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 |
|---|---|---|---|---|
2,085,622 | <p>The question I am working through states:</p>
<p><em>Three digits are selected at random from the digits 1 through 10, without replacement. Find the probability that at least 2 digits are prime.</em></p>
<p>I feel as though I am close, but not quite hitting the mark with my solution:</p>
<p>My total number of sel... | Kiran | 82,744 | <p>P(exact $1$ prime) = $\dfrac{\dbinom{4}{1}\dbinom{6}{2}}{\dbinom{10}{3}}$<br>
P(No prime) = $\dfrac{\dbinom{6}{3}}{\dbinom{10}{3}}$</p>
<p>Answer <br>
$1-\dfrac{\dbinom{4}{1}\dbinom{6}{2}}{\dbinom{10}{3}}-\dfrac{\dbinom{6}{3}}{\dbinom{10}{3}}$</p>
|
1,815,918 | <p>Let $G$ be a group and $|G|=p^n$ for some prime $p$. If $f:G\to H$ is a surjective homomorphism, how do I know $H=f(G)$ also has cardinality a power of $p$?</p>
| Bérénice | 317,086 | <p>Let $E=\{f:[-1,1]\rightarrow \mathbb{R} | f \text{ is even}\}$ and $O=\{f:[-1,1]\rightarrow \mathbb{R} | f \text{ is odd}\}$. In order to show that $X$ is the direct sum of $E$ and $O$. You need to show two things :</p>
<ul>
<li>$X=E+O$ : Let $f \in X$, let $f_1(x)=\frac{f(x)-f(-x)}{2}$ and $f_2(x)=\frac{f(x)+f(-x)... |
3,066,762 | <p>This is a very interesting identity but I don't know how to prove this,
note that <span class="math-container">$$A_1,\ldots,A_J,B \in \mathbb{R}^{n \times n}$$</span>
and <span class="math-container">$m$</span> is the number of block diagonal in <span class="math-container">$\mathbf{A}$</span> ,so consider
<span cl... | Claude Leibovici | 82,404 | <p><em>Interesting problem, for sure.</em></p>
<p>Using another CAS, I played with the precision. For Mathematica, your first command would be </p>
<blockquote>
<blockquote>
<p>f[a_,b_,n_]:=NIntegrate[BesselJ[0, 2 a Sinh[x/2]] Sin[b x], {x, 0, Infinity},
WorkingPrecision -> n]</p>
</blockquote>
</block... |
2,505,216 | <blockquote>
<p>How can I evaluate<br>
<span class="math-container">$$
\lim_{n \rightarrow \infty} \frac{n\sin n}{2n^2- 1}?
$$</span></p>
</blockquote>
<hr>
<p>Unsuccessful <strong>attempt</strong>:</p>
<p>In the expression <span class="math-container">$\frac{n\sin n}{2n^2 - 1}$</span>, I divided the numerator... | Clement C. | 75,808 | <p>Since $-1\leq \sin x\leq 1$ for all $x$, we have
$$
\left\lvert\frac{n\sin n}{2n^2-1}\right\rvert
= \frac{n\lvert\sin n\rvert}{2n^2-1}
\leq \frac{n}{2n^2-1}\,;
$$
and since $2n^2-1\geq n^2$ for all $n\geq 1$,
$$
\left\lvert\frac{n\sin n}{2n^2-1}\right\rvert
\leq \frac{n}{n^2} = \frac{1}{n} \xrightarrow[n\to\infty]{... |
2,223,788 | <p>$R$ (1 is not assumed to be in $R$) is a prime right Goldie ring (finite dimensional and ACC on right annihilators) which contains a minimal right ideal. Show that $R$ must be a simple Artinian ring. </p>
<p>This appeared in a past paper, the first 2 parts which I managed to prove were "Every essential right ideal ... | Henry | 326,325 | <p>Let $E(R)$ be the right socle of $R$. $E(R)$ is an ideal of $R$, so it is essential as a right ideal and therefore contains a regular element, $c$ say.</p>
<p>$R$ is isomorphic to $cR$, via the isomorphism $r\rightarrow cr$. (If $cr=0$ then $r(c)=0$ implies $r=0$, injective. Surjective obvious.)</p>
<p>$cR$ is ins... |
2,444,591 | <p>How do I prove that: $f(S ∩ T) \neq f(S) ∩ f(T)$ unless $f$ is one-to-one?</p>
<p>I tried finding contradicting example like considering the function $f(x)=x^2$ with $S = \{-2,-1,0,1,2\}$ and $T=\{0,1,2,3,4\}$ or using the function $f(x) = |x|$, but, it seems to be true. But, my book says that In general the afore... | Arthur | 15,500 | <p>Just to be clear, the statement that you're asking about is really this: given a function $f:X\to Y$, the following two statements are equivalent:</p>
<ol>
<li>$f$ is one-to-one</li>
<li>For any two sets $S, T\subseteq X$, we have $f(S\cap T) = f(S)\cap f(T)$</li>
</ol>
<p>Or, to be more specific, you are looking ... |
333,265 | <p>Can $|-x^2 | < 1 $ imply that $-1<x<1$?
My steps are as follows?
$$| -x^2| < 1 $$
$$-1<(-x^2)< 1 $$
$$-1<(-x^2)< 1 $$
$$-1<x^2< 1 $$
$$\sqrt{-1}<x< \sqrt 1 $$</p>
<p>I'm actually looking for the radius of convergence for the power series of $\frac{1}{1-x^2}$:</p>
<p>$$\fra... | Pedro | 23,350 | <p>You want to prove that $|-x^2|<1$ implies $|x|<1$. Note this is the same as $|x^2|<1$ implies $|x|<1$. </p>
<p>Let's prove the <a href="http://en.wikipedia.org/wiki/Contraposition" rel="nofollow">contrapositive</a>. If $|x|\geq 1$ then $|x^2|\geq 1$. If $x=1,-1$ this is clear. Since $|x|\geq 0$, the arg... |
1,318,884 | <blockquote>
<p>Let S be a piecewise smooth oriented surface in <span class="math-container">$\mathbb{R}^3$</span> with positive oriented piecewise smooth boundary curve <span class="math-container">$\Gamma:=\partial S$</span> and <span class="math-container">$\Gamma : X=\gamma(t), t\in [a,b]$</span> a rectifiable para... | Bill Dubuque | 242 | <p><strong>Hint</strong> <span class="math-container">$\,\ n,m\mid j \!\iff\! nm\mid nj,mj\!$</span> <span class="math-container">$\overset{\ \rm\color{darkorange}U}\iff\! nm\mid (nj,mj) \overset{\ \rm \color{#0a0}D_{\phantom |}}= (n,m)j\!$</span> <span class="math-container">$\iff\! nm/(n,m)\mid j$</span></p>
<p>where... |
724,823 | <p>I would like to know if there is some way to imagine the case when a 3D subspace intersects with a 2D plane in a 4D space.
For example, let's have a 3D space in 4D
$$A = \left(\begin{array}{c}1 & 2 & 3 & 4\\7 & 9 & 0 & 5\\ 3 & 3 & 4 & 1\end{array}\right)$$
and
$$B=\left(\begin{arr... | vadim123 | 73,324 | <p>The key result you need is that for a finite-dimensional vector space $V$, with subspaces $A,B$, we have $$\dim(A)+\dim(B)=\dim(A+B)+\dim(A\cap B)$$</p>
<p>Hence $5=\dim(A+B)+\dim(A\cap B)$. Since $B\subseteq A+B$, we have $\dim(A+B)\in\{3,4\}$. If $3$, then $A\cap B=A$. If $4$, then $\dim(A\cap B)=1$.</p>
<hr>... |
2,042,476 | <p>In this question, I'm assuming the definition of Riemann Integrability.<br>
a) Produce an example of a sequence $f_n \rightarrow 0$ pointwise on $[0,1]$ where $\lim_{n\to\infty}\int_0^1f_n$ does not exist.<br>
b) Produce an example of a sequence $g_n$ with $\int_0^1g_n\to0$ but $g_n(x)$ does not converge to zero for... | babakks | 349,014 | <p>I think part (a) is fine. But for part (b) how about this
$$0\leq g_n(x)=\begin{cases}1+(-1)^n & x\in\{0,1\}\\1 &\frac k{2n}\leq x\leq\frac k{2n}+2^{-(n+1)}, k\in \Bbb{N}\\ 0 &(\mathrm{otherwise})\end{cases}$$</p>
<p>For each $n$, $g_n(x)$ equals $1$ over the intervals of the form $$\left[\frac k{2n},\f... |
100,433 | <p>I am trying to export some largish integer arrays to HDF5 and know that every entry in them would fit into an unsigned 8 bit integer array. As the default "DataFormat" for export that Mathematica is using is a 32bit integer array the resulting files are unnecessarily large. Does anyone know the correct syntax to exp... | rafalc | 51,110 | <p>Thie syntax has slightly changed in Mathematica <strong>12</strong>. Now it can be done like this:</p>
<pre><code>In[2]:= Export["test.h5", "path/to/dataset" -> {
"Data" -> RandomInteger[{2^8 - 1}, {100, 100}],
"DataFormat" -> "UnsignedInteger8"
}]
Out[2]= "test.h5"
In[4]:= Import["test.h5", {{"... |
3,646,694 | <p>Let <span class="math-container">$M$</span> be an elliptic element of <span class="math-container">$SL_2(\mathbb R)$</span>. Then it is conjugate to a rotation <span class="math-container">$R(\theta)$</span>. Note that we can calculate <span class="math-container">$\theta$</span> in terms of the trace of <span class... | Community | -1 | <p>Call the roots <span class="math-container">$\alpha, \alpha', \alpha''$</span> with <span class="math-container">$\alpha' = \alpha^2 + 2\alpha - 4$</span>.</p>
<p><span class="math-container">$$p(x) = x^3 + 2x^2 - 5x + 1 = (x - \alpha)(x - \alpha')(x - \alpha'') = q(x)(x - \alpha'')$$</span></p>
<p>with <span clas... |
3,341,784 | <p>How do you get to the general equation of <span class="math-container">$y = ax^2 + bx + c$</span> for a parabola?</p>
<p>I have not found any resources that show how to get to that equation.</p>
<p>This equation is shown 1 minute 30 into this video on Simpsons Rule <a href="https://www.youtube.com/watch?v=vpfy3sGw... | Somos | 438,089 | <p>One way to show that the general quation<span class="math-container">$\,y=ax^2+bx+c\,$</span> is a parabola
is to show that any such equation (with <span class="math-container">$a\ne 0$</span>) is transformed by a unique
<a href="https://en.wikipedia.org/wiki/Homothetic_transformation" rel="nofollow noreferrer">homo... |
4,264,856 | <p>I filled in from the definition of a Stirling number of the second kind that the following holds.
