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 |
|---|---|---|---|---|
264,740 | <p>On Hilbert spaces, the following is true:</p>
<p>Let $T$ be a densely-defined linear operator with non-empty resolvent set, then $T$ is closed.</p>
<p>The obvious proof I see to show this uses explicitly the Hilbert space structure which is why I would like to ask:</p>
<p>Is the same result true for operators on ... | Convexity | 86,275 | <p>STANDING ASSUMPTIONS: Let <span class="math-container">$T:D_T\rightarrow X$</span> be a linear operator, where <span class="math-container">$X$</span> is a normed space (or metrizable TVS) and <span class="math-container">$D_T\subset X$</span>.</p>
<p>DEFINITION: Then <span class="math-container">$\lambda$</span> b... |
1,515,900 | <p>In many books and papers on analysis I met this equality without proof:</p>
<p>$$\sup \limits_{t\in[a,b]}f(t)-\inf \limits_{t\in[a,b]}f(t)=\sup \limits_{t,s\in[a,b]}|f(t)-f(s)|$$</p>
<p>Can anyone show strict and nice proof of that equality?</p>
<p>I would really grateful foe your help!</p>
| Lutz Lehmann | 115,115 | <p>For every $ε>0$ there are $s$ and $t$ such that
$$
\inf f\le f(s)\le \inf f+ε
$$
and
$$
\sup f -ε \le f(t) \le \sup f
$$
Combine this to
$$
\sup f - \inf f-2ε \le f(t)-f(s)\le \sup f - \inf f
$$
to see that indeed the claimed identity holds.</p>
|
830,599 | <p>The function $f$ is defined as follows:
$$f(x):=\sum_{j=1}^{\infty} \frac{x^j}{j!} e^{-x}$$</p>
<p>It's easy to see that $f(0)=0$. But I am interested in the value
$$\lim_{x \rightarrow 0^+} f(x).$$</p>
<p>Even <a href="https://www.wolframalpha.com/input/?i=lim_%28x-%3E0%29+%28sum_%28i%3D1%29%5Einfinity+x%5Ej%2F%... | Eff | 112,061 | <p>$$ f(x) =\sum\limits_{j=1}^\infty \frac{x^j}{j!}e^{-x} = e^{-x}\sum\limits_{j=1}^\infty \frac{x^j}{j!} = e^{-x}\cdot (e^x-1) = 1-e^{-x}.$$</p>
<p>Here we have used that
$$e^x = \sum\limits_{j=0}^\infty\frac{x^j}{j!} = 1+\sum\limits_{j=1}^{\infty}\frac{x^j}{j!} $$
So that
$$ \sum\limits_{j=1}^{\infty}\frac{x^j}{j!}... |
2,423,055 | <p>I am sort of baffled by this thing, already real number has every thing in it why is this concept of $\Bbb R^2$ ? What does it mean? What is its advantage?</p>
| G Tony Jacobs | 92,129 | <p>We usually use $\mathbb{R}$, the set of real numbers, to refer to what we picture as the number line.</p>
<p>Thus, $\mathbb{R}^2$, the set of pairs of real numbers, is what we picture as the $xy$-plane, coordinatized by two number lines.</p>
<p>If you want to talk about space with $n$ dimensions, then you want a "... |
516,244 | <p>My professor gave us this example on her notes:</p>
<p>$$\sum_{n = 1}^\infty \left(\frac{3}{n(n+3)}+\frac{1}{2^n}\right)$$</p>
<p>So I know we're supposed to find the partial fraction, which ends up being</p>
<p>$$\left(\frac{3}{n(n+3)}=\frac{A}{n}+\frac{B}{n+3}=
\frac{1}{n}-\frac{1}{n+3}\right)$$</p>
<p>So ba... | Piquito | 219,998 | <p>You know that <strong>AB × AC</strong> is a vector perpendicular to the plane ABC such that |<strong>AB × AC</strong>|= Area of the parallelogram ABA’C. Thus this area is equal to ½ |AB × AC|.</p>
<p><a href="https://i.stack.imgur.com/3oDbh.png" rel="noreferrer"><img src="https://i.stack.imgur.com/3oDbh.png" alt="en... |
1,022,485 | <p>I'm working on a homework problem for my discrete math class, and I'm stuck. (Note: I made a post about this earlier, but I read the problem incorrectly, thus the work was wrong, so I deleted the post.)</p>
<p>Prove by mathematical induction that for every integer n $\ge$ 0, $12\mid8^{2n+1}+2^{4n+2}$</p>
<p>I sta... | ir7 | 26,651 | <p>$$
8^{2n+1}\cdot63+2^{4n+2}\cdot 15=8^{2n}\cdot(8\cdot63)+2^{4n}\cdot(4\cdot15).
$$</p>
|
2,376,315 | <p>So I'm trying to solve the problem irrational ^ irrational = rational. Here is my proof
Let $i_{1},i_{2}$ be two irrational numbers and r be a rational number such that $$i_{1}^{i_{2}} = r$$
So we can rewrite this as $$i_{1}^{i_{2}} = \frac{p}{q}$$
Then by applying ln() to both sides we get $$i_2\ln(i_1)... | Jens Renders | 131,972 | <p>How do you know that the values you end up with are irrational? If you are going to use the irrationality of e and natural logarithms, you might aswel just use $e^{\ln (2)}=2$ as a counterexample and you're done. The famous proof given in another answer uses only $\sqrt{2}$ because it's easy to prove that that is an... |
893,959 | <p>I have a series of problems in inequalities that I can not solve,please help me if you can.</p>
<p>problem 1 :$a,b,c \geq 0$ such that $\sqrt{a^2+b^2+c^2}=\sqrt[3]{ab+bc+ca} $
prove that $a^2b+b^2c+c^2a+abc \le \frac{4}{27}$</p>
<p>problem 2 : $a,b,c\geq0$ and $a+b+c = 1$ prove that </p>
<p>1, $ \sqrt{a+\frac{(b... | math110 | 58,742 | <p>We have $$\frac{\sqrt b+\sqrt c}{\sqrt 2}=\sqrt{b+c-\frac{(\sqrt b-\sqrt c)^2}{2}}\le \sqrt{b+c-\frac{( b- c)^2}{4}}$$
therefore from this and Cauchy-Schwarz it follows that
$$\left( \sqrt{a+\frac{(b-c)^2}{4}}+\sqrt{b}+\sqrt{c}\right)^2\le \left(a+\frac{(b-c)^2}{4}+\frac{(\sqrt b+\sqrt c)^2}{ 2}\right)(1+2)\le 3\l... |
2,090,059 | <p>Let $M = \bigcup\limits_{n=1}^\infty M_n$, $M_n$ be a measurable sets on $\mathbb R$ and $f$ be a measurable function on $\mathbb R$. Then I want to know how the equality
$$
\lim\limits_{N\to \infty}\int_{M\setminus \bigcup\limits_{n=1}^NM_n} |f(x)| \,dx = 0
$$
holds. It seems something like monotone convergence th... | Gabriel Romon | 66,096 | <p>The map $\nu:A\mapsto \int_A |f|$ is a measure.</p>
<p>Note that $A_N:=M\setminus \bigcup_{n=1}^NM_n$ is a decreasing sequence that converges to $\emptyset$.</p>
<p>Assuming $\int |f|<\infty$, we have $\lim_N\nu(A_N)=\nu(\emptyset)=0$.</p>
|
293,110 | <p>How do I prove that homomorphism $\phi : \; \mathrm{Mod}(S_g)\to \mathrm{Sp}(2g, \mathbb{Z})$ (induced by the action of mapping class group of a surface on integer homologies of a surface) is an epimorphism? My idea was to work with generators, but I was not able to prove it this way. </p>
<p>I would love to get de... | Zuhair Al-Johar | 95,347 | <p>The following is actually an answer to whether the theory presented in this post can interpret $\text{NBG}$. </p>
<p>The answer is to the positive! Moreover, it can interpret Morse-Kelley class theory $\text{MK}$.</p>
<p>The axioms of the theory are all those present in this post with the only exception of "Set Un... |
3,713,206 | <p>Spivak (3rd edition) proposes solving the integral <span class="math-container">$$\int \frac{1+e^x}{1-e^x} dx$$</span> by letting <span class="math-container">$u=e^x$</span>, <span class="math-container">$x=\ln(u)$</span>, and <span class="math-container">$dx=\frac{1}{u}du$</span>. This results in the integral <span... | peek-a-boo | 568,204 | <p>Note that
<span class="math-container">\begin{align}
\int f(g(x))\cdot g'(x) \dfrac{1}{g'(x)}\ dx
&= \int \underbrace{f(g(x))\cdot\dfrac{1}{(g' \circ g^{-1})(g(x))}}_{h(g(x))}\ \underbrace{g'(x)\ dx}_{du} \\
\end{align}</span>
where I defined <span class="math-container">$h(t):= f(t) \cdot \dfrac{1}{(g' \circ g^... |
