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 |
|---|---|---|---|---|
162,360 | <p>First, the definition of a connected set:</p>
<blockquote>
<p><strong>Definition:</strong> A topological space is connected if, and only if, it cannot be divided in two
nonempty, open and disjoint subsets, or, similarly, if the empty set and the whole set are the
only subsets that are open and closed at the ... | Arthur | 15,500 | <p>If not $s = a$, then $a<s$ because $s$ is the infimum of a set of elements all larger than $a$.</p>
<p>The entire interval $(a, s)$ must be in $A$ because of how $s$ was chosen. The number $s$ has been chosen so that it is smaller than any element in $B$ greater than $a$, so there can't be any elements of $B$ in... |
344,994 | <p>Are there any decimals that times to make 1? I know that 0.25 x 4 = 1 but it's a decimal and a number. I need two decimals. Any ideas? I'm not an expert btw so... thanks</p>
| Abhijit | 27,809 | <p>Assuming you mean to find the pair of Non Integer Rational Numbers whose product is equal to 1, then the best way is to express it as a fraction and then convert it to a decimal.
Remember</p>
<p>$$\frac{p}{q}\cdot \frac{q}{p} = 1$$</p>
<p>So plug in any $p$ and $q$ to get the desired decimal fractional number.</p>... |
2,714,738 | <p>If $R$ is a noetherian ring then also $R[x]$ is a noetherian ring, i.e. $R[x]$ is noetherian as $R[x]$-module. Is $R[x]$ also noetherian as $R$-module?</p>
| Pierre-Yves Gaillard | 660 | <p>Let $R$ be a commutative ring and $X$ an indeterminate. Then $R[X]$ is a noetherian $R$-module if and only if $R$ is the zero ring.</p>
<p>The other answers contain a proof (but not a statement) of this fact.</p>
<p>Edit. In the first version, I had written "$R[X]$ is a noetherian $R$-module if and only if $R$ is ... |
3,600,868 | <p>Ellipse can be
<a href="https://math.stackexchange.com/q/3594700/122782">perfectly packed with <span class="math-container">$n$</span> circles</a>
if </p>
<p><span class="math-container">\begin{align}
b&=a\,\sin\frac{\pi}{2\,n}
\quad
\text{or equivalently, }\quad
e=\cos\frac{\pi}{2\,n}
,
\end{align}</span> <... | Blue | 409 | <p>We can consider of a broad family of inellipses that includes <a href="https://en.wikipedia.org/wiki/Steiner_inellipse" rel="nofollow noreferrer">Steiner's</a> and <a href="https://en.wikipedia.org/wiki/Mandart_inellipse" rel="nofollow noreferrer">Mandart's</a>.</p>
<p>Given <span class="math-container">$\triangle ... |
2,653,401 | <p>I have a question about the following partial fraction:</p>
<p>$$\frac{x^4+2x^3+6x^2+20x+6}{x^3+2x^2+x}$$
After long division you get:
$$x+\frac{5x^2+20x+6}{x^3+2x^2+x}$$
So the factored form of the denominator is
$$x(x+1)^2$$
So
$$\frac{5x^2+20x+6}{x(x+1)^2}=\frac{A}{x}+\frac{B}{x+1}+\frac{C}{(x+1)^2}$$</p>
<p>Wh... | BCLC | 140,308 | <p>Explanation 1: If you have only $(x+1)$, your denominators are too weak:</p>
<p>$$\frac{A}{x}+\frac{B}{x+1}+\frac{C}{x+1} = \frac{A(x+1)}{x(x+1)}+\frac{Bx}{x(x+1)}+\frac{Cx}{x(x+1)}$$</p>
<p>Thus, you your denominator is quadratic whereas the desired denominator is $x(x+1)^2$, cubic. I mean, $\frac{C}{x+1}$ is red... |
901,161 | <p>Let's consider the manifold $S^1$</p>
<p>It is well known that we need two charts to cover this manifold.</p>
<p>Nonetheless, we can cover the full space using a single coordinate $\theta$ which is just the angle from the center.</p>
<p>Now, is this a general feature? I mean, is it always possible to have in ever... | Ted Shifrin | 71,348 | <p>No, your "single coordinate" is only a smooth (continuous) function on $S^1-(1,0)$.</p>
|
901,161 | <p>Let's consider the manifold $S^1$</p>
<p>It is well known that we need two charts to cover this manifold.</p>
<p>Nonetheless, we can cover the full space using a single coordinate $\theta$ which is just the angle from the center.</p>
<p>Now, is this a general feature? I mean, is it always possible to have in ever... | Andrew D. Hwang | 86,418 | <p>Perhaps you're "really" asking whether or not every $n$-manifold $M$ has $\mathbf{R}^{n}$ as a covering space, i.e., whether there's a single (smooth) map $\pi:\mathbf{R}^{n} \to M$ whose restrictions to suitably small subsets define coordinate charts on $M$.</p>
<p>If so, the answer is "no"; even among spheres, on... |
1,305,661 | <p>How can I solve this Question ? </p>
<ul>
<li>Find all solutions exactly for $2\sin^2 x - \sin x = 0$</li>
</ul>
<p>my answer was $( 2k\pi,\pi+2k\pi, 11\pi/6, 7\pi/6)$
but my teacher answer was $( 2k\pi,\pi+2k\pi, \pi/6 +2k\pi, 5\pi/6 +2k\pi )$</p>
| Dr. Sonnhard Graubner | 175,066 | <p>HINT: solve the equation $$\sin(x)(2\sin(x)-1)=0$$</p>
|
1,305,661 | <p>How can I solve this Question ? </p>
<ul>
<li>Find all solutions exactly for $2\sin^2 x - \sin x = 0$</li>
</ul>
<p>my answer was $( 2k\pi,\pi+2k\pi, 11\pi/6, 7\pi/6)$
but my teacher answer was $( 2k\pi,\pi+2k\pi, \pi/6 +2k\pi, 5\pi/6 +2k\pi )$</p>
| Ivo Terek | 118,056 | <p><strong>Hint:</strong> $$2\sin^2x-\sin x = 0 \implies \color{red}{2}\sin^2x + \color{blue}{(-1)}\sin x + \color{green}{0} = 0,$$ so we have $$\sin x = \frac{-\color{blue}{(-1)}\pm \sqrt{\color{blue}{(-1)}^2 - 4\cdot \color{red}{2}\cdot\color{green}{0}}}{2\cdot \color{red}{2}} = \frac{1 \pm 1}{4} = \begin{cases}\frac... |
1,305,661 | <p>How can I solve this Question ? </p>
<ul>
<li>Find all solutions exactly for $2\sin^2 x - \sin x = 0$</li>
</ul>
<p>my answer was $( 2k\pi,\pi+2k\pi, 11\pi/6, 7\pi/6)$
but my teacher answer was $( 2k\pi,\pi+2k\pi, \pi/6 +2k\pi, 5\pi/6 +2k\pi )$</p>
| graydad | 166,967 | <p>We have solutions to $2y^2-y=0$ as $$y = \frac{1 \pm \sqrt{1-0}}{4} \implies y = 0,\frac{1}{2}$$ With each pass around the unit circle we know $y = \sin(x)$ takes the value $0$ twice, and the value $\frac{1}{2}$ twice. Can you see why your teacher is correct now?</p>
|
221,130 | <p>Let $P$ be a simple convex polytope in $\mathbb{R}^d$ (that is, any vertex belongs to exactly $d+1$ facets). Given the collection of outer normals to facets of $P$, combinatorics of $P$ may be different, of course. But is this information enough to reconstruct number of faces of $P$ of all dimensions? If yes, what i... | Joseph O'Rourke | 6,094 | <p>Not an answer; just an illustration:
<hr />
<a href="https://i.stack.imgur.com/ya7nk.jpg" rel="noreferrer"><img src="https://i.stack.imgur.com/ya7nk.jpg" alt="fvector"></a>
<br />
<sup>
$f$-vector left & right: $(V,E,F)=(6,9,5)$.