<span class="math-container">$${2n\brace 2} = \frac{1}{2} \sum_{i=0}^{2} (-1)^i \binom{2}{i} (2-i)^{2n}$$</span></p>
<p>And I've visually confirmed in Desmos that the following equality 'appears' to hold.</p>
<p><span c... | HJS | 973,557 | <p>The definition of <span class="math-container">${2n \brace 2}$</span> is dividing set <span class="math-container">$A= \{ a_1,...,a_{2n} \}$</span> into two sets which is not an empty set.</p>
<p>First, lets say that we are dividing set A to set B and set C.
For an integer <span class="math-container">$i$</span> suc... |
686,296 | <p>Let $S$ be a set of rational numbers that is closed under addition and multiplication, and having the property that for every rational number $r$ exactly one the following three statements is true: $r\in S$, $-r\in S$, $r=0$.</p>
<p>There are a couple of questions I have to prove but I am having trouble with the fi... | amWhy | 9,003 | <p><em>Exactly</em> here signifies "one and only one" of the three statements is true for each element in $S$. </p>
<p>$0\notin S$ because if it were an element of $S$, all statements are true, simultaneously, and so it violates the "one and only one" restriction.</p>
<p>To prove $0 \notin S$, you can start by assum... |
1,900,365 | <p>Apparently this is a true statement, but I cannot figure out how to prove this. I have tried setting </p>
<p>$$(m)(m + 1)(m + 2)(m + 3) = (m + 4)^2 - 1 $$</p>
<p>but to no avail. Could someone point me in the right direction? </p>
| Brian Tung | 224,454 | <p>If you have trouble approaching this, try some examples: Note that (for instance)</p>
<p>$$
1 \times \color{red}{2 \times 3} \times 4 = 24 = 5^2-1
$$</p>
<p>$$
2 \times \color{red}{3 \times 4} \times 5 = 120 = 11^2-1
$$</p>
<p>$$
3 \times \color{red}{4 \times 5} \times 6 = 360 = 19^2-1
$$</p>
<p>and observe that... |
1,900,365 | <p>Apparently this is a true statement, but I cannot figure out how to prove this. I have tried setting </p>
<p>$$(m)(m + 1)(m + 2)(m + 3) = (m + 4)^2 - 1 $$</p>
<p>but to no avail. Could someone point me in the right direction? </p>
| Wildcard | 276,406 | <p>This has been very adequately answered according to modern methods; however I feel I may add something by explaining it in the plain English prose approach used in older mathematics texts. My statements can be translated into equations if you so wish and will approximate the proofs already given; however by stating... |
1,900,365 | <p>Apparently this is a true statement, but I cannot figure out how to prove this. I have tried setting </p>
<p>$$(m)(m + 1)(m + 2)(m + 3) = (m + 4)^2 - 1 $$</p>
<p>but to no avail. Could someone point me in the right direction? </p>
| alephzero | 223,485 | <p>You can make the algebra simpler by starting from a better place than $m(m+1)(m+2)(m+3)$. Think symmetry!</p>
<p>$$\begin{align}
(m - \tfrac 3 2)(m - \tfrac 1 2)(m + \tfrac 1 2)(m + \tfrac 3 2)
&= (m^2 - \tfrac 9 4)(m^2 - \tfrac 1 4) \\
&= m^4 - \tfrac 5 2 m^2 + \tfrac{9}{16} \\
&= (m^2 - \tfrac 5 4)^2 ... |
171,602 | <p>In all calculus textbooks, after the part about successive derivatives, the $C^k$ class of functions is defined.
The definition says : </p>
<blockquote>
<p>A function is of class $C^k$ if it is differentiable $k$ times and the $k$-th derivative is continuous.</p>
</blockquote>
<p>Wouldn't be more natural to defi... | bartgol | 33,868 | <p>Well, first of all, if the derivative of order $k$-$1$ is not continuous, then the $k$-th derivative does not even exist.</p>
<p>As for the last one (the $k$-th one), you have a point: why do we need that derivative to be continuous. If we drop this requirement, wouldn't we get a broader class of functions, maybe a... |
36,083 | <p>I'm trying to find a way to prove this:</p>
<p>EDIT: without using LHopital theorem.
$$\lim_{x\rightarrow 1}\frac{x^{1/m}-1}{x^{1/n}-1}=\frac{n}{m}.$$
Honestly, I didn't come with any good idea.</p>
<p>We know that $\lim_{x\rightarrow 1}x^{1/m}$ is $1$.</p>
<p>I'd love your help with this.</p>
<p>Thank you.</p>
| Bill Dubuque | 242 | <p><strong>HINT</strong> $\ $ If you change variables $\rm\ z = x^{1/n} $ then the limit reduces to a very simple first derivative calculation. See also some of my <a href="https://math.stackexchange.com/search?q=user%3A242+derivative+limit">prior posts</a> for further examples of limits that may be calculated simply ... |
1,547,057 | <p>I've been working on the following problems, and I know how to integrate functions,but I do not know how to find the value of "c" in the examples below when finding the antiderivative. Any idea what to do? Cheers</p>
<ol>
<li><p>A particle travels in a straight line such that its acceleration at time <em>t</em> sec... | Ctx | 293,338 | <p>Let me give you a hint for your second problem:</p>
<p>Again, the problem is to find the constant C after integrating from the acceleration to the velocity equation. To achieve this, you have to integrate the velocity equation again, to get the displacement equation.
