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 |
|---|---|---|---|---|
1,602,312 | <p>Can we prove that $\sum_{i=1}^k x_i$ and $\prod_{i=1}^k x_i$ is unique for $x_i \in R > 0$?<br>
I stated that conjecture to solve CS task, but I do not know how to prove it (or stop using it if it is false assumption). </p>
<p>Is the pair of Sum and Product of array of k real numbers > 0 unique per array? I do ... | Christian Blatter | 1,303 | <p>The following is pretty intuitive:</p>
<p>A list $(a_1,a_2,\ldots,a_r)$ of vectors $a_k\in V$ is <em>linearly independent</em> iff none of the $a_k$, $1\leq k\leq r$, is a linear combination of its predecessors in the list. This implies that for each $k$ one has $${\rm dim}\bigl({\rm span}(a_1,\ldots, a_k)\bigr)={... |
3,499,862 | <p>What is the equivalent of <span class="math-container">$\neg (\forall x) (P(x) \vee Q(x))$</span>? Will <span class="math-container">$P(x) \vee Q(x)$</span> be negated too? Or is just <span class="math-container">$\forall x$</span> negated?</p>
| Bram28 | 256,001 | <p>the Quantifier Negation Law says that:</p>
<p><span class="math-container">$$\neg (\forall x ) \varphi(x) \Leftrightarrow (\exists x) \neg \varphi(x)$$</span></p>
<p>for any formula <span class="math-container">$\varphi(x)$</span></p>
<p>Hence:</p>
<p><span class="math-container">$$\neg (\forall x)(P(x) \lor Q(x... |
4,580,717 | <p>I know I can express "everyone is A" as:</p>
<p>P: is a person
<span class="math-container">$$ \forall x (Px \implies Ax) $$</span></p>
<p>And I can express "everyone who's A is B" as:</p>
<p><span class="math-container">$$ \forall x ((Px \land Ax) \implies Bx) $$</span></p>
<p>But how can I expr... | Bertrand Wittgenstein's Ghost | 606,249 | <p>Okay, so this is where we are at: How to say, "everyone who is <span class="math-container">$A$</span>". Everyone who is <span class="math-container">$A$</span> is not really a proposition in and of itself. It's a collection of objects in some particular domain of discourse.</p>
<p>So, we can define the fo... |
435,936 | <p>Does anyone know when
$x^2-dy^2=k$ is resoluble in $\mathbb{Z}_n$ with $(n,k)=1$ and $(n,d)=1$ ?
I'm interested in the case $n=p^t$</p>
| Najib Idrissi | 10,014 | <p>Yes, it is possible. What follows isn't a completely rigorous proof, but you should be able to make one from that.</p>
<p>The basic idea is to construct a set $A$ such that along one subsequence, $\lim |A \cap I_{a_n}|/a_n = 1$ and along another subsequence, $\lim |A \cap I_{b_n}|/b_n = 0$. We'll do this by adding ... |
239,653 | <p>It is usually told that Birch and Swinnerton-Dyer developped their famous conjecture after studying the growth of the function
$$
f_E(x) = \prod_{p \le x}\frac{|E(\mathbb{F}_p)|}{p}
$$
as $x$ tends to $+\infty$ for elliptic curves $E$ defined over $\mathbb{Q}$, where the product is defined for the primes $p$ where $... | Jeremy Rouse | 48,142 | <p>In regards to question 2, in 1982 Goldfeld proved that if $f_{E}(x) \sim C (\log x)^{r}$, then (i) $L(E,s)$ has no zeroes with ${\rm Re}(s) > 1$, and (ii) the order of vanishing at $L(E,s)$ is equal to $r$. I do not know if the converse is true (even assuming GRH for $L(E,s)$), as I don't have a copy of Goldfeld'... |
239,653 | <p>It is usually told that Birch and Swinnerton-Dyer developped their famous conjecture after studying the growth of the function
$$
f_E(x) = \prod_{p \le x}\frac{|E(\mathbb{F}_p)|}{p}
$$
as $x$ tends to $+\infty$ for elliptic curves $E$ defined over $\mathbb{Q}$, where the product is defined for the primes $p$ where $... | Myshkin | 43,108 | <p>I recommend you to look at Birch and Swinnerton-Dyer's second paper (Notes on elliptic curves II). They explain beautifully the background for their conjecture.</p>
<p>The basic idea is that if you can prove that $L_E$ has an analytic continuation beyond $\Re(s)=3/2$ (Hasse-Weil conjecture), then</p>
<p>$$L_E(1)=\... |
239,653 | <p>It is usually told that Birch and Swinnerton-Dyer developped their famous conjecture after studying the growth of the function
$$
f_E(x) = \prod_{p \le x}\frac{|E(\mathbb{F}_p)|}{p}
$$
as $x$ tends to $+\infty$ for elliptic curves $E$ defined over $\mathbb{Q}$, where the product is defined for the primes $p$ where $... | Community | -1 | <p>If $E$ is an elliptic curve over $\mathbf{Q}$, define $L_p(E,s) = \frac{1}{1-ap^{-s} + p^{1-2s}}$, where $a$ is $p-(|E_p(\mathbf{F}_p)| - 1)$. The L-function of E is defined as the product over the $L_p(E,s)$ over all primes not dividing $2\Delta$ (called “good primes”):
$$ L(E,s) = \prod_{p} L_p(E,s)$$
Since we wis... |
4,431,185 | <p>Let <span class="math-container">$ABC$</span> be a triangle and denote its area by <span class="math-container">$k = \mathrm{area}(ABC)$</span>. I want to divide <span class="math-container">$ABC$</span> into two sub-triangles <span class="math-container">$ABE$</span> and <span class="math-container">$AEC$</span> su... | Vasili | 469,083 | <p><span class="math-container">$A_{\triangle ABC}=\frac{1}{2}BC\cdot h=k \implies BC=\frac{2k}{h}$</span> (<span class="math-container">$h$</span> is a height from point <span class="math-container">$A$</span>)<p>
<span class="math-container">$A_{\triangle ABE}=\frac{1}{2}BE\cdot h=t \implies BE=\frac{2t}{h}$</span><p... |
3,040,110 | <p>What is the Range of <span class="math-container">$5|\sin x|+12|\cos x|$</span> ?</p>
<p>I entered the value in desmos.com and getting the range as <span class="math-container">$[5,13]$</span>.</p>
<p>Using <span class="math-container">$\sqrt{5^2+12^2} =13$</span>, i am able to get maximum value but not able to fi... | Community | -1 | <p>If <span class="math-container">$f(x) = 5|\sin x| + 12 |\cos x|$</span>, then</p>
<p><span class="math-container">\begin{align*}
f(x) &= \sqrt{f(x)^2} \\
&= \sqrt{25 \sin^2 x + 144 \cos^2 x + 60 |\sin x \cos x|} \\
&= \sqrt{25 + (144 - 25) \cos^2 x + 60 |\sin x \cos x|} \\
&\ge 5
\end{align*}</span>... |
254,126 | <p>If 0 < a < b, where a, b $\in\mathbb{R}$, determine $\lim \bigg(\dfrac{a^{n+1} + b^{n+1}}{a^{n} + b^{n}}\bigg)$</p>
<p>The answer (from the back of the text) is $\lim \bigg(\dfrac{a^{n+1} + b^{n+1}}{a^{n} + b^{n}}\bigg) = b$ but I have no idea how to get there. The course is Real Analysis 1, so its a course o... | Community | -1 | <p>Here's a proof using induction that $H^2(M_d) = 0$ for all $d \geq 1$. Now for $d = 1$ this is clear. Then by Mayer Vietoris we have</p>
<p>$$\ldots \leftarrow H^3\left(M_{d-1} \cup( \Bbb{R}^2 -\{p_d\};\Bbb{R}\right)\leftarrow H^2\left(M_{d-1}\cap ( \Bbb{R}^2 -\{p_d\});\Bbb{R}\right) \leftarrow \\ H^2(M_{d-1};\Bbb{... |
3,846,339 | <p>Suppose I have the inequality <span class="math-container">$(\frac{A}{B})^X < (\frac{C}{D})\cdot(\frac{E}{F})^Y$</span> and I want X by itself.</p>
<p>Can I do this <span class="math-container">$X\cdot \log(\frac{A}{B}) < \log(\frac{C}{D})\cdot(Y\cdot \log(\frac{E}{F}))$</span>?
Am I breaking any rules on the ... | meet2410shah | 698,925 | <p>Yes!!! You need to apply the rule of addition for logarithmic functions.</p>
<p>Right hand side should be log(C / D) + (Y * log(E / F)).</p>
|
3,846,339 | <p>Suppose I have the inequality <span class="math-container">$(\frac{A}{B})^X < (\frac{C}{D})\cdot(\frac{E}{F})^Y$</span> and I want X by itself.</p>
<p>Can I do this <span class="math-container">$X\cdot \log(\frac{A}{B}) < \log(\frac{C}{D})\cdot(Y\cdot \log(\frac{E}{F}))$</span>?
