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 |
|---|---|---|---|---|
3,071,076 | <p>Let <span class="math-container">$ABC$</span> be an acute angled triangle whose inscribed circle touches <span class="math-container">$AB$</span> and <span class="math-container">$AC$</span> at <span class="math-container">$D$</span> and <span class="math-container">$E$</span> respectively. Let <span class="math-con... | Dr. Richard Klitzing | 518,676 | <p>Let
<span class="math-container">$$A=(0,\ a)\\
B=(-b,\ 0)\\
C=(b,\ 0)$$</span>
and thus
<span class="math-container">$$Z=(0,\ 0)$$</span>
then we get
<span class="math-container">$$\tan(\angle ABC)=\frac ab$$</span>
and thus
<span class="math-container">$$\tan(\frac 12\ \angle ABC)=\frac{a/b}{1+\sqrt{1+a^2/b^2}}=\f... |
3,587,387 | <p>Assume draw that you draw a card from a standard deck.Find the probability of drawing a heart Given that your drew a face card (JQK) Using probability formulas how do I figure this out
Given in this equations mean what exactly??</p>
| fleablood | 280,126 | <p>I think the proposition/Corollary will help you that:</p>
<blockquote>
<p>Proposition: Every subset of a finite set is finite.</p>
<p>Corolary: If <span class="math-container">$A$</span> is an infinite subset of <span class="math-container">$B$</span> then <span class="math-container">$B$</span> is also infinite</p... |
2,957,611 | <p>Let <span class="math-container">$A=\{(x_n)_{n\in\mathbb{N}}\in\mathbb{R}^\mathbb{N}|\exists M\in\mathbb{N} ,\forall n>M, x_n=0 \}\subset\mathbb{R}^\mathbb{N}$</span>, series of real numbers that are zero from some point forward.</p>
<p>Let <span class="math-container">$X$</span> be <span class="math-container">... | Scientifica | 164,983 | <p>That's right. If <span class="math-container">$x=ys+(1-y)r$</span> then <span class="math-container">$E(x^2)=E((ys+(1-y)r)^2)$</span>. You can expand this and use properties like: if <span class="math-container">$x$</span> is a random variable and <span class="math-container">$a,b\in \mathbb R$</span> then <span cla... |
3,022,921 | <p>If 6 divides x and 8 divides x how do you deduce 24 divides x</p>
| NL1992 | 621,833 | <p>In general, this is due to prime factorisation. If <span class="math-container">$6, 8$</span> both divide <span class="math-container">$x$</span>, then <span class="math-container">$3$</span> divides <span class="math-container">$x$</span> (as <span class="math-container">$3$</span> divides <span class="math-contain... |
978,392 | <p>Assume we have a vector $u= (u_1.u_2, u_3) \in R^3$ </p>
<p>My problem is to find vectors $\vec w, \vec v$ such that $u= v \times w$
All vectors should be orthonormal. </p>
<p>If $u= (u_1, u_2, u_3)$ ,is there a way to express these vectors $\vec w, \vec v$ with respect to $\vec u$. </p>
<p>If there is, I would l... | Travis Willse | 155,629 | <p>Pick any vector $v$ orthonormal to $u$; one way to do this is to pick any vector $v'$ not parallel to $u$ and then apply the <a href="http://en.wikipedia.org/wiki/Gram%E2%80%93Schmidt_process" rel="nofollow">Gram-Schmidt algorithm</a> to the pair $(u, v')$. Then, set $w := u \times v$, so that $(u, v, w)$ is an orie... |
978,392 | <p>Assume we have a vector $u= (u_1.u_2, u_3) \in R^3$ </p>
<p>My problem is to find vectors $\vec w, \vec v$ such that $u= v \times w$
All vectors should be orthonormal. </p>
<p>If $u= (u_1, u_2, u_3)$ ,is there a way to express these vectors $\vec w, \vec v$ with respect to $\vec u$. </p>
<p>If there is, I would l... | Gerry Myerson | 8,269 | <p>If $u_1=0$, let $v=(1,0,0)$ and let $w=u\times v$. </p>
<p>If $u_2=0$, let $v=(0,1,0)$, etc. </p>
<p>Otherwise, let $v=(u_2,-u_1,0)/\sqrt{u_1^2+u_2^2}$, etc. </p>
|
2,203,907 | <p>I am trying to show that:
$$\mathcal{L}\{erfc( \frac{k}{2\sqrt t})\} = \frac{1}{s}e^{-k\sqrt s}$$
The hint given for this question is the Laplace Transform of an integral (from convolution):
