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 |
|---|---|---|---|---|
2,396,287 | <p>The product has only positive factors so it has zero as lower bound. Also the product is decreasing as all its factors are less than one. In conclusion the series must have a limit. I also compute the first 150 values of the product and I got around 0.297. I believe that the product converges very, very, slowly to z... | Robert Israel | 8,508 | <p>$$\ln \left(\dfrac{\ln(2n)}{\ln(2n+1)}\right) \sim \ln\left(1 - \frac{1}{2 n \ln(n)}\right) \sim - \frac{1}{2 n \ln(n)}$$
and $\sum_n \frac{1}{n \ln(n)}$ diverges, so the limit is $0$.</p>
|
27,126 | <p>$$e^{\pi i} + 1 = 0$$</p>
<p>I have been searching for a convincing interpretation of this. I understand how it comes about but what is it that it is telling us? </p>
<p>Best that I can figure out is that it just emphasizes that the various definitions mathematicians have provided for non-intuitive operations (com... | gowers | 1,459 | <p>I'd like to add something to the visual answer above (or below, or wherever it ends up). It was not until I was into my 40s that I realized that there was an intuitive way of understanding that $e^{i\pi}=-1$, as opposed to the power-series derivation that seems a bit too formal somehow. (What I'm about to say comes ... |
27,126 | <p>$$e^{\pi i} + 1 = 0$$</p>
<p>I have been searching for a convincing interpretation of this. I understand how it comes about but what is it that it is telling us? </p>
<p>Best that I can figure out is that it just emphasizes that the various definitions mathematicians have provided for non-intuitive operations (com... | Bo Peng | 4,782 | <p>Consider the exponential map for the Lie group $U(1)$. </p>
<p>The formula $e^{\pi i}$ means: starting from the identity element $1$, keep traveling "in the direction of $i$" (Which is "going up" at first, because the imaginary axis is of course going up. But this direction "changes" on the $\mathbb{R}^2$-plane as ... |
1,336,419 | <p>What are the three final numbers of $2003^{2003}$ and $2003^{2003^{2003}}$? </p>
<p>Do I use the Chinese Remainder Theorem here, and if so, how?</p>
| ajotatxe | 132,456 | <p><a href="https://en.wikipedia.org/wiki/Euler%27s_theorem" rel="nofollow">Euler's theorem</a> guarantees that, if $\gcd(a,1000)=1$ then
$$a^{400}\equiv 1\pmod {1000}$$
that is, the three last digits of $a^{400}$ will be $001$.</p>
<p>Then, since $2003\equiv 3\pmod {1000}$ and $2003\equiv 3\pmod{400}$, then
$$2003^{2... |
256,298 | <p>I am looking at an example problem in my text:</p>
<p>"Determine whether these system specifications are consistent:</p>
<p>'The diagnostic message is stored in the buffer or it is re-transmitted.'</p>
<p>'The diagnostic message is not stored in the buffer.'</p>
<p>'If the diagnostic message is stored in the buf... | user595454 | 595,454 | <p>Pretty confusing!
Let p denote “The diagnostic message is not stored in the buffer.” Let q denote “The diagnostic message is retransmitted” The specification can be written as p ∨ q, p→ q, and negation of p </p>
<p>When p is false and q is true all three statements are true. So the specification is must be consist... |
1,541,623 | <p>So I'm just getting the grasp of set theory and I have this question.</p>
<blockquote>
<p>Let $|A| = m$ and $|B| = n$. What is the cardinality of the set $A \times B
$?</p>
</blockquote>
<p>I put $\{1,1\}$ as the answer however I wasn't totally sure what the two vertical bars between set $A$ and set $B$ mean. If... | Eric Wofsey | 86,856 | <p>The notation $|A|$ is just a shorthand for "the cardinality of $A$".</p>
|
1,986,333 | <p>Given that in a finite field $K$ the equation $x^2 = 1$ has zero or two solutions depending if $1 \neq -1$, is it true that $x^2 = a$ has at most two roots for a given $a \in K$?</p>
| Ben Grossmann | 81,360 | <p>That's right. If $a$ has some square root $b$, we can write
$$
x^2=b^2\implies (x-b)(x+b)=0
$$
Recall that fields do not have zero divisors.</p>
|
53,073 | <p>Suppose $X$ is a non-explosive diffusion with dynamics</p>
<p>$dX_t = \mu(X_t)dt + \sigma(X_t)dW_t$, </p>
<p>where $W$ is a standard Brownian motion. My intuition about $X$ is that if $\mu$ and $\sigma$ are sufficiently nice, then the sample paths of $X$ are in some sense "deformed" sample paths of $W$. Is there a... | Leonid Petrov | 979 | <p>You should probably look at the Girsanov's theorem <a href="http://en.wikipedia.org/wiki/Girsanov_theorem" rel="nofollow">http://en.wikipedia.org/wiki/Girsanov_theorem</a></p>
<p>The process $X$ is a probability distrubution on the space of continuous functions, so is the Wiener process $W$. Girsanov' theorem state... |
53,073 | <p>Suppose $X$ is a non-explosive diffusion with dynamics</p>
<p>$dX_t = \mu(X_t)dt + \sigma(X_t)dW_t$, </p>
<p>where $W$ is a standard Brownian motion. My intuition about $X$ is that if $\mu$ and $\sigma$ are sufficiently nice, then the sample paths of $X$ are in some sense "deformed" sample paths of $W$. Is there a... | Piotr Miłoś | 1,302 | <p>Another way to approach the problem is as follows.</p>
<p>One can notice that $X$ is a semimartingale (probably under some mild assumptions on $\sigma,\mu$). The martingale $M$ part of $X$ can be represented as</p>
<p>$$M_t = B_{< M,M>_t},$$
where $B$ is some Brownian motion. This is know as "Dambis, Dubins-... |
786,596 | <p>So I'm trying to solve this practice exam question, </p>
<blockquote>
<p>Let $G$ be a planar graph with at least two edges and does not contain $K_{3}$ as a subgraph. Prove that $|E|\leq 2|V|-4$.</p>
</blockquote>
<p>Now I started doing this by induction, but it seems to me like the base-case is a counter-examp... | Bill Dubuque | 242 | <p><strong>Hint</strong> $\ $ Suppose that $\ \color{#c00}{11^n = 2^n\! + 9k}.\ $ Then</p>
<p>$\qquad \begin{eqnarray} 11^{n+1}&=\,&\quad 11\cdot \color{#c00}{11^n}\\ &=& (2\!+\!9)(\color{#c00}{2^n\!+9k})\\ &=&\quad 2^{n+1}\! + 9(\cdots)\end{eqnarray} $</p>
<p>which yields the induction step.... |
786,596 | <p>So I'm trying to solve this practice exam question, </p>
<blockquote>
<p>Let $G$ be a planar graph with at least two edges and does not contain $K_{3}$ as a subgraph. Prove that $|E|\leq 2|V|-4$.</p>
</blockquote>
<p>Now I started doing this by induction, but it seems to me like the base-case is a counter-examp... | A S D | 621,315 | <p>One of numerous possible approaches using Binomial theorem</p>
<p><span class="math-container">$$11^n -2^n=(10+1)^n -\sum_{j=0}^{n}\binom{n}{j} =\\
\sum_{j=0}^{n}\binom{n}{j}10^{n-j}-\sum_{j=0}^{n}\binom{n}{j} \\
=\sum_{j=0}^{n}\binom{n}{j}\left(10^{n-j}-1 \right) \equiv 0\,(mod\,9)$$</span></p>
<p>The identity <spa... |
3,042,433 | <blockquote>
<p>Rosanne drops a ball from a height of 400 ft. Find the ball's
average height and its average velocity between the time it is
dropped and the time it strikes the ground.</p>
</blockquote>
<p>My trial...</p>
<p>So, I tried to use average value theorem for integrals. I took acceleration as positive... | Sameer Baheti | 567,070 | <p>Let <span class="math-container">$H=400\text{ft}$</span>.
<span class="math-container">$$h(t)=H-\frac12gt^2$$</span>
Average height = particular height <span class="math-container">$\times$</span> time during which it was on that height/ total time.