Middle pol... |
376,772 | <blockquote>
<p>Find a polynomial $q(a)$ of degree less than equal to $2$ that saitsifies the condition $q(a_0)=b_0, q'(a_0)=b'_0, \ \text{and} \ q'(a_1)=b'_1,$ where $a_0,a_1,b_0,b'_0,b'_1\in \mathbb{R}$, where $a_0\ne a_1$. And give a formula of the form $q(a)=b_0k_0(a)+b'_0k_1(a)+b'_1k_2(a).$</p>
</blockquote>
<p... | Ross Millikan | 1,827 | <p>A polynomial $q(a)$ of degree less than $2$ is linear, so you can write $q(a)=ca+d$. You will fail in this unless $b_0'=b_1'$. Probably you are asked for a polynomial of degree less than or equal to $2$ and should write $q(a)=ca^2+da+e$ Now plug in the restrictions you have on $q$ and solve for $c,d,e$</p>
|
3,900,679 | <p>So there's this one thing I can't wrap my head around.</p>
<p>The problem is:</p>
<p>Use the change of variables <span class="math-container">$x = \sinh u$</span> to compute the indefinite integral
<span class="math-container">\begin{equation*}
\int \frac{dx}{\sqrt{1+x^2}}
\end{equation*}</span></p>
<p>Solution:... | Will Jagy | 10,400 | <p>CW</p>
<p>positivity is immediate when <span class="math-container">$x>1$</span></p>
<p>Relation to zeta</p>
<p><a href="https://i.stack.imgur.com/lRS2m.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/lRS2m.png" alt="enter image description here" /></a></p>
<p>alright, there is a functional equ... |
3,057,200 | <blockquote>
<p>Does there exist an analytic function <span class="math-container">$f:D\rightarrow\mathbb C$</span>, where <span class="math-container">$D$</span> is a domain containing the unit disk, such that <span class="math-container">$f(z) = e^{i \text{Im} z}$</span> on the unit circle <span class="math-contain... | Greg Martin | 16,078 | <p>Sequence of hints:</p>
<ul>
<li>If <span class="math-container">$f$</span> and <span class="math-container">$g$</span> are two such functions, show they must be equal.</li>
<li>If <span class="math-container">$f(z)$</span> is such a function, show that <span class="math-container">$g(z) = \overline{f(\bar z)}$</spa... |
3,494,820 | <p>Yesterday I learned about dual spaces when reading about spaces of linear maps. The concept of a linear map and why linear maps form a vector space is clear to me. But there are some details about the dual space and its basis that I could not fully understand.</p>
<p>The text I am reading states the following:</p>
... | Claude Leibovici | 82,404 | <p>A possible approach would be to prove that
<span class="math-container">$$f(x)=e^{\frac{x^2}{2}} (x+1)-e^x$$</span> is positive if <span class="math-container">$x >0$</span>.</p>
<p>Using Taylor expansion built at <span class="math-container">$x=0$</span>, you would have
<span class="math-container">$$f(x)=\frac... |
2,917,024 | <p>Let $H$ be a Hilbert space and let ${e_n} ,\ n=1,2,3,\ldots$ be an orthonormal basis of $H$. Suppose $T$ is a bounded linear oprator on $H$. Then which of the following can not be true?
$$(a)\quad T(e_n)=e_1, n=1,2,3,\ldots$$
$$(b)\quad T(e_n)=e_{n+1}, n=1,2,3,\ldots$$
$$(c)\quad T(e_n)=e_{n-1} , n=2,3,4,\ldots ... | Angina Seng | 436,618 | <p>Let's look at (a).</p>
<p>If $T$ satisfies $T(e_n)=e_1$ for all $n$, then
$$T(e_1+e_2+\cdots+e_n)=ne_1.$$
But $\|e_1+e_2+\cdots+e_n\|=\sqrt n$ and $\|ne_1\|=n$.</p>
|
338,155 | <p>I've problem with this surface integral:
$$
\iint\limits_S {\sqrt{ \left(\frac{x^2}{a^4}+\frac{y^2}{b^4}+\frac{z^2}{c^4}\right)}}{dS}
$$, where
$$
S = \{(x,y,z)\in\mathbb{R}^3: \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2}= 1\}
$$</p>
| Ron Gordon | 53,268 | <p>What an interesting integral. I had to resort to referring to the <a href="http://mathworld.wolfram.com/Ellipsoid.html">first differential form</a> of the spherical parametrization, but doing that, I am amazed at how this turns out.</p>
<p>We parametrize in the usual way:</p>
<p>$$x=a\sin{u} \cos{v}$$
$$y=b\sin{u... |
81,349 | <p>My current research took me to the realm of PDE's (which for the long time used to be terra incognita for me as I'am a probabilist). Equations that I'am working with are mostly of second order or Hamilton-Jacobi equations. It's no suprise that I'am dealing with various notions of weak solutions (viscosity solutions ... | Jeff | 18,406 | <p>You might also be interested in the general theory of viscosity solutions to second order PDE <a href="http://www.arxiv.org/pdf/math/9207212" rel="nofollow">http://www.arxiv.org/pdf/math/9207212</a></p>
<p>I would have posted as a comment but do not have the privilege yet. </p>
|
81,349 | <p>My current research took me to the realm of PDE's (which for the long time used to be terra incognita for me as I'am a probabilist). Equations that I'am working with are mostly of second order or Hamilton-Jacobi equations. It's no suprise that I'am dealing with various notions of weak solutions (viscosity solutions ... | Pawel | 19,379 | <p>Just to make everything clear: the answer for my question is "yes, it is possible".</p>
<p>Appropriate example is in paper under link provided by pgassiat in comments.</p>
|
3,873,438 | <blockquote>
<p>A particle of unit mass moves under the action of <span class="math-container">$n$</span> forces directed towards <span class="math-container">$n$</span> fixed points <span class="math-container">$A_1,A_2,...,A_n$</span>. The force towards <span class="math-container">$A_i$</span> is of magnitude <span ... | red whisker | 833,796 | <p>Treat all points as vectors in three space. At <span class="math-container">$B$</span>, the total force is <span class="math-container">$\sum_{i=1}^nk_i(A_i-B) = 0$</span> and at <span class="math-container">$C$</span>, the total force is <span class="math-container">$\sum_{i=1}^nk_i(A_i-C) = \vec{f}$</span> where <... |
2,225,932 | <p>If given a convex function $f: \mathbb{R} \to \mathbb{R}$, then the conjugate function $f^*$ is defined as $$f^*(s) = \sup_{t \in \mathbb{R}} (st-f(t))$$</p>
<p>Now i want to understand what is the physical interpretation of this conjugate function? What is its exposition? Please help me. </p>
| Amin | 264,002 | <p>Suppose <span class="math-container">$f: \mathbb{R}^n \to \mathbb{R}$</span>, then the conjugate function is:
<span class="math-container">$$f^*(\mathbf{y}) \triangleq \sup_{x\in \text{dom} f} [\mathbf{y}^ \mathsf{T}\mathbf{x}-f(\mathbf{x})],$$</span>
where: <span class="math-container">$\text{dom}f^* = \{\textbf{y}... |
106,838 | <p>Suppose $n > 1$ is a natural number. Suppose $K$ and $L$ are fields such that the general linear groups of degree $n$ over them are isomorphic, i.e., $GL(n,K) \cong GL(n,L)$ as groups. Is it necessarily true that $K \cong L$?</p>
<p>I'm also interested in the corresponding question for the special linear group i... | Jim Humphreys | 4,231 | <p>The answer to the question is yes, though I don't have all the old literature at my fingertips. This kind of question for various classes of linear groups has a long history in the study of homomorphisms and isomorphisms of classical groups and then other algebraic groups (van der Waerden, Dieudonne, ...) The most... |
109,292 | <p>Let $\kappa$ be a regular, uncountable cardinal. Let $A$ be an unbounded set, i.e. $\operatorname{sup}A=\kappa$. Let $C$ denote the set of limit points $< \kappa$ of $A$, i.e. the non-zero limit ordinals $\alpha < \kappa$ such that $\operatorname{sup}(X \cap \alpha) = \alpha$. How can I show that $C$ is unboun... | azarel | 20,998 | <p>Fix $\xi\in \kappa$, since $A$ is unbounded there is a $\alpha_0\in A$ so that $\xi<\alpha_0$. Now, construct recursively a strictly increasing sequence $\langle \alpha_n: n\in \omega\rangle$. Let $\alpha=\sup\{\alpha_n: n\in \omega\}.$ Since $\kappa$ is regular and uncountable, we have $\alpha<\kappa.$ It is ... |
3,007,044 | <p>Sequence <span class="math-container">$a_n$</span> is defined as <span class="math-container">$a_n = \ln (1 + a_{n-1})$</span>, where <span class="math-container">$a_0 > -1$</span>. Find all <span class="math-container">$a_0$</span> for which the sequence converges.</p>
<p>Using arithmetic properties of limits, ... | user | 505,767 | <p><strong>HINT</strong></p>
<p>We can show that as <span class="math-container">$a_0 >0$</span></p>
<ul>
<li><span class="math-container">$a_n$</span> is decreasing</li>
<li><span class="math-container">$a_n$</span> is bounded below</li>
</ul>
|
4,189,417 | <p>The book I am currently reading, is written in a local language, uses the symbol <span class="math-container">$$\gtreqqless$$</span></p>
<p>Noted usage in the book (translated):</p>
<blockquote>
<p>Let <span class="math-container">$f(x) = x^3e^{-3x}$</span> for x > 0. Then what is the maximum value of <span class... | Troposphere | 907,303 | <p>It's a shorthand notation for
<span class="math-container">$$
1-x > 0 \iff 1 > x
\quad\text{and}\quad 1-x = 0 \iff 1 = x
\quad\text{and}\quad 1-x < 0 \iff 1 < x \, .
