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 |
|---|---|---|---|---|
3,051,207 | <p>In the very first chapter in my class "mathematical analysis 1" I've seen something called <strong>the triangle inequality</strong>, which is <span class="math-container">$||a| - |b|| \leq |a \pm b| \leq |a| + |b|$</span>. Now the thing is that I do understand why this is true, but i fail to see what this actually h... | Lee Mosher | 26,501 | <p>In Euclidean space with Cartesian coordinates, let <span class="math-container">$+$</span> denote vector addition, and let <span class="math-container">$O$</span> denote the origin. Choose points <span class="math-container">$a,b$</span>, and form a triangle with vertices <span class="math-container">$a,b,O$</span>.... |
4,245,475 | <p>Evaluate <span class="math-container">$ \, \displaystyle \int _{0}^{4} \frac{1}{(2x+8)\, \sqrt{x(x+8)}}\, dx. $</span>
<br /><br /><span class="math-container">$My\ work:-$</span><br />
by completing the square and substitution i.e. <span class="math-container">$\displaystyle \left(\begin{array}{rl}x+4 & = 4\sec... | Mark Saving | 798,694 | <p>Take a basis <span class="math-container">$B \subseteq W$</span>. For each <span class="math-container">$b \in B$</span>, pick <span class="math-container">$s_b \in V$</span> such that <span class="math-container">$T(s_b) = b$</span>.</p>
<p>Then <span class="math-container">$S = \{s_b \mid b \in B\}$</span> must sp... |
4,245,475 | <p>Evaluate <span class="math-container">$ \, \displaystyle \int _{0}^{4} \frac{1}{(2x+8)\, \sqrt{x(x+8)}}\, dx. $</span>
<br /><br /><span class="math-container">$My\ work:-$</span><br />
by completing the square and substitution i.e. <span class="math-container">$\displaystyle \left(\begin{array}{rl}x+4 & = 4\sec... | baharampuri | 50,080 | <p>Let <span class="math-container">$v \neq 0$</span> be such that <span class="math-container">$T(v)=0$</span>. Extend this to a basis of <span class="math-container">$V$</span>. Let's call that basis <span class="math-container">$B$</span>. Now as <span class="math-container">$T$</span> is surjective <span class="mat... |
1,817,542 | <p><strong>Problem:</strong> Let $(X, Y)$ be uniformly distributed on the unit disk $\{ (x,y) : x^2 + y^2 \le 1\}$. Let $R = \sqrt{X^2 + Y^2}$. Find the CDF and PDF of $R$.</p>
<p><strong>Attempted Solution:</strong> First note that $r \in R = \sqrt{X^2 + Y^2}$ represents a point on $\mathbb{R}^2$ with radius $r$ abou... | Natalio | 499,705 | <p>I know that this question is quite old <a href="https://math.stackexchange.com/questions/2712583/find-the-distribution-of-r-sqrtx2y2-where-x-y-is-uniform-on-the-un/2717764#2717764">but I asked the same some days ago</a> and now I think I have an answer which involves more calculus. Since this question is more popu... |
253,359 | <p>I'm trying to prove by induction the following statement without success:<br>
$$\forall n \ge 2, \;\forall d \ge 2 : d \mid n(n+1)(n+2)...(n+d-1) $$</p>
<p>For the base case: $n = 2$, $d = 2$<br>
$2\mid 2(2+1)$ which is true.<br></p>
<p>Now, the confusion begins! I assume I would need to use the second induction p... | Hagen von Eitzen | 39,174 | <p>As abbreviation, define $f(n,d)=n\cdot (n+1)\cdot\ldots\cdot (n+d-1)$.</p>
<p>Base case $n=2$:
$$\forall d\ge 2\colon d|f(2,d) = 2\cdot 3\cdot\ldots \cdot d$$
This is true because $d$ occurs among the factors on the rihght.</p>
<p>$n\to n+1$:</p>
<p>Assume that we know about $n$ that
$$\tag1\forall d\ge 2\colon ... |
2,455,408 | <p>I am encountering questions like this below.</p>
<p>$$\frac{dP}{dt}=(a-b\cos t ) \left(P+ \frac{P^2}{M}\right)$$</p>
<p>Then there is information stating $M$ is a positive integer and $a$ and $b$ are positive. That when $t=0$, $P=P_0$.</p>
<p>It wants me to solve the differential equation and show the process.</p... | Mathemagical | 446,771 | <p>The area you need is 8 times the shaded area. Observe the equations carefully to see the symmetries. <a href="https://i.stack.imgur.com/549TS.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/549TS.png" alt="enter image description here"></a></p>
<p>How to find it? Well you can see that (0,0) is wh... |
499,476 | <p>Use mathematical induction to prove that the derivative of $f(x)=\sin(ax+b)$ is given by</p>
<p>$f^{(n)}(x)= (-1)^ka^n\sin(ax+b)$ if $n=2k$, and $(-1)^ka^n\cos(ax+b)$ if $n=2k+1$</p>
<p>for a number $k=0,1,2,3,...$</p>
<p>I have done som proofs by induction, but I seem to struggle as soon as trig functions appea... | Cameron Buie | 28,900 | <p>You'll need to use the Chain Rule, together with the fact that $$\frac{d}{du}\sin(u)=\cos(u)$$ and $$\frac{d}{du}\cos(u)=-\sin(u).$$</p>
|
1,698,039 | <p>Alright, so let's say I have $$\frac{x^{-6}}{-x^{-4}}$$ The answer is $\dfrac{1}{x^2}$, but why isn't it $\dfrac{1}{-x^2}$?</p>
| BLAZE | 144,533 | <p>It might help you see what's going on if you write them as two story fractions: $$\frac{x^{-6}}{-x^{-4}}=\frac{\left(\dfrac{1}{x^6}\right)}{\left(-\dfrac{1}{x^4}\right)}=\underbrace{\color{red}{\frac{1}{x^6}\times -\frac{x^4}{1}}}_{\large \color{blue}{\text{by the reciprocal rule}}}=-\frac{1}{x^2}=\frac{1}{-x^2}$$ S... |
315,844 | <p>What is the probability P(X>Y) given that X,Y are Uniformly distributed between [0,1]?</p>
| Seyhmus Güngören | 29,940 | <p>Assuming $X$ and $Y$ are <em>independent</em>, Let $Z=X-Y$, Then were looking for $P(Z>0)$. To obtain the distribution of $Z$, you can convolve $X$ by $-Y$. The result will be a triange in the range $-1<x<1$. Then it it is obvious to see that $P(Z>0)=0.5$.</p>
|
866,654 | <p>Reading various betting forum I came across different threads claiming <strong><em>betting multiple is worse than betting on single events</em></strong>.</p>
<p>Could you explain why?</p>
<p>[Clairification for the ones not familiar with betting:
Betting on a single event: predict the outcome of a single match.
