong>Hint:</strong></p>
<p>The sum is a polynomial with no constant term and its handling is not a problem.</p>
<p>Then</p>
<p><span class="math-container">$$\frac{\eta}{\eta_c-\eta}=1-\frac1{1-\dfrac\eta{\eta_c}}=\frac\eta{\eta_c}+\frac{\eta^2}{\eta_c^2}+\frac{\eta^3}{\eta_c^3}+\cdots$$</span></p>
|
3,187,793 | <p>We know that <span class="math-container">$221 = 17*13$</span>. So we can check if the system has roots to both of those equations separately, which it does:</p>
<p><span class="math-container">$x^{5} \equiv 2$</span> mod <span class="math-container">$13$</span> has the solution <span class="math-container">$6 + 13... | dan_fulea | 550,003 | <p>We work all the time modulo <span class="math-container">$221$</span> from now on tacitly. Since <span class="math-container">$2$</span> is relatively prime to <span class="math-container">$221$</span>, a/the solution <span class="math-container">$x$</span> of <span class="math-container">$x^5=2$</span> (modulo <sp... |
2,408,141 | <p>Let $X\to Y$ so that $f(x)=x^2+4$ </p>
<p>$X=\{6,9,2,8,5\}$ and $Y=\{27,85,40,8,12,29,63,68,17\}$</p>
<p>a) state the domain of $f$</p>
<p>b) state the range of $f$</p>
<p>I have calculated that the domain of $f(x)$ is all real numbers and that the range of $f(x)$ is $f\ge4$.</p>
<p>Therefore I thought that the... | Callus - Reinstate Monica | 94,624 | <p>I think I understand what you're missing. And I think this is what Eric explained, too, in some very general way, but I'm going to be more hand-wavy. To be more precise, I'd need to know what definition you're working with for a field extension generated by an element $e$, or more generally by a set $S$. </p>
<p... |
3,621,836 | <p>Let's say I have a large determinant with scalar elements:</p>
<p><span class="math-container">$$\begin{vmatrix} x\cdot a & x\cdot b & x\cdot c & x\cdot d \\ x\cdot e & x\cdot f & x\cdot g & x\cdot h \\ x\cdot i & x\cdot g & x\cdot k & x\cdot l \\ x\cdot m & x\cdot n & x\... | s1mple | 697,936 | <p>Clearly <span class="math-container">$\omega^3=1$</span> and <span class="math-container">$1+\omega+\omega^2=0$</span>. So, the problem reduces to:</p>
<p><span class="math-container">$$(2-\omega)(2-\omega^2)(2-\omega)(2-\omega^{2})=(2-\omega)^2(2-\omega^2)^2=(4+\omega^2-4\omega)(4+\omega^4-4\omega^2)$$</span>
<spa... |
3,621,836 | <p>Let's say I have a large determinant with scalar elements:</p>
<p><span class="math-container">$$\begin{vmatrix} x\cdot a & x\cdot b & x\cdot c & x\cdot d \\ x\cdot e & x\cdot f & x\cdot g & x\cdot h \\ x\cdot i & x\cdot g & x\cdot k & x\cdot l \\ x\cdot m & x\cdot n & x\... | Matteo | 686,644 | <p>Using modular arhitmetic and the fact that <span class="math-container">$\omega^3=1$</span>, we can write:
<span class="math-container">$$(2-\omega)
\left(2-\omega^2\right)
\left(2-\omega^{19}\right)
\left(2-\omega^{23}\right) =(5-2\omega^2-2\omega)\cdot(5-2\omega^2-2\omega)$$</span>
Now, rearranging this express... |
3,238,316 | <p>I think there is a glitch in the <a href="https://en.wikipedia.org/w/index.php?title=Jensen%27s_inequality&oldid=887772941#Proof_1_(finite_form)" rel="nofollow noreferrer">proof by induction</a>. The proof is still valid, but they add an unnecessary assumption:</p>
<p>In the induction step, they choose one of t... | Botond | 281,471 | <p>Of course, <span class="math-container">$\lambda_1=0$</span> is valid, but not so interesting:
<span class="math-container">$$ \varphi\left(\sum_{i=2}^{n+1}\lambda_i x_i \right) ~\leqslant~ \varphi\left( \sum_{i=2}^{n+1} \lambda_i x_i \right)$$</span> </p>
|
3,420,960 | <p>The problem is as follows:</p>
<blockquote>
<p>A protein sample spins in the counterclockwise direction in a
centrifuge seen from the top as shown in the diagram from below. The
radius of the centrifuge es <span class="math-container">$R=2\,m$</span>. The magnitude of its speed
changes. At a certain instant... | IamWill | 389,232 | <p>You have to decompose the acceleration vector into two components, one which is tangential to the orbit and the other which points towards the center of the circle. Note that the tangential component is paralel to the velocity <span class="math-container">$\vec{v}$</span> but it points in the opposite direction , s... |
2,738,138 | <p>I know it's probably a stupid question, but I'm confused. I have a set {$x\in\mathbb R, \frac{1}{x} \le 1$} that I want to represent as interval/s.</p>
<p>Thinking about it logically, I know that the set is $x\in]-\infty, 0[$U$[1, +\infty[$.</p>
<p>However, when trying to solve the inequality, I can't seem to get ... | Community | -1 | <p>In your second analysis you must intersect the conditions within each case.</p>
<p>In 1. you got $x>0$ and $x\geq 1$. The conjuction of these two is $x\geq 1$.</p>
<p>In 2. you got $x<0$ and $x\leq 1$. The conjunction of these two is $x<0$. The idea is that the solution $x\leq 1$ must be taken into accoun... |
77,290 | <p>How would you determine all integers $m$ such that the following is true? </p>
<p>$$\frac{1}{m}=\frac{1}{\lfloor 2x \rfloor}+\frac{1}{\lfloor 5x \rfloor} .$$</p>
<p>Note that $\lfloor \cdot \rfloor$ means the greatest integer function. Also, $x$ must be a positive real number.</p>