$$</span></p>
|
4,200,434 | <p>I am having difficulty with what should be a routine question, Exercise 2.2.2 (b) of <em>Understanding Analysis</em> by Stephen Abbott (2015).</p>
<blockquote>
<p><strong>Exercise 2.2.2.</strong> Verify, using the definition of convergence of a sequence, that the following sequences converge to the proposed limit.</... | Community | -1 | <p>Let <span class="math-container">$\frac{1}{2}\varepsilon>\frac{1}{n}>0$</span> by the Archimedean property (I believe this is defined in Abbott's book).</p>
<p>Now, observe the following for <span class="math-container">$n\in \mathbb{N}$</span>:</p>
<p><span class="math-container">$\Big|\frac{2n^2}{n^3+3}-0\Bi... |
1,063,334 | <p>I'm really stumped on this problem and don't know how to go about it. It says $g(x)$ = $|f(x)|$ and to show that if $f(c) = 0$ and g is differentiable at c, then one must have $D(f)(c) = 0$. Everywhere I look on the internet it says that |x| is not differentiable at 0 even there is nothing on absolute value of funct... | Community | -1 | <p>Generally it is not true. If $\chi_{\mathbb{Q}} $ is a charakteristic function of rationals then the function $g(t) =\left|\chi_{\mathbb{Q}} (t) -\frac{1}{2} \right|t^2 $ is differentiable everywhwre but the function $f(t) =\left(\chi_{\mathbb{Q}} (t) -\frac{1}{2} \right)t^2$ is equal zero only at zero.</p>
|
1,762,078 | <p>show that
$$e^x-\ln{(x+2)}>\dfrac{1}{6}\tag{(1)}$$</p>
<p>I know
$$e^x>x+1,\ln{(x+2)}<x+1$$
so I have only prove
$$e^x-\ln{(x+2)}>0$$
But How to prove $(1)$?</p>
| Christian Blatter | 1,303 | <p>A look at the graphs of $y=e^x$ and $y=\log(x+2)$ shows that the line $y=1+x$ is a tangent to both of them, and that it suffices to prove the inequality in question for the interval $-1<x<0$.</p>
<p>We start with the inequalities
$$e^{-t}>1-t+{t^2\over2}-{t^3\over 6},\qquad\log(1+t)<t-{t^2\over2}+{t^3... |
1,394,753 | <p>I came across this puzzle recently which I hope people might enjoy.</p>
<p>Let $S(n)$ be the set of positive integers less than $n$ which do not have a $2$ in their decimal representation and let $\sigma(n)$ be the sum of the reciprocals of the numbers in $S(n)$, so for example $\sigma(5) = 1 + 1/3
+ 1/4$ . </p>
... | Brian Tung | 224,454 | <p><strong>Hint for the second part.</strong> Consider the number of elements of $S(10^n)$ larger than $10^{n-1}$, and observe that they all contribute less than $1/10^{n-1}$. Obtain the appropriate geometric series with ratio $9/10$.</p>
|
1,632,990 | <p>I'm having a bit of confusion here.</p>
<p>What are the solutions of</p>
<p>$\begin{pmatrix} 0&1&0 \\ 0&0&1 \\ 0&0&3\\ \end{pmatrix}x=0$</p>
<p>Clearly,</p>
<p>$x_2=0$<br>
$x_3=0$<br>
$3x_3=0$</p>
<p>$x_1=s\in\mathbb{R}$, because there are no constraints for $x_1$.<br></p>
<p>then e.g.<... | Arnaud D. | 245,577 | <blockquote>
<p>But since this is from a matrix having double eigenvalue on this matrix, then there should be two solutions.</p>
</blockquote>
<p>If you mean that the characteristic polynomial has a double root, then this does not mean that you can find two linearly independant eigenvectors.</p>
<p>For a matrix $A\... |
1,632,990 | <p>I'm having a bit of confusion here.</p>
<p>What are the solutions of</p>
<p>$\begin{pmatrix} 0&1&0 \\ 0&0&1 \\ 0&0&3\\ \end{pmatrix}x=0$</p>
<p>Clearly,</p>
<p>$x_2=0$<br>
$x_3=0$<br>
$3x_3=0$</p>
<p>$x_1=s\in\mathbb{R}$, because there are no constraints for $x_1$.<br></p>
<p>then e.g.<... | Bernard | 202,857 | <p>There is no other (independent) solution. $0$ is an eigenvalue of the matrix with <em>algebraic</em> multiplicity $2$, but the <em>geometric</em> multiplicity of $0$ (the dimension of the kernel) is $1$..</p>
<p>Generally speaking, if $\lambda$ is an eigenvalue of a matrix $A$, we must distinguish between its alge... |
104,186 | <p>I'm having difficulty understanding how to express text in <code>Epilog</code> that is dynamically updated using <code>Log[b, x]</code>. <em>Mathematica</em> changes this to base $e$, but I would like it to be <code>Log[b, x]</code> in traditional format with base $b$, and I can't seem to make it work. I'm guessin... | J. M.'s persistent exhaustion | 50 | <p>The code in the OP will work right off the bat if the following (sadly undocumented) setting is first executed:</p>
<pre><code>SetSystemOptions["SimplificationOptions" ->
{"AutosimplifyTwoArgumentLog" -> False}]
</code></pre>
<p>After running that line, the <code>Manipulate[]</code> works as... |
349,147 | <p>What is known regarding which hyperbolic groups are cubulated?</p>
<p>I take it the usual definition of cubulated is acting properly and cocompactly on a CAT(0)-cube complex.</p>
<p>My impression is that not all of them are, but I didn't manage to find references with a counterexample.</p>
<p>Are there known ways... | HJRW | 1,463 | <p>As @AGenevois says in his answer, the standard examples of non-cubulated hyperbolic groups are those with Kazhdan's property (T), such as quaternionic hyperbolic lattices.</p>
<p>Complex hyperbolic lattices (in dimension >2) provide a more delicate class of examples. On the one hand, they are not cubulable, by a t... |
1,818,260 | <p>I'm studying mathematical physics and working on numerical solutions of partial differential equations. I am having trouble understanding the way we solve partial dif. equations, e.g.,</p>
<p>$\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2}+\frac{2}{x}.\frac{\partial u}{\partial x}$</p>
<p>$\frac... | user1820964 | 315,800 | <p>The advection term, $\frac{\partial u}{\partial x}$ is always tricky part to solve PDE in a numerical way. It has a certain direction as you said. </p>
<p>There are many techniques to treat this term. Instead of explaining all things, I give you a link that has good notation and explanation. </p>
<p><a href="http:... |
3,259,073 | <p>Let <span class="math-container">$A$</span> be a commutative ring with <span class="math-container">$1$</span>, and let <span class="math-container">$a, b, c \in A$</span>. Suppose there exist <span class="math-container">$x, y, z ∈ A$</span> such that <span class="math-container">$ax+by +cz = 1$</span>. Then there ... | Gabriel Soranzo | 51,400 | <p>I think I have it (for the first)!</p>
<p>For the first: <span class="math-container">$\text{Tor}_i(M,N)$</span> is annihilated by <span class="math-container">$\text{Ann}(M)$</span> and <span class="math-container">$\text{Ann}(N)$</span> because element of <span class="math-container">$\text{Tor}$</span> are nothi... |
4,132,124 | <p>I am not able to find this answer anywhere its about discrete mathematics.
Plz Help </p>
| CoveredInChocolate | 901,866 | <p>Did you try using truth tables? Do you agree with the tables I wrote down?</p>
<p><span class="math-container">$$
\begin{array}{cc|c}
p & q & p\wedge q \\
\hline
T & T & T \\
T & F & F \\
F & T & F \\
F & F & F
\end{array}
\quad\quad\quad
\begin{array}{cc|c}
p & q & p... |
4,132,124 | <p>I am not able to find this answer anywhere its about discrete mathematics.
