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 |
|---|---|---|---|---|
194,569 | <blockquote>
<p>How can we find an integer $n$ such that $\mathbb{Z}[\frac{1}{20},\frac{1}{32}]=\mathbb{Z}[\frac{1}{n}]$?</p>
</blockquote>
<p>Thanks in advance!</p>
| Bill Dubuque | 242 | <p><strong>Hint</strong> $\ $ Let $\rm\,Z = \Bbb Z$ or any Bezout domain, i.e. a domain with gcds which are linear combinations $\rm\,(a,b) = ja+kb,\ j,k\in Z.\:$ Every ring $\rm\,R\,$ between $\rm\,Z\,$ and its fraction field $\rm\,Q\,$ is equal to the subring generated by $\rm\,Z\,$ and inverses of primes of $\rm\,Z\... |
2,714,262 | <p>I've been trying to prove that $\pi > 3$ by using the following definition:</p>
<p>$$\pi = 2\int_{-1}^1{ {\sqrt{1-t^2}}}\, dt$$</p>
<p>Which comes from finding what the area of the unit circle is. (This path can be found in Spivak's Calculus, in case someone wants to read about this topic)</p>
<p>I've done it ... | Yiorgos S. Smyrlis | 57,021 | <p>In the OP $\pi$ is defined as the area of the unit circle, and it is thus larger than the area of any inscribed, in the unit circle, canonical polygon.</p>
<p>We shall show that the area $A_{12}$ of the canonical 12-gon, inscribed in the unit circle is EXACTLY 3, and hence $\pi>3$.</p>
<p>Indeed,
$$
A_{12}=12\t... |
3,141,004 | <p>I am just a high school student, so please excuse me if this question sound silly to you.</p>
<p>How do I graph this : <span class="math-container">$\dfrac{9}{x} - \dfrac{4}{y} = 8$</span></p>
<p>I could do trial and error, but is there a more systematic way that I can use to graph this equation.</p>
| Dr. Sonnhard Graubner | 175,066 | <p>I would write <span class="math-container">$$\frac{9}{x}-8=\frac{4}{y}$$</span> so,
<span class="math-container">$$\frac{9-8x}{x}=\frac{4}{y}$$</span> and so, <span class="math-container">$$y=\frac{4x}{9-8x}$$</span> and then use a function plotter from the internet.</p>
|
1,588,827 | <p>I'm trying to understand an algorithm, to solve $Ax = b$ linear equations.</p>
<p>But there is an equation, which I can not understand: $e^{-iAt}$
What does it mean, to calculate the $-iAt$ power of $e$ ?</p>
| John Gowers | 26,267 | <p>Use the <a href="https://en.wikipedia.org/wiki/Matrix_exponential" rel="nofollow">matrix exponential</a>:</p>
<p>$$
e^A = I + A + \frac12 A^2 + \frac 1{3!} A^3 + \cdots
$$</p>
<p>You can check for yourself that it satisfies the important property:</p>
<p>$$
\frac{d}{dt} e^{tA} = Ae^{tA} = e^{tA}A
$$</p>
|
2,304,168 | <p>Let's say we have a random point inside a circle . My intuition tells me that that If I draw vectors from all the circumference points to that point and find their sum the result will be zero.</p>
<p>Am I correct ? How can we prove this ?</p>
<p>Edit: I am asking about the sum of the 'vectors'</p>
| fosho | 166,258 | <p>In your attempt, line $(1)$ and line $(2)$ conclude the same thing. </p>
<p>Proceed like this: let $x\in \bar{A}\setminus A$. We want to show $x\in \text{bd}{A}$. Recall $\text{bd} A = \bar{A}\cap \overline{A'}$. We already know that $x\in \bar{A}$. </p>
<p>To show that $x\in \overline{A'}$ we must show that $B(x,... |
3,948,378 | <p>For <span class="math-container">$x\in(-\frac{\pi}{2}, -\frac{\pi}{2})\setminus \{0\}$</span> we can say that since the area of sector of circle is greater than the triangle (ABC) area I have <span class="math-container">$2|\sin{x}|<2|x|$</span>. But what's about x in the 2nd and 3rd quadrant?
Look at the figure ... | Sage Stark | 745,622 | <p><strong>Hints:</strong> Arclength (with units of distance) is the product of the radius <span class="math-container">$1$</span> and the angle <span class="math-container">$x$</span>. Show how this arclength is longer than the altitude by showing how on a Euclidean plane, a line segment is the shortest path between t... |
1,177,111 | <p>Let $(a_n)$ be defined by $a_1=1$, $a_{n+1}=a_n+{1\over a_n}$. Prove there exists $\lim_\limits{n\to \infty}{a_n}$ and find $\lim_\limits{n\to \infty}{a_n\over n}$. </p>
<p>I tried many things but they wouldn't work. I would appreciate your help.</p>
| Gabriel Romon | 66,096 | <p>As suggested in the comments, $\lim_{n\to\infty}a_n= \infty$.</p>
<p>Note that $a_{n+1}^2=a_n^2+2+\frac{1}{a_n^2}$</p>
<p>Hence $\lim_{n\to\infty} (a_{n+1}^2-a_n^2)=2$</p>
<p>By Cesaro theorem, $\displaystyle \lim_{n\to\infty}\frac{\sum_{k=1}^n a_{k+1}^2-a_{k}^2}{n}=2$, that is to say $\lim_{n\to\infty}\frac{a_n^... |
1,177,111 | <p>Let $(a_n)$ be defined by $a_1=1$, $a_{n+1}=a_n+{1\over a_n}$. Prove there exists $\lim_\limits{n\to \infty}{a_n}$ and find $\lim_\limits{n\to \infty}{a_n\over n}$. </p>
<p>I tried many things but they wouldn't work. I would appreciate your help.</p>
| Sangchul Lee | 9,340 | <p>Just for a reference, we can prove that</p>
<p>$$ a_n = \sqrt{2n} + \frac{\log n}{4\sqrt{2n}} + O \left( \frac{1}{\sqrt{n}} \right). $$</p>
<p>The idea is essentially the same as @LeGrandDODOM's proof, but in this case we successively improve the error term.</p>
|
44,983 | <p>(I had no idea how to tag this, any help would be great) :)</p>
<p>I am a programmer with little math experience beyond high school - this will be demonstrated by my explanation. </p>
<p>I am developing an application that needs a formula. The formula involves calculating what amounts of food to eat based on facto... | Luboš Motl | 10,599 | <p>For Acme widgets, you may order either $0$ or $1$ or $2$ or $3$ boxes; this integer will be called $M$. It's just four possibilities and all of them are small numbers of boxes and it is easy to manually check which of these four values will you the best results.</p>
<p>Most of the adjustment is done by and most of ... |
10,352 | <p>I teach high school geometry and see many of my students fall in to the trap of "it looks like it, so it's true" -- a <a href="https://en.wikipedia.org/wiki/Van_Hiele_model" rel="noreferrer">Van Hiele Level 0 to 1</a> thought process. For instance, when talking about parallel line angle relationships, once they real... | Ieuan Stanley | 6,023 | <p>The argument has been made that this is sort of a misappropriation of the terms, because the levels are meant to define levels of understanding rather than levels of detail. I'm assuming what you're aiming for is that you can say to your student(s) "I'm looking for a level 2 explanation here", and they wou... |
2,205,688 | <p>"What is the difference between
⊨
(semantic consequence) and
⊢
(syntactic consequence)?" was a question that has been <a href="https://math.stackexchange.com/questions/286077/implies-vs-entails-vs-provable">posted</a>, but I am wanting a more specific answer. For example, this <a href="https://www.yout... | Mauro ALLEGRANZA | 108,274 | <p>$\vDash$ means: <a href="https://en.wikipedia.org/wiki/Logical_consequence" rel="nofollow noreferrer">logical consequence</a>.</p>
<p>The general definition of it is:</p>
<blockquote>
<p>A sentence $\varphi$ is said to be a <em>logical consequence</em> of a set of sentences $\Gamma$ (in symbols: $\Gamma \vDash \... |
2,205,688 | <p>"What is the difference between
⊨
(semantic consequence) and
⊢
(syntactic consequence)?" was a question that has been <a href="https://math.stackexchange.com/questions/286077/implies-vs-entails-vs-provable">posted</a>, but I am wanting a more specific answer. For example, this <a href="https://www.yout... | Bram28 | 256,001 | <p><span class="math-container">$\vdash$</span> is used to make statement about formal proof systems, which include rules of inference, that say:</p>
<p>"If you have a (or two) statement(s) that look like such-and-so, then you can write down a new statement that looks like this-and-that". </p>
<p>For example, many fo... |
3,264,330 | <p>I am really struggling with a contour integration question, which I am revising for an exam. I want to show that the contour integral of </p>
<p><span class="math-container">$$\int_\Gamma e^{t(z+z^{-1})}z^{-2} dz=\sum_{m=0}^\infty b_mt^{2m+1}$$</span> where the <span class="math-container">$b_m$</span> are constant... | Felix Marin | 85,343 | <p><span class="math-container">$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\exp... |
3,672,378 | <p>Consider the following Primal LP </p>
<p><span class="math-container">$$
\min_{x_1,x_2,x_3} x_1+x_2+x_3 $$</span>
subject to constraints,
<span class="math-container">$$
x_1+x_2 \ge a \\
x_1+x_3 \ge b \\
x_2+x_3 \ge c\\
x_1 \ge d \\
x_2 \ge e \\
x_3 \ge f
$$</span></p>
<p>Here <span class="math-container">$$a,b,c... | Siong Thye Goh | 306,553 | <p>For your particular problem, </p>
<p>We have <span class="math-container">$x_1+x_2+x_3 \ge a + f$</span>. Hence it must be bounded and can't to go <span class="math-container">$-\infty$</span>.</p>
<p>To construct a feasible solution, we just need the variables to be large enough to be feasible. <span class="math-... |
4,361,604 | <p>I am studying the <strong>the basic theorem of field extensions</strong> with <em>Charles C. Pinter</em>'s book: <em>A book of Abstract Algebra</em>. The very same theorem that is being discussed here: <a href="https://math.stackexchange.com/questions/2272238/example-of-basic-theorem-of-field-extensions">Example of ... | Wuestenfux | 417,848 | <p>Here <span class="math-container">$J=\langle p(x)\rangle$</span> is the ideal generated by the underlying irreducible polynomial.</p>
<p>The result basically says that <span class="math-container">$\bar x = x+J$</span> is a zero of <span class="math-container">$p(x)$</span> which lies in the extension field <span cl... |
2,057,788 | <p><strong>Is the following true?</strong></p>
<p>If $k \geq 0$...</p>
<p>$x^2 > k \implies$</p>
<p>$|x| > \sqrt k \implies$</p>
<p>$x > \sqrt k$ or $x < -\sqrt k$</p>
<p>If $k < 0$...</p>
<p>$x^2 > k \implies$</p>
<p>$x^2 \geq 0$</p>
| student forever | 328,820 | <p>Let $y=x+h$ with $h\ne 0$. Then $$0 \le \left| \frac{f(x+h)-f(x)}{h}\right|<|h|$$ and so $$-|h| \le\frac{f(x+h)-f(x)}{h} \le |h|.$$Take limit for $h$ to zero then $$f^\prime(x)=0.$$</p>
|
1,783,841 | <p>If a function $f(x)$ has a derivative $f'(x)$ then where $f'(x_0) = 0$ there is an extreme point at $x=x_0$.