| Brian M. Scott | 12,042 | <p>You can solve it by cases. Let $x=n+r$, where $n$ is an integer and $r=\lfloor x\rfloor$. Then $\lfloor 2x\rfloor = \lfloor 2n+2r\rfloor = 2n+\lfloor 2r\rfloor$ and $\lfloor 5x\rfloor = \lfloor 5n+5r\rfloor = 5n+\lfloor 5r\rfloor$, and the original equation can be written as $$\frac1m = \frac1{2n+\lfloor 2r\rfloor}+... |
77,290 | <p>How would you determine all integers $m$ such that the following is true? </p>
<p>$$\frac{1}{m}=\frac{1}{\lfloor 2x \rfloor}+\frac{1}{\lfloor 5x \rfloor} .$$</p>
<p>Note that $\lfloor \cdot \rfloor$ means the greatest integer function. Also, $x$ must be a positive real number.</p>
| Yuval Filmus | 1,277 | <p>We will assume that $x,m > 0$. Our results show (unless there is some mistake in the case analysis) that the only solutions (up to a value of $x$) are given by
$$ \frac{1}{10C} = \frac{1}{14C} + \frac{1}{35C} $$
and the exceptional solution
$$ \frac{1}{3} = \frac{1}{\lfloor 2\cdot 2.4\rfloor} + \frac{1}{\lfloor 5... |
2,406,049 | <p>I would appreciate help understanding what $\aleph_1$ is according to this definition:</p>
<blockquote>
<blockquote>
<p>If $\alpha$ is an ordinal, then $\aleph_{\alpha}$ is the unique infinite cardinal such that:
$\{\kappa:\kappa\text{ is an infinite cardinal and }\kappa\lt\aleph_{\alpha}\}$
is isomor... | Asaf Karagila | 622 | <p>Note that $1$ is just $\{0\}$. So a well-ordered set is isomorphic to $1$ if and only if it has exactly one element.</p>
<p>In the case of $\aleph_1$, it is the unique transfinite cardinal $\kappa$ which has exactly one transfinite cardinal smaller than it. In other words, it would be exactly the smallest cardinal ... |
87,740 | <p>There appear to be questions perhaps tangentially related to this that have been asked already. If so a reference and a close would be heartily appreciated.</p>
<p>Give some category $\mathcal{C}$ with the modifiers listed in the title of the question, and an object $A$ in that category, how does one define the "f... | Yemon Choi | 763 | <p>The <strong>2nd edition</strong> of Mac Lane's <em>Categories for the Working Mathematician</em> has a description/construction of the free monoid on a given object in a monoidal category satisfying some side conditions, see Section 7.3. I am not sure off the top of my head if those conditions are met for every symm... |
87,740 | <p>There appear to be questions perhaps tangentially related to this that have been asked already. If so a reference and a close would be heartily appreciated.</p>
<p>Give some category $\mathcal{C}$ with the modifiers listed in the title of the question, and an object $A$ in that category, how does one define the "f... | Todd Trimble | 2,926 | <p>Very simply, if a closed symmetric monoidal category has countable coproducts, and if monoid means monoid with respect to the monoidal product, then the free monoid on an object $A$ can be constructed as the "geometric series" </p>
<p>$$F(A) = \sum_{n \geq 0} A^{\otimes n}.$$ </p>
<p>The key fact needed to prove t... |
1,393,955 | <p>I have the following evolution equations realted to mean curavture flow, with the induced metric $g=\{g_{ij}\}$, measure $d\mu$ and second fundamental form $A=\{h_{ij}\}$:</p>
<p>1)$\frac{\partial}{\partial t}g_{ij}=-2Hh_{ij}$</p>
<p>2)$\frac{\partial}{\partial t}d\mu=-H^2d\mu$</p>
<p>3)$\frac{\partial}{\partial ... | Alex | 78,161 | <p>I've found this to be the best way to do it. I won't post a full solution but a rough sketch,</p>
<p>$\partial_tP=\sum(h\cdots h)_t$</p>
<p>$=\sum\sum(h_t h\cdots h)$</p>
<p>$=\sum\sum((\Delta h) h\cdots h)+\sum\sum|A|^2h\cdots h)$</p>
<p>$=\sum\sum((\Delta h)h\cdots h)+\alpha|A|^2P$</p>
<p>Now $\Delta P=g^{ij}... |
2,536,206 | <p>Let $M$ be a smooth manifold of dimension $n$, and let $T(M)$ be the tangent bundle of $M$. Let $F$ be a smooth vector field, which is to say a smooth section of the canonical map $T(M) \rightarrow M$. If I understand the manifold structure on $T(M)$ correctly, smoothness means that for any chart $(U,\phi)$ of $M$... | Mastrem | 253,433 | <p>Let $m,n\ge 3$ be given and suppose that $m\neq n$. Without loss of generality, we can say that $n>m$.</p>
<p>Assume there are primes between $m$ and $n$. Let $p$ be the largest prime number with $n>p\ge m$. It is clear that $p\not\in P_m$. If $p\not\in P_n$, we must have $p\mid n$, so $n\ge 2p$. However, Bet... |
673,946 | <p>I have the function $f(x)=\frac {1}{2} \mathbf x^T Q \mathbf x$. </p>
<p>I want to use the steepest descent algorithm where $Q$ is the diagonal matrix $\begin{bmatrix}1 & 0\\0 & 20\end{bmatrix}$ and $\mathbf x = \begin{bmatrix}0.7\\-0.2\end{bmatrix}$.</p>
<p>I want to implement the ideal line search algori... | Royi | 33 | <p>You can solve it by seeing that $ d = -Q x $ namely the gradient descent.</p>
<p>Solving you equation will yield:</p>
<p>$$ \alpha = \frac{ {x}^{T} {Q}^{T} Q x }{ 2 {x}^{T} {Q}^{T} Q Q x } $$</p>
|
3,001,860 | <p>I am looking for a <span class="math-container">$u(x)$</span> and an <span class="math-container">$f$</span> that make the answer to <span class="math-container">$$u(x)''''-u(x)''+u(x)=f(x)$$</span> where <span class="math-container">$$u'(-1)=u'(1)=u(1)=u(-1)=0$$</span> reasonably short.</p>
