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 |
|---|---|---|---|---|
222,105 | <p>I want to use geometric shapes in Mathematica to build complex shapes and use my raytracing algorithm on it. I have a working example where we can get the intersections from a combination of a <code>Cone[]</code> and <code>Cuboid[]</code>, e.g </p>
<pre><code>shape1 = Cone[];
shape2 = Cuboid[];
(* add shapes in thi... | Frei Zhang | 79,635 | <p>This issue could be solved by "sphere tracing", it's similar to Tim's particle advancer" and reasonable fast which will converge in a very small amount of steps.</p>
<pre><code>RayIntersect[ray_, region_, maxIteration_, maxRadius_,
radiusThreshold_] := Module[{ rnf, RegionMarcher, result},
rnf = ... |
2,237,963 | <p>One-point compactification of $S_{\Omega}$ is homeomorphic with $\bar S_{\Omega}$.</p>
<p>Let $X$ be a topological space. Then the One-point compactification of $X$ is a certain compact space $X^*$ together with an open embedding $c : X \to X^*$ such that the complement of $X$ in $X^*$ consists of a single point, ... | Michael Rozenberg | 190,319 | <p>Since $\sin$ is a concave function on $[0,\pi]$ and sum of concave functions is a concave function, </p>
<p>we have
$$\min_{[0,\pi]}f=\min\{f(0),f(\pi-1),f(\pi-2),f(\pi)\}=f(\pi-1)=2\sin1>\frac{8}{5}$$</p>
|
183,881 | <p>I want to replace an element of a matrix by a function of its current value. The code I'm using is as follows</p>
<pre><code>T = {{2, 3}, {3, 2}, {1, 4}}
KNeeds = Transpose[T][[2]]
pos = Flatten[{Position[KNeeds, Min[KNeeds]], 2}]
ReplacePart[T, pos -> T[[pos[[1]]]] - Epsilon]]
</code></pre>
<p>I was hoping to ... | kirma | 3,056 | <p>You can also acquire a symbolic solution to this question with <code>Total</code> of <code>Boole</code> (truth value to 0/1) values of membership on parametric regions:</p>
<pre><code>FullSimplify@
Total[Function[{u, v, w},
Evaluate@
Boole@RegionMember[
RegionIntersection[
Cylinder[{{3, 0,... |
1,706,310 | <p>Suppose X and Y are compact and they are Hausdorff topological spaces. Let $f : X \rightarrow Y$ be a continuous surjective function. Prove that any $U \subset Y$ is open if and only if $f^{−1}(U)$ is open.</p>
<p>I got stuck here because I don't know how to use compactness. Can someone help please?
Thanks</p>
| carmichael561 | 314,708 | <p>If $U$ is open then $f^{-1}(U)$ is open by the definition of continuity.</p>
<p>On the other hand, if $f^{-1}(U)$ is open, then its complement $X\setminus f^{-1}(U)$ is closed, hence compact because $X$ is compact.</p>
<p>Therefore $f(X\setminus f^{-1}(U))$ is compact because $f$ is continuous, and since $Y$ is Ha... |
1,785,044 | <p>Looking at the picture below, it's easy to see why the perimeter of a polygon inscribed in a circle is an underestimation of the circle's perimeter. This follows from the triangle inequality: Any side (say $AB$) of the polygon is shorter than the circular arc with the same endpoints ($\stackrel{\frown}{AB}$). Summin... | marty cohen | 13,079 | <p>This expands
Yves Daoust's comment.</p>
<p>Call the point that
the tangent from $D$
touches the circle
$P$,
and the point where
$DE$ intersects the circle
$Q$.</p>
<p>Then
$DEQ$ is a right triangle.</p>
<p>Let
$t = \angle DEP$.</p>
<p>Then
$\tan(t)
=\dfrac{DP}{PE}
$
so
$DP
=PE \tan(t)
$.</p>
<p>We also have
the... |
4,320,849 | <p>I had this problem in an exam I recently appeared for:</p>
<blockquote>
<p>Find the range of
<span class="math-container">$$y =\frac{x^2+2x+4}{2x^2+4x+9}$$</span></p>
</blockquote>
<p>By randomly assuming the value of <span class="math-container">$x$</span>, I got the lower range of this expression as <span class="m... | Spectre | 799,646 | <p>As a follow-up to @NikolaAlfredi's answer:</p>
<p><span class="math-container">$ y = \frac{x^2 + 2x + 4}{2x^2 + 4x + 9} = \frac{2x^2 + 4x + 8}{2(2x^2 + 4x + 9)} = \frac{2x^2+4x+9 - 1}{2(2x^2+4x+9)} = \frac{1}{2}(1-\frac{1}{2x^2+4x+9}) \implies 2y = 1 - \frac{1}{2x^2+4x+9}$</span>. Now find the extremes of the range ... |
794,301 | <p>I am trying to find out the sum (I just derived these from 2 + 0.5 + 0.125 + 0.03125 + ...):</p>
<p>$$\sum_{n=0}^{\infty} \frac{5^{2n-1}}{10^{2n -1}}$$</p>
<p>It's confusing me because it doesn't match $${ar}^{n-1}$$ the power by which $r$ is raised. </p>
| nadia-liza | 113,971 | <p>$$\sum_{n=0}^{\infty} \frac{5^{2n-1}}{10^{2n -1}}=\sum_{n=0}^{\infty} (\frac{5}{10})^{2n-1}=2\sum_{n=0}^{\infty} (\frac{1}{2})^{2n}= 2 \sum_{n=0}^{\infty} (\frac14)^n=2 \frac{1}{1-\frac14}$$</p>
|
1,513,078 | <blockquote>
<p>Suppose $b\in\mathbb Z$. Find all possible remainders of $b^3$ divided by $7$. </p>
</blockquote>
<p>I know $b^6=1\bmod7$ which means $(b^2)^3=1\bmod7$ so all square numbers^3 leave $1$ as a remainder, but how to continue?</p>
| Patrick Stevens | 259,262 | <p>Notice that this depends only on the value of $b$ mod $7$, since $(b+7k)^3 = b^3 + (\text{multiple of 7})$. So we can work wlog with $b = 0, 1, \dots, 6$.</p>
<p>Your method is also faulty: just because $b^6 \equiv 1 \pmod{7}$, that doesn't mean $b^2 \equiv 1 \pmod{7}$. For example, $3^2 \equiv 2 \pmod{7}$.</p>
<h... |
265,377 | <p>Let $k$ be a finite field, and let $G$ be the absolute Galois group of $k$, which is isomorphic to $\widehat{\mathbb{Z}}$. Let $\mathcal{C}$ be the category of $G$-modules. Then, we have the following:</p>
<p>For a finite $G$-module $N$, we have
$$
Ext^r_{\mathcal{C}}(N, \mathbb{Z}) \simeq H^{r-1}(G,~N^D),
$$
wher... | R. van Dobben de Bruyn | 82,179 | <p>Consider the short exact sequence
$$0 \to \mathbb Z \to \mathbb Q \to \mathbb Q/\mathbb Z \to 0.$$
Note that $\operatorname{Ext}^i_\mathcal C(N,\mathbb Q) = 0$ for all $i$: it is torsion since $N$ is torsion, but multiplication by $n \in \mathbb Z_{>0}$ is an isomorphism since it is so on $\mathbb Q$. Thus, the s... |
1,728,910 | <p>Whenever I get this question, I have a hard time with it. </p>
<p>An example of a problem:</p>
<p>In the fall, the weather in the evening is <em>dry</em> on 40% of the days, <em>rainy</em>
on 58% of days and <em>snowy</em> 2% of the days. </p>
<p>At noon you notice clouds in the sky. </p>
<p>Clouds appear
at noo... | Τίμων | 319,592 | <p>Maybe you are looking for <a href="https://en.wikipedia.org/wiki/Bayes%27_theorem" rel="nofollow">Bayes' theorem</a>, which states</p>
<p>$$P(S\mid C) = \frac{P(C \mid S) \, P(S)}{P(C)}$$</p>
<p>But you need $P(C)$ too.</p>
|
226,534 | <p>I have the following function for which I want to know the range</p>
<pre><code>FunctionRange[{1/
32 (8 - Sqrt[(-8 - 36 Abs[y] (1 + 2 Sqrt[Abs[z]]) -
27 (Abs[y] + 2 Abs[y] Sqrt[Abs[z]])^2)^2 -
64 (1 + Abs[y] + 2 Abs[y] Sqrt[Abs[z]])] +
36 Abs[y] (1 + 2 Sqrt[Abs[z]]) +
27 (Abs[y] + 2 Ab... | user64494 | 7,152 | <p>This can be done by the change <code>Abs[y] -> a^2, Abs[z] -> b^2</code> in order to obtain a polynomial in <code>a</code> and <code>b</code> as follows.</p>
<pre><code>Maximize[{1/
32 (8 - Sqrt[(-8 - 36 Abs[y] (1 + 2 Sqrt[Abs[z]]) -
27 (Abs[y] + 2 Abs[y] Sqrt[Abs[z]])^2)^2 -
64 (1 + Abs[y] + 2 A... |
4,169,445 | <p>I'm getting stuck on perhaps a simple step in the Hille-Yosida theorem from 13.37 in Rudin's functional analysis.