Be... | Thanos Darkadakis | 105,049 | <p>I'm afraid you're wrong. Betting on multiple events is worse. This happens because of the commissions. I will try to explain it with numbers and not with formulas, because it's easier to be understood.</p>
<p>Assume you want to bet on 2 events with P(win)=0.5. Initially you have €100.</p>
<p><strong>Case 1: No com... |
2,406,043 | <p>Let the triangle $\triangle ABC$ have sides $a,b,c$ and be inscribed in a circle with radius $R$. If $p=\frac{a+b+c}{2}$ The radius of the circle can be expressed as</p>
<p>a) $$R=\frac{\sqrt{p(p-a)(p-b)(p-c)}}{4abc}$$</p>
<p>b) $$R=\frac{4\sqrt{p(p-a)(p-b)(p-c)}}{abc}$$</p>
<p>c) $$R=\frac{abc}{4\sqrt{p(p-a)(p-b... | Michael Rozenberg | 190,319 | <p>Also, we can use SOS here:
$$a^3+b^3+c^3-3abc=\sum_{cyc}(a^3-abc)=\sum_{cyc}a(a^2-bc)=$$
$$=\frac{1}{2}\sum_{cyc}a((a-b)(a+c)-(c-a)(a+b))=$$
$$=\frac{1}{2}\sum_{cyc}(a-b)(a(a+c)-b(b+c))=$$
$$=\frac{1}{2}\sum_{cyc}(a-b)(a^2-b^2+ac-bc)=\frac{1}{2}\sum_{cyc}(a-b)^2(a+b+c)\geq0.$$</p>
|
3,636,667 | <blockquote>
<p>Evaluate
<span class="math-container">$$\lim_{n\to\infty}\frac{1}{n^{p+1}}\cdot \sum_ \limits{i=1}^{n} \frac{(p+i)!}{i!} , p \in N$$</span> </p>
</blockquote>
<p>Now, I found this problem while doing some practice and I am curious on how to solve it . I have no good ideas yet, so I will appreciate... | bernat tobella | 745,671 | <p>Did you try by factoring the inside of summatory? Also, you can solve it for different values of p. (this should be a comment but i don't have enough reputation)</p>
|
3,636,667 | <blockquote>
<p>Evaluate
<span class="math-container">$$\lim_{n\to\infty}\frac{1}{n^{p+1}}\cdot \sum_ \limits{i=1}^{n} \frac{(p+i)!}{i!} , p \in N$$</span> </p>
</blockquote>
<p>Now, I found this problem while doing some practice and I am curious on how to solve it . I have no good ideas yet, so I will appreciate... | CHAMSI | 758,100 | <p>First of all, <span class="math-container">$ \left(\forall x\in\mathbb{R}_{+}\right),\ \mathrm{e}^{x}-1=x\int_{0}^{1}{\mathrm{e}^{xy}\,\mathrm{d}y}\leq x\, \mathrm{e}^{x} \cdot $</span></p>
<p>Let <span class="math-container">$ n,p $</span> be positive integers, we have the following : <span class="math-container">... |
4,274,600 | <p><span class="math-container">$(x^3+x+1)^{-1} \mod (x^4+x+1)$</span> over <span class="math-container">$\text{GF}(2)$</span></p>
<p>I understand well how to solve the equation without inverse but don't know how to solve it with inverse.</p>
| Bernard | 202,857 | <p>Just perform the <em>extended Euclidean algorithm</em> in <span class="math-container">$\mathbf F_2$</span> to obtain a Bézout's relation between <span class="math-container">$X^4+X+1$</span> and <span class="math-container">$X^3+X+1$</span>:
<span class="math-container">\begin{array}{r| lll}
r(X) & u & v &a... |
3,682,987 | <blockquote>
<p>Let <span class="math-container">$a$</span>, <span class="math-container">$b$</span>, and <span class="math-container">$c$</span> be positive real numbers. What is the smallest possible value of <span class="math-container">$(a+b+c)\left(\frac{1}{a+b}+\frac{1}{a+c}+\frac{1}{b+c}\right)$</span>?</p>
</... | Anas A. Ibrahim | 650,028 | <p>Well, by Cauchy-Schwarz,
<span class="math-container">$$((a+b)+(a+c)+(b+c))\left(\frac{1}{a+b}+\frac{1}{a+c}+\frac{1}{b+c}\right) \ge (1+1+1)^2=9$$</span>
<span class="math-container">$$\iff (a+b+c)\left(\frac{1}{a+b}+\frac{1}{a+c}+\frac{1}{b+c}\right) \ge \frac{9}{2} $$</span></p>
|
1,309,670 | <p>Suppose $D \subset \mathbb{R}$ is open, $f : D \to \mathbb{R}$ is a smooth (not necessarily real analytic) function, $x_0 \in D$, and $T_n$ is the degree $n$ Taylor polynomial of $f$ centered at $x_0$. Let $S=\{ x \in D : f(x)=T_n(x) \}$. It is not hard to see that $S$ is closed and contains $x_0$. What else can be ... | Robert Israel | 8,508 | <p>Outside a neighbourhood of $x_0$, $T_n - f$ is an arbitrary smooth function.
So essentially you're asking what can be said about the zero set of an arbitrary smooth function. That can be any closed subset of $\mathbb R$.</p>
|
997,602 | <blockquote>
<p>Prove that the function <span class="math-container">$x \mapsto \dfrac 1{1+ x^2}$</span> is uniformly continuous on <span class="math-container">$\mathbb{R}$</span>.</p>
</blockquote>
<p>Attempt: By definition a function <span class="math-container">$f: E →\Bbb R$</span> is uniformly continuous iff for ... | Michael Hardy | 11,667 | <p>According to the mean value theorem,
$$
\left|\frac1{1 + x^2} - \frac1{1 + a^2}\right| = f'(c)|x-a|
$$
where $f(x)=\dfrac 1 {1+x^2}$ and $c$ is somewhere between $x$ and $a$. But $|f'(c)|\le\max |f'|$, the absolute maximum value of $|f'|$. In order for this to make sense, you need to show that $|f'|$ does have an ... |
86,800 | <p>I am curious about how the Heegaard genus changes after a finite covering. </p>
<p>Is there anyone constructing an closed hyperbolic 3-manifold $N$ such that </p>
<p>the Heegaard genus of a finite covering of $N$ is less than the Heegaard genus of $N$? </p>
<p>Thank you!</p>
<p>Note: Heegaard genus of a 3-manifo... | Michael Siler | 5,413 | <p>There are examples like this. Check out section 4.5 of Shalen's paper "Hyperbolic volume, Heegaard genus and ranks of groups." It's here: <a href="http://arxiv.org/abs/0904.0191">http://arxiv.org/abs/0904.0191</a></p>
<p>He gives a reference for a genus 3 example by Alan Reid and a sketch of a technique for produci... |
2,449,443 | <p>Set of numbers $\ x_1, \ldots, x_m , y_1, \ldots, y_n $ where $\ x_i=0 $ for $i = 1,\ldots, m$ and $\ y_i=1 $ for $i = 1,\ldots, n$</p>
<p>Show that mean $M$ of this set is given by $\frac{n}{m+n}$ and the standard deviation $S$ by $\frac{ \sqrt{mn}} {m+n} $</p>
<p>I know the definitions of the mean and standard ... | Donald Splutterwit | 404,247 | <p>Calculate the zero, first and second moments
\begin{eqnarray*}
\sum 1 = ? \\
\sum z_j = ? \\
\sum z_j^2 = ? \\
\end{eqnarray*}
Then use the formulea
\begin{eqnarray*}
\mu&=&\frac{\sum z}{\sum 1 } \\
\sigma^2 &=& \frac{\sum z^2}{\sum 1 }- \frac{(\sum z)^2}{(\sum 1)^2 }.
\end{eqnarray*}</p>
<blockquot... |
801,081 | <p>I was doing some school work and got bored so I started messing with k-gonal numbers. I started with the triangular numbers, square numbers and looked for patterns. I noticed something.</p>
<p>Let $n^{(k)}$ denote the $n$-th $k$-gonal number. For example, $3^{(3)}$ is the third triangular number, 6.</p>
<p>I f... | Janaka Rodrigo | 1,043,137 | <p>Method- (ii) <br/>
3-gonal numbers
1,3,6,10,15,..
4-gonal numbers
1,4,9,16,25,...
5-gonal numbers <br/>
1,5,12,22,35,... <br/>
6-gonal numbers <br/>
1,6,15,28,45,... <br/>
Let u(k,n) be the n th k-gonal number <br/>
In above sequences second difference is a constant, 3-gonalsequence it is 1, 4- gonal sequence it... |
2,199,222 | <p>I have the feeling of being stuck or missing something trying to prove
$$ \lim_{N\to\infty}\sum_{k=1}^{N} \frac{1}{N+k} =\int_{1}^{2} \frac{1}{x} dx = ln(2)$$</p>
<p>Using Riemann-Sums I have shown that $$\int_{1}^{a} \frac{1}{x} dx=\lim_{N\to\infty}\sum_{k=1}^{N} (a^{1/N}-1)=\lim_{N\to\infty}N(a^{1/N}-1)=\lim_{h\t... | Riley | 320,172 | <p>I believe you are thinking of a skew-symmetric matrix, but this requires the diagonal to be 0 as well. It is skew-Hermitian if you require the diagonal to be imaginary and all other entries to be real. If the diagonal must have real values, then I don't believe there is an appropriate term for this, but you can call... |
1,618,753 | <p>Trying to expand $f(x)=\cot(x)$ to Taylor series (Maclaurin, actually).