<p>I have some numerica... | David G. Stork | 210,401 | <p>If <span class="math-container">$u(t) = (t-1)^2 (t+1)^2$</span>, then</p>
<p><span class="math-container">$f(t) = 29 - 14 t^2 + t^4$</span></p>
<p>Seems "simple enough."</p>
<p>Here's the solution for <span class="math-container">$f(t) = 0$</span>:</p>
<p><span class="math-container">$$-\frac{c_1 e^{-\frac{\sqrt... |
28,348 | <p>Thomson et al. provide a proof that <span class="math-container">$\lim_{n\rightarrow \infty} \sqrt[n]{n}=1$</span> in <a href="http://classicalrealanalysis.info/documents/TBB-AllChapters-Landscape.pdf#page=95" rel="noreferrer">this book (page 73)</a>. It has to do with using an inequality that relies on the binomial... | Community | -1 | <p>Take <span class="math-container">$n=2^m$</span></p>
<p><span class="math-container">$$\lim\limits_{n \to \infty} \sqrt[n]{n} = \lim\limits_{m \to \infty} \sqrt[2^m]{2^m}= \lim\limits_{m \to \infty} 2^{\frac{m}{2^m}}=2^{\lim\limits_{m \to \infty} \frac{m}{2^m}}=2^0=1$$</span></p>
<p>This is inverted and maybe a mo... |
3,477,532 | <p>In the cyclic quadrilateral <span class="math-container">$ABCD$</span>, <span class="math-container">$AB:BC:CD:DA=1:9:9:8$</span>, <span class="math-container">$AC$</span> intersects <span class="math-container">$BD$</span> at <span class="math-container">$P$</span>, what is <span class="math-container">$S_{\triangl... | peterwhy | 89,922 | <p>Consider <span class="math-container">$\triangle PAB$</span> and <span class="math-container">$\triangle PDC$</span>. These pairs of inscribed angles are equal:</p>
<p><span class="math-container">$$\begin{align*}
\angle CAB &=\angle BDC\\
\angle ABD &= \angle DCA
\end{align*}$$</span></p>
<p>So <span clas... |
3,536,120 | <p>Textbook example says that:</p>
<p><span class="math-container">$$\sum \limits_{i=k+1}^{\infty}(1-p)^{i-1}p = \frac{(1-p)^k p}{1-(1-p)}=(1-p)^k~~~~~k=1,2,3,\cdots$$</span></p>
<p>I'm told that a geometric series identity used to obtaining the above result is:</p>
<p><span class="math-container">$$\sum \limits_{n=... | Henno Brandsma | 4,280 | <p>Of course the smallest <span class="math-container">$\sigma$</span>-algebra containing <span class="math-container">$\Bbb K$</span> contains <span class="math-container">$\tau$</span> (as all open sets are (countable) unions of base elements. This holds already in hereditarily Lindelöf general spaces and certainly i... |
2,240,616 | <p>Let $f:[0,1] \to \mathbb{R}$ be a function such that for every $a \in [0,1)$ and $b \in (0,1]$ the one-sided limits $$f(a^+)=\lim _{x\to a^+}f(x) \in \mathbb{R}$$ $$f(b^-)=\lim _{x \to b^-} f(x) \in \mathbb {R}$$ exist. </p>
<p>A) Show that $f$ is bounded. </p>
<p>B) Does $f$ necessarily achieve its maximum at som... | Hagen von Eitzen | 39,174 | <p>A) Assume $f$ is not bounded from above. Then there exists a sequence $(x_n)_{n\in\Bbb N}$ such that $f(x_n)\to+\infty$. This sequence has a limit point $x\in[0,1]$. Then a subsequence of $(x_n)$ converges to $x$, either from the right or from the left. But then $f(x_n)\to \infty$ contradicts the existence of $f(x+)... |
2,240,616 | <p>Let $f:[0,1] \to \mathbb{R}$ be a function such that for every $a \in [0,1)$ and $b \in (0,1]$ the one-sided limits $$f(a^+)=\lim _{x\to a^+}f(x) \in \mathbb{R}$$ $$f(b^-)=\lim _{x \to b^-} f(x) \in \mathbb {R}$$ exist. </p>
<p>A) Show that $f$ is bounded. </p>
<p>B) Does $f$ necessarily achieve its maximum at som... | Paramanand Singh | 72,031 | <p>Let's extend $f$ to whole of $\mathbb{R}$ by letting $f(x) = f(0)$ for $x < 0$ and $f(x) = f(1)$ for $x > 1$. Then $f$ has left and right hand limits at each point of $[0, 1]$. Therefore for each point $x \in [0, 1]$ there is an open interval $I_{x}$ containing $x$ such that $f$ is bounded on $I_{x}$. By Heine... |
2,663,370 | <p>Consider the following integral:</p>
<p>$$
I(k):= \int_k^{+\infty} \frac{e^{-1/x} \log(x)}{x^2} dx\\
=-\int_0^{1/k} e^{-t} \log t \, dt,
$$</p>
<p>where $k>0$. It is known that $\lim_{k \to 0} I(k)=\gamma$, where $\gamma$ denotes the Euler-Mascheroni constant. Henceforth, it must be that $\lim_{k\to +\infty}I(k... | Micah | 30,836 | <p>If $k$ is large, $t$ is near zero, so $I(k)$ is roughly (to $0$th order) equal to
$$-\int_0^{1/k} \log t \,dt = -\frac{1}{k}\left(\log\left(\frac{1}{k}\right) - 1\right)=\frac{1}{k}(\log k + 1)$$</p>
<p>That is, $I(k)$ is $\Theta\left(\frac{\log k}{k}\right)$.</p>
|
405,830 | <p>Are there any cases of finite subgroups of <span class="math-container">$O_n(\mathbb{Q})$</span> <s>not contained in</s> <i>not isomorphic to any subgroup of</i> <span class="math-container">$O_n(\mathbb{Z})$</span>?</p>
| YCor | 14,094 | <p>Let <span class="math-container">$n=m^2$</span> be a square. Then the vector <span class="math-container">$(1,\dots,1)\in\mathbf{Q}^n$</span> has norm <span class="math-container">$m$</span>, as does <span class="math-container">$(0,\dots 0,m)$</span>. Hence by Witt's theorem there exists an element <span class="mat... |