I wonder if someone has had this same difficulty before or knows how to get around it -</p>
<p><strong>Setup:</strong> <span class="math-container">$A$</span> is a densely defined operator with domain <s... | Kman3 | 641,945 | <p>Firstly, you have the wrong formula. The area of the semicircle in Cartesian coordinates would be</p>
<p><span class="math-container">$$\int_{-r}^r \sqrt{r^2-x^2} \ dx$$</span></p>
<p>or, as a double integral,</p>
<p><span class="math-container">$$\int_{-r}^r \int_0^{\sqrt{r^2-x^2}} \,dy \ dx$$</span></p>
<p>Secondl... |
1,348,046 | <p>I ran into this sum $$\sum_{n=3}^{\infty} \frac{3n-4}{n(n-1)(n-2)}$$
I tried to derive it from a standard sequence using integration and derivatives, but couldn't find a proper function to describe it.
Any ideas?</p>
| Zain Patel | 161,779 | <p><strong>Hint:</strong></p>
<p>$$\frac{3n-4}{n(n-1)(n-2)} \equiv \frac{1}{n-1} + \frac{1}{n-2} - \frac{2}{n}$$ </p>
|
1,348,046 | <p>I ran into this sum $$\sum_{n=3}^{\infty} \frac{3n-4}{n(n-1)(n-2)}$$
I tried to derive it from a standard sequence using integration and derivatives, but couldn't find a proper function to describe it.
Any ideas?</p>
| robjohn | 13,854 | <p>Using the <a href="https://en.wikipedia.org/wiki/Heaviside_cover-up_method" rel="nofollow">Heaviside Method of Partial Fractions</a> to solve
$$
\frac{3n-4}{n(n-1)(n-2)}=\frac{A}{n}+\frac{B}{n-1}+\frac{C}{n-2}
$$
we get</p>
<p>multiply by $n$ and set $n=0\implies\frac{3\cdot\color{#C00000}{0}-4}{(\color{#C00000}{0}... |
259,751 | <p>I'm playing this video game where people can get kills, deaths, and assists , and all this is recorded on a stats website. The stats website gives you a rating by directly manipulating these numbers.</p>
<p>In the first entry, I have 26 kills, 5 deaths, and 19 assists. The KDA ratio the website gave me was 29.8. At... | Mario Carneiro | 50,776 | <p>I thought I'd give an overview of the prime-generating functions in <a href="http://www.sonoma.edu/math/colloq/primes_sonoma_state_9_24_08.pdf" rel="nofollow">Is there a formula that Generates Prime Numbers?</a> and <a href="http://mathworld.wolfram.com/PrimeFormulas.html" rel="nofollow">MathWorld: Prime Formulas</a... |
1,206,528 | <p>Find the matrix $A^{50}$ given</p>
<p>$$A = \begin{bmatrix} 2 & -1 \\ 0 & 1 \end{bmatrix}$$ as well as for $$A=\begin{bmatrix} 2 & 0 \\ 2 & 1\end{bmatrix}$$</p>
<p>I was practicing some questions for my exam and I found questions of this form in a previous year's paper.</p>
<p>I don't know how to ... | Joel | 85,072 | <p>You can also compute it with the following method. First note that $50 = 32 + 16 + 2$. Let $$A=\left(\begin{array}{cc}2&0\\2&1\end{array}\right).$$</p>
<p>Then $$A^{50} = A^{32} A^{16} A^2$$.</p>
<p>We can compute $A^{2^n}$ easier than $A^{50}$,</p>
<p>$$A^2 = \left( \begin{array}{cc} 4 & 0\\ 6 & ... |
1,206,528 | <p>Find the matrix $A^{50}$ given</p>
<p>$$A = \begin{bmatrix} 2 & -1 \\ 0 & 1 \end{bmatrix}$$ as well as for $$A=\begin{bmatrix} 2 & 0 \\ 2 & 1\end{bmatrix}$$</p>
<p>I was practicing some questions for my exam and I found questions of this form in a previous year's paper.</p>
<p>I don't know how to ... | vudu vucu | 215,476 | <p>Hint: Find a matrix $P$ such that $P^{-1}AP=D$ where $D=diag(2,1)$. Then $P^{-1}A^{50}P=D^{50}=diag(2^{50},1).$</p>
|
167,891 | <p>I'm working through the James Stewart Calculus text to prep for school. I'm stuck at this particular point. </p>
<p>How would you sketch the graph for the parametric equations:
$x = \cos t$, $y = \sin t$, and $z = \sin 5t$? I understand that if it were the case that $z=t$, I'd merely get a helix around the $z$-axis... | robjohn | 13,854 | <p><strong>Hint:</strong> Note that the $x$ and $y$ coordinates trace out a circle. As they do, the $z$ coordinate goes through $5$ sinusoidal cycles.</p>
|
2,815,900 | <p>Consider the proposition $∀x (P (x) → Q (x))$, where $P (x)$ and $Q (x)$ are predicates on a domain $U$. Prove or disprove the following statement, justifying the answer.</p>
<p>If the statement $∀x (P (x) → Q (x))$ is FALSE then it's FALSE also $∀x P (x)$</p>
<p>How can I solve this exercise?</p>
| Bram28 | 256,001 | <p>It's typically a good strategy to try to come up with a counterexample to the given statement: if you can find a counterexample, then the statement is false, and if you can't, then you probably get some idea why you can;t, and that might translate into a proof why the statement is in fact true.</p>
<p>Now, to find ... |
1,721,055 | <p>This afternoon I've been studying the pythagorean identities & compound angles. I've got a problem with a question working with 2 sets of compound angles:</p>
<blockquote>
<p>Solve, in the interval $0^\circ \leq \theta \leq 360^\circ$, $$\cos(\theta + 25^\circ) + \sin(\theta +65^\circ) = 1$$ </p>
</blockquote... | Jalejo | 97,487 | <p>"Solve, in the interval 0≤θ≤360, cos(θ+25)+sin(θ+65)=1"</p>
<p>$\cos(\theta+25)+\cos(25-\theta)=1$</p>
<p>∵ $\sin(\theta+65)=\cos(90-(\theta+65))=\cos(25-\theta)$</p>
<p>$\cos(\theta)\cos(25)- \sin(\theta)\sin(25)+\cos(25)\cos(\theta)+\sin(\theta)\sin(25)=1$</p>
<p>$2\cos(\theta)\cos(25)=1$</p>
<p>$\cos(\theta)... |
89,638 | <p>I'm trying to solve an exercise as it follows:</p>
<p>$\alpha, \beta \in \mathbb{C}$ such that $a^{37}=2=\beta^{17}$. Note that both are prime.</p>
<p>a) Find the minimum polynomials of $\alpha$ and $\beta$ over $\mathbb{Q}$.</p>
<p>Well, I guess that $(x^{37}-2)$ and $(x^{17}-2)$ are irreducible over $\mathbb{Q}... | Dilip Sarwate | 15,941 | <p>Suppose that $\psi(t)$ is the characteristic function corresponding to $f(x)$. Using the properties of Fourier transforms, we have that </p>
<ul>
<li><p>since $f(x)$ is a real-valued function, $\psi(t)$ has <em>conjugate symmetry:</em>
$\psi(-t) = [\psi(t)]^*.$</p></li>
<li><p>the characteristic function correspon... |
1,571,083 | <p>Given the 2 terms
$$ \frac{k + a}{k + b}$$
and $$\frac{a}{b}$$
with $a, b, k \in \mathbb{R^+}$ and $a > b$</p>
<p>I want to show, that the first term is always bigger than the second one.</p>
<p><strong>My try</strong>
$$
\frac{k + a}{k + b} > \frac{a}{b} \\
\frac{k + b + (a - b)}{k + b} > \frac{b + ... | Colm Bhandal | 252,983 | <p>Assuming the events are independent and everyone has the same chance of winning, then the odds of the same person winning twice is:</p>
<p>$$\frac{2}{31} \times \frac{1}{25} = \frac{2}{775}$$</p>
<p>In other words, they have a $2$ in $775$ or roughly $0.26\%$ chance of winning both. Unlikely, but not impossible.</... |
1,571,083 | <p>Given the 2 terms
$$ \frac{k + a}{k + b}$$
and $$\frac{a}{b}$$
with $a, b, k \in \mathbb{R^+}$ and $a > b$</p>
<p>I want to show, that the first term is always bigger than the second one.</p>
<p><strong>My try</strong>
$$
\frac{k + a}{k + b} > \frac{a}{b} \\
\frac{k + b + (a - b)}{k + b} > \frac{b + ... | fleablood | 280,126 | <p>The probability of winning the first raffle is: $1/31 + 30/31*1/30 = 2/31$.</p>
<p>The probability of winning the second raffle is: $1/25$.</p>
<p>The probability of her winning both: $2/31 * 1/25 = 2/775$ but that's the same probability of <em>any</em> outcome.</p>
<p>The probability of one of the two winners of... |
2,389,581 | <p>A town has 2017 houses. Of these 2017 houses, 1820 have a dog, 1651 have a cat, and 1182 have a turtle. If x is the largest possible number of houses that have a dog, a cat, and a turtle, and y is the smallest possible number of houses that have a dog, a cat, and a turtle, then what is the value of x−y.</p>
| fleablood | 280,126 | <p>The trick is to think if the set of houses with all kinds of pets isn't maximal or minimal what we can do to maximize or minimize it and what result that will be.</p>
<p>If there are any houses with only a turtle in it, or only a turtle and one other pet it int we can add the other pets from houses with pets but no... |
186,395 | <p>i want write a module to find the integer combination for a multi variable fomula. For example</p>
<p>$8x + 9y \le 124$</p>
<p>The module will return all possible positive integer for $x$ and $y$.Eg. $x=2$, $y=12$.