<span class="math-container">$$\frac12gT^2=H\implies T=\sqrt{\frac... |
1,258,198 | <p>Suppose $a > 1$. I want to compare
$$\int_0^{\infty} \frac{e^{-ax}}{1+x^2}\,\,dx$$ and $$\int_0^{\infty} \frac{e^{-2ax}}{1+x^2}\,\,dx$$</p>
<p>My instinct suggests that after a certain value of $a$, $$\int_0^{\infty} \frac{e^{-2ax}}{1+x^2}\,\,dx < e^{-a}\int_0^{\infty} \frac{e^{-ax}}{1+x^2}\,\,dx$$</p>
<p>bu... | lvb | 1,403 | <p>One can show that your integral behaves like $1/a$ as $a\to\infty$. In particular the integral does not decay exponentially, and your claim does not hold.</p>
<p>We can use the substitution $y=ax $ to rewrite the integral:
$$\int_0^\infty \frac {e^{-ax}}{1+x^2}dx= \int_0^\infty \frac {a e^{-y}}{a^2+y^2}dy.$$</p>
<... |
1,028,448 | <p>I am working on a problem to find the orbits of the general linear group $\mathrm{GL}_n(\mathbb{R})$, acting on $\mathbb{R}^n$, with the invertible matrix $A$ acting on a column vector $x \in \mathbb{R}^n$ by taking it to the vector $Ax$.</p>
<p>I have already verified that this is a group action, but I'm not sure ... | Jim | 56,747 | <p>The two orbits are $\{0\}$ and $\mathbb R^n \setminus \{0\}$. It's pretty clear the first is an orbit, so for the second you have to prove that if $v, w \neq 0$ then there exists an $A$ such that $Av = w$. To do this I would suggest thinking of $A$ as a linear transformation of the vector space $\mathbb R^n$.</p>
|
1,028,448 | <p>I am working on a problem to find the orbits of the general linear group $\mathrm{GL}_n(\mathbb{R})$, acting on $\mathbb{R}^n$, with the invertible matrix $A$ acting on a column vector $x \in \mathbb{R}^n$ by taking it to the vector $Ax$.</p>
<p>I have already verified that this is a group action, but I'm not sure ... | Geoff Robinson | 13,147 | <p>This is equivalent to Jim's perfectly good answer, but it helps to think in terms of linear algebra, and things work just as well over any field. Let $V$ be an $n$-dimensional vector space over a field $K$. The group ${\rm GL}(V)$ of invertible linear transformations from $V$ to $V$ is defined independently of choic... |
195,176 | <p>Find the values of the real constants $c$ and $d$ such that</p>
<p>$$\lim_{x\to 0}\frac{\sqrt{c+dx}-\sqrt{3}}{x}=\sqrt{3}$$</p>
<p>I really have no clue how to even get started.</p>
| Artes | 21,946 | <p><strong>Method I</strong></p>
<p>Since we have a fraction going to a non-vanishing value given its denominator is going to <code>0</code> we have to assume that its numerator also tends to <code>0</code>, therefore we should solve :</p>
<pre><code>Reduce[Limit[Numerator[(Sqrt[c + d*x] - Sqrt[3])/x], x -> 0] == ... |
121,645 | <p>I have a (presumably simple) Laplace Transform problem which I'm having trouble with:</p>
<p>$$\mathcal L\big\{t \sinh(4t)\big\} = ?$$</p>
<p>How would I go about solving this? Would you please show working if possible, or alternatively point me in the right direction regarding how to go about solving this?</p>
<... | chemeng | 25,845 | <p>You got $t\operatorname{sinh}{4t}=t\frac{e^{4t}-e^{-4t}}{2}$ , so:
$$\mathcal{L}\big\{t\operatorname{sinh}4t\big\}=\tfrac{1}{2}\left(\mathcal{L}\left\{{te^{4t}}\right\}+\mathcal{L}\left\{te^{-4t}\right\}\right)$$</p>
<p>We know that $\mathcal{L}\big\{e^{at}f(t)\big\}=F(s-a)$. Furthermore $\mathcal{L}\left\{t\right\... |
1,724,881 | <p>I'm having a bit of trouble with this problem:</p>
<p>Let $β=\{e^{2x}, xe^{2x}, e^{x}\}$ and define $V=\mbox{span}(\beta)$. Let $T=D-2$ where $D=d/dx$. Show that $\beta$ is a Jordan basis for $T$.</p>
<p>How do I show that $\{e^{2x}, xe^{2x}, e^{x}\}$ is a Jordan basis for $D-2$?</p>
| Sensei | 328,071 | <p>If the vector space is over a field K, then a basis is a jordan basis if and only if all eigenvalues of the matrix lie in K. </p>
<p>Using this to solve your specific problem will simply involve finding the eigenvalues and showing that they are in the given field.</p>
<p>Have fun with the quiz. ;)</p>
|
3,676,879 | <p>I'm trying to derive a reduction formula for the following integral:</p>
<blockquote>
<p><span class="math-container">$$I_n=\int _0^1 \left(1+x^2\right)^n \mathrm{d}x$$</span></p>
</blockquote>
<p>So far, I have tried applying integration by parts and have reached till:</p>
<p><span class="math-container">$$I_n... | Lai | 732,917 | <p>I would like to investigate their difference which help us evaluate the integral easily.
<span class="math-container">$$
\begin{aligned}
I_{n}-I_{n-1}
&=\int_{0}^{1} x^{2}\left(1+x^{2}\right)^{n-1} d x \\
& \stackrel{IBP}{=} \frac{1}{2 n} \int_{0}^{1} x d\left(1+x^{2}\right)^{n} \\
&=\left[\frac{x\left(... |
1,305,151 | <p>I want to prove this without using any of the properties about the field of algebraic numbers (specifically that it is one). Essentially I just want to find a polynomial for which $\cos\frac{2\pi}{n}$ is a root.</p>
<p>I know roots of unity and De Moivre's theorem is clearly going to be important here but I just ca... | Gregory Grant | 217,398 | <p>We know $\cos(2\pi/n)+i\sin(2\pi/n)$ is algebraic ($n$th root of unity). And the complex conjugate of algebraic is algebraic (in fact it's just another $n$th root of unity), so add it to its conjugate because also the sum of two algebraics is algebraic. Then divide by two.</p>
|
2,018,703 | <p>Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function with the property that $\lim_{x \rightarrow \infty} f(x)$ and $\lim_{x \rightarrow -\infty} f(x)$ exist and are equal. Prove that $\forall d > 0$ there exists $x_1, x_2 \in \mathbb{R}$ such that $x_1 - x_2 = d$ and $f(x_1) = f(x_2)$.</p>
<p>I a... | Hugo Berndsen | 385,957 | <p>Since you can't use Rolle's theorem because your function is not continuous, maybe we can try something else. The idea is this: you take a point $x_0$ and calculate $x_0+d$. Now you are certain that the two points are at the right distance from each other. </p>
<p>Then calculate $f(x_0)$ and $f(x_0+d)$. Then define... |
2,631,733 | <blockquote>
<p>Let $E = \mathcal{C}^0([a,b],\mathbb{R})$, provided with the $||\cdot ||_{\infty}$ norm. Let $\phi: \mathbb{R} \rightarrow \mathbb{R}$ that is $\mathcal{C}^1$. Show that the function given by $\Psi:E \rightarrow \mathbb{R}$:
$$ \Psi(f) = \int^{b}_{a}\phi(f(x))dx $$ is differentiable.</p>
</blockquo... | user284331 | 284,331 | <p>I guess that is to look for a linear operator $A_{f}:C^{0}[a,b]\rightarrow{\bf{R}}$ such that
\begin{align*}
\lim_{h\rightarrow 0}\dfrac{|\Psi(f+h)-\Psi(f)-A_{f}(h)|}{\|h\|_{\infty}}=0.