405,830 | <p>Are there any cases of finite subgroups of <span class="math-container">$O_n(\mathbb{Q})$</span> <s>not contained in</s> <i>not isomorphic to any subgroup of</i> <span class="math-container">$O_n(\mathbb{Z})$</span>?</p>
| David E Speyer | 297 | <p>A theorem which you might be looking for is that, if <span class="math-container">$G$</span> is any finite subgroup of <span class="math-container">$O_n(\mathbb{Q})$</span>, then there is a lattice <span class="math-container">$\Lambda$</span> in <span class="math-container">$\mathbb{Q}^n$</span> which is preserved ... |
4,517,063 | <p>Prove that there are exactly <span class="math-container">$8100$</span> different ways of distributing <span class="math-container">$4$</span> indistinguishable black marbles and <span class="math-container">$6$</span> distinguishable coloured marbles ( none of them black) into <span class="math-container">$5$</span... | Graham Kemp | 135,106 | <p>Count ways to assign black balls to the boxes – partitioned by how many boxes contain two black balls – then count ways to arrange distinguishable balls among the remaining spaces — remembering that ball placement inside a box is not ordered.</p>
<p>Eg: The count of ways to assign two boxes with double-black, and ar... |
2,392,841 | <p>Assuming the series of functions $\{\phi_{n}(x)\}$ and the function $\phi(x)$ are in Schwartz space $S(R)$,
now my question is this,</p>
<p>Does $\lim_{n\to\infty} \phi_{n}(x)\to \phi(x)$ (in Hilbert space $L^2[-\infty,\infty]$)
imply $\lim_{n\to\infty}\phi_{n}(x)\to \phi(x)$ (in semi-norms topology of Schwartz spa... | D_S | 28,556 | <p>Let $R$ be a ring whose prime spectrum $X$ is finite and discrete. Then $R$ must be a semilocal ring with maximal ideals $\mathfrak m_1, ... , \mathfrak m_n$, with no prime ideals. </p>
<p>I think the geometric argument you suggested will work. I don't know how much algebraic geometry you know, so I'll state the ... |
1,775,378 | <p>So this question is inspired by the following thread: <a href="https://forums.factorio.com/viewtopic.php?f=5&t=25008">https://forums.factorio.com/viewtopic.php?f=5&t=25008</a></p>
<p>In it, the poster is examining an $8$-belt balancer (more on that to come) which he shows fails to satisfy a desirable proper... | David Ketcheson | 20,368 | <p>UTU = Universally throughput unlimited</p>
<p><strong>There exist UTU balancers for 3, 4, and 5 belts, and likely for any number of belts</strong>. Examples are given in <a href="https://gist.github.com/ketch/b0e30aae8452716233969479d6b6ca5d" rel="noreferrer">this Jupyter notebook</a>, along with a Python implement... |
83,921 | <p>Consider $J = \sum_{i=0}^{N}y_{i-1}x_{i}y_{i+1}$ where $+$ and $-$ in the indices are mod $N+1$. Let $x_{i} = 1 - y_{i} \in \{0,1\}$. What are some of the tools useful and relaxation techniques available to maximize $J$ or any other symmetric multivariate polynomial?</p>
| bnaul | 19,008 | <p>In case anyone else comes across this: the model I ended up using is by León, Massé, and Rivest in the Journal of Multivariate Analysis (see <a href="http://www.sciencedirect.com/science/article/pii/S0047259X05000382" rel="nofollow">here</a>). They give a distribution on the space of skew-symmetric matrices that giv... |
3,426,316 | <blockquote>
<p>If <span class="math-container">$f(x)\leq a$</span> then can we say <span class="math-container">$$\int f(x) dx\leq a\int dx?$$</span></p>
</blockquote>
<p>I want to know if this is true or not and can we prove the answer. if it's right can you provide a good simple explanation?
thanks in advance.</p... | Dietrich Burde | 83,966 | <p>An <span class="math-container">$L$</span>-module <span class="math-container">$V$</span> is just a Lie algebra representation <span class="math-container">$\phi\colon L\rightarrow \mathfrak{gl}(V)$</span>, where <span class="math-container">$x.v=\phi(x)(v)$</span>. By the universal property of <span class="math-con... |
2,843,337 | <blockquote>
<p>If $0 \lt x \lt \dfrac{\pi}{2}$, prove that
$$x^{3/2}\sin x + \sqrt{9-x^3}\cos x \leq 3$$</p>
</blockquote>
<p>This question must be done without calculus. First, I tried splitting it into the intervals $(0,\pi/4)$ and $(\pi/4, \pi/2)$, hoping that, $\sin x$ was bound tightly enough on the interva... | Aaron Meyerowitz | 84,560 | <p>It is just trig: Any expression $A\sin(\theta)+B\cos(\theta)=C\sin(\theta+\omega)$ for a phase shift $\omega$ with $\tan(\omega)=\frac{B}{A}$ and $C^2=A^2+B^2.$ Here $C=3.$</p>
|
1,387,216 | <p>Let $X$ be a metric space, and let $E$ and $K$ be two sets such that $E\subset K$.</p>
<p>I want to prove:</p>
<p>If $p$ is a limit point of $E$, then $p$ is a limit point of $K$.</p>
<p>Proof: if every neighborhood of $p\in E$ contains a point $q\neq p$ such that $q\in E$, then this $q$ is also in $K$, then $p$ ... | OKPALA MMADUABUCHI | 98,218 | <p>I dont think so. To proceed, let $p$ be a limit pointn of $E$. Then any neighbourhood of $N\setminus\{p\}\cap E\neq \emptyset.$ Now let $N$ be a neighbourhood of $\{p\}$ in $K$. Then $N\cap E$ is a neighbourhood of $p$ in $E$. So $(N\cap E\setminus \{p\})\cap E\neq \emptyset$ and therefore $N\cap K\neq \emptyset$. ... |
2,426,394 | <p>I'm sorry to bother you with this easy problem. But I'm working alone and totally confused.