$$\mathcal{L}\{\int_{0}^{t}f(u) \, du \} = \frac{1}{s} \mathcal{L}\{f(u)\} \tag{1}$$</p>
<p>I have read in a different text t... | Jack D'Aurizio | 44,121 | <p>With the definition</p>
<p>$$ \text{erfc}\left(\alpha\right)=\frac{2}{\sqrt{\pi}}\int_{\alpha}^{+\infty} e^{-x^2}\,dx \tag{1}$$
we have:
$$ \text{erfc}\left(\frac{k}{2\sqrt{t}}\right)=\frac{2}{\sqrt{\pi}}\int_{\frac{k}{2\sqrt{t}}}^{+\infty}e^{-x^2}\,dx =\frac{1}{\sqrt{\pi}}\int_{\frac{k^2}{2t}}^{+\infty}\frac{e^{-x... |
3,416,019 | <p>If a function <span class="math-container">$f:A\longrightarrow B\times C, f(x)=(g(x),h(x))$</span> is injective, does it imply that <span class="math-container">$h(x)$</span> and <span class="math-container">$g(x)$</span> are also injective? I think this is straight forward but just want to confirm. </p>
<p>Suppose... | fleablood | 280,126 | <p>It's not a contradiction. If <span class="math-container">$g(x) = g(y)$</span> but <span class="math-container">$h(x) \ne h(y)$</span> then <span class="math-container">$f(x)= (g(x),h(x)) \ne (g(x), h(y)) = f(y)$</span>.</p>
<p>That's possible.</p>
<p>.....</p>
<p>Example: Let <span class="math-container">$g(x)... |
2,113,596 | <p>Questions with likely obvious answers, but I don't have the required intuition to go with the flow.</p>
<p>Consider $a+be^x + ce^{-x} = 0$. To solve it for the constants, we can try out different values of $x$ to get a system of $3$ equations and simply use a calculator (given technique). Why are we allowed to stic... | scott | 330,966 | <p>Your statement that an increase of one degree Fahrenheit is equivalent to $\frac{5}{9}$ degree Celsius is accurate. The two temperature scales have a proportional relationship.</p>
<p>The equation you provide can be thought of as a function. You input a number that represents the temperature in Fahrenheit, and the ... |
2,801,406 | <blockquote>
<p>Find the coordinates of the points where the line tangent to the curve $$x^2-2xy+2y^2=4$$ is parallel to the $x$-axis, given that $$\frac{dy}{dx}=\frac{y-x}{2y-x}$$</p>
</blockquote>
<p>By letting $dy/dx = 0$ I get $y=x$ which is no help... what do I do?</p>
<p>Thanks</p>
| Henno Brandsma | 4,280 | <p>All points on the curve obey the equation $x^2 - 2xy + 2y^2 = 4$. Having $y=x$ when $\frac{dy}{dx} = 0$ reduces this to $x^2 - 2x^2 + 2x^2 = 4$ or $x^2= 4$ or $x=2$ or $x=-2$, which you can find the possible $y$-coordinates for by using the equation again (substitute $x=2$ and solve for $y$, then the same for $x=-2$... |
3,940,447 | <p>Disclaimer: I believe this proof is wrong, but I'm asking because I can't find what's wrong with it, which means I must have some basic misunderstanding of the concepts involved.</p>
<p>First, some definitions. Recursively, we define an ordinal <span class="math-container">$\alpha$</span> to be <span class="math-con... | ForeignVolatility | 129,455 | <p>Let <span class="math-container">$X$</span> be the number of required children. Notice that, after the first child is born, the remaining number of children follows a Geometric distribution with a parameter depending on the first child's gender.</p>
<p>Let <span class="math-container">$G$</span> denote the gender of... |
1,435,590 | <p>Suppose I have a statement like this:</p>
<p>(~p ^ ~q) V (p ^ q)</p>
<p>If I understand this correctly, I can apply the law to both sides separately while leaving the OR in the middle intact. Leaving this:</p>
<p>(p V q) V (~p V ~q)</p>
<p>Is this valid? (as opposed to taking the negation of the entire statement... | Fly by Night | 38,495 | <p>The usual way to do this is to use the <a href="https://en.wikipedia.org/wiki/Factor_theorem" rel="nofollow">Factor Theorem</a>.</p>
<p>A polynomial $\mathrm{f}(x)$ is divisible by $(x-a)$ if $\mathrm{f}(a)=0$.</p>
<p>In your case, you have $\mathrm{f}(x) = x^3 - x^2 - 5x - 3$, and you need to find an $a$ for whic... |