\end{align*}
Set $A_{f}(h)=\displaystyle\int_{a}^{b}\phi'(f(x))h(x)dx$, then
\begin{align*}
|\Psi(f+h)-\Psi(f)-A_{f}(h)|&=\le... |
4,558,977 | <p>I have a circle. I know the radius (800) and I know the point coordinates (0, -800) under the circle. I double the point and move this one to the right. And a second point now has coordinates (500, -800). I have to define y (z coordinates according to my screenshot) value like a point is located on the circle and de... | Lee Mosher | 26,501 | <p>Every closed 2-manifold <span class="math-container">$M$</span> that can be covered by two simply connected charts is homeomorphic to <span class="math-container">$S^2$</span>.</p>
<p>I do know some fancy language for this fact: the <a href="https://en.wikipedia.org/wiki/Lusternik%E2%80%93Schnirelmann_category" rel=... |
11,618 | <p>I'm teaching a preparatory course on mathematics at a university. The content is mostly calculus, manipulating expressions and solving equations and inequalities. I show a couple of simple derivations/proofs and ask the students to occasionally prove some simple equality, so the course is by no means rigorous. Most ... | James S. Cook | 128 | <p>But, a <strong>geometric</strong> definition for the exponential function is the following:</p>
<blockquote>
<p>The function $f$ for which $f(0)=1$ and is such that its value is equal to its slope at each point is <strong>the</strong> exponential function. Or, in the language of differential equations, it is the ... |
175,661 | <blockquote>
<p>Prove that the series $\sum_{n=1}^{\infty}\left\Vert x\right\Vert ^{n} $, $x\in\mathbb{R}^{n} $, does not converge uniformly on the unit ball $\left\{ x\in\mathbb{R}^{n}\mid\left\Vert x\right\Vert <1\right\} $. </p>
</blockquote>
<p>I am not sure how to show this. What I got to is that the given s... | Alex Becker | 8,173 | <p>This is true. Suppose $\{U_i\}_{i\in I}$ is an open cover of $f(K)$. Then $\{f^{-1}(U_i)\}_{i\in I}$ is an open cover of $K$, so has a finite subcover, say $f^{-1}(U_1),\ldots,f^{-1}(U_n)$. Then $U_1,\ldots,U_n$ is a finite subcover of $\{U_i\}_{i\in I}$, thus $f(K)$ is compact.</p>
|
617,598 | <p>Does anyone know any examples of $f$'s for which $-\triangle u(x) = k f(u(x))$ has an explicit solution (i.e. a formula for the solution, not a numerical approximation scheme) in terms of $k$?</p>
<p>I am interested in examples where $f\geq 0$ is neither constant nor linear. Optimally I would be interested in a smo... | Obinna Nwakwue | 307,490 | <p>Let's start from the very beginning here. Your expression is $$\frac {x^2 - 16y^2}{x} \cdot \frac {x^2 + 4xy}{x - 4y}$$
You can multiply the numerators and denominators to get this: $$\frac {x^4 + 4x^3y - 16x^2y^2 - 64xy^3}{x^2 - 4xy}$$
When you divide via the polynomial long division method, you get $$x^2 + 8xy + 1... |
3,418,811 | <p>Let <span class="math-container">$A$</span> be an integral, finitely-generated algebra over some field <span class="math-container">$k$</span>, of dimension <span class="math-container">$\text{dim}(A)\geq2$</span> such that <span class="math-container">$A = \cap_Q A_Q$</span> where <span class="math-container">$Q$</... | KReiser | 21,412 | <p>To clear this from the unanswered queue, <span class="math-container">$\varphi$</span> is indeed the restriction. As you say, <span class="math-container">$\varphi$</span> is injective since <span class="math-container">$X$</span> is integral. The idea behind talking about elements of <span class="math-container">$\... |
2,728,317 | <p>As I know when you move to "bigger" number systems (such as from complex to quaternions) you lose some properties (e.g. moving from complex to quaternions requires loss of commutativity), but does it hold when you move for example from naturals to integers or from reals to complex and what properties do you lose?</p... | marty cohen | 13,079 | <p>Comparing naturals to integers,
there is a smallest natural
(0 or 1) which often
makes solving problems easier.</p>
<p>Comparing reals to complex,
you can always compare reals
but there is no complete ordering
of the complex numbers.</p>
|
2,941,421 | <p>A study is conducted to test the hypothesis that people with glaucoma have higher variability in systolic blood pressure(SBP). The study includes 41 people with glaucoma whose mean SBP is 140 mmHg with a standard deviation of 25 mmHg. If the population standard deviation is 20 mmHg, verify the claim at 1% significan... | Michael Hardy | 11,667 | <p>The only mistake I find in your posting is your assertion that <span class="math-container">$63.691<62.5.$</span></p>
<p>Since the value of the test statistic is less than the critical value, the null hypothesis is not rejected.</p>
<p>Likewise, since the p-value is more than <span class="math-container">$0.01$... |
1,008,253 | <p>The Wronskian for $\sin^2x, \cos^2x$ is</p>
<p>\begin{align}
& \left| \begin{array}{cc} \sin^2 x & \cos^2 x \\ 2\sin x\cos x & -2\cos x\sin x \end{array} \right| \\[8pt]
= {} & -2\sin^2x \cos x \sin x - 2 \cos^2 x \sin x \cos x,
\end{align}
with $x = \frac{π}{6},$ this is $=$
$$
-\sqrt{\frac{3}{2}}... | hmakholm left over Monica | 14,366 | <p>No calculus needed --</p>
<p>If $a\sin^2 x + b\cos^2 x=0$, then $a$ must be $0$ because that's the only way to make the sum $0$ at $x=\pi/2$ where $\cos^2 x=0$.</p>
<p>Similarly $b=0$ is the only way to make the sum zero at $x=0$.</p>
|
2,460,003 | <p>I need some help showing that these are equivalent. I made a couple attempts to get this right but so far the following work is as far as I've gotten.</p>
<p>Here is the question in its entirety:</p>
<blockquote>
<p>Let n be a natural number. Give a combinatorial proof of the following:
$\binom{2n+2}{n+1} = \b... | Donald Splutterwit | 404,247 | <p>\begin{eqnarray*}
\binom{2n+2}{n+1}= \color{red}{\binom{2n+1}{n+1}}+\color{blue}{\binom{2n+1}{n}}=\color{red}{\binom{2n}{n+1}+\binom{2n}{n}}+\color{blue}{\binom{2n}{n}+\underbrace{\binom{2n}{n-1}}_{\binom{2n}{n-1}=\binom{2n}{n+1}}}=2 \left(\binom{2n}{n+1}+\binom{2n}{n}\right)
\end{eqnarray*}</p>
|
2,460,003 | <p>I need some help showing that these are equivalent. I made a couple attempts to get this right but so far the following work is as far as I've gotten.</p>
<p>Here is the question in its entirety:</p>
<blockquote>
<p>Let n be a natural number. Give a combinatorial proof of the following:
$\binom{2n+2}{n+1} = \b... | Sarvesh Ravichandran Iyer | 316,409 | <p>Consider picking $n+1$ objects from a set of $2n+2$ objects, where two of these are "special", and the other $2n$ are unspecial.</p>
<p>One way is simple : to pick $n+1$ objects from $2n+2$ objects, the number of ways are $\binom {2n+2}{n+1}$ clearly. </p>
<p>The other way. We split in three cases.</p>
<p>One can... |
4,087,134 | <p>I have the polynomial <span class="math-container">$ p(x) = ax^3 + bx^2 + cx + d $</span>. I have to show that:</p>
<p><span class="math-container">$ \int_{-1}^{1} p(x) = p(- \frac{1}{\sqrt{3}}) + p (\frac{1}{\sqrt{3}}) $</span></p>
<p>I'm kind of stuck. My idea so far is to use "proof by symmetry" and the... | Cm7F7Bb | 23,249 | <p>The variance <span class="math-container">$\operatorname{Var}X$</span> is not necessarily equal to the variance <span class="math-container">$\operatorname{Var}|X|$</span> and <span class="math-container">$\operatorname EX$</span> is not necessarily equal to <span class="math-container">$\operatorname E|X|$</span> (... |
233,846 | <p>This may be related to <a href="https://mathematica.stackexchange.com/q/98147">How to discretize a BezierCurve?</a>, but this question deals with <code>BSplineCurve</code>s with specific <code>SplineWeights</code>, so I don't think the answers there will help here.</p>
<hr />
<p><strong>Background</strong></p>
<p>I ... | Carl Woll | 45,431 | <p>To avoid the issue mentioned by kglr where points are repeated, you can just add another layer of list:</p>
<pre><code>segments = {
BSplineCurve[{{1,0},{1,1},{0,1}},SplineWeights->{1,1/Sqrt[2],1}],
BSplineCurve[{{0,1},{-1,1},{-1,0}},SplineWeights->{1,1/Sqrt[2],1}]
};
Graphics[{{Red,segments}, JoinedCur... |
2,040,678 | <p>I have a confusion regarding the symmetry of the volume in the following question. </p>
<p>Find the volume common to the sphere $x^2+y^2+z^2=16$ and cylinder $x^2+y^2=4y$.</p>
<p>The author used polar coordinates $x=rcos\theta$ snd $y=rsin\theta$ and does something like this:</p>
<p>Required volume $V=4\int_0^{π/... | Arnaldo | 391,612 | <p>The volume can be cut into 4 identical parts. See that both volume are simetric for each quadrant. That's why the angle variation is from $0$ to $\pi/2$. </p>
|
261,361 | <ol>
<li>In a group of 200 people, number of people having at least primary education (assuming - <em>Category I</em>): number of people having at least middle school education (<em>Category II</em>): number of people having at least high school education (<em>Category III</em>) are in the ratio 7 : 3 : 1</li>
<li>Out ... | miracle173 | 11,206 | <p>An exercise in formatting</p>
<p><em>Variables</em></p>
<p>Let's call the category that contains all people the category 0 and we forget category 4. this smplifies the wording of the following definitions:</p>
<ul>
<li>$f_i$: number of persons playing football only and $i$ is the highest category they are member ... |
3,680,124 | <p>I'm trying to integrate the following:</p>
<p><span class="math-container">$$\int \frac {dx}{x\sqrt{x^2-49}}\,$$</span></p>
<p>using the substitution <span class="math-container">$x=7\cosh(t)$</span></p>
<p>This is as far as I've gotten:</p>
<p><span class="math-container">$\int \frac {dx}{x\sqrt{x^2-49}}\,$</sp... | Toby Mak | 285,313 | <p>Hint: </p>
<p><span class="math-container">$\cosh^2 t - \sinh^2 t =1$</span> so <span class="math-container">$\sinh t = \sqrt{\cosh^2 t- 1}$</span>.</p>
<p>Then refer back to the substitution <span class="math-container">$x = 7 \cosh t$</span>.</p>
|
3,956,400 | <blockquote>
<p>Arrange in ascending order <span class="math-container">$\tan45^\circ,\tan80^\circ$</span> and <span class="math-container">$\tan100^\circ.$</span></p>
</blockquote>
<p>We know that <span class="math-container">$\tan45^\circ=1$</span> because a right triangle with angle equal to <span class="math-contai... | herb steinberg | 501,262 | <p><span class="math-container">$\frac{1}{n}-\frac{1}{n+1}=\frac{1}{n(n+1)}\gt\frac{1}{(2n+1)(n+1)}$</span>. Therefore that extra term is taken care of.</p>
|
820,614 | <p>In a scalene triangle,does there exist three cevians which are equal in length,where length is measured between the corresponding vertex and the intersection point of the cevian with the corresponding side? This is just a question I had in my mind. </p>
| poolpt | 150,343 | <p>The distance from the top of the cone to the center of mass is given by:</p>
<p>$$\frac{3}{h^{3}}\int_{0}^{h}x^{3}dx$$</p>
|
1,883,443 | <p>I need to compare (asymptoticly) between $\left ( \frac{\ln n +4 \ln\ln n}{\ln n} \right )^{\ln n}$ and $16^{\ln\ln n}$. The options are $\Theta , \omega, o$.