Now, you can use your knowledge for the displace... |
4,338,558 | <p>I am not sure if my proof here is sound, please could I have some opinions on it? If you disagree, I would appreciate some advice on how to fix my proof. Thanks</p>
<p><span class="math-container">$X_1, X_2, ..., X_n$</span> are countably infinite sets.</p>
<p>Let <span class="math-container">$X_1 = \{{x_1}_1, {x_1}... | Ninad Munshi | 698,724 | <p>We can rewrite a few things to make the computations a bit nicer. First denote <span class="math-container">$G$</span> as</p>
<p><span class="math-container">$$F(x,y) = x^{\frac{5}{3}}y^{\frac{5}{3}}G(x,y)$$</span></p>
<p>which means <span class="math-container">$G$</span> now encompasses all of the change between t... |
215,125 | <p>Suppose $A(x)$ is a $\Delta_0$ formula defining a non-empty set of natural numbers. It's an easy theorem that there is a primitive recursive function $f:\mathbb{N} \rightarrow \mathbb{N}$ such that $Range(f) = \{n \in \mathbb{N} \mid A(n)\}$. I'm wondering if it's known whether either of the following strengthenings... | Connor Mooney | 16,659 | <p>I think that $H_0$ grows like $e^{A}$, which is optimal in light of the example $u(x,t) = e^{-At}\cos(x)$ which solves $u_t - A\Delta u = 0$.</p>
<p>Say we fix $C_i$ and the corresponding Harnack constant is $H$, and that $u$ is positive in $B_2 \times [-4,0]$. The proof is by applying the Harnack inequality to the... |
439,750 | <p>I have to check if $\int_{0}^\infty \mathrm 1/(x\ln(x)^2)\,\mathrm dx $ is convergent or divergent.</p>
<p>My approach was to integrate the function , hence : $\int_{0}^\infty \mathrm 1/(x\ln(x)^2)\,\mathrm dx=-\lim_{x \to \infty} 1/\ln(x)+ \lim_{x \to 0} 1/\ln(x)=0 $</p>
<p>Still my book says that it is diverg... | Ali | 78,741 | <p>Let's change the variables:</p>
<p>$$y=\ln {x} \\ dy =\frac{dx}{x}$$</p>
<p>Now the integral becomes:</p>
<p>$$I=\int_0^\infty \frac{dx}{x(\ln{x})^2}=\int_{-\infty}^\infty \frac{dy}{y^2}.$$</p>
<p>Note that the integrand is an even function, ergo(the main goal is to avoid the singularity):</p>
<p>$$I=2\int_0^{\... |
3,772,371 | <p>Write the polynomial of degree <span class="math-container">$4$</span> with <span class="math-container">$x$</span> intercepts of <span class="math-container">$(\frac{1}{2},0), (6,0) $</span> and <span class="math-container">$ (-2,0)$</span> and <span class="math-container">$y$</span> intercept of <span class="math-... | E. Z. L. | 811,580 | <p>Both forms are equivalent:
<span class="math-container">$$
-6(x-\frac{1}{2})^2(x+2)(x-6)\\
=-\frac{3}{2}\cdot 2^2(x-\frac{1}{2})^2(x+2)(x-6)\\
=-\frac{3}{2}(2(x-\frac{1}{2}))^2(x+2)(x-6)\\
=-\frac{3}{2}(2x-1)^2(x+2)(x-6)
$$</span></p>
|
2,102,619 | <p><strong>What is the total running time of counting from 1 to n in binary if the time needed to add 1 to the current number i is proportional to the number of bits in the binary expansion of i that must change in going from i to i + 1?</strong></p>
<p>Having trouble getting my head a around this one because I cannot... | Teh Rod | 389,818 | <p>$y=f(x)$ set $v$ equal to $y'=\frac{dy}{dx}$ and then rewrite the equation as $v'=(1+v^2)^{3/2}$ so now by separation we have $\frac{1}{\left (1+v^2\right )^{3/2}}\text{d}v=\text{d}x$ The right side of this is trivial, as for the left side making the substitution $v=\tan u\to \text{d}v=\sec^2 u\text{d}u\implies \int... |
345,652 | <p>I try to understand why by definition </p>
<ol>
<li>$[c_0,c_1,\ldots,c_n]=[c_0,[c_1,\ldots,c_n]]$ and also </li>
<li>$[c_0,c_1,\ldots,c_n]=[c_0,c_1,\ldots,c_{n-2},[c_{n-1},c_n]]$ .</li>
</ol>
<p>Those are continued fractions, and $1$ and $2$ are notes I have in the lecture summery.</p>
<p>But we can add bracke... | DonAntonio | 31,254 | <p>By definition:</p>
<p>$$[c_0,c_1,\ldots,c_n]:=c_0+\cfrac{1}{c_1+\cfrac{1}{c_2+\cfrac{1}{\ddots+\cfrac{1}{c_n}}}}=c_0+\cfrac{1}{[c_1,c_2,\ldots,c_n]}=[c_0,[c_1,\ldots,c_n]]\;,\;etc.$$</p>
<p>So yes: you can put the brackets where you want <em>on the right</em>.</p>
|
2,755,091 | <p>I know that I have to prove this with the induction formula. If proved the first condition i.e. $n=6$ which is divisible by $6$. But I got stuck on how to proceed with the second condition i.e $k$.</p>
| José Carlos Santos | 446,262 | <p><strong>Hint:</strong> $7^{n+1}-1=7^{n+1}-7^n+7^n-1=6\times 7^n+7^n-1$.</p>
|
4,008,277 | <p><a href="https://en.wikipedia.org/wiki/Vandermonde%27s_identity#Generalized_Vandermonde%27s_identity" rel="noreferrer">Vadermonde's identity</a> states that</p>
<p><span class="math-container">$$ \sum_{k=0}^{r} {m \choose k}{n \choose r-k} = {m+n \choose r} $$</span></p>
<p>I wonder if we can use this formula or oth... | Trevor | 493,232 | <p>The approach above can be simplified to following general case.</p>
<p>Let <span class="math-container">$f(n):= \sum_{i=1}^{n}{(n\bmod i)}$</span> as described.</p>
<p>Given any pair of natural numbers <span class="math-container">$(n,k)$</span>, we have <span class="math-container">$f(n)=f(n+k)$</span> if and only ... |
3,420,960 | <p>The problem is as follows:</p>
<blockquote>
<p>A protein sample spins in the counterclockwise direction in a
centrifuge seen from the top as shown in the diagram from below. The
radius of the centrifuge es <span class="math-container">$R=2\,m$</span>. The magnitude of its speed
changes. At a certain instant... | Alexander Geldhof | 560,477 | <p>The tangential acceleration has opposite direction from the tangential velocity so there will be tangential deceleration D.</p>
<p>The centripetal acceleration is <span class="math-container">$50 \frac{m}{s^2}$</span>; as @guest stated, this leads to a velocity of <span class="math-container">$\sqrt{50 \frac{m}{s^2... |
2,738,138 | <p>I know it's probably a stupid question, but I'm confused. I have a set {$x\in\mathbb R, \frac{1}{x} \le 1$} that I want to represent as interval/s.</p>
<p>Thinking about it logically, I know that the set is $x\in]-\infty, 0[$U$[1, +\infty[$.</p>
<p>However, when trying to solve the inequality, I can't seem to get ... | Eff | 112,061 | <p>Consider the two cases $x>0$ and $x<0$ separately.</p>
<p>If $x>0$ then the direction of the inequality remains if you multiply by $x$, hence
$$\frac1x \geq 1 \implies \frac1x\times x\geq 1\times x\implies 1\geq x. $$</p>
<p>If $x<0$, then the equality changes direction if you multiply by $x$, hence
$$... |
1,308,872 | <p>Let $H$ be a Hilbert space, and let $T \colon H \to H$ be a bounded linear operator. Then how to show the following?</p>
<p>The range of $T$ is finite-dimensional if and only if $T$ can be represented in the form
$$Tx = \sum_{j=1}^n \langle x, v_j \rangle w_j \ \ \ [ v_j, w_j \in H].$$</p>
<p>My effort: </p>
<p>... | Emilio Novati | 187,568 | <p>Hint:</p>
<p>Let $M,N$ two points on the common tangent at opposite sides with respect to $A$, than $\angle DAM = \angle DEA$ because they subtend the same arc $DA$ ( $\angle DAM$ is a ''limit'' angle, being the side $AM$ tangent, but, if you don't like the concept of ''limit'' angle, you can proof the same claim ... |
1,308,872 | <p>Let $H$ be a Hilbert space, and let $T \colon H \to H$ be a bounded linear operator. Then how to show the following?</p>
<p>The range of $T$ is finite-dimensional if and only if $T$ can be represented in the form
$$Tx = \sum_{j=1}^n \langle x, v_j \rangle w_j \ \ \ [ v_j, w_j \in H].$$</p>
<p>My effort: </p>
<p>... | Archimedesprinciple | 289,804 | <p><a href="https://i.stack.imgur.com/9z9HG.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/9z9HG.jpg" alt="Problem Figure." /></a></p>
<p>Let the common tangent <span class="math-container">$L$</span> be drawn.
Then, by the Alternate Segment Theorem:
<span class="math-container">$$\angle DEA =\angle... |
1,677,141 | <p>Is there a standard symbol for <strong>percentile</strong> in mathematics, much like <code>%</code> is used for <em>percentage</em>? I have trying to get the right answer but only getting conflicting answers and logic.</p>
| Raniz | 249,806 | <p>As <a href="https://math.stackexchange.com/users/10576/neil">Neil</a> mentions in his comment $P_i$ is a common notation to denote the $i$-th percentile.</p>
<p>The <a href="https://en.wikipedia.org/wiki/Percentile" rel="noreferrer">Wikipedia page on Percentile</a> doesn't actually mention the notation as far as I ... |
1,677,141 | <p>Is there a standard symbol for <strong>percentile</strong> in mathematics, much like <code>%</code> is used for <em>percentage</em>? I have trying to get the right answer but only getting conflicting answers and logic.</p>
| Candace Agonafir | 714,145 | <p>If you want to specify the vector (i.e. vector C) and the percentile (i.e. 95), then you may write: percentile(C,95).</p>
|
4,065,655 | <p>If <span class="math-container">$A B = C$</span> where matrix <span class="math-container">$C$</span> is stochastic (all entries are positive and all rows add to <span class="math-container">$1$</span>) then is it necessary that both <span class="math-container">$A$</span> and <span class="math-container">$B$</span>... | nullUser | 17,459 | <p>Indeed, split it up into an integral away from <span class="math-container">$0$</span> and an integral near <span class="math-container">$0$</span>, say <span class="math-container">$(1,\infty)$</span> and <span class="math-container">$[0,1]$</span>. First consider <span class="math-container">$(1,\infty)$</span>.