But I keep "adding up" infinities when using the formula. (Because of $\cot(0)=\infty$) Could you perhaps give me a hint on how to proceed?</p>
| latorrefabian | 306,438 | <p>The correct answer is that x = 0 is not in the domain of cot(x). Continuity is a property of elements of the domain of the function.</p>
|
1,749,730 | <p>What is the maximum number of faces of totally convex solid that one can "see" from a point? </p>
<p>...and, more importantly, how can I ask this question better? (I'm a college student with little experience in asking well formed questions, much less answering them.) </p>
<p>By "see" I mean something like this: y... | shardulc | 140,607 | <p>As @almagest has pointed out, the absolute maximum number of faces you can see of a polyhedron with $n$ faces is $n-1$. This is achieved in the case of a right pyramid with a base and $n-1$ sides; if you view the pyramid from above the apex, you can see all the sides except the base. This is perhaps true for non-rig... |
1,749,730 | <p>What is the maximum number of faces of totally convex solid that one can "see" from a point? </p>
<p>...and, more importantly, how can I ask this question better? (I'm a college student with little experience in asking well formed questions, much less answering them.) </p>
<p>By "see" I mean something like this: y... | G Cab | 317,234 | <p>Not only for a pyramid, but also for a "hemi-spheric diamond" cut with any number of flat faces, when looked from far enough, will show $n-1$ faces.<br>
If you consider regular polyhedron only, than the answer may vary.<br>
Coming back to a general method, for a general convex polyhedron and a "common" camera, then ... |
1,690,715 | <p>I have this space $E=\mathcal{C}([0,1],\mathbb{R})$ and the inner product $d(f,g)=\int_0^1 |f(x)-g(x)|\,{\rm d}x$.</p>
<p>Who have an idea about a simple sequence $\{f_n\}_{n=1}^\infty$ which is Cauchy but not convergent in $(E,d)$?</p>
| Community | -1 | <p>Hint: consider a sequence of functions like:</p>
<p>$$f_n(x)=\begin{cases} 0 \text{ for } x\in [0,\frac{1}{2}-\frac{1}{n}]\\
\text{linear} \text{ for } x\in [\frac{1}{2}-\frac{1}{n},\frac{1}{2}+\frac1n]\\
1 \text{ for } x\in [\frac{1}{2}+\frac{1}{n},1]
\end{cases}$$</p>
|
266,832 | <p>Let $p(x) = \sum_{k \geq 0} a_k x^k$ where the $a_k$'s are IID random variables taken from a mean-zero random variable taking finitely many values in $\mathbb{R}$; it clearly converges for $-1<x<1$. Is it a.s. true that the sign of $p(x)$ oscillates infinitely often as $x \rightarrow 1^-$? That is, is it the c... | Igor Rivin | 11,142 | <p><em>Edelman, Alan; Kostlan, Eric</em>, <a href="http://dx.doi.org/10.1090/S0273-0979-1995-00571-9" rel="nofollow noreferrer"><strong>How many zeros of a random polynomial are real?</strong></a>, Bull. Am. Math. Soc., New Ser. 32, No.1, 1-37 (1995). <a href="https://zbmath.org/?q=an:0820.34038" rel="nofollow noreferr... |
4,510,795 | <p>I need to find the % difference between two numbers. One person told me to use <span class="math-container">$\frac{x-y} {x} $</span>, another told me to use <span class="math-container">$\frac x y$</span> <span class="math-container">$- 1$</span> . Who is right?</p>
<p>Example:
Today's price: <span class="math-c... | Yaroslav Nikitenko | 186,740 | <p>The original equation is equivalent to a system with <span class="math-container">$n+1$</span> unknowns:</p>
<p><span class="math-container">$$ \eqalign{
\sum a_i r_i & = 0, \cr
r_1^2 - f_1(x) & = 0, \cr
&... \cr
r_n^2 - f_n(x) & = 0,
}
$$</span></p>
<p>with additional requirements all <span class="m... |
3,631,648 | <p>Suppose <span class="math-container">$X_1, ..., X_n \stackrel{iid}{\sim}$</span> Exponential(rate = <span class="math-container">$\lambda$</span>) independent of <span class="math-container">$Y_1, ..., Y_n \stackrel{iid}{\sim}$</span> Exponential<span class="math-container">$(1)$</span>. </p>
<p>Define <span class=... | qp212223 | 428,839 | <p>Would this be <span class="math-container">$$\prod_{\{i: Y_i = Z_i\}} \frac{1}{\lambda +1} \prod_{\{i: Y_i > Z_i\}} e^{-Y_i}\lambda e^{-\lambda Z_i} $$</span></p>
<p>where we just have the point mass/probability of equality contributing when <span class="math-container">$Y_i = Z_i$</span> and the joint density c... |
3,631,648 | <p>Suppose <span class="math-container">$X_1, ..., X_n \stackrel{iid}{\sim}$</span> Exponential(rate = <span class="math-container">$\lambda$</span>) independent of <span class="math-container">$Y_1, ..., Y_n \stackrel{iid}{\sim}$</span> Exponential<span class="math-container">$(1)$</span>. </p>
<p>Define <span class=... | Henry | 6,460 | <p>I would guess that the useful information is in the values of <span class="math-container">$Z_i$</span> and how often <span class="math-container">$Y_i=Z_i$</span> or not (perhaps call this <span class="math-container">$Q$</span>); the actual values of <span class="math-container">$Y_i$</span> may not help beyond th... |
208,802 | <p>Is there a continuous increasing function $ f : [0, \pi] \to [0, e] $ such that $ f(0) = 0, f(\pi) = e $ and $ f (q ) \in \mathbb{Q} $ for $ q \in \mathbb{Q} $ and $ f (q ) \in \mathbb{Q}^c $ for $ q \in \mathbb{Q}^c $? I think there should be, but I am unable to construct one. </p>
| Ross Millikan | 1,827 | <p>The <a href="http://en.wikipedia.org/wiki/Back-and-forth_method" rel="nofollow">back-and-forth method</a> that shows the isomorphism of dense countable linear orders gets you there. Let $a_i$ be an enumeration of the rationals in $(0,\pi)$ and $b_j$ be an enumeration of the rationals in $(0,e)$. Set $f(a_1)$ to th... |
3,484,052 | <p>Let's say you have a series that looks like <span class="math-container">$\sum^\infty_{n=N}f(n)$</span>, where <span class="math-container">$f(n)$</span> is some <span class="math-container">$n$</span>-dependent thing. If you take the limit of this series as <span class="math-container">$N$</span> approaches infinit... | user284331 | 284,331 | <p>Because of the Cauchy-criterion of the convergent sequence. For the existence of <span class="math-container">$\displaystyle\sum_{n=1}^{\infty}a_{n}$</span>, it simply means that the sequence <span class="math-container">$s_{n}=\displaystyle\sum_{k=1}^{n}a_{k}$</span> is convergent, then it is Cauchy. Then for each... |
4,302,213 | <blockquote>
<p>Let <span class="math-container">$R,S$</span> be rings and <span class="math-container">$\varphi : R\to S$</span> be a ring homomorphism. Verify that</p>
<ol>
<li><span class="math-container">$\varphi(na) = n\varphi(a)$</span> for all <span class="math-container">$n\in\mathbb Z$</span> and <span class="... | Matt E. | 948,077 | <p>I agree with Wuestenfux's answer, and have some feedback to give. There is some error in your answer to part <span class="math-container">$(1)$</span>, which is similar to one in part <span class="math-container">$(2)$</span> as well. How do you know that <span class="math-container">$\phi((n-1)a)=(n-1)\phi(a)$</spa... |
1,722,964 | <p>Expression :$$(p\rightarrow q)\leftrightarrow(\neg q\rightarrow \neg p)$$
What does the symbol $\leftrightarrow$ mean ? Please explain by drawing the truth table for this expression and also with other examples if possible. <strong>I'm in a desperate situation so I'd really appreciate a quick response !</strong></p>... | Paul Evans | 190,670 | <p>The expression $A\iff B$ means <em>if and only if</em>. </p>
<p>For $\iff$, we get the following truth table
$$\begin{array}{c|c|c|}
& B\text{ is true}& B\text{ is false}\\\hline
A\text{ is true} & \text{true} & \text{false}\\\hline
A\text{ is false} & \text{false} & \text{true}\\\hline
\en... |
2,347,820 | <p>What is the solution to $\log_{10} x -x=2?$</p>
<p>I have tried to solve it but I couldn't. I've got to $x^x =200$.</p>
| Integral | 33,688 | <p>Solving $\log_{10}x - x = 2$ is equivalent to solving $x = 10^{x+2}$. Now note that $x = 0 < 10^{0+2}$ and that the function $10^{x+2}$ grows faster than the function $x$ for $x \geq 0$. From this we can conclude that $x < 10^{x+2}$ for all $x \geq 0$. Therefore, the equality can't be valid.</p>
<p>Another wa... |
1,709,713 | <p>How do you make the jump from:</p>
<p>$$\frac{1-(\frac{4}{25})^{21}}{1-\frac{4}{25}}$$</p>
<p>To:</p>
<p>$$\frac{25^{21}-4^{21}}{25^{21}-4(25^{20})}$$</p>
| user146925 | 146,925 | <p>Have
$$\sum_{k=0}^{20}(\frac{4}{25})^k=\frac{1-(\frac{4}{25})^{21}}{1-\frac{4}{25}}=\frac{\frac{1}{25}\cdot(\frac{1}{25})^{20}(25^{21}-4^{21})}{\frac{1}{25}(25-4)}=\frac{25^{21}-4^{21}}{21\cdot25^{20}}$$</p>
|
997,587 | <p>The first sequence given is 3, 7, 16, 41, 77,....