Plz Help </p>
| Floridus Floridi | 841,808 | <p>A fallacy has to be a <em><strong>reasoning</strong></em> , that is a sequence of sentences with an inference word such as "therefore" before the last sentence.</p>
<p>What you wrote is a <em><strong>sentence</strong></em> , an assertion, not a reasoning.</p>
<p>What goes wrong in this sentence is that it ... |
2,290,295 | <p>I do not know if the following is true: </p>
<p>Question: If $\Omega\subseteq\mathbb{R}^n$ is open and connected, then there exists a covering by balls $\{B_j\}_{j=1}^{\infty}$ with the property $B_j\subseteq\Omega$ and $B_j\cap B_{j+1}\neq\emptyset$, for all $j\geq1$.</p>
<p>Motivation: I have a function $u$ that... | Henno Brandsma | 4,280 | <p>There is in fact a useful characterisation of connectedness which is what you want:</p>
<p>A space $X$ is connected iff for every open cover $\mathcal{U}$ of $X$, and for all $x,y\in X$ we have a chain from $\mathcal{O}$ from $x$ to $y$.</p>
<p>I wrote a proof <a href="https://math.stackexchange.com/a/44938">in th... |
554,869 | <p>I want to show that the following conditions are equivalent for a nonzero element $a$ in a Boolean algebra $\mathcal{B}$:</p>
<p>1) for all $x\in\mathcal{B},a\leq x$ or $a\leq x'$</p>
<p>2) for all $x,y\in\mathcal{B},a\leq x\sqcup y\Rightarrow a\leq x$ or $a\leq y$</p>
<p>3) $a$ is minimal among nonzero elements ... | Cameron Buie | 28,900 | <p>For (1)$\implies$(2), suppose that $a\not\le x$ and $a\not\le y,$ and use (1) to show that we must have $a\not\le x\sqcup y,$ thus proving the contrapositive of (2), so proving (2).</p>
<p>For (2)$\implies$(1), take any $x\in\mathcal B$ and observe that $x\sqcup x'=1,$ so....</p>
<p>For (1)$\implies$(3), suppose $... |
1,990,033 | <p>Suppose I have a point $P(x_1, y_1$) and a line $ax + by + c = 0$. I draw a perpendicular from the point $P$ to the line. The perpendicular meets the line at point $Q(x_2, y_2)$. I want to find the coordinates of the point $Q$, i.e., $x_2$ and $y_2$.</p>
<p>I searched up for similar questions where the coordinates ... | Steven Alexis Gregory | 75,410 | <p>\begin{align}
\sin 3x
&= \sin(2x + x) \\
&= \sin 2x \; \cos x + \cos 2x \; \sin x \\
&= 2 \sin x \; \cos^2 x + \cos 2x \; \sin x \\
&= \sin x \; (2 \cos^2 x + \cos 2x) \\
&= \sin x \; (2 \cos 2x + 1)
\end{align}</p>
<p>\begin{align}
4 \sin 2x \; \sin 4x
&= 2(\cos 4x ... |
314,246 | <p>In the case that $L:B_1 \rightarrow B_2 $ is a linear mapping of Banach spaces and $L$ is a isometric isomorphism (bijection and $\|Lx\|_{B_1} = \|x\|_{B_2} $) can I say that $L\overline{L}= 1 $ is trivial ? (the bar denotes the complex conjugate); </p>
<p>TIA</p>
| Lance | 405,830 | <p>I do not think this is true.</p>
<p>As an example take $B_1 = B_2 = \mathbb{C}^3$ equipped with euclidean norm. Choose a basis $\{e_1, e_2, e_3 \}$ of $\mathbb{C}^3$ and define $L = \begin{bmatrix}
0 & 0 & 1 \\
1 & 0 & 0 \\
0 & 1 & 0 \\
\end{bmatrix}$.</p>
<p>Note that $L$ permutes the ... |
84,735 | <p>Unlike using</p>
<pre><code>Unprotect[In,Out];
Clear[In,Out];
Protect[In,Out];
</code></pre>
<p>to clean the occupied memory of the whole notebook, I want to clean the memory dynamically and accurately. That is, I want to clean out the specific memory I assign whenever I need.</p>
<p>For example, I wrote a loop a... | Bichoy | 6,770 | <p>You can use <code>Block</code> for this purpose.
From Mathematica <a href="https://reference.wolfram.com/language/ref/Block.html" rel="noreferrer">documentation</a>:</p>
<blockquote>
<p>When you execute a block, values assigned to x, y, ... are
cleared. When the execution of the block is finished, the original
... |
3,342,761 | <p>I'm trying to get an equation for a solid angle of a segment of octahedron in the same vein as described in this article <a href="http://www.rorydriscoll.com/2012/01/15/cubemap-texel-solid-angle/" rel="nofollow noreferrer">cubemap-texel-solid-angle</a>. I ended up having to integrate
<span class="math-container">$$\... | David K | 139,123 | <p>The general problem I think you have described is to find the solid angle subtended at one vertex of a tetrahedron.</p>
<p>If we label the vertex in question <span class="math-container">$O$</span> and put it at the center of a unit sphere, then project the opposite face onto the sphere, we get a spherical triangle... |
1,349,002 | <blockquote>
<p>Let <span class="math-container">$g(z)= z^4+iz^3 +1$</span>. How many zeros does <span class="math-container">$g$</span> have in <span class="math-container">$\{z\in \Bbb{C}: \text{Re }(z), \text{Im }(z)>0\}$</span>?</p>
</blockquote>
<p>I tried comparing the number of zeros of <span class="math-c... | J.E.M.S | 73,275 | <p><strong>Hint:</strong> surely zeroes are inside some big quartercircle with radius R, than use argument principle.</p>
|
1,010,820 | <p>I tried the following :</p>
<p>\begin{align}\sec\theta + \tan\theta&=4\\
\frac1{\cos\theta} + \frac{\sin\theta}{\cos\theta}&=4\\
\frac{1+\sin\theta}{\cos\theta}&=4\\
\frac{1+\sin\theta}4&=\cos\theta\end{align}</p>
<p>now don't know how to evaluate further ?</p>
| Felix Marin | 85,343 | <p>$\newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle}
\newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack}
\newcommand{\ceil}[1]{\,\left\lceil\, #1 \,\right\rceil\,}
\newcommand{\dd}{{\rm d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcomma... |
1,557,359 | <p>I've been doing Project-Euler just as a way to increase my competency in computer science. I'm currently a Pure and Applied Math major who recently adopted computer science as a minor in order to apply to grad schools in comp. sci.</p>
<p>Some built-in Sage functions that are extremely fast and I do not have compa... | learner | 228,313 | <p>From the perspective of a programmer/coder, it isn't a bad practice to use inbuilt functions when you can (in fact, people are lazy, if there's an inbuilt function, why do the work of manually defining another?)</p>
<p>Now, from the perspective of a mathematician, the aim of the problems there is to motivate you to... |
4,586,330 | <p>How do I prove the following equality?
<span class="math-container">$$\sqrt[5]{\frac{5\sqrt5+11}{2}}+\sqrt[5]{\frac{5\sqrt5-11}{2}}=\sqrt{5} $$</span></p>
<p>My approach was to notice that the first term equals the golden ratio and the second term equals the reciprocal of the golden ratio, and adding them up would g... | Wang YeFei | 955,668 | <p>You can raise both sides to the <span class="math-container">$5^{th}$</span> power and cancel out like terms of both sides until you see that both sides look the same. That's called brute force. We can save some times off by noting some relations between the terms of the sum on the left. Specifically, put <span clas... |
299,576 | <p>Suppose $X = \displaystyle\bigsqcup_{i \in I} X_i$ is the disjoint union of infinitely many continua. The components of the Stone-Cech remainder $X^*$ can be described as follows. Treat $I$ as a discrete topological space and consider the continuous map $F:X_i \to I$ that sends each $X_i$ to $i \in I$. The Stone Cec... | D. S. Lipham | 95,718 | <p>The conjecture is true, but I'll leave it to others to address the other parts of your question (or maybe I'll edit this post later). </p>
<p>Let $Y$ be Tychonoff and $p\in \beta Y$.</p>
<p>Using the facts that (1) $A=\overline{A\cap Y}$ for every clopen $A\subseteq \beta Y$, and (2) every clopen subset of $Y$ ... |
1,508,753 | <p>Show that for $x,y\in\mathbb{R}$ with $x,y\geq 0$, the arithmetic mean-quadratic mean inequality
$$\frac{x+y}{2}\leq \sqrt{\frac{x^2+y^2}{2}}$$
holds.</p>
<p>After my calculations I'll get: </p>
<p>$$-x^2+2xy-y^2$$
which can't be $\leq 0$.</p>
| Pacciu | 8,553 | <p>@Garsa : The $\varepsilon-\delta$ argument cannot be used to evaluate a limit; instead, it can only be used to prove that a conjectured value $l$ is the limit of a function at a given point.</p>
<p>In order to guess a possible value for the limit $\displaystyle \lim_{x\to 2} \frac{1}{1-x}$, one possible way is to p... |
1,508,753 | <p>Show that for $x,y\in\mathbb{R}$ with $x,y\geq 0$, the arithmetic mean-quadratic mean inequality
$$\frac{x+y}{2}\leq \sqrt{\frac{x^2+y^2}{2}}$$
holds.</p>
<p>After my calculations I'll get: </p>
<p>$$-x^2+2xy-y^2$$
which can't be $\leq 0$.</p>