My work so far:</p>
<p>I denoted $t_n=\ln n$ to make things cleaner.</p>
<p>The first sequence is
$\left ( 1+\frac{4 \ln t_n}{t_n} \right )^{t_n}$. It looks... | Ian | 83,396 | <p>No, the negation of "you know nothing" is "you know something". "You know nothing" is of the form $(\forall x) \: \neg P(x)$, where $P(x)$ is "you know $x$". So its negation is $(\exists x) \: P(x)$, which is "you know something" or slightly more precisely "you know at least one thing".</p>
|
1,883,443 | <p>I need to compare (asymptoticly) between $\left ( \frac{\ln n +4 \ln\ln n}{\ln n} \right )^{\ln n}$ and $16^{\ln\ln n}$. The options are $\Theta , \omega, o$.
My work so far:</p>
<p>I denoted $t_n=\ln n$ to make things cleaner.</p>
<p>The first sequence is
$\left ( 1+\frac{4 \ln t_n}{t_n} \right )^{t_n}$. It looks... | celtschk | 34,930 | <p>Imagine someone would say to you "you know nothing!" Now you want to refute that. What would you say?</p>
<p>If you answer "I know everything" you would at best be laughed at. Maybe he'd reply with something like "oh, so you know what I dreamed this night? Let's hear!"</p>
<p>No, what you'd likely answer is: "It's... |
720,504 | <p>Why is $\displaystyle \lim_{x \to \infty} \ x^{2/x} = 1$ since this is an indeterminate form $\infty^{0}$ and I can't see any manipulation that would suggest this result?</p>
| Frank | 59,738 | <p>$$x^{2/x} = (e^{\log(x)})^{2/x} = \exp(\frac{2 \log(x)}{x}) \rightarrow \exp(0) = 1 \quad \textrm{as} \quad x \to \infty$$</p>
<p>where in the limit I use the fact that $x$ dominates $\log(x)$ as $x \to \infty$. </p>
|
720,504 | <p>Why is $\displaystyle \lim_{x \to \infty} \ x^{2/x} = 1$ since this is an indeterminate form $\infty^{0}$ and I can't see any manipulation that would suggest this result?</p>
| Community | -1 | <p>Let $ y=x^{2/x} $. Then, $\ln y=\dfrac {2\ln x}{x} $. Take the limit as $ x $ goes to infinity. Then, finally, take the exponential ($ e^\cdot $) of your resulting answer. This is the value of your limit.</p>
|
924,551 | <p>$$\displaystyle \lim_{x \to \infty}\dfrac{8-\sqrt{x}}{8+\sqrt{x}}$$ </p>
<p>I tried rationalizing the numerator: </p>
<p>$$\lim_{x \to \infty}\dfrac{8-\sqrt{x}}{8+\sqrt{x}} \times \dfrac{(8-\sqrt{x})}{(8-\sqrt{x})}$$ </p>
<p>$$\lim_{x \to \infty}\dfrac{64-16\sqrt{x}+x}{64-x}$$</p>
<p>Is this correct? how do I p... | Hypergeometricx | 168,053 | <p>Another approach, although this may not be rigorous:</p>
<p>As $x\to \infty$, $\sqrt{x}$ dominates, being the highest power in both the numerator and denominator, hence
$$\require{cancel}\lim_{x\to\infty}\frac{8−\sqrt{x}}{8+\sqrt{x}}=\lim_{x\to\infty}\frac {-\cancel{\sqrt{x}}}{+\cancel{\sqrt{x}}}=−1$$</p>
|
924,551 | <p>$$\displaystyle \lim_{x \to \infty}\dfrac{8-\sqrt{x}}{8+\sqrt{x}}$$ </p>
<p>I tried rationalizing the numerator: </p>
<p>$$\lim_{x \to \infty}\dfrac{8-\sqrt{x}}{8+\sqrt{x}} \times \dfrac{(8-\sqrt{x})}{(8-\sqrt{x})}$$ </p>
<p>$$\lim_{x \to \infty}\dfrac{64-16\sqrt{x}+x}{64-x}$$</p>
<p>Is this correct? how do I p... | patatahooligan | 169,288 | <p>I'll add two more ways of solving.</p>
<ol>
<li><p>L'Hôpital's rule states that if $\lim_{x \to \infty}{f(x)} = \lim_{x
\to \infty}{g(x)} = \infty$ then</p>
<p>$$ \lim_{x \to \infty} \frac{f(x)}{g(x)} = \lim_{x \to \infty}
\frac{f'(x)}{g'(x)} $$</p>
<p>In this case</p>
<p>$$
\lim_{x \to \infty} \frac... |
581,497 | <p>Case $1$: $4$ games: Team A wins first $4$ games, team B wins none = $\binom{4}{4}\binom{4}{0}$</p>
<p>Case $2$: $5$ games: Team A wins $4$ games, team B wins one = $\binom{5}{4}\binom{5}{1}-1$...minus $1$ for the possibility of team A winning the first four.</p>
<p>Case $3$: $6$ games: Team A wins $4$ games, team... | Ross Millikan | 1,827 | <p>For another approach, if the series ends before the seventh game, extend it to seven games by having the losing team win the rest. The series will now be four games to three. There are $2$ ways to choose the losing team, and ${7 \choose 3}$ ways to choose which game the losing team wins. The total is then $2{7 \c... |
1,752,577 | <p>Let X be a topological space. All that I know is Borel $\sigma$ algebra on X is the smallest $\sigma$ algebra generated by $T_X$ i.e. set of all open sets in X. Is there any other characterization of Borel $\sigma$ algebra on X? or we can show the above assertion using the definition only.</p>
| Chris Apostol | 245,658 | <p>You can show that the set $\{A\times B: A\in \mathbb{B}(X),B\in \mathbb{B}(Y)\}$ is $\sigma$-algebra that contains the open sets of $A\times B$.</p>
|
194,123 | <blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="https://math.stackexchange.com/questions/150482/probability-of-a-random-binary-string-containing-a-long-run-of-1s">Probability of a random binary string containing a long run of 1s?</a> </p>
</blockquote>
<p><strong><em>EDIT</strong>: Cocopuffs b... | mjqxxxx | 5,546 | <p>The solution is <em>not</em> the right one; in fact, you have a vanishingly small probability of ever catching your friend as $L$ becomes large.</p>
<p>Let $X_{i}$ for $i=1,2,3,\ldots$ be i.i.d. discrete random variables representing the lengths of your non-zero runs of heads. (Note that there are a.s. infinitely ... |
1,090,418 | <p>I just started reading D. Eisenbud <em>Commutative algebra with a view towards algebraic geometry</em> and I wonder about a theorem on page 42:</p>
<p>If $M$ is a finitely generated graded module over $k[x_1,...,x_r]$ then $H_M(s)$ agrees, for large $s$, with a polynomial of degree $\leq r-1$, where $H_M(s):=\dim_k... | davidlowryduda | 9,754 | <p>It means that $H_M(s) = p(s)$ for some polynomial $p$ of degree $\leq r-1$ for $s$ sufficiently large.</p>
|
189,119 | <p>How to calculate my age given date of birth? I have read other questions and answers but is not satisfied with answers. My question is how do I calculate my age if my date of birth is 18.03.2000?</p>
| Bill Watts | 53,121 | <p>Try this. Use exact numbers in the equations</p>
<pre><code>fr = {-(29/(10^16 x^4)) + 1/x^2 - 1/(y - x)^2 - 1/(z - x)^2 ==
0, -(29/(10^16 y^4)) + 1/y^2 + 1/(y - x)^2 - 1/(z - y)^2 ==
0, -(29/(10^16 z^4)) + 1/z^2 + 1/(z - x)^2 + 1/(z - y)^2 == 0}
</code></pre>
<p>Then solve</p>
<pre><code>sol = Solve[fr,... |
2,010,158 | <p>Let $b >0$ , let $B= \{ f \in C^r([-b,b]) : f(x) = f(-x) for \ \ 0\leq x\leq b\}$, and let $A$ be the set of all polynomials that contain only terms of even degree (with domains restricted to $[-b,b]$). Then the uniform closure of $A$ is $B$.</p>
<p>I am not getting any clue how to solve the problem. Help Neede... | Christian Blatter | 1,303 | <p>We apply Weierstrass' approximation theorem to the auxiliary function
<span class="math-container">$$g(t):=f\bigl(\sqrt{t}\bigr)\qquad(0\leq t\leq 1)\ ,$$</span>
which is continuous on <span class="math-container">$[0,1]$</span>: For any given <span class="math-container">$\epsilon>0$</span> we can find a polynom... |
1,365,268 | <p>Part A is in the title, Part B is here:
Is it true that $(k, n+k)= d$ if and only if $(k, n)=d$?</p>
<p>I am still working on the Part A. </p>
<p>What I have so far:</p>
<p>if $(k, n)= 1$ then $1|k$, $1|n$ and $1|(n-k)$</p>
<p>if $(k, n+k)=1$ then $1|k$, $1|n+k$ and $1|((n+k)- k) \to 1|n$</p>
<p>I was under the... | Thomas Andrews | 7,933 | <p>First, this is equivalent to $$\left(1+\frac xn\right)^n \geq 2^{x}$$</p>
<p>Letting $y=\frac{x}{n}$, this is equivalent to, for $y\in[0,1]$ and any $n>0$:</p>
<p>$$(1+y)^n \geq 2^{yn}$$ or:</p>
<p>$$1+y\geq 2^y$$</p>
<p>Now we use that $f(x)=2^y$ is convex - that is, $f(at+b(1-t))\leq tf(a)+(1-t)f(b)$ for an... |
2,365,932 | <p>I am trying to analytically solve a simple looking integral equation:
\begin{align*}