Am I breaking any rules on the ... | rash | 650,763 | <p>It should be
<span class="math-container">$$X\log \left(\frac{A}{B}\right)=\log \left(\frac{C}{D}\right)+Y\log \left(\frac{E}{F}\right)$$</span>
You did not break the log into addition as <span class="math-container">$\log xy =\log x +\log y$</span></p>
|
231,583 | <p>Although there already exists active research area, so-called, automated theorem proving, mostly work on logic and elementary geometry. </p>
<p>Rather than only logic and elementary geometry, are there existing research results which by using machine learning techniques(possibly generative learning models) to disco... | joro | 12,481 | <p>Check the answers in <a href="https://mathoverflow.net/questions/92148/interesting-conjectures-discovered-by-computers-and-proved-by-humans">Interesting conjectures “discovered” by computers and proved by humans?</a>.</p>
<p>An example in graph theory is the software <a href="http://www.math.illinois.edu/~dwest/reg... |
1,980,510 | <p>$$ \lim_{(x,y) \to (1,0)} \frac{(x-1)^2\ln(x)}{(x-1)^2 + y^2}$$</p>
<p>I tried L' Hospitals Rule and got now where. Then I tried using $x = r\cos(\theta)$ and $y = r\sin(\theta)$, but no help. How would I approach this? T</p>
| Ahmed S. Attaalla | 229,023 | <p>$$|\frac{(x-1)^2\ln(x)}{(x-1)^2 + y^2}| \leq |\frac{(x-1)^2 \ln (x)}{(x-1)^2}| \stackrel{x \neq 1}{=} |\ln x|$$</p>
<p>But $\ln x \to 0$ as $x \to 1$.</p>
<p>Hence, by squeeze theorem:</p>
<p>$$ \lim_{(x,y) \to (1,0)} \frac{(x-1)^2\ln(x)}{(x-1)^2 + y^2}=0$$</p>
|
3,297,190 | <p>Let <span class="math-container">$\{z_i\}_{i=1}^n$</span> and <span class="math-container">$\{w_i\}_{i=1}^n$</span> be two collections of vectors in <span class="math-container">$\mathbb R^p$</span>. Let <span class="math-container">$A$</span> be a real positive definite <span class="math-container">$p\times p$</spa... | greg | 357,854 | <p>Rather than solve for the Cholesky factor directly, find a solution in terms of a less <em>structured</em> matrix, <span class="math-container">$M$</span>. Let a colon denote the matrix inner product, i.e.
<span class="math-container">$$\eqalign{
A:B &= {\rm Tr}(AB^T) \cr
M:M &= {\rm Tr}(MM^T) &= \frac{1... |
3,297,190 | <p>Let <span class="math-container">$\{z_i\}_{i=1}^n$</span> and <span class="math-container">$\{w_i\}_{i=1}^n$</span> be two collections of vectors in <span class="math-container">$\mathbb R^p$</span>. Let <span class="math-container">$A$</span> be a real positive definite <span class="math-container">$p\times p$</spa... | dineshdileep | 41,541 | <p>Define the <span class="math-container">$p\times n$</span> matrices <span class="math-container">$Z=[z_1,\dots,z_n]$</span> and <span class="math-container">$W=[w_1,\dots,w_n]$</span> (such that given vectors are respectively their columns). Convince yourself that you can rewrite your optimization problem as
<span c... |
1,393,154 | <p><span class="math-container">$4n$</span> to the power of <span class="math-container">$3$</span> over <span class="math-container">$2 = 8$</span> to the power of negative <span class="math-container">$1$</span> over <span class="math-container">$3$</span></p>
<p>Written Differently for Clarity:</p>
<p><span class="m... | layman | 131,740 | <p>So the real problem we are trying to complete is to solve $4n^{\frac{3}{2}} = 8^{-\frac{1}{3}}$.</p>
<p>The way to do this is to first divide both sides by $4$ and get:</p>
<p>$n^{\frac{3}{2}} = \dfrac{8^{-\frac{1}{3}}}{4}$</p>
<p>Now, since $8 = 2^{3}$ and $4 = 2^{2}$, we can rewrite this as:</p>
<p>$n^{\frac{3... |
259,795 | <p>Consider the function $f \colon\mathbb R \to\mathbb R$ defined by
$f(x)=
\begin{cases}
x^2\sin(1/x); & \text{if }x\ne 0, \\
0 & \text{if }x=0.
\end{cases}$</p>
<p>Use $\varepsilon$-$\delta$ definition to prove that the limit $f'(0)=0$.</p>
<p>Now I see that h should equals to delta; and delta should eq... | Qiaochu Yuan | 232 | <p>Any finite-dimensional (edit: Hausdorff) topological real vector space has the usual Euclidean topology (exercise), and any surjective linear transformation between finite-dimensional topological real vector spaces is isomorphic to projection to some subset of coordinates on $\mathbb{R}^n$ (exercise), so this follow... |
2,843,560 | <p>If $\sin x +\sin 2x + \sin 3x = \sin y\:$ and $\:\cos x + \cos 2x + \cos 3x =\cos y$, then $x$ is equal to</p>
<p>(a) $y$</p>
<p>(b) $y/2$</p>
<p>(c) $2y$</p>
<p>(d) $y/6$</p>
<p>I expanded the first equation to reach $2\sin x(2+\cos x-2\sin x)= \sin y$, but I doubt it leads me anywhere. A little hint would be... | lesnik | 121,451 | <p>Draw 3 unit vectors $e_1$, $e_2$ and $e_3$. The angle between these each of these vectors and $x$ axis is $x$, $2x$ and $3x$ correspondingly.</p>
<p>$x$ coordinates of this vectors would be $cos(x)$, $cos(2x)$ and $cos(3x)$.</p>
<p>To find the sum of these coordinates you can first add up the vectors and than the ... |
2,949,011 | <blockquote>
<p>If <span class="math-container">$b_n\to\infty$</span> and <span class="math-container">$\{a_n\}$</span> is such that <span class="math-container">$b_n>a_n$</span> for all <span class="math-container">$n$</span>, then <span class="math-container">$a_n\to\infty$</span>.</p>
</blockquote>
<p>We are t... | Jannik Pitt | 355,418 | <p>I assume you meant <span class="math-container">$a_n \geq b_n$</span>. A sequence <span class="math-container">$(x_n)_{n \in \mathbb{N}}$</span> diverges to <span class="math-container">$\infty$</span> if for every <span class="math-container">$K$</span> there exists an <span class="math-container">$N \in \mathbb{N}... |
529,708 | <p>A is prime greater than 5, B is A*(A-1)+1,if B is prime,</p>
<p>then digital root of A and B must the same.(OEIS A065508)</p>
<p>Sample: 13*(13-1)+1 = 157
13 and 157 are prime and have same digital root 4</p>
| Calvin Lin | 54,563 | <p><strong>hint</strong>: consider mod 9. What happens in each of these cases?</p>
<p>Use the fact that B us prime, and in particular not a multiple of 3</p>
|
3,609,906 | <p>I need to compute <span class="math-container">$$\lim_{n \to \infty}\sqrt{n}\int_{0}^{1}(1-x^2)^n dx.$$</span>
I proved that
for <span class="math-container">$n\ge1$</span>,
<span class="math-container">$$\int_{0}^{1}(1-x^2)^ndx={(2n)!!\over (2n+1)!!},$$</span>
but I don't know how to continue from here.</p>
<p>I a... | Gary | 83,800 | <p>Answer to your second question. Since
<span class="math-container">$$
\frac{{\frac{{(2n)!!}}{{(2n + 1)!!}}}}{{\frac{{(2n + 2)!!}}{{(2n + 3)!!}}}} = \frac{{(2n)!!(2n + 3)!!}}{{(2n + 1)!!(2n + 2)!!}} = \frac{{n + \frac{3}{2}}}{{n + 1}}
$$</span>
and
<span class="math-container">$$
\frac{{n + 2}}{{n + 1}} < \left( {... |
3,609,906 | <p>I need to compute <span class="math-container">$$\lim_{n \to \infty}\sqrt{n}\int_{0}^{1}(1-x^2)^n dx.$$</span>
I proved that
for <span class="math-container">$n\ge1$</span>,
<span class="math-container">$$\int_{0}^{1}(1-x^2)^ndx={(2n)!!\over (2n+1)!!},$$</span>
but I don't know how to continue from here.</p>
<p>I a... | CHAMSI | 758,100 | <p>You won't need a close form for the integral. Here is an easy way to do it :</p>
<p>Denoting <span class="math-container">$ \left(\forall n\in\mathbb{N}\right),\ W_{n}=\displaystyle\int_{0}^{\frac{\pi}{2}}{\sin^{n}{x}\,\mathrm{d}x} : $</span></p>
<p>We have : <span class="math-container">\begin{aligned} \left(\for... |
1,897,538 | <p>Next week I will start teaching Calculus for the first time. I am preparing my notes, and, as pure mathematician, I cannot come up with a good real world example of the following.</p>
<p>Are there good examples of
\begin{equation}
\lim_{x \to c} f(x) \neq f(c),
\end{equation}
or of cases when $c$ is not in the doma... | John Wayland Bales | 246,513 | <p>A $10$ cm length of spring steel wire with a breaking tension of 1 kg is formed into a spring of length $1$ cm.</p>
<p>With the spring suspended vertically, a $100$ gm weight attached to the end of the spring stretches it $1$ cm beyond its natural length. Assume Hooke's law holds until the spring is completely stra... |
3,283,606 | <p>Good Evening,</p>
<p>I know this is a basic question, but I haven't been able to find a clear explanation for how to simplify the follow equation:
<span class="math-container">$$n\log_2n=10^6$$</span>