7,064 | <p>The <a href="http://en.wikipedia.org/wiki/Long_division" rel="nofollow">Wikipedia article</a> on long division explains the different notations. I still use the European notation I learned in elementary school in Colombia. I had difficulty adapting to the US/UK notation when I moved to the US. However, I did enjoy s... | Juan S | 2,219 | <p>US notation - which is taught in Australia</p>
|
7,064 | <p>The <a href="http://en.wikipedia.org/wiki/Long_division" rel="nofollow">Wikipedia article</a> on long division explains the different notations. I still use the European notation I learned in elementary school in Colombia. I had difficulty adapting to the US/UK notation when I moved to the US. However, I did enjoy s... | Community | -1 | <p>This is so interesting, especially the idea of language dictating direction! I grew up in Argentina and attended a bilingual school...so I learned both European and US notation but am stronger in the US because we used it more often. Now I'm teaching language and will do a lesson on the different "formats" for sim... |
2,421,421 | <blockquote>
<p>Evaluate
$$ \lim_{x\to \pi/2} \frac{\sqrt{1+\cos(2x)}}{\sqrt{\pi}-\sqrt{2x}}$$ </p>
</blockquote>
<p>I tried to solve this by L'Hospital's rule..but that doesn't give a solution..appreciate if you can give a clue.</p>
| Robert Z | 299,698 | <p>The given limit does not exists. However we are able to evaluate the right and the left limit (without using <a href="https://en.wikipedia.org/wiki/L%27H%C3%B4pital%27s_rule" rel="nofollow noreferrer">L'Hospital's rule</a>).</p>
<p>Note that
$$\sqrt{1+\cos(2x)}=\sqrt{1+\cos^2(x)-\sin^2(x)}=\sqrt{2}|\cos(x)|=\sqrt{... |
99,617 | <p>How can I animate a point on a polar curve? I have used <code>Animate</code> and <code>Show</code> together before in order to get the curve and the moving point together on the same plot, but combining the polar plot and the point doesn't seem to be working because point only works with Cartesian coordinates.</p>
... | David G. Stork | 9,735 | <pre><code>Animate[
Show[
PolarPlot[Sin[3 t], {t, 0, π}],
Graphics[{Red, PointSize[0.02],
Point[Sin[3 θ] {Cos[θ], Sin[θ]}]}]],
{θ, 0, π}]
</code></pre>
|
2,280,323 | <p>I am given a linear transformation
$$T:\mathbb{R^3} \rightarrow \mathbb{R^2}$$
$$T((x,y,z)) = (x+y,-y+z)$$
The task if to find a basis in $\mathbb{R^3}$, let's call it $B=\{e_1, e_2, e_3\}$ and $\mathbb{R^2}$, let's call it $B'=\{f_1, f_2\}$ such that $A$ is the matrix of this transformation with respect to the foun... | Bill | 418,485 | <p>You can find the transformation matrix by standard basis of $\mathbb R^3$ and $\mathbb R^2$ called standared basis. $\{ e_1,e_2,e_3\}$ for $\mathbb R^3$ and $\{ e_1,e_2\}$ for $\mathbb R^2$. now</p>
<p>$T(e_1)=T(1,0,0)=(1,0)=e_1$</p>
<p>$T(e_2)=T(0,1,0)=(1,-1)=e_1-e_2$</p>
<p>$T(e_3)=T(0,0,1)=(0,1)=e_2$</p>
<p>... |
3,585,111 | <p>Given that <span class="math-container">$x^{\frac{2}{3}}=({x^2})^{\frac{1}{3}}$</span>, I thought that <span class="math-container">$f(x)$</span> is continuous in <span class="math-container">$\mathbb{R}$</span> but when I plot this function, I get something like this in wolfram.</p>
<p><a href="https://i.stack.img... | Sujit Bhattacharyya | 524,692 | <p>I think it's maybe the fault of the plotting program (maybe the software understood something else), I've used <a href="https://www.desmos.com/calculator" rel="nofollow noreferrer">Desmos Graphic Calculator</a> and find that this function is perfectly continuous in <span class="math-container">$\Bbb R$</span>.</p>
... |
381,313 | <p>Solve the limits of below:</p>
<p>(1) $\lim\limits_{n \to \infty} \int_0^n (1+\frac{x}{n})^n e^{-2x}dx$.</p>
<p>(2) $\lim\limits_{n \to \infty} \int_0^n (1-\frac{x}{n})^n e^{\frac{x}{2}}dx$. </p>
<p>(3) $\int_0^{\infty} \frac{e^{-x}\sin^2 x}{x}dx$</p>
<p>(Hint: show $f(x,y)=e^{-x}\sin2xy$ is integrable on $[0,... | Neal | 20,569 | <p>Consider your question for principal $G$-bundles over an honest space $M$. Note that the structure group of a principal $G$-bundle is $G$. </p>
<p>Isomorphism classes of principal $G$-bundles over $M$ are in one-to-one correspondence with homotopy classes of maps $M\to BG$, the classifying space of $G$. So in parti... |
206,636 | <p>What's the fastest way to find the local maxima of a 2D list? <em>E.g.</em></p>
<pre><code>nx = ny = 100;
dat = Table[Sin[2. \[Pi] x/nx] (0.1 + Cos[2. \[Pi] y/ny]), {y, 0, ny}, {x, 0, nx}];
ListPlot3D[dat]
</code></pre>
<p><img src="https://i.stack.imgur.com/MyGBl.png" alt="Mathematica graphics"></p>
<p>This (u... | Niki Estner | 242 | <blockquote>
<p>Is there any faster way available?</p>
</blockquote>
<p>There is, if you're willing to to use a different definition of "local maximum". If a local maximum is any value at least as high as any value in a 11x11 neighborhood, then you can use <code>UnitStep[dat - Dilation[dat, 5]]</code>, which is abou... |
144,709 | <p>Recently, I was informed that we can verify the famous formula about <strong>$\mathrm{lcm}(a,b)$</strong> and <strong>$\gcd(a,b)$</strong> which is $$\mathrm{lcm}(a,b)=\frac{|ab|}{\gcd(a,b)} $$ via group theory. </p>
<p>The least common multiple of two integers $a$ and $b$, usually denoted by <strong>$\mathrm{lcm}(... | lhf | 589 | <p>Consider the canonical map $\mathbb Z \to \mathbb Z/(a) \times \mathbb Z/(b)$ given by $x\mapsto (x \bmod a, x \bmod b)$.</p>
<p>What is the kernel? What is the image?</p>
|
3,809,026 | <p>I'm trying to solve this problem in the implicit differentiation section of the book I'm going through:</p>
<blockquote>
<p>The equation that implicitly defines <span class="math-container">$f$</span> can be written as</p>
<p><span class="math-container">$y = \dfrac{2 \sin x + \cos y}{3}$</span></p>
<p>In this probl... | Javi | 554,663 | <p>Yes. If the proposition <span class="math-container">$P(n) $</span> holds true for <span class="math-container">$n \geq n_0$</span>, and you want to prove it by induction, you could try to prove <span class="math-container">$P (n_0)$</span> is true, <span class="math-container">$P (n_1) $</span> is true, and so on. ... |
3,907,519 | <p>Prove that for all <span class="math-container">$a,b \in\mathbb Z$</span>, if <span class="math-container">$a+b$</span> is even then <span class="math-container">$a-b$</span> is even.</p>
<p>I started off by assuming that a and <span class="math-container">$b$</span> are both odd integers, <span class="math-containe... | DMcMor | 155,622 | <p>So far you've only proved that the sum of two odd integers is even. Try dropping that assumption and see what you can do, i.e. suppose <span class="math-container">$a + b = 2n$</span>, for some integers <span class="math-container">$a,b,n$</span>. Can you get from that equation to an equation involving <span class... |
58,920 | <p>I have a csv that encodes the results of plugging certain parameters (say, <code>A</code> and <code>B</code>) into certain functions (say, <code>f</code> and <code>g</code>). Think matrix with (say, 4) columns and a million rows with the columns corresponding to <code>A, B, C = f(A, B)</code>, and <code>D = g(A, B)<... | hieron | 16,373 | <p>C, D, E, I, K, N, O are symbols already in use by Mathematica <br>(ONCE KID)<br>