| Kashi | 76,422 | <p>Let $f(x)=1$ for every $x\in \mathbb {R}$ and $g(x)=1-x$ for every $x\in \mathbb {R}$. Notice that $\lim_{x\rightarrow 2}f(x)=1$; to see the $\epsilon-\delta$ argument for this notice that for this function $f(x)$, choose $\delta =\epsilon$ so that we have, $|f(x)-1|=0\leq |x-2|\leq \delta=\epsilon.$</p>
<p>Further... |
48,679 | <p>I've been going through Fermats proof that a rational square is never congruent. And I've stumbled upon something I can't see why is. Fermat says: ''If a square is made up of a square and the double of another square, its side is also made up of a square and the double of another square'' Im having difficulties unde... | Laie | 6,013 | <p>Feynman's path integral in quantum field theory. It involves integration over spaces of fields, using measures that have not been made rigorous.</p>
|
48,679 | <p>I've been going through Fermats proof that a rational square is never congruent. And I've stumbled upon something I can't see why is. Fermat says: ''If a square is made up of a square and the double of another square, its side is also made up of a square and the double of another square'' Im having difficulties unde... | Eric Zaslow | 1,186 | <p>Finally, a Math Overflow question that addresses my specialty: non-rigor!</p>
<p>Here are a few examples of non-rigor as applied to evidence for dualities:</p>
<ol>
<li><p>Heterotic-Type II. In earlier times, the best evidence for heterotic-Type-II duality was a) counting the number of supersymmetries of the the... |
2,886,502 | <p>In the book of <em>Function of One Complex Variable</em> by Conway, at page 31, it is given that</p>
<p><a href="https://i.stack.imgur.com/Ff0nB.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Ff0nB.png" alt="enter image description here"></a></p>
<p>However, normally, if $z$ was a real number, ... | what a disgrace | 569,485 | <p>Just write your complex numbers in polar form — are you familiar with the polar representation of complex numbers?</p>
<p>Write $z=re^{i\theta}=r(\cos\theta+i\sin\theta)$ with $r=|z|$. Then $z^{n}=r^{n}e^{in\theta}$; $|e^{in\theta}| = 1$ for all real $n,\theta$ thus</p>
<p>$$|z^{n}|=|r^{n}|\to0$$</p>
|
3,014,144 | <p>Say we have a closed interval <span class="math-container">$[a,b]$</span>. How can I prove <span class="math-container">$|[a,b]|$</span> equal to <span class="math-container">$c$</span>, where <span class="math-container">$c$</span> is the continuum.</p>
<p>I already proved <span class="math-container">$|(a,b)| = |... | Marco | 582,590 | <p>Let <span class="math-container">$g(x)=f(1/x)$</span> for <span class="math-container">$x\in (1,\infty)$</span>. The hypotheses on <span class="math-container">$f(x)$</span> imply that
<span class="math-container">$$\lim_{x \rightarrow \infty} g(x)=0~\mbox{and}~|(x^2g'(x))'|<C,$$</span>
since <span class="math-c... |
1,227,610 | <p>I need to calculate derivative of the following function with respect to the matrix X:</p>
<p>$f(X)=||diag(X^TX)||_2^2$</p>
<p>where $diag()$ returns diagonal elements of a matrix into a vector. How can I calculate $\frac {\partial f(X)} {\partial X}$? Please help me.</p>
<p>Thanks in advance!</p>
| Community | -1 | <p>Here you get a scalar function of the matrix $X$. The derivative of such a scalar function is defined as the matrix of partial derivatives wrt each entry of the matrix, as <a href="http://en.wikipedia.org/wiki/Matrix_calculus#Scalar-by-matrix" rel="nofollow">here</a>.</p>
<p>You can expand out the expression and ca... |
4,024,071 | <p>'A or B' means one or the other or both.
<br/><br/>So I think <span class="math-container">$\{x | x = u$</span> or <span class="math-container">$ x = v\}$</span> can be equal to <span class="math-container">$\{u\}$</span> or <span class="math-container">$\{v\}$</span> or <span class="math-container">$\{u, v\}$</span... | Arturo Magidin | 742 | <p>The price per pants must be a <strong>divisor</strong> of each of the amounts of money made in each of the months, as presumably he sold an integral number of pants. Thus, the price per pant must divide the greatest common common divisor of 6804 (money made in September), of 10800 (money made in October), and of 793... |
1,197,056 | <p>I tried to evaluate the following limits but I just couldn't succeed, basically I can't use L'Hopital to solve this... </p>
<p>for the second limit I tried to transform it into $e^{\frac{2n\sqrt{n+3}ln(\frac{3n-1}{2n+3})}{(n+4)\sqrt{n+1}}}$ but still with no success...</p>
<p>$$\lim_{n \to \infty } \frac{2n^2-3}{... | Mark Fischler | 150,362 | <p>For the first limit, it breaks into 2 factors with finite limits.<br>
$$\lim{n \to \infty} \frac{2n^2-3}{7-n^2} = \frac{2n^2}{-n^2} =-2\\
\lim{n \to \infty} \frac{3^n-2^{n-1}}{3^{n+2}+2^n2} = \frac{3^n}{3^{n+2}} = \frac{1}{9}
$$
so the answer is $-\frac{2}{9}$.</p>
<p>For the second, rewrite it as
$$
\left(\frac{(3... |
1,100,266 | <p>I begin with this:</p>
<blockquote>
<p>If $f:[a,b]\rightarrow [a,b]$ is continuous and $|f(x)-f(y)|<|x-y|$ for all $x\neq y$, then the equation $f(x)=x$ has exactly one root.</p>
</blockquote>
<p>The problem is not very hard and I solved it. Then I wonder if this problem is true or not:</p>
<blockquote>
<p... | Harald Hanche-Olsen | 23,290 | <p>You can construct a counterexample by setting $f(x)=x-g(x)$ where $g$ is any positive, strictly increasing function that doesn't increase too fast. Something like $g(x)=\frac\pi2+\arctan x$ should work well.</p>
<p>If you have $|f(x)-f(y)|\le L|x-y|$ for a constant $L<1$, there will alays be a fixed point of $f$... |
625,746 | <p>If the nth term of a series is given by $T(n)=T(n-1) + T(n-1) \times C$ where $C$ is a given constant and $T(1)=A$ and $n \ge 2$ . I need to tell whether its value will be greater than $G$ or not before its mth term.</p>
<p>EXAMPLE : Say A that is first term is $2$ and say $C$ is also $2$ . If we need to check wea... | mathlove | 78,967 | <p>Since
$$T(n)=(1+C)T(n-1)$$
we have
$$T(n)=(1+C)^{n-1}T(1)=(1+C)^{n-1}A.$$</p>
<p>Hence, you can use the followings (we suppose that $A\gt0$) :</p>
<p>$$T(n)\ge G\iff (1+C)^{n-1}A\ge G\iff(1+C)^{n-1}\ge\frac{G}{A}\iff n\ge\log_{1+C}\frac GA+1,$$
$$T(n)\lt G\iff n\lt \log_{1+C}\frac{G}{A}+1.$$</p>
<p>Here, we suppo... |
2,309,527 | <p>I was sincerely hoping someone could explain to me how for the function found below I would determine its standard matrix and whether or not the function is 1-to-1 and onto. </p>
<blockquote>
<p>The linear function $T:\mathbb{R}^2 \to \mathbb{R}^3$ is given by $$ T(x,y) = \begin{pmatrix} x-y \\ 5x+3y \\ 2x+4y \en... | copper.hat | 27,978 | <p>Note that $T_n(x) \in \operatorname{sp} \{ e_1,...,e_n\}$, hence
$T_n$ has finite dimensional range.</p>
|
694,098 | <p>$$\lim\limits_{x \to 3} \frac{3-x}{5-\sqrt{x^2+16}}$$</p>
<p>The professor says we can't use l'hopital's rule and must solve algebraically.</p>
| amWhy | 9,003 | <p>Try multiplying the numerator and denominator by the conjugate of the denominator:</p>
<p>Multiply by $$\dfrac{5+ \sqrt{x^2 +16}}{5+\sqrt{x^2 + 16}}$$</p>
<hr>
<p>$$\lim\limits_{x \to 3} \frac{3-x}{5-\sqrt{x^2+16}} \cdot \dfrac{5+ \sqrt{x^2 +16}}{5+\sqrt{x^2 + 16}} = \lim_{x\to 3}\dfrac{(3-x)(5 + \sqrt {x^2 + 16)... |
694,098 | <p>$$\lim\limits_{x \to 3} \frac{3-x}{5-\sqrt{x^2+16}}$$</p>
<p>The professor says we can't use l'hopital's rule and must solve algebraically.</p>
| Bill Dubuque | 242 | <p><strong>Hint</strong> $\ \ f\bar f = (a\!-\!x)(a\!+\!x)\ \Rightarrow\ \dfrac{a-x}f \,=\, \dfrac{\bar f}{a+x}\ \ $ where $\ \ \bar f,\,f\, =\, 5\pm \sqrt{x^2+16}$</p>
<p><strong>Remark</strong> $\ $ This is a special case of <em>rationalizing the denominator</em>, which <a href="http://www.google.com/search?q=site... |
4,301,673 | <p>Suppose we have chosen <span class="math-container">$n$</span> random points in a line segment <span class="math-container">$[0, t]$</span>, <span class="math-container">$n\leq t+1$</span>. What is the probability that distance between each pair of adjacent points is > 1? Or more formally, let <span class="math-c... | Roah | 2,481 | <p>@MXXZ answer is toooo beautiful to be the only solution :)</p>
<h6>The denominator</h6>
<p>Our <span class="math-container">$n$</span> points are distributed uniformly on <span class="math-container">$[0;t]$</span>, so the denominator is <span class="math-container">$t^n$</span>.</p>
<h6>The nominator</h6>