It does not necessary be exactly $124$, could be any number less or equal to $124$. Must be as close... | David Nehme | 16,013 | <p>You are trying to solve</p>
<p>$${\rm maximize} 8x + 9y$$
subject to
$$8x + 9y \le 124$$
$$ x, y \in \mathbb{Z}^+$$</p>
<p>Which is essentially a special case of the <a href="http://en.wikipedia.org/wiki/Knapsack_problem#Multiple_Knapsack_Problem" rel="nofollow">knapsack problem</a>.
$${\rm maximize} \sum_i a_i x_... |
1,429,000 | <p>Prove that for any integer $n$, the integer $n^2 + 7n + 1$ is odd.</p>
<p>I have $n=2k+1$ for some $k\in Z$</p>
<p>I really do not how to do this problem. any help in understanding would be greatly appreciated.</p>
| angryavian | 43,949 | <p>Hint: if $n$ is odd, then is $n^2$ odd or even? what about $7n$? Ask the same questions if $n$ is even.</p>
|
1,429,000 | <p>Prove that for any integer $n$, the integer $n^2 + 7n + 1$ is odd.</p>
<p>I have $n=2k+1$ for some $k\in Z$</p>
<p>I really do not how to do this problem. any help in understanding would be greatly appreciated.</p>
| Cameron Buie | 28,900 | <p><strong>Hint</strong>: You should be able to fill in the the following blanks and justify each claim made. We start by noting that $$n^2+7n=n(n+7).$$ If $n$ is an integer, then either $n$ or $n+7$ must be _____ and the other must be _____. Hence, $n(n+7)$ must be _____, and so....</p>
|
850,162 | <p>How can i prove that there exist some real $a >0$ such that $\tan{a} = a$ ? </p>
<p>I tried compute $$\lim_{x\to\frac{\pi}{2}^{+}}\tan x=\lim_{x\to\frac{\pi}{2}^{+}}\frac{\sin x}{\cos x}$$ </p>
<p>We have the situation " $\frac{1}{0}$ " which leads us " $\infty$ " </p>
<p>$$\lim_{x\to\frac{\pi}{2}^{-}}\tan x=\... | Adam Hughes | 58,831 | <p>Use the intermediate value theorem on $\tan x -x$ after plugging in $x={3\pi\over 4}$ and $x={35\pi\over 24}$.</p>
<p>I chose those specific values because (in theory) you can compute them by hand using half-angle formulae and it's more constructive, but one can use the definition of unboundedness near $3\pi/2$ and... |
301,318 | <p>As the title stated , what is the meaning of infinitely many ? When we say a set contains infinitely many elements, does this mean we cannot finish counting all the elements in the set ? Does infinitely many same as $\forall$ ? </p>
| Sigur | 31,682 | <p><strong>Infinitely many</strong> means that you can find a subset and a bijection with $\mathbb{N}$, that is, you have an infinite elements.</p>
|
3,767,421 | <p>This is a rather soft question.</p>
<p><strong>My understanding:</strong></p>
<p>Suppose we have <span class="math-container">$x \in ℝ$</span> and <span class="math-container">$x^2 = -1$</span> [in the normal interpretation].</p>
<p>Then the statement "<em>there exists <span class="math-container">$r \in ℝ$</sp... | lemontree | 344,246 | <p>First off, "There exist <span class="math-container">$x \in \mathbb{R}$</span> such that <span class="math-container">$x^2 = -1$</span>" is by itself not at all contradictory. It is just not true in the actual world by our usual understanding of the symbols <span class="math-container">$\mathbb{R}, -x$</sp... |
1,283,541 | <p>Find the least value of $f(x)=3^{-x+1} + e^{-x-1}$.</p>
<p>I tried to use the maxima/minima concept but it was of no use. Please help.</p>
| Barry Cipra | 86,747 | <p>Trigonometric identities and other such properties of trig functions are independent of whether angles are measured in radians, degrees, or what have you. But the squeeze theorem for $\sin x\over x$ and other such relations <em>do</em> depend on $x$ being measured in radians, so the underlying geometry is inescapab... |
89,622 | <p>I'm having a hard time trying to understand a proof of the Principle of Local Reflexivity. I'm following the proofs from </p>
<p>1) Topics in Banach Space Theory (by Fernando Albiac, Nigel J. Kalton)
2) <a href="http://www.math.tamu.edu/~schlump/sofar.pdf" rel="nofollow">http://www.math.tamu.edu/~schlump/sofar.pdf<... | Keaton Hamm | 22,500 | <p>I cannot comment above, so I have to write another answer. To clarify for you Rafael why such an $i$ exists in Bill's comment. Take the same setup, $\xi\in\mathbb{K}^N$ such that
\begin{equation}\underset{j=1}{\overset{N}{\sum}}\xi_jx_j^{\*\*}=0 ;\quad\underset{j=1}{\overset{N}{\sum}}\xi_jx_j\neq0 \end{equation}<b... |
214,520 | <p>I'm trying to make tables like in the image to calculate max and min of the expressions.
Whenever I change the max and min of <code>a</code> and <code>b</code> in the upper table then the <code>MixEx</code>and <code>MaxEx</code> are updated automatically.</p>
<p>How can I do that in Mathematica?
<a href="https://i.... | kglr | 125 | <p>A few customizations for the <code>IntervalSlider</code> and <code>InputField</code> controls:</p>
<pre><code>ClearAll[thumb, intSlider, inpField]
thumb = Graphics[{#, Text[Style["▲", #, 16], Offset[{0, -20}, {0, 0}]],
Text[Style[#2, 12], Offset[{0, -35}, {0, 0}]]}, ImageSize -> 20] &;
intSlider[Dynam... |
3,004,767 | <p>If <span class="math-container">$\displaystyle\lambda_n = \int_{0}^{1} \frac{dt}{(1+t)^n}$</span> for <span class="math-container">$n \in \mathbb{N}$</span>. Then prove that <span class="math-container">$\lim_{n \to \infty} (\lambda_{n})^{1/n}=1.$</span></p>
<p><span class="math-container">$$\lambda_n=\int_{0}^{1} ... | Jack D'Aurizio | 44,121 | <p>In general, if <span class="math-container">$f(x)$</span> is a continuous and non-negative function on <span class="math-container">$[0,1]$</span>, </p>
<p><span class="math-container">$$ \lim_{n\to +\infty}\sqrt[n]{\int_{0}^{1}f(x)^n\,dx} = \max_{x\in[0,1]}f(x) $$</span>
by the inequality between means.</p>
|
2,118,266 | <p>In Order Theory, what is the exact definition of a dual statement? And what is the duality principle for posets/lattices? I haven't been able to find an exact statement or definition in this regard.</p>
| Hagen von Eitzen | 39,174 | <p>If $(X,\le)$ is a poset, then so is $(X,\preceq)$ with $a\preceq b\iff b\le a$. Any statement about the former translates into a statement of the latter</p>
|
2,345,142 | <p>Can you use Skolemization to reduce a formula to the variables you want it to be about? I was trying to think of a nice algorithmic way to do it but only ended up having problems. </p>
<p>Say you have a formula $G$ in prenex normal form, i.e. something looking like this:</p>
<p>$$G \equiv Q_1x_1...Q_nx_n\;F$$</p>
... | Noah Schweber | 28,111 | <p>Your version of Skolemization is incorrect, and we can see this at the very first step. "$\forall xL(x)$" is <strong>not</strong> equivalent to "$\neg L(x/c)$:" while the former implies the latter, the latter doesn't imply the former (how do we know $c$ wasn't "chosen badly"?) Similarly, "$\exists x L(x)$" is not eq... |
804,963 | <p>Let $f,g:\mathbb{R}\longrightarrow \mathbb{R}$ be Lebesgue measurable. If $f$ is Borel measurable, then $f\circ g$ is Lebesgue mesuarable. In general, $f\circ g$ is not necessarily Lebesgue measurable. Is there any counterexample?</p>
| Etienne | 80,469 | <p>Let $g:\mathbb R\to \mathbb R$ be any Borel function with the following property: there exists a Lebesgue-measurable set $B$ such that $g^{-1}(B)$ is not Lebesgue-measurable. Such functions do exist (see below).</p>
<p>Now take $f:=\mathbf 1_{B}$. Then $f$ is Lebesgue-measurable because $B$ is, but $f\circ g=\mathb... |
1,317,974 | <p>Consider the affine toric variety $V \subset k^{5}$ parametrized by $$\Phi(s,t,u) = (s^{4},t^{4},u^{4},s^{8}u,t^{12}u^{3}) \in k^{5}$$ where k is an algebraically closed field of characteristic 2. This is problem 1.1.8 from Cox, Little, and Schenck, but my question regards a more general notion: How exactly does one... | Community | -1 | <p>The following items illustrate an algorithm to compute saturation of ideals. Let $I\subset S := k[x_1,\dots,x_n]$ be an ideal and fix $f\in S$. Then the <strong>saturation</strong> of $I$ w.r.t. $f$ is the set
$$I:f^\infty=\{g\in S|f^m g\in I \text{ for some } m>0\}.$$</p>
<ul>
<li>$I:f^\infty$ is an ideal.</li>... |
43,640 | <p>Consider the following problem:</p>
<p>Let ${\mathbb Q} \subset A\subset {\mathbb R}$, which of the following must be true?</p>
<p>A. If $A$ is open, then $A={\mathbb R}$</p>
<p>B. If $A$ is closed, then $A={\mathbb R}$</p>
<p>Since $\overline{\mathbb Q}={\mathbb R}$, one can immediately get that B is the answer... | Jonas Meyer | 1,424 | <p>The question boils down to whether there are nonempty subsets of $\mathbb{R}\setminus \mathbb{Q}$ that are closed in $\mathbb{R}$. The easiest examples are finite sets, as <a href="https://math.stackexchange.com/questions/43640/open-set-in-bf-r/43642#43642">Luboš Motl noted</a>. An easy infinite example is $\sqrt{... |
43,640 | <p>Consider the following problem:</p>
<p>Let ${\mathbb Q} \subset A\subset {\mathbb R}$, which of the following must be true?</p>
<p>A. If $A$ is open, then $A={\mathbb R}$</p>
<p>B. If $A$ is closed, then $A={\mathbb R}$</p>
<p>Since $\overline{\mathbb Q}={\mathbb R}$, one can immediately get that B is the answer... | Michael Hardy | 11,667 | <p>"Why A is not necessarily true? What can be a counterexample?"</p>
<p>I'm surprised at the complexity of some answers given to this. Here's a counterexample:
$$
(-\infty,\pi)\cup (\pi,\infty).