I really am quite stuck on this because I can't seem to find any relationship between one term and the terms prior to it. I first noticed that it seemed like we were adding a perfect square to each one, since 3+4=7, 7+9=16, etc. But we skipped over adding the perfect... | Community | -1 | <p>For the second problem,</p>
<p>$s_n=3s_{n-1}+2 \implies s_{n-1}=3s_{n-2}+2$</p>
<p>$\therefore s_n-4s_{n-1}+3s_{n-2}=0$</p>
<p>Therefore the <a href="http://en.wikipedia.org/wiki/Recurrence_relation#Theorem" rel="nofollow">characteristic equation</a> is $t^2-4t+3=0$. Can you take it from here?</p>
|
2,153,340 | <p>Let $G$ be a group, and $C$ a set of proper subgroups of $G$.</p>
<p>Each subgroup in $C$ is normal subgroup of $G$.</p>
<p>For $G_1 , G_2\in C$, if $G_1 \ne G_2$ then $G_1\cap G_2=\{e_G\}$</p>
<p>$\bigcup\limits_{H\in C}H= G$.</p>
<p>Need to prove that G is Abelian group, hint someone?</p>
| Andreas Caranti | 58,401 | <p>First note that elements from two different normal subgroups in the family $\mathcal{C}$ commute. If $a \in A$, $b \in B$, with $A, B$ <em>different</em> normal subgroups in $\mathcal{C}$, we have
$$
[a, b] = a^{-1} b^{-1} a b = a^{-1} a^{b} = (b^{-1})^{a} b \in A \cap B = 1.
$$</p>
<p>Now let $x, y \in A$, with ... |
4,269,898 | <p>I've a question concerning inverse limits, since I don't usually work with them this extensively.</p>
<p>I'm considering the inverse limit of the following "bi-inverse system" of <span class="math-container">$R$</span>-modules and black arrows <span class="math-container">$f_{\bullet,\bullet}$</span>, and ... | Kevin Arlin | 31,228 | <p>Just for the sake of argument, it is indeed easy to give a more concrete approach here. The limit of your double system <span class="math-container">$A$</span> is a submodule <span class="math-container">$\prod_{i,j} A_{i,j}$</span> defined by those tuples <span class="math-container">$(a_{i,j})$</span> such that <s... |
88,565 | <p>Today I had an argument with my math teacher at school. We were answering some simple True/False questions and one of the questions was the following:</p>
<p><span class="math-container">$$x^2\ne x\implies x\ne 1$$</span></p>
<p>I immediately answered true, but for some reason, everyone (including my classmates and ... | Samuel Muldoon | 697,677 | <h2>Introduction</h2>
<p>You asked whether the following statement is true or not:</p>
<blockquote>
<p><span class="math-container">$x^2 \ne x \implies x \ne 1$</span></p>
</blockquote>
<p>Another way to write the above is as:</p>
<blockquote>
<p>if <span class="math-container">$(x^2 \ne x)$</span> then <span class="ma... |
3,085,181 | <p>I have to cope with a constraint of the form (1) in the following problem: </p>
<p><span class="math-container">$$\begin{align}\max\quad& x+y\\
\text{s.t.}\quad&
x + y \leq \max \{x,y\} &(1)\\
&0 \leq x \leq U_x&(2)\\
&0 \leq y \leq U_y&(3)\\
\end{align}$$</span></p>
<p>In the follow... | Jorge Cordova | 966,421 | <p>Hello I hope it is not too late; the penalty could be representative like this:</p>
<blockquote>
<p>max x + y<br />
x <=s1 + s2<br />
y <=s1 + s2<br />
s1 >= x - y<br />
s2 >= y - x<br />
s1 - bigM<em>b <= 0<br />
s2 - bigM</em>(1-b) <= 0<br />
0 <= x <= Ux<br />
0 <= y <= Uy</p>
</bloc... |
3,051,480 | <p>Now we have the equation
<span class="math-container">$$\sum_{i}(x_i-\hat x_i)^2,$$</span>
where <span class="math-container">$x_i$</span> is the observed value of a data sample <span class="math-container">$S$</span>. Here is the question:</p>
<blockquote>
<p>Why does this expression get its minimum value when <... | Noble Mushtak | 307,483 | <p>Let's take the function:</p>
<p><span class="math-container">$$f(\hat x)=\sum_{i=1}^n (x_i-\hat x)^2$$</span></p>
<p>Here, we want to find the value of <span class="math-container">$\hat x$</span> which minimizes <span class="math-container">$f(\hat x)$</span>. Now, even though there are multiple variables of this... |
2,251,998 | <p>Here is a question that I am working on:</p>
<blockquote>
<p>If $G$ is a group such that <em>every</em> non-identity element has order $2$, show that $G$ is abelian (commutative).</p>
</blockquote>
<p><strong>My attempt</strong></p>
<p>Suppose that for all $a \in G$, we have $$a^2 = Id$$</p>
<p>My goal is to s... | Carl Schildkraut | 253,966 | <p>As far as I know there isn't a system like OEIS, but as vrugtehagel said, WolframAlpha is pretty good. Also, if you know it's algebraic, it might be feasible to search for a minimal polynomial with some computer algebra system.</p>
|
1,162,161 | <p>A patient would like to take a test to determine if he has a nasty disease. Let the variable A denote that
the patient has the disease and the variable B denote a positive test. The following assumptions apply:
• The probability that the test is positive given the patient has the disease is 99%.
• The probability th... | cactus314 | 4,997 | <p>Using geometric series we know exactly what your series should be</p>
<p>$$ \sum_{n=1}^\infty n x^{n-1} = \frac{1}{(1-x)^2}$$</p>
<p>Uniform convergence asked about the remaining terms and you can try to give uniform estimate</p>
<p>$$ \sum_{n=N+1}^\infty n x^{n-1} = x^N \left( \frac{1}{(1-x)^2}
+ \frac{N}{1-x} \... |
189,074 | <p>the function f defined by $f(x)=(x^3+1)/3$ has three fixed points say α,β,γ where
$-2<α<-1$, $0<β<1$, $1<γ<2$.
For arbitrarily chosen $x_{1}$, define ${x_{n}}$ by setting $x_{n+1}=f(x_{n})$
If $α<x_{1}<γ$, prove that $x_{n}\rightarrow β$ as $n \rightarrow \infty$</p>
<p>I think I must prove ... | Robert Israel | 8,508 | <p>Hints: Since there are only those three fixed points, we either have $f(x) < x$ for all $x$ with $\alpha < x < \beta$ or $f(x) > x$ for all those $x$. Check one point $x$ in that interval to see which it is. Similarly for $\beta < x < \gamma$. </p>
<p>Since $f$ is an increasing function, if $... |
3,495,852 | <p>I couldn't find an example or explanation why the following sentence is correct </p>
<p>If a Transformation is linear, and vectors <span class="math-container">$u_1$</span>,<span class="math-container">$u_2$</span>,<span class="math-container">$u_3$</span> are dependent then
<span class="math-container">$T(u_1)$</... | Kavi Rama Murthy | 142,385 | <p>It is not assumed that <span class="math-container">$(\theta_n)$</span>'s exist on the same space. It is rarely necessary to construct random variables on a particular space to prove theorems in Probability Theory. What Kallenberg wants is some probability space on which random variables <span class="math-container"... |
70,603 | <p>We were shown in class this next calculation: (Here, $V_n(RB^n)$ is the volume of an $n$ dimensional ball of radius $R$, likewise $S_{n-1}$ is the surface area of the $n$ dimensional sphere in $\mathbb{R}^n$. $rS^{n-1}$ denotes the $n$ dimensional sphere of radius $r$ and integrating $d\textbf{S}$ means a surface in... | Ross Millikan | 1,827 | <p>As an example, if we let $n=3$, the area of a two-sphere is proportional to $r^2$, not $r^3$. You haven't changed the $dr$ integral, which still goes from $0$ to $R$.</p>
|
475,863 | <p>Let $f,P,Q$ three analytic functions. Here $P$ is a polynomial.</p>
<p>I want to solve this equation: $$f(s)=P(s)\exp(Q(s)).$$</p>
<p>The unknown here are $P, Q$ and $f$ is known. </p>
| Daron | 53,993 | <p>$P(s)$ is an arbitrary polynomial. Then $Q(s) = \log(\frac{f(s)}{P(s)})$.</p>
|
1,146,802 | <p>Often seen similar systems of equations. Usually consider such systems in which decisions no. Such as there. <a href="https://math.stackexchange.com/questions/1146460/is-there-a-b-c-d-in-mathbb-n-so-that-a2b2-c2-b2c2-d2">Is there $a,b,c,d\in \mathbb N$ so that $a^2+b^2=c^2$, $b^2+c^2=d^2$?</a></p>
<p>I think it wo... | individ | 128,505 | <p>If you solve the system of equations:</p>
<p>$$\left\{\begin{aligned}&a^2+b^2+c^2=q^2\\&c^2+q^2=w^2\end{aligned}\right.$$</p>
<p>When the standard approach solution and using a replacement.</p>
<p>$$p=9t^2-10tk+5k^2$$</p>
<p>$$s=5t^2-10tk+9k^2$$</p>
<p>$$x=7t^2-10tk+7k^2$$</p>
<p>$$y=4(t^2-k^2)$$</p>
... |
1,535,731 | <p>I am currently self-learning about matrix exponential and found that determining the matrix of a diagonalizable matrix is pretty straight forward :).</p>
<p>I do not, however, know how to find the exponential matrix of a non-diagonalizable matrix.</p>
<p>For example, consider the matrix
$$\begin{bmatrix}1 & 0... | Dr. Sundar | 1,040,807 | <p>Laplace Transforms approach is an useful for finding <span class="math-container">$e^{A t}$</span>, whether <span class="math-container">$A$</span> is diagonalizable or non-diagonalizable. It is a very useful method in Electrical