\int_{0}^1 e^{(1-t)x} \varphi(t) dt = 1, \hspace{0.2cm} \forall x \in [-1,0],
\end{align*}
but could neither say whether it is solvable and what function $\varphi(t)$ solves the equation.</p>
<p>In the case that this is not solvabl... | Felix Klein | 457,386 | <p>Suppose that there is an integrable function $\phi$ on $[0,1]$ such that
$$\forall\,x\in[-1,0],\qquad \int_0^1e^{-xt}\phi(t)dt=e^{-x}\tag{1}$$
Taking successive derivatives and evaluating the result for $x=0$ we see that
$$\forall, n\in\mathbb{N},\quad \int_0^1 t^n \phi(t)dt=1$$
This implies that for every polynomia... |
2,365,932 | <p>I am trying to analytically solve a simple looking integral equation:
\begin{align*}
\int_{0}^1 e^{(1-t)x} \varphi(t) dt = 1, \hspace{0.2cm} \forall x \in [-1,0],
\end{align*}
but could neither say whether it is solvable and what function $\varphi(t)$ solves the equation.</p>
<p>In the case that this is not solvabl... | Community | -1 | <p>As I mentioned in a comment (and as Felix Klein showed in more detail), a solution of the equation should behave like the distribution $\delta(t-1),$ so it's easy to find approximate solutions:<br>
If $$\varphi(t)=\frac1\epsilon\chi_{[1-\epsilon,1]}(t),$$ the integral becomes $$\int_{0}^1 e^{(1-t)x} \varphi(t) dt = ... |
392,580 | <p>How to evaluate the following
$$\int_0^{\infty} \frac{\sin (ax)}{e^{\pi x} \sinh(\pi x)} dx $$
Given hints says to construct a rectangle $0\to R\to R+i\to i \to 0$ and consider $\displaystyle f(z):=\frac{e^{iaz}}{e^{2\pi z}-1} $ and evaluate around it but that does not help.</p>
<p><strong>ADDED::</strong> I need ... | Ron Gordon | 53,268 | <p>For the contour you describe in your text, you have to indent about the poles at $z=0$ and $z=1$. In that case, the contour integral</p>
<p>$$\oint_C dz \frac{e^{i a z}}{e^{2 \pi z}-1}$$</p>
<p>is split into $6$ segments:</p>
<p>$$\int_{\epsilon}^R dx \frac{e^{i a x}}{e^{2 \pi x}-1} + i \int_{\epsilon}^{1-\epsil... |
468,344 | <p>Suppose $0\leq \alpha, \beta, \gamma\leq \pi$ and $\cos^2\alpha+\cos^2\beta+\cos^2\gamma = 1$, then what is the maximum and minimum of $\alpha+\beta+\gamma$.</p>
| Robert Mastragostino | 28,869 | <p>Just some geometric intuition: if $\alpha,\beta,\gamma$ are the angles that a vector makes with the 3 positive coordinate axes, these formulas will hold. What Calvin said, about "increasing the linear sum", is saying that there's no point in considering all acute angles: if you fix two angles, the valid options for ... |
468,344 | <p>Suppose $0\leq \alpha, \beta, \gamma\leq \pi$ and $\cos^2\alpha+\cos^2\beta+\cos^2\gamma = 1$, then what is the maximum and minimum of $\alpha+\beta+\gamma$.</p>
| Christian Blatter | 1,303 | <p>Put
$$\alpha:={\pi\over2}+x,\quad \beta:={\pi\over2}+y,\quad \gamma:={\pi\over2}+z\ .$$
Then maximizing/minimizing $\alpha+\beta+\gamma$ under the constraints
$$\cos^2\alpha+\cos^\beta+\cos^2\gamma=1,\qquad\alpha,\ \beta,\ \gamma\in[0,\pi]$$ is the same as maximizing $$s(x,y,z):=x+y+z$$
under the constraints
$$g(x,y... |
3,117,260 | <p>So according to the commutative property for multiplication:</p>
<p><span class="math-container">$a \times b = b \times a$</span> </p>
<p>However this does not hold for division</p>
<p><span class="math-container">$a \div b \neq b \div a$</span> </p>
<p>Why is it that in the following case:</p>
<p><span class="... | Community | -1 | <p>Hope this makes sense.</p>
<p><span class="math-container">$$ a\times b ÷ c $$</span></p>
<p><span class="math-container">$$=a\times b\times\dfrac{1}{c}$$</span></p>
<p><span class="math-container">$$=(a\times\dfrac{1}{c})\times b$$</span></p>
<p><span class="math-container">$$=\dfrac{a}{c}\times b$$</span></p>
... |
1,025,642 | <p>Let $X$ be a non-empty set. Suppose that $d_1$ and $d_2$ are two possibly different metrics on $X$. Let $\tau_i$ denote the topology generated by the metric $d_i$ ($i\in\{1,2\}$).</p>
<p>The following are known:</p>
<ul>
<li>$\tau_1=\tau_2\equiv\tau$;</li>
<li>$(X,d_1)$ is a totally bounded metric space;</li>
<li>... | Claude Leibovici | 82,404 | <p>Multiply the numerator by the denominator; so $$a_n=\frac{1\times3\times5\times ... \times(2n-1)}{ 2\times4\times6\times...\times(2n)}=\frac{1\times2\times3\times ... \times(2n)}{\Big(2\times4\times6\times...\times(2n)\Big)^2}=\frac{(2n)!}{4^n(n!)^2}$$ If now you use Stirling approximation $$m! \approx \sqrt{2 \pi }... |
1,025,642 | <p>Let $X$ be a non-empty set. Suppose that $d_1$ and $d_2$ are two possibly different metrics on $X$. Let $\tau_i$ denote the topology generated by the metric $d_i$ ($i\in\{1,2\}$).</p>
<p>The following are known:</p>
<ul>
<li>$\tau_1=\tau_2\equiv\tau$;</li>
<li>$(X,d_1)$ is a totally bounded metric space;</li>
<li>... | pointer | 121,270 | <p>Let's prove using induction that $$a_n\le\frac{1}{\sqrt{3n+1}}.$$ For $n=1$ it is true. Now we just need to prove that$$\frac{(2n+1)^2}{(2n+2)^2}\le\frac{3n+1}{3n+4}$$or $$(4n^2+4n+1)(3n+4)\le(4n^2+8n+4)(3n+1)$$or$$12n^3+28n^2+19n+4\le12n^3+28n^2+20n+4.$$</p>
|
1,447,089 | <p>I am trying to prove that (0.1) is uncountable given that R is uncountable.</p>
<p>I start by assuming that (0.1) is countable. </p>
<p>Then there exists a bijective map between (0.1) and N.</p>
<p>I guess then we can construct bijective map for (1.2) also.</p>
<p>this shows that each (i-1,i) for i=intergers is ... | Redundant Aunt | 109,899 | <p>We have:
$$
\sqrt{c}-\sqrt{c-1}≥\sqrt{c+1}-\sqrt{c}\iff\\
\\
\left(\sqrt{c}-\sqrt{c-1}\right)\frac{\sqrt{c}+\sqrt{c-1}}{\sqrt{c}+\sqrt{c-1}}≥\left(\sqrt{c+1}-\sqrt{c}\right)\frac{\sqrt{c+1}+\sqrt{c}}{\sqrt{c+1}+\sqrt{c}}\\
\\
\frac{1}{\sqrt{c}+\sqrt{c-1}}≥\frac{1}{\sqrt{c+1}+\sqrt{c}}\iff\\
\\
\sqrt{c+1}+\sqrt{c}≥\s... |
1,447,089 | <p>I am trying to prove that (0.1) is uncountable given that R is uncountable.</p>
<p>I start by assuming that (0.1) is countable. </p>
<p>Then there exists a bijective map between (0.1) and N.</p>
<p>I guess then we can construct bijective map for (1.2) also.</p>
<p>this shows that each (i-1,i) for i=intergers is ... | peterwhy | 89,922 | <p>Consider $$B = \sqrt c - \sqrt{c-1},\quad D = \sqrt {c+1} - \sqrt{c}$$
By mean value theorem, there exist $$b\in (c-1, c),\quad d\in (c, c+1)$$ that satisfy
$$\frac1{2\sqrt b} = B,\quad \frac1{2\sqrt d} = D$$
and since $b\le d$, $$\begin{align*}
\dfrac1{2\sqrt b} &\ge \dfrac1{2\sqrt d}\\
\sqrt c - \sqrt{c-1} &am... |
1,447,089 | <p>I am trying to prove that (0.1) is uncountable given that R is uncountable.</p>
<p>I start by assuming that (0.1) is countable. </p>
<p>Then there exists a bijective map between (0.1) and N.</p>
<p>I guess then we can construct bijective map for (1.2) also.</p>
<p>this shows that each (i-1,i) for i=intergers is ... | nonuser | 463,553 | <p>Rewrite as: <span class="math-container">$$2\sqrt{ c} \geq \sqrt{ c + 1} +\sqrt{c-1}$$</span> and since both sides are nonegative it is equaivalent if we square it:</p>
<p><span class="math-container">$$4c\geq c+1+2\sqrt{c^2-1}+c-1$$</span> or <span class="math-container">$$c\geq \sqrt{c^2-1}\iff c^2\geq c^2-1$$</s... |
443,099 | <p>I remember hearing someone say "almost infinite" on one of the science-esque youtube channels. I can't remember which video exactly, but if I do, I'll include it for reference.</p>
<p>As someone who hasn't studied very much math, "almost infinite" sounds like nonsense. Either something ends or it doesn't, there rea... | k170 | 161,538 | <p>This is a common misuse of the word infinite. Anything that is infinite or approaching infinity, is not quantifiable, regardless of the context. </p>
<p>There exists no such quantity that can ever get close to infinity. Therefore, it will never make sense to say that a quantity is "almost infinite". </p>