I... |
4,065,655 | <p>If <span class="math-container">$A B = C$</span> where matrix <span class="math-container">$C$</span> is stochastic (all entries are positive and all rows add to <span class="math-container">$1$</span>) then is it necessary that both <span class="math-container">$A$</span> and <span class="math-container">$B$</span>... | VIVID | 752,069 | <p>According to subadditivity of definite integrals, we split the integral into two as follows
<span class="math-container">$$\int_0^{\infty}x^{t-1}e^{-x}dx = \underbrace{\int_0^1 x^{t-1}e^{-x}dx}_{(1)} + \underbrace{\int_1^\infty x^{t-1}e^{-x}dx}_{(2)}$$</span></p>
<hr />
<ul>
<li>For the first integral, we note that ... |
1,597,332 | <p>Get the following quadratic form:</p>
<p>$$Q(x)=x_1^{2}+x_3^2+4x_1x_2-4x_1x_3 $$</p>
<p>to obtain the canonical form I tried the following: </p>
<p>$$Q(x)=x_1^{2}+x_3^{2}+4x_1x_2-4x_1x_3=4(x_1^{2})+(x_3^{2})-4x_1x_3-3x_1x_1+4x_1x_2-(4/3)x_2x_2+(4/3)x_2x_2=(2x_1-x_3)^2-... $$</p>
<p>There I stopped because I reme... | levap | 32,262 | <p>Since $x_1^2$ appears in $Q(x)$, you should start by writing</p>
<p>$$ Q(x) = (ax_1 + bx_2 + cx_3)^2 + \star $$</p>
<p>for $a, b, c \in \mathbb{R}$ in such a way that $\star$ won't involve $x_1$ at all. Since the terms </p>
<p>$$ x_1^2, 4x_1x_2, -4x_1x_3 $$</p>
<p>appear in $Q$, we choose $a = 1, b = 2, c = -2$ ... |
28,348 | <p>Thomson et al. provide a proof that <span class="math-container">$\lim_{n\rightarrow \infty} \sqrt[n]{n}=1$</span> in <a href="http://classicalrealanalysis.info/documents/TBB-AllChapters-Landscape.pdf#page=95" rel="noreferrer">this book (page 73)</a>. It has to do with using an inequality that relies on the binomial... | Aryabhata | 1,102 | <p>Here is one using $AM \ge GM$ to $1$ appearing $n-2$ times and $\sqrt{n}$ appearing twice.</p>
<p>$$\frac{1 + 1 + \dots + 1 + \sqrt{n} + \sqrt{n}}{n} \ge n^{1/n}$$</p>
<p>i.e</p>
<p>$$\frac{n - 2 + 2 \sqrt{n}}{n} \ge n^{1/n}$$</p>
<p>i.e.</p>
<p>$$ 1 - \frac{2}{n} + \frac{2}{\sqrt{n}} \ge n^{1/n} \ge 1$$</p>
<... |
2,694,122 | <blockquote>
<p>Given that a function $f$ is defined as $$f(x)=1+2x+3x^2+4x^3+...$$. We have to prove that $f$ is continuous on $[0,\frac{1}{8}]$ and evaluate $$\int_{0}^{\frac{1}{8}}f(x)dx$$ </p>
</blockquote>
<p>I am not sure in a particular area. I have proved $f$ is continuous on $[0,\frac{1}{8}]$. But the evalu... | José Carlos Santos | 446,262 | <p>The first mistake that I was able to detect was that you series should start at $1$, not at $0$:\begin{align}\int_0^{\frac18}f(x)\,\mathrm dx&=\sum_{n=1}^\infty\left(\frac18\right)^n\\&=\frac{\frac18}{1-\frac18}\\&=\frac17.\end{align}Another mistake lies in the fact that the answer should be a number; th... |
2,268,225 | <p>I want to investigate the convergence behavior of $$\int_{0}^{\infty} \cos(x^r)\, dx \hspace{5mm} \textrm{and} \hspace{5mm} \int_{\pi}^{\infty} \left(\frac{\cos x}{\log x}\right)\arctan\lfloor x\rfloor dx.$$ My theoretical tools are Abel's test and Dirichlet's test:
Say I have an integral of the form $$\int_{a}^{b... | egreg | 62,967 | <p>The series is
$$
\frac{1}{4}\sum_{n\ge1}nx^{n-1}
$$
where $x=1/2$. Does this remind you about something?</p>
|
602,789 | <p>Find the fundamental group of the surface of a cube with interior of all edges removed (i.e. the space which consists of the vertices and interior of the faces of the cube. </p>
<p>Can I deformation retract the interior of each face into a cross meeting in the middle of the face made up of two diagonal lines which ... | rewritten | 43,219 | <p>It is probably the free product $\mathbb{Z}^{\ast11} = \mathbb{Z}\ast\dots\ast\mathbb{Z}$, as the loops do not commute.</p>
<p>An easier way is that you can "continuously" (not exactly an equivalence but in this case it doesn't matter) deform your set into a cube with $12$ points removed, which is topologically equ... |
4,620,379 | <blockquote>
<p>Let <span class="math-container">$f: [0,∞)→[0,∞)$</span> be a differentiable function, such that <span class="math-container">$ \lim\limits_{x\to\infty}\big(f(x) + f'(x)\big) = 5$</span>. Prove that <span class="math-container">$f$</span> is uniformly continuous.</p>
</blockquote>
<p>Now, both my tutor ... | Hagen von Eitzen | 39,174 | <p>Consider <span class="math-container">$f(x)-5$</span> instead of <span class="math-container">$f$</span> so that the given limit is <span class="math-container">$0$</span> instead of <span class="math-container">$5$</span>, just to simplify things.
By the limit, there exists <span class="math-container">$x_0$</span>... |
4,620,379 | <blockquote>
<p>Let <span class="math-container">$f: [0,∞)→[0,∞)$</span> be a differentiable function, such that <span class="math-container">$ \lim\limits_{x\to\infty}\big(f(x) + f'(x)\big) = 5$</span>. Prove that <span class="math-container">$f$</span> is uniformly continuous.</p>
</blockquote>
<p>Now, both my tutor ... | Thomas Andrews | 7,933 | <p>We will prove if <span class="math-container">$f(x)+f'(x)\to 0$</span> then <span class="math-container">$f(x)\to 0.$</span> We can then apply this to <span class="math-container">$f(x)-5$</span> in the original problem.</p>
<p>This is a little complicated, and Hagen's answer to show <span class="math-container">$f$... |
324,307 | <p><a href="http://en.wikipedia.org/wiki/Tucker%27s_lemma" rel="nofollow">Tucker's Lemma is here.</a></p>
<p>Let's stay within the 2D case for now. A standard proof is constructive:</p>
<p>(1) Pick an arced edge on the boundary of the circle. Note its labeling (for example, (1, 2)).</p>
<p>(2) Walk into the circle... | 2'5 9'2 | 11,123 | <p>Identify $x\mapsto sx+t$ with $\begin{bmatrix}s&t\\0&1\end{bmatrix}$, and you will be able to prove that composition of the affine functions corresponds to multiplication of the corresponding matrices. </p>
<p>Now the given information that $fg=gf$ corresponds to $\begin{bmatrix}f\end{bmatrix}\begin{bmatrix... |
718,901 | <p>I have for a very long time been thinking about certain notations in differential calculus, for instance it might sometimes be sufficient to carry out a variable substitution in integrals. For example we put $t=x^2$ and then write $\frac {dt}{dx} = 2x \Leftrightarrow dt = (2x)dx$</p>
<p>I dont quite understand the ... | DeepSea | 101,504 | <p>dt is the linear functional in term of dx, and it is defined as: dt = t'(x)dx = 2xdx</p>
|
54,088 | <p>I am studying KS (Kolmogorov-Sinai) entropy of order <em>q</em> and it can be defined as</p>
<p>$$
h_q = \sup_P \left(\lim_{m\to\infty}\left(\frac 1 m H_q(m,ε)\right)\right)
$$</p>
<p>Why is it defined as supremum over all possible partitions <em>P</em> and not maximum? </p>
<p>When do people use supremum and w... | Tom Au | 12,506 | <p>A maximum is the largest number WITHIN a set. A sup is a number that BOUNDS a set. A sup may or may not be part of the set itself (0 is not part of the set of negative numbers, but it is a sup because it is the least upper bound). If the sup IS part of the set, it is also the max.</p>
|
657,992 | <p>I understand how to explain but can't put it down on paper.</p>
<p>$\displaystyle \frac{a}{b}$ and $\displaystyle \frac{c}{d}$ are rational numbers. For there to be any rational between two numbers I assume $\displaystyle \frac{a}{b} < \frac{c}{d}$.</p>
<p>I let $\displaystyle x = \frac{a}{b}$ and $\displaystyl... | William Ballinger | 79,615 | <p>One natural way to complete your proof is by contradiction.</p>
<p>You noticed that, between any two rational numbers, there is a third.</p>
<p>Now, pick any two rational numbers $x < y$, and assume that there are a finite number of (say, only $n$) rational numbers between them. Call these numbers $a_0 < a_1... |
657,992 | <p>I understand how to explain but can't put it down on paper.</p>
<p>$\displaystyle \frac{a}{b}$ and $\displaystyle \frac{c}{d}$ are rational numbers. For there to be any rational between two numbers I assume $\displaystyle \frac{a}{b} < \frac{c}{d}$.</p>
<p>I let $\displaystyle x = \frac{a}{b}$ and $\displaystyl... | Cameron Buie | 28,900 | <p>You've got the idea. Let $a_1=\frac{x+y}2,$ and $a_{n+1}=\frac{x+a_n}2.$ A quick induction proof will show that each $a_n$ is rational, that $x<a_1<y$ and that $x<a_{n+1}<a_n$ for all $n.$ Thus, you have explicitly determined infinitely-many rational numbers between $x$ and $y.$</p>
<p>More explicitly, ... |
657,992 | <p>I understand how to explain but can't put it down on paper.</p>
<p>$\displaystyle \frac{a}{b}$ and $\displaystyle \frac{c}{d}$ are rational numbers. For there to be any rational between two numbers I assume $\displaystyle \frac{a}{b} < \frac{c}{d}$.</p>
<p>I let $\displaystyle x = \frac{a}{b}$ and $\displaystyl... | CopyPasteIt | 432,081 | <p>Slight adaption to Steven Stadnicki linear method (see his comment).</p>
<p>Let $p$ and $q$ be any two rational numbers with $p \lt q$. Now<br></p>
<p>$\tag 1 p + 1/n \lt q \text{ iff } 1/n \lt q - p \text{ iff } n \gt \frac{1}{q-p}$.</p>
<p>By Archimedean property, we can find an $n_0$ so that (1) is true, and... |
60,606 | <p>Ok, this will probably be a silly question, but I can't get my head around it.