$$
You can construct lots of others similar to that but more complicated if need be.</p>
|
3,346,543 | <p>I am aware this is a pretty big topic, but the attempts at layman's explanations I have seen either barely provide commentary on the formal proofs, or fail to provide an explanation (e.g "it gets too complex" does not really say anything)</p>
<p>Is there a good intuitive explanation as to why we fail to obtain a ge... | NBeneparte | 709,516 | <p>Responding to Dmitry Ezhov's comment above: the "finite combination" is necessary, since with an infinite number of operations one <em>can</em> solve an arbitrary quintic. For example, consider <span class="math-container">$x^5 - x - 1$</span>. Galois theory tells us the roots of this cannot be expressed in terms of... |
497,687 | <p>How would I integrate the following.</p>
<p>$$
\int_{\ln2}^{\ln3}\frac{e^{-x}}{\sqrt{1-e^{-2x}}}\,dx
$$ </p>
<p>I think I have to use the $\arcsin(x)$ formula.</p>
<p>Which means would I use $y=u^2$ with $u=e^{-x}$ but that does not seem to work.</p>
| Mercy King | 23,304 | <p>Setting
$$
t=e^{-x},\ t=\sin\theta
$$
we have
\begin{eqnarray}
\int_{\ln2}^{\ln3}\frac{e^{-x}}{\sqrt{1-e^{-2x}}}\,dx
&=&\int_{\frac13}^{\frac12}\frac{1}{\sqrt{1-t^2}}\,dt=\int_{\arcsin\frac13}^{\arcsin\frac12}\frac{\cos\theta}{\sqrt{1-\sin^2\theta}}\,d\theta\\
&=&\int_{\arcsin\frac13}^{\frac\pi6}\,d\... |
1,630,831 | <p>I just woke up this and I was extremely curious if there were any subspaces of a vector space such that $X\cap (Y + Z) \neq (X \cap Y) + (X \cap Z)$?</p>
<hr>
<p>Just a curious person. Thanks in advances. </p>
| DylanSp | 308,461 | <p>$5 \frac{(x +5)(6x - 1)}{(7 - x)(8x + 2)} = \frac{5[(x +5)(6x - 1)]}{(7 - x)(8x + 2)}$. Just use FOIL to multiply the binomials in the numerator, then use the distributive law to multiply the resulting trinomial by 5.</p>
|
1,630,831 | <p>I just woke up this and I was extremely curious if there were any subspaces of a vector space such that $X\cap (Y + Z) \neq (X \cap Y) + (X \cap Z)$?</p>
<hr>
<p>Just a curious person. Thanks in advances. </p>
| Narasimham | 95,860 | <p>No need to FOIL. Divide numerator and denominator with $x^2,$ and let x go to infinity.</p>
<p>$$ 5 \dfrac{(x +5)(6x - 1)}{(7 - x)(8x + 2)} = \dfrac{5\,(1 +5/x)(6 - 1/x)}{(-1 +7/x)(8 + 2/x)} = \frac{30}{-8} = - 3.75 $$</p>
|
1,587,007 | <p>I have the following matrix $\mathbf{U}$ which is in echelon form. The strange to me is that I havent met a matrix with first column zero. </p>
<p>$$\mathbf{U} = \begin{bmatrix}
0 & 5 & 4 & 3 \\
0 & 0 & 2 & 1 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
\end{bmatrix}$$</p>
<... | janmarqz | 74,166 | <p>The three sets $B_1,B_2,B_3$ serve as a basis for a unique two dimensional subspace of $\Bbb R^4$ because each set has two linearly independent vectors.</p>
|
382,935 | <p>I reading my textbook and I don't understand the concept of distributions or number of solutions to an equation. It's explained that this problem is 1/4 types of sampling/distributions problems. An example is provided to illustrate:</p>
<blockquote>
<p>In how many ways can 4 identical jobs (indistinguishable ba... | Alex Wertheim | 73,817 | <p>Like so many problems in combinatorics, the key to this problem is understanding precisely what kind of things you are counting. You confusion seems to be arising, at least in part, from the fact that you are trying to count distributions of jobs to urns, and your authors are counting something seemingly different a... |
1,627,381 | <p>There is a subset of sigma field $G_2$, say $G_1 \subset G_2$. $G_1$ is proven to be a sigma field.
Does this necessarily imply that $G_1 = G_2$?</p>
| BrianO | 277,043 | <p>Not at all: the Borel sets of $\Bbb R$ form a $\sigma$-field $\mathcal{B}$, and the power set $\mathcal{P}(\Bbb R)$ of $\Bbb R$ is a $\sigma$-field, but $\mathcal{B} \subsetneqq \mathcal{P}(\Bbb R)$, as not every set of reals is Borel.</p>
|
30,789 | <p>I'm trying to identify the frequencies in my time history samples, and I can see a frequency in the time history, but can't see it in its Fourier transform.
Here it is :</p>
<p>the sample data:</p>
<pre><code>dt = 0.01;(*0.01 second per sample*)
ls={7.18384,9.08503,7.13301,9.03243,7.23692,8.82911,7.48153,8.50053,7... | bill s | 1,783 | <p>What is happening in your second example (with the single sine wave giving the "beating") is that you have exceeded the <a href="http://en.wikipedia.org/wiki/Nyquist_frequency" rel="nofollow noreferrer">Nyquist frequency</a>: what you are seeing is called <a href="http://en.wikipedia.org/wiki/Aliasing" rel="nofollow... |
1,232,439 | <p><strong>EDIT:</strong> This is for a production scheduling problem with quadratic production and linear inventory costs. </p>
<p>The goal is to \begin{equation*}
\max_{u} \int_{0}^{T} -(c_{1}u^{2} + c_{2}y) dt
\end{equation*}
subject to
\begin{align*}
y' &= u,\\
y(0) &= 0,\\
y(T) &=B.\\
\end{a... | Anurag A | 68,092 | <p>Let $n$ be the total number of people at the meeting before Mr. L arrived. Then the number of handshakes will be $\dfrac{n(n-1)}{2}$. Suppose $m$ people left before Mr. L arrived .So he shakes $n-m$ hands. Then
\begin{align*}
\frac{n(n-1)}{2}+n-m & =100\\
n^2+n-2m & = 200\\
m & = \frac{n(n+1)}{2}-100.