388,225 | <p>If we have a random graph $G \in g(n,\frac{1}{2})$ how do we show that the expected number of edges is $\frac{1}{2} {{n}\choose{2}}$</p>
<p>Thanks in advance</p>
| Community | -1 | <p>HINT: You have $^nC_2$ edges with probability of connecting = $\frac{1}{2}$. So the result.</p>
|
2,694,525 | <p>I came across this exercise</p>
<p>$f(x,y)= \lim_{y\to\infty}{{1-y\sin{\pi x\over y}}\over \arctan x}$</p>
<p>The result I get is ${1-\pi x \over \arctan x}$, which depends on the value of $x$.</p>
<p>However, the question I have is that whatever $x$ is, since it's in the $\sin()$, which is a bounded function, sh... | Rohan Shinde | 463,895 | <p>Hint </p>
<p>Let $\frac 1y=t$
Then as $y\to \infty$ hence $t\to 0$
Hence $$\lim_{y\to \infty} y\sin \frac {\pi x}{y}=\lim_{y\to \infty} \frac {\sin \frac {\pi x}{y}}{\frac 1y}=\lim_{t=0} \frac {\sin \pi xt}{t}=\pi x$$</p>
|
1,641,922 | <p>I've came accros this excersize:<br>
Suppose that $D=\{z:|z| \le 1\}\subset \mathbb C$ and $$f:D\rightarrow\mathbb C$$
suppose that for every $z\in D$ such that $|z|<1$ $$|f(z)-\bar z|<0.9$$ where $\bar z$ is the complex conjugate of $z$. Prove that $f$ cannot be analytic in $D$.<br>
I started with assuming th... | Community | -1 | <p>First you have to formulate the given information in maths. We write $C$ and $J$ for the ages. Since Catherine is twice as old as Jason we have
$$C = 2J.$$
Moreover we know that 6 years ago, Catherine was five times older then Jason. This can be formulated as
$$ C - 6 = 5 \cdot (J - 6) = 5J - 30.$$
This equation is ... |
2,492,206 | <p>Let $(X, \mathcal{X})$ and $(Y,\mathcal{Y})$ be measurable spaces, and $f: X \to Y$ a measurable function. Let $A \in \mathcal{X}$ be a measurable subset of $X$.</p>
<p>Is it guaranteed that $f_{\mid A}: A \to Y$ is measurable?</p>
<p>The measurable space on $A$ is $(X, \mathcal{X})$ restricted to $A$. Formally, t... | James Garrett | 457,432 | <p>That’s true. If $f$ is measurable then $E_a=f^{-1}((a,+\infty))$ is a measurable set, for every $-\infty<a<+\infty$. Hence, the set $E_a \cap A$ is measurable. Noticing that $f_{|A}(a,+\infty)=E_a \cap A$, we are done. You are correct!</p>
|
238,659 | <p>Let $f : [a, b]\to R$ be a continuous function such that $[a,b] \subset [f(a), f(b)]$. Prove that there exists $x\in [a,b]$ such that $f(x) = x$.</p>
<p>My attempt:
I said let there be a $\delta > 0 $and defined $c$ and $d$ to be $x + \delta$ and $x-\delta$ respectively. From here since $f$ is continuous $[f(c)... | Per Erik Manne | 33,572 | <p>Your approach is not correct, since you are assuming what you are supposed to show. You cannot define $c$ and $d$ to be something which depends on $x^*$ before you have shown that there is such a number as $x^*$. </p>
<p>A better approach would be to consider the function $g(x)=f(x)-x$. Argue that $g$ is continuous... |
2,877,578 | <p>Yesterday, I asked the question: <a href="https://math.stackexchange.com/questions/2876740/prove-that-if-a-b-are-closed-then-exists-u-v-open-sets-such-that-u-cap?noredirect=1#comment5938458_2876740">Prove that if $A,B$ are closed then, $ \exists\;U,V$ open sets such that $U\cap V= \emptyset$</a>. </p>
<p>Here is th... | DanielWainfleet | 254,665 | <p>Sets $A, B$ are called completely separated iff there are disjoint open sets $U,V$ with $A\subset U$ and $B\subset V.$ The definition of a normal ($T_4$) space is a $T_1$ space in which every disjoint pair of closed sets is completely separated. It appears you are trying to prove that a metric space is a normal sp... |
1,665,533 | <p>Let $\mathcal{E}_1, ...,\mathcal{E}_n$ be collections of measurable sets on $(\Omega,\mathcal{F},P)$, each closed under intersection. Suppose
\begin{align*}
P(A_1\cap...\cap\ A_n)=P(A_1)\cdot ... \cdot P(A_n),
\end{align*}
for all $A_i \in \mathcal{E}_i$ for $1 \leq i \leq n$. </p>