And where $f''(x_0)=0$ there is an inflection point at $x=x_0$.</p>
<p>I am asking are there any significance or applications where $f(x_0)=f'(x_0)$ at some point $x=x_0$?</p>
<p>I am referring to a case li... | Micah | 30,836 | <p>Say this happens at the point $(a,b)$. Then the slope of the tangent line that point is also $b$, so the tangent line at that point is
$$
y-b=b(x-a)
$$
which can be rewritten as
$$
y=b(x-(a-1))
$$
That is, the tangent line at $(a,b)$ is the line of slope $b$ which intersects the $x$-axis at the point $(a-1, 0)$.</p>... |
375,348 | <p>Prove that $\mathbb Z_{m}\times\mathbb Z_{n} \cong \mathbb Z_{mn}$ implies $\gcd(m,n)=1$.</p>
<p>This is the converse of the Chinese remainder theorem in abstract algebra. Any help would be appreciated.</p>
<p>Thanks!</p>
| amWhy | 9,003 | <p>Suppose $\gcd(m, n) = d\gt 1$. </p>
<p>Then $\frac{mn}{d}$ is divisible by both $\,m \text { and}\;n.\;$ Therefore, for any $(r, s) \in \mathbb Z_m \times \mathbb Z_n$ we have that $$\underbrace{(r, s) + (r, s) + \cdots + (r, s)}_{\Large \frac{mn}{d} \,\lt\, mn \;\;\text{summands}} = (0, 0).$$ Hence no element $\,... |
1,063,638 | <p>A student is standing in each cell of an $n\times n$ grid, looking at one of the four directions: up, down, left, right. It turns out that no student is at the border and looking out of the grid, and no student directly sees the face of another student (every student sees either the ear or the back of another studen... | heropup | 118,193 | <p>Consider a slight modification of your arrangement: $$\begin{matrix}
\downarrow&\downarrow&\downarrow&\downarrow&\downarrow\\
\downarrow&\downarrow&\downarrow&\downarrow&\downarrow\\
\downarrow&\downarrow&\downarrow&\downarrow&\color{red}\downarrow\\
\color{red}\downa... |
3,808,774 | <p>I have this limit I've been trying to solve, and so far I can only solve it using the property of geometric series.</p>
<p>So we get
<span class="math-container">$$\lim_{n \to \infty}\frac{1}{3^2}+\frac{1}{3^3}+...+\frac{1}{3^n}=\sum_{n=2}^{\infty}\left(\frac{1}{3}\right)^n=\frac{1}{1-\frac{1}{3}}=\frac{3}{2}$$</spa... | user | 505,767 | <p>We have that</p>
<p><span class="math-container">$$\sum_{\color{red}{n=0}}^{\infty}\left(\frac{1}{3}\right)^n=1+\frac13+\frac{1}{3^2}+\frac{1}{3^3}+...+\frac{1}{3^n}+\ldots=\frac{1}{1-\frac{1}{3}}=\frac{3}{2}$$</span></p>
<p>therefore</p>
<p><span class="math-container">$$\sum_{n=2}^{\infty}\left(\frac{1}{3}\right)^... |
2,220,801 | <p>I am having trouble verifying stokes theorem for the following surface and vector field. Let $F$ be $z\vec{i}+xy^2\vec{j}+x^2z\vec{k}$ and let $z=x^2+y^2$ be the surface, where $0 \leq z \leq 6$. So I took the top of this bowl (in the plane $z=6$)to be a disc of radius equal to $\sqrt{6}$. Stokes theorem says that t... | Stella Biderman | 123,230 | <p>$24=8\cdot 3$ so we need to show that $8$ and $3$ divide that number.</p>
<p>First, let's consider $(a+b)(a-b)$. Since $a,b$ are odd, both of these are even. Furthermore, we can check by going through the possible values of $a,b\pmod{4}$ that at least one is a multiple of $4$. $1-1,1+3,3+1,$ and $3-3$ are all $0\pm... |
2,220,801 | <p>I am having trouble verifying stokes theorem for the following surface and vector field. Let $F$ be $z\vec{i}+xy^2\vec{j}+x^2z\vec{k}$ and let $z=x^2+y^2$ be the surface, where $0 \leq z \leq 6$. So I took the top of this bowl (in the plane $z=6$)to be a disc of radius equal to $\sqrt{6}$. Stokes theorem says that t... | zwim | 399,263 | <p><em>Here is a method exploiting the fact that powers are often reductible to lesser exponents, especially when working modulo $2$ or $3$.</em></p>
<p><br/>
Let's have $c=ab(a+b)(a-b)=a^3b-ab^3$</p>
<p><br/>
We always have $x^3\equiv x \pmod 3\quad$ </p>
<p>So $c \pmod 3\equiv ab-ab\equiv 0$.</p>
<p><br/>
Now let... |
2,554,060 | <p>The book said: "if $f$ is a bounded linear functional then ker $f$ is a closed subspace of the Hilbert space $H$."</p>
<p>But I do not know why, could anyone clarify this for me please? </p>
| Fghj | 458,544 | <p>Whenever cyclic coefficients of $x^2, x, 1 $ are observed,
Like $ (a-b),(b-c),(c-a) $, then $ x= 1$ must be a solution.</p>
<p>Now $ x=1 $, since roots of this quadratic are equal, so
Roots are $ z_1=z_2=1 $</p>
<p>Product of roots $ = z_1z_2 = \frac{c(a-b)}{a(b-c)} $</p>
<p>$ \implies 1 = \frac{c(a-b)}{a(b-c)... |
3,174,732 | <p>Let <span class="math-container">$(X,\mathcal{A}, \mu)$</span> be of finite measure and <span class="math-container">$f$</span> integrable on the measure space. Show that: </p>
<p><span class="math-container">$\lim_{n \to \infty} \int_{X}|f|^{\frac{1}{n}}d\mu=\mu(\{f\neq0\})$</span></p>
<p>My ideas:</p>
<p>I init... | Yanko | 426,577 | <p>Hint:</p>
<p>Use the dominated convergence theorem.</p>
<p>If you can make the <span class="math-container">$\lim$</span> inside the integral you can use the fact that <span class="math-container">$\lim_{n\rightarrow\infty} \sqrt[n]{|x|} = 1$</span> whenever <span class="math-container">$x\not = 0$</span>. </p>
|
3,313,655 | <p><span class="math-container">$$a^2 (a + 2 b) (a + b^2 + b)^{\frac{3}{2}}>b^3 (a^2 + a + b)^{\frac{3}{2}}$$</span></p>
<p>This is true with <span class="math-container">$a>b>0$</span>, <a href="https://www.wolframalpha.com/input/?i=a%5E2%20(a%20%2B%202%20b)%20(a%20%2B%20b%5E2%20%2B%20b)%5E(3%2F2)%3Eb%5E3%20... | Amy Ngo | 692,535 | <p>Try moving all the terms raised to the power of 3/2 onto one side of the equation only, and move all other terms to the other side of the equation. Then square both sides so that the power of 3/2 becomes 3, and use algebra from there.</p>
|
3,313,655 | <p><span class="math-container">$$a^2 (a + 2 b) (a + b^2 + b)^{\frac{3}{2}}>b^3 (a^2 + a + b)^{\frac{3}{2}}$$</span></p>
<p>This is true with <span class="math-container">$a>b>0$</span>, <a href="https://www.wolframalpha.com/input/?i=a%5E2%20(a%20%2B%202%20b)%20(a%20%2B%20b%5E2%20%2B%20b)%5E(3%2F2)%3Eb%5E3%20... | Piquito | 219,998 | <p>COMMENT.-The points <span class="math-container">$(a,b)=(1,t)$</span> for <span class="math-container">$t\ge4$</span> does not satisfy the inequality.</p>
|
468,437 | <p>If $A\times B$ is closed in $X\times Y$, then are $A$ and $B$ closed in $X$ and $Y$ respectively?</p>
| Ahaan S. Rungta | 85,039 | <p>You don't need a computer to put $ \left( \dfrac {1}{10^7} \right)^{455} $. This is just $ \dfrac {1}{10^{7 \cdot 455}} = \boxed {\dfrac {1}{10^{3185}} = 10^{-3185}} $. </p>
|
1,710,095 | <p>I know that, for $x_i ∈ \mathbb R$, </p>
<p>$ 0 = \sum_{i=0}^n x_i^2$ </p>
<p>iff $x_0=x_1=x_2=...=x_n=0$</p>
<p>However, if we consider $z_i ∈ \mathbb C$, does the same principle hold? Or rather, under what conditions would that principle hold?</p>