you do not follow the recommended convention that user variables are written lowercase.</p>
<p>You can see that when typing the letter it is colored black, and not blue.</p>
<p>most probably the <code>Evaluate</code> statement solves ... |
13,478 | <p>This may be a silly question - but are there interesting results about the invariant: the minimal size of an open affine cover? For example, can it be expressed in a nice way? Maybe under some additional hypotheses?</p>
| Emerton | 2,874 | <p>In fact this is a difficult question in general, I believe. For example, if $X$ is a closed subvariety of ${\mathbb P}^n$, then asking for the minimal number of affine opens that cover
the complement ${\mathbb P}^n\setminus X$ is the same as asking for the minimal number of hypersurfaces
whose (set-theoretic) inter... |
1,028,301 | <p>What I've first done is show that $Aut(\mathbb Z / 24\mathbb Z)$ is isomorphic to $\mathbb Z_2\oplus\mathbb Z_2\oplus\mathbb Z_2$ due to it having 8 members and the greatest rank out of all of them is 2.</p>
<p>Now $M_2(\mathbb Z/3\mathbb Z)$ is isomophic to $\mathbb Z_3\oplus\mathbb Z_3\oplus\mathbb Z_3\oplus\math... | Stahl | 62,500 | <p><strong>Hint:</strong> If $\phi : M_2(\Bbb Z/(3))\to\operatorname{Aut}\Bbb Z/(24)$ is a homomorphism and $e\neq M\in M_2(\Bbb Z/(3))$, you must have $e = \phi(e) = \phi(M^{ord(M)}) = \phi(M)^{ord(M)}$. Hence, $ord(M)\mid \#\operatorname{Aut}\Bbb Z/(24) = 8$. But...</p>
|
618,166 | <p>Does there exists a real valued function which is everywhere continuous and differentiable at exactly one point on the real line?</p>
| ncmathsadist | 4,154 | <p>You know that you can produce a function $f$ so that $f$ is continuous and differentiable nowhere. Pick $a\in\mathbb{R}$; put $g(x) = (x - a)^2 f(x).$ The function $g$ is differentiable exactly at $a$ and nowhere else.</p>
|
1,382,003 | <p>I'm trying to solve problem 7 from the IMC 2015, Blagoevgrad, Bulgaria (Day 2, July 30). Here is the problem</p>
<blockquote>
<p>Compute
$$\large\lim_{A\to\infty}\frac{1}{A}\int_1^A A^\frac{1}{x}\,\mathrm dx$$</p>
</blockquote>
<p>And here is my approach:</p>
<p>Since $A^\frac{1}{x}=\exp\left(\frac{\ln A}{x}\... | Erick Wong | 30,402 | <p>The key to showing that this limit converges to <span class="math-container">$1$</span> is to show that the function <span class="math-container">$A^{1/x}$</span> decreases quickly towards <span class="math-container">$1$</span> as <span class="math-container">$x$</span> increases beyond <span class="math-container"... |
732,548 | <p>The proof for this using Dependent choice, is easy. I forgot how to prove this with only AC$_\omega$. Could someone help?</p>
<p>I feel really sorry that i keep asking these basic questions.. This would be my last one. Thank you in advance</p>
| Jason Zimba | 132,296 | <p>Rename $x$ as $\Delta x$. That may help you to see that your limit is $\int_0^\infty{f(x)\,dx}$, where $f(x) = {2\over x^2+1}$. (You should recognize your limit as the limit of a Riemann sum.)</p>
|
3,653,016 | <p>I am solving this equation: </p>
<p><img src="https://i.stack.imgur.com/WN7ce.png" alt="Image of the equation"><br>
My issue arises on line 2 where we have (n + 1 - 2 + 1)(n + 1 + 2)/2. This is what i understand. We have the formula n! = n(n+1)/2. Subbing values into the equation yields us with 10[(n+1)(n+1 + 1)/2]... | user781829 | 781,829 | <p>in line 2 the <span class="math-container">$\frac{(n+1+2)}{2}$</span> part is the average of the terms being summed. In this problem this is calculated by adding the first and last term and dividing by 2. The <span class="math-container">$(n+1-2+1)$</span> part is the number of terms being summed. This is calculated... |
404,954 | <p>I've never seen phrases like "$\sqrt{5}$ people" or "a set with $\pi$ many elements". Are there sets with cardinality, say, $\frac{1}{2}$?</p>
<p><strong>Edit:</strong> As Brian M. Scott pointed out, the only real numbers that are cardinalities of sets are the non-negative integers. Could you explain why is it so?<... | Nate Eldredge | 822 | <p>Fundamentally, cardinals and real numbers are different things. You can think of the cardinality of a set as some abstract object ("cardinal") assigned to it, in such a way that two sets get assigned the same cardinal if and only if there is a bijection between them.</p>
<p>But there is a natural way to identify t... |
2,898,616 | <p>My Math teacher told me that $\frac{dx}{dy} = \frac{1}{\frac{dy}{dx}}$. I asked for a proof and he gave me the result by using the chain rule. I don't understand why this is the case for every function because the function $y=x^2$ doesn't have an inverse function. If this is the case then why is $\frac{dx}{dy}$ no... | J.G. | 56,861 | <p>The branch choice doesn't matter, since if $x=-y^{1/2}$ then $\frac{dx}{dy}=-\frac{1}{2}y^{-1/2}=\frac{1}{2x}$. A general proof that $\frac{dx}{dy}=\frac{1}{\frac{dy}{dx}}$ is easy if you write each side as a limit by definition.</p>
|
4,417,576 | <p>Say we divide 1858.54 cm by 5 -- where 5 is an exact number. We get the quotient 371.708</p>
<p>According to the rules of significant figures, we would answer write 371.708 cm, since "1858.54" is 6 significant figures, and 5, being an exact number doesn't count.</p>
<p>My question is, how do you make sense... | David K | 139,123 | <p>According to you, significant figures don't matter. What matters is whether you have hundredths or thousandths. (Or, presumably, tenths or hundredths, etc.)</p>
<p>So let's apply your logic in the other direction.</p>
<p>Let's try a simpler example.
Take a measurement of <span class="math-container">$102.3$</span> a... |
119,492 | <p>First I define a function, just a sum of a few sin waves at different angular frequencies:</p>
<pre><code>ubdat = 50;
ws = 10*{2, 5, 10, 20, 40}
fn = Table[Sum[Sin[w*x], {w, ws}], {x, 0, ubdat, .001}];
pts = Length@fn
ListPlot[fn, Joined -> True, PlotRange -> {{0, 1000}, All}]
{20, 50, 100, 200, 400}
</code>... | Community | -1 | <p>I choose a simple Butterworth filter and show, based on the spectra, how it works. Cutoff frequency and filter order can be set.</p>
<pre><code>ubdat = 50;
ws = 10*{2, 5, 10, 20, 40};
fn = Table[Sum[Sin[w*x], {w, ws}], {x, 0, ubdat, 0.001}];
Manipulate[
filter = ButterworthFilterModel[{"Lowpass", filterOrder, cut... |
168,926 | <p>How much of ring theory arises simply from applying group-theoretic results to the additive structure, or semigroup/monoid-theoretic results to the multiplicative structure? To be more precise, can anyone describe <em>how much</em> rigidity is imposed on the structure of rings by the distributive axiom which "connec... | Martin Brandenburg | 1,650 | <blockquote>
<p>How much of ring theory arises simply from applying group-theoretic results to the additive structure, or semigroup/monoid-theoretic results to the multiplicative structure?</p>
</blockquote>
<p>Almost nothing. And in any case this would not be called ring theory. For example, a finite subgroup of th... |