<p>Imagin... |
1,728,524 | <p>Classical texts for control theory show the linear system $\dot x=A \,x$, is stable if the real parts of the eigenvalues are negative.</p>
<p>Does the same criteria apply for a system of the following form:
$$ \left[
\begin{array}{cc|c}
\dot x\\
0
\end{array}
\right] = B \left[ \begin{array... | Robert Israel | 8,508 | <p>The conditions of symmetry and not having zero elements are not relevant. A necessary and sufficient condition is that the column space of $A$ contains the column space of $B$. If so, you can take $C = A^+ B$
where $A^+$ is the <a href="https://en.wikipedia.org/wiki/Moore%E2%80%93Penrose_pseudoinverse" rel="nofoll... |
3,757,038 | <h2>The problem</h2>
<p>So recently in school, we should do a task somewhat like this (roughly translated):</p>
<blockquote>
<p><em>Assign a system of linear equations to each drawing</em></p>
</blockquote>
<p>Then, there were some systems of three linear equations (SLEs) where each equation was describing a plane in t... | Marco Camurri | 780,467 | <p>A line in the Euclidean space can be described by a system of two equations describing a plane.</p>
<p>The equations will be in the form:
<span class="math-container">$$
\begin{cases} ax + by + cz + d = 0 \\
a'x + b'y +c'z + d' = 0
\end{cases}
$$</span>
Another way to express a line is in parametric form:
<span clas... |
3,757,038 | <h2>The problem</h2>
<p>So recently in school, we should do a task somewhat like this (roughly translated):</p>
<blockquote>
<p><em>Assign a system of linear equations to each drawing</em></p>
</blockquote>
<p>Then, there were some systems of three linear equations (SLEs) where each equation was describing a plane in t... | twosigma | 780,083 | <p>If you know some linear algebra, this problem becomes easier to describe and answer. The concepts and terminology from linear algebra capture the relevant ideas well.</p>
<p>As long as the <strong>null space</strong> of the system of three <strong>planes</strong> has <strong>dimension 1</strong>, the planes will for... |
1,448,062 | <p>Find $\overline{\lim}\limits_{n\to\infty}\left(\frac{1}{n}-\frac{2}{n}+\frac{3}{n}-...+(-1)^{n-1}\frac{n}{n}\right)$</p>
<p>$\frac{1}{n}-\frac{2}{n}+\frac{3}{n}-...+(-1)^{n-1}\frac{n}{n}=\frac{1}{n}\sum\limits_{k=1}^{n}(-1)^{k-1}k$</p>
<p>How to evaluate sum $\sum\limits_{k=1}^{n}(-1)^{k-1}k$?</p>
| true blue anil | 22,388 | <p>I would use David Quinn's method, but since the logic of the book answer is bothering you, let me explain.</p>
<p>Leaving aside the glued together $(SY)$, there are 10 letters with 3 repeats, which can be permuted in $\dfrac{10!}{3!}$ ways.</p>
<p>Now, the $(SY)$ can be inserted into any of the <strong>11 gaps (in... |
3,123,681 | <p>let us consider following problem taken from book</p>
<p><em>An appliance store purchases electric ranges from
two companies. From company A, 500 ranges are
purchased and 2% are defective. From company B,
850 ranges are purchased and 2% are defective.
Given that a range is defective, find the probability
that it... | robjohn | 13,854 | <p>Note that <span class="math-container">$\arctan(x)=x+O\!\left(x^3\right)$</span>, then
<span class="math-container">$$
\begin{align}
\sum_{k=1}^n\arctan\left(\frac1{n+k}\right)
&=\sum_{k=1}^n\left[\frac1{n+k}+O\!\left(\frac1{(n+k)^3}\right)\right]\\
&=\sum_{k=1}^n\frac1{1+k/n}\frac1n+O\left(\frac1{n^2}\right... |
14,568 | <p>The Poisson summation says, roughly, that summing a smooth $L^1$-function of a real variable at integral points is the same as summing its Fourier transform at integral points(after suitable normalization). <a href="http://en.wikipedia.org/wiki/Poisson_summation_formula">Here</a> is the wikipedia link.</p>
<p>For m... | coudy | 6,129 | <p>The Poisson formula is just the decomposition in Fourier series of the Dirac measure on the circle. </p>
<p>It happens that the theory of distributions on the circle is far more simpler than the one on the real line since everything is compactly supported. In his book, Laurent Schwartz starts the chapter about the ... |
519,224 | <p>Using the Squeeze Theorem, how do I find: $$\lim_{x\to 3} (x^2-2x-3)^2\cos\left(\pi \over x-3\right)$$
I thought I knew the Squeeze Theorem, but I haven't encountered anything like this yet, so I honestly have no idea how and where to start.</p>
<p>I would appreciate any help I get because I really want to be able ... | André Nicolas | 6,312 | <p><strong>Hint:</strong> The absolute value is between $0$ and $(x^2-2x-3)^2$.</p>
|
3,814,502 | <p>I came across the following question:</p>
<blockquote>
<p>Show that <span class="math-container">$(a, b:a^3 = 1, b^2= 1, ba=a^2b)$</span> gives a group of order <span class="math-container">$6$</span>. Show that it is non abelian. Is it the only non abelian group of order <span class="math-container">$6$</span> up t... | ccroth | 793,822 | <p>In general, for given <span class="math-container">$n \in \mathbb{N}$</span>, there is not some simple way to figure out all non-isomoprhic groups of order <span class="math-container">$n$</span> (though it's been deduced for special cases of <span class="math-container">$n$</span> and a number of 'small' <span clas... |
3,651,173 | <p>I am trying to use generating function to find closed form formula for this expression:
<span class="math-container">$$\sum_{n \geq 0} \frac{n^2+4n+5}{n!}$$</span>
but I don't how to start this. any suggestion or hints. thank you</p>
| Deepak | 151,732 | <p><span class="math-container">$$\begin{align}\sum_{n \geq 0} \frac{n^2+4n+5}{n!} &= \sum_{n \geq 0} \frac{n^2}{n!} + \frac{4n}{n!} + \frac 5{n!} \\&= \sum_{n \geq 1}\frac n{(n-1)!} + 4\sum_{n \geq 1}\frac 1{(n-1)!} + 5\sum_{n \geq 0}\frac 1{n!}\\&=\sum_{n \geq 1}\frac{n-1+1}{(n-1)!} + 4\sum_{n \geq 1}\fra... |
3,106,784 | <p>I've recently posted another question regarding natural deduction proofs and I've definitely made some progress, but I'm now stuck with a proof which seems like it could be flawed.</p>
<p><img src="https://i.stack.imgur.com/5sMPp.png" alt="The proof"></p>
<p>Now as you can see, it looks like I've got it all figure... | Frank Hubeny | 312,852 | <p>The justification for line 13 should be indirect proof (IP) rather than negation introduction. Here is a correct proof using what you have done and the same proof checker:</p>
<p><a href="https://i.stack.imgur.com/x1qHS.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/x1qHS.png" alt="enter image ... |
1,210,194 | <p>for example: I have a series <img src="https://i.stack.imgur.com/hV9QG.jpg" alt="enter image description here"></p>
<p>is there numerical computation method to find it ? thanks</p>
| Lutz Lehmann | 115,115 | <p>Your terms are proportional to <span class="math-container">$\frac1{16·36·n^6}$</span>, accordingly the error of the n-th partial sum is about <span class="math-container">$\frac1{16·36·5·n^5}$</span>, so the first 100 terms, summed from smallest to largest, should give a valid numerical result.</p>
<p>You might als... |
177,915 | <p>I have two inequalities that I can't prove:</p>
<ol>
<li>$\displaystyle{n\choose i+k}\le {n\choose i}{n-i\choose k}$</li>
<li>$\displaystyle{n\choose k} \le \frac{n^n}{k^k(n-k)^{n-k}}$</li>
</ol>
<p>What is the best way to prove them? Induction (it associates with simple problems, but sometimes I find it difficult... | joriki | 6,622 | <p>For the first one, if you write out all the binomial coefficients and cancel identical factors on both sides, you're left with</p>
<p>$$\frac1{(i+k)!}\le\frac1{i!}\frac1{k!}\;,$$</p>
<p>which is clearly true.</p>
<p>For the second one, you can use <a href="http://en.wikipedia.org/wiki/Stirling%27s_approximation" ... |
475,151 | <blockquote>
<blockquote>
<p>Determine the volume of
$$
M:=\left\{(x,y,z)\in\mathbb{R}^3: z\in [0,2\pi],(x-\cos(z))^2+(y-\sin(z))^2\leq\frac{1}{4}\right\}
$$</p>
</blockquote>
</blockquote>
<p>My idea is to use the principle of Cavalieri, i.e. to set
$$
M_z:=\left\{(x,y)\in\mathbb{R}^2: (x,y,z)\in M\right... | 2'5 9'2 | 11,123 | <p>You are making a significant mistake at the point "So I used Fubini...". Your double integral is over a square, when it should be over a circle. The inner integral should have limits that depend on the outer integral's variable.</p>
|
1,752,106 | <p>You roll $n$ fair dice. Let X be the number of pairs of dice that sum to $7$.