<p><strong>EDIT</strong>: Just realized that... </p>
<p>For th... | Christian Blatter | 1,303 | <p>Since the right hand side of the given DE is a real analytic function of the variables $y$ and $t$ it follows from general principles that any solution is real analytic, in particular $C^\infty$, on its domain of definition.</p>
<p>The $C^\infty$ part can be proven by induction: If $\phi: t\mapsto y(t)$ is a solut... |
1,121,492 | <p>We define e to be a number which satisfies the following condition</p>
<p>$$\lim _{a \to 0} \frac{e^a-1}{a}=1. $$</p>
<p>How did we arrive to the following from above equation</p>
<p>$$e=\lim _{n \to \infty} \bigg(1+\frac{1}{n}\bigg)^n ? $$</p>
<p>so that we get the value of n</p>
| rubik | 2,582 | <p>It's easier to start from the first equation and prove that it's the same as the second one.</p>
<p>From
$$L := \lim _{a \to 0} \frac{e^a-1}{a}$$
set $e^a - 1 = t$, so that $a = \ln(1 + t)$:
$$L = \lim_{t \to 0} \frac{t}{\ln(1 + t)} = \lim_{t \to 0} \left(\frac{\ln(1 + t)}{t}\right)^{-1} = \lim_{t \to 0} \left(\ln\... |
28,670 | <p><strong>The problem:</strong> Sometimes, an user begins acting adversarially and is unresponsive to improvement of their post. This is bad behaviour in general and part of the community feels urged to address it.</p>
<p>We know that, in practice, the close reason "off-topic because ... missing context or other ... | D.W. | 14,578 | <p>I don't recommend adopting a close reason based on unresponsiveness.</p>
<p>If you think a response is needed, presumably that's because you think there is something unsuitable about the question if no change is made: maybe it is too broad, maybe it is unclear what is being asked, maybe it appears to be out of scop... |
278,511 | <p>What is the best unconditional upper bound for $p_{n^2}-p_{(n-1)^2}$ such that $p_n$ is the $n$-th prime number?</p>
<p>Asymptotics suggest it's somewhere near $4 n \ln n$, but how to prove this?</p>
<p>Edit: it's known that there is a prime between $[x,x+x^{13/23}]$, so one can bound $p_{n^2}-p_{(n-1)^2} \leq 2 n... | Terry Tao | 766 | <p>By the results of </p>
<p><em>Baker, R.C.; Harman, G.; Pintz, J.</em>, <a href="http://dx.doi.org/10.1112/plms/83.3.532" rel="nofollow noreferrer"><strong>The difference between consecutive primes. II</strong></a>, Proc. Lond. Math. Soc., III. Ser. 83, No.3, 532-562 (2001). <a href="https://zbmath.org/?q=an:1016.11... |
4,236,301 | <p>Taking derivatives both sides with respect to <span class="math-container">$a$</span> in the definition of homogeneous distribution equation <span class="math-container">$$a^{-d}\int f(x)\phi(\frac{x}{a})=a^k\int f(x)\phi(x)$$</span> for all <span class="math-container">$a>0$</span> and the test function <span cl... | Romulus Augustulus | 865,041 | <p>You proved that
<span class="math-container">$$(k+d)\langle f,\varphi\rangle = -\langle f ,x\cdot\nabla\varphi\rangle.$$</span>
Now the derivative on the RHS must be brought on <span class="math-container">$f$</span>:
<span class="math-container">$$\langle f ,x\cdot\nabla\varphi\rangle = \sum_i \left\langle f ,x_i\f... |
243,990 | <p>Suppose we have the following data list</p>
<pre><code>data = {{-1, 2}, {0, 3}, {6, 6}, {1}, {-5, 1}, {0, 0}, {3}, {-3, 3}, {1, 2}, {1},
{0, -1}, {2, 2}, {7}, {5, 5}, {2, 2}, {3, -3}, {4, -4}, {2, -2}, {7}}
</code></pre>
<p>The data are formatted as follows: first, we have N rows of (a,b) elements and then ... | Ben Izd | 77,079 | <p>This answer is inspired from the answer given by <a href="https://mathematica.stackexchange.com/users/125/kglr">kglr</a> in your other <a href="https://mathematica.stackexchange.com/questions/243902/create-a-new-list-from-given-data">question</a>:</p>
<pre><code>data = {{-1, 2}, {0, 3}, {6, 6}, {1}, {-5, 1}, {0, 0},... |
243,990 | <p>Suppose we have the following data list</p>
<pre><code>data = {{-1, 2}, {0, 3}, {6, 6}, {1}, {-5, 1}, {0, 0}, {3}, {-3, 3}, {1, 2}, {1},
{0, -1}, {2, 2}, {7}, {5, 5}, {2, 2}, {3, -3}, {4, -4}, {2, -2}, {7}}
</code></pre>
<p>The data are formatted as follows: first, we have N rows of (a,b) elements and then ... | kglr | 125 | <pre><code>split = Split[data, Length @ # != 1 &];
Column @ split
</code></pre>
<p><a href="https://i.stack.imgur.com/t3YgH.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/t3YgH.png" alt="enter image description here" /></a></p>
<p>You can use <code>GroupBy</code> to get an <code>Association</co... |
894,969 | <p>Broadly speaking, when <em>numerically</em> minimizing a <em>d</em>-dimensional objective function:</p>
<ul>
<li><p>Gradient descent generally requires more iterations, but each iteration is fast<br />
(we only need to compute 1st derivatives)</p>
</li>
<li><p>Newton's method generally requires fewer iterations, but... | wonko | 163,865 | <p>I think this depends a lot on the structure of the function you are optimizing (duh). In general non-convex cases, both algorithms have the same worst-case complexity for the number of iterations taken to drive the norm of the gradient below some given tolerance. Not sure what this means in terms of actual computati... |
1,339,041 | <p>The Fourier coefficients are defined (in our course) as: $$\hat{f(n)}={1\over 2\pi}\int_{0}^{2\pi}{f(t)e^{-int}dt}$$</p>
<p>I am asked to express g's coefficients as a combination of f's ones, given $g(x)=f(x+c)$. The answer says: $$\hat{g(n)}=e^{2\pi inc}\hat{f(n)}$$ </p>
<p>But I don't understand why. What I did... | NoseKnowsAll | 180,054 | <p>I believe that your domain of interest is incorrect. As far as I can tell, you can only get this answer if you're considering the function on the domain $x\in [0,1]$. Starting from the definition of the nth Fourier coefficient on this domain:</p>
<p>$$f_n:=\int_{0}^{1} f(t) e^{-2\pi int}dt$$</p>
<p>Then, for $g(x)... |
1,348,714 | <p>Is there always a square between two consecutive cubes? </p>
<p>I just thought of this question. It seems really simple and the answer is probably yes.</p>
<p>Edit: I should have given this more than 2 seconds of thought.</p>
| André Nicolas | 6,312 | <p>We show that if $a^3$ and $(a+1)^3$ are two consecutive <em>positive</em> cubes, then there is a perfect square strictly between $a^3$ and $(a+1)^3$. Let $x=\sqrt{a^3}$. If $x$ is an integer, let $b=x+1$. If $x$ is not an integer, let $b=\lceil x\rceil$ (the ceiling function). We will show that $b^2\lt (a+1)^3$.</p>... |
31,113 | <p>Zagier has a very short proof (<a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1041893" rel="noreferrer">MR1041893</a>, <a href="https://www.jstor.org/stable/2323918" rel="noreferrer">JSTOR</a>) for the fact that every prime number $p$ of the form $4k+1$ is the sum of two squares.