Engineering.</p>
<p>Basically, we use the formula
<span class="math-container">$$
e^{A ... |
3,668,101 | <p>I know that if <span class="math-container">$n \bmod k \le k-1$</span> then this sum is converge then it has finite sum, I just guess it's <span class="math-container">$\ln(k)$</span> because when <span class="math-container">$k=1$</span> sum is <span class="math-container">$0=ln(1)$</span>. I really don't know how ... | xpaul | 66,420 | <p>Note that <span class="math-container">$n=mk+r$</span>, <span class="math-container">$r=0,1,\cdots k-1$</span>. So
<span class="math-container">\begin{eqnarray}
&&\sum_{n=1}^{\infty }\frac{(n)\mod(k)}{n(n+1)}=\sum_{m=0}^{\infty }\sum_{r=1}^{k-1}\frac{(mk+r)\mod(k)}{(mk+r)(mk+r+1)}\\
&=&\sum_{m=0}^{\i... |
3,668,101 | <p>I know that if <span class="math-container">$n \bmod k \le k-1$</span> then this sum is converge then it has finite sum, I just guess it's <span class="math-container">$\ln(k)$</span> because when <span class="math-container">$k=1$</span> sum is <span class="math-container">$0=ln(1)$</span>. I really don't know how ... | Gary | 83,800 | <p>Continuing xpaul's answer
<span class="math-container">\begin{align*}
& \sum\limits_{r = 1}^{k - 1} {\frac{r}{k}\left[ {\psi ^{(0)} \left( {\frac{{r + 1}}{k}} \right) - \psi ^{(0)} \left( {\frac{r}{k}} \right)} \right]} = \sum\limits_{r = 1}^{k - 1} {\frac{r}{k}\psi ^{(0)} \left( {\frac{{r + 1}}{k}} \right)} -... |
1,951 | <p>In <a href="https://matheducators.stackexchange.com/a/1949/704">this answer</a>, user <a href="https://matheducators.stackexchange.com/users/942/robert-talbert">Robert Talbert</a> stated that</p>
<blockquote>
<p>There are some amazing things you can do pedagogically with clickers.</p>
</blockquote>
<p>I'd like t... | Brian S | 328 | <p>Several times in my high school Computer Science classes, my teacher used devices like this to administer quizzes <em>in parallel</em> with a lecture. That is, as he covered material, he would put up quiz questions on a projector, and we would answer with the clickers. The questions in general tested comprehension o... |
3,258,642 | <blockquote>
<p>If the roots of quadratic equation <span class="math-container">$$x^2 − 2ax + a^2 + a – 3 = 0$$</span>
are real and less than <span class="math-container">$3$</span>, find the range of <span class="math-container">$a$</span>.</p>
</blockquote>
<p>The roots are <span class="math-container">$a... | nonuser | 463,553 | <p>Since <span class="math-container">$x_1,x_2<3$</span> we have <span class="math-container">$3-x_i>0$</span> so their product is positive:<span class="math-container">$$ 0<9-3(x_1+x_2)+x_1x_2$$</span>
thus <span class="math-container">$$0<9-6a+a^2+a-3 = a^2-5a+6=(a-3)(a-2)$$</span></p>
<p>So <span class=... |
459,374 | <p>Let $X$ be the random variable which denotes the number of times a die has been rolled till each side has appeared. The order does not matter.
We are trying to find $E[X]$.</p>
<p>Let $X_i$ be a random variable which denotes how many times a die has to be rolled till side i has appeared.</p>
<p>So,</p>
<p>$$E[X]=... | ronno | 32,766 | <p>Let the vertices be <span class="math-container">$a_1,a_2,a_3$</span>. Then any <span class="math-container">$x,y \in T$</span> can be written as convex combinations of these, so let <span class="math-container">$x = \sum \lambda_i a_i$</span> and <span class="math-container">$y = \sum \mu_i a_i$</span> with <span c... |
459,374 | <p>Let $X$ be the random variable which denotes the number of times a die has been rolled till each side has appeared. The order does not matter.
We are trying to find $E[X]$.</p>
<p>Let $X_i$ be a random variable which denotes how many times a die has to be rolled till side i has appeared.</p>
<p>So,</p>
<p>$$E[X]=... | achille hui | 59,379 | <p>Given any triangle $T$, in fact any bounded closed convex subset of $\mathbb{R}^2$.
The map</p>
<p>$$T^2 \ni (x,y) \mapsto |x-y|^2 \in \mathbb{R}$$</p>
<p>is a continuous function on $T^2$ bounded from above. Since $T^2$ is compact, the map achieves its maximum on some $(u, v) \in T^2$. i.e.</p>
<p>$$\sup \{\;|x-... |
2,558,870 | <p>Suppose $f:[0,1]\to \mathbb{R}$ is uniformly continuous, and $(p_n)_{n\in\mathbb{N}}$ is a sequence of polynomial functions converging uniformly to $f$.</p>
<p>Does it follow that $\mathcal{F}=\{p_n\mid n\in\mathbb{N}\}\cup \{f\}$ is equicontinuous?</p>
<p>Also, if $C_n$ are the Lipschitz constants of the polynomi... | QED | 91,884 | <p>If $(A-pI)=-p(I-A/p)$ is invertible then $[-p(I-A/p)]^{-1}=-\frac{1}{p}\left[I-\frac{A}{p}+\frac{A^2}{p^2}+\frac{A^3}{p^3}+\cdots\right]$. If $v$ is an eigenvector of $A$ corresponding to the eigenvalue $q$, then $(A-pI)^{-1}v=-\frac{1}{p}\left[1-\frac{q}{p}+\frac{q^2}{p^2}+\frac{q^3}{p^3}+\cdots\right]v=(q-p)^{-1}v... |
847 | <p>Apologies in advance if this is obvious.</p>
| moonface | 513 | <p>Not a satisfying argument: We can, first of all, find a basis in which the entries lie in some algebraic number field <span class="math-container">$K$</span>. Let <span class="math-container">$\mathcal{O}$</span> be the ring of integers of <span class="math-container">$K$</span>.
Then there is a locally free <span c... |
2,569,557 | <p>I'm still confused by the use of $\Rightarrow$ in (ε,δ)-definition of limit. <br/>
Take for example the definition of $\underset{x\rightarrow x_{0}}{\lim}f\left(x\right)=l$ :<br/></p>
<blockquote>
<p>$$\forall\varepsilon>0,\;\exists\delta>0\quad\mathrm{such\:that\quad}\forall x\in\mathrm{dom}\,... | user | 505,767 | <p>You can think about it in this way:</p>
<blockquote>
<p>first set <span class="math-container">$\epsilon$</span> and then you have to find <span class="math-container">$\delta$</span> such that the inequality:</p>
<p><span class="math-container">$$\left|f\left(x\right)-l\right|<\varepsilon$$</span></p>
<p>is sati... |
2,669,292 | <p>$g_n(x) = \frac{\ln(1+x/n)}{n}$ on $\mathbb{R}$.
Don't they all converge to 0?</p>
| Ian | 83,396 | <p>Sure (though the domain can't be all of $\mathbb{R}$, presumably you intend $[0,\infty)$ or maybe $(-n,\infty)$). But for any fixed $n$, the whole thing still goes to infinity as $x$ goes to infinity, so $\| g_n - g \|_\infty=+\infty \not \to 0$. </p>
<p>On the other hand the convergence is uniform on compact subse... |
4,481,314 | <p>This is an exercise in Tristan Needham's <em>Visual Differential Geometry and Forms</em>. He uses the term <em>ultimate equality</em> to mean roughly the same thing as first order approximation, which he says is motivated by Newton's Principia. The book is dedicated to Needham's longtime personal friend Roger Penr... | Z Ahmed | 671,540 | <p><span class="math-container">$\lim_{x\rightarrow 0 } \sin\left(\dfrac{\pi}{x}\right)$</span></p>
<p>Think of two sequences <span class="math-container">$x_n=\frac{1}{(2n+1/2)}$</span> and <span class="math-container">$x'_n=\frac{1}{(n+1)}$</span>
such that when <span class="math-container">$n \rightarrow \infty$</sp... |
3,995,492 | <p>I have no clue how to do this, I manage to get I get that <span class="math-container">$11^{36} \equiv 1 \hspace{0.1cm} \text{mod} (13)$</span> but I can't get anywhere from there.</p>
| Dietrich Burde | 83,966 | <p>If <span class="math-container">$11^{36}=1$</span> in <span class="math-container">$\Bbb Z/13$</span>, then <span class="math-container">$11^{35}=11^{-1}$</span>. But since <span class="math-container">$6\cdot 11=66=1$</span> in <span class="math-container">$\Bbb Z/13$</span>, we have <span class="math-container">$1... |
3,995,492 | <p>I have no clue how to do this, I manage to get I get that <span class="math-container">$11^{36} \equiv 1 \hspace{0.1cm} \text{mod} (13)$</span> but I can't get anywhere from there.</p>
| jacopoburelli | 530,398 | <p>What were your efforts ? However, if you proved that <span class="math-container">$11^{36} \equiv 1 \hspace{0.1cm} (13)$</span> then you have that <span class="math-container">$11^{35}\cdot 11 \equiv 1 \hspace{0.1cm}(13)$</span>. In this line it is written that <span class="math-container">$11^{35}$</span> is the in... |
3,695,127 | <p>Before the moderators close my question, I cant think of any starting approach to the question. </p>
<p>Another question of the similar type I am having trouble with is: 12 balls are distributed at random among 3 boxes. What is the probability that the first box will contain 3 balls?