Solving this equation is part of the solution for Problem 1-1 from the Intro. to Algorithms book by CLRS:
<a href="... | Parcly Taxel | 357,390 | <p><span class="math-container">$$n\log_2n=10^6$$</span>
<span class="math-container">$$n\ln n=10^6\ln 2$$</span>
<span class="math-container">$$\ln n=\frac1n10^6\ln 2$$</span>
<span class="math-container">$$n=e^{(10^6\ln 2)/n}$$</span>
<span class="math-container">$$10^6\ln2=\frac{10^6\ln2}ne^{(10^6\ln2)/n}$$</span>
N... |
937,912 | <p>I'm looking for a closed form of this integral.</p>
<p>$$I = \int_0^1 \frac{\operatorname{Li}_2\left( x \right)}{\sqrt{1-x^2}} \,dx ,$$</p>
<p>where $\operatorname{Li}_2$ is the <a href="http://mathworld.wolfram.com/Dilogarithm.html" rel="noreferrer">dilogarithm function</a>.</p>
<p>A numerical approximation of i... | Ali Shadhar | 432,085 | <p>From writing <span class="math-container">$$\operatorname{Li}_2(x)=-\int_0^1\frac{x\ln u}{1-xu}du$$</span></p>
<p>It follows that </p>
<p><span class="math-container">$$-I=-\int_0^1\frac{\operatorname{Li}_2(x)}{\sqrt{1-x^2}}dx=\int_0^1\ln u\left[\int_0^1\frac{x}{(1-ux)\sqrt{1-x^2}}dx\right]du$$</span></p>
<p><spa... |
907,851 | <p>I am really new into math, why is $-2^2 = -4 $ and $(-2)^2 = 4 $? </p>
| Labba | 171,314 | <p>Because $-2^2$ means that you take the number $2$, raise it to the second power ($^2$), and then you consider its additive inverse ($-$). So, $2$ raised to the second power is $4$, whose additive inverse is $-4$. This is because exponentiation has a higher priority and it is the first thing you have to do; hence you... |
251,028 | <p>I want to make <span class="math-container">$S[\{,\cdots, \}]$</span> as follows</p>
<p>First input of <span class="math-container">$S$</span> is given list <span class="math-container">$\{1,2,3,\cdots, n\}$</span> and it produces <span class="math-container">$s_{123\cdots n}$</span></p>
<p>Further, if the ordering ... | LouisB | 22,158 | <p>This can be done with a pure function, like this:</p>
<pre><code>ClearAll[S]
S = Subscript[s, Row[{##} // Flatten // Sort, " "]] &;
S[{1, 2, 3, 4}]
S[1, 3, 2]
</code></pre>
<p><a href="https://i.stack.imgur.com/T0hhe.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/T0hhe.png" alt="e... |
251,028 | <p>I want to make <span class="math-container">$S[\{,\cdots, \}]$</span> as follows</p>
<p>First input of <span class="math-container">$S$</span> is given list <span class="math-container">$\{1,2,3,\cdots, n\}$</span> and it produces <span class="math-container">$s_{123\cdots n}$</span></p>
<p>Further, if the ordering ... | Lukas Lang | 36,508 | <p>Using <a href="https://reference.wolfram.com/language/ref/Format.html" rel="noreferrer"><code>Format</code></a>, <a href="https://reference.wolfram.com/language/ref/Interpretation.html" rel="noreferrer"><code>Interpretation</code></a>, and <a href="https://reference.wolfram.com/language/ref/Orderless.html" rel="nore... |
10,666 | <p>My question is about <a href="http://en.wikipedia.org/wiki/Non-standard_analysis">nonstandard analysis</a>, and the diverse possibilities for the choice of the nonstandard model R*. Although one hears talk of <em>the</em> nonstandard reals R*, there are of course many non-isomorphic possibilities for R*. My question... | Andreas Blass | 6,794 | <p>Under a not unreasonable assumption about cardinal arithmetic, namely $2^{<c}=c$ (which follows from the continuum hypothesis, or Martin's Axiom, or the cardinal characteristic equation t=c), the number of non-isomorphic possibilities for *R of cardinality c is exactly 2^c. To see this, the first step is to dedu... |
4,604,730 | <p>Given a function <span class="math-container">$f$</span> from vectors to scalars, and a vector <span class="math-container">$\vec v$</span>, the directional derivative of <span class="math-container">$f$</span> with respect to <span class="math-container">$\vec v$</span> is defined as <span class="math-container">$\... | P. Lawrence | 545,558 | <p>Take two distinct points <span class="math-container">$P_0$</span> and <span class="math-container">$P_1$</span> in <span class="math-container">$\mathbb{R^n}$</span>.We'll assume that <span class="math-container">$f:B \to \mathbb R$</span> where <span class="math-container">$B$</span> is an open ball in <span class... |
1,290,111 | <p>How one can prove the following statement:</p>
<p>$k(n-1)<n^2-2n$ for all odd $n$ and $k<n$</p>
<p><em>Tried so far</em>: induction on $n$, graphing, and rewriting $n^2−2n$ as $(n−1)^2−1$.</p>
| Alufat | 198,706 | <p>Look, you should do something like that</p>
<p>\begin{align} &k(n-1) < n^2-2n \\
\iff &k(n-1) < (n-1)^2-1 \quad\text{(subtract $k(n-1)$ from both sides)}\\
\iff &0 < (n-1-k)(n-1)-1 \\
\iff &1 < (n-1-k)(n-1) \quad\text{(divide both sides by $(n-1)$)} \\
\iff &(n-1)^{-1} < (n-1-k)
... |
1,988,517 | <p>$$\binom{1}{0},\binom{1}{1},\binom{1}{2}$$
What does this mean, and how do I achieve an numerical value when trying to solve a proof or problem in this form? </p>
| Davis Yoshida | 113,908 | <p>The definition of ${n}\choose{k}$ is $\frac{n}{k!(n-k)!}$. To use them numerically you can simply compute those values. Notice that there's som cancellation between the numerator and denominator so you could instead compute:</p>
<p>$\frac{n(n-1)(n-2)...(n-k + 1)}{k!}$ or $\frac{n(n-1)(n-2)...(k + 1)}{(n-k)!}$</p>
|
1,947,082 | <p>Let us have sum of sequence (I'm not sure how this properly called in English): $$X(n) = \frac{1}{2} + \frac{3}{4}+\frac{5}{8}+...+\frac{2n-1}{2^n}$$</p>
<p>We need</p>
<p>$$\lim_{n \to\infty }X(n)$$</p>
<p>I have a solution, but was unable to find right answer or solution on the internet.</p>
<p>My idea:</p>
<... | DonAntonio | 31,254 | <p>$$\frac12+\frac34+\ldots+\frac{2n-1}{2^n}=\left(1-\frac12\right)+\left(1-\frac14\right)+\left(\frac34-\frac18\right)+\ldots\left(\frac n{2^{n-1}}-\frac1{2^n}\right)=$$</p>
<p>$$1+1+\frac34+\ldots+\frac n{2^{n-1}}-\frac12-\frac14-\ldots-\frac1{2^n}\xrightarrow[n\to\infty]{}$$</p>
<p>$$\to\sum_{k=1}^\infty\frac k{2^... |
1,816,807 | <blockquote>
<p>If $x,y,z\in \mathbb{R}$ and $x+y+z=4$ and $x^2+y^2+z^2=6\;,$ Then range of $xyz$</p>
</blockquote>
<p>$\bf{My\; Try::}$Using $$(x+y+z)^2=x^2+y^2+z^2+2(xy+yz+zx)$$</p>
<p>So we get $$16=6+2(xy+yz+zx)\Rightarrow xy+yz+zx = -5$$ and given $x+y+z=4$</p>
<p>Now let $xyz=c\;,$ Now leyt $t=x,y,z$ be the ... | colormegone | 71,645 | <p>I thought it might be worth showing the Lagrange-multiplier method applied to this problem, largely for illustrating how the system of "Lagrange equations" can be handled, and for showing the interesting character of the solution.</p>
<p>With the constraints $ \ x^2 \ + \ y^2 \ + \ z^2 \ = \ 6 \ $ and $ \ x \ + \ y... |
820,015 | <p>Suppose that you have $k$ dice, each with $N$ sides, where $k\geq N$. The definition of a straight is when all $k$ dice are rolled, there is at least one die revealing each number from $1$ to $N$. </p>
<p>Given the pair $(k,N)$, what is the probability that any particular roll will give a straight?</p>
| sayantankhan | 47,812 | <p>I'm assuming that you're familiar with the inclusion-exclusion principle. In this particular problem, you need to determine the probability of a straight, or in other words, the complement of the event that atleast $1$ number does not appear in the throws. Let that probability be $P$. Also, let $p(j)$ be the probabi... |
820,015 | <p>Suppose that you have $k$ dice, each with $N$ sides, where $k\geq N$. The definition of a straight is when all $k$ dice are rolled, there is at least one die revealing each number from $1$ to $N$. </p>
<p>Given the pair $(k,N)$, what is the probability that any particular roll will give a straight?</p>
| Tony | 155,506 | <p>What about simply $\dfrac{^k\text{P}_n(k-n)!}{n^k}$, since $k \ge n$. If $k=n$ then we simply get $\dfrac{n!}{n^k}$?</p>
<p>Since there are only $n$ sides to the dice and if $k=n$, we are asking the total number of permutations of $n$ different values which is $^n\text{P}_n=n!$ divided by all the possible outcomes ... |
2,130,699 | <p>I have the coordinates of potentially any points within a 3D arc, representing the path an object will take when launched through the air. X is forward, Z is up, if that is relevant.