It is the 1378th Problem from "Problems in Mathematical Analysis" written by Demidovich. The standard answer is
$$\frac{60(1+x)^{99}(1+6x)}{(1-2x)^{41}(1+2x)^{61}}$$</p>
<p>But when I followed the rutine
$$\frac{d(P(x)/Q(x))... | neonpokharkar | 477,567 | <p>You can write those equations as
$$\Biggr(\frac{1+x}{1-2x}\Biggr)^{40}\Biggr(\frac{1+x}{1+2x}\Biggr)^{60}$$
And use multiplication rule</p>
<p>It is easier this way</p>
<blockquote class="spoiler">
<p>$$\color{green}{\frac{d}{dx} }\Biggr(\color{red}{\Biggr(\frac{1+x}{1-2x}\Biggr)^{40}}\color{blue}{\Biggr(\frac{1... |
2,903,259 | <p>I am working on one example from Munkres book "Topology"and I would like to clarify one question.</p>
<p><strong>Example:</strong> Consider the set $[0,1)$ of real numbers and the set $\mathbb{Z}_+$ of positive integers, both in their usual orders; give $\mathbb{Z}_+\times [0,1)$ the dictionary order. This set has ... | Chessanator | 363,017 | <p>The order type of the dictionary order depends on which way round the two factors are, because one of the factors has a much greater effect on the ordering than the other. Going by the convention used in your example (so that $\mathbb{Z}_+\times [0,1)$ is isomorphic to the non-negative reals) we see that $[0,1) \tim... |
4,292,822 | <blockquote>
<p>Prove linear independence of <span class="math-container">$1+x^3-x^5,1-x^3,1+x^5$</span> in the Vector Space of Polynomials</p>
</blockquote>
<p>The attempts I found online all are quite easy. You just substitute something in for <span class="math-container">$x$</span> into the equation <span class="mat... | José Carlos Santos | 446,262 | <p>In order to deal with this problem, you deal with expressions of the type<span class="math-container">$$a(1+x^3-x^5)+b(1-x^3)+c(1+x^5).\tag1$$</span>Now, take <span class="math-container">$\alpha\in\Bbb R$</span>. Then the map<span class="math-container">$$\begin{array}{rccc}\operatorname{ev}_\alpha\colon&\Bbb R... |
68,147 | <p>While I was investigating some specific types of prime numbers I have faced with the following infinite sequence :</p>
<p>$1,2,8,9,15,20,26,38,45,65,112,244,303,393,560,....$</p>
<p>I tried to find recursive formula using Maple and it's listtorec command, so up to $393$ I got the next output:</p>
<p>$ f(n+3) = ((... | Juan S | 2,219 | <p>Almost certainly the first thing you should do is try <a href="http://oeis.org/search?q=1%2C2%2C8%2C9%2C15%2C20&language=english&go=Search" rel="nofollow">OEIS</a></p>
<p>Doing so gives in your case gives that the sequence is numbers $n$ such that $6\times10^n+1$ is prime.</p>
|
1,991,600 | <p><a href="https://i.stack.imgur.com/2L9j9.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/2L9j9.jpg" alt="enter image description here" /></a></p>
<p>I have worked out the areas as <span class="math-container">$\pi/3$</span> for the circle and <span class="math-container">$2/\sqrt3$</span> for the ... | David Quinn | 187,299 | <p>If the radius of the circle is $1$ then the area of the circle is $\pi$ and the area of the triangle is $3\sqrt{3}$</p>
<p>Therefore the percentage is $$\frac{\pi}{3\sqrt{3}}\times100\simeq\frac{100}{\sqrt{3}}\simeq\sqrt{\frac{10000}{3}}\simeq\sqrt{3300}\simeq60%$$</p>
|
1,137 | <p>In a freshman course in mathematics for biology students, I have had the issue that basic algebra (e.g. simplifying $\frac{a/b}{c/d}$) is far from being mastered. On the one hand, it seems ill-suited to teach more advanced matters to student failing on the basics. On the other hand, if we take too much time to cover... | paul garrett | 63 | <p>In addition to other useful answers... this sort of problem will recur forever, since students (even with good attitudes) chronically misjudge the importance of background courses, and chronically misjudge the degree of facility and competence really desirable for later work. E.g., sometimes students see a passing g... |
1,137 | <p>In a freshman course in mathematics for biology students, I have had the issue that basic algebra (e.g. simplifying $\frac{a/b}{c/d}$) is far from being mastered. On the one hand, it seems ill-suited to teach more advanced matters to student failing on the basics. On the other hand, if we take too much time to cover... | William | 581 | <p>You should give your students an assessment test to see what they have not mastered.