Write $X$ as a sum of indicator variables and find $E[X]$</p>
<p>This is how I approached the problem</p>
<p>Let $S$ be all the strings length n on [6].</p>
<p>Let $X(S)$ be the number of ways to add up the numbers on the n dice that ad... | André Nicolas | 6,312 | <p>There are $N=\binom{n}{2}$ (unordered) pairs of dice. Label these $P_1$ up to $P_N$.</p>
<p>For every pair $P_i$, define random variable $Y_i$ by $Y_i=1$ if the pair $P_i$ has sum $7$, and by $Y_i=0$ otherwise. </p>
<p>Then $X=Y_1+\cdots+Y_N$, and by the linearity of expectation we have $$E(X)=E(Y_1)+\cdots+E(Y_N... |
2,078,496 | <p>Given the sequence of functions</p>
<p>$f_n:[0,1]\rightarrow\mathbb{R}$</p>
<p>$x\mapsto\begin{cases}2nx, & x\in[0,\frac{1}{2n}] \\ -2nx+2, & x\in (\frac{1}{2n},\frac{1}{n}) \\ 0, & \text{otherwise} \end{cases}$</p>
<hr>
<p>The functions look like this:</p>
<p><a href="https://i.stack.imgur.com/cE7h... | qualcuno | 362,866 | <p>This function converges pointwise to the zero function. It is trivial that $f_n(0) = 0 \ \forall n \in \mathbb{N}$. If $\alpha \in (0,1]$, by archimedianity there is some $n$ for wich $\alpha$ lies in the interval $[\frac{1}{n},1]$ and therefore for that $n$ and every larger integer, $f_n(\alpha) = 0$. This means ex... |
306,732 | <p>Let $R$ and $S$ be rings and $h$ and $g$ be homomorphisms from $R$ into $S$. </p>
<p>Let $T=\{r| r\in R$ and $h(r)=g(r)\}$</p>
<p>Prove that $T$ is a subring of $R$. </p>
<p>I understand what the question is asking but I am a little confused on how to get it started. I know that I have to prove $T\neq \emptyset$;... | chubbycantorset | 51,426 | <p>$T$ is just the set of those elements in the domain, $R$, that take the same value when mapped by the homomorphisms $g$ and $h$.</p>
<p>Hope this makes it clearer.</p>
|
2,861,179 | <p>Here's the question:</p>
<blockquote>
<p>Evaluate $\iint_{S} \boldsymbol{F} \cdot \boldsymbol{\hat{n}}$ if
$\boldsymbol{F} = (x+y) \boldsymbol{\hat{i}} + x \boldsymbol{\hat{j}}
+z \boldsymbol{\hat{k}}$ and $S$ is the surface of the cube bounded by the planes $x=0$,$x=1$,$y=0$, $y=1$, $z=0$ and $z=1$.</p>
</blo... | user | 505,767 | <p>By direct calculation keep attention on the sign of the flux, we need to assume a consistent direction for the normal.</p>
<p>As an alternative by <a href="https://en.wikipedia.org/wiki/Divergence_theorem" rel="nofollow noreferrer"><strong>divergence theorem</strong></a> we have</p>
<p>$$div\boldsymbol{F} = 2\impl... |
2,861,179 | <p>Here's the question:</p>
<blockquote>
<p>Evaluate $\iint_{S} \boldsymbol{F} \cdot \boldsymbol{\hat{n}}$ if
$\boldsymbol{F} = (x+y) \boldsymbol{\hat{i}} + x \boldsymbol{\hat{j}}
+z \boldsymbol{\hat{k}}$ and $S$ is the surface of the cube bounded by the planes $x=0$,$x=1$,$y=0$, $y=1$, $z=0$ and $z=1$.</p>
</blo... | Michael Burr | 86,421 | <p>The problem, is that you are not using <em>outward pointing</em> normal vectors. Every plane has two unit normal vectors, but only one of them is pointing outward. For a flux integral, we use the convention that the flux is the flow out of an object.</p>
<p>In your case, the outward pointing normal vector for $S_... |
3,423,840 | <p><a href="https://i.stack.imgur.com/batlm.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/batlm.jpg" alt="enter image description here"></a><br>
I used the ratio test for this problem and I messed up somewhere in the algebra. Sorry for the messy writing. <a href="https://i.stack.imgur.com/hRMUF.png... | Doug M | 317,176 | <p>The denominator can be written as <span class="math-container">$4^nn!$</span></p>
<p><span class="math-container">$n!$</span> grows faster than <span class="math-container">$x^n$</span> for all <span class="math-container">$x$</span> and <span class="math-container">$x^n$</span> grows faster than any polynomial if ... |
4,354,663 | <p>I'm trying to prove that if <span class="math-container">$x$</span> is a multiple of <span class="math-container">$3$</span>, then <span class="math-container">$x + 1$</span> is not a multiple of <span class="math-container">$3$</span>. This is a rather obvious fact, but I don't think I understand the solution.</p>
... | Sourav Ghosh | 977,780 | <p>Suppose that <span class="math-container">$ x $</span> is a multiple of <span class="math-container">$3 $</span> but, for the sake of contradiction, that <span class="math-container">$x+1$</span> is likewise a multiple of <span class="math-container">$3$</span> . Then <span class="math-container">$x=3k $</span> for... |
545,380 | <p>Let $F$ be a field such that $|F|=3^{2n+1}$ and $r=3^{n+1}$. I want to find the number of $x\in F$ that satisfies the equation $x^{r+1}=1$.</p>
| arctic tern | 296,782 | <p>Say we want to count the solutions to $x^k=e$ in a cyclic group of order $m$. We may as well write everything additively and work in $\mathbb{Z}/m\mathbb{Z}$. Writing $\overline{k}=k/\gcd(m,k)$ and $\overline{m}=m/\gcd(m,k)$,</p>
<p>$$kx\equiv 0~\bmod m ~\iff~ \overline{k}x\equiv 0\bmod\overline{m} ~\iff~ x\equiv 0... |
1,922,417 | <p>This is not for the Maths part of the General GRE. This is for the GRE <a href="https://www.ets.org/gre/subject/about" rel="noreferrer">Subject Test</a> in <a href="https://www.ets.org/gre/subject/about/content/mathematics" rel="noreferrer">Maths</a>. Feel free to add or comment.</p>
<ul>
<li><p><a href="https://ma... | operatorerror | 210,391 | <p>Schaum's outlines "Advanced Mathematics for Engineers and Scientists" has some excellent ODE, linear algebra, and complex problems that mimic the ones I have seen on practice exams. </p>
|
275,937 | <p>Choose a random polynomial $P\in\mathbb{Z}[x]$ of degree $n$ and coefficients $\leq n$ and $\geq-n$. </p>
<p>Let $r_1,\ldots,r_n$ be the roots of $P$ and consider $$G=\operatorname{Gal}(\mathbb{Q}(r_1,\ldots,r_n)/\mathbb{Q})$$</p>
<p>What is the probability, as $n\to\infty,$ that $G$ is solvable? (I assume 0.) Who... | MCT | 92,774 | <p>Yes, the probability is zero since it equals the symmetric group with probability one:</p>
<p>If $Q_d(N)$ denotes the set of degree $d$ polynomials with coefficients $|a_i|\le N$ with Galois group not equal to $S_d$, then
$$
|Q_d(N)|\ll N^{d-1/2}\log N
$$</p>
<p>This bound is sufficient to prove your result. This ... |
1,371,075 | <p>$$3^x = 3 - x$$</p>
<p>I have to prove that only one solution exists, and then find that one solution.</p>
<p>My approach has been the following:</p>
<p>$$\log 3^x = \log (3 - x)$$</p>
<p>$$x\log 3 = \log (3 - x)$$</p>
<p>$$\log 3 = \frac{\log (3 - x)}{x}$$</p>
<p>And this is where I get stuck. Any help will b... | Olivier Oloa | 118,798 | <p>You may consider the function given by
$$
f(x)=3^x-(3-x),\quad x \in \mathbb{R}.