2,845,033 | <p>Empirically this equality holds:</p>
<p>$\gcd(10,8) = 2 $ and $\gcd(\operatorname{mod}(10,8),8) = \gcd(2,8) = 2 $</p>
<p>$\gcd(18,9) = 9 $ and $\gcd( \operatorname{mod}(18,9),9) = \gcd(0,9) = 9 $</p>
<p>I am stuck on how to prove it,though , and do not understand why this holds true.</p>
| Robert Lewis | 67,071 | <p>For any planar system of the form</p>
<p>$\dot x = X(x, y), \tag 1$</p>
<p>$\dot y = Y(x, y), \tag 2$</p>
<p>the <a href="https://en.wikipedia.org/wiki/Bendixson%E2%80%93Dulac_theorem" rel="nofollow noreferrer">Bendixson-Dulac criterion</a> states that there are no periodic orbits in a region $\Omega$ where</p>
... |
856,556 | <p>Let $f_n(x)=\frac{1}{n}\chi_{[0,n]}(x)$, $x\in\mathbb{R}$, $n\in\mathbb{N}$ and $\chi$ is the characteristic/indicator function. Now it is clear that $f_n\rightarrow 0$, but in the text I am using it says we can't apply the Dominated Convergence theorem as there is no function to dominate $f_n$</p>
<p>I have troubl... | Michael Hardy | 11,667 | <p>The smallest thing that could possibly dominate the functions $f_n$, $n=1,2,3,\ldots$ is the function
$$
x\mapsto g(x) = \sup_n |f_n(x)| = \begin{cases} \dfrac{1}{\lceil x \rceil} & \text{if }x>0, \\[10pt] 0 & \text{if }x<0. \end{cases}
$$
The problem is that
$$
\int_{[0,\infty)} g(x)\,dx = \sum_{n=1}^... |
2,493,802 | <p>If $A \subset \mathbb R$, we have the distance between a point in $\mathbb R$ and $A$ as</p>
<p>$$d(x,A)=\inf\{|x-a| \mid a \in A\}.$$</p>
<p><strong>For $\varepsilon>0$, if we define $A(\varepsilon) = \{x \in \mathbb R \mid d(x,A) < \varepsilon \}$, how do we show this set is open?</strong></p>
<hr>
<p>I ... | qualcuno | 362,866 | <p>If you know that a function is continous if and only if preimages of open sets are open, you can just prove that $d(x,A)$ is continous, and then observe that if $f(x) = d(x,A)$,</p>
<p>$$
A(\epsilon) = f^{-1}(-\infty, \epsilon)
$$</p>
<p>which is in fact the preimage via a continous function of the open set $(-\in... |
3,930,373 | <p>Hi so im working on a question about finding the <span class="math-container">$rank(A)$</span> and the <span class="math-container">$dim(Ker(A))$</span> of a 7x5 Matrix. Without being given an actual matrix to work from.</p>
<p>I have been told that that the homogeneous equation <span class="math-container">$A\vec x... | Barry Cipra | 86,747 | <p>In <em>complex</em> analysis, the assertion <span class="math-container">$\lim_{z\to0}\sqrt{-z^2}=0$</span> is defensibly true even if you don't bother to specify the sign of the square root for nonzero numbers. But as a <em>real</em>-valued function of a <em>real</em> variable, the expression <span class="math-cont... |
3,259,910 | <p>My question is: if G:= Group of polynomials under addition with elements from <span class="math-container">$\{0,1,2,3,4,5,6,7,8,9\}$</span>
<span class="math-container">$f(x)=7x^2+5^x+4$</span> and <span class="math-container">$g(x)=4x^2+8^x+6$</span> ,<span class="math-container">$f(x)+g(x)=x^2+3x$</span>
what ... | Key Flex | 568,718 | <p>Let <span class="math-container">$f(x)=x^3+bx^2+cx+1$</span></p>
<p><span class="math-container">$f(0)=1>0$</span></p>
<p><span class="math-container">$f(-1)=(b-c)<0$</span></p>
<p>So, <span class="math-container">$\alpha$</span> lies between <span class="math-container">$f(0)$</span> and <span class="math-... |
1,899,740 | <p>Let $(\Omega,\Sigma,\nu)$ be a measurable space with $\nu$ a positive $\sigma$-additive measure on $\Sigma$ and le $f:\Omega\to \mathbb{R}$ be a function such that,
$$
\int_{\Omega}f(x)d\nu(x)=0
$$
Does this imply that $f=0\quad \nu-\ a.e.$ ?</p>
<p>Any hint will be appreciated :) thank you for your time !</p>
| Ant | 66,711 | <p>How about $ f(x)= x $ on $\Omega = (-1,1)$ and $\nu $ the lebesgue measure?</p>
|
4,541,392 | <p>I have an integral that looks like the following:
<span class="math-container">$$
\int_a^b \frac{1}{\left(1 + cx^2\right)^{3/2}} \mathrm{d}x
$$</span>
I have seen a method of solving it being to substitute <span class="math-container">$x = \frac{\mathrm{tan}(u)}{\sqrt{c}}$</span>; however, this seems somewhat sloppy... | math and physics forever | 879,009 | <p>let <span class="math-container">${(1+cx^2)}^{-1/2}=u \rightarrow 1 $</span></p>
<p>so you have <span class="math-container">$du= \frac{-1}{2} \frac{2x}{(1+cx^2)^{-3/2}}dx$</span></p>
<p>from 1</p>
<p><span class="math-container">$x=\sqrt{\frac{1}{u^2} -c}$</span></p>
<p>so the new integrand is <span class="math-co... |
163,654 | <p>Do You know any successful applications of the geometric group theory in the number theory? GTG is my main field of interest and I would love to use it to prove new facts in the number theory.</p>
| HenrikRüping | 3,969 | <p>Here is a counterexample, which is probably not "nice". Let $X$ be the Warsaw-circle. Let $X_n$ be the obtained from the Warsaw-circle by thickening the limit inverval by $1/n$. The intersection of all the $X_n$'s is the Warsaw-circle, and its first homology vanishes. </p>
<p>Each $X_n$ is homotopy equivalent to $S... |
119,527 | <p>I'm trying to prove the following:</p>
<blockquote>
<p>Let $(a_n)_{n\in\mathbb{N}}$ and $(b_n)_{n\in\mathbb{N}}$ be two sequences such that $(a_n)_{n\in\mathbb{N}}$ converges and $(b_n)_{n\in\mathbb{N}}$ is bounded. If $a=\lim_{n\to\infty} a_n$, prove that $$\limsup_{n\to\infty} (a_n+b_n) = a +\limsup_{n\to\infty... | anon | 11,763 | <p>Show that</p>
<p>$$\forall\epsilon>0:\quad \limsup\, (a_n+b_n)\ge \limsup\, \big((a-\epsilon)+b_n\big)=(a-\epsilon)+\limsup\,b_n $$</p>
|
1,169,591 | <p>For all finite vector spaces and all linear transformations to/from those spaces, <strong>how can you prove/show from the definition of a linear transformation</strong> that all linear transformations can be calculated using a matrix.</p>
| Mathemagician1234 | 7,012 | <p>The most general way to show this is to show the dual space of linear transformations on a finite dimensional vector space V over a field F is isomorphic to the vector space of m x n matrices with coeffecients in the same field F. However, although the proof isn't difficult,it needs considerably more machinery then ... |
3,251,928 | <p><span class="math-container">$$\frac{1}{a(a-b)(a-c)} + \frac{1}{b(b-a)(b-c)} + \frac{1}{c(c-a)(c-b)} $$</span></p>
<p>I tried to get everything to the same denominator, and then simplify numerators first but it is very complicated and long if I just use brute force, to multiply all the expressions given from the pr... | lab bhattacharjee | 33,337 | <p>The common denominator is <span class="math-container">$$-abc(a-b)(b-c)(c-a)$$</span></p>
<p>The numerator is</p>
<p><span class="math-container">$$bc(b-c)+ca(c-a)+ab(a-b)=bc(b-c)+c^2a-ca^2+a^2b-ab^2$$</span></p>
<p><span class="math-container">$$=(b-c)(bc+a^2-a(b+c))$$</span></p>
<p><span class="math-container"... |
2,248,303 | <p>Evaluate the integral $ \int_{0}^{\infty} {\frac{\sin ^3 (x)}{x} }\:dx$ </p>
| Angina Seng | 436,618 | <p>$$4\sin^3 x=3\sin x-\sin 3x$$
Your integral is
$$\frac34\int_0^\infty\frac{\sin x}x\,dx
-\frac14\int_0^\infty\frac{\sin 3x}x\,dx.