\e... |
1,160,454 | <p>I missed two classes in calculus and we're on a subject that I do not understand at all. If someone could just walk me through this problem I could probably begin to comprehend the rest.</p>
<p>The base of a solid elliptical cylinder is given by $ (x/5)^2 + (y/3)^2 = 1.$ A solid is formed by cutting off or removin... | MonkeysUncle | 217,283 | <p>First imagine how you'd do this problem in 2 dimensions. You need to set up a double integral over some x interval and then some y interval. The equation you're working with is in the form of an ellipse centered at the origin. Sections perpendicular to the x axis have constant x value, so you want x to be your last ... |
527,799 | <p>In Hatcher's Algebraic Topology section 1.3, Cayley complexes are explained. The book states that we get a Cayley complex out of a Cayley graph by attaching a 2-cell to each loop. There is an example showing the Cayley complex for $\mathbb{Z}\times\mathbb{Z}$ (the fundamental group of the torus). We attach one 2-cel... | Benjamin Steinberg | 85,907 | <p>This is essentially the same answer as user32240 but I will try to explain it differently. Hatcher's description is a bit sloppy. The correct thing to say is that if $R$ is the set of defining relators for $G$, then each element $r\in R$ labels a loop based at every vertex of the Cayley graph. To each of these bas... |
2,745,436 | <p>For some reason I have been struggling with this problem for the past couple hours.</p>
<p><a href="https://i.stack.imgur.com/diFLe.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/diFLe.png" alt="enter image description here"></a></p>
<p>I believe I have solved part a.</p>
<p>Since there are 6 ... | Michael Hardy | 11,667 | <p>From state $s_1$ you have $1/6$ chance of remaining in that state and a $1/6$ chance of transitioning to any of the other states.</p>
<p>From state $s_2$ you have a $2/6$ chance of remaining there (by getting either a $1$ or a $2$) anad a $1/6$ chance of transitioning to each of $3,4,5,6.$</p>
<p>From state $s_3$ ... |
2,343,216 | <p>Let $\operatorname{gd}(x)$ the Gudermannian function, defined as in this <a href="http://mathworld.wolfram.com/Gudermannian.html" rel="nofollow noreferrer">MathWorld's article</a>, and implemented in Wolfram Language as <em>Gudermannian[x]</em>. </p>
<p>This afternoon I've spent an hour playing with <a href="https:... | Raffaele | 83,382 | <p>You considered $\log(1+x)\approx x$ forgetting that gd(x) goes from 0 to 0.6 circa on the given interval, that is too much and gives a huge error</p>
<p>Compare $\log(1+0.4)\approx 0.33;\;\log(1+0.5)\approx 0.4$ and $\log(1+0.6)\approx 0.47$ </p>
<p>This causes your approximation being too high. Taking integratio... |
4,373,262 | <p>Let G be a graph with order 9 so that the degree of each vertex is either 5 or 6. Prove
that there are either at least 5 vertices of degree 6 or at least 6 vertices of degree 5</p>
| Community | -1 | <p>Let the graph have <span class="math-container">$m$</span> vertices of degree <span class="math-container">$6$</span> and <span class="math-container">$n$</span> vertices of degree <span class="math-container">$5$</span>.</p>
<p>Then <span class="math-container">$6m+5n$</span> is twice the number of edges and so <sp... |
1,665,107 | <p>I must use a generating function to solve this question: </p>
<p>In how many ways can you collect six dollars from eight people if six people give either $0$ or $1$ dollars and the other two people each give $0$, $1$, or $5$ dollars?</p>
<p>Here is what I have so far.</p>
<p>The generating function is $(1+x)^6(1+... | Archis Welankar | 275,884 | <p>Hint its multinomial theorem where $n=x_1+x_2+...x_n$ and coefficient is given by ${n\choose x_1,x_2,x_3..x_n}$</p>
|
1,665,107 | <p>I must use a generating function to solve this question: </p>
<p>In how many ways can you collect six dollars from eight people if six people give either $0$ or $1$ dollars and the other two people each give $0$, $1$, or $5$ dollars?</p>
<p>Here is what I have so far.</p>
<p>The generating function is $(1+x)^6(1+... | choco_addicted | 310,026 | <p>In general,
$$
(a+b+c)^2=a^2+b^2+c^2+2(ab+bc+ca).
$$
Thus
\begin{align}
(1+x+x^5)^2&=1+x^2+x^{10}+2(x+x^6+x^5)\\
&=1+2x+x^2+2x^5+2x^6+2x^{10}
\end{align}
and so the coefficient of $x^6$ is
\begin{align}
\binom{6}{6}+2\binom{6}{5}+\binom{6}{4}+2\binom{6}{1}+\binom{6}{0}&=1+2\cdot 6+15+ 2\cdot 6+1\\
&=... |
3,517,722 | <p>Let <span class="math-container">$\alpha:= \sqrt[7]{2},\omega:= e^{\frac{2\pi i }{7}}\in \Bbb C$</span>. We set <span class="math-container">$E:=\Bbb Q(\alpha,\omega)$</span> and <span class="math-container">$B:=\Bbb Q(\omega+\omega^2+\omega^4)\leq E$</span>. </p>
<p>Thus, we have the Tower of Fields
<span class="m... | reuns | 276,986 | <p>You meant <span class="math-container">$\alpha=\sqrt[7]{2}$</span>. Let <span class="math-container">$K=\Bbb{Q}(\omega)$</span>.</p>
<p>We get <span class="math-container">$E/K/B/\Bbb{Q}$</span> where <span class="math-container">$E/\Bbb{Q}$</span> and <span class="math-container">$K/B$</span> are Galois.</p>
<p>W... |
2,851,609 | <p>I need to find the solution to the inequality $(x - y)(x + y -1) > z$, where $x,y,z \geq 0$ and $x,y,z \leq 1$. As $z$ is positive, then the inequality holds whenever (i) $x - y > 0$ and $x + y - 1 > 0$ OR (ii) $x - y < 0$ and $x + y - 1 < 0$. I can solve the cases (i) and (ii) on their own, but I don... | Luca Bressan | 89,879 | <p>If we multiply and rearrange the terms in the inequality, we get:
<span class="math-container">$$z < x^2 - y^2 - x + y$$</span>
which is equivalent to:
<span class="math-container">$$z < \left (x - \frac 1 2 \right)^2 - \left (y - \frac 1 2 \right)^2.$$</span>
The associated equation represents a hyperbolic pa... |
223,509 | <p>Let $\ulcorner \cdot \urcorner$ be a fixed encoding of formulas by numbers, e.g.
let $\ulcorner \varphi \urcorner$ denote the <a href="https://en.wikipedia.org/wiki/G%C3%B6del_numbering" rel="nofollow">Godel number</a> of $\varphi$.
Let $T$ be a first-order arithmetic theory, e.g. PA.
Let $\Phi$ be a class of close... | Bjørn Kjos-Hanssen | 4,600 | <p>At the risk of stating the obvious, in modal logic the axiom schema
$$\Box\varphi\rightarrow\varphi$$
is called the schema T.</p>
|
8,756 | <p>I'd like to suggest that after a Question receives an Accepted Answer, some consideration be given to revising the title (if appropriate) to reflect what the real issue turned out to be.</p>
<p>It seems to me users often pick titles when first posting a question that are uninformative and are worth revisiting once ... | Brett Frankel | 22,405 | <p>I think we should be more conservative about changing titles than retagging.</p>
<p>It's reasonable to suppose that someone searching for an answer to a similar question will be thinking along the lines of the OP's original wording, and not necessarily in the language of the answer. I can't see why anybody would ob... |
8,756 | <p>I'd like to suggest that after a Question receives an Accepted Answer, some consideration be given to revising the title (if appropriate) to reflect what the real issue turned out to be.</p>
<p>It seems to me users often pick titles when first posting a question that are uninformative and are worth revisiting once ... | Alexander Gruber | 12,952 | <p>I'm for it. I have been making an effort lately to do this myself. I am sick of seeing titles like "<a href="https://math.stackexchange.com/q/328718/12952">Proof about a semigroup</a>" (an example taken from the front page right at this moment). This provides absolutely no information, and there's no reason why t... |
2,037,428 | <p>Let $D$ be a compact, connected $Jordan$ domain in $R^n$ with positive volume, and suppose that the fuction $f:D →R$ is continuous. Show that there is a point $x$ in $D$ in which
$f(x)=(1/volD)$$\int$$f$ over $D$.
(Mean Value Property for integral)</p>
<p>I can prove it when $D$ is pathwise connected</p>
<p>But ... | Julián Aguirre | 4,791 | <p>No. The standard example is the <em>Topologist sine curve</em>:
$$
\Bigl\{\Bigl(x,\sin\frac{1}{x}\Bigr):0<x\le1\Bigr\}\bigcup\{0\}\times[-1,1].
$$
To prove the MVT all you need is that the continuous image of a connected set is connected.</p>
|
991,377 | <p>I have to evalute this integral.