For the second question I can f... | E. KOW | 443,898 | <p><span class="math-container">$S^1 \times S^1$</span> is homeomorphic to <span class="math-container">$\left[-1, 1\right]^2 / \sim$</span> where <span class="math-container">$\left(x,-1\right) \sim \left(x, 1\right), \left(-1,y\right) \sim \left(1, y\right)$</span>. There is a deformation retract of <span class="math... |
366,249 | <p>$3xy^2dx+2x^3dy$
where is the boundary of the region between the circles $x^2+y^2=25$ and $x^2+y^2=64$ having positive orientation.</p>
<p>Not quite sure how to evaluate this...</p>
| Mikasa | 8,581 | <p>Hints:</p>
<p>First of all set $(x+1)=t$ to have $t^2y''+ty'-y=0, ~~t\neq -1$ instead. Then, solve the auxiliary equation $$am^2+(b-a)m+c=0$$ wherein </p>
<p>$a=1$ (the coffecient of $t^2$), </p>
<p>$b=+1$ (the cofficient of $t$ above) and $c=-1$</p>
<p>for finding the possible $m$'s.</p>
<p>If $m_1,m_2$ are di... |
1,274,317 | <blockquote>
<p>Let $f:R \longrightarrow S$ a surjective ring homomorphism. Is the inverse image $f^{-1}(M)$ a maximal left ideal of $R$ for any maximal
left ideal $M$ of $S$?</p>
</blockquote>
<p><strong>Comments:</strong> I tied something like this: if $M$ is maximal then</p>
<p>$M \neq S$ and if $J$ is a le... | user 1 | 133,030 | <p>Hint.<br>
Let $K:=ker f$. Then $R/ K \cong S$. So maximal ideals of $S$ correspond to maximal ideals of $R/ K$. On the other hand maximal ideals of $R/K$ correspond to <em>those maximal ideals of $R$ which contain $K$</em>.</p>
|
4,309,812 | <p>Recently I knew about <a href="https://en.m.wikipedia.org/wiki/Heron%27s_formula" rel="nofollow noreferrer">Heron's formula</a> for the area of some triangle, and its generalizations to quadrilaterals by Bretschneider's formula. According to Wikipedia there are also generalizations for pentagons and hexagons inscrib... | Sidvhid Hsinynjad | 866,977 | <p>Hint:</p>
<p>You may use the following identity</p>
<p><span class="math-container">$$(y+x)^2-(y-x)^2=4xy$$</span></p>
|
134,987 | <blockquote>
<p>$$3x^2 + 2y^4 = z^4$$</p>
</blockquote>
<p><em>How do I solve this??</em> I would like to use so-called "elementary number theory", not abstract algebra (e.g. $\mathbb{Z} ( \sqrt d)$) or elliptic curves.</p>
<p>Note: I'm not asking <em>what</em> the solutions are, but rather <em>how</em> to find the... | Dayo Adeyemi | 29,565 | <p>Supposing we did have a solution lets consider the equation modulo $3$, since a square (hence a fourth power) must be congruent to $0$ or $1$ so the LHS is congruent to $0$ or $2$ and the RHS is $0$ or $1$ we see $3$ must divide both $y$ and $z$ thus $3^3$ must divide $x^2$ so $3^2$ divides $x$ hence $3^4$ divides t... |
134,987 | <blockquote>
<p>$$3x^2 + 2y^4 = z^4$$</p>
</blockquote>
<p><em>How do I solve this??</em> I would like to use so-called "elementary number theory", not abstract algebra (e.g. $\mathbb{Z} ( \sqrt d)$) or elliptic curves.</p>
<p>Note: I'm not asking <em>what</em> the solutions are, but rather <em>how</em> to find the... | user55514 | 55,514 | <p>The Chaz said:</p>
<p>""- use <a href="http://www.alpertron.com.ar/QUAD.HTM" rel="nofollow">Alpern's solver</a> - which seemed to indicate that there are no solutions (though I might have made an illegal substitution, so to speak) </p>
<p>I was able to identify $A = 6, B = 3, C = 6$ as solutions of $ \ 3A + 2B \... |
4,187,932 | <p>Is there a general <strong>algebraic</strong> form to the integral <span class="math-container">$$\int_{k_1}^{k_2} x^2 e^{-\alpha x^2}dx?$$</span> I know that if this integral is an improper one, then the integral can be calculated quite easily (i.e. is a well known result). However, when these bounds are not impose... | Kyle Miller | 172,988 | <p>Expanding @Intelligenti pauca's comment, the power series for <span class="math-container">$\sqrt{x^2+y^2}$</span> in terms of <span class="math-container">$y$</span> centered at <span class="math-container">$0$</span> starts as
<span class="math-container">$$\lvert x\rvert + \frac{y^2}{2\lvert x\rvert} + \frac{y^4}... |
2,746,153 | <p>Assume $m\ \mathrm{and}\ n\ \mathrm{are\ two\ relative\ prime\ positive\ integers.}$</p>
<p>Given $x \equiv a\ \pmod m$ and $x \equiv a\ \pmod n$.</p>
<p>Prove that $x \equiv a\ \pmod {mn}\ \mathrm{by\ using\ Chinese\ Remainder\ Theorem}.$<br/></p>
<p>And I did the following:
<br>
$$ \mathrm {M_1 = }\ n\ \ and\... | Joffan | 206,402 | <p>We have:<br>
$n \mid (x-a)$, and<br>
$m \mid (x-a)$</p>
<p>and $n$ and $m$ have no common factors, so<br>
$nm \mid (x-a)$</p>
|
122,770 | <p>Given that $A$ is an open set in $\mathbb R^n$ and $f:A \to \mathbb R^n$ is differentiable, and its derivative is non-singular at every point in $A$, prove that $f(A)$ is open in $\mathbb R^n$</p>
<p>Note $f$ is differentiable, <em>not</em> continuously differentiable. </p>
| Leandro | 633 | <p>The answer is given in this excellent post by Terence Tao </p>
<p><a href="https://terrytao.wordpress.com/2011/09/12/the-inverse-function-theorem-for-everywhere-differentiable-maps/" rel="nofollow">https://terrytao.wordpress.com/2011/09/12/the-inverse-function-theorem-for-everywhere-differentiable-maps/</a></p>
|
3,045,677 | <p>I am trying to create a program for a school project where I need to plot points of a triangle given all 3 side lengths (a=10, b=20, c=30).