The proof defines an inv... | Christian Elsholtz | 7,673 | <p>As the answers above linked to an old paper of mine (in German, and a somewhat
different English preprint), some readers might like to know that an updated version is to appear very soon and is now linked on my webpage:</p>
<p><a href="http://www.math.tugraz.at/~elsholtz/WWW/papers/papers30nathanson-new-address3.pd... |
2,907,487 | <p>Trying to better understand how an "interpretation" works in logic: <a href="https://en.wikipedia.org/wiki/Interpretation_(logic)" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Interpretation_(logic)</a></p>
<p>For the most part it seems to be the idea that we have some syntactical symbol and we can assig... | nikola | 213,926 | <p>By definition, interpretation is valuation of given sentence when variables are given to be - true or false. So, if you have a sentence with variables $p,q,r,...$ those $p,q,r,....$ can be either true or false.</p>
|
4,295,884 | <p>The challenge is to solve this equation <span class="math-container">$2^{x}+7^{y}=9^{z}$</span> in positive integers. The obvious solution is <span class="math-container">$x=y=z=1$</span>. Using brute force, I found <span class="math-container">$3$</span> possible solutions:
<span class="math-container">\begin{eqnar... | Servaes | 30,382 | <p><strong>Hint 1:</strong> If <span class="math-container">$x>1$</span> then reducing mod <span class="math-container">$4$</span> shows that <span class="math-container">$y$</span> is even, say <span class="math-container">$y=2v$</span>, and so
<span class="math-container">$$2^x=(3^z+7^v)(3^z-7^v).$$</span>
<strong... |
4,295,884 | <p>The challenge is to solve this equation <span class="math-container">$2^{x}+7^{y}=9^{z}$</span> in positive integers. The obvious solution is <span class="math-container">$x=y=z=1$</span>. Using brute force, I found <span class="math-container">$3$</span> possible solutions:
<span class="math-container">\begin{eqnar... | Max Alekseyev | 147,470 | <p>It is easy to verify computationally that considering the equation modulo <span class="math-container">$17043520$</span> implies that it has the only solutions:
<span class="math-container">$$(x,y,z)\in \{ (1, 1, 1),\ (3, 0, 1),\ (5, 2, 2)\}.$$</span></p>
<p>I refer to the <a href="https://artofproblemsolving.com/co... |
275,558 | <p>I recently had a homework problem asking to prove that the direct product of rings (or rings with identity) are still rings (with identity), and it seemed really silly to go through all the steps in order to conclude what was extremely trivial. </p>
<p>My question is, are there instances when this is not the case? ... | Clayton | 43,239 | <p>A direct product of integral domains is never an integral domain since $(1,0)\cdot(0,1)=0$. You can also consider PIDs; $\Bbb Z$ is a PID while $\Bbb Z\times\Bbb Z$ is no. The direct product of fields is not a field, say $\Bbb Q$ and $\Bbb Q\times\Bbb Q$. </p>
<p>There is also a problem of checking that if you don'... |
2,029,326 | <p>In the latest episode of Survivor there were 6 people and they had to draw rocks from a bag. In the bag were 6 rocks, 5 white rocks and 1 black rock.
First person drew a rock and kept it in his hand.
Then the second person drew a rock.
Last person took the only rock that left. </p>
<p>Then they all showed what the... | BruceET | 221,800 | <p>Yes, the probability of drawing the black rock is 1/6 for each person.
This is obvious for the first person. Intuitively "by symmetry," it
should be clear that everyone has the same probability 1/6 of drawing
the black rock.</p>
<p>One more formal way to look at it is that there are $6! = 720$ orders
in which the s... |
1,989,764 | <blockquote>
<p>Let $f,g$ be continuous functions on $[a,b]$ and suppose $g(x) \geq 0
$ for all $x \in [a,b]$. Prove there exists $\xi \in [a,b]$ such that </p>
<p>$$ \int\limits_a^b f(x) g(x) \mathrm{d} x = f( \xi ) \int\limits_a^b
g(x) \mathrm{d} x $$</p>
</blockquote>
<p>My attempt:</p>
<p>Mean value theo... | Doug M | 317,162 | <p>$\begin{bmatrix}1&0&2\\ 0&1&3\\ 0&0&1\end{bmatrix}
\begin{bmatrix}
\cos 30& -\sin 30& 0\\
\sin 30& \cos 30& 0\\
0& 0& 1
\end{bmatrix}
\begin{bmatrix}1&0&-2\\ 0&1&-3\\ 0&0&1\end{bmatrix}
\begin{bmatrix}x\\ y\\ 1\end{bmatrix}$</p>
<p>$\begin{bmat... |
2,998,479 | <p>Solving the integration problem:</p>
<p><span class="math-container">$$\int \frac{4^x-5^x}{7^x} \, dx$$</span></p>
<p>Not sure how to solve this problem, I don't see a way to incorporate integration by parts into this problem or even some kind of substitution.</p>
<p>Seems that taking the natural log of each elem... | clathratus | 583,016 | <p><span class="math-container">$$I=\int\frac{a^x-b^x}{c^x}dx$$</span>
<span class="math-container">$$I=\int\frac{a^x}{c^x}dx-\int\frac{b^x}{c^x}dx$$</span>
<span class="math-container">$$I=\int \bigg(\frac ac\bigg)^xdx-\int\bigg(\frac bc\bigg)^xdx$$</span>
Each integral is in the form
<span class="math-container">$$G... |
2,148,196 | <p>The below question, I am looking at the 80, They translated it as LXXX, but I can also write it as XXC right? </p>
<p><a href="https://i.stack.imgur.com/yvWQY.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/yvWQY.png" alt="enter image description here"></a></p>
| Reese Johnston | 351,805 | <p>Generally, in Roman numerals, you can't subtract more than one thing. For example, IV is $4$, but $3$ can't be written IIV. So XC would be 90, but XXC is meaningless.</p>
|
834,451 | <p>Simple question from a calc 3 beginner. Visually I cannot imagine the span of two vectors, what does this necessarily mean? For example my text mentions if two vectors are parallel their span is a line, otherwise a plane. Can anyone elaborate?</p>
| user157107 | 157,107 | <p>In $R^n$:</p>
<p>If two vectors are linearly dependent their span is the line determined by the vectors (the line made by a vector starting at the origin).</p>
<p>If two vectors are linearly independent their span is the plane.</p>
<p>For three linearly independent vectors the span is the entire three dimensional... |
2,410,118 | <p>Edit: Excuse me, the numerator of the given solution should've been a rising factorial. That said, I still don't understand where it comes from?</p>
<p>The question is as follows :</p>
<blockquote>
<p>n friends visit k exhibitions, each person visits only 1 exhibition. Find the number of possibilities if</p>
<p>b) O... | Hypergeometricx | 168,053 | <p>From observation,
$$\begin{aligned}
&\begin{cases}T_{2m}&=mk\\\\
T_{2m+1}&=mk+1\end{cases}\\
\\
\Longrightarrow
&\begin{cases}\text{For even $n$:} \;\;\quad T_n&=\displaystyle\frac n2k\\
\text{For odd $n$:}\qquad T_n&=\displaystyle\frac {n-1}2k+1\;\;=\frac n2k+(1-\frac k2)\end{cases}\end{a... |
3,137,775 | <p>Let <span class="math-container">$Z$</span> be the matrix </p>
<p><span class="math-container">$$Z =
\begin{pmatrix} 1 & 2\\ 3 & 4 \end{pmatrix}$$</span></p>
<p>And the set <span class="math-container">$\text{Commu}(Z) = \{B ∈ M_{2\times2} | ZB = BZ\}$</span>.</p>
<p>What is the set <span class="math-cont... | Robert Z | 299,698 | <p>Alternative solution (without trigonometry). Note that by integration by parts
<span class="math-container">$$\begin{align}
\int \frac{\sqrt{x^2-1}}{x^4}dx&=-\frac{\sqrt{x^2-1}}{3x^3}+\frac{1}{3}\int \frac{1}{x^2\sqrt{x^2-1}}dx\\
&=-\frac{\sqrt{x^2-1}}{3x^3}+\frac{1}{6}\int \frac{D(1-1/x^2)}{\sqrt{1-1/x^2}}d... |
2,154,295 | <p>Consider $(1)$</p>
<blockquote>
<p>$$\int_{0}^{\infty}{\sin^2bx\over x^2}\cdot{\mathrm dx\over e^{2ax}}=I\tag1$$
$(a,b)>0$</p>
</blockquote>
<p>How does one show that $$I=\color{blue}{b\tan^{-1}\left({b\over a}\right)-{a\over 2}\ln{\left(a^2+b^2\over a^2\right)}}?$$</p>
<p>An attempt:</p>
<p>Using $\sin^2... | Felix Marin | 85,343 | <p>$\newcommand{\bbx}[1]{\,\bbox[8px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\n... |
1,179,154 | <blockquote>
<p>Let <span class="math-container">$f$</span> be a twice continuously differentiable function from <span class="math-container">$[0,1]$</span> into <span class="math-container">$\mathbb R$</span>. Given that