I'm making a simple module that, given a list and two integers, will swap the position of the elements in the list.(probably there's a function that does it, but I'm interested in doing it myself)
This is what I have so far:</p>
<pre><co... | RunnyKine | 5,709 | <p>The problem here is that in <em>Mathematica</em>, parameters of a function are not local variables. So trying to modify a parameter of a function inside it's body will lead to an error. The reason is that function arguments are evaluated, when the function is called so that it is actually the result of this evaluati... |
60,606 | <p>Ok, this will probably be a silly question, but I can't get my head around it.
I'm making a simple module that, given a list and two integers, will swap the position of the elements in the list.(probably there's a function that does it, but I'm interested in doing it myself)
This is what I have so far:</p>
<pre><co... | eldo | 14,254 | <p>If you want to change the original list "directly" (inplace) you can do it with <code>HoldFirst</code>:</p>
<pre><code>SetAttributes[Exchange, HoldFirst];
Exchange[list_, a_, b_] := list[[{a, b}]] = list[[{b, a}]]
list1 = {1, 2, 3, 4};
Exchange[list1, 3, 1];
list1
</code></pre>
<blockquote>
<p>{3, 1, 2, 4}</... |
1,775,378 | <p>So this question is inspired by the following thread: <a href="https://forums.factorio.com/viewtopic.php?f=5&t=25008">https://forums.factorio.com/viewtopic.php?f=5&t=25008</a></p>
<p>In it, the poster is examining an $8$-belt balancer (more on that to come) which he shows fails to satisfy a desirable proper... | hintss | 172,700 | <p>This isn't quite a math-y answer, but it sounds like you're reinventing the <a href="https://en.wikipedia.org/wiki/Nonblocking_minimal_spanning_switch" rel="noreferrer">nonblocking minimal spanning switch</a> and <a href="https://en.wikipedia.org/wiki/Clos_network" rel="noreferrer">Clos networks</a>. In this case, a... |
3,426,316 | <blockquote>
<p>If <span class="math-container">$f(x)\leq a$</span> then can we say <span class="math-container">$$\int f(x) dx\leq a\int dx?$$</span></p>
</blockquote>
<p>I want to know if this is true or not and can we prove the answer. if it's right can you provide a good simple explanation?
thanks in advance.</p... | Captain Lama | 318,467 | <p>You are right to say that those two definitions do not match: a <span class="math-container">$\mathfrak{g}$</span>-module wher <span class="math-container">$\mathfrak{g}$</span> is seen as a Lie algebra is <em>not</em> a module where <span class="math-container">$\mathfrak{g}$</span> is seen as a nonassociative alge... |
2,843,337 | <blockquote>
<p>If $0 \lt x \lt \dfrac{\pi}{2}$, prove that
$$x^{3/2}\sin x + \sqrt{9-x^3}\cos x \leq 3$$</p>
</blockquote>
<p>This question must be done without calculus. First, I tried splitting it into the intervals $(0,\pi/4)$ and $(\pi/4, \pi/2)$, hoping that, $\sin x$ was bound tightly enough on the interva... | Community | -1 | <p>For all $0\le x\le 3^{2/3}$ (which is $\ge\frac\pi2$) \begin{align}&x^{3/2}\sin x+(9-x^3)^{1/2}\cos x=\\&=\sqrt{(x^{3/2})^2+9-x^3}\left(\frac{x^{3/2}}{\sqrt{(x^{3/2})^2+9-x^3}}\sin x+\frac{(9-x^3)^{1/2}}{\sqrt{(x^{3/2})^2+9-x^3}}\cos x\right)=\\&=3\left(\frac{x^{3/2}}{3}\sin x+\frac{(9-x^3)^{1/2}}{3}\cos... |
2,843,337 | <blockquote>
<p>If $0 \lt x \lt \dfrac{\pi}{2}$, prove that
$$x^{3/2}\sin x + \sqrt{9-x^3}\cos x \leq 3$$</p>
</blockquote>
<p>This question must be done without calculus. First, I tried splitting it into the intervals $(0,\pi/4)$ and $(\pi/4, \pi/2)$, hoping that, $\sin x$ was bound tightly enough on the interva... | Hari Shankar | 351,559 | <p>By Cauchy Schwarz $$ 9 = \left(x^3+(9-x^3) \right) \left(\sin^2 x + \cos^2 x\right) \ge \left(x^{3/2}\sin x + \sqrt{9-x^3}\cos x\right)^2$$ and the result follows</p>
|
407,289 | <p>Is it true that in the category of connected smooth manifolds equipped with a compatible field structure (all six operations are smooth) there are only two objects (up to isomorphism) - <span class="math-container">$\mathbb{R}$</span> and <span class="math-container">$\mathbb{C}$</span>?</p>
| M.G. | 1,849 | <p>Here is a series of standard arguments.</p>
<p>Let <span class="math-container">$(\mathbb{F},+,\star)$</span> be such a field. Then <span class="math-container">$(\mathbb{F},+)$</span> is a finite-dimensional (path-)connected abelian Lie group, hence <span class="math-container">$(\mathbb{F},+) \cong \mathbb{R}^n \t... |
1,143,161 | <p>My question is inspired by the structure of Royden's <em>Real Analysis</em>, which introduces measure theory and Lebesgue integration for $\mathbb{R}$ in its Part I and then reconstructs large portions of those mathematical apparatuses in greater generality by constructing (versions of) them for general metric space... | user169845 | 169,845 | <p>As per Section 9.5 of the text mentioned in the original question, the following theorem, which the authors call the Characterization of Compactness for Metric Spaces theorem, holds for general metric spaces:</p>
<blockquote>
<p>For a metric space $X$, the following three assertions are equivalent:</p>
<p>(i... |
2,426,394 | <p>I'm sorry to bother you with this easy problem. But I'm working alone and totally confused.
It is the 1378th Problem from "Problems in Mathematical Analysis" written by Demidovich. The standard answer is
$$\frac{60(1+x)^{99}(1+6x)}{(1-2x)^{41}(1+2x)^{61}}$$</p>
<p>But when I followed the rutine
$$\frac{d(P(x)/Q(x))... | Piquito | 219,998 | <p>HINT.-Take first derivative of $\dfrac{(1+x)^a}{(1-2x)^b(1+2x)^c}$ and replace the values of $a,b,c$ in your result. This way could be better for your calculations.</p>
|
2,903,259 | <p>I am working on one example from Munkres book "Topology"and I would like to clarify one question.</p>
<p><strong>Example:</strong> Consider the set $[0,1)$ of real numbers and the set $\mathbb{Z}_+$ of positive integers, both in their usual orders; give $\mathbb{Z}_+\times [0,1)$ the dictionary order. This set has ... | José Carlos Santos | 446,262 | <ol>
<li>No. You checked that $\mathbb{Z}_+\times[0,1)$ has the same order type as the set of nonnegative reals.</li>
<li>Right.</li>
<li>Right, for every element of that set.</li>
</ol>
|
1,044,005 | <p>so i'm trying to find the general formula of a sequence, $${18,30,46,66,90,118}$$</p>
<p>i found the first difference, $${12,16,20,24,24}$$</p>
<p>second difference, $${4,4,4,4}$$</p>
<p>If the second difference is 4, you start with $2n^2$</p>
<p>Then i have $${18,30,46,66,90,118}$$</p>
<p>-$${2,8,18,32,50,72}$... | Zubin Mukerjee | 111,946 | <p>If you <em>know</em> that the pattern is a quadratic, then you can use the method of finite differences. Otherwise, there are an infinity of formulas/patterns that will give you those six numbers in order. </p>
<hr>
<p>Anyway, Let your quadratic be $$Q(x) = ax^2 + bx + c$$ with coefficients $a,b,c \in \mathbb{R}$.... |
4,292,822 | <blockquote>
<p>Prove linear independence of <span class="math-container">$1+x^3-x^5,1-x^3,1+x^5$</span> in the Vector Space of Polynomials</p>
</blockquote>
<p>The attempts I found online all are quite easy. You just substitute something in for <span class="math-container">$x$</span> into the equation <span class="mat... | Mark Bennet | 2,906 | <p>Evaluating a polynomial at a point <span class="math-container">$p$</span> is a ring homomorphism whose kernel is the ideal generated by <span class="math-container">$x-p$</span>.</p>
<p>A ring homomorphism always takes zero to zero, so if you have a polynomial equal to zero, and apply the evaluation map, the result... |
1,991,600 | <p><a href="https://i.stack.imgur.com/2L9j9.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/2L9j9.jpg" alt="enter image description here" /></a></p>
<p>I have worked out the areas as <span class="math-container">$\pi/3$</span> for the circle and <span class="math-container">$2/\sqrt3$</span> for the ... | Momo | 384,029 | <p>Just replace $\pi$ with 3.14 and $\sqrt3{}$ with 1.73 (do the students today know how to do the square root by hand any longer?), and calculate approximately. Than see what percentage is closest.</p>
|
228,283 | <p>Could someone give a reference or construct an example of closed subspace of $Y\subset L_1[0,1]$ such that $\operatorname{dist}(x,Y)$ is not attained of for any $x\notin Y$.</p>
<p>I read somewhere that $Y$ is necessarily of infinite dimension and codimension.</p>
| Mikhail Ostrovskii | 37,822 | <p>Consider any functional $f$ which does not attain its norm on
$L_1[0,1]$ (such $f$ exists by James's theorem, but in this case
one can find it without, as an $L_\infty$-function with essential
supremum equal to $1$, which is not attained on a set of non-zero
measure) and let $Y$ be the kernel of $f$. Let $x\notin Y$... |
1,137 | <p>In a freshman course in mathematics for biology students, I have had the issue that basic algebra (e.g. simplifying $\frac{a/b}{c/d}$) is far from being mastered. On the one hand, it seems ill-suited to teach more advanced matters to student failing on the basics. On the other hand, if we take too much time to cover... | Community | -1 | <p>Some strategies could be useful for improving students' basic information without wasting time in the main process of the course.</p>
<ul>
<li>Use a TA and ask him/her to work on students' basic information during some sessions.</li>
<li>Surely there are some students in your class who know the basics better than o... |
1,137 | <p>In a freshman course in mathematics for biology students, I have had the issue that basic algebra (e.g. simplifying $\frac{a/b}{c/d}$) is far from being mastered. On the one hand, it seems ill-suited to teach more advanced matters to student failing on the basics. On the other hand, if we take too much time to cover... | user570 | 570 | <p>I've previously written online algebra tutorials for 2nd year bio students, with online quizzes at the end. It seemed to help. The advantage is you can base the learning materials on the course content to some extent. Also you can let students know early that they may have a problem before its too late to pass or wi... |
2,787,114 | <p>Before going to my question, let me give two prelimilary definitions</p>
<blockquote>
<p><strong>Definition 1.</strong> Let $S\subseteq\mathbb{R}^n$ be a non=empty open set in $\mathbb{R}^n$ under the usual topology on $\mathbb{R}^n$ and $f:S\to \mathbb{R}^m$. Let $\mathbf{c}\in S$ and $g:U(\subseteq \mathbb{R})\... | Fimpellizzeri | 173,410 | <p>Suppose $f$ were not continuous at $c$.