|
443,099 | <p>I remember hearing someone say "almost infinite" on one of the science-esque youtube channels. I can't remember which video exactly, but if I do, I'll include it for reference.</p>
<p>As someone who hasn't studied very much math, "almost infinite" sounds like nonsense. Either something ends or it doesn't, there rea... | Lucian Mihail | 198,360 | <p>Transfinite numbers are numbers that are "infinite" in the sense that they are larger than all finite numbers, yet not necessarily absolutely infinite. The term transfinite was coined by Georg Cantor, who wished to avoid some of the implications of the word infinite in connection with these objects, which were never... |
2,906,832 | <p>I want to write $\csc$ and $\tan$ and terms of classical trigonometric functions like $\sin$ and $\cos$. I know about the identity $\sin(x)^2+\cos(x)^2=1$. But I am not sure where to go from here. </p>
| Ross Millikan | 1,827 | <p>$$\tan x=\frac {\sin x}{\cos x}=\frac {\pm \sqrt{1-\cos^2x}}{\cos x}$$
for example. There a number of forms. Which one you want depends on what you want to do with it.</p>
|
3,997,992 | <p>Taken from <a href="https://artofproblemsolving.com/wiki/index.php/1970_Canadian_MO_Problems/Problem_10" rel="nofollow noreferrer">https://artofproblemsolving.com/wiki/index.php/1970_Canadian_MO_Problems/Problem_10</a></p>
<p>Problem <br>
Given the polynomial <span class="math-container">$f(x)=x^n+a_{1}x^{n-1}+a_{2}... | José Carlos Santos | 446,262 | <p>The orthogonal projection of <span class="math-container">$C$</span> on the segment <span class="math-container">$AB$</span> is a point <span class="math-container">$P$</span> of the form <span class="math-container">$(t,1-t,0)$</span>. The value of <span class="math-container">$t$</span> must be chosen so that <spa... |
137,691 | <p>What is the eigenvalue/eigenvector relationship between matrix A,B and AB?</p>
| Robert Israel | 13,650 | <p>I assume this is over $\mathbb C$.
If they don't commute, about all you can say is that the determinant (which is the product of the eigenvalues, counted by algebraic multiplicity) of $AB$ is the product of the determinants of $A$ and $B$.</p>
|
106,265 | <p>I am a beginner in <em>Mathematica</em>, so take this into account. I could not find anything similar in the internet so far.</p>
<p>I am trying to find a closed form solution for <code>A</code> in the equation below as a function of <code>n</code>, where <code>n</code> is countable and grows large. Potentially is ... | m_goldberg | 3,066 | <p>I have rewritten your equations as </p>
<pre><code>Sum[λ[t]*((a + b)/2 + β[t] - d - e)^2, {t, 1, n}] ==
Sum[λ[t]*((b + c)/2 + β[t] - d - e)^2, {t, 1, n}] &&
Sum[λ[t], {t, 1, n}] == 1
</code></pre>
<p>because identifiers <code>C</code>, <code>D</code>, and <code>E</code> are reserved for system use in <... |
2,871,419 | <p>Let $n$ be an integer greater than $1$. The notation $f^n$ is notoriously ambiguous: it means either the $n$-th iterate of $f$ or its $n$-th power.</p>
<p>I was wondering when the two interpretations are in fact the same. In other words, if we write $f^n(x)$ for $f(f(\dotsb f(x) \dotsb))$ and $f(x)^n$ for $f(x) \cd... | Community | -1 | <p>If for all $x$, $f(\cdots f(f(x)))=(f(x))^n$ then $f(\cdots f(y))=(y)^n$, and the function $f$ is a functional $n-1^{th}$ root of the $n^{th}$ power function.</p>
<p>In particular, for $n=2$, $f$ is the square function. For $n=3$, a solution is $f(x)=x^{\sqrt 3}$ and more generally</p>
<p>$$f(x)=x^{\sqrt[n-1]n}.$$... |
319,663 | <p>I need to determine whether this matrix is injective
\begin{pmatrix}
2 & 0 & 4\\
0 & 3 & 0\\
1 & 7 & 2
\end{pmatrix}</p>
<p>Using gaussian elimination, this is what I have done:
\begin{pmatrix}
2 & 0 & 4 &|& 0\\
0 & 3 & 0 &|& 0\\
1 & 7 & 2 &|&... | Dominic Michaelis | 62,278 | <p>The formal definition of injective is, that a function is injective, if $f(x)=f(y)\implies x=y$. Maybe it is at first not very intuitive that for linear functions it is the same as the triviality of the null space. But in fact it is, as $f(0 \cdot x)=0 \cdot f(x)=0$ we know that a linear function which is injective... |
1,234,820 | <p>I was working on this problem </p>
<blockquote>
<p>Find all ring homomorphisms from $M_3(\mathbb{R})$ into $\mathbb{R}$.</p>
</blockquote>
<p>My attempt:-
I found that if we have any ring homomorphism $\phi$, then $\ker(\phi)$ should be either zero or the entire ring (since $M_3(\mathbb{R})$ is simple) and in ca... | user26857 | 121,097 | <p>There is no injective ring homomorphism $\phi:M_3(\mathbb R)\to\mathbb R$ for simple reasons: consider the matrix $A=\pmatrix{0 & 1 & 0\\ 0 & 0 & 1\\ 0 & 0 & 0}$. Notice that $A^3=0$. Then $\phi(A)^3=0$, so $\phi(A)=0$. Since $\phi$ is injective we get $A=0$, false.</p>
|
454,622 | <p>I am trying to solve a particular probability question. </p>
<p>I have a fair 10-sides die, whose sides are labelled 1 through 10. I am trying to find the probability of rolling a multiple of 5 or an odd number. </p>
<p>I find the probability as: </p>
<p>P(multiple of 5) OR P(odd number)=P(multiple of 5) + P(odd... | Jared | 65,034 | <p>It might seem odd, but in fact the two events of rolling a multiple $5$ and rolling an odd number are independent.</p>
<p>To see this, let $A$ be the event that a multiple of $5$ is rolled, and $B$ be the event that an odd number is rolled. What is $P(A)$? What is $P(A|B)$? Since the two numbers are the same, it... |
634,510 | <p>Consider a uniformly selected random vector $V= (v_1,v_2,\dots, v_n)$ where $v_i \in \{0,1\}$. Let us define $V_1 = (v_1,v_2,\dots, v_n)$, $V_2 = (v_n, v_1,v_2,\dots, v_{n-1})$, $V_3 = (v_{n-1}, v_n, v_1,v_2,\dots,v_{n-2})$ and so on.</p>
<p>I am trying to work out the probability that there exists $i \ne j$ such ... | bof | 111,012 | <p>Let's try it with smaller numbers. I want to select $3$ men from $4$ men. This can be done easily in $_4\text{C}_3=4$ ways. (Equivalently, I can choose to <em>omit</em> one of the $4$ men in $_4\text{C}_1=4$ ways.) This is clearly the right answer.</p>
<p>Now let's try it another way. First I select $1$ man from $4... |
3,077,882 | <p>Simplify the expression
<span class="math-container">$$\sin\left(\tan^{-1}(x)\right)$$</span>
Using a triangle with an angle <span class="math-container">$\theta$</span>, opposite is x and adjacent is 1 meaning the hypo. is <span class="math-container">${\sqrt {x^2+1}}$</span> </p>
<p>Now because the problem has s... | Fred | 380,717 | <p>A solution with differential equations: let <span class="math-container">$f(x):=\sin(\tan^{-1} (x))$</span> and <span class="math-container">$g(x):= \frac{x}{\sqrt {x^2+1}}.$</span></p>
<p>Then it is easy to see that <span class="math-container">$f$</span> and <span class="math-container">$g$</span> are solutions o... |
2,575,439 | <p>How to attach pen tool to endpoint of a segment (or curve) in Geogebra such that by moving segment, the pen draw a curve? for example I want to draw animated <a href="https://en.wikipedia.org/wiki/Cycloid" rel="nofollow noreferrer">Cycloid</a> in Geogebra but I don't know how to do it?</p>
| Community | -1 | <p>You can actually derive (1) from (2), if (2) is proven for <em>all</em> n.</p>
<p>Namely, if $z_k=r^{\frac {1}{n}}e^{i\frac {2k\pi+x}{n}}$ then $\frac {z_k}{z_l}=e^{(k-l)\frac {2i\pi}{n}}=\zeta_n^{k-l} $ for $0\le l\le k\lt n $. Thus, it suffices to show that $\zeta_n^k=1$ (where $0\le k\lt n $) only for $k=0$.</p>... |
258,071 | <p>Let $P(x),Q(x),R(x)$ be the statements $x$ is a clear explanation,$x$ is satisfactory,$x$ is an excuse,respectively. Suppose that the domain for $x$ consists of all the English text. Express each of these statements using quantifiers, logical connectives and $P(x),Q(x),R(x)$.</p>