<span class="math-container">$$f(0)+2f(\frac{1}{2})+f(1)=0,$$</span>
show that
<span class... | lingdanda | 649,872 | <p>Let <span class="math-container">$g(x)$</span> be the piecewise differentiable function defined as,
<span class="math-container">$$g(x)=\begin{cases}
g_1(x),\quad x\in\left[0,\frac{1}{2}\right)\\
g_2(x),\quad x\in\left[\frac{1}{2},1\right]
\end{cases}$$</span>
By integration by parts, we find
<span class="math-conta... |
2,079,496 | <p>Let $D$ be a finite integral domain with unity and let $x \in D$. Is it true that then $x^{-1} = x^n$ for some $n \in N$, and thus every element has an inverse. This was my solution to proving that $D$ is also a field. I am thinking of the analogue proof in groups (See Grillet's abstract algebra corollary 2.4) and s... | Bernard | 202,857 | <p>Yes, it's true: since $D$ is finite, the set of positive powers of a non-zero element $a$ is finite, hence there exist distinct positive powers $r$ and $s$ (say $s>r$ such that $a^r=a^s$. We can simplify both sides by $a^r$, since we're in a domain, hence $a^{s-r}=1$.</p>
<p>To prove a finite integral domain is ... |
564,918 | <p>$$\sec(2x)=\frac{\sec^2x+\sec^4x}{2+\sec^2x-\sec^4x}$$</p>
<p>I have no idea how to verify this. I've tried changing it into cosine but it doesn't work. </p>
| Andrey Sokolov | 30,727 | <p>You can also use Wolfram Alpha <a href="http://www.wolframalpha.com" rel="nofollow">http://www.wolframalpha.com</a> to verify it </p>
<pre><code>Simplify[(Sec[x]^2+Sec[x]^4)/(2+Sec[x]^2-Sec[x]^4)]
</code></pre>
|
973,771 | <p>I am trying to find </p>
<p>$$ \lim_{x\to 0} \frac{e^{\alpha x} -e^{\beta x}}{\sin(\alpha x) + \sin(\beta x) }$$</p>
<p>but dont have a clue where to start, could someone give me a hint please ?</p>
| Petite Etincelle | 100,564 | <p>Since $\alpha \neq -\beta$</p>
<p>$$e^{\alpha x} = 1 + \alpha x + o(x)$$
$$e^{\beta x} = 1 + \beta x + o(x)$$
$$\sin(\alpha x) = \alpha x + o(x)$$
$$\sin(\beta x) = \beta x + o(x)$$
So
$$ \frac{e^{\alpha x} -e^{\beta x}}{\sin(\alpha x) + \sin(\beta x) } = \dfrac{1 + \alpha x - 1- \beta x +o(x)}{\alpha x + \beta x +... |
3,322,760 | <p>I am trying to solve the non linear ODE</p>
<p><span class="math-container">$$\displaystyle \frac{d^2y} {dx^2}=a^2(y+y^3)$$</span></p>
<p>With the boundary conditions that it vanish at <span class="math-container">$\pm\infty$</span>.
I am thinking that it might be better to deal with it an integral equation</p>
<... | fleablood | 280,126 | <p>You are correct about 1). Any irrational + rational will always be an irrational so <span class="math-container">$\mathbb Q + \mathbb {IR}\subset \mathbb {IR} \subsetneq \mathbb R$</span>. </p>
<p>(I'm using the notation <span class="math-container">$\mathbb {IR}:= \mathbb R\setminus \mathbb Q$</span> only becaus... |
1,751,531 | <p>I have a question concerning the component test for conservative fields. So the component test tells us that the vector field is conservative if the following three conditions are met.
$$
P_y = N_z \\
M_z = P_x \\
N_x = M_y
$$
I am having trouble memorizing, because I can't seem to find why these values are used... | Robert Israel | 8,508 | <p>Since your $P(x)$ is cubic, $Q(x)$ and $G(x)$ are linear. So
$$(x^2 - x - 3) (q_1 x + q_0) + (13 x - 2) = (x^2 - 2 x + 5)(g_1 x + g_0) + (-1 - 7 x)$$
Expanding and equating coefficients of like powers, you get four equations in the four unknowns $q_0, q_1, g_0, g_1$.</p>
<p>Alternatively, find the remainder (in t... |
245,464 | <p>Here's the statement that I am to prove/disprove:</p>
<blockquote>
<p>If the minimum polynomial of $A$ is $A^5 = 5 A^2$, then $A$ is not
diagonalizable.</p>
</blockquote>
<p>First, I'm confused about how this can actually be a minimum polynomial. From my understanding, the minimum polynomial is a specific poly... | DonAntonio | 31,254 | <p>If you meant (a guess) that $\,m_A(x):=x^5-5x^2\,$ is the minimal polynomial of a matrix $\,A\,$, then since $\,m_A(x)=x^2(x^3-5)\,\,,\,\,A$ is not diagonalizable since it is not the product of different linear factors (never mind whether the field of definition of $\,A\,$ contains or not the roots of the factor $\,... |
2,041 | <p>I'd like to process data like the following, from my most recent seven-question exam.</p>
<pre><code> Item 1 2 3 4 5 6 7
PtsPossible 9 13 11 12 12 11 12
Median 8 9 8 7 10 10 12
Mean 7.10 9.10 7.25 6.65 9.27 9.39 10.57... | Adrienne | 1,207 | <p>I've got no suggestions for item analysis on your presented data; stop reading if the rest is irritating because it doesn't answer your question.</p>
<p>I've read a fair bit on test design, and the vast majority of it is based upon analysis of multiple-choice questions... which you clearly do not have. And, sadly, ... |
871,221 | <blockquote>
<p>Let $F$ be a field. Show that the field of quotients of $F$ is ring isomorphic to $F$.</p>
</blockquote>
<p><strong>Attempt:</strong> Let $F'$ be the field of Quotients of the field $F$. </p>
<p>Let $\Phi:F \rightarrow F'$ such that $\Phi(x)=x/1$ </p>
<p>$x/1$ refers to the equivalence class contai... | Disintegrating By Parts | 112,478 | <p>For a given $b$, the equation $A^{T}Ax=A^{T}b$ always has a solution $x$, but this solution may not be unique. In fact, $x$ is unique iff $\mathcal{N}(A)=\{0\}$. If both $x_{1}$ and $x_{2}$ are solutions, then $Ax_{1}=Ax_{2}$ and
$$
|Ax_{1}-b|^{2}=|Ax_{2}-b|^{2},
$$
which means that both $x_{1}$ and $x... |
4,321,423 | <p>Let <span class="math-container">$(X, \mathcal{A}, \mu)$</span> be a finite measure space. Fix <span class="math-container">$1 < p, q < \infty$</span> with <span class="math-container">$\frac{1}{p} + \frac{1}{q} = 1$</span>. Let <span class="math-container">$\{f_n\}$</span> be a sequence in <span class="math-c... | Lazy | 958,820 | <p>Do you know the representation theorem for <span class="math-container">$L_p$</span>? This one says that for <span class="math-container">$1\leq p < \infty$</span> each element of the dual space of <span class="math-container">$L_p$</span> can be written as <span class="math-container">$f\to \int fg \,d\mu$</span... |
1,414,717 | <p>Given a probability space $(\Omega,\cal F,\Bbb P)$, when $\sigma$-algebra $\cal F_0$$\subseteq \cal F$ is finite (which is generated by a finite partition $\Gamma \subseteq \cal F_0$), the conditional expectation $\Bbb{E}(X\mid\cal F)$ is a function that maps each set $A$ in $\Gamma$ to the mean value of $X$ on $A$,... | Hari Shankar | 351,559 | <p>Let the centroid be <span class="math-container">$G(h,k)$</span> and the vertices be <span class="math-container">$P(t_1), Q(t_2), R(t_3)$</span> in parametric form.</p>
<p>Since <span class="math-container">$G$</span> is also the circumcenter, we have that <span class="math-container">$P,Q,R$</span> lie on the circ... |
980,470 | <p>How to find the continued fraction of $\sqrt{n}$, for an integer $n$?</p>
<p>I saw a site where they explained it, but it required a calculator. Is it possible to do it without a calculator? </p>
| Grey Matters | 176,262 | <p>The easiest way to find a continued fraction for $\sqrt{n}$ is to use a generalized continued fraction (continued fractions where the numerator is something besides 1).</p>
<p>Starting from $\sqrt{n}$, partition ${n}$ as the sum of the perfect square which is less than or equal to ${n}$, ${s^{2}}$, and a remainder,... |
164,007 | <p>I didn't find it in any book, although it seems that this should be standard: Endow the space $C^\infty_c(\mathbb{R})$ of compactly supported functions with the inductive topology coming from the embeddings
$$ \mathcal{D}_K \longrightarrow C^\infty_c(\mathbb{R}).$$
(Here $\mathcal{D}_K$ is the set of all smooth func... | PhoemueX | 59,219 | <p>Here is a more direct argument than that of the other answer:
Let us consider the function
<span class="math-container">$$
\Phi : C_c^\infty (\mathbb{R}) \to \mathbb{R},
f \mapsto \sum_{n=0}^\infty |f^{(n)} (0)| \cdot |f(n)| .