$\displaystyle\iint\limits_{D}(2+x^2y^3 - y^2\sin x)\,dA$
$$D=\left \{ (x, y):\left | x \right |+\left | y \right | \leq 1\right \}$$</p>
<p>At first, I evaluated simply by putting $-1\leq x\leq 1, -1\leq y\leq 1$, thus making
$$
\int_{-1}^{1}\int_{-1}^{1}(2+x^2y^3 - y^2\sin x)\,... | MvG | 35,416 | <p>This is a coordinate-based approach, making heavy use of tools from projective geometry.</p>
<p>Without loss of generality, you can choose the coordinate system in such a way that the inscribed circle is the unit circle. On that you can use a rational parametrization, i.e. choose $a,b,c\in\mathbb R$ such that $A'=(... |
2,282,359 | <p>I am trying to calculate this limit:
$$\lim_{x \to \infty} x^2(\ln x-\ln (x-1))-x$$
The answer is $1/2$ but I am trying to verify this through proper means. I have tried L'Hospital's Rule by factoring out an $x$ and putting that as $\frac{1}{x}$ in the denominator (indeterminate form) but it becomes hopeless afterwa... | CY Aries | 268,334 | <p>\begin{align*}
\lim_{x \to \infty} [ x^2(\ln x-\ln (x-1))-x]&=\lim_{x \to \infty} \frac{\displaystyle\ln x-\ln (x-1)-\frac{1}{x}}{\displaystyle\frac{1}{x^2}}\\
&=\lim_{x \to \infty} \frac{\displaystyle\frac{1}{x}-\frac{1}{x-1}+\frac{1}{x^2}}{\displaystyle\frac{-2}{x^3}}\quad (\text{ L'Hopital Rule})\\
&=... |
390,718 | <p>I am trying to prove something that seemed simple to me at first sight but apparently it is giving me a hard time. Here is <a href="https://math.stackexchange.com/questions/4108555/bounded-sequence-in-spatial-tensor-product-and-boundedness-of-simple-tensor-summ">the same question on MSE</a>.</p>
<p>Let <span class="... | Mateusz Wasilewski | 24,953 | <p>One way to see it is to use the dual basis of <span class="math-container">$\{x_1,\dots,x_n\}$</span>. Let <span class="math-container">$\{\varphi_1,\dots,\varphi_n\} \subset E^{\ast}$</span> be functionals such that <span class="math-container">$\varphi_i(x_j)=\delta_{ij}$</span>. Because functionals are automatica... |
390,718 | <p>I am trying to prove something that seemed simple to me at first sight but apparently it is giving me a hard time. Here is <a href="https://math.stackexchange.com/questions/4108555/bounded-sequence-in-spatial-tensor-product-and-boundedness-of-simple-tensor-summ">the same question on MSE</a>.</p>
<p>Let <span class="... | Matthew Daws | 406 | <p>Here's an approach motivated by considerations of Banach space tensor products; it owes a debt to the approach taken by Takesaki in his book, Volume 1, Chapter IV, Sections 2 and 4. Another good book is <a href="https://link.springer.com/book/10.1007/978-1-4471-3903-4" rel="nofollow noreferrer">Ryan's book</a>, &qu... |
11,315 | <p>I just answered this question:</p>
<p><a href="https://math.stackexchange.com/questions/528880/boolean-formula">Boolean formula over 64 Boolean variables X</a></p>
<p>By the time I had posted my answer, another user had edited the question so as to remove all the mathematical interest.</p>
<p>What is going on her... | Rob Arthan | 23,171 | <p>Thanks to everyone for their comments and to Andres Caicedo for undoing the irresponsible edit. I conclude that I was bamboozled into solving a homework question. When I gave an answer the student or students tried to cover up what had happened by conspiring to delete the mathematical content of the question.</p>
<... |
48,237 | <p>I have found by a numerical experiment that first such primes are:
$2,5,13,17,29,37,41$. But I cannot work out the general formula for it.<br>
Please share any your ideas on the subject.</p>
| GEdgar | 442 | <p>In a number theory course, you would probably come to the <em>Legendre symbol</em> after a while... <a href="http://en.wikipedia.org/wiki/Legendre_symbol" rel="nofollow noreferrer">http://en.wikipedia.org/wiki/Legendre_symbol</a> </p>
|
4,415,037 | <p>I would like to prove upper and lower bounds on <span class="math-container">$|\cos(x) - \cos(y)|$</span> in terms of <span class="math-container">$|x-y|$</span>. I was able to show that <span class="math-container">$|\cos(x) - \cos(y)| \leq |x - y|$</span>. I'm stuck on the lower bound. Does anyone know how to appr... | jjagmath | 571,433 | <p>Assume there exists a function <span class="math-container">$f:\Bbb R^+ \to \Bbb R$</span> such that <span class="math-container">$\left|\cos x - \cos y\right| \ge f(\left|x-y\right|)$</span> for all <span class="math-container">$x, y$</span>.</p>
<p>Choose any <span class="math-container">$a \ge 0$</span>.</p>
<p>S... |
873,183 | <p>Are variables logical or non-logical symbols in a logic system?
I understand constants are 0-ary logical operation symbols. I think variables are non-logical symbols.</p>
<p>But here are two contrary examples:</p>
<p>It seems that variables are logical symbols in a propositional logic system, according to <a href=... | Mauro ALLEGRANZA | 108,274 | <p>We can agree that there is some "variability" in the practice, regarding the definition (if any) of <em>logical symbols</em> in <em>first-order</em> logic.</p>
<p>According to the definition in Herbert Enderton, <a href="https://books.google.it/books?id=dVncCl_EtUkC&pg=PA69" rel="nofollow">A Mathematical Introd... |
2,707,744 | <h2>The Problem</h2>
<p>I am really struggling to wrap my head around the definition of the integral in measure theory. According to <a href="http://measure.axler.net/" rel="nofollow noreferrer">Axler</a>, the integral of a nonnegative function is defined as follows:</p>
<blockquote>
<p>Suppose $(X, \mathcal S,\mu)... | nonuser | 463,553 | <p>You can not do that since $L$ is not linear transformation. </p>
<p>If $L$ is linear, then $L(\vec{0}) = \vec{0}$ but in your case $L(\vec{0}) = a\ne \vec{0}$</p>
|
3,968,429 | <p>I am working through the Stewart Calculus text independently and am stuck on one of the practice problems (edition 7e - problem 29 in section 1.7).</p>
<p>I am confused by particularly by hint #3 which says:</p>
<p>Why is <span class="math-container">$\lvert x^2 -4x + 4 \rvert = \lvert x + 2 \rvert \lvert x - 2 \rve... | user0102 | 322,814 | <p><strong>Solution</strong></p>
<p>Suppose that <span class="math-container">$|x - 2| \leq \delta$</span>. Then one has that
<span class="math-container">\begin{align*}
|f(x) - f(2)| & = |x^{2} - 4x + 5 - 1|\\\\
& = |x^{2} - 4x + 4|\\\\
& = |x-2|^{2} \leq \delta^{2} = \varepsilon
\end{align*}</span></p>
<... |
3,968,429 | <p>I am working through the Stewart Calculus text independently and am stuck on one of the practice problems (edition 7e - problem 29 in section 1.7).</p>
<p>I am confused by particularly by hint #3 which says:</p>
<p>Why is <span class="math-container">$\lvert x^2 -4x + 4 \rvert = \lvert x + 2 \rvert \lvert x - 2 \rve... | VIVID | 752,069 | <p>Regarding your question, I think there was a typo in that hint #3; that should be <span class="math-container">$$|x^2-4x+4| = |x\color{red}-2||x-2|$$</span> instead.</p>
|
3,104,210 | <p>Is there a commonly accepted notation for a vector whose first <span class="math-container">$k$</span> entries are <span class="math-container">$1$</span>'s, with <span class="math-container">$0$</span>'s afterwards? I have seen <span class="math-container">$\mathbf{e}_i$</span> for a vector with a <span class="mat... | J.G. | 56,861 | <p>Given any unary predicate <span class="math-container">$\phi(u)$</span>, comprehension implies <span class="math-container">$\exists p\forall u (u\in p\iff u\in z\land\phi(u))$</span>, with <span class="math-container">$z$</span> defined as per your usage of pair. For this inference to coincide with pair#, it suffic... |