Then you may wish to give them some basic material for practice on their own time.</p>
<p>Once a professor asked me "Didn't you take a course in differential equations"?</p>
<p>I said "Of course, twice".</p>
|
183,237 | <blockquote>
<p>Does</p>
<p><span class="math-container">$$\int_{-\infty}^\infty \text{e}^{\ a\ (x+b)^2}\ \text dx=\int_{-\infty}^\infty \text{e}^{\ a\ x^2}\ \text dx\ \ \ \ \ ?$$</span></p>
<p>hold, even if the imaginary part of <span class="math-container">$b$</span> is nonzero?</p>
</blockquote>
<p>What I really wan... | DonAntonio | 31,254 | <p>Put
$$(x+b)^2=u^2\Longrightarrow 2(x+b)dx=2udu\Longrightarrow dx= du$$ so we get</p>
<p>$$\int_{-\infty}^\infty e^{a(x+b)^2}dx=\int_{-\infty}^\infty e^{au^2}du$$
and your question's answered in the affirmative.</p>
<p><strong>Added:</strong> You may want to divide the integral in two rays $\,(-\infty,0)\,,\,(0,\i... |
481,017 | <blockquote>
<p>Find all $f(x)$ satisfying $f(f(x)) = x^2 - 2$.</p>
</blockquote>
<p>Presumably $f(x)$ is supposed to be a function from $\mathbb R$ to $\mathbb R$ with no further restrictions (we don't assume continuity, etc), but the text of the problem does not specify further. </p>
<p><strong>Possibly Helpful L... | Gottfried Helms | 1,714 | <p>There is another way to arrive by a meaningful generalization to a result. <em>(Btw., this is essentially the same result which I got using the Carlemanmatrix; if we recenter the function $w(x)$ of my previous posting we get $f(x)$ with all displayed digits correct.)</em> </p>
<p>The resulting function has the f... |
3,489,376 | <p>Let <span class="math-container">$C$</span> be a category. Then by definition, for very ordered triple <span class="math-container">$A,B,C$</span> of objects, there is a law of composition of morphisms, i.e., a map <span class="math-container">$$Hom_C(A,B)\times Hom_C(B,C)\longrightarrow Hom_C(A,C)$$</span> where <s... | Berci | 41,488 | <p>No, that is not possible:</p>
<p>If we have a function <span class="math-container">$X\to\emptyset$</span>, then <span class="math-container">$X=\emptyset$</span>, because for every <span class="math-container">$x\in X$</span> there should be an assigned element in the codomain.</p>
<p>If, however, <span class="ma... |
2,393,130 | <blockquote>
<p><strong>Problem:</strong> <span class="math-container">$ABCD$</span> is a rectangle. A point <span class="math-container">$P$</span> is <span class="math-container">$11$</span> from <span class="math-container">$A$</span>, <span class="math-container">$13$</span> from <span class="math-container">$B$</s... | Michael Rozenberg | 190,319 | <p>Just $x^2+13^2=7^2+11^2$ by Pythagoras theorem.</p>
<p>Let $PK$, $PL$, $PM$ and $PN$ be altitudes of $\Delta DPA$, $\Delta DPC$, $\Delta CPB$ and $\Delta APB$ respectively.
Thus,
$$x^2+13^2=DK^2+KP^2+PM^2+MB^2=PN^2+KP^2+PM^2+PL^2=11^2+7^2$$ </p>
|
16,477 | <p>What is meant when one says that one has chosen a basis of fields on the manifold with ``anolonomy"?</p>
<p>I get the feeling that it is a choice of basis with non-trivial structure constants say $C^{k}_{ij}$ </p>
<p>Now in some papers people seem to state that once such a basis is chosen one can write the Christo... | jvkersch | 3,909 | <p>You are correct, a nonholonomic (or anholonomic) basis of the tangent bundle is a set of vector fields $X_i$, $i = 1, \ldots, n$, with nonvanishing structure constants: $[X_i, X_j] = C_{ij}^k X_k$. I'm not sure where this terminology comes from, since in classical mechanics nonholonomicity means exactly the opposit... |
16,477 | <p>What is meant when one says that one has chosen a basis of fields on the manifold with ``anolonomy"?</p>
<p>I get the feeling that it is a choice of basis with non-trivial structure constants say $C^{k}_{ij}$ </p>
<p>Now in some papers people seem to state that once such a basis is chosen one can write the Christo... | José Figueroa-O'Farrill | 394 | <p>Maybe I should elaborate on my comment. On a riemannian manifold $(M,g)$ there exists a unique metric-compatible torsion-free affine connection. It's the Levi-Civita connection and one can prove its existence and uniqueness constructively by giving a formula, known as the Koszul formula.