$$</p>
|
2,248,303 | <p>Evaluate the integral $ \int_{0}^{\infty} {\frac{\sin ^3 (x)}{x} }\:dx$ </p>
| Bérénice | 317,086 | <p>We can linearise $\sin^3(x)$ :
$${\displaystyle\int}\dfrac{\sin^3\left(x\right)}{x}\,\mathrm{d}x=-\class{steps-node}{\cssId{steps-node-1}{\dfrac{1}{4}}}{\displaystyle\int}\dfrac{\sin\left(3x\right)-3\sin\left(x\right)}{x}\,\mathrm{d}x=-\class{steps-node}{\cssId{steps-node-1}{\dfrac{1}{4}}}{\displaystyle\int}\dfrac{\... |
3,148,623 | <p>I want to find all prime ideals in <span class="math-container">$\mathbb{Z}/n\mathbb{Z}$</span> where <span class="math-container">$n>1$</span>.</p>
<p>I think I have to use the following theorem (because they asked me to prove it right before this exercise, which wasn't too complicated):</p>
<blockquote>
<p>... | Bargabbiati | 352,078 | <p>An ideal <span class="math-container">$I$</span> in a commutative ring <span class="math-container">$A$</span> is prime if and only if <span class="math-container">$A/I$</span> is an integral domain; <span class="math-container">$\mathbb Z/n \mathbb Z$</span> is a finite ring, so a the quotient by a prime ideal has ... |
549,438 | <p>Need to take the limit:
$$\lim_{x \to 1}\left(\frac{x}{x-1}-\frac{1}{\ln(x)}\right) = \lim_{x \to 1}\left(\frac{x\cdot \ln(x)-x+1}{(x-1)\cdot \ln(x)}\right)=(0/0)$$
Now I can use L'Hospital's rule:
$$\lim_{x \to 1}\left(\frac{1\cdot \ln(x)+x\cdot \frac{1}{x}-1}{1\cdot \ln(x)+(x-1)\cdot\frac{1}{x}}\right)= \lim_{x \t... | pi37 | 46,271 | <p>You can apply L'Hopital's rule again on the new limit:
$$
\lim_{x \to 1}\frac{x\cdot ln(x)}{x-1+x\cdot ln(x)}=\lim_{x\to 1}\frac{ln(x)+1}{1+ln(x)+1}=\frac{0+1}{0+1+1}=\frac{1}{2}
$$</p>
<hr>
<p>Alternatively, note that
$$
\lim_{x \to 1}\frac{x\cdot ln(x)}{x-1+x\cdot ln(x)}=\lim_{x\to 1}\displaystyle\frac{1}{1+\fra... |
2,070,353 | <p>Let there be two arbitrary functions $f$ and $g$, and let $g \circ f$ be their composition.</p>
<ol>
<li><p>Suppose $g \circ f$ is one-one. Then prove that this implies $f$ is also one-one.</p></li>
<li><p>Suppose $g \circ f$ is onto. Then prove that this implies $g$ is also onto.</p></li>
</ol>
<p>I tried to do b... | Community | -1 | <p>Find out the common difference denoted by $d_1$. </p>
<p>Let the first term be represented by $A$ and let the last term be represented by $L$. </p>
<p>Calculate number of terms $T$ using the formula :
$T = \frac{L-A}{d_1} + 1$. We thus get $T = n-2$ using $L = 2n+1$ and $A= 7$. </p>
<hr>
<p>As to why the... |
3,547,768 | <p>Given a function <span class="math-container">$f:\mathbb{R}\rightarrow\mathbb{R}$</span> which is differentiable and given that <span class="math-container">$f(0)=2$</span> and <span class="math-container">$(e^x + 1)f'(x) = e^x f(x)$</span> for every <span class="math-container">$x \in \mathbb{R}$</span>, prove that... | user247327 | 247,327 | <p>That's a pretty straightforward "differential equation". From <span class="math-container">$(e^x+ 1)f'(x)= (e^x+ 1)\frac{df}{dx}= e^xf(x)$</span> we have <span class="math-container">$\frac{df}{f}= \frac{e^x}{e^x+ 1} dx$</span>.</p>
<p>Now integrate both sides. On the left we have <span class="math-container">$ln... |
2,657,112 | <p>I am currently reading the paper "PRIMES is in p" and have come across some notation that I don't quite understand in this following sentence</p>
<blockquote>
<p>Consider a prime $q$ that is a factor of $n$ and let $q^k || n$. Then...</p>
</blockquote>
<p>What does the notation $q^k || n$ mean here?</p>
<p>The ... | Randall | 464,495 | <p>Typically, that $q^k$ divides $n$ but no higher power of $q$ does.</p>
|
1,755,107 | <p>Let $(X,d)$ be a metric space and let $A \subseteq X$. We define the distance from a point $x \in X$ to $A$ by $d(x,A)= \inf \{ d(x,a) : a \in A \} $.</p>
<p>What will be the value of $d(x, \emptyset )$? I am confused between $+ \infty $ and $- \infty$. Also, is it possible to find $ \min \{ d(x,a) : a \in \emptyse... | Jack | 206,560 | <p>If $A$ is empty, then $\{ d(x,a) : a \in A \}$ is empty. Then inf$\{\emptyset \} = \infty$. </p>
<p>The proof that inf$\{\emptyset \} = \infty$ can be found in this post:</p>
<p><a href="https://math.stackexchange.com/a/432308">https://math.stackexchange.com/a/432308</a></p>
|
851,112 | <p>My goal is to obtain a reasonable approximation of the Gini index of a company (UBS).
I need to obtain an estimate of the salaries distribution from publicly available data:</p>
<ul>
<li>Nuber of employees=60205 </li>
<li>total compensation paid=15.182E9 CHF</li>
<li>min salary=50000 CHF</li>
<li>median salary = 10... | Alijah Ahmed | 124,032 | <p>Starting from the expression which you have just derived, we can remain in the complex exponential domain and simplify further in this domain as follows:-
$$\displaystyle \frac{(e^{i\theta}+e^{i\phi})(e^{i(\theta+\phi)}-e^{i0})}{(e^{i\theta}-e^{i\phi})(e^{i(\theta+\phi)}+e^{i0})}=\frac{e^{i(2\theta+\phi)}-e^{i\theta... |
494,141 | <p>I'm slightly confused about representations of Simple functions. I take it that simple functions are of the form $\phi(x) = \sum_{k=1}^N a_k \chi_{A_k}(x)$ where $\chi$ is the indicator function and that $A_k$'s are measurable sets.</p>
<p>Now on building the theory of integration using simple functions, it appears... | Community | -1 | <p>If there are two sets $A_1$ and $A_2$ such that $A_1 \cap A_2 \ne \emptyset$, we may simply rewrite the sum to include the terms</p>
<p>$$a_1 \chi_{A_1 \setminus A_2} + (a_1 + a_2) \chi_{A_2 \cap A_1} + a_2 \chi_{A_2 \setminus A_1}$$</p>
<p>Now it <em>is</em> true that intersections and differences of measurable s... |
297,686 | <p>I want to show the following inequality is true: $|\int_{a}^{b}fg|^2\leq \int_{a}^{b}f^2\int_{a}^{b}g^2$. </p>
<p>My first thought was to use the tagged partition definition of a Riemann integral combined with the Schwarz inequality.</p>
<p>So let $P=\{a=x_0<x_1<...<x_n=b\}$ and $S_p=\{t_1,...,t_n\}$ whe... | Alex Youcis | 16,497 | <p>This is the $p=q=2$ case of Holder's inequality, which <a href="http://planetmath.org/ProofOfHolderInequality.html" rel="nofollow">has a very simple proof</a>.</p>
|
297,686 | <p>I want to show the following inequality is true: $|\int_{a}^{b}fg|^2\leq \int_{a}^{b}f^2\int_{a}^{b}g^2$. </p>
<p>My first thought was to use the tagged partition definition of a Riemann integral combined with the Schwarz inequality.</p>
<p>So let $P=\{a=x_0<x_1<...<x_n=b\}$ and $S_p=\{t_1,...,t_n\}$ whe... | Robert Israel | 8,508 | <p>You seem to know the "sum" version of the Cauchy-Schwarz inequality. The standard proof of the "integral" version is exactly the same as that for the "sum" version: just change sums to integrals. It's really a theorem about inner products.</p>
|
2,107,164 | <p>Let $G$ be gorup and $H$ be normal subgroup of $G$.</p>
<p>Suppose that $H \cong \mathbb{Z}$ ($\mathbb{Z}$:integer) and $G/H \cong \mathbb{Z}/ n \mathbb{Z} (\mathbb{Z} \ni n \geq 2 )$.</p>
<p>Then I'm stuck in next problems.</p>
<p>(1)If $n$ is odd number, G is abelian group.</p>
<p>(2)Classfy the group G up to... | Servaes | 30,382 | <p><strong>Hint:</strong> Classify the homomorphisms $\Bbb{Z}/n\Bbb{Z}\ \longrightarrow\ \operatorname{Aut}(\Bbb{Z})$.</p>
|
4,631,014 | <blockquote>
<p>Finding range of function</p>
</blockquote>
<blockquote>
<p><span class="math-container">$\displaystyle f(x)=\cos(x)\sin(x)+\cos(x)\sqrt{\sin^2(x)+\sin^2(\alpha)}$</span></p>
</blockquote>
<p>I have use Algebric inequality</p>
<p><span class="math-container">$\displaystyle -(a^2+b^2)\leq 2ab\leq (a^2+b^... | Anne Bauval | 386,889 | <p>Let <span class="math-container">$a=|\sin\alpha|.$</span></p>
<p>First notice that (since <span class="math-container">$|\sin x|\le\sqrt{a^2+\sin^2x}$</span>) when the maximum value <span class="math-container">$f_{\max}$</span> is attained the <span class="math-container">$\sin$</span> and <span class="math-contain... |
227,732 | <p>Are there any general relations between the eigenvalues of a matrix $M$ and those of $M^2$?</p>
| Muphrid | 45,296 | <p>Clearly, if $\underline M(u) = \lambda u$, then $\underline M^2(u) = \lambda^2 u$.</p>
<p>However, $\underline M^2$ may have eigenvectors that weren't eigenvectors of the original linear operator. Rotation operators come to mind. A 90 degree rotation has no eigenvectors*, but chaining two of these together makes ... |
227,732 | <p>Are there any general relations between the eigenvalues of a matrix $M$ and those of $M^2$?</p>
| copper.hat | 27,978 | <p>A useful way to view an eigenspace is that the matrix $M$ just becomes multiplication on the eigenspace.</p>
<p>So, suppose $Mv = \lambda v$. Applying $M$ again just results in another multiplication by $\lambda$, as in $M(M v) = M (\lambda v) = \lambda M v = \lambda^2 v$. Repeating gives $M^kv = \lambda^k v$.</p>
... |
406,252 | <p>How to define the amount of additions. E.g. $1+2+3+4+5+6+7+8+9$</p>
<p>Are there $9$ additions, because of the nine numbers that are added together.