$$ We have
$$
f'(x)=3^x \cdot\ln3+1>0,\quad x \in \mathbb{R},
$$ Thus the function is <strong>strictly increasing</strong> on $\mathbb{R}$.</p>
<p>We have
$$
\begin{align}
f(0)&=1-(3-0)=-2<0\\\\
f(1)&=3-(3-1)=1>0
\end{... |
2,909,036 | <p>I'm reading a book and there is some strange proof (strange for me) of the theorem that within each interval, no matter how small, there are rational points. Proof: we need only take a denominator $n$ large enough so that the interval $[0, \frac{1}{n}]$ is smaller than the interval $[A, B]$ in question; then at leas... | Arthur | 15,500 | <p>Given a natural number $n$, if no fraction with denominator $n$ is contained in $[A,B]$, then there is a fraction $\frac mn$ which is the largest one which is <em>below</em> $[A,B]$ (meaning $\frac mn< A$), and in addition we have that the next fraction with denominator $n$, which is $\frac{m+1}n$, must be <em>ab... |
2,984,534 | <p>I've gone ahead and split up <span class="math-container">$4000$</span> into <span class="math-container">$2^{5} 5^{3}$</span> and solved each solution separately - as in applied Hensel's lemma for mod 2 and mod 5 solutions separately, I just don't understand how I would combine these solutions.</p>
<p>For mod <sp... | Sri-Amirthan Theivendran | 302,692 | <p>Suppose that <span class="math-container">$f$</span> is not constant everywhere. In particular, if <span class="math-container">$f(x)=c$</span> for <span class="math-container">$x\in A$</span>, then, there exists <span class="math-container">$x_0\notin A$</span> such that <span class="math-container">$f(x_0)\neq c$<... |
3,315,622 | <p>Given <span class="math-container">$\textbf{a}$</span> , <span class="math-container">$\textbf{h}$</span> <span class="math-container">$\in$</span> <span class="math-container">$\mathbb{R}^n$</span> and a function from some subset of <span class="math-container">$\mathbb{R}^n$</span> to <span class="math-container">... | Wassilis | 693,418 | <p>If <span class="math-container">$t\to 0$</span> then <span class="math-container">$tv\to 0$</span></p>
<p>To do this formaly try and rewrite the increment parameter h as a function <span class="math-container">$h(t)$</span>
of the real parameter <span class="math-container">$t$</span>(which you basicly already did ... |
105,413 | <p>I know: I'm going to make a poor showing, but really I can't understand this:</p>
<p><code>a</code> is an expressione whose FullForm is</p>
<pre><code>Power[Plus[Subscript[u,x],Times[Complex[0,-1],Subscript[u,y]],Times[Complex[0,1],Subscript[v,x]],Subscript[v,y]],2]
</code></pre>
<p>Why does the following code re... | Jason B. | 9,490 | <p>Assuming this is what you are going for,</p>
<p><a href="https://i.stack.imgur.com/KrEgF.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/KrEgF.png" alt="enter image description here"></a></p>
<p>You can get there two ways,</p>
<pre><code>Replace[a, {I x_ :> H x, -I x_ :> - H x}, Infinity]... |
1,560,192 | <ol>
<li><p>Compute the determinant of \begin{pmatrix}
1 & 2 & 3 & ...& n\\
-1 & 0 & 3 & ...& n\\
-1 & -2 & 0 & ...& n\\
...& ...& ...& ...& \\
-1 & -2 & -3 & ...& n
\end{pmatrix}
after some elementary raw operation, one can reach:
\... | bartgol | 33,868 | <p>Yes. The derivative of $x^2$ tells you how fast the function $x^2$ increases. The second derivative of $x^2$ tells you how fast the derivative of $x^2$ increases. The second derivative is 2, meaning that the speed at which the rate of growth of $x^2$ increases is fixed. Therefore, the difference of the difference of... |
2,242,865 | <p>I have seen proofs using the delta-epsilon definition of continuity, and they make perfect sense, but I have not found one proof using the sequential definition of continuity. </p>
<p>For example, when given functions, $f$ and $g$ that are continuous on [$a,b$], prove that the function $h=f+g$ is also continuous on... | Mark Twain | 60,908 | <p>Yes. </p>
<p>Let $f$, $g \colon D \to \mathbb{R}$, where $D$ is some nonempty subset of $\mathbb{R}$. Let $a \in D$. Suppose both of $f$ and $g$ are continuous at $a$. We can show that $f+g$ is continuous at $a$ using the sequential criterion. </p>
<p>Let $(x_n)$ be a sequence in $D$ such that $\lim x_n = a$. Sinc... |
252,049 | <p>I have seen on the nLab that we can <a href="https://ncatlab.org/nlab/show/loop+space+object" rel="nofollow">view the loop space as a particular homotopy pullback</a>, and that it is even the way a "loop space object" is defined in general (when it exists). </p>
<p>Can someone give me some intuition of why it is so... | Emily | 130,058 | <p>This can be seen from the following general formula<span class="math-container">$^1$</span> for homotopy pullbacks of topological spaces.</p>
<p><strong>Proposition.</strong> Let
<span class="math-container">$$X\xrightarrow{f}Z\xleftarrow{g}Y$$</span>
be a diagram of topological spaces. Then
<span class="math-conta... |
3,334,910 | <p>Question. <span class="math-container">$\square ABCD$</span> is a square with <span class="math-container">$AB = 10$</span>. Circle <span class="math-container">$O$</span> inscribes the <span class="math-container">$\square ABCD$</span>. The center of the arc is <span class="math-container">$A$</span>. What is the a... | N. S. | 9,176 | <p><strong>Hint</strong><a href="https://i.stack.imgur.com/52jNe.png" rel="noreferrer"><img src="https://i.stack.imgur.com/52jNe.png" alt="Pic"></a></p>
<p>The two yellow areas are equal.</p>
<p>You can easily calculate <span class="math-container">$$2 \mbox{yellow}+\mbox{blue}+\mbox{purple}=\mbox{Area} ABCD-\frac{1}... |
1,456,591 | <p>It is known that if $$|a+b|=|a|+|b|$$ then we can find the solution by simply observing that we can instead solve the inequality $$a b \geq 0$$</p>
<p>My question is, if $|a+b+c|=|a|+|b|+|c|$, then what would be the '3 degree version' of the above?</p>
| Umberto P. | 67,536 | <p>The third degree (or higher) version is "all nonnegative" or "all nonpositive".</p>
|
203,009 | <p>Given $k,c \in \mathbb{N}$, let $P(k,c)$ be the minimum $n$ such that no matter how we color the edges of the complete graph $K_n$ with $c$ colors, there is always a monochromatic path of length $k$.</p>
<p>What are the best known upper and lower bounds for $P(k,c)$?</p>
| Tony Huynh | 2,233 | <p>For $c=2$, it is a theorem of <a href="http://www.renyi.hu/~gyarfas/Cikkek/01_GerencserGyarfas_OnRamseyTypeProblems.pdf">Gerencsér and Gyárfás</a> that $P(k,2)=\lfloor (3k-2)/2 \rfloor$. </p>
<p>For $c=3$, <a href="http://web.cs.wpi.edu/~gsarkozy/Cikkek/27.pdf">Gyárfás, Ruszinkó, Sárközy and Szemerédi</a> proved t... |
203,009 | <p>Given $k,c \in \mathbb{N}$, let $P(k,c)$ be the minimum $n$ such that no matter how we color the edges of the complete graph $K_n$ with $c$ colors, there is always a monochromatic path of length $k$.</p>
<p>What are the best known upper and lower bounds for $P(k,c)$?</p>
| Jan Kyncl | 24,076 | <p>For the multi-color Ramsey numbers of even cycles, <a href="https://dx.doi.org/10.1002/jgt.20572" rel="nofollow">Luczak, Simonovits and Skokan</a> proved that $R(C_k;c)\le ck+o(k)$ for fixed number $c$ of colors and $k\rightarrow \infty$.</p>
<p>For odd cycles, <a href="http://dx.doi.org/10.1016/S0095-8956(73)80005... |
1,572,065 | <p>I came across this question in Probability & Statistics by Rohatgi Exercise 1.6.9.</p>
<p>Let $P(A|B) = P(A|B\cap C)P(C) + P(A|B\cap C^c)P(C^c)$ with $P(A|B\cap C) \neq P(A|B)$.</p>
<p>If all the three events have non-zero probability, prove that $B$ and $C$ are independent.</p>
<p>After several efforts, I c... | ki3i | 202,257 | <p>Since ${\bf P}[B], {\bf P}[A\,\vert\,B]>0$ then, using the given equation, note that we can relate two ways of computing $\,{\bf P}[A \cap B]$: </p>
<p>$${\bf P}[A \,\vert\, B \cap C ]\,{\bf P}[ B \cap C ] + {\bf P}[ A \,\vert\, B \cap C^c ]\,{\bf P}[ B \cap C^c ] \\= {\bf P}[A \cap B ] \\= {\bf P}[A \,\vert\, B... |
341,728 | <p>Trivially, for any Lie Algebra (LA) g, g':=[g,g] is an ideal. What's wrong with the following argument?</p>
<p>Be g a simple LA, then it has to be g'=g by definition of simple LA. But [g,g]=g seems to be an alternative way of characterizing a semi simple LG. Furthermore, for sl(2) it doesn't seem to be true z=[x,y]... | Derek Holt | 2,820 | <p>There is a general computer algorithm for such calculations (Schreier-Sims), but it is tedious to do it by hand.</p>
<p>Carrying on with DonAntonio's calculations, you might notice that $(ab)^2 = (1,4)(3,5)(2,6)(7,8)$ commutes with both $a$ and $b$, so the subgroup $H = \langle (ab)^2 \rangle$ of $G$ has order 2 an... |
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