1,833,406 | <p>Now I have a topological space $X$ that is $C_2$ and $T_4$, and $U$ is an open set in it, I want to show that $U$ can be expressed as $\cup_{i\in\Bbb Z_+} F_i$ where $F_i$ are closed sets, <strong>without the aid of any metrisation theorem</strong> (but Uryshon's theorem about normal spaces and the equivalent Tietze... | Empy2 | 81,790 | <p>Simply multiply each term in $\arctan(x)$ by $x^2+1$, then collect like terms.</p>
|
1,833,406 | <p>Now I have a topological space $X$ that is $C_2$ and $T_4$, and $U$ is an open set in it, I want to show that $U$ can be expressed as $\cup_{i\in\Bbb Z_+} F_i$ where $F_i$ are closed sets, <strong>without the aid of any metrisation theorem</strong> (but Uryshon's theorem about normal spaces and the equivalent Tietze... | C. Dubussy | 310,801 | <p>The key here is that you want the same power of $x$ for all the terms of the sum.</p>
<p>\begin{align*}
(x^2+1)\arctan(x) & = (x^2+1) \sum_{k=0}^{+\infty}(-1)^k\frac{x^{2k+1}}{2k+1}\\
& = \sum_{k=0}^{+\infty}(-1)^k\frac{x^{2k+3}}{2k+1}+\sum_{k=0}^{+\infty}(-1)^k\frac{x^{2k+1}}{2k+1}\\
& = \sum_{k=0}^{+... |
1,955,509 | <p>There's this exercise in Hubbard's book:</p>
<blockquote>
<p>Let $ h:\Bbb R \to \Bbb R $ be a $C^1$ function, periodic of period $2\pi$, and define the function $ f:\Bbb R^2 \to \Bbb R $ by
$$f\begin{pmatrix}r\cos\theta\\r\sin\theta \end{pmatrix}=rh(\theta)$$</p>
<p>a. Show that $f$ is a continuous real-v... | Olivier Oloa | 118,798 | <p>One may start with the standard <strong>finite</strong> evaluation:
$$
1+x+x^2+...+x^n=\frac{1-x^{n+1}}{1-x}, \quad |x|<1. \tag1
$$ Then by differentiating $(1)$ we have
$$
1+2x+3x^2+...+nx^{n-1}=\frac{1-x^{n+1}}{(1-x)^2}+\frac{-(n+1)x^{n}}{1-x}, \quad |x|<1, \tag2
$$ by differentiating once more one gets
$$
... |
1,955,509 | <p>There's this exercise in Hubbard's book:</p>
<blockquote>
<p>Let $ h:\Bbb R \to \Bbb R $ be a $C^1$ function, periodic of period $2\pi$, and define the function $ f:\Bbb R^2 \to \Bbb R $ by
$$f\begin{pmatrix}r\cos\theta\\r\sin\theta \end{pmatrix}=rh(\theta)$$</p>
<p>a. Show that $f$ is a continuous real-v... | robjohn | 13,854 | <p>Using the formula for the sum of a geometric series:
$$
\sum_{k=0}^\infty x^k=\frac1{1-x}\tag{1}
$$
Taking the derivative of $(1)$ and tossing the terms which are $0$:
$$
\sum_{k=1}^\infty kx^{k-1}=\frac1{(1-x)^2}\tag{2}
$$
Taking the derivative of $(2)$ and tossing the terms which are $0$:
$$
\sum_{k=2}^\infty k(k-... |
2,253,768 | <p>I am currently working on a small optimization problem in which I need to find an optimal number of servers for an objective function that incorporates the Erlang Loss formula. To this end, I have been searching for an expression for the first order difference of the Erlang Loss formula with respect to the number of... | egreg | 62,967 | <p>Consider the homomorphism
$$
R[x]\to R/M,\qquad f(x)\mapsto f(0)+M
$$
and compute its kernel and image.</p>
|
923,871 | <p>Let $Y=\{(a,b)∈ \Bbb R\times \Bbb R∣ a≠0\}$. Given $(a,b),(c,d)\in Y$, define $(a,b)*(c,d)=(ac,ad+b)$. Prove that $Y$ is a group with the operation $*$.</p>
<p>I already do the proof of ∗ is an operation on Y. And proof is associative like this:
$$\{(A,B)*(C,D)\}*(E,F)=(A,B)*\{(C,D)\}*(E,F)\}$$</p>
<p>$$(AC,AD+B)(... | Community | -1 | <p><strong>Hint</strong></p>
<p>Prove the associativity of $*$, find the neutral element and prove that each element has an inverse.</p>
|
3,373,576 | <p>What are the non-trivial solutions of
<span class="math-container">$$\tan x = \arctan x$$</span></p>
<p>Can these solutions be expressed e.g. in terms of <span class="math-container">$\pi$</span> or in radicals?
I mean are they some "nice" numbers?</p>
<p>E.g. do we know if these solutions are irrational, ... | hamam_Abdallah | 369,188 | <p>Put
<span class="math-container">$$a=x^2-1=(x-1)(x+1)$$</span>
and
<span class="math-container">$$b=x^2-4x+3=(x-1)(x-3).$$</span></p>
<p>observe that when <span class="math-container">$ x \to 1 $</span>, <span class="math-container">$ a $</span> and <span class="math-container">$b $</span> go to zero. so</p>
<p><s... |
1,361,948 | <p>$\frac{df}{dx} = 2xe^{y^2-x^2}(1-x^2-y^2) = 0.$</p>
<p>$\frac{df}{dy} = 2ye^{y^2-x^2}(1+x^2+y^2) = 0.$</p>
<p>So, $2xe^{y^2-x^2}(1-x^2-y^2) = 2ye^{y^2-x^2}(1+x^2+y^2)$.</p>
<p>$x(1-x^2-y^2) = y(1+x^2+y^2)$</p>
<p>$x-x^3-xy^2 = y + x^2y + y^3$</p>
<p>Is the guessing the values of the variables the only way of so... | Community | -1 | <p><strong>Without computing</strong>:</p>
<p>The denominator of $z$ has non-trivial roots (when the argument of the exponential is $i2k\pi$); not the denominator of $\Re(z)$.</p>
|
2,319,119 | <p>Been given a question and find it to be too vague to know what's going on. </p>
<p>The question is: </p>
<p>$f(x) = 2x + 2$. Define $f(x)$ recursively.</p>
<p>I'm just quite puzzled as there is no $f(0)$, $f(1)$ or $f(x-1)$ function to go by other than the original function.</p>
<p>Supposed to be in the form of ... | DanielWainfleet | 254,665 | <p>(1). If $X$ is a normed linear space and $S\subset X$ then $Y$ is dense in $X$ iff $v+S=\{v+y:y\in S\}$ is dense in $X$ for every $v\in X.$</p>
<p>(2). Let $v_1(x)=-5$ for all $x\in [0,1].$ So $S$ is dense in $C[0,1]$ iff $v_1+S=\{f\in C[0,1]:\int_0^1 f(x)dx=0\}$ is dense.</p>
<p>(3). The function $\psi(f)=\int_0^... |
4,095,715 | <p>I know how to do these in a very tedious way using a binomial distribution, but is there a clever way to solve this without doing 31 binomial coefficients (with some equivalents)?</p>
| Ankit Saha | 876,128 | <p>You can try using the multinomial theorem
<span class="math-container">$$(1+x^2+x^4+x^6)^n = \sum_{0 \le a,b,c,d \le n} \dfrac{n!}{a!~b!~c!~d!}~1^a~x^{2b}~x^{4c}~x^{6d} $$</span>
<span class="math-container">$$ a+b+c+d=n ~~~~;~~~ a,b,c,d \in \mathbb{W}$$</span>
For the coefficient of <span class="math-container">$x... |
313,298 | <p>For example, let's say that I have some sequence $$\left\{c_n\right\} = \left\{\frac{n^2 + 10}{2n^2}\right\}$$ How can I prove that $\{c_n\}$ approaches $\frac{1}{2}$ as $n\rightarrow\infty$?</p>
<p>I'm using the Buchanan textbook, but I'm not understanding their proofs at all.</p>
| Tom Oldfield | 45,760 | <p>In general, we say that a sequence $x_n \rightarrow x$ if $|x_n - x| \rightarrow 0$. This means that for all $\epsilon > 0$ there exists some $N \in \mathbb{N}$ such that for all $n \geq N$, $|x_n - x| < \epsilon$.</p>
<p>We can also use results about sequences that will be proved in any good introductory tex... |
53,989 | <p>The broad, generic and badly posed question may be formulated in this way:</p>
<blockquote>
<p>Let <span class="math-container">$X$</span> be a compact Kälher manifold (or even a projective one). If one considers the Hodge decomposition <span class="math-container">$H^k(X, \mathbb{C}) = \bigoplus_{p+q=k} H^{p,q}(X)$... | Donu Arapura | 4,144 | <p>Marco,</p>
<p>Since I don't fully understand what you are asking, let me instead discuss
what you say is motivating you. If it isn't relevant, you can say so.