I tried the solution from the other topic and it didn't work since the result produced was C(20,0) and that cant be right since one of the sides is already placed on the axis a... | fleablood | 280,126 | <blockquote>
<p>I am trying to create a program for a school project where I need to plot points of a triangle given all 3 side lengths (a=10, b=20, c=30).</p>
</blockquote>
<p>By the triangle inequality <span class="math-container">$a + b > c$</span>. In this case you have <span class="math-container">$a + b =c... |
2,055,878 | <p>If$$ x^2+x+1=0$$ find the value of $$8x^{282}+1799x^{183}+87x^{51}+124x^{-3}+1$$</p>
<p>Solving this equation gives imaginary solutions. </p>
<p>Is there an easy way to do this ?</p>
| Dietrich Burde | 83,966 | <p>With $x^3-1=(x-1)(x^2+x+1)$ we can use $x^{3}=1$ and then reduce the polynomial to $f(x)=8+1799+87+124+1=2019$. This is almost $3$ years ahead, though. </p>
|
2,206,247 | <p><strong>Question:</strong> Consider the following non linear recurrence relation defined for $n \in \mathbb{N}$:</p>
<p>$$a_1=1, \ \ \ a_{n}=na_0+(n-1)a_1+(n-2)a_2+\cdots+2a_{n-2}+a_{n-1}$$</p>
<p>a) Calculate $a_1,a_2,a_3,a_4.$</p>
<p>b) Use induction to prove for all positive integers that:</p>
<p>$$a_n=\dfra... | Deepak | 151,732 | <p>Yes it's true, and it's easily proven by squaring both sides. This operation is allowed (without worrying about the direction of the inequality) because all terms are non-negative.</p>
|
1,238,210 | <p>How we can solve that $\lim _{_{x\rightarrow \infty }}\int _0^x\:e^{t^2}dt$ ?</p>
<p>P.S: This is my method as I thought:
$\int _0^x\:\:e^{t^2}dt>\int _1^x\:e^tdt=e^x-e$ which is divergent, so all your answers, helped me to think otherwise, maybe my method help something else :D</p>
| davidlowryduda | 9,754 | <p>This function diverges extremely fast. Notably, $e^{t^2}$ is monotone increasing with limit $\infty$ as $t \to \infty$. Thus your integral diverges (and it get very, very large very, very quickly).</p>
|
64,643 | <blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="https://math.stackexchange.com/questions/4467/a1-2-is-either-an-integer-or-an-irrational-number">$a^{1/2}$ is either an integer or an irrational number</a> </p>
</blockquote>
<p>I know how to prove $\sqrt 2$ is an irrational number. Who can tell ... | lhf | 589 | <p>A well-known variant of the usual proof may be clearer. If $\sqrt{3}=\frac{a}{b}$, then $a^2=3b^2$. Recall the Fundamental Theorem of Arithmetic and consider the exponent of $3$ in the prime factorization of both sides. On the left you have an even exponent. On the right you have an odd exponent, contradiction. This... |
3,244,866 | <p>How can i prove that <span class="math-container">$$2^n\not \in O(n^2)$$</span> by formal definition and not using limits?</p>
<p>With:</p>
<p><a href="https://i.stack.imgur.com/id9gx.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/id9gx.png" alt="enter image description here"></a></p>
| J. W. Tanner | 615,567 | <p><strong>Hint:</strong></p>
<p>for <span class="math-container">$n>2,$</span></p>
<p><span class="math-container">$
n^3+\dfrac{n-1}2<\sqrt{n^6+n^4+1}<n^3+\dfrac n2.$</span></p>
|
4,249,573 | <p>I'm studying Set Theory in my own, with Goldrei's textbook. The chapter I'm reading is on order-isomorphism and well-ordering. One exercise asks (i) to argue that, in general, a collection of well-ordered sets order-isomorphic to a given well-ordered set is a proper class (rather than a set). The proof, IMHO, is fai... | Community | -1 | <p>The class of the empty well-order is the only such class that is also a set. For every other class <span class="math-container">$[(X,\le)]$</span> the class-function sending <span class="math-container">$(Y,\preceq)\mapsto Y$</span> would be onto the class of sets that are in bijection with <span class="math-contain... |
208,615 | <p>Maybe I'm not using the best programming practices in Mathematica, but my notebooks usually contain a mixture of "definitions" and "computations" with these definitions. Say,</p>
<pre><code>f[x_] := x^2 (* Definition *)
f[10] (* Computation *)
g[x_] := f[x] - 1/f[x] (* Definition *)
g[100] (* Computation *)
</code>... | Carl Woll | 45,431 | <p><strong>Update</strong></p>
<p><em>(I added a gray background when in the "Initialization" screen style to remind the user to switch back after evaluating code)</em></p>
<p>Here's a stylesheet solution. The basic idea is to add an "Initialization" screen style environment (like "Working", "SlideShow" and "Presenta... |
308,329 | <p>I need help with writing $\sin^4 \theta$ in terms of $\cos \theta, \cos 2\theta,\cos3\theta, \cos4\theta$.</p>
<p>My attempts so far has been unsuccessful and I constantly get developments that are way to cumbersome and not elegant at all. What is the best way to approach this problem?</p>
<p>I know that the answe... | Ron Gordon | 53,268 | <p>Write</p>
<p>$$\sin^4{\theta} = \left ( \frac{e^{i \theta} - e^{-i \theta}}{2 i} \right )^4$$</p>
<p>and use the binomial theorem.</p>
<p>$$\begin{align}\left ( \frac{e^{i \theta} - e^{-i \theta}}{2 i} \right )^4 &= \frac{1}{16} (e^{i 4 \theta} - 4 e^{i 2 \theta} + 6 - 4 e^{-i 2 \theta} + e^{-i 4 \theta}) \\ ... |
1,918,408 | <p>Let $C(B)$ be a $\infty$-order polynomial: $$ C(B) = \sum_{k=0}^\infty \alpha_k B^k$$</p>
<p>Show that $$C(B) = C(1) + (1-B)C^*(B)$$ where $C^*(B)$ is a another $\infty$-order polynomial.</p>
<p>This comes from the prove of the Engle-Granger Representation Theorem in their <a href="http://www.uta.edu/faculty/crowd... | Community | -1 | <p>For instance, when $\inf A>0$ and $\sup B<0$. Since all the products will be negative, you want to minimise the absolute values of each factor.</p>
|
39,423 | <ul>
<li><p>case1</p>
<pre><code>Options[f] = {"t" -> "0"};
f[___, OptionsPattern[]] := StringReplace["content", "t" :> OptionValue["t"]]
f[]
(*
con0en0
*)
</code></pre></li>
<li><p>case2</p>
<pre><code>rule = {"t" -> OptionValue["t1"]};
Options[gg] = {"t1" -> "T1", "t2" -> "1"};
gg[___, OptionsPat... | Chris Degnen | 363 | <p>One solution is to keep <code>OptionValue</code> inside the function :-</p>
<pre><code>rule = {"t" -> "t1"};
Options[gg] = {"t1" -> "T1", "t2" -> "1"};
gg[___, OptionsPattern[]] := StringReplace["content", #1 -> OptionValue[#2] & @@@ rule]
gg[1]
</code></pre>
<blockquote>
<p>"conT1enT1"</p>
</b... |
1,807,479 | <blockquote>
<p>I recently took a test and was confused about a question. I feel that
the answer is B. Could anyone please elucidate it. Thanks!</p>
</blockquote>
<p>The point $(−4, 3)$ is on the terminal side of angle $\theta$ as sketched below. Find $\cos\theta$.</p>
<p><a href="https://i.stack.imgur.com/BiOiI.... | Vladhagen | 79,934 | <p>It is pretty easy to show (and might even be given in most textbooks) that
$$\cos(\pi - \alpha) = -\cos(\alpha)$$ for any angle $\alpha$. </p>
<p>Note that the angle opposite $\theta$ in your diagram is found by computing $\pi-\theta$. Since it is pretty easy to get that $\cos(\pi-\theta) = \frac{-4}{5}$ from the d... |
108,010 | <p>It is not necessarily true that the closure of an open ball $B_{r}(x)$ is equal
to the closed ball of the same radius $r$ centered at the same point $x$. For a quick example, take $X$ to be any set and define a metric
$$
d(x,y)=
\begin{cases}
0\qquad&\text{if and only if $x=y$}\\
1&\text{otherwise}
\end{case... | Lyapunov | 25,683 | <blockquote>
<p>Let $(X,\|\cdot\|)$ be a normed linear space. Then $\overline{B_1(0)}=\bar{B}_1(0)$.</p>
</blockquote>
<p>Proof. Observe that $\overline{B_1(0)}$ is the smallest closed set containing $B_1(0)$ and $B_1(0)\subset \bar{B}_1(0)$, so trivially $\overline{B_1(0)}\subset\bar{B}_1(0)$. Now to show
$\bar{B}... |
1,987,317 | <blockquote>
<p>Prove that the system $$A^T A x = A^T b$$ always has a solution. The matrices and vectors are all real. The matrix $A$ is $m \times n$. </p>
</blockquote>
<p>I think it makes sense intuitively but I can't prove it formally.</p>
| Taha Akbari | 331,405 | <p>Sorry for bumping but I had a solution which I liked to share.</p>
<p>Using the <a href="https://en.wikipedia.org/wiki/Fundamental_theorem_of_linear_algebra" rel="nofollow noreferrer">fundamental theorem of linear algebra</a> We can decompose our matrix <span class="math-container">$b$</span> into two matrices <span... |
3,832,484 | <p>Title's all there is to say.
I'm very new to linear algebra and haven't wrapped my head around determinant rules yet.