Or can you also say that there are $8$ additions, because there are only $8$ '$+$' signs.</p>
| Tony Stark | 80,093 | <p>$$
\sum_{i=1}^n a_i = a_1 + \cdots + a_n
$$
There are $n$ terms in the addition but what if there are minuses? Hence counting terms are kinda less ambiguous. Also we can do:
$$
\sum_{i=1}^n a_i = 0+ a_1 + \cdots + a_n
$$
and we have $n$ additions now.</p>
|
432,291 | <p>This question is related to <a href="https://mathoverflow.net/questions/421450/homotopy-type-theory-how-to-disprove-that-0-operatornamesucc0-in-the-ty">Homotopy type theory : how to disprove that <span class="math-container">$0=\mathrm{succ}(0)$</span>
in the type <span class="math-container">$\mathbb N$</span></a>.... | L. Garde | 110,166 | <p>As Daniel explained, the induction principle associated with <span class="math-container">$\mathbb{N}$</span> allows you, by definition, to define <span class="math-container">$code$</span> in this way, even if <span class="math-container">$0$</span> and <span class="math-container">$succ(0)$</span> were judgmentall... |
3,687,521 | <p>Recently I came accross this integral :
<span class="math-container">$$ \int \:\frac{1}{\sqrt[3]{\left(1+x\right)^2\left(1-x\right)^4}}dx $$</span>
I would evaluate it like this, first start by the substitution:
<span class="math-container">$$ x=\cos(2u) $$</span>
<span class="math-container">$$ dx=-2\sin(2u)du $$</... | Claude Leibovici | 82,404 | <p>Another way (more complex than @Quanto's one)
<span class="math-container">$$I=\int\dfrac{dx}{\sqrt[3]{\left(1-x\right)^4\left(x+1\right)^2}}=\int\dfrac{dx}{\left(x-1\right)^\frac{4}{3}\left(x+1\right)^\frac{2}{3}}$$</span>
<span class="math-container">$$u=\dfrac{1}{\sqrt[3]{x-1}}\implies I=-3\int\dfrac{du}{\left(\f... |
6,056 | <p>Are there any good places for me to sell off my mathematics books online especially Springer and Dover books?</p>
<p>(I thought perhaps this was off topic, but then I thought everybody in math probably has the problem of having bought pricey books that they would prefer to get some money back for, at one point or a... | jericson | 1,352 | <p>You can also list your books on amazon to sell rather than have amazon buy them from you</p>
|
6,056 | <p>Are there any good places for me to sell off my mathematics books online especially Springer and Dover books?</p>
<p>(I thought perhaps this was off topic, but then I thought everybody in math probably has the problem of having bought pricey books that they would prefer to get some money back for, at one point or a... | Mathemagician1234 | 7,012 | <p>I've been a seller on EBAY for years and they are out of control on fees. Also,people go to EBAY looking for a deal and you probably won't be able to make a good return on serious currently used textbooks on EBAY.You've got a much better chance of a good profit on Amazon.
<strong><em>DON'T try and sell your books at... |
23,980 | <p>I am calculating huge data files with an external program. I would like then to import the data into Mathematica for analysis. The files are 2 columns and up to many millions of rows. </p>
<p>So for small data files I have just been using:</p>
<pre><code>dataTable = Import["data.txt", "Table"];
</code></pre>
... | Michael Hale | 5,568 | <p>You'll want to use <a href="http://en.wikipedia.org/wiki/Reservoir_sampling" rel="nofollow">Algorithm R for reservoir sampling</a> to get a truly random sample (as opposed to periodic) with only one scan of the input and no need to know the length of the input in advance. Basically, you take the first items and then... |
241,138 | <p>I think I didn't pay attention to uniform distributions because they're too easy.
So I have this problem</p>
<blockquote>
<ol>
<li>It takes a professor a random time between 20 and 27 minutes to walk from his home to school every day. If he has a class at 9.00 a.m. and he leaves home at 8.37 a.m., find the prob... | Mike | 17,976 | <p>Your problem is $e^{4x}$ is part of your homogeneous solution, which explains why you get $0$ when you try $Ae^{4x}$ as a particular solution.</p>
<p>I would try $Ax^2e^{4x}$ as a particular solution.</p>
<p>Just to show it works, let me show you another way to solve the problem. Let</p>
<p>$$z=y'-4y,z'=y''-4y'$... |
3,401,055 | <blockquote>
<p>If the distance of any point <span class="math-container">$(x,y)$</span> from the origin is defined as <span class="math-container">$d(x,y)=max\{|x|,|y|\}$</span> where <span class="math-container">$|.|$</span> represents the absolute value function, and <span class="math-container">$d(x,y)=a$</span> wh... | mechanodroid | 144,766 | <p>I'm assuming <span class="math-container">$V$</span> is a complex vector space.</p>
<p>Notice that from <span class="math-container">$\langle Av,v\rangle \le -\|v\|^2, \forall v \in V$</span> follows that <span class="math-container">$A$</span> is hermitian. Indeed, we can write <span class="math-container">$A = B+... |
1,564,750 | <p>Imagine that your country's postal system only issues 2 cent and 5 cent stamps. Prove that it possible to pay for postage using only these stamps for any amount n cents, where n is at least 4.</p>
<hr>
<p>My attempt (using Strong induction, I know we can use induction but since they say you can can apply strong in... | Christian Blatter | 1,303 | <p>You can do all even amounts, and you can do all odd amounts $\geq5$. It follows that the only positive amounts you cannot do are $1$ and $3$.</p>
|
1,836,236 | <p>For which $a$ and $b$ is this matrix diagonalizable?</p>
<p>$$A=\begin{pmatrix} a & 0 & b \\ 0 & b & 0 \\ b & 0 & a \end{pmatrix}$$</p>
<p>How to get those $a$ and $b$? I calculated eigenvalues and eigenvectors, but don't know what to do next?</p>
| user115350 | 334,306 | <p>Matrix A has 3 eigenvalues a-b, a+b and b. In order for them to be distinguishable, following conditions should be met. $$b \ne 0$$ $$a \ne 0$$ $$a \ne 2b$$</p>
|
3,453,626 | <p>I was playing around with the concepts of injectivity, surjectivity, and bijectivity recently. I used these three Wikipedia articles as references:</p>
<p><a href="https://en.wikipedia.org/wiki/Injective_function" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Injective_function</a></p>
<p><a href="https... | Dr. Sonnhard Graubner | 175,066 | <p>Use that <span class="math-container">$$m_a=\frac{1}{2}\sqrt{2(b^2+c^2)-a^2}$$</span> etc.</p>
|
3,453,626 | <p>I was playing around with the concepts of injectivity, surjectivity, and bijectivity recently. I used these three Wikipedia articles as references:</p>
<p><a href="https://en.wikipedia.org/wiki/Injective_function" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Injective_function</a></p>
<p><a href="https... | Quanto | 686,284 | <p>Note,</p>
<p><span class="math-container">$$m_a^4+m_b^4+m_c^4 = (m_a^2+m_b^2+m_c^2 )^2 - 2(m_a^2m_b^2+m_b^2m_c^2+m_c^2m_a^2) $$</span></p>
<p>Then, evaluate,
<span class="math-container">$$m_a^2+m_b^2+m_c^2 = \frac{2b^2+2c^2-a^2}{4} +\frac{2c^2+2a^2-b^2}{4} +\frac{2a^2+2b^2-c^2}{4} = \frac34 (a^2+b^2+c^2)$$</span>... |
2,646,749 | <p>I'm solving the following problem:</p>
<blockquote>
<p>Identify all maximal ideals in the ring $\mathbb R[x]/(x^2-3x+2)$.</p>
</blockquote>
<p>I want to solve it in two ways.</p>
<p><strong>First method.</strong> Consider the surjective quotient homomorphism $\mathbb R[x]\rightarrow \mathbb R[x]/(x^2-3x+2)$ wit... | johnnycrab | 171,304 | <p>For the first method, it is the usual proof that for any field $K$ and $a \in K$ we have $K[X]/(X-a) \cong K$, as by the third isomorphism theorem $(K[X]/I)/(\mathfrak a/I) \cong K[X]/\mathfrak a$ for $I \subset \mathfrak a \subset K[X]$. For this define the ring morphism:</p>
<p>$$K[X] \to K, f \mapsto f(a).$$</p>... |
873,755 | <p>Let $f(x) = x^{10}+5x^9-8x^8+7x^7-x^6-12x^5+4x^4-8x^3+12x^2-5x-5. $</p>
<p>Without using long division (which would be horribly nasty!), find the remainder when $f(x)$ is divided by $x^2-1$.</p>
<p>I'm not sure how to do this, as the only way I know of dividing polynomials other than long division is synthetic div... | ReverseFlow | 109,908 | <p>Plug in $1$ and $-1$ to get two values of $r(x)$ which is linear. From there you can get what $a,b$ are in $ax+b.$ </p>
<p>Since
$$f(x)=g(x)(x+1)(x-1)+r(x)$$</p>
<p>we have</p>
<p>$$ f(1)=g(1)(1+1)(1-1)+r(1)=r(1)=-10$$
$$ f(-1)=g(1)(-1+1)(-1-1)+r(-1)=r(-1)=16$$</p>
<p>We know the remainder is of degree $1$, so<... |
2,174 | <p>As a teenager I was given this problem which took me a few years to solve. I'd like to know if this hae ever been published. When I presented my solution I was told that it was similar to one of several he had seen.</p>
<p>The problem:</p>
<p>For an <span class="math-container">$n$</span> dimensional space, develo... | Doug Chatham | 273 | <p>Section 3 of <a href="http://www.macalester.edu/~bressoud/pub/Art_of_Counting/Art_of_Counting.pdf" rel="noreferrer">http://www.macalester.edu/~bressoud/pub/Art_of_Counting/Art_of_Counting.pdf</a> discusses the problem for $n=3$ and poses the problem of generalizing to higher dimensions. This article states that the... |