$$</span>
It is easy to see that <span class="math-container">$\Phi$</span> is ... |
30,339 | <p>I need to construct a vector similar to:</p>
<pre><code>v[x_]:={0, 0, x, 0, 0, 2*x, 0, 0, 3*x, ....., 0, 0, n*x};
</code></pre>
<p>where <code>n=10^9</code>. How can I make such a vector wisely?</p>
| image_doctor | 776 | <p><strong>In Memory:</strong></p>
<p>Table is handy.</p>
<pre><code>v[x_, n_] := Flatten@Table[{0, 0, x i}, {i,n}]
v[x,5]
</code></pre>
<blockquote>
<p>{0, 0, x, 0, 0, 2 x, 0, 0, 3 x, 0, 0, 4 x, 0, 0, 5 x}</p>
</blockquote>
<p>Or</p>
<pre><code>{0, 0, x #} & /@ Range@5 // Flatten
</code></pre>
<p><strong... |
1,068,183 | <blockquote>
<p>If $x,y \in \mathbb Z$ , then find all the solutions of<br>
$$y^2=3x^4+3x^2+1$$ </p>
</blockquote>
<p>I was asked this question by my friend who said that he encountered this while solving another problem. I have tried several things but am unable to solve this question. Moreover, this has to be... | Samuel Hambleton | 16,825 | <p>This does not immediately answer your question but here is how I would approach it. Let $X = 4 y$ and $Y = 2 x^2 + 1$. Then there exists integers $x, y$ satisfying $y^2 = 3 x^4 + 3 x^2 + 1$ if there exists an integer solution $(X, Y)$ to $$X^2 - 12 Y^2 = 4$$ satisfying $4 \mid X$ and $Y$ is odd of the form $2 x^2 +1... |
1,597,483 | <p>I'm trying to prove that for every $x,y>0$, $x>y$, and $n \in
\mathbb N$, $x^{1/n}>y^{1/n}$. Just some direction will be fine. We haven't learned derivatives yet. Thanks in advance.</p>
| Patrick Da Silva | 10,704 | <p>Since $x > y > 0$, for any $a > 0$, you have $a \cdot x > a \cdot y > 0$. Therefore
$$
x^2= x \cdot x > x \cdot y = y \cdot x > y \cdot y.
$$
(In the first inequality, $a=x$, but in the second, $a=y$.) You can repeat this trick for $3$ as the exponent instead of $2$ :
$$
x^3 = x \cdot x^2 >... |
2,748,162 | <blockquote>
<p>An elevator with capacity for 10 people is designed to support a maximum load of 750kg.If the weights of people that use the elevator are distribuited with mean 70kg and standard deviation 8kg. Which is the probability that a group of 10 people exceed the maximum load ? </p>
</blockquote>
<p>Attempt... | BruceET | 221,800 | <p>The total weight $T$ of ten such people is distributed as
$\mathsf{Norm}(\mu = 700,\, \sigma = \sqrt{640} = 25.2982).$</p>
<p>You seek $P(X > 750) = 1 - P(X \le 750) = 0.0241.$ You can evaluate this
probability using software (I used R statistical software below) or
by standardizing and using printed tables of ... |
1,771,845 | <p>The problem is to find the number of real solutions of a cubic equation. This exercise is in a book, in the chapter about functions, limits and continuity. This chapter is before the chapter about derivatives therefore I assume that they cannot be used to solve the exercise.</p>
<p>$$f(x) = x^3-9x^2+24x-17 = 0$$</p... | mvw | 86,776 | <p>What about using the <a href="https://en.wikipedia.org/wiki/Cubic_function#The_nature_of_the_roots" rel="nofollow">discriminant</a> and following case distinctions?
$$
\Delta = 18abcd -4b^3d + b^2c^2 - 4ac^3 - 27a^2d^2
$$
with $a = 1$, $b=-9$, $c=24$, $d=-17$ we get
$$
\Delta = 1682613 > 0
$$
which means three re... |
1,463,152 | <p>Let $$f(x)= \begin{cases}
e^{-1/x} & x>0 \\
0 & x\le 0
\end{cases}
$$
I want to show that $f^{(n)}(0)=0$ for every $n\in\mathbb{N}$.</p>
<p>My idea is to do this by induction on $n$. I can compute the first derivative by taking the limit of $$\frac{f(x)-f(0)}{x-0}=\frac{e^{-1/x}}{x}$$ as ... | user84413 | 84,413 | <p>Since $r^2-6r+9=0\implies(r-3)^2=0\implies r=3$,</p>
<p>we have that $a_n=c(3^n)+d(n3^n)$ for some constants $c$ and $d$ since 3 is a double root.</p>
<p>Then $a_1=-1$ gives $3c+3d=-1,\;\;$ and $a_2=1$ gives $9c+18d=1$.</p>
<p>Then $c+d=-\frac{1}{3}$ and $c+2d=\frac{1}{9}$, so $d=\frac{4}{9}$ and $c=-\frac{7}{9}$... |
350,239 | <p>How to prove that if polynomial $W(x)$ has $n$ real roots then
$\forall a \in \mathbb{R}$</p>
<p>$a W(x)+W'(x)$ has more than $n-1$ roots</p>
<p>I have no idea how to solve. Please some hint.</p>
| Julien | 38,053 | <p>Let $r_1<\ldots<r_l$ denote these $n$ real roots without multiplicity in increasing order, and let $m_k$ denote their multiplicity. So $n=m_1+\ldots+m_l$.</p>
<p>Every root of $W$ of multiplicity $m\geq 2$ is a root of $aW+W'$ with multiplicity $m-1$. This yields already $(m_1-1)+\ldots+(m_l-1)=n-l$ roots for... |
1,385,197 | <p>For $n\in\mathbb N$,
$$a_{n+1}=2a_n+\frac{1}{a_n},\quad a_1=1.
$$
Can any one give an explicit formula for all $a_n$? If such an explicit general formula doesn't exist, please explain it. I've tried to figure out the $n$-iterated function $f^{(n)}$ where $f(x)=2x+1/x$ or even $f(\tan(t))$. But in either cases, I fai... | Bumblebee | 156,886 | <p>We can transform this equation in to $$\dfrac{b_{n+1}}{2}=b_n+\dfrac1{b_n}$$ by substituting $b_n=\sqrt2a_n.$<br>
Note that $$b_{n+1}-4=\dfrac{2}{b_n}(b_n-1)^2$$ and $$b_{n+1}+4=\dfrac{2}{b_n}(b_n+1)^2$$ Now $$\dfrac{b_{n+1}/4-1}{b_{n+1}/4+1}=\left(\dfrac{b_n-1}{b_n+1}\right)^2$$ Continuing this processes we can obt... |
383,259 | <p>We know that the real ordered field can be characterized up to isomorphism as a complete ordered field. However this is a second order characterization. That raises the following question. Consider the following theory. We take as axioms the axioms for ordered fields, and then add an axiom schema that states that ev... | Spencer Dembner | 140,821 | <p>EDIT 2: This wrong proof at least shows that the upper bound property holds in <span class="math-container">$K(\pi)$</span>, <em>when <span class="math-container">$\phi$</span> is <span class="math-container">$\Sigma_1$</span></em>. Indeed, if <span class="math-container">$\phi$</span> is <span class="math-container... |
383,259 | <p>We know that the real ordered field can be characterized up to isomorphism as a complete ordered field. However this is a second order characterization. That raises the following question. Consider the following theory. We take as axioms the axioms for ordered fields, and then add an axiom schema that states that ev... | Alex Kruckman | 2,126 | <p><strong>Edit:</strong> As Emil Jeřábek has pointed out in the comments, there is an error in the paper linked below, and the example does not work. I'll leave the answer here to document the error and add to the interesting list of attempted answers gathered here. An erratum has now appeared: <a href="https://rml.tc... |
4,185,702 | <p>I could find three good approaches to find maximum of the function <span class="math-container">$y=x\sqrt{100-x^2}$</span>. I will explain them briefly :</p>
<p>First: Finding <span class="math-container">$x$</span> satisfies <span class="math-container">$y'=0$</span> then plugging it in the function.</p>
<p>Second:... | user0102 | 322,814 | <p>Let <span class="math-container">$f(x) = x\sqrt{100-x^{2}}$</span>. Then <span class="math-container">$x\in[-10,10]$</span>.</p>
<p>In order to find the maximum value, we can consider that <span class="math-container">$x\geq 0$</span> (otherwise <span class="math-container">$f(x) \leq 0$</span>).</p>
<p>For such val... |
4,185,702 | <p>I could find three good approaches to find maximum of the function <span class="math-container">$y=x\sqrt{100-x^2}$</span>. I will explain them briefly :</p>
<p>First: Finding <span class="math-container">$x$</span> satisfies <span class="math-container">$y'=0$</span> then plugging it in the function.</p>
<p>Second:... | dxiv | 291,201 | <p><a href="https://i.stack.imgur.com/PVdTS.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/PVdTS.jpg" alt="enter image description here" /></a></p>
<p><span class="math-container">$\,\\\;\;\;\;\;\;\;\; 2\,S \;=\; x \cdot \sqrt{100-x^2} \;=\; h \cdot 10 \;\le\; 5 \cdot 10$</span></p>
|
152,707 | <p>I have a simple equation (if you are interested it is originated from the field of hydrology):</p>
<pre><code>S^0.5 (1-(1-S^6)^0.166667)^2 == 1/1000
</code></pre>
<p>with <code>S</code> certainly being between 0 and 1. The LHS of the equation is strictly growing function of <code>S</code>. So, the equation can be ... | Rom38 | 10,455 | <p>It can be done in terms of the <code>PolarPlot</code> too. However, the kglr's shading mechanism is still needed:</p>
<pre><code>p1 = PolarPlot[{2*(1 + Cos[θ]), 2*(1 - Cos[θ])}, {θ, 0, 2 π}];
p2 = Show[
PolarPlot[{2*(1 - Cos[θ]), {θ, -π/2, π/2}],
PolarPlot[{2*(1 + Cos[θ])}, {θ, π/2, 3 π/2}]}];
Show[p1,Graph... |
152,707 | <p>I have a simple equation (if you are interested it is originated from the field of hydrology):</p>
<pre><code>S^0.5 (1-(1-S^6)^0.166667)^2 == 1/1000
</code></pre>
<p>with <code>S</code> certainly being between 0 and 1. The LHS of the equation is strictly growing function of <code>S</code>. So, the equation can be ... | matrix42 | 5,379 | <pre><code>Show[
PolarPlot[{2 (1 - Cos[θ]), 2 (1 + Cos[θ])}, {θ, 0, 2 Pi}],
RegionPlot[
With[{r = Sqrt[x^2 + y^2], θ = ArcTan[x, y]},
2 (1 - Cos[θ]) > r && 2 (1 + Cos[θ]) > r
],
{x, -5, 5}, {y, -5, 5}, PlotPoints -> 60
],
PlotRange -> All
]
</code></pre>
<p>or</p>
<pre><code>Sho... |
282,382 | <p>Let $s\text{Ring}$ denote the category of simplicial commutative rings.