<p>a. All clear explanations are sa... | Karolis Juodelė | 30,701 | <p>It should be $\wedge$ in both b and c. It might be more intuitive to write (for example, b) $\exists x~ \neg (R(x) \to Q(x))$.</p>
|
284,444 | <p>For a positive integer $n$ put $\omega(n)$ for the number of distinct prime divisors of $n$. It is a well-known theorem of Erdős and Kac that the probability distribution for the quantity</p>
<p>$\displaystyle \frac{\omega(n) - \log \log n}{\sqrt{\log \log n}}$ </p>
<p>is the standard normal distribution. In other... | Igor Rivin | 11,142 | <p>This is answered in:</p>
<p><em>Mehrdad, Behzad; Zhu, Lingjiong</em>, <a href="http://dx.doi.org/10.1093/qmath/hav035" rel="noreferrer"><strong>Moderate and large deviations for the Erdős-Kac theorem</strong></a>, <a href="https://zbmath.org/?q=an:06553541" rel="noreferrer">ZBL06553541</a>.</p>
<p>(can be found o... |
1,558,934 | <p>How many permutations, ρ, are there in $S_9$(the group of permutations of order 9!) whose decomposition into disjoint
cycles consists of three 2-cycles (transpositions) and one 3-cycle? In other words, how
many permutations are there in $S_9$ whose decomposition into disjoint cycles is of the form
$(a_1a_2)(a_3a_4)(... | user293517 | 293,517 | <p>Step 1: Choose, out of 9 possible numbers, the first two for the first 2-cycle. (I believe the number of ways you can do this is $\binom{9}{2}$)</p>
<p>Step 2: Choose, out of the remaining 7 possible numbers, the next two for the second 2-cycle. (I believe the number of ways you can do this is $\binom{7}{2}$)</p>
... |
1,653,934 | <p>I'm a software engineering and mathematics student, I was searching for disciplines of mathematics that would go well with my engineering degree, and found a lot of people recommended that software engineers should learn at least a bit of linear algebra, giving book recomendations and else, but I couldn't find any r... | DaveNine | 17,243 | <p>If you decide to take up any classes in regards to numerical analysis for differential equations, you'll see that in both finite difference methods and finite element methods, things seem to always come down to solving a massive system of linear equations. Obviously diff eq's have their place all over engineering.</... |
3,614,864 | <p>I noticed that most equations that I've encountered in physics and engineering classes are formulated as differential equations. Some examples I can think of on top of my head are Newton's 2nd law, the wave equation, Maxwell's equations, etc. My question is, what's so special about differential equations that make t... | Empy2 | 81,790 | <p>We can make progress with many differential equations. We rarely ask about the lengths of curves because they don't have nice answers.</p>
|
2,448,082 | <p>Trying to solve $$\int 27x^3(9x^2+1)^{12} dx$$
I know the process and formula of integration by parts. When I set $u = 9x^2 + 1$, $du = 18x dx$. I am stuck on the next step as 18x does not line up with the $27x^3$. </p>
| eg123 | 484,627 | <p>With integration by parts, formula is
$$\int u dv = uv - \int v du$$
When you set $u = 9x^2+1$, in order to apply integration by parts, you must let $dv = \left(27x^{3}(9x^2+1)^12\right)/(9x^2+1)$. In this case, it might be easier to let $u$ be something else.</p>
<p>Regardless, once you've chosen your values of $u... |
2,841,102 | <p>I'm asked to find the minimum and maximum values of $f(x, y, z) = x^2+y^2+z^2$ given the constraints $x+2y+z=5$ and $x-y=6$. </p>
<p>I have successfully computed the following:
$x = \frac{57}{11}, y = \frac{-9}{11}, z= \frac{16}{11}$. </p>
<p>I was then able to obtain $f(\frac{57}{11}, \frac{-9}{11}, \frac{16}{11}... | Donald Splutterwit | 404,247 | <p>Rewrite the constraints and substitute
\begin{eqnarray*}
x&=& y+6 \\
z&=& -1-3y \\
f(x,y,z)&=& (y+6)^2+y^2+(-1-3y)^2=11y^2+18y+37= 11 \left(y+\frac{9}{11} \right)^2+\frac{326}{11}.
\end{eqnarray*}
So you have found the minimum ... now what can you say about the maximum ?</p>
|
203,995 | <p>Let $\mathbb{C}[x,y]$ be the polynomial ring with variables $x,y$ and coefficient in $\mathbb{C}$.</p>
<p>Let $f,g\in \mathbb{C}[x,y]$. </p>
<p>Let $(f,g)$ be the ideal of $\mathbb{C}[x,y]$ generated by $f,g$. </p>
<p>Given $h\in \mathbb{C}[x,y]$, how to determine whether $h\in (f,g)$ or not? </p>
<p>I have trie... | user 1 | 47,763 | <p>You should use "<strong>Gröbner basis</strong>", (Groebner) . see the book by "Cox D., Little J., O'Shea D.": named "<em>Ideals, Varieties, and Algorithms</em>", for example. In page.82 they have: </p>
<p>Corollary.2. Let $G = \{g_1, \cdots , g_t\}$ be a <em>Groebner basis</em> for an ideal $I \subset k[x_1, \cdot... |
484,589 | <p>a) $ \bigcap_{n=1}^{\infty}(-\frac{1}{n},\frac{1}{n}) $</p>
<p>b) $ \bigcap_{n=1}^{\infty}(-\frac{1}{n}, 1+\frac{1}{n})$</p>
<p>c) $\bigcup_{n=1}^{\infty}(-\frac{1}{n}, 2+\frac{1}{n})$</p>
<p>Could anyone please explain how to do this problems? I'm having a hard time trying to come up with the intervals for thes... | user71352 | 71,352 | <p>Assuming that $A,B$ are integers then $x^{2}+Ax+B$ has a rational root implies the root, $r$, takes the form $r=\frac{p}{q}$ where $p$ divides $B$ and $q$ divies $1$ (the leading coefficient). This means $q=\pm1$.</p>
|
1,026,018 | <p>Say i want a power series for a function such as $$\frac{(2x+2)(x)}{(2x)(3x+1)}$$ at $x=0$.
How would one go about this? I have acquired the second, third and fourth terms, but am struggling getting the first term since f(0) is undefined.</p>
<p>Can one just assume that for very small $x%$, i.e $\lambda << 1$... | Community | -1 | <p>The first term is the limit at the point. </p>
<p>We get $$\frac{(2x+2)(x)}{(2x)(3x+1)}=\frac{x+1}{3x+1}=\frac{1}{3}+\frac{2}{3}\frac{1}{3x+1}=\frac{1}{3}+\frac{2}{3}\sum_{n=0}^{\infty}(-1)^n3^nx^n$$</p>
|
1,026,018 | <p>Say i want a power series for a function such as $$\frac{(2x+2)(x)}{(2x)(3x+1)}$$ at $x=0$.
How would one go about this? I have acquired the second, third and fourth terms, but am struggling getting the first term since f(0) is undefined.</p>
<p>Can one just assume that for very small $x%$, i.e $\lambda << 1$... | John Hughes | 114,036 | <p>In this case, you might factor a 2 out of the first factor of the top, and then cancel $2x$ on top and bottom. That gets you a new function $x/(3x+1)$ that equals yours almost everywhere, hence they have the same MacLaurin series. The series for YOUR function at 0 isn't really well-defined, but if it were, it'd have... |
2,980,366 | <p>Euler's theorem was expanded to encompass polyhedrons homeomorphic to not only spheres but also <span class="math-container">$g$</span>-holed toruses. I've tried to understand proofs about how <span class="math-container">$2-2g$</span> is a topological invariant but have always had trouble with the use of planar gra... | Deane | 10,584 | <p>The idea is to use the standard affine charts of <span class="math-container">$G(k,n)$</span>. Start with the <span class="math-container">$k$</span>-plane <span class="math-container">$P \subset \mathbb{R}^n$</span> (say, the one spanned by <span class="math-container">$e_1, \dots, e_k$</span>) and
a <span class="m... |
258,598 | <pre><code>Solve[Sin[Tan[x]] - Tan[Sin[x]] == 0, x]
</code></pre>
<p>It says it runs out of methods. After a very long time. Plotting the expression shows there are periodic solutions and some very nasty bits, too. NSolve does no better.</p>
| Richard Fateman | 14,181 | <p>One way I found (using Maxima, not Mathematica) but should be possible to do in Mathematica as well... convert the expression to complex exponentials, and factor it.