Any help would be appreciated.</p>
| jacopoburelli | 530,398 | <p>Solution : <span class="math-container">$\det(A + A) = \det (2A) = 2^3\det(A)$</span> since <span class="math-container">$A$</span> is <span class="math-container">$3\times3$</span>, so <span class="math-container">$\det(A + A) = 2^3 4 = 32.$</span></p>
|
200,322 | <p>Is there a compact topological space $(X,\tau)$ such that for no cardinal $\kappa$ there is a surjective continuous map $e:\{0,1\}^\kappa \to X$? </p>
<p>(We assume that $\{0,1\}$ is endowed with the discrete topology, and $\{0,1\}^\kappa$ has the product topology.)</p>
| Włodzimierz Holsztyński | 8,385 | <p>I'd say that the simplest counter-example is the $1$-point compactification of a non-countable discrete space (of course any compactification of any non-countable discrete space would do).</p>
|
65,223 | <p>The least rational number greater than $\sqrt{2}$ that can be written as a ratio of integers $x/y$ with $y\le10^{100}$ can be found in a moment using a little Python program. Can anyone write a program that finds, in hours rather than centuries, the least rational greater than $\sqrt{2}$ of the form $x/y^2$ with $y^... | Victor Miller | 2,784 | <p>You should try the algorithms in Elkies' paper (from 2000) "Rational points near curves ..." <a href="http://arxiv.org/abs/math/0005139" rel="nofollow">http://arxiv.org/abs/math/0005139</a> . His idea is to cover the curve with a bunch of small rectangles, and use lattice basis reduction within each such region. H... |
658,078 | <p>I'm embarrassed to ask this question, but my child has the following homework question:</p>
<p>"Use absolute value to describe the relationship between a negative credit card balance and the amount owed."</p>
<p>I'm not sure for what it is they're looking. Clearly a <code>-$25</code> balance means you have <code>... | John Habert | 123,636 | <p>You have identified the identity element $e=-5$. To show that you have inverses, you need to prove that for every $a \in \mathbb{Z}$ there is some $b \in \mathbb{Z}$ such that $a*b = b*a = e$. Assuming you proved $*$ is commutative as mentioned in the comments, then it suffices to show there is a $b$ such that $a*b=... |
1,234,093 | <p>Given that $ e= \frac{a^2-b^2}{b^2} $ , and $L$ is the length of the perimeter, which equals $4aE(e, \pi/2)$, find the length of the perimeter up to $e^2$ in terms of $a$ and $b$.</p>
<p>How does one begin this?</p>
| Christian Blatter | 1,303 | <p>Note that the eccentricity of the ellipse is defined by $e:=\sqrt{a^2-b^2}/a$, whereby $a$ is the longer semiaxis.</p>
<p>A beginning: Use the parametrization
$$x=a\cos \phi,\quad y=b\sin\phi\qquad(0\leq\phi\leq2\pi)$$
in order to set up the integral for $L$. In the resulting expression substitute $b^2:=a^2(1-e^2)$... |
2,138,916 | <p>My question read: </p>
<p>Show that $S_{10}$ contains elements of orders $10,20,30$. Does it contain an element of order $40$? </p>
<p>I am not too sure what the question is asking. Would I have to explicitly write out all the permutations in $S_{10}$ first and then find the orders for all of them? </p>
<p>Update... | Chappers | 221,811 | <p>It's easy to see that your integral is the same as
$$ I(a) = \int_0^{\infty} \frac{\log{(x^2+a^2)}}{x^2+b^2} \, dx $$
Now, we can do the case $a=0$ fairly easily, by setting $x=b^2/y$:
$$ \begin{align}
I(0) &= 2\int_0^{\infty} \frac{\log{x}}{x^2+b^2} \, dx \\
&= 2\int_0^{\infty} \frac{\log{(b^2/y)}}{b^2/y^2+... |
59,567 | <p>I am looking for a way to add a legend showing the identity of various atoms (with different colours) to this picture. Any Clues?</p>
<pre><code>Import["ExampleData/1PPT.pdb", "Rendering" -> "BallAndStick"]
</code></pre>
<p><img src="https://i.stack.imgur.com/FSFoH.png" alt="enter image description here"></p>
| Jason B. | 9,490 | <p>To show the colors associated with atoms, use the <code>Automatic</code> setting for the PlotLegends option:</p>
<pre><code>MoleculePlot3D[
Molecule @ "1-(amino-dimethyl-silyl)-2,3,4,5,6-pentafluoro-benzene",
PlotLegends -> Automatic
]
</code></pre>
<p><a href="https://i.stack.imgur.com/8ugWk.png" rel=... |
3,878,723 | <blockquote>
<p>Find the value of <span class="math-container">$k$</span> if the curve <span class="math-container">$y = x^2 - 2x$</span> is tangent to the line <span class="math-container">$y = 4x + k$</span></p>
</blockquote>
<p>I have looked at the solution to this question and the first step is the "equate the... | Bernard | 202,857 | <p>It's rather simple: equating the functions yields the equation for the abscissæ of the intersection points of the parabola and the line, and the curve & the line are tangent if and only if this equation has a double root, i.e., as we have a quadratic equation, if and only if its (reduced) discriminant is <spa... |
317,547 | <p>Let $X$ and $Y$ have joint mass function</p>
<p>$f(j,k)=\frac {c(j+k)a^{j+k}}{j!k!}$, $j,k\geq 0$</p>
<p>where $a$ is a constant. Find $c$</p>
<p>This sum seems hard to to. How to complete this sum?</p>
| Ross Millikan | 1,827 | <p>Hint: It looks like $\frac c{j!k!} \frac d{da} a^{j+k+1}$ so binomial identities have a lot to say.</p>
|
145,785 | <p>Let $V$ be a $\mathbb{C}$-vector space of finite dimension. Denote its $d$-th symmetric power by $V^{\odot d}$. I am looking for a proof that $V^{\odot d}$ is generated by the elements $v^{\odot d}$ for $v\in V$. </p>
<p>A different way to look at it is the following: Consider the polynomial ring $R=\mathbb{C}[x_1,... | anon | 11,763 | <p>In <a href="https://math.stackexchange.com/a/137911/11763">my answer here</a> I note that symmetric tensors, as multilinear functionals, descend to linear maps on the symmetric power of the underlying vector space. I then reason that if we could show that $\mathrm{Sym}^n V$ is generated by $n$th powers of elements f... |
3,583,117 | <p>I would like to understand clearly why the following equality is true</p>
<p><span class="math-container">$P[X+Y \leq z] = E_Y[P[X+Y] \leq z | Y]]$</span></p>
<p>I wrote the left part of the equation as follows:</p>
<p><span class="math-container">$E_Y[P[X+Y] \leq z | Y]] = \sum_y y P[X+y \leq z]P(y)$</span></p>
... | drhab | 75,923 | <p>In situations like this where a probability is involved it is often handsome to convert these probabilities to expectations by means of: <span class="math-container">$$P[A]=\mathbb E[1_A]$$</span></p>
<p>Doing so we find by means of the general rule <span class="math-container">$\mathbb E[Z]=\mathbb E[\mathbb E[Z\m... |
24,186 | <p>Consider the code below:</p>
<pre><code>s = Solve[(3 - Cos[4*x])*(Sin[x] - Cos[x]) == 2, x, InverseFunctions -> True];
Select[s[[All, 1, 2]], Element[#, Reals] &]
</code></pre>
<p>In MMA 8.0, I get </p>
<pre><code>{-\[Pi], \[Pi]/2, \[Pi]}
</code></pre>
<p>but in MMA 9.0, I get an empty set { }</p>
<p>Ass... | bill s | 1,783 | <p>I know for me, I spent years using Matlab (or should I say, a toolbox-based computational system), where there is a trick called vectorization: you turn almost everything (ifs, ands, sums, products...) into simple vector commands. Doing this with your function is pretty natural since you've already defined the entri... |
1,579,521 | <p>Find the value of
<span class="math-container">$$ \iint_{\Sigma} \langle x, y^3, -z\rangle. d\vec{S} $$</span>
where <span class="math-container">$ \Sigma $</span> is the sphere <span class="math-container">$ x^2 + y^2 + z^2 = 1 $</span> oriented outward by using the divergence theorem.</p>
<p>So I calculate <span ... | N74 | 288,459 | <p>You need to evaluate: $$\int_0^1 3 p^4 \ dp\int_0^{2\pi} \sin^2\theta \ d\theta \int_0^\pi \sin^3\phi \ d\phi $$</p>
|
3,043,039 | <p>Let <span class="math-container">$f:(0,1) \to \mathbb{R}$</span> be a given function. Explain how the following definition is not equivalent to the definition of the limit</p>
<p><span class="math-container">$\lim\limits_{x \to x_0} f(x) = L$</span></p>
<p>of <span class="math-container">$f$</span> at <span class=... | hamam_Abdallah | 369,188 | <p>Your definition implies the known definition but the converse is not true.</p>
<p>Take <span class="math-container">$f$</span> defined by :</p>
<p><span class="math-container">$$f(x)=0 \;\; \text{ if } \;\; x<\frac 12$$</span>
and
<span class="math-container">$$f(x)=\color{red}{4}\;\; \text{ if } \;\; x\ge \fra... |
2,030,739 | <p>Find <span class="math-container">$\frac{d^2y}{dx^2}$</span> of:</p>
<blockquote>
<p><span class="math-container">$$4y^2+2=3x^2$$</span></p>
</blockquote>
<h2>My Attempt</h2>
<p>I attempted the probelm my first solving for the first derivative:</p>
<blockquote>
<p><span class="math-container">$8y*y'=6x$</span><br>
<... | Senex Ægypti Parvi | 89,020 | <p>For future reference,<br>
$F(x,y)=0$<br>
$\frac{d^2[F(x,y)]}{dx^2}=-\frac
{\frac{\partial^2 F}{\partial x^2}\left(\frac{\partial F}{\partial y}\right)^2
-2·\frac{\partial^2 F}{\partial x\partial y}·\frac{\partial F}{\partial y}·\frac{\partial F}{\partial x}
+\frac{\partial^2 F}{\partial y^2}\left(\frac{\partial F}{\... |
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