Then</p>
<p>$$\exists \epsilon > 0,\,\forall \delta > 0,\,\exists x\in S \,\text{ with }\, \lVert x - c\rVert < \delta,\, \lVert f(x) - f(c)\rVert \geq \epsilon\tag{$*$}$$</p>
<p>For $n\in\mathbb N$, let $x_n$ be obtained from $(*)$ by taking $\delta = \frac1n$.
Thi... |
481,017 | <blockquote>
<p>Find all $f(x)$ satisfying $f(f(x)) = x^2 - 2$.</p>
</blockquote>
<p>Presumably $f(x)$ is supposed to be a function from $\mathbb R$ to $\mathbb R$ with no further restrictions (we don't assume continuity, etc), but the text of the problem does not specify further. </p>
<p><strong>Possibly Helpful L... | Gottfried Helms | 1,714 | <p>Hmm, if we look at $g(x) = f(f(x)) = x^2-2$ and find the fixpoints $t_0=-1,t_1=2$ then we might assume a solution $h(x)$ for $g(x) = h(x-2)+2 $ and then from this we search for the half-iterate of $h(x)$ defining $h(x-2)+2 = w(w(x-2))+2$ and shall have a solution of the problem by $f(x)=w(x+2)-2$, hopefully having ... |
799,602 | <p>If I have</p>
<pre><code>(¬q ⇒ r)
</code></pre>
<p>and use implication on this does it give me:</p>
<pre><code>(¬¬q v r)
</code></pre>
<p>from where I can then use double negation,</p>
<p>or does it give me </p>
<pre><code>¬(¬q v r)
</code></pre>
<p>I am thinking the former.</p>
<p>Thank you in advance.</p>... | gebruiker | 145,141 | <p>IMHO the authors of the Stacks project do a good job in proving certain theorems on CSA's and Brauer groups. The proofs are very short and to the point. <a href="http://stacks.math.columbia.edu/download/brauer.pdf" rel="nofollow">Here</a>'s a link to the chapter on Brauer groups. </p>
<p>In your particular case th... |
28,166 | <p>With the initial conditions: $a>b>0$;</p>
<p>I need to find $$\lim_{n\to\infty}\sqrt[n]{a^n-b^n}.$$</p>
<p>I tried to block the equation left and right in order to use the Squeeze (sandwich, two policemen and a drunk, choose your favourite) theorem.</p>
| user 1591719 | 32,016 | <p><strong>METHOD I</strong></p>
<p>Since $a>b>0$, our limit boils down to:</p>
<p>$$\lim_{n\to\infty}\sqrt[n]{a^n-b^n}=\lim_{n\to\infty}\sqrt[n]{a^n \left(1-({\frac{b}{a})^{n}}\right)} = \lim_{n\to\infty}\sqrt[n]{a^n}\lim_{n\to\infty}\sqrt[n]{1-(\frac{b}{a})^{n}}=a.$$ </p>
<p><strong>METHOD II</strong></p>
<... |
28,166 | <p>With the initial conditions: $a>b>0$;</p>
<p>I need to find $$\lim_{n\to\infty}\sqrt[n]{a^n-b^n}.$$</p>
<p>I tried to block the equation left and right in order to use the Squeeze (sandwich, two policemen and a drunk, choose your favourite) theorem.</p>
| Zarrax | 3,035 | <p>Another way: Note that $a^n - b^n = (a - b)(a^{n-1} + a^{n-2}b + ... + ab^{n-2} + b^{n-1})$. Since $b < a$, each term in the sum is bounded by $a^{n-1}$ and we have
$$a^{n-1} + a^{n-2}b + ... + ab^{n-2} + b^{n-1} < na^{n-1}$$
Since the sum is at least the first term, we also have
$$a^{n-1} + a^{n-2}b + ... + a... |
1,314,802 | <p>True.</p>
<p>Since $f$ is continuous (because all uniformly continuous function is continuous), we can assume:</p>
<p>$$ f\left(\lim_{n\to\infty} \frac{1}{n}\right) $$</p>
<p>Since $ \lim_{n\to\infty} \frac{1}{n} $ is bounded in $ (0,1] $ and $ I \subset (0,1] $, we have by hypothesis $f$ uniformly continuous the... | Community | -1 | <p>Note that we can write
\begin{equation*}
\lim_{x\to 0}\exp(\frac{\ln(1+\frac{1-\cos(x)}{x})}{x}).
\end{equation*}
Applying L'Hopital's rule twice gives us
\begin{equation*}
\exp(\lim_{x\to 0}\frac{x\cos(x)}{1+2x-\cos(x)+x\sin(x)}) \\
=\exp(\lim_{x\to 0}\frac{x}{1+2x-\cos(x)+x\sin(x)}).
\end{equation*}
Applying L'Ho... |
1,840,176 | <p>I got a task in front of me but I don't really understand it. If someone could explain, I think I would be able to solve it myself.</p>
<blockquote>
<p>$P(x) = \sum_{k=0}^{\infty}a_{k}x^{k}$ is a power series. There exists a $k_{0} \in \mathbb{N}$ with $a_{k} \neq 0$ for all $k \geq k_{0}$.</p>
<p>Proof that... | user247327 | 247,327 | <p>"What is meant by $k_0$ ? It's just any unknown variable which seems to be smaller or equal k, right? Oh and it cannot be smaller than zero."
No. $k_0$ is a constant, not a variable. This is saying that $a_k$ is non-zero for all $a_k$ in the sequence with k larger than $k_0$.</p>
<p>"What is $a_k$? It's just any... |
4,486,425 | <blockquote>
<p>Suppose that we are given reduced rational numbers <span class="math-container">$\,\dfrac{a}{k},\
\dfrac{b}{\ell},\ \dfrac{c}{q},\,$</span> i.e.
<span class="math-container">$\gcd(a,k)=\gcd(b,\ell)=\gcd(c,q)=1$</span> such that
<span class="math-container">$$\frac{c}{q}=\frac{a}{k}-\frac{b}{\ell}.$$</s... | John Omielan | 602,049 | <p>For simpler algebra, let <span class="math-container">$g = \gcd(k,\ell)$</span>, which means <span class="math-container">$\ell = g\ell'$</span> and <span class="math-container">$k = gk'$</span>. This gives that</p>
<p><span class="math-container">$$\begin{equation}\begin{aligned}
\frac{a\ell - bk}{k\ell} & = \f... |
4,486,425 | <blockquote>
<p>Suppose that we are given reduced rational numbers <span class="math-container">$\,\dfrac{a}{k},\
\dfrac{b}{\ell},\ \dfrac{c}{q},\,$</span> i.e.