Using any of these points, I need to create a bezier curve that follows this arc. The curve requires the tangent values for the very s... | Mariano Suárez-Álvarez | 274 | <p>The ring $\mathbb R[X]/(X^2)$ is a real algebra with zero divisor and just trivial idempotents.</p>
|
72,854 | <p>Hi everybody,</p>
<p>Does there exist an explicit formula for the Stirling Numbers of the First Kind which are given by the formula
$$
x(x-1)\cdots (x-n+1) = \sum_{k=0}^n s(n,k)x^k.
$$</p>
<p>Otherwise, what is the computationally fastest formula one knows?</p>
| Igor Rivin | 11,142 | <p>Since the Stirling numbers are the coefficients of a polynomial of degree $n$ which is already factored, it can be evaluated at the roots of unity in $O(n\log n)$ multiplications. Then, by Fourier transform, the coefficients can be found in another $O(n\log n)$ multiplications, of roughly $O( n)$ bit numbers. This w... |
1,733,970 | <p>I am watching continuous probabilities lectures (having almost no calc background) and I cannot understand solution for the following exercise:</p>
<p>Let $X$ be continuous random variable with a PDF of the form:
$f_X(x) = c(1-x),$ if $x \in [0,1]$ and $f_X(x) = 0 $ otherwise.</p>
<p>Find the following values:</p>... | browngreen | 321,445 | <p>Before getting into the calculus details, the idea behind this problem is that the integral of a PDF across its domain has to equal one. This is because the sum of the probabilities of all possible outcomes for a random variable has to be one. That being said, we set the integral equal to one in order to solve for ... |
1,733,970 | <p>I am watching continuous probabilities lectures (having almost no calc background) and I cannot understand solution for the following exercise:</p>
<p>Let $X$ be continuous random variable with a PDF of the form:
$f_X(x) = c(1-x),$ if $x \in [0,1]$ and $f_X(x) = 0 $ otherwise.</p>
<p>Find the following values:</p>... | John | 7,163 | <p>If you don't have a calculus background, then this will (rightfully) seem like black magic.</p>
<p>The statement</p>
<p>$$\int_{-\infty}^{\infty} f_X(x)dx = 1$$</p>
<p>in essence tells you "the sum of all probabilities is $1$", but in a continuous way.</p>
<p>From the problem, you're told that the valid values o... |
2,492,107 | <p>The question asks $\text{span}(A1,A2)$</p>
<p>$$A1 =\begin{bmatrix}1&2\\0&1\end{bmatrix}$$
$$A2 = \begin{bmatrix}0&1\\2&1\end{bmatrix}$$</p>
<p>I began by calculating $c_1[A1] + c_2[A2]$ then converting it into a matrix and row reducing. I found the restrictions where the stuff after the augment m... | Guy Fsone | 385,707 | <p>Set $$\color{blue}{a_k= \frac{1}{k^2 -k +1} \implies a_{k+1}= \frac{1}{(k+1)^2 -k } =\frac{1}{k^2+ k +1}} $$</p>
<p>But since $ (k^2 -k +1)(k^2 +k +1) =k^4 +k^2 +1$ we have,</p>
<p>$$ a_{k+1} -a_k = \frac{1}{k^2 +k +1}- \frac{1}{k^2 -k +1} = \frac{2k}{(k^2 -k +1)(k^2 +k +1)} =\frac{2k}{k^4 +k^2 +1}$$</p>
<p>Th... |
465,001 | <p>$\mathbf{h}_i\in\mathbb{C}^{M}$ are column vectors $\forall i=\{1, 2, \cdots, K\}$.</p>
<p>$q_i\in\mathbb{R}_+$ are scalars $\forall i=\{1, 2, \cdots, K\}$</p>
<p>$\lvert\bullet\rvert$ denotes determinant of a square matrix or Euclidean norm of a vector according to the context. </p>
<p>From Sylvester's theorem, ... | Community | -1 | <p>Recall that
$$\lim_{x\to a}f(x)=\ell\in\overline{\mathbb R}\iff \forall(x_n)\to a,\; f(x_n)\to\ell$$
so take
$$x_n=\frac{1}{n\pi}$$
to show that the limit doesn't exist.</p>
|
1,250,755 | <p>I'm sure there's something I'm missing here; probably a naive confusion of mathematics with metamathematics. Regardless, I've come up with what looks to me like a proof that (first-order) ZF+AD is inconsistent:</p>
<p>Assume otherwise. Then it cannot be shown that ZF+AD is consistent relative to ZF. However, ZFC is... | Asaf Karagila | 622 | <p>All of this is problematic. You claim that the theory is inconsistent. Okay. So by contradiction it is not inconsistent, therefore the working assumption is that the theory is consistent. </p>
<p>But a theory being consistent does not mean consistent relative to $\sf ZF$. it means consistent. Moreover the fact ther... |
1,250,755 | <p>I'm sure there's something I'm missing here; probably a naive confusion of mathematics with metamathematics. Regardless, I've come up with what looks to me like a proof that (first-order) ZF+AD is inconsistent:</p>
<p>Assume otherwise. Then it cannot be shown that ZF+AD is consistent relative to ZF. However, ZFC is... | Burak | 137,499 | <p>In the second line of your second paragraph, you assume the statement $Con(ZF+AD)$ and then get a set model of $ZF+AD$ by Gödel's completeness theorem (in $ZFC$ or $ZF$, as our meta theory). Then, you claim that $ZF+AD$ is consistent relative to $ZFC$. Why would the statement $Con(ZFC) \rightarrow Con(ZF+AD)$ follow... |
298,029 | <p>how can we convert sin function into continued fraction ?</p>
<p>for example </p>
<p><a href="http://mathworld.wolfram.com/EulersContinuedFraction.html">http://mathworld.wolfram.com/EulersContinuedFraction.html</a></p>
<p>how can we convert sin to simmilar continued fraction ?? </p>
<p>and what about sinh and co... | Mahkoe | 124,475 | <p>I learned how to do this from this document (look for Theorem I)
<a href="https://people.math.osu.edu/sinnott.1/ReadingClassics/continuedfractions.pdf" rel="nofollow">https://people.math.osu.edu/sinnott.1/ReadingClassics/continuedfractions.pdf</a></p>
<p>Interestingly, this looks to be a translation of Euler's orig... |
1,878,806 | <p>I am a graduate school freshman.</p>
<p>I did not take a probability lecture.</p>
<p>So I don't have anything about Probability.</p>
<p>Could you suggest Probability book No matter What book level?</p>
| Community | -1 | <p>Art of Problem Solving's Intermediate Counting and Probability is a book for students with solid basics. If you would like something easier, Introduction to Counting and Probability(also by AoPS) is perfect for beginners, no matter what age. AoPS textbooks are written by the nation's best mathmeticians and contain... |
848,415 | <p>If the limit of one sequence $\{a_n\}$ is zero and the limit of another sequence $\{b_n\}$ is also zero does that mean that $\displaystyle\lim_{n\to\infty}(a_n/b_n) = 1$?</p>
| Wonder | 27,958 | <p>Given $b_n$ whose limit is 0, we can choose $a_n$ to make the limit:</p>
<ul>
<li><p>1: set $a_n = b_n$</p></li>
<li><p>0: set $a_n = b_n^2$, or even $a_n = 0$</p></li>
<li><p>Some other constant c: set $a_n = cb_n$</p></li>
<li><p>Undefined: $a_n = (-1)^nb_n$</p></li>
<li><p>Infinite: $a_n = \sqrt{|b_n|}$</p></li>... |
3,537,226 | <p>A deck contains six cards, one pair labelled '1', another pair labelled '2' and the last labelled '3'. The deck is shuffled and you a pair of cards at a time until there are no cards left. A pair of cards <span class="math-container">$(i,j)$</span> is called acceptable if <span class="math-container">$|i-j|\leq1$</s... | R. Burton | 614,269 | <p>Not a complete answer, but it might help.</p>
<hr>
<p>Let <span class="math-container">$[n]=\{1,2,\ldots,n\}$</span> . The deck of cards is represented by the multiset <span class="math-container">$D_n=[n]\cup[n]$</span>. For each <span class="math-container">$m$</span>, let <span class="math-container">$S_m^n$</s... |
209,892 | <p>From the values:</p>
<pre><code>{57.02, 71.04, 87.03, 97.05, 99.07, 101.05, 103.01, 113.08, 114.04, 115.03,
128.06, 128.09, 129.04, 131.04, 137.06, 147.07, 156.10, 163.03, 186.08}
</code></pre>
<p>I would like to find all possible combinations of 3 values that have the sum of roughly 344.25 (+/- 0.05 would be ok)... | kglr | 125 | <pre><code>list = {57.0, 71.0, 87.0, 97.1, 99.1, 101.1, 103.0, 113.1, 114.0,
115.0, 128.1, 128.1, 129.0, 131.0, 137.1, 147.1, 156.1, 163.0, 186.1};
subsets = DeleteDuplicates @ Select[Subsets[list, {3}], 344.2 <= Total[#] <= 344.3 &]
</code></pre>
<blockquote>
<p>{{57., 101.1, 186.1},<br>
{87., 101... |
2,052,826 | <p>I've been working on some quantum mechanics problems and arrived to this one where I have to deal with subscripts. I got stuck doing this:
I have $\epsilon_{imk}\epsilon_{ikn}=\delta_{mk}\delta_{kn}-\delta_{mn}\delta_{kk}$. But then I went to check and $\delta_{mk}\delta_{kn}-\delta_{mn}\delta_{kk}$ is equal to $\de... | Mark Viola | 218,419 | <p>Another approach is to write $$\epsilon_{ijk}=\hat x_i\cdot(\hat x_j\times \hat x_k)$$Then, we have</p>
<p>$$\begin{align}
\epsilon_{imk}\epsilon_{ikn}&=\hat x_i\cdot(\hat x_m\times \hat x_k)x_i\cdot(\hat x_k\times \hat x_n)\\\\
&=(\hat x_m\times \hat x_k)\cdot(\hat x_k\times \hat x_n)\\\\
&=\hat x_m \c... |
192,636 | <p>Suppose I have some 3D points, e.g. <code>{{0, 0, 1}, {0, 0, 1.3}, {0, 1, 0}, {1.2, 0, 0}}</code>. Now I want to find the smallest and largest distance between two points.</p>
<p>A trivial way is to find all possible distances, then look for the smallest and largest number.This becomes very much time-consuming for ... | Gladaed | 57,652 | <p>You can find out all the distances by calculating the <code>DistanceMatrix</code>. Using <code>Min</code> and <code>Max</code> you can find the smallest and largest values. </p>
<pre><code>dm = {{0, 0, 1}, {0, 0, 1.3}, {0, 1, 0}, {1.2, 0, 0}} // DistanceMatrix
closest = Min@dm (* is 0 since the point is infinitly c... |
2,787,227 | <p>I'm trying to compute two integrals involving the Dirac delta, namely
\begin{align}
I_1&=\int^{1}_0\!\!\!\! dx_1\!\cdots\!\int^{1}_0\!\!\!\! dx_{8}\,\delta(x_1-x_2+x_3-x_4+x_5-x_6+x_7-x_8)\,,\\
I_2&=\int^{1}_0\!\!\!\! dx_1\!\cdots\!\int^{1}_0\!\!\!\! dx_{8}\,\delta(x_1-x_2+x_4-x_5)\delta(x_3-x_4+x_6-x_7)\,\d... | hypernova | 549,945 | <p>Let us figure out an expression of $\left(I_n+\alpha\mathbf{x}\mathbf{x}^{\top}\right)^{1/2}$ at first. Denote
$$
\mathbf{v}_1=\frac{\mathbf{x}}{\left\|\mathbf{x}\right\|},
$$
and let
$$
\left\{\mathbf{v}_1,\mathbf{v}_2,\cdots,\mathbf{v}_n\right\}
$$
be an orthonormal basis for $\mathbb{R}^n$. We will show that each... |
2,445,855 | <p>1)How is this letter or symbol pronounced mathematically?</p>
<p>$$\overline k$$</p>
<p>2) $'$ is this sign just a symbol of derivative? For example: </p>
<p>$$k'$$ Do we only understand this as a derivative?</p>
| Community | -1 | <p>It would depend on what mathematical area you were studying when you saw it probably. Just like x isn't always referring to the $x$-axis or $y$ to the $y$- axis on the real plane it could be any Cartesian product etc. $\circ$ could be used as function composition or other compositions as well as in place of the degr... |
114,487 | <p>I have a stack of images (usually ca 100) of the same sample. The images have intrinsic variation of the sample, which is my signal, and a lot of statistical noise. I did a principal components analysis (PCA) on the whole stack and found that components 2-5 are just random noise, whereas the rest is fine. How can I ... | Anton Antonov | 34,008 | <p>This answer compares two dimension reduction techniques SVD and Non-Negative Matrix Factorization (NNMF) over a set of images with two different classes of signals (two digits below) produced by different generators and overlaid with different types of noise.</p>
<p>Note that question states that the images have on... |
3,741,222 | <p>I'm reading the Thompson's book about lattices and sphere packing and got stuck by a sentence of a kind of <span class="math-container">$Z_8$</span> he introduced to reach 2 pages later the full <span class="math-container">$E_8$</span> lattice.
You can find this lattice defined at pages 73-74 and it's basically. To... | Heterotic | 117,522 | <p>Putting in 2 in each slot creates a legitimate basis, but for a different lattice! The lattice as introduced on page 73 is not the collection of vectors with all coordinates even. That would indeed be generated by 2xId<span class="math-container">$_8$</span>. However, the lattice the author wants to study is the col... |
477,483 | <p>I'm trying to solve this question but yet I don't manage to find an answer...</p>
<p>Does it exist a topology with cardinality $\alpha, \> \forall \> \alpha \ge 1$ ?</p>
| Ross Millikan | 1,827 | <p>The basic idea is that quadrilaterals average $90^\circ$ angles. If four of them meet at every corner, you wont have the required $720^\circ$ deficit to make a closed sphere. You need eight three-way vertices to get it to close. This is like the requirement that using hexagons and pentagons you need 12 pentagons ... |
3,987,357 | <p>Let
<span class="math-container">$$
\psi(x)=
\left\{
\begin{array}{cll}
x \sin\Big(\dfrac{1}{x}\Big) & \text{if} & x\in (0,1],\, \\
0 & \text{if} & x=0,
\end{array}
\right.
$$</span>
and let <span class="math-container">$f:[-1,1]\rightarrow \mathbb{R}$</span> be Riemann integrable.</p>
<p>How can I s... | Yiorgos S. Smyrlis | 57,021 | <p><strong>Note.</strong> If <span class="math-container">$f$</span> is Riemann integrable and <span class="math-container">$\varphi$</span> continuous, then their composition <span class="math-container">$f\circ\varphi$</span> is NOT necessarily Riemann integrable. (See <a href="https://mathoverflow.net/questions/2004... |
3,336,742 | <p>Can we evaluate <span class="math-container">$\displaystyle\sum_{n=1}^\infty\frac{H_n^2}{n^32^n}$</span> ?</p>
<p>where <span class="math-container">$H_n=\sum_{k=1}^n\frac1n$</span> is the harmonic number.</p>
<p>A related integral is <span class="math-container">$\displaystyle\int_0^1\frac{\ln^2(1-x)\operatorna... | Ali Shadhar | 432,085 | <p>We proved <a href="https://math.stackexchange.com/questions/3366039/a-group-of-important-generating-functions-involving-harmonic-number">here</a></p>
<p><span class="math-container">$$\frac{\ln^2(1-x)}{1-x}=\sum_{n=1}^\infty x^n\left(H_n^2-H_n^{(2)}\right)\tag{1}$$</span></p>
<p>multiply both sides by <span class=... |
1,844,894 | <p>To explain my question, here is an example.</p>
<p>Below is an AP:</p>
<p>2, 6, 10, 14....n</p>
<p>Calculating the nth term in this sequence is easy because we have a formula. The common difference (d = 4) in AP is constant and that's why the formula is applicable, I think.</p>
<p>But what about this sequence:</... | Hypergeometricx | 168,053 | <p>It's a cascasded form of the summation along a diagonal of Pascal's triangle, otherwise known as the <a href="https://math.stackexchange.com/questions/1490794/proof-of-the-hockey-stick-identity-sum-t-0n-binom-tk-binomn1k1">hockey stick identity</a>, i.e.
$$\sum_{r=0}^m \binom ra=\binom {m+1}{a+1}$$
Assume $p$ level... |
26,259 | <p>I've been reading <em>generationgfunctionology</em> by Herbert S. Wilf (you can find a copy of the second edition on the author's page <a href="http://www.math.upenn.edu/~wilf/DownldGF.html" rel="nofollow">here</a>).</p>
<p>On page 33 he does something I find weird. He wants to shuffle the index forward and does so... | Jonas Meyer | 1,424 | <p>If a power series $g(X)$ has zero constant term, then it is true that it is not invertible, so you cannot divide by it in general. However, it may happen that you can divide <em>some</em> power series by $g(X)$, namely those that have $g(X)$ as a factor. If $f(X)$ is a power series and $g(X)$ is a nonzero power se... |
15,124 | <p>Let $X \geq 1$ be an integer r.v. with $E[X]=\mu$. Let $X_i$ be a sequence of iid rvs with the distribution of $X$. On the integer line, we start at $0$, and want to know the expected position after we first cross $K$, which is some fixed integer. Each next position is determined by adding $X_i$ to the previous posi... | Yuval Filmus | 1,277 | <p>Edit: This answer shows what happens when $X$ could be zero (contrary to the question).</p>
<p>Let $K = 1$, and $X = M$ ($M > 1$) with probability $p$ and $0$ otherwise. Thus $E[Y_\tau - K] = M-1$ whereas $E[X] = pM$. Choosing $M = n+1$ and $p = 1/n(n+1)$ we get $E[X] = 1/n$ and $E[Y_\tau - K] = n$, so a bound of $... |
1,511,733 | <p>B = matrix given below. I is identity matrix.</p>
<pre><code> [1 2 3 4]
[3 2 4 3]
[1 3 2 4]
[5 4 3 7]
</code></pre>
<p>So What will be the relation between the matrices A and C if AB = I and BC = I?</p>
<p>I think that A = C because both AB and BC have B in common and both of their product is an identity matri... | charles | 286,740 | <p>$B$ is invertible.