This formula is given by
$... |
54,815 | <p>A notorious question with prime numbers is estimating the gaps between consecutive primes. That is, if $(p_n)_{n \geq 1}$ is the canonical enumeration of the primes, then set $g_n = p_{n+1} - p_n$. It is shown that $g_n > \frac{c \log(n) \log \log(n) \log \log \log \log(n)}{(\log \log \log(n))^2}$ infinitely ofte... | Micah Milinovich | 3,659 | <p>Let $q_n$ denote the $n^{\text{th}}$ number that is a product of exactly two distinct primes. It is known that
$$\liminf_{n\to \infty} \ (q_{n+1}-q_n) \le 6.$$
This is a result of Goldston, Graham, Pintz, and Yildirim.</p>
<p><a href="http://arxiv.org/abs/math/0609615" rel="noreferrer">http://arxiv.org/abs/math/06... |
120,281 | <p>Does this outline of a proof work?</p>
<p>Consider the ball and the bidisc in $\mathbb{C}^2$. Give each space its Bergman metric. To show that the ball and the bidisc are not holomorphic, it is enough to show that they are not isometric. </p>
<p>One way to distinguish the two spaces is their sectional curvature.... | ahmed sulejmani | 31,537 | <p>You can look at the holomorphic sectional curvature. For the ball it is constant. Actually the constancy of the holomorphic sectional curvature of the Bergman metric distinguishes the ball (and domains biholomorphic to it) by an old theorem of Lu Qi Keng. </p>
|
1,405,486 | <p>My question concerns division by zero. Let's say there are two functions, $g(x)$ and $f(x)$ that approach $0$, as $x\to t$, also assume their derivative w.r.t. $x$ is finite as $x\to t$. Using L'Hopital's Rule: $\frac{f(x)}{g(x)} = \frac{f'(x)}{g'(x)}$ $x\to t$, Can you say $\frac{f'(x)}{f(x)} = \frac{g'(x)}{g(x)}$?... | Dodo | 263,266 | <p>Arguably, $\frac10$ can take on any or no value. Therefore it is incorrect, though this may seem strange, to say that $\frac10=\frac10$ in any (real) case. Of course, any value <em>approaching</em> zero but not zero can be compared.</p>
<p><strong>Edit:</strong> In any other real case, $\frac1x=\frac1x$, because va... |
3,994,280 | <p>Show that <span class="math-container">$(\lnot(p\lor(\lnot p\land q))$</span> is logically equivalent to <span class="math-container">$\lnot p \land \lnot q$</span>.</p>
<p>I am wondering what I did is correct. Very new to learning simple logic.</p>
<p><span class="math-container">$$(\lnot(p\lor(\lnot p\land q)) \eq... | SagarM | 142,677 | <p>Only your last step is wrong:
<span class="math-container">$$\neg(p \lor q) \equiv (\neg p \land \neg q)$$</span></p>
|
2,136,183 | <p>Find integers x,y such that the repeating decimal 0.712341234.... = x/y.</p>
<p>I would actually do this problem if the 7 was not there. If the 7 was not there, my proof would be as follows.</p>
<p>proof:</p>
<p>Let z = 0.12341234...</p>
<p>Then 10^4z = 1234.1234</p>
<p>10^4z - z = 1234</p>
<p>z = 1234/(10^4 -... | Henricus V. | 239,207 | <p>$$0.7\overline{1234} = 0.7 + 0.0\overline{1234}
$$</p>
|
90,006 | <p>I have two problem collections I am currently working through, the "Berkeley Problems in Mathematics" book, and the first of the three volumes of Putnam problems compiled by the MAA. These both contain many problems on basic differential equations.</p>
<p>Unfortunately, I never had a course in differential equation... | David Mitra | 18,986 | <p>Let $\epsilon=1/4$. Let $N$ be given. Then
$${n+1\over 2n+3}\le {n \over 2n}\le{1\over2}\quad\Rightarrow|{n+1\over 2n+3} -1|\ge 1/2.$$
This is true for all $n\ge N$. </p>
<p>So, there is no $N$ such that $|{n+1\over 2n+3} - 1|<\epsilon$
for all $n\ge N$. This shows the sequence does not converge to 1.</p>
|
2,831,469 | <p>How can we solve $$y'''(t)+a(y''(t))^2+b(y'(t))^3=0$$</p>
<p>Could one make some kind of least common denominator argument to decide possible substitutions? Since the chain rule will come into play, I suppose a substitution both for the variable $t$ and the function $y$ could be useful. Possibly some powers of them... | Julián Aguirre | 4,791 | <p>As Adrian suggests, let $y'=z$, to get the second order equation
$$
z''+a\,(z')^2+b\,z^3=0.
$$
Since the independent variable $t$ does not appear explicitly in the equation (I am assuming $a$ and $b$ are constants), we let
$$
z'=p,\quad z''=\frac{dp}{dt}=\frac{dp}{dz}\,\frac{dz}{dt}=p\,\frac{dp}{dz}.
$$
This gives t... |
2,831,469 | <p>How can we solve $$y'''(t)+a(y''(t))^2+b(y'(t))^3=0$$</p>
<p>Could one make some kind of least common denominator argument to decide possible substitutions? Since the chain rule will come into play, I suppose a substitution both for the variable $t$ and the function $y$ could be useful. Possibly some powers of them... | user577215664 | 475,762 | <p>$$y'''(t)+a(y''(t))^2+b(y'(t))^3=0$$
Substitute $z=y'$
$$z''(t)+a(z'(t))^2+bz^3=0$$
Substitute $p=z'$
$$\frac {dp}{dz}p+ap^2+bz^3=0$$
$$\frac 12(p^2)'+ap^2+bz^3=0$$
Finally substitute $w=p^2$
$$\frac 12w'+aw+bz^3=0$$
Bernouilli's equation</p>
|
477,250 | <p>Let $u=\{u_\alpha\}$ be an open cover of $[a,b], a <b $. Let $S=\{r \in [a,b]$ such that $[a,r]$ is covered by some finite collection of open sets belonging to $u\}$. Is $S$ non-empty ? How ?</p>
| bradhd | 5,116 | <p>Let $a_{ij}$ be the $(i,j)^{th}$ entry of $A$ and $b_{ij}$ the $(i,j)^{th}$ entry of $A^{-1}$. Then from $AA^{-1}=I$ we have</p>
<p>$$\sum_{k=1}^5 a_{ik}b_{kj} = \delta_{ij},$$ and the given condition on the rows of $A$ is that