When $X$ is a compact Kaehler manifold
then what Simpson calls the Betti moduli space $M_B(X)$
is the set of semisimple representations in $Hom(\pi_1(X),G... |
53,989 | <p>The broad, generic and badly posed question may be formulated in this way:</p>
<blockquote>
<p>Let <span class="math-container">$X$</span> be a compact Kälher manifold (or even a projective one). If one considers the Hodge decomposition <span class="math-container">$H^k(X, \mathbb{C}) = \bigoplus_{p+q=k} H^{p,q}(X)$... | David E Speyer | 297 | <p>This is really a comment on Donu Arapura's answer, but it seemed large enough to deserve it's own post. Working again in the case of $GL_1$, Simpson considers three spaces:</p>
<p>$M_{betti}$: The space of $\mathbb{C}^*$-local systems on $X$. If you like, you can think of this as smooth complex line bundles with an... |
523,376 | <p>Suppose $a_i$ is a sequence of positive integers. Define $a_1 = 1$, $a_2 = 2$ and $a_{n+1} = 2a_n + a_{n-1}$. Does it follow that </p>
<p>$$ \gcd(a_{2n+1} , 4 ) = 1 $$ ???</p>
<p>Im trying to see this by induction assuming above holds, we need to see that $\gcd(a_{2n+3} , 4 ) = 1$.</p>
<p>But, $\gcd(a_{2n+3} , 4 ... | Michael Hoppe | 93,935 | <p>The normal vector field has to be differentiable, in this case it's not even continuous.</p>
|
106,708 | <p><strong>My fragile attempt:</strong> Note that if $1987^k-1$ ends with 1987 zeros, that means $1987^k$ has last digit 1 (and 1986 "next" ones are zeros). For this to be satisfied, $k$ has to be in form $k=4n$, where $n\in N$. This means out number can be written in form </p>
<p>$$ [(1987^n)^2+1][1987^n+1][1987^n-1]... | Michael Sink | 26,102 | <p>Hint: You've already done the hard part, you have identified how the formula can be factored. Don't look for powers of 10, as all 3 terms are divisible by 2. Focus on powers of 5. </p>
|
738,435 | <p>I am having a problem understanding how to determine if a function is one to one.</p>
<p>The problem is: Show that the function f(x) = 3x+4 is one-to-one.</p>
<p>Also, I'm being thrown off by the notation x[subset 1] = x[subset 2], what does that mean, loosely speaking? "In my eyes" that would mean two different v... | Brian J. Fink | 125,131 | <p>Actually, the question of whether a function is one-to-one is answered by whether there is one and only one value of every $f(x)$ for every $x$. Since it can be demonstrated that $f(x)=3x+4$ is linear and continuous (there are no gaps), there is both one and only one value of $3x+4$ for every $x \in \mathbb{R}$. The... |
2,107,990 | <p>I'm looking for suggestions for textbooks covering multivariable calculus. I am looking for two textbooks, one which covers the theory and the other which covers the computational aspects. I have already taken a (not so taught well) first course in multivariable calculus, but I'd ideally like to to keep a computatio... | Batominovski | 72,152 | <p>If $a$, $b$, and $c$ are required to only be positive integers and some of them is $1$, then we have a unique solution $(a,b,c)=(1,1,1)$. For solutions with $a,b,c>1$, note that
$$(a-1)(b-1)(c-1)=abc-bc-ca-ab+a+b+c-1=a+b+c-3\,.$$
Set $x:=a-1$, $y:=b-1$, and $z:=c-1$. Therefore,
$$xyz=x+y+z\,.$$
Without loss of g... |
1,758,194 | <p>Consider the function f: {-1, +1} -> R defined by</p>
<p>$f(x)= \arcsin (\frac{1+x^2}{2x})$.</p>
<p>Due to the following two inequalities :</p>
<p>(i) $1+x^2 \geq 2x$</p>
<p>(ii)$1+x^2 \geq -2x$ , </p>
<p>the function can only be defined at $x=1$ and $x=-1$. I have learnt that the epsilon delta definition only ... | Christian Blatter | 1,303 | <p>It seems that $x$ is tacitly assumed to be real. In this case the given expression is defined only on the two-element set $S=\{-1,1\}\subset{\mathbb R}$. The set $S$ inherits from ${\mathbb R}$ the metric $d(x,y):=|x-y|$, and as $|1-(-1)|=2$ it follows that $S$ is a discrete metric (resp., topological) space. It fo... |
1,668,653 | <blockquote>
<p>If $\sum a_n$ is convergent and $a_n>0$, then $\sum (-1)^n a_n$ is convergent.</p>
</blockquote>
<p>So far I've tried convergence/divergence tests and also I tried to prove this using partial sums. But tests do not work because the latter series <em>includes negative terms</em>. Maybe the Alternat... | frosh | 211,697 | <blockquote>
<p><strong>Theorem:</strong> If $\sum |a_n|$ is convergent, then $\sum a_n$ is convergent.</p>
</blockquote>
<p><strong>Proof:</strong> Let $b_n=|a_n|+a_n \Rightarrow b_n=0 \: \: \mbox{or} \: \: b_n=2|a_n| \Rightarrow 0\le b_n \le 2|a_n| \stackrel{\mbox{DCT}}{\Rightarrow} \sum b_n$ is convergent. Theref... |
4,079,842 | <p>Consider the below function.
<span class="math-container">$$f(x,y) =
\begin{cases}
1 & xy \neq 0 \\
0 & xy = 0
\end{cases}$$</span>
Suppose i want to calculate the below repeated limit.
<span class="math-container">$$\lim_{x\to0}\lim_{y\to0}f(x,y)$$</span>.
In general textbooks in India for BS courses,the a... | José Carlos Santos | 446,262 | <p>The limit <span class="math-container">$\lim_{y\to0}f(x,y)$</span> is <span class="math-container">$0$</span> when <span class="math-container">$x=0$</span> and <span class="math-container">$1$</span> otherwise. But when you compute<span class="math-container">$$\lim_{x\to0}\left(\lim_{y\to0}f(x,y)\right),\tag1$$</s... |
492,125 | <p>How many different equivalence relations can be defined on a set of five elements?</p>
| azarel | 20,998 | <p>Hint: In how many ways can you partition a five element set?</p>
|
384,090 | <p>Find all real numbers $x$ for which $$\frac{8^x+27^x}{12^x+18^x}=\frac76$$</p>
<p>I have tried to fiddle with it as follows: </p>
<p>$$2^{3x} \cdot 6 +3^{3x} \cdot 6=12^x \cdot 7+18^x \cdot 7$$
$$ 3 \cdot 2^{3x+1}+ 2 \cdot 3^{3x+1}=7 \cdot 6^x(2^x+3^x)$$
Dividing both sides by $6$ gives us
$$2^{3x}+3^{3x}=7 \cdot ... | Community | -1 | <p><span class="math-container">$\Rightarrow $</span> <span class="math-container">$6(8^x+27^x)=7(12^x+18^x)$</span></p>
<p>Divide by<span class="math-container">$12^x$</span></p>
<p><span class="math-container">$\Rightarrow $$ 6((\frac{2}{3}) ^x+(\frac{3}{2})^{2x}) =7(1+(\frac{3}{2})^{x})$</span></p>
<p><span class="m... |
4,381,737 | <p><span class="math-container">$\mathbb{Z}[i] = \{a + bi : a, b \in \mathbb{Z}\}$</span> are the Gaussian integers. I want to show that there is no ideal <span class="math-container">$I \subseteq \mathbb{Z}[i]$</span> such that the quotient <span class="math-container">$\mathbb{Z}[i]/I$</span> is a field of size <span... | Claudius | 218,931 | <p>Since you asked whether there are other solutions:</p>
<p>If there existed a ring homomorphism <span class="math-container">$\mathbb{Z}[i] \to \mathbb{F}_3$</span>, then the image of <span class="math-container">$i$</span> in <span class="math-container">$\mathbb{F}_3$</span> would be a square root of <span class="m... |
3,397,834 | <blockquote>
<h2><span class="math-container">$$49y″−98y′+48y= 0 \quad\quad\quad \,\, y(2)=3,y′(2)=9.$$</span></h2>
</blockquote>
<p>When I solved, I got that my <span class="math-container">$r_1= \frac67$</span> and <span class="math-container">$r_2= \frac87.$</span> Then I got that <span class="math-container">$y... | Robert Lewis | 67,071 | <p>Starting with</p>
<p><span class="math-container">$y^5 + x^5 = 8, \tag 1$</span></p>
<p>we differentiate once to obtain</p>
<p><span class="math-container">$5y^4 y' + 5x^4 = 0, \tag 2$</span></p>
<p>from which we may isolate <span class="math-container">$y'$</span>:</p>
<p><span class="math-container">$y' = -\d... |
3,791,350 | <p>An example question is:</p>
<p>In radian measure, what is <span class="math-container">$\arcsin \left(\frac{1}{2}\right)$</span>?</p>
<p>Select one:</p>
<p>a. <span class="math-container">$0$</span></p>
<p>b. <span class="math-container">$\frac{\pi}{6}$</span></p>
<p>c. <span class="math-container">$\frac{\pi}{4}$</... | Air Mike | 802,358 | <p>The function <span class="math-container">$\arcsin$</span> is the inverse of <span class="math-container">$\sin$</span>.</p>
<p>So to compute <span class="math-container">$\arcsin(\frac{1}{2})$</span> we have to see “where” does <span class="math-container">$\sin$</span> of some angle equals <span class="math-contai... |
507,494 | <p>$\begin{align*}\frac{n!}{e}-!n&=n!\sum_{k=n+1}^\infty \frac{(-1)^k}{k!}\\&=\frac{(-1)^{n+1}}{n+1}\left(1-\frac1{n+2}+\frac1{(n+2)(n+3)}-\cdots\right)\end{align*}$</p>
<p>and hence</p>
<p>$$\frac1{n+2} \leq \left|\frac{n!}{e}-!n\right| \leq \frac1{n+1}$$</p>
<p>I can understand how$$ \left|\frac{n!}{e}-!n\... | Silent | 94,817 | <p>I think $$\frac1{n+2} \leq \left|\frac{n!}{e}-!n\right|$$ is because $$\frac{1}{n+1}-\frac{1}{(n+1)(n+2)}=\frac{1}{n+2}$$ and there are many many terms to add in $\frac{1}{n+2}$ so it is less than $\left|\frac{n!}{e}-!n\right|$.</p>
<p>Is my answer correct?</p>
<p>Please verify!</p>
<p>Thank you.</p>
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.