Since $AB=I$, $A=IB^{-1}$,
and since $BC=I$, $C=B^{-1}I$.
Therefore $A = C$.</p>
|
821,768 | <p><img src="https://i.stack.imgur.com/bS4PE.png" alt="enter image description here"></p>
<p>In the rectangle ABCD,
$$1. \, BE = EF = FC = AB$$
$$2. \, \angle AEB = \beta , \angle AFB = \alpha , \angle ACB = \theta. $$
Prove that $\alpha + \theta = \beta$.</p>
<p>I have so far obtained that - $$1. \cos\beta = \sin \... | ajotatxe | 132,456 | <p>It is clear that the three angles are acute. Moreover,</p>
<p>$$\tan\beta=2\tan\alpha=3\tan\theta$$</p>
<p>$$\begin{align}
\tan(\alpha+\theta)&=\frac{\tan\alpha+\tan\theta}{1-\tan\alpha\tan\theta}\\
&=\frac{\frac56\tan\beta}{1-\frac{\tan^2\beta}6}\\
&=\frac{5\tan\beta}{6-\tan^2\beta}
\end{align}$$</p>
... |
3,156,962 | <blockquote>
<p>Find general term of <span class="math-container">$1+\frac{2!}{3}+\frac{3!}{11}+\frac{4!}{43}+\frac{5!}{171}+....$</span></p>
</blockquote>
<p>However it has been ask to check convergence but how can i do that before knowing the general term. I can't see any pattern,comment quickly!</p>
| Daniel Schepler | 337,888 | <p>Note that <span class="math-container">$\alpha^3 - 3 \alpha \beta^2$</span> and <span class="math-container">$3\beta\alpha^2 - \beta^3$</span> are the real and imaginary parts respectively of <span class="math-container">$(\alpha + i \beta)^3$</span>; this suggests that it will be useful to work in the ring of Gauss... |
215,864 | <p>I run to the following problem which says if you have a smooth curve that is evolving over time (say finite length at the beginning) then </p>
<p>$$\frac{d}{dt}(curve \; length \; at \; time \; t)=-\int_{curve} k\cdot v \; ds,$$</p>
<p>where $k$ is curvature of the curve and $v$ is velocity of point on curve. $ds... | Joseph O'Rourke | 237 | <p>This is not as comprehensive an answer as John Mangual's.
Consider it, rather, supplemental information to provide intuition,
drawn from the textbook, <a href="http://cs.smith.edu/~orourke/DCG/" rel="nofollow noreferrer"><em>Discrete and Computational Geometry</em></a>:
<img src="https://i.stack.imgur.com/AlS2R.png"... |
877,687 | <p>So my textbook's explanation of the derivative of e is very sketchy. They used lots of approximations and plugging things into the calculator. Basically I want to know how you can work out as h approaches 0</p>
<p>$$
\lim_{h\to0}\frac {10^{x+h}-10^x }h
$$</p>
| Community | -1 | <p>The curve you see is by <a href="http://en.m.wikipedia.org/wiki/B%C3%A9zier_curve#Quadratic_B.C3.A9zier_curves" rel="nofollow">definition</a> a quadratic bézier curve which is always a segment of a parabola. </p>
|
1,518,697 | <blockquote>
<p>For this homework exercise, we are asked to show that the ideal $I=(3,1+\sqrt{-5})$ is a flat $\mathbb{Z}[\sqrt{-5}]$-module. The hint is to show that $I$ becomes principal (and thus free as a module) when we invert $2$ <strong>or</strong> $3$, so that $I$ is locally flat.</p>
</blockquote>
<p>I'm ha... | user26857 | 121,097 | <p>$I\mathbb Z[\sqrt{-5}]_3=\mathbb Z[\sqrt{-5}]_3$, and $I\mathbb Z[\sqrt{-5}]_2=(1+\sqrt{-5})\mathbb Z[\sqrt{-5}]_2\simeq\mathbb Z[\sqrt{-5}]_2$, so in both cases we get a free module. Now use that $2$ and $3$ are coprime in $\mathbb Z[\sqrt{-5}]$.</p>
<p><strong>Remark.</strong> In fact, $\mathbb Z[\sqrt{-5}]$ is a... |
1,811,581 | <p>When discussing the order relation on $\mathbb{C}$, it is said that such a statement as $z_1 < z_2$ where $z_1, z_2 \in \mathbb{C}$ is meaningless, unless $z_1$ and $z_2$ are real.</p>
<p>My question is, when will a complex number $z$ be real? I know that if $\bar{z}$ is the conjugate of $z$, then</p>
<p>$$z + ... | Felicity | 332,295 | <p>$\mathrm{span}\left\{\begin{bmatrix}
1\\0
\end{bmatrix},\begin{bmatrix}
0\\1
\end{bmatrix}\right\}$ is the set of linear combinations of the two vectors $\begin{bmatrix}
1\\0
\end{bmatrix}$
and $\begin{bmatrix}
0\\1
\end{bmatrix}$.</p>
<p>In other words, $\mathrm{span}\left\{\begin{bmatrix}
1\\0
\end{bmatrix},\begi... |
1,170,602 | <p>How to evaluate the integral </p>
<p>$$\int \sqrt{\sec x} \, dx$$</p>
<p>I read that its not defined.<br>
But why is it so ? Does it contradict some basic rules ?
Please clarify it .</p>
| Aaron Maroja | 143,413 | <p>First notice that $\cos x = 1 - 2\sin^2 \Big(\frac{x}{2}\Big)$ then </p>
<p>$$\int \frac{1}{\sqrt{\cos x}} \, dx = \int \frac{1}{\sqrt{1 - 2\sin^2 \Big(\frac{x}{2}\Big)}} \, dx = \color{red}{2F\Big(\left.\frac{x}{2}\right\vert 2\Big)} + C$$</p>
<p>where $F(\left.x\right\vert m)$ is an <a href="http://mathworld.wo... |
1,502,309 | <p>The initial notation is:</p>
<p>$$\sum_{n=5}^\infty \frac{8}{n^2 -1}$$</p>
<p>I get to about here then I get confused.</p>
<p>$$\left(1-\frac{3}{2}\right)+\left(\frac{4}{5}-\frac{4}{7}\right)+...+\left(\frac{4}{n-3}-\frac{4}{n-1}\right)+...$$</p>
<p>How do you figure out how to get the $\frac{1}{n-3}-\frac{1}{n-... | Mark Viola | 218,419 | <p>The correct way to analyze this is to write</p>
<p>$$\begin{align}
\sum_{n=5}^N\frac{2}{n^2-1}&=\sum_{n=5}^{N}\left(\frac{1}{n-1}-\frac{1}{n+1}\right)\\\\
&=\left(\frac14-\frac16\right)+\left(\frac15-\frac17\right)+\left(\frac16-\frac18\right)+\cdots \\\\
&+\left(\frac1{N-3}-\frac1{N-1}\right)+\left(\fr... |
3,501,052 | <p>I want to find the number of real roots of the polynomial <span class="math-container">$x^3+7x^2+6x+5$</span>.
Using Descartes rule, this polynomial has either only one real root or 3 real roots (all are negetive). How will we conclude one answer without doing some long process?</p>
| Kori | 229,036 | <p>If you actually want the roots of a cubic, you can use the <a href="https://www.purplemath.com/modules/rtnlroot.htm" rel="nofollow noreferrer">rational root test</a> to find one of the roots. After you have one of the roots you can easily do a long division to find the resulting second degree polynomial from there y... |
430,654 | <p>Show that this sequence converges and find the limit.