$$\sum_{i=1}^5 a_{ij}=1$$ for each $j$. We wish to deduce that
$$\sum_{i,j}b_{ij}=5.$$</... |
107,653 | <p>Suppose you have the set of all possible $n$ x $n$ square adjacency matrices where $n$={1,2,3,4...}. For each matrix, compute the logarithm of the largest eigenvalue. Is it true that the set of logarithms you obtain is dense in $\mathbb{R}$? How do you begin to prove/disprove this?</p>
| Douglas Lind | 8,112 | <p>I think you mean dense in $[0,\infty)$, since the spectral radius of a nonnegative integer matrix must be at least 1 (the product of all nonzero eigenvalues must be a nonzero integer). You are effectively asking whether Perron numbers are dense in $[1,\infty)$, and this is easy to see. For example, let $A_n$ be the ... |
3,036,489 | <blockquote>
<p>For <span class="math-container">$k, l \in \mathbb N$</span>
<span class="math-container">$$\sum_{i=0}^k\sum_{j=0}^l\binom{i+j}i=\binom{k+l+2}{k+1}-1$$</span>
How can I prove this?</p>
</blockquote>
<p>I thought some ideas with Pascal's triangle, counting paths on the grid and simple deformation ... | Anubhab | 602,341 | <p><span class="math-container">$$\displaystyle\sum_{i=0}^k\sum_{j=0}^l\binom{i+j}i=\sum_{i=0}^k\sum_{j=i}^{i+l}\binom{j}i=\sum_{i=0}^k\binom{i+l+1}{i+1} \ ^{[1]}$$</span></p>
<p><span class="math-container">$$=\sum_{i=0}^k\binom{i+l+1}{l}=\sum_{i=l}^{k+l+1}\binom{i}{l}−1=\binom{k+l+2}{k+1}-1\ ^{[1]}$$</span></p>
<p>... |
3,036,489 | <blockquote>
<p>For <span class="math-container">$k, l \in \mathbb N$</span>
<span class="math-container">$$\sum_{i=0}^k\sum_{j=0}^l\binom{i+j}i=\binom{k+l+2}{k+1}-1$$</span>
How can I prove this?</p>
</blockquote>
<p>I thought some ideas with Pascal's triangle, counting paths on the grid and simple deformation ... | Felix Marin | 85,343 | <p><span class="math-container">$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\exp... |
1,607,108 | <p>Using the ratio test:
$$\frac{1}{3}\lim\limits_{n\to\infty}\left|\frac{(n+1)^3(\sqrt{2}+(-1)^{n+1})^{n+1}}{n^3(\sqrt{2}+(-1)^n)^n}\right|$$</p>
<p>Without evaluating the limit, the numerator is greater then denominator and the series is divergent. Is there an easier method for checking the convergence of this seque... | πr8 | 302,863 | <p>If $2\vert x^2-1$, then $x$ must be odd, so write $x=2y+1, y\in\mathbb{Z}$. </p>
<p>Then $x^2-1=4y^2+4y=8\frac{y(y+1)}{2}$. </p>
<p>Note that at least one of $y,y+1$ is even, so the fraction on the right is always an integer. </p>
<p>Thus we have that $8\vert x^2-1$.</p>
|
2,127,110 | <p>I am working on this differetiation problem:</p>
<p>$ \frac{d}{dx}x(1-\frac{2}{x})$</p>
<p>and I am currently stuck at this point:</p>
<p>$1\cdot \left(1-\frac{2}{x}\right)+\frac{2}{x^2}x$</p>
<p>Symbolab tells me this simplifies to $1$ but I do not understand how. I am under the impression that;</p>
<p>$1\cdot... | Community | -1 | <p><strong>Do not get confused by fractions and exponents.</strong>
You should remember that $\frac {k}{n} = kn^{-1}$ and not $k^{-n} $ and that $\frac {k}{n} l = kln^{-1}$ and not $kl^{-n} $.</p>
<p>We have $$1\times (1-\frac {2}{x}) = 1-\frac {2}{x} \tag {1}$$ and then $$\frac {2}{x^2}x = \frac {2}{x^2} \times x = \... |
2,617,555 | <p>So the cost per hour of running a cruiser is $\$ \left(\frac {V^2}{40} + 10\right)$, where $V $is the speed in knots. So I’ve answered the first question showing the cost would be $\$\frac DV \left(\frac {V^2}{40} + 10\right)$. And then they asked me to find the most economical speed for running the cruiser, and I h... | Nash J. | 439,920 | <p>The <em>most economical speed</em> will be the one at which the cost ($C$) is minimum. So the problem boils down to
\begin{align}
& \dfrac{dC}{dV} = 0 \\
\implies & \dfrac{d}{dV} \left[ \dfrac{D}{V} \left( \dfrac{V^{2}}{40} + 10\right)\right] = 0 \\
\implies & D \dfrac{d}{dV} \left( \dfrac{V}{40} + \dfr... |
1,937,515 | <p>If a function is analytic on $B(0,1)$ and continuous as well as real valued on the boundary (i.e. $|z| = 1$),then show that the function is constant.</p>
| TZakrevskiy | 77,314 | <p>Hint: apply the maximum principle for harmonic functions to imaginary part of your function.</p>
<p><strong>Edit</strong></p>
<p>As usual, we say that complelx variable $z$ writes $$z=x+iy$$
and the function $f$ has real and imaginary parts $u$ and $v$ respectively:
$$f(x+iy) = u(x,y)+iv(x,y).$$
It is a standard r... |
259,431 | <p>In the book of Richard Hammack, I come accross with the following question:</p>
<blockquote>
<p>There are two different equivalence relations on the set $A = \{a,b\}$.
Describe them.</p>
</blockquote>
<p>OK, I found that the solution is,</p>
<p>$$R_1 = \{(a,a),(b,b),(a,b),(b,a)$$
and
$$R_2 = \{(a,a),(b,b)\}$$<... | joriki | 6,622 | <p>Because they're not reflexive. An equivalence relation is reflexive, i.e. it contains all pairs of the form $(x,x)$.</p>
|
653,887 | <p>Why is the following set linearly independent for all x on ($-\infty$, $\infty$)?</p>
<p>$$\{1+x, 1-x, 1-3x\}$$</p>
<p>The Wronskian is $0$, but Wolfram Alpha says it is still linear independent? Why is this?</p>
<p>Thanks!</p>
<p><a href="http://www.wolframalpha.com/input/?i=Is+%7B1-x%2C+1%2Bx%2C+1-3x%7D+linea... | user127.0.0.1 | 50,800 | <p>You have <strong>three</strong> polynomials of degree $1$ hence they cannot be linear independent.</p>
|
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