<span class="math-container">$\gcd(a,k)=\gcd(b,\ell)=\gcd(c,q)=1$</span> such that
<span class="math-container">$$\frac{c}{q}=\frac{a}{k}-\frac{b}{\ell}.$$</s... | Bill Dubuque | 242 | <p><span class="math-container">$\ \,\dfrac{c}q \,=\, {\dfrac{a}k-\dfrac{b}\ell} \,=\!\!\!\!\!\!\!\!\overbrace{\dfrac{a\ell -bk}{k\ell}}^{\!\textstyle\small {\rm cancel}\,\ g \!=\! (k,l) \Rightarrow }\!\!\!\!\!\!\!\!= {\dfrac{\color{#0a0}{a\ell' -bk'}}{\color{#0a0}g\:\!\color{#c00}{k'\ell'}}},\ $</span> for <span clas... |
3,165,494 | <blockquote>
<p>Consider the function <span class="math-container">$\phi(t,x)$</span> where <span class="math-container">$x=(x_1,x_2) \in \mathbb{R}^2$</span>. How can we check if <span class="math-container">$\phi(t,x)$</span> is a flow for a vector field? <span class="math-container">$$\phi(t,x)=(x_1\cos(r^2t)+x_2\si... | Tsemo Aristide | 280,301 | <p><span class="math-container">$\phi_t$</span> is the flow of a vector field if and only <span class="math-container">$\phi_{t+t'}=\phi_t\circ \phi_{t'}$</span>.</p>
<p>If <span class="math-container">$\phi_{t+t'}=\phi_t\circ \phi_{t'}$</span>, write <span class="math-container">$X(x)=lim_{\rightarrow 0}{d\over {dt}}... |
1,405,486 | <p>My question concerns division by zero. Let's say there are two functions, $g(x)$ and $f(x)$ that approach $0$, as $x\to t$, also assume their derivative w.r.t. $x$ is finite as $x\to t$. Using L'Hopital's Rule: $\frac{f(x)}{g(x)} = \frac{f'(x)}{g'(x)}$ $x\to t$, Can you say $\frac{f'(x)}{f(x)} = \frac{g'(x)}{g(x)}$?... | Community | -1 | <p>You cannot say any of those things. Division by $0$ is <strong>ALWAYS</strong> undefined (in the real numbers before anyone says anything).</p>
|
1,405,486 | <p>My question concerns division by zero. Let's say there are two functions, $g(x)$ and $f(x)$ that approach $0$, as $x\to t$, also assume their derivative w.r.t. $x$ is finite as $x\to t$. Using L'Hopital's Rule: $\frac{f(x)}{g(x)} = \frac{f'(x)}{g'(x)}$ $x\to t$, Can you say $\frac{f'(x)}{f(x)} = \frac{g'(x)}{g(x)}$?... | Paramanand Singh | 72,031 | <p>When you talk of $f(x)/g(x)$ then you assume that $g(x) \neq 0$ as $x \to t$. At the same time you assume that $f(x) \to 0, g(x) \to 0$ as $x \to t$. This is the typical use-case where L'Hopital's Rule can be applied provided $f'(x)/g'(x)$ tends to a limit as $x \to t$.</p>
<p>Now note that if $f'(x)/g'(x) \to L$ a... |
2,840,091 | <blockquote>
<p>Consider the linear map$:T:\mathbb{R}^3 → \mathbb{R}$ with
$$T\left(\begin{bmatrix} x \\ y \\ z \end{bmatrix}\right)=x-2y-3z$$
Find the basis of its kernel.</p>
</blockquote>
<p><strong>My try</strong></p>
<p>Since the plane is the nullspace of the matrix
$$A=\begin{bmatrix} 1 & -2 & -3 ... | Mohammad Riazi-Kermani | 514,496 | <p>The Kernel of $T$ is a two dimensional space.</p>
<p>You need to find two linearly independent vectors satisfying $$ x-2y-3z=0 $$</p>
<p>Finding such vectors is achieved by choosing $x$ and $y$ values and solving for $z$</p>
<p>I have found $(3,0,1)$ and $(0,3,-2)$ to serve as a basis for the kernel of T. </p>
|
2,702,329 | <blockquote>
<p>Let $A$ be an $m\times n$ matrix, let $v \in \mathbb{R}^m$, and let $S$ be the set of least squares solutions to the system $Ax=b$. Show that there exists a unique <em>minimal least square solution</em> to $Ax=b$; that is, there exists some $x\in S$ such that $||x|| \le||y||$ for all $y\in S$.</p>
</b... | Siong Thye Goh | 306,553 | <p>We can't have $$\exists x \in S, \|x\| < \|y \|, \forall y \in S$$</p>
<p>because this would imply that $\|x\| < \|x\|$ such $x \in S$.</p>
<p>However, we can write </p>
<p>$$\exists x \in S, \|x\|<\|y\|, \forall y \in S \setminus \{x\}.$$</p>
<p>A minimal point need not exists in general, for example ... |
1,501,736 | <p>Let $x_1 = 1$ and for $n \ge2, x_n = \sqrt{3 + x_{n-1}}$, I just want to know how to determine the limit. </p>
| Lutz Lehmann | 115,115 | <p><strong>Hint:</strong> If you know the Banach fixed-point theorem, then $f(x)=\sqrt{3+x}$ has Lipschitz constant $1/4$ on $[1,\infty)$.</p>
|
1,773,060 | <p>Is every (non-trivial) quotient of a Boolean algebra isomorphic to a subalgebra of that Boolean algebra? And conversely is every subalgebra isomorphic to a quotient algebra?</p>
| Robert Furber | 184,596 | <p>For the first question, the answer is no, there are quotient algebras that cannot be embedded as subalgebras. Let's take <span class="math-container">$\newcommand{\powerset}{\mathcal{P}}\newcommand{\N}{\mathbb{N}}A = \powerset(\N)$</span>, the ideal <span class="math-container">$I \subseteq A$</span> to be the set o... |
2,136,183 | <p>Find integers x,y such that the repeating decimal 0.712341234.... = x/y.</p>
<p>I would actually do this problem if the 7 was not there. If the 7 was not there, my proof would be as follows.</p>
<p>proof:</p>
<p>Let z = 0.12341234...</p>
<p>Then 10^4z = 1234.1234</p>
<p>10^4z - z = 1234</p>
<p>z = 1234/(10^4 -... | fleablood | 280,126 | <p>Let $x=.712341234.... $</p>
<p>$10000x=7123.41234..... $</p>
<p>$9999x =7122.7$</p>
<p>$x =\frac {71227}{99990} $</p>
<p>May be able to reduce. Or maybe not.</p>
<p>The irregular 7 doesn't change anything seriously. </p>
<p>$1/n $ will reduce to a purely repeating pattern but most $m/n $ will not.</p>
<p>...o... |
90,006 | <p>I have two problem collections I am currently working through, the "Berkeley Problems in Mathematics" book, and the first of the three volumes of Putnam problems compiled by the MAA. These both contain many problems on basic differential equations.</p>
<p>Unfortunately, I never had a course in differential equation... | Zev Chonoles | 264 | <p>Let $X$ be some number <em>other</em> than $\frac{1}{2}$. Let's define $d=|X-\frac{1}{2}|$, and because $X\neq\frac{1}{2}$ we have $d>0$. </p>
<p>The <a href="http://en.wikipedia.org/wiki/Triangle_inequality#Reverse_triangle_inequality" rel="nofollow">reverse triangle inequality</a> says that for any $a$ and $b$... |
2,497,799 | <h2>Question</h2>
<p>While Solving a recursive equation , i am stuck at this summation and unable to move forward.Summation is </p>
<blockquote>
<p>$$\sum_{j=0}^{n-2}2^j (n-j)$$</p>
</blockquote>
<h2>My Approach</h2>
<blockquote>
<p>$$\sum_{j=0}^{n-2}2^j (n-j) = \sum_{j=0}^{n-2}2^j \times n-\sum_{j=0}^{n-2}
2^... | Leucippus | 148,155 | <p>In general:
\begin{align}
\sum_{k=0}^{n} x^{k} &= \frac{1 - x^{n+1}}{1-x} \\
\sum_{k=0}^{n} k \, x^{k} &= \frac{x \, (1 - (n+1) x^{n} + n x^{n+1})}{(1-x)^{2}}
\end{align}
Now
\begin{align}
\sum_{k=0}^{n} k \, \frac{1}{x^{k}} &= \frac{n - (n+1) x + x^{n+1}}{x^{n} \, (1-x)^{2}} \\
\sum_{k=0}^{n} k \, x^{n-... |
4,230,409 | <p>I recently stumbled upon Edwin Moise's proof of the area of a square, but there is something that is bothering me in his proof - he claims that since the inequalities <span class="math-container">$(1)$</span> and <span class="math-container">$(5)$</span> are true, then <span class="math-container">$a =\sqrt{S_a}$</s... | Sayan Dutta | 943,723 | <p>Here's another mathematical proof of the given fact. This is not really radically different from the last one, but it's more geometric in the sense that it makes the picture much more clear.</p>
<p>We will use <span class="math-container">$b=\sqrt{\alpha S_a}$</span> in this argument.</p>
<p>If possible let <span cl... |
3,036,489 | <blockquote>
<p>For <span class="math-container">$k, l \in \mathbb N$</span>
<span class="math-container">$$\sum_{i=0}^k\sum_{j=0}^l\binom{i+j}i=\binom{k+l+2}{k+1}-1$$</span>
How can I prove this?</p>
</blockquote>
<p>I thought some ideas with Pascal's triangle, counting paths on the grid and simple deformation ... | Robert Z | 299,698 | <p>Your idea about a <strong>combinatorial proof</strong> which is related to counting paths in a grid is a good one!</p>
<p>The binomial <span class="math-container">$\binom{i+j}i$</span>
counts the paths on the grid from <span class="math-container">$(0,0)$</span> to <span class="math-container">$(i,j)$</span> movi... |
11,435 | <p><strong>Question</strong>: What is the correct notion of a <em>product</em> of integral (or rational) polytopes which induces a factorization of its Ehrhart (quasi-)polynomial into two primitive Ehrhart (quasi-)polynomials corresponding to its constituent polytopes, viz., $L_{P \times Q}(t) = L_{P}(t) L_{Q}(t)$? </p... | Robin Chapman | 226 | <p>Isn't this obvious (or have I missed something?)</p>
<p>The appropriate product of rational polytopes $P\subseteq\mathbb R^m$
and $Q\subseteq\mathbb R^n$ is surely their Cartesian product
$P\times Q\subseteq\mathbb R^{m+n}$. For a positive
integer $t$, the integer points in $t(P\times Q)$ are those of the form
$(a... |
2,118,230 | <blockquote>
<p>Let $G$ be a group and let $H,K \le G$ be subgroups of finite index, say $|G:H| = m$ and $|G:K|=n$. Prove that $\mathrm{lcm}(m,n) \le |G:H \cap K| \le mn$. </p>
</blockquote>
<p>I was able to establish the upper bound, but I am having difficulty establishing the lower bound, so I consulted <a href="h... | Stefan4024 | 67,746 | <p><strong>HINT:</strong> $$[G:H \cap K] = [G:H][H:H \cap K]$$</p>
|
1,515,645 | <p>The question is to find the values of a real number $\lambda$ for which the following equation is satisfied for all real values of $\alpha$ which are not integral multiples of $\pi/2$
$${\sin\lambda\alpha\over \sin\alpha}-{\cos\lambda\alpha\over \cos\alpha}=\lambda-1$$</p>
<p>All I could do was to guess some values... | some one | 284,056 | <p>$1111111111111111=10^{15} + ... + 1 = \large{(10^{16} -1)\over9}$</p>
<p>$10^2 \equiv -2\pmod {17}$</p>
<p>$10^{16} - 1 = (-2)^8 - 1 = 256 -1 \equiv 1 -1 = 0\pmod {17}$</p>
<p>$11111111= \large{(10^8 -1 )\over9} $</p>
<p>$10^2 \equiv -2\pmod {17}$</p>
<p>$10^8 - 1 \equiv (-2)^4 - 1 = 15 \pmod {17}$</p>
|
1,515,645 | <p>The question is to find the values of a real number $\lambda$ for which the following equation is satisfied for all real values of $\alpha$ which are not integral multiples of $\pi/2$
$${\sin\lambda\alpha\over \sin\alpha}-{\cos\lambda\alpha\over \cos\alpha}=\lambda-1$$</p>
<p>All I could do was to guess some values... | 2'5 9'2 | 11,123 | <p>Do you know that $a^{p-1}-1$ is always divisible by $p$ when $p$ is prime and $a$ is not divisible by $p$? It's a famous result. Take $a=10$ and $p=17$ to get $9999999999999999$. And dividing out $9$ won't change divisibility by $17$.</p>
<p>$11111111$ is just too easy to outright factor. It's clearly divisible by ... |
259,431 | <p>In the book of Richard Hammack, I come accross with the following question:</p>
<blockquote>
<p>There are two different equivalence relations on the set $A = \{a,b\}$.