$a_1 = 0$, $a_{n+1} = \sqrt{5+2a_{n} }$ </p>
| Angela Pretorius | 15,624 | <p>For convergence, show that $\displaystyle\left|\sqrt{5+2(a_n+\delta)}-a_n\right|<\delta$ for sufficiently small delta and $a_n$ close to the limit. The limit is given by $\displaystyle a=\sqrt{5+2a}$, $a>0$ (since $a_1>0$ and $a_{n+1}$ has the same sign as $a_n$ thereafter) or $\displaystyle a=1+\sqrt{6}$.<... |
941,632 | <p>Is the Set $$S=\{e^{2x},e^{3x}\}$$ linearly independent?? And answer says Linearly independent over any interval $(a,b)$,only when $0$ doesnot belong to $(a,b)$</p>
<p>How do I proceed??</p>
<p>Thanks for the help!!</p>
| egreg | 62,967 | <p>I assume that you're working on the vector space of continuous real maps over the interval $(a,b)$.</p>
<p>So let's consider the two functions $f(x)=e^{2x}$ and $g(x)=e^{3x}$ and suppose that
$$
\alpha f+\beta g = 0.
$$
This means that, for every $x\in(a,b)$, we have
$$
\alpha f(x)+\beta g(x)=0
$$
and, in particula... |
3,278,797 | <p>I tried to solve it and I got answer '3'. But that is just my intuition.I don't have concrete method to prove my answer .I did like this, in order to maximize the fraction, we need to minimize the denominator .So if plug in '1' in expression, denominator becomes '1'.Now denominator is minimalized,the result of ex... | David K | 139,123 | <p>You have the correct result but the wrong reasons.</p>
<p>First of all, you assume that a ratio can be maximized by minimizing the denominator. This is true if the numerator is constant, but the numerator here is not constant.</p>
<p>Second, you decided that the minimum value of the denominator is <span class="mat... |
3,492,856 | <p><a href="https://i.stack.imgur.com/FHCP2.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/FHCP2.jpg" alt="This is the question"></a> My solution:-</p>
<p>Since <span class="math-container">$OA=AB$</span>, let us find OA first.
<span class="math-container">$OA=\sqrt{(18-0)^2+(3-0)^2}=3\sqrt{37}$</s... | Math1000 | 38,584 | <p>Let <span class="math-container">$T_1:V\to W$</span> and <span class="math-container">$T_2:W\to U$</span> be linear maps. Then <span class="math-container">$$\ker T_1 = \{v\in V : T_1v=0\}\subset V,$$</span> whereas <span class="math-container">$$\ker T_2\circ T_1 = \{v\in V: T_2(T_1(v)) = 0\}\subset V.$$</span> Ind... |
5,897 | <p>The following creates a button to select a notebook to run. When the button is pressed it seems that Mathematica finds the notebook but cannot evaluate it. The following error occurs</p>
<blockquote>
<p>Could not process unknown packet "1"</p>
</blockquote>
<pre><code>Button["run file 1",
NotebookEvaluate[... | celtschk | 129 | <p>To make it work, use</p>
<pre><code>Button["run file 1", NotebookEvaluate["/../file1.nb"], Method->"Queued"]
</code></pre>
|
467,301 | <p>I'm reading Intro to Topology by Mendelson.</p>
<p>The problem statement is in the title.</p>
<p>My attempt at the proof is:</p>
<p>Since $X$ is a compact metric space, for each $n\in\mathbb{N}$, there exists $\{x_1^n,\dots,x_p^n\}$ such that $X\subset\bigcup\limits_{i=1}^p B(x_i^n;\frac{1}{n})$. Let $K=\frac{2p}... | Prahlad Vaidyanathan | 89,789 | <p>By compactness, there are finitely many points $x_1, x_2, \ldots, x_n$ such that
$$
X = \cup_{i=1}^n B(x_i, 1)
$$
Now for any two points $x, y \in X$, choose $x_i$ and $x_j$ such that
$$
d(x, x_i) < 1 \qquad d(y,x_j) < 1
$$
Then
$$
d(x,y) \leq d(x,x_i) + d(x_i, x_j) + d(x_j, y) < 2 + d(x_i, x_j)
$$
So choos... |
40,709 | <p>Wolfram's MathWorld website, at the page on <a href="http://mathworld.wolfram.com/Function.html" rel="nofollow">functions</a>, makes the following claim about the notation $f(x)$ for a function:</p>
<blockquote>
<p>While this notation is deprecated by professional mathematicians, it is the more familiar one for m... | Andrew Stacey | 45 | <p>Vote for this answer if you <strong>agree</strong> with the statement:</p>
<blockquote>
<p>This house believes that the notation $f(x)$ to refer to a function has value in professional mathematics and that there is no need to apologise or feel embarrassed when using it thus.</p>
</blockquote>
<p>(Note: the answ... |
3,336,506 | <p>Let <span class="math-container">$V=\mathbb R^3$</span> be an inner product space with the standard inner product (that means <span class="math-container">$\langle(x_1,y_1,z_1),(x_2,y_2,z_2)\rangle=x_1y_1+x_2y_2+x_3y_3$</span> ).<br>
<span class="math-container">$U=span\{(1,2,3),(1,2,1)\}\subseteq V$</span> </p>
<... | reuns | 276,986 | <p>It is not a distribution on <span class="math-container">$\Bbb{R}^n$</span> because <span class="math-container">$$ \lim_{r \to 0}\int_{\mathbb{R}^n} \phi(x)\frac{1_{|x|> r}}{|x|^n} d^n x $$</span> diverges whenever <span class="math-container">$\phi(0) \ne 0$</span>. What is a (tempered) distribution is <span cl... |
1,204,864 | <blockquote>
<p>$$\text{Find }\,\dfrac{d}{dx}\Big(\cos^2(5x+1)\Big).$$</p>
</blockquote>
<p>I have tried using the rules outlined in my standard derivatives notes but I've failed to find the point of application.</p>
| abel | 9,252 | <p>you can also use $$2y = 2\cos^2(5x+1) = \cos(10x + 2) + 1$$ so that $$2\frac{dy}{dx} = -10\sin(10x+2)\to \frac{dy}{dx} = -5\sin(10x+2) $$</p>
|
1,102,668 | <p><a href="https://math.stackexchange.com/q/67994/198434">This question</a> shows how dividing both sides of an equation by some $f(x)$ may eliminate some solutions, namely $f(x)=0$. Naturally, all examples admit $f(x)=0$ as a solution to prove the point.</p>
<p>I tried to find a simple example of an equation that co... | Kyle Delaney | 250,589 | <p>Consider this differential equation:</p>
<p>$\frac{df}{dx} = kf$</p>
<p>You'd have to divide both sides by $f$ to get all your $f$ terms on the same side, right?</p>
|
1,746,782 | <p>This is what I've done:</p>
<p>Let $s < t$ and $F_t$ be a filtration adapted to $W(t)$
$$E[e^{t/2}\cos(W(t))|F_s] = e^{t/2} E[\cos(W(t)) - \cos(W(s)) + \cos(W(s))|F_s]$$
$$= e^{t/2} [E[\cos(W(t)) - \cos(W(s))|F_s] + \cos(W(s))]$$
Because of the independence of the increments:
$$= e^{t/2} [E[\cos(W(t)) - \cos(W(s... | Cavents | 322,026 | <p>To find $\mathbb{E}[\cos(W(t))]$ you could try integrating the density function. However, that might turn out to be messy. Instead, recall that
$$\cos x = \frac{e^{ix}+e^{-ix}}{2},$$
and hence
\begin{align}
\mathbb{E}[\cos(W(t))] = \mathbb{E}\left[\frac{e^{iW(t)}+e^{-iW(t)}}{2}\right],
\end{align}
where $\mathbb{E}[... |
1,333,637 | <p>Where X is a space obtained by pasting the edges of a polygonal region together in pairs. </p>
<p>Alternatively: Show that X is homeomorphic to exactly one of the spaces in the following list: $S^2,T_n, P^2, K_m, P^2\#K_m$, where $K_m$ is the m-fold connected sum of $K$(Klein bottle) with itself and $M \geq 0$.</p>... | Gerry Myerson | 8,269 | <p>I think all you need to know is that the sum of three projective planes is homeomorphic to the sum of a torus and a projective plane. </p>
|
2,555,463 | <p>Given a line $l$ and two points $p_1$ and $p_2$, identify the point $v$ which is equidistant from $l$, $p_1$, and $p_2$, assuming it exists.</p>
<p>My idea is to: (1) identify the parabolas containing all points equidistant from each point and the line, then (2) intersect these parabolas. As $v$ is equidistant from... | user | 505,767 | <p>Assuming such point <span class="math-container">$Q$</span> exists it must lie on the <strong>Bisector Line b</strong> of <span class="math-container">$P_1$</span> and <span class="math-container">$P_2$</span> i.e. the line through the midpoint of <span class="math-container">$P_1$</span> and <span class="math-conta... |
1,840,485 | <p>I am an undergraduate really passionate about the mathematics and microbiology. I have few big problems in learning which I would like to seek your advice. </p>
<p>Whenever I study mathematical books (Rudin, Hoffman/Kunze, etc.), I always try to prove every theorem, lemma, corollary, and their relationships in t... | Bhavani Chandra | 350,346 | <p>Mathematics is kind of a subject that is fun as well as scary. I always know the concept and do one or two problems per concept. As solving each & every problem is time consuming and is of no use.
My advice is practice all the concepts and theorems, and do two or 3 different types of problems which based on sam... |
1,482,205 | <p>Show that $\sigma(AB) \cup \{0\} = \sigma(BA) \cup \{0\}$ in general, and that $\sigma(AB) = \sigma(BA)$ if $A$ is bijective. </p>
<p>I studied the associative statement of this somewhere but it did not include the zeroth element.