We endow it with the model structure defined by declaring that fibrations, trivial fibrations and weak equivalences are, respectively, those maps inducing fibrations, trivial fibrations and weak equivalences on underlying simplicial sets.</p>
<... | Charles Rezk | 437 | <p><a href="https://arxiv.org/abs/math/0003065" rel="noreferrer">This paper</a> proves some things about left properness for categories of simplicial algebras. The context of the paper is in terms of "algebras for a simplicial algebraic theory", which certainly includes the case of simplicial objects $s\mathcal{A}$ in... |
121,852 | <p>I am creating a Java game with collisions. I found myself stuck on the following problem.</p>
<p>I have got two known lines: $y$ and $i.$ $i$ is the inbound direction and $o$ the outbound direction, therefore both angles are the same. From $y$ and $i,$ I have calculated $\alpha.$ I also know the coordinates of the ... | joriki | 6,622 | <p>I'll use capital letters for the lines to keep the notation a bit clearer. (You used the same variable name, lowercase $y$, for one of the lines and for one of the coordinates; that's a bad idea.)</p>
<p>As I understand the question, you have lines $Y$ and $I$ given by $y=a_Yx+b_Y$ and $y=a_Ix+b_I$, respectively, a... |
121,852 | <p>I am creating a Java game with collisions. I found myself stuck on the following problem.</p>
<p>I have got two known lines: $y$ and $i.$ $i$ is the inbound direction and $o$ the outbound direction, therefore both angles are the same. From $y$ and $i,$ I have calculated $\alpha.$ I also know the coordinates of the ... | Community | -1 | <p>You should generally try to avoid dealing directly with angles and lengths: use vector arithmetic when possible, or matrix transformations describing geometric operations.</p>
<p>In this case, you're <em>really</em> doing is computing a reflection across the line $Y$, right? The matrix that computes this is surely ... |
1,807,070 | <p>Suppose company {X} represents clients in buying inventory during automated (online) auctions. The client chooses the bid value beforehand, and {X} repeatedly makes that bid in the automated auctions which are constantly occurring throughout the day. (Bid value is constant). Consider that for any given auction, {X} ... | Jack D'Aurizio | 44,121 | <p>I am getting:</p>
<p>$$\begin{eqnarray*}\sum_{j=0}^{s}\binom{s}{j}(n-s+j)!(s-j)!&=&\iint_{(0,+\infty)^2}\sum_{j=0}^{s}\binom{s}{j}x^{s-j} y^{n-s+j} e^{-(x+y)}\,dx\,dy\\&=&\iint_{(0,+\infty)^2} x^s y^n e^{-(x+y)}\left(\frac{1}{x}+\frac{1}{y}\right)^s\,dx\,dy\\&=&\iint_{(0,+\infty)^2} y^{n-s} ... |
1,807,070 | <p>Suppose company {X} represents clients in buying inventory during automated (online) auctions. The client chooses the bid value beforehand, and {X} repeatedly makes that bid in the automated auctions which are constantly occurring throughout the day. (Bid value is constant). Consider that for any given auction, {X} ... | Brian M. Scott | 12,042 | <p>I use the notation $x^{\underline k}=x(x-1)\ldots(x-k+1)$ for the <a href="https://en.wikipedia.org/wiki/Falling_and_rising_factorials">falling factorial</a>. First note that</p>
<p>$$\sum_{j=0}^s(n-s+j)!\binom{s}j(s-j)!=\sum_{j=0}^s(n-s+j)!s^{\underline{s-j}}=\sum_{j=0}^ss^{\underline j}(n-j)!\;.\tag{1}$$</p>
<p>... |
1,532,331 | <p>Define $f:[0,1]\to \Bbb R$ by</p>
<p>$$f(x)=\sum_{n=1}^{\infty}x^2(1-x^2)^{n-1}$$</p>
<p>Check that $f$ is continuous or not.</p>
<p>Attempt: $$f(x)=\sum_{n=1}^{\infty}x^2(1-x^2)^{n-1}\\
=\lim\int_{0}^{1}x^2(1-x^2)^{n-1}$$
Now, Putting $x=\sin\theta$, then $\mathrm dx=\cos\theta\mathrm d\theta$, therefore integra... | Matematleta | 138,929 | <p>The series converges pointwise to $f$ on $[0,1]$. Note that $f(0)=0$. </p>
<p>By the $M$-Test, $\sum_{n=1}^{\infty }x^2(1-x^2)^{n}$ converges uniformly on $[a,1];\ a\neq 0$ and so is continuous there. </p>
<p>But $\sum_{n=1}^{\infty }a^2(1-a^2)^{n}=a^2\sum_{n=1}^{\infty }(1-a^2)^{n}=\frac{a^{2}}{1-(1-a^{2})}=1$ a... |
79,991 | <p>I'm attempting to MapThread a function of two lists that requires the index of the list values. For example,</p>
<pre><code>MapThread[#1*i+#2*j&,{{a,b,c},{e,f,g}}]
</code></pre>
<p>Where #1 represents the value from list 1, #2 the value from list 2, i the index of list 1, and j the index of list 2. The expecte... | LLlAMnYP | 26,956 | <p>At face value there is this solution:</p>
<pre><code>IndexedMapThread[list1_,list2_] :=
MapThread[(#1*#3+#2*#4 &),{list1,list2,Range@Length@list1,Range@Length@list2}]
IndexedMapThread[{a, b, c}, {d, e, f}]
(* {a + d, 2 b + 2 e, 3 c + 3 f} *)
</code></pre>
|
3,339,105 | <p>Can anyone explain to me how the expectation of geometric distribution can be used here?</p>
| k170 | 161,538 | <p>Here's one way to do it
<span class="math-container">$$\sum^{n}_{k=1}\frac{(-1)^k}{k}$$</span>
<span class="math-container">$$=\sum^{n}_{k=1}(-1)^k\int_0^1\frac{t^k}{t+1}+\frac{t^{k-1}}{t+1}\ \mathrm dt$$</span>
<span class="math-container">$$=\sum^{n}_{k=1}(-1)^k\int_0^1\frac{t^k+t^{k-1}}{t+1}\ \mathrm dt$$</span>
... |
233,465 | <p>Which function verified that: $f(x)+f(\frac1x)=\frac1a; a$-constant value?</p>
| martini | 15,379 | <p>Let $g \colon (1, \infty) \to \mathbb R$ be any function. Define $f \colon (0, \infty) \to \mathbb R$ by
\[
f(x) = \begin{cases} g(x) & x > 1\\[2mm] \frac 1{2a} & x = 1\\[2mm] \frac 1a -g\left(\frac 1x\right) & x < 1 \end{cases}
\]
Then $f$ is as wished.</p>
|
233,465 | <p>Which function verified that: $f(x)+f(\frac1x)=\frac1a; a$-constant value?</p>
| nofrills | 48,637 | <p>$f(x)=\frac{1}{2a}+\log{x^n}$ is a solution.</p>
|
1,579,077 | <p>This expression has appeared in my work:</p>
<p>$$P_n(x)=\sum _{j=0}^n\binom{n}{j}\, j^n x^j $$</p>
<p>It seems too simple not to be already somewhere (perhaps in a different form). Has anyone seen it before?</p>
| Robert Israel | 8,508 | <p>The exponential generating function of these polynomials is
$$ \eqalign{G(x,s) &=\sum_{n=0}^\infty \dfrac{1}{n!}\sum_{j=0}^n {n \choose j} j^n x^j s^n
= \sum_{j=0}^\infty \sum_{n=j}^\infty \dfrac{1}{n!}{n \choose j} j^n x^j s^n\cr
&= \sum_{j=0}^\infty \dfrac{(jxs e^s)^j}{j!} \cr }$$
Now
$$ \sum_{j=1}^\infty ... |
1,366,737 | <p>Demonstrate by induction: $3^{2n} + 7 = 4k$ is true, for any $n\in \mathbb N$. I need to demonstrate this using the induction principle.</p>
<p>So far I have:</p>
<p><strong>$n = 1$</strong></p>
<p>$$3^{2\cdot 1} + 7 = 4\cdot k $$
$$9 + 7 = 4k$$
$$16 = 4k$$
$$k = 4$$</p>
<p>So it checks for $n=1$.</p>
<p><stron... | Théophile | 26,091 | <p>Be careful not to assume what you're trying to prove. After the induction hypothesis, i.e.,
$$3^{2h} + 7 = 4k \quad \textrm{for some } k,\tag1$$
you can't jump to
$$3^{2(h+1)} + 7 = 4k' \quad \textrm{for some } k',\tag2$$
because that's what you must <em>prove</em>. Instead, start with just the left-hand side of $(2... |
4,455,969 | <p>The following question is part of a chemistry problem that came in the Dhaka University admission exam 2013-14.</p>