Describe them.</p>
</blockquote>
<p>OK, I found that the solution is,</p>
<p>$$R_1 = \{(a,a),(b,b),(a,b),(b,a)$$
and
$$R_2 = \{(a,a),(b,b)\}$$<... | Isomorphism | 16,823 | <p>Read the reflexive property again. </p>
<p>Reflexive property is not conditional.</p>
|
3,612,316 | <p>I am trying to interpret an equation, but can't understand how the less-than signs work:</p>
<p><span class="math-container">$\sum _{n=1}^{\infty } \frac{1}{n^2}<1+\sum _{n=1}^{\infty } \frac{1}{n (n+1)}=1+\sum _{n=1}^{\infty } \left(\frac{1}{n}-\frac{1}{n+1)}\right)=1+1=2$</span></p>
<p>The first part returns ... | Wouter | 89,671 | <p>Your statement "unpacks"</p>
<p>First
<span class="math-container">$$\sum _{n=1}^{\infty } \frac{1}{n^2}<1+\sum _{n=1}^{\infty } \frac{1}{n (n+1)}$$</span>
second
<span class="math-container">$$1+\sum _{n=1}^{\infty } \frac{1}{n (n+1)}=1+\sum _{n=1}^{\infty } \left(\frac{1}{n}-\frac{1}{n+1)}\right)$$</span>
thir... |
2,598,848 | <p>I have a project I'm doing in desmos and want to know if this is possible.</p>
<p>I want to be able to set some parameter, a, equal to the solution of f(x) = 0. I plugged my specific equation into Maple and it could not arrive at an explicit solution for x.</p>
<p>However, when I type in the equation f(x) = 0 into... | Mostafa Ayaz | 518,023 | <p>Consider the picture below:</p>
<p><a href="https://i.stack.imgur.com/n285G.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/n285G.jpg" alt="enter image description here"></a></p>
<p>The description of two ellipses in polar coordinates is:
$$r^2={1\over {{\cos^2\theta\over{a^2}}+{\sin^2\theta\ove... |
1,583,540 | <p>Is it possible to find an example of an one-sided inverse of a function? other than matrix?</p>
<p>I am trying to find such an example but having no luck. Anybody got an idea about it?</p>
| Tim Raczkowski | 192,581 | <p>Here's another example:
Let $f:\Bbb R\to[0,\infty)$ be defined by $f(x)=x^2$, and $g:[0,\infty)\to\Bbb R$ by $g(x)=\sqrt x$.</p>
<p>Now,
$$(f\circ g)(x)=\sqrt x^2=x,$$
but
$$(g\circ f)(x)=\sqrt{x^2}=|x|.$$</p>
|
244,048 | <p>A couple days ago I posted this on MSE (<a href="https://math.stackexchange.com/questions/1853774/determining-a-function-is-harmonic-from-mean-value-property-for-just-three-ra">here</a>) but in retrospect it might be more appropriate for this site.</p>
<p>This theorem is well-known (maybe it can be called Morera's ... | Dave Witte Morris | 68,305 | <p>Yes, in fact you only need two radii. More precisely, a theorem on page 167 of <a href="http://dx.doi.org/10.2307/2321600">this Monthly paper of Zalcman</a> says any two radii $r_1$ and $r_2$ will work unless the quotient $r_1/r_2$ is a quotient of zeros of a certain explicit function. The author says this result "w... |
602,353 | <p>Let $f: \mathbb{R} \to \mathbb{R}$ be given by $$f(x)= \begin{cases}
-x-1, &\text{if }x≥0\\
1, &\text{if }x<0.
\end{cases}$$</p>
<p>Denote by</p>
<ul>
<li>$U$ the <em>usual</em> topology on $\mathbb{R}$.</li>
<li>$H$ the <em>half-open interval</em> topology on $\mathbb{R}$.</li>
<li>$C$ is the <em>open ... | user642796 | 8,348 | <p>We should start by looking at what the inverse images of various basic open sets look like. This will help us determine the continuity of the function. (And plotting the graph of the function is quite helpful.)</p>
<p>Let's start out with $U$, the usual topology. The basic open sets in this topology are just the ... |
1,335,734 | <blockquote>
<p>A library leases its photocopier. One monthly bill was 750 dollars for
12,000 copies. Another month, the bill was $862.50 for 16,500 copies.
How much does the library pay for each copy?</p>
</blockquote>
<p>How should I start set up the equation for this problem?
How exactly do I find the linear... | SamC | 139,011 | <p>Total derivative: ${df \over dx}={\partial f \over \partial x}{dx \over dx} + {\partial f \over \partial y}{dy \over dx}$ and ${dx \over dx }=1$</p>
<p>So,</p>
<p>$${d \over{dx}}3(x^2+y^2)^2 = {\partial \over \partial x}3(x^2+y^2)^2 + {\partial \over \partial y}3(x^2+y^2)^2 {dy \over dx} $$</p>
<p>You should be a... |
187,511 | <p>There are two questions:</p>
<ol>
<li><p>How to prove that in general </p>
<p>$[\hat{A}(\mathbb HP^m)]_{4m} = 0$</p>
<p>It is possible to verify it for low values of $m$.</p></li>
<li><p>How to prove that in general</p>
<p>$\left[\frac{\hat{A}(\mathbb HP^m)} { \hat{M}(\mathbb HP^m) }\right]_{4m} = 0$</p>
<p... | Johannes Ebert | 9,928 | <p>Since $HP^n$ is a spin manifold and since the homogeneous metric has positive scalar curvature, the $\hat{A}$-genus of $HP^n$ is zero by the index theorem and the Weitzenboeck-Lichenrowicz formula.</p>
<p>I cannot at the moment answer the question for the quotient by the Mayer class. But there is a direct computati... |
2,111,579 | <p>A question:</p>
<p>If $\displaystyle{m}^{2}={1}-{m},{\quad\text{and}\quad}{n}^{2}={1}-{n},{\quad\text{and}\quad}{n}\ne{m};$</p>
<p>Proof that $\displaystyle{m}^{7}+{n}^{7}+{30}={1}$</p>
<p>Without finding the roots of equation $\displaystyle{x}^{2}+{x}-{1}={0}$.</p>
<p>Is there such a shortcut solution?</p>
| Jean Marie | 305,862 | <p>I would like to present a very simple solution by interpretation of these matrices as operators on $\mathbb{R^n}$ (which will surprise nobody...). Triangular matrix $A$ acts as a <strong>discrete integration operator</strong>: </p>
<p>For any $x_1,x_2,x_3,\cdots x_n$:</p>
<p>$$\tag{1}A (x_1,x_2,x_3,\cdots x_n)^T=(... |
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