2,706,834 | <p>I have the next exercise: </p>
<p>Let $E$ be a normed space, $n\in \mathbb{N}$ and $\{x_1,\dots,x_n\}\in E$ a family of independent vectors. Show that for all $\alpha_1,\dots,\alpha_n\in \mathbb{K}$ there is a linear form $T\in E'$ such that $T(x_i)=\alpha_i$, for all $i=1,\dots,n$.</p>
<p>My solution:</p>
<p>I u... | Aweygan | 234,668 | <p>The result you are trying to prove requires $\alpha_1,\ldots,\alpha_n$ be arbitrary, but at the end of your proof you have $\alpha_i=\|x_i\|$. Thus your solution is incorrect.</p>
<p>In fact, the result your using (the corollary of the Hahn-Banach theorem), while it is a great one, isn't strong enough to show what... |
2,784,531 | <p>Let <span class="math-container">$X = (X_1, X_2, \dots, X_n)$</span> be jointly Gaussian with mean vector <span class="math-container">$\mu$</span> and covariance matrix <span class="math-container">$\Sigma$</span>. Let <span class="math-container">$S$</span> be their sum.</p>
<p>I know that the distribution of eac... | Maxim | 491,644 | <p>The distribution of <span class="math-container">$(\boldsymbol X | S = s)$</span> is still jointly normal but degenerate. Let <span class="math-container">$\boldsymbol T = (1, 1, \dots, 1)^t$</span> and let <span class="math-container">$\boldsymbol X$</span> and <span class="math-container">$\boldsymbol \mu$</span> ... |
147,441 | <p>I've just been looking through my Linear Algebra notes recently, and while revising the topic of change of basis matrices I've been trying something:</p>
<p>"Suppose that our coordinates are $x$ in the standard basis and $y$ in a different basis, so that $x = Fy$, where $F$ is our change of basis matrix, then any m... | Christian Blatter | 1,303 | <p>Your statements a) – d) are correct as stated.</p>
<p>Concerning the sentence "Also, just to check, are the entries in the matrix $F^{-1}AF$ still written in terms of the standard basis?", one can say the following: The entries in this matrix are just numbers. The given matrix $A$ describes a certain linear map act... |
4,285,189 | <p>I am aware of the following identity</p>
<p><span class="math-container">$\det\begin{bmatrix}A & B\\ C & D\end{bmatrix} = \det(A)\det(D - CA^{-1}B)$</span></p>
<p>When <span class="math-container">$A = D$</span> and <span class="math-container">$B = C$</span> and when <span class="math-container">$AB = BA$</... | user1551 | 1,551 | <p>Your second attempt almost works. You may use the fact that <span class="math-container">$\det\pmatrix{A&B\\ C&D}=\det(AD-BC)$</span> whenever <span class="math-container">$A,B,C,D$</span> are square matrices of the same sizes and <span class="math-container">$CD=DC$</span>:
<span class="math-container">$$
\... |
4,179,720 | <p>I'm starting to study triple integrals. In general, I have been doing problems which require me to sketch the projection on the <span class="math-container">$xy$</span> plane so I can figure out the boundaries for <span class="math-container">$x$</span> and <span class="math-container">$y$</span>. For example, I had... | true blue anil | 22,388 | <p>Same thing, possibly shortest expression, <span class="math-container">$\frac{8!!}{8!} = \frac1{105}$</span></p>
<p>(Ladies sit 1 by 1 in available seats, reserving opposite end for spouse)</p>
|
267,121 | <h2>UPDATE on FINAL RESULT</h2>
<p>Thanks to @SquareOne effort I generated higher-resolution videos with smoothing transitions that can be seen here:</p>
<ul>
<li><p><a href="https://www.linkedin.com/feed/update/urn:li:activity:6926902980323512320/" rel="noreferrer">https://www.linkedin.com/feed/update/urn:li:activity:... | SquareOne | 19,960 | <p><a href="https://i.stack.imgur.com/TFygI.gif" rel="noreferrer"><img src="https://i.stack.imgur.com/TFygI.gif" alt="enter image description here" /></a></p>
<p><a href="https://drive.google.com/file/d/1eaVxxExRI-b14jKKJmaqEQIqZVZuv9JM/view?usp=sharing" rel="noreferrer">Here</a> is a link to a mp4 video (400 pixel wid... |
80,787 | <p>Hello,</p>
<p>I am very new to the field of approximation theory, and since
an extended search on the Internet did not provide answers for
two rather basic questions, I decided to ask them here. </p>
<p>1) From my understanding upper bounds for</p>
<p>$$ \inf_{q} \int_{-1}^{1} |f(x) - q(x)|^{2p} dt $$</p>
<p>wi... | Anatoly Kochubei | 12,205 | <p>Results on the $L^p$-approximation theory can be found in the basic books on the subject:</p>
<p>Timan, A.F. Theory of approximation of functions of a real variable. Oxford: Pergamon Press. 1963.</p>
<p>Achieser, N.I. Theory of approximation. New York: Frederick Ungar Publishing Co. (1956). </p>
|
2,711,189 | <p>Is their anyway you can integrate a wedge of cylinder (picture below)
not by adding up the parts cut in parallel lines but
by adding up parts cut in pivotal lines to the center of the half circle?</p>
<p><a href="https://i.stack.imgur.com/BrucD.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Br... | Cye Waldman | 424,641 | <p>The answer to your question is yes, but it's a little trickier than than when the cuts are parallel. Let's consider in the simplest terms</p>
<p>$$V=\int dV$$</p>
<p>If we wish to reduce this to an integral over $\theta$ only, then we need to reduce $dV$ in terms of the area. In order to do that we need recourse t... |
1,088,687 | <p>I have this example here:</p>
<p>$\lim_{n \to \infty} \frac{n^{2}+1}{n^{2}+n+1} = 1$</p>
<p>so I have to prove that, $\forall \epsilon > 0, \exists n_{0}\in N, \forall n(n \in N) \geq n_{0} \Rightarrow |\frac{n^{2}+1}{n^{2}+n+1} - 1| < \epsilon$</p>
<p>I have done in these steps <br/> <br/>
$|\frac{n^{2}+... | paw88789 | 147,810 | <p>Sometimes you don't want to be too precise with your estimates. A loose estimate with a nice form that gets the job done is often easiest.</p>
<p>Here's a start:</p>
<p>$|\frac{n^2+1}{n^2+n+1}-1|=\frac{n}{n^2+n+1}<\frac{n}{n^2}=\frac1n$</p>
|
336,170 | <p>Create a rational function with vertical asymptotes $x=\pm1$ and oblique asymptote of $y=2x-3$ and a $y$-intercept of $4$.</p>
| Pedro | 23,350 | <p>You can go by parts, a la Jack.</p>
<p>First, you want your function to have vertical asymptotes at $x=1,-1$. So we really want $$\tag 1 \frac{1}{(1-x)(1+x)}$$</p>
<p>Now, as $x\to\infty$; this has a "oblique" asymptote of $0$, so we can just add the asymptote we want: $$\frac{1}{(1-x)(1+x)}+2x-3$$</p>
<p>Now, l... |
1,509,443 | <p>Prove that for $x\geq 1$ the following inequality is true: $$\dfrac{1}{x}\leq \ln(2x+1)-\ln(2x-1).$$</p>
<p>Can anyone help with this problem? I thought about some hours but any ideas</p>
| Thomas Andrews | 7,933 | <p>More generally, show that if $f(t)$ is a convex differentiable function on $[a,b]$ then $$(b-a)f\left(\frac{a+b}{2}\right) \leq \int_{a}^b f(t)\,dt$$</p>
<p>In the above case, $a=2x-1,b=2x+1, f(t)=\frac{1}{t}$.</p>
|
2,744,981 | <blockquote>
<p>Let $X$ be a Banach space. Let $T\in \mathbb{B}(X)$. If $T$ is an
isometry and not invertible, prove that $\sigma(T) = \overline{\mathbb{D}}$.</p>
</blockquote>
<p>I can show that $\sigma(T) \subset \overline{\mathbb{D}}$. Since $T$ is not invertible, then $0 \in \sigma(T)$. Suppose $\sigma(T) \neq... | Balaji sb | 213,498 | <p>Just completing the argument i.e., prove if <span class="math-container">$\lambda \in \partial \sigma(T)$</span> with <span class="math-container">$|\lambda|<1$</span> then <span class="math-container">$\lambda \in \sigma_{ap}(T)$</span> :</p>
<p><span class="math-container">$T-\lambda I$</span> is not invertible... |
1,927,599 | <p>In first-order logic, there is normally a formal distinction between <em>constants</em> and <em>variables</em>. Namely:</p>
<ul>
<li><p>A <em>constant symbol</em> is a $0$-ary function symbol in a language $\mathcal{L}$.</p></li>
<li><p>A <em>variable</em> is one of countably many special symbols used for first-or... | Stephen A. Meigs | 160,805 | <p>On a fundamental level (not relative to the theory under consideration) Bourbaki does not define constants as separate from variables. To him, a <i>constant in a theory</i> is just a variable that does not occur (freely) in axioms of the theory. Clearly if you don't fundamentally distinguish constants from variables... |
2,350,256 | <p>Solve:
$$y^{y\sqrt {y}}=(y\sqrt {y})^{y}$$</p>
<p>My Attempt:
$$y^{y\sqrt {y}}=(y\sqrt {y})^{y}$$
$$y^{y^{\dfrac {3}{2}}}=(y^{\dfrac {3}{2}})^y$$</p>
<p>How do I solve further?</p>
| Michael Rozenberg | 190,319 | <p>The domain is $y>0$.</p>
<p>We need to solve
$$y^{y\sqrt{y}}=y^{\frac{3}{2}y},$$
which gives $y=1$ or
$$y\sqrt{y}=\frac{3}{2}y,$$
which is $y=\frac{9}{4}$ and we get the answer: $\{1,\frac{9}{4}\}$.</p>
|
2,350,256 | <p>Solve:
$$y^{y\sqrt {y}}=(y\sqrt {y})^{y}$$</p>
<p>My Attempt:
$$y^{y\sqrt {y}}=(y\sqrt {y})^{y}$$
$$y^{y^{\dfrac {3}{2}}}=(y^{\dfrac {3}{2}})^y$$</p>
<p>How do I solve further?</p>
| Grown pains | 417,854 | <p>Use logarithm?</p>
<p>y = 1 is a solution.</p>
<p>Let $y \not = 1$.</p>
<p>\[
y^{3/2}\log y = 3y/2 \log y
\]
or dividing by $y \log y$,
\[
y^{1/2} = 3/2.