One factor is exp(exp(I x)) + exp(exp(- I x)).
setting this to zero and solving gives x=I log(-1) or x=0. I think this corresponds to a complete ... |
2,178,698 | <p>I am looking to construct an example of sequence of functions such that </p>
<p>$s_k =0$ outside of $[0,1]$, $\lim s_k = 0$ and $\lim \int s_k d\lambda$ = $\infty$ </p>
<p>In my attempt I came up with the following sequence </p>
<p>$f_k(x)=\begin{cases}
k, &\quad\text{if } - \frac{1}{k} \leq x \leq \fr... | MAS | 182,585 | <p>By the mean value version of Taylor's theorem we have:
\begin{align}
f(y) &=f(x)+f'(x)(y-x)+\dfrac{1}{2}f''(z)(y-x)^2, \text{for some }z\in[x,y].\\
\end{align}</p>
|
4,498,547 | <p>(Foreword: Sorry to bother this platform with this extremely simple confusion, but I've been stuck on this trivial question for a good 3.5 hours, leading to anguish/frustrated to the point that again (and in the past countless times) want to change another self-study textbook, or even give up (although it will not c... | Prem | 464,087 | <p>I have not gone over all the calculations, but I have a concern here:</p>
<hr />
<p><a href="https://i.stack.imgur.com/lpKhN.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/lpKhN.png" alt="Q Part" /></a></p>
<hr />
<p>I think here the first term is fine , <strong>the middle term is missing <span c... |
75,517 | <p>I'm writing an importer for the <a href="http://www.nitrc.org/projects/gifti/" rel="noreferrer">GIFTI file format</a>. The details of the format are not particularly important, but the basic idea is that it is a relatively simple XML file which includes binary arrays of 32-bit floating point numbers that are represe... | Shadowray | 47,416 | <p>A simple, efficient, but undocumented way to use <code>zlib</code> <code>deflate</code> algorithm in <em>Mathematica</em> is to utilize the functions <code>Developer`RawCompress[]</code> and <code>Developer`RawUncompress[]</code>.</p>
<p>They have the following syntax:</p>
<pre><code>zlibStreamBytes = Developer`Ra... |
1,129,150 | <p>I know the eigenvalues for a matrix. Let's say they are 2 and 1. How can I find the matrix A for them (all members of A are not null) ?</p>
| Bill Dubuque | 242 | <p>Below we <em>derive</em> that it is <em>equivalent</em> to $\, a(a\!+\!1)\,$ is even.</p>
<p><strong>Lemma</strong> $\,\ ab(a\!+\!b)+ac(a\!+\!c)+bc(b\!+\!c)$ even $\,\iff ab(a\!+\!b)\,$ even</p>
<p><strong>Proof</strong> $\ \ (\Rightarrow)\ \ $ Let $\,c = 0.\quad (\Leftarrow)\ \ $ Permute $\,a,b,c\,$ and add.</p>
... |
2,006,626 | <p>I was just wondering if the following proof makes logical sense and is set out in a manor which is easy to read and understand, to a mathematician.</p>
<p>Prove the $\lim_\limits{x\to2}\,\, 2x^2-6x+7=3$.</p>
<p>$\underline {Proof}:$</p>
<p>Let $\epsilon \gt 0$ be given.</p>
<p>$\underline{Observation}:$</p>
<p>... | Gono | 384,471 | <p>For technical reasons you have to choose $\delta=min\{1,\,\,\frac{\epsilon}{4}\}$ to make sure the step "Let a suitable interval for $x$ be $1<x<3$." won't fail. Because otherwise the condition $|x-2| < \delta$ wouldn't ensure that $x\in(1,3)$ if $\delta$ is greater then 1.</p>
<p>Otherwise the $\varepsilo... |
2,331,657 | <p>On the generalization of a <a href="https://math.stackexchange.com/questions/2329248/to-compute-frac12-pi-i-int-mathcalc-1zz22-dz-where-mathcalc">recent question</a>, I have shown, by analytic and numerical means, that</p>
<p>$$\frac{1}{2\pi i}\int_\mathcal{C} |1+z+z^2+\cdots+z^{2n}|^2~dz =2n$$</p>
<p>where $\math... | Yiorgos S. Smyrlis | 57,021 | <p>First of all
$$
\int_{|z|=1}\overline{z}\,dz=\int_{|z|=1}\frac{dz}{z}=2\pi i,
$$
since $z\overline{z}=1$. Meanwhile, for $k>1$
$$
\int_{|z|=1}\overline{z}^k\,dz=\int_{|z|=1}\frac{dz}{z^k}=0.
$$</p>
<p>Now, when $|z|=1$, we have $\overline{z}=z^{-1}$ and hence
$$
|1+z+\cdots+z^{2n}|^2=(1+z+\cdots+z^{2n})(1+\over... |
1,470,476 | <p>Using Stokes' theorem, the line integral of a vector field gives a surface integral of the curl of the vector field, and after that, if we apply Gauss' divergence theorem in that, it gives a volume integral of the divergence of the curl of that vector field. But we know the divergence of the curl of a vector field i... | Fabrice NEYRET | 277,841 | <p>For Stokes, your curved line is closed and bounds a surface.
For Gauss, your surface is closed and bounds a volume.
Both basically say the same thing, relating a border to the inside.
But the out of the first does not fit the in of the second, even if both are "surfaces" ;-) </p>
|
206,390 | <p>I have a list of strings:</p>
<pre><code>lis = {"a","b","c","12","d","q","r","X","s"}
</code></pre>
<p>I'd like to delete list members starting with "X" moving backwards through the list from "X" until a list member that's a digit character is found, to get:</p>
<pre><code>res = {"a","b","c","12","s"}
</code></pr... | MelaGo | 63,360 | <pre><code>rlis = Reverse[lis];
xpos = First[Flatten[Position[StringMatchQ[rlis, "X"], True]]];
Reverse[Drop[rlis, {xpos, xpos + LengthWhile[rlis[[xpos + 1 ;;]],
StringMatchQ[#, NumberString] == False &]}]]
</code></pre>
<blockquote>
<p>{"a", "b", "c", "12", "s"}</p>
</blockquote>
|
1,229,194 | <p>Problems with the following limits:</p>
<p>$$
1. \quad \quad \lim_{x\to0^+} e^{1/x} + \ln x \, .
$$</p>
<p>Substitutions such as $e^{1/x}=t$ and $1/x = t$ don't yield any useful results. </p>
<p>Pretty much the same with
$$
2. \quad \quad \lim_{x\to 0^+} e^{1/x} - 1/x \, ,
$$
Common denominator doesn't help much... | abel | 9,252 | <p>first let us show that $$ \lim_{u \to \infty} \frac{\ln u}{e^u} = 0$$ this follows because $$ \ln u < u < \frac{u^2}{2} < e^u \text{ so that } \frac{\ln u}{e^u} < \frac{2u}{u^2} \rightarrow 0 \text{ as } u \to \infty. $$
let us make a change of variable $u = \frac 1 x, x = \frac 1 u.$ with that we nee t... |
1,638,267 | <p>Can someone please point out where I am (If I am) going wrong during the solution process of the following question:</p>
<p>I have been presented with the following :</p>
<p>$$4sinh(2ln(2))-cosh(ln2)$$</p>
<p>and told by my tutor the solution is 10. however I cannot obtain this value, the steps I take are as foll... | Chappers | 221,811 | <p>You're correct:
$$\begin{align} 4\sinh{2\log{2}} - \cosh{\log{2}} &= \cosh{(\log{2})} \left( 8\sinh{(\log{2})}-1 \right) \\
&= \frac{1}{2}(2+1/2)(4 \cdot 2 - 4 \cdot 1/2-1) \\
&= \frac{5}{4}(8-2-1) \\ &= \frac{25}{4} \end{align}$$</p>
|
2,012,564 | <p>We had to prove that if</p>
<p>$$\lim_{n\to\infty}(a_n\cdot b_n)=0$$ </p>
<p>Then either $\lim_{n\to\infty}a_n$ or $\lim_{n\to\infty}b_n$ HAS to be equal to $0$.</p>
<p>My hypothesis is that since</p>
<p>$$\lim_{n\to\infty}(a_n\cdot b_n)=\lim_{n\to\infty}a_n\cdot \lim_{n\to\infty}b_n$$</p>
<p>Then for $\lim_{n\... | mfl | 148,513 | <p>The claim is false. Consider $$a_n=\begin{cases}0, & n\:\:\mathrm{even}, \\1, & n\:\:\mathrm{odd}, \end{cases}$$ and $$b_n=\begin{cases}1, & n\:\:\mathrm{even}, \\0, & n\:\:\mathrm{odd}. \end{cases}$$ </p>
<p>We have $a_nb_n=0,\forall n$ and thus $\lim_{n\infty} a_nb_n=0.$ But $\lim_{n\to\infty}a_n$... |
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