If you assume the bijection, how can you show the first part?</p>
<h2>My attempt</h... | Marcos | 39,676 | <p>You got the addition wrong, it's (a+b)/b, not ab/b.</p>
|
3,399,586 | <p>Suppose that <span class="math-container">$f$$:$$\mathbb{R}\to\mathbb{R}$</span> is differentiable at every point and that </p>
<p><span class="math-container">$$f’(x) = x^2$$</span></p>
<p>for all <span class="math-container">$x$</span>. Prove that </p>
<p><span class="math-container">$$f(x) = \frac{x^3}{3} + C$... | Quanto | 686,284 | <p>Here is to use the Mean-Value-Theorem (MVT) for the derivation. Let <span class="math-container">$\epsilon$</span> be small positive variable and observe that,</p>
<p><span class="math-container">$$x^2+\frac13\epsilon^2= \frac{6\epsilon x^2+2\epsilon^3}{6\epsilon}
=\frac{(x+\epsilon)^3 - (x-\epsilon)^3}{3(2\epsilon... |
3,346,676 | <blockquote>
<p><strong>Question.</strong> Find a divergent sequence <span class="math-container">$\{X_n\}$</span> in <span class="math-container">$\mathbb{R}$</span> such that for any <span class="math-container">$m\in\mathbb{N}$</span>,
<span class="math-container">$$\lim_{n\to\infty}|X_{n+m}-X_n|=0$$</span></p>
... | N. S. | 9,176 | <p><span class="math-container">$X_n= \ln(n)$</span>.</p>
<p>Then, for each <span class="math-container">$m$</span> you have
<span class="math-container">$$\lim_{n\to\infty}|X_{n+m}-X_n| =\lim_{n\to\infty}|\ln(\frac{m+n}{n}) |= \lim_{n\to\infty}|\ln(1+\frac{m}{n}) | = \ln(1)=0$$</span></p>
|
192,821 | <p>I am using <a href="https://reference.wolfram.com/language/ref/TransformedField.html" rel="noreferrer"><code>TransformedField</code></a> to convert a system of ODEs from Cartesian to polar coordinates:</p>
<pre><code>TransformedField[
"Cartesian" -> "Polar",
{μ x1 - x2 - σ x1 (x1^2 + x2^2), x1 + μ x2 - σ x2... | Itai Seggev | 4,848 | <p>Mathematica's answer is correct and consistent with your expectations, but you are not accounting for the basis of the vector field. </p>
<p><code>TransformedField</code> transforms a vector field between two coordinate systems <em>and</em> bases. In this case, it is converting from <span class="math-container">$f... |
545,634 | <p>Consider a function $f:\Omega\subset\mathbb{R}^n\rightarrow\mathbb{R}^m$, for which the Jacobian matrix </p>
<p>$J_f(x_1,...,x_n)= \left( \begin{array}{ccc}
\frac{\partial f_1}{\partial x_1} & ... & \frac{\partial f_1}{\partial x_n} \\
\vdots & & \vdots \\
\frac{\partial f_m}{\partial x_1} & .... | Olivier | 45,622 | <p>For a starter, the pdf of $X$ you posted is not a valid pdf. $\int_{-\infty}^{\infty} f_X(x) dx = \frac{3}{2}$. This also results in an incorret plot of $g(X)$. Which pdf do you mean?</p>
<p>Edit: Excuse me, I misread your definition. </p>
|
3,006,046 | <p>How to find the Newton polygon of the polynomial product <span class="math-container">$ \ \prod_{i=1}^{p^2} (1-iX)$</span> ?</p>
<p><strong>Answer:</strong></p>
<p>Let <span class="math-container">$ \ f(X)=\prod_{i=1}^{p^2} (1-iX)=(1-X)(1-2X) \cdots (1-pX) \cdots (1-p^2X).$</span></p>
<p>If I multiply , then we... | sigmatau | 77,518 | <p><em>Partial Answer</em>: regarding the coefficients of the polynomial: </p>
<p>Fix one term in the brackets, say <span class="math-container">$Y=(1-5X)$</span>. In order for the coefficient <span class="math-container">$5$</span> to contribute to <span class="math-container">$a_j$</span>, we have to multiply <span ... |
482,102 | <p>The problem comes from Alan Pollack's Differential Topology, pg. 5.
Suppose that X is a k-dimensional manifold. Show that every point in X has a neighborhood diffeomorphic to all of $\Bbb{R}^k$.</p>
<p>I have already shown that $\Bbb{R}^k$ is diffeomorphic to $B_a$ (part (a) of the question) the open ball of radius... | Rhys | 47,565 | <p>Given a point $p \in \mathbb{R}^k$, and an open neighbourhood $U$ of $p$, there exists an open set $V$ with $p \in V \subset U$ such that $V$ is diffeomorphic to the $k$-dimensional unit ball. Just take the Euclidean metric on $\mathbb{R}^k$, and let $V$ be a sufficiently small open ball centred on $p$.</p>
|
843,634 | <p>I am wondering whether for any two lines $\mathfrak{L}, \mathfrak{L'}$ and any point $\mathfrak{P}$ in $\mathbf{P}^3$ there is a line having nonempty intersection with all of $\mathfrak{L}, \mathfrak{L'}$, $\mathfrak{P}$. I don't really know how to approach this, because I was never taught thinking about such a prob... | Charles | 1,778 | <p>The polynomial $x^2+x$ is always divisible by 2, but as polynomials $x^2+x\not\equiv0\pmod2$ -- for one thing, the degree on the left is different from the degree on the right.</p>
<p>Similarly, even though $x^p-x$ is zero mod $p$ for any $x$, <em>as polynomials</em>, $x^p$ and $x$ are different.</p>
<p>Basically ... |
2,073,230 | <p>I thought I'd might use induction, but that seems too hard, then I tried to take the derivative and show that that's positive $\forall$n. But I can't figure out how to do that either, I've tried induction there too.</p>
| Mark | 310,244 | <p>Let's think about it via the derivative. Let:
$$f(x) = \left(1+\frac{a}{x}\right)^x$$
When $a = 0.6$ and $x\in\mathbb N$ we have your function.</p>
<p>We can't take the derivative of this yet, as we can't use the power rule (the exponent is a variable), and can't use the exponential rule (the base is variable). W... |
3,337,210 | <p>I am struggling with the following equation, which I need to proof by induction:</p>
<p><span class="math-container">$$\sum_{k=1}^{2n}\frac{(-1)^{k+1}}{k}= \sum_{k=n+1}^{2n}\frac{1}{k}$$</span></p>
<p><span class="math-container">$n\in \mathbb{N}$</span>.<br/> I tried a few times and always got stuck.</p>
<p>Help... | fleablood | 280,126 | <p>The cosine proof can be made intuitive.</p>
<p>We know <span class="math-container">$b > d= b-r$</span> and <span class="math-container">$c > h$</span>. And we know by the pythagorean th. and <span class="math-container">$d$</span> and <span class="math-container">$h$</span> are sides of a right triangle wit... |
1,584,594 | <p>Find all $n$ for which $n^8 + n + 1$ is prime. I can do this by writing it as a linear product, but it took me a lot of time. Is there any other way to solve this? The answer is $n = 1$.</p>
| lab bhattacharjee | 33,337 | <p>HINT:</p>
<p>If $w$ is a complex cube root of unity and $f(x)=x^8+x+1$</p>
<p>$f(w)=(w^3)^2\cdot w^2+w+1=0$</p>
<p>So $(x^2+x+1)|(x^8+x+1)$</p>
|
726,574 | <p>Ant stands at the end of a rubber string which has 1km of length. Ant starts going to the other end at speed 1cm/s. Every second the string becomes 1km longer. </p>
<p>For readers from countries where people use imperial system: 1km = 1000m = 100 000cm</p>
<p><strong>Will the ant ever reach the end of the string? ... | DonAntonio | 31,254 | <p>$$y=mx+c=\frac12x-3=\frac{x-6}2\implies 2y=x-6\implies 2y-x+6=0$$</p>
<p>You forgot to multiply $\;C\;$ by two...</p>
|
2,382,058 | <p>Penrose's paper <a href="http://www.iri.upc.edu/people/ros/StructuralTopology/ST17/st17-05-a2-ocr.pdf" rel="nofollow noreferrer"><em>On the cohomology of impossible figures</em></a> suggests me that we can't draw such an impossible figure on a contractible part $Q$ of a sheet of paper so that it completely fills it,... | Arthur | 15,500 | <p>The drawing domain of most impossible figures is a circle or an annulus. The reason they're impossible is that there is no single "height function" that covers all of it, even though there is a "steepness function".</p>
<p>In other words, there exists a function that looks like it should be a derivative / gradient,... |
2,382,058 | <p>Penrose's paper <a href="http://www.iri.upc.edu/people/ros/StructuralTopology/ST17/st17-05-a2-ocr.pdf" rel="nofollow noreferrer"><em>On the cohomology of impossible figures</em></a> suggests me that we can't draw such an impossible figure on a contractible part $Q$ of a sheet of paper so that it completely fills it,... | mma | 63,290 | <p>I think, I misunderstood the role of the annulus. My conclusion is that we can draw an impossible figure on any domain of a sheet. The annulus (or any non-contractible open set) is not a constraint. It is only a tool for testing the possibility of the picture. The test is that the cocycle $d_{ij}$ described in Penro... |
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