\]</p>
|
2,350,256 | <p>Solve:
$$y^{y\sqrt {y}}=(y\sqrt {y})^{y}$$</p>
<p>My Attempt:
$$y^{y\sqrt {y}}=(y\sqrt {y})^{y}$$
$$y^{y^{\dfrac {3}{2}}}=(y^{\dfrac {3}{2}})^y$$</p>
<p>How do I solve further?</p>
| Baris Aytekin | 450,397 | <p>$y^{y^3\over 2}={y^y} {y^{y\over 2}}$</p>
<p>$y^{y^3\over 2}={y^3y\over 2}$</p>
<p>$\frac{y^3} 2 =\frac {3y} 2 $</p>
<p>$2{y^3\over 2}=3y$</p>
<p>$4{y^3}=9y^2$</p>
<p>$4{y^3}-9{y^2}=0$</p>
<p>${y^2}(4y-9)=0$</p>
<p>$SS={\{0,0,{9\over 4}}\}$</p>
<p>However if you define $0^0$ as an indeterminate form the only... |
3,969,679 | <p><strong>The letters in the word GUMTREE and KOALA are rearranged to form a 12-letter word where KOALA appears precisely in order but not necessarily together. How many ways can this happen?</strong></p>
<p>So I attmepted it via this method:</p>
<p>Firstly arrange like this since KOALA must be in order but not necess... | ryang | 21,813 | <p>Probably simplest to frame the problem as arranging 12 letters, in which 2 are identical of one kind (EE in GUMTREE), and 5 are identical of another kind (XXXXX substituting for KOALA, which is treated as a black box since its letters have already been pre-arranged). So, <span class="math-container">$$\frac{12!}{2!5... |
3,969,679 | <p><strong>The letters in the word GUMTREE and KOALA are rearranged to form a 12-letter word where KOALA appears precisely in order but not necessarily together. How many ways can this happen?</strong></p>
<p>So I attmepted it via this method:</p>
<p>Firstly arrange like this since KOALA must be in order but not necess... | Will Orrick | 3,736 | <p>As I understand it, you are doing <span class="math-container">$7$</span> trials, one for each letter of GUMTREE. In the first trial, you decide which of the six slots, indicated by full stops, to use for G, in the second you decide which to use for U, and so on.</p>
<p>Here's the problem with that approach: let's s... |
3,969,679 | <p><strong>The letters in the word GUMTREE and KOALA are rearranged to form a 12-letter word where KOALA appears precisely in order but not necessarily together. How many ways can this happen?</strong></p>
<p>So I attmepted it via this method:</p>
<p>Firstly arrange like this since KOALA must be in order but not necess... | true blue anil | 22,388 | <p>This might appeal to you as a simple way</p>
<p>First arrange <span class="math-container">$-K-O-A-L-A-$</span> with gaps between letters.</p>
<p>All we need to do is to count how many ways GUMTREE can be interleaved.</p>
<p><span class="math-container">$G$</span> can be inserted in <span class="math-container">$6$... |
758,158 | <p>I am trying to use <span class="math-container">$f(x)=x^3$</span> as a counterexample to the following statement. </p>
<p>If <span class="math-container">$f(x)$</span> is strictly increasing over <span class="math-container">$[a,b]$</span> then for any <span class="math-container">$x\in (a,b), f'(x)>0$</span>. <... | Thomas | 26,188 | <p>You want to show that the function $f(x) = x^3$ is strictly increasing on $\mathbb{R}$.</p>
<p>Maybe you can just use the definition. That is, let $a < b$. Assume that $0<a$ (you can do the other cases I am sure). Let $h = b - a < 0$. You want to show that $f(a) < f(b)$.</p>
<p>So
$$\begin{align}
f(a) ... |
696,900 | <p>This should be trivial but for some reason I cannot think of a formula which enlarges a number in proportion to another number decreasing in the negative direction</p>
<p>Example:</p>
<p>if value 1 = $-0.1$, value 2 should be set to $0.9$<br>
if value 1 = $-0.2$, value 2 should be set to $0.8$<br>
if value 1 = $-0... | Zev Chonoles | 264 | <p>How about $$\mathsf{(\text{value 2})}=1+\mathsf{(\text{value 1})}\quad ?$$
It fits all the examples you gave.</p>
|
3,112,263 | <p>Given that there are three integers <span class="math-container">$a, b,$</span> and <span class="math-container">$c$</span> such that <span class="math-container">$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{6}{7}$</span>, what is the value of a+b+c?</p>
<hr>
<p>Immediately, I see that I should combine the left hand... | Community | -1 | <p>Your argument isn't correct as stated due to the issue pointed out by Dan Uznanski in the comments. But following your idea of combining things, we have</p>
<p><span class="math-container">$$6abc = 7(ab + ac + bc)$$</span></p>
<p>so that <span class="math-container">$7 | abc$</span> and <span class="math-container... |
2,362,500 | <p>When you are solving an integral, and you have a square root of a square, for instance</p>
<p>$$ \int \sqrt{1+\cos x} = \int \sqrt{\frac{\sin^2x}{1-\cos x}}$$</p>
<p>Do I have to take the absolute value of $\sin x$, or can I just take the positive value? </p>
<p>$$= \int \frac{|\sin x|}{\sqrt{1-\cos x}}$$ OR
$$= ... | ProgSnob | 456,185 | <p>Using same nomenclature as you have used, there are 4 cases:</p>
<ol>
<li>P5 to P9 are selected = 1 case</li>
<li>P1 and P2 both selected, choose any three from P5-P9 = 1*(5C3) = 10 cases</li>
<li>P1 and P2 both selected, one of (P3, P4) is selected, and choose any two from P5-P9 = 1*(2C1)*(5C2) = 20 cases</li>
<li... |
1,912,660 | <p>I am trying to really understand why the gradient of a function gives the direction of steepest ascent intuitively.</p>
<p>Assuming that the function is differentiable at the point in question,<br>
a) I had a look at a few resources online and also looked at this
<a href="https://math.stackexchange.com/questions/2... | user326210 | 326,210 | <p>The question is <em>how you would measure the steepness of ascent</em>. For one-dimensional functions, steepness is defined in terms of the derivative:</p>
<p>$$g^\prime(x) \equiv \lim_{h \rightarrow 0}\frac{f(x+h)-f(x)}{h}$$</p>
<p>By this limit definition, steepness is measured by computing the slope between the... |
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