<hr />
<blockquote>
<p>What is <span class="math-container">$-\log\left({\sqrt{1.8\times 10^{-5}\times 0.1}}\right)$</span>? <br> <br>
(a) 2.672 <br> <br>
(b) 2.772 <br> <br>
(c) 2.872 <br> <br>
(d) 2.... | JMP | 210,189 | <p>The <span class="math-container">$\sqrt{1.8}$</span> isn't insignificant. in fact it is essential to the answer.</p>
<p>The sum approximates to <span class="math-container">$-\log_{10} 1.35 + 3$</span>, and the answers are then effectively</p>
<ul>
<li><span class="math-container">$10^{0.33}\approx 10^{\frac13}>2... |
153,668 | <p>The following code (from the Mathematica documentation) produces a simple density graph:</p>
<pre><code>DensityPlot[Sin[x]Sin[y],{x,-4,4},{y,-3,3}]
</code></pre>
<p><a href="https://i.stack.imgur.com/WXfhA.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/WXfhA.png" alt="enter image description he... | zhk | 8,538 | <p>For this, use <code>PlotRangePadding</code></p>
<pre><code>DensityPlot[Sin[x] Sin[y], {x, -4, 4}, {y, -3, 3}, PlotRange -> {{-4, 4}, {-3, 3}},
PlotRangePadding -> 0.1]
</code></pre>
<p><a href="https://i.stack.imgur.com/Bo9zZ.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Bo9zZ.jpg" alt=... |
195,413 | <p>I have tried to write a code to check whether a line is parallel to the plane in Mathematica.
Using that a plane has the normal vector of <span class="math-container">$\vec{n}=(a,b,c)$</span> and a line has a direction vector <span class="math-container">$\vec{L}=(l,m,n),$</span> the condition is <span class="math-c... | Carl Woll | 45,431 | <p>You could use <a href="http://reference.wolfram.com/language/ref/RegionDisjoint" rel="nofollow noreferrer"><code>RegionDisjoint</code></a>:</p>
<pre><code>RegionDisjoint[plane, line]
</code></pre>
<blockquote>
<p>False</p>
</blockquote>
|
2,132,863 | <p>I am studying trigonometry in my school and learned that in a triangle the side opposite to angle theta should be taken as perpendicular side - hypotenuse remains the same and the third remaining side is the base.</p>
<p>The same is required for calculating the sine / cosine etc of the angle theta for below formula... | David K | 139,123 | <p>While the idea of measuring the sine of a right angle using a degenerate triangle (hypotenuse coincident with the opposite leg, and a base of zero length) may be intuitive to some, not everyone finds it satisfactory, as you can see from the other answers.</p>
<p>And what about the sine and cosine of angles between ... |
2,074,785 | <p>I am looking for a formula for the volume $V_i(r,L)$ of the intersection between a sphere of radius $r$ and a cube of edge $L$ with the same center.</p>
<p>For $r<\frac L 2$, $V_i$ is simply equal to the volume of the sphere:</p>
<p>$$V_i(r,L) = \frac 4 3 \pi r^3 \ \ \ \ r<\frac L 2$$</p>
<p>For $r>\frac... | Andrew D. Hwang | 86,418 | <p>As John Hughes notes, you can fix the side of the cube or radius of the sphere and work in terms of the ratio. The strategy here is to fix the cube's edge length to be $2$, calculate the area of plane slices parallel to one face of the cube, then apply Cavalieri's theorem.</p>
<p><a href="https://i.stack.imgur... |
1,852,826 | <p>Let $P_n$ be the $(n+1) \times (n+1)$ matrix that contains the numbers of Pascal's triangle in the upper triangle. For example in the case of $n=3$
$$
P_3 =
\begin{pmatrix}
1 & 1 & 1 & 1 \\
0 & 1 & 2 & 3 \\
0 & 0 & 1 & 3 \\
0 & 0 & 0 & 1 \\
\end{pmatrix}
$$
or in gene... | orangeskid | 168,051 | <p>With Jyrki's idea, the map of <span class="math-container">$V_n$</span>
<span class="math-container">$$T_{\lambda} f(t) \mapsto f(t+\lambda)$$</span>
satisfies
<span class="math-container">$$T_{\lambda} (t^k) = (t+\lambda)^k = \sum_{l=0}^k \binom{k}{l} \lambda^{k-l} t^l$$</span>
so the matrix of <span class="math-co... |
2,444,641 | <p>A student asked a very insightful question in my Calculus class this morning. I did not know how to answer him. (Admittedly, I am not an analyst by trade: once I passed my qualifiers I never looked back.) I would like to know if anyone here can give a precise answer, and if that can be massaged into an answer un... | seed | 294,587 | <p>Without additional assumptions on $a_i$, there can be no such theorem. Suppose $x_n$ is a series that, no matter how slowly, converges to zero. Then the sum of the series $\{x_1, -x_1, x_2, -x_2, \ldots \}$ converges to zero, too.</p>
|
2,225,818 | <p>If a dice is rolled 1 time,the probability of getting a 4 is 1/6.what is the probability of getting exactly 4 once if the die is rolled thrice</p>
| Siong Thye Goh | 306,553 | <p>Define rolling a $4$ as a success, this is a binomial distribution with $n=3$ and $p = \frac16$.</p>
<p>$$X \sim Bin(3, \frac16)$$ where $X$ denotes the number of $4$.</p>
<p>$$P(X=1)= 3 \left(\frac16 \right)\left(\frac56\right)^2$$</p>
|
2,324,718 | <p>To find the inverse of a function, you switch the spots of $x$ and $y$ in the function and solve for the new $y$ output. For example:</p>
<p>$y = 3 + \sin\ x$</p>
<ul>
<li>Switch positions of $x$ and $y$</li>
</ul>
<p>$x = 3 + \sin\ y$</p>
<ul>
<li>Solve for this new $y$</li>
</ul>
<p>$y = \sin^{-1}(x-3)$</p>
... | scjorge | 65,878 | <p>Yes, I have an explanation about where that method comes from.</p>
<p>As we know, a function <span class="math-container">$f$</span> and its inverse function <span class="math-container">$f^{-1}$</span> fulfill the following:
<span class="math-container">$$ f\left(f^{-1}\left(x\right)\right) = x = f^{-1}\big(f\left... |
1,377,633 | <p>In a mathematical methods problem, where the $c$ is a the unit circle around the origin and in counterclockwise, I need to use a step that I'm not so sure about (Because I don't know how to develop it)</p>
<p>How can I affirm that:
$$\oint_c{\dfrac{1}{z^{2n+1}}(z^2+1)^{2n}}=\binom {2n} {n} 2\pi i$$</p>
<p>I know t... | whacka | 169,605 | <p>Employ linearity of the integral (briefly, $\int(f+g)=\int(f)+\int(g)$):</p>
<p>$$\oint_c\frac{1}{z^{2n+1}}\sum_{k=0}^{2n}\binom{2n}{n}z^{2k}\,{\rm d}z=\sum_{k=0}^{2n}\binom{2n}{n}\oint_c z^{2k-2n-1}\,{\rm d}z$$</p>
<p>Do you know how to evaluate $\oint_c z^m\,{\rm d}z$? It's $2\pi i$ if $m=-1$ and $0$ otherwise.<... |
2,964,040 | <p>"Consider the following open disks:</p>
<p><span class="math-container">$D(0,1) = \{z ∈ \mathbb{C} : \lvert z\rvert < 1\}$</span></p>
<p><span class="math-container">$D(0,p) = \{z ∈ \mathbb{C} : \lvert z\rvert < p\}$</span> with <span class="math-container">$0<p<1$</span></p>
<p>i) Prove that <span cl... | Fab's | 606,512 | <p>Suppose <span class="math-container">$A\times B\neq\varnothing$</span>. Hence, by AoC, there exists an element <span class="math-container">$(a,b)\in A\times B$</span>, so, by definition, <span class="math-container">$a\in A$</span> and <span class="math-container">$b\in B$</span>, but this is not true, because B is... |
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