<p>Now I want to show that the $... | Jimmy R. | 128,037 | <p>Since <span class="math-container">$\mathcal E_i$</span> are <span class="math-container">$π$</span>-systems and independent, then you know (if not, there is a sketch of the proof in the end of this answer) that also the induced Dynkin systems <span class="math-container">$δ(\mathcal E_i)$</span> are independent. No... |
1,679,615 | <p>From what I have read about a transitive relation is that if xRy and yRz are both true then xRz has to be true. </p>
<p>I'm doing some practice problems and I'm a little confused with identifying a transitive relation. </p>
<p>My first example is a "equivalence relation"
$S=\{1,2,3\}$ and $R = \{(1,1),(1,3),(2,2),... | Graham Kemp | 135,106 | <p>A relation $R$ of the set $S$ is transitive if:
$$\forall a{\in} S~\forall b{\in}S~\forall c{\in}S: \Big(\big((a,b){\in}R\wedge(b,c){\in}R\big)\to (a,c){\in}R\Big)$$</p>
<p>That definition is equivalent to:</p>
<p>$$\neg \exists a{\in}S~\exists b{\in}S~\exists c{\in}S: \Big(\big((a,b){\in}R\wedge(b,c){\in}R\big)\w... |
2,161,911 | <p>Find the limit :</p>
<p>$$\lim_{ n\to \infty }\sqrt[n]{\prod_{i=1}^n \frac{1}{\cos\frac{1}{i}}}=\,\,?$$</p>
<p>My try :</p>
<p>$$\lim_{ n\to \infty }\sqrt[n]{a} =1\,\, \text {for} \,\,a>0$$</p>
<p>and;</p>
<p>$$\prod_{i=1}^n \frac{1}{\cos\frac{1}{i}}>0$$</p>
<p>so :</p>
<p>$$\lim_{ n\to \infty }\sqrt[n... | RRL | 148,510 | <p>You got the right answer for the wrong reason, eg. $(2^n)^{1/n} \to 2 \neq 1$.</p>
<p>Cauchy's second limit theorem states</p>
<p>$$\lim_{n \to \infty} (a_n)^{1/n} = \lim_{n \to \infty} \frac{a_{n+1}}{a_n},$$</p>
<p>if the limit on the RHS exists.</p>
<p>This case reduces to </p>
<p>$$\lim_{n \to \infty} \frac{... |
6,637 | <p>I'm reading Madsen and Tornehave's "From Calculus to Cohomology" and tried to solve this interesting problem regarding knots. </p>
<p>Let $\Sigma\subset \mathbb{R}^n$ be homeomorphic to $\mathbb{S}^k$, show that $H^p(\mathbb{R}^n - \Sigma)$ equals $\mathbb{R}$ for $p=0,n-k-1, n-1$ and $0$ for all other $p$. Here $1... | Tom Church | 250 | <p>To apply the Mayer-Vietoris sequence, you need subspaces whose <em>interiors</em> cover your space (see e.g. <a href="http://en.wikipedia.org/wiki/Mayer%E2%80%93Vietoris_sequence">Wikipedia</a>, or <a href="http://www.math.cornell.edu/~hatcher/AT/ATpage.html">Hatcher</a>, p. 149). This is not true in your example, b... |
1,666,295 | <p>I was wondering when we add partial pivoting to an $LU$ factorization to a matrix $A$ it supposedly changes the data structure but improves the overall algorithm since we get better numerical stability. I am curious to why this is? </p>
<p>Any feedback is appreciated, my apologies for not formally introducing the m... | Carl Christian | 307,944 | <p>Consider the operation which we do by hand, i.e. an in-place implemention of Gaussian elimination with partial pivoting which overwrites the matrix $A$ with the LU factorization. </p>
<p>If a pivot is small, then the linear update of the lower right hand submatrix will almost certainly be done with some componentwi... |
1,781,269 | <p>What's the general method to find the slope of a curve at the origin if the derivative at the origin becomes indeterminate. For Eg--</p>
<p>What is the slope of the curve <span class="math-container">$x^3 + y^3= 3axy$</span> at origin and how to find it because after following the process of implicit differentiatio... | thecat | 338,383 | <p>Use l'Hopital's rule.$ $ This rule, the proof of which is very complex, states that if you get 0/0 or infinity/infinity for some limit function $h(x)=f(x)/g(x)$, you can take the derivative of the top and bottom functions and recalculate. Thus, you would do $f'(x)/g'(x)$. If this result is 0/0 or infinity over infin... |
124,498 | <p>When I produce a simple plot like so:</p>
<pre><code>Plot[Sin[x], {x, 0, 2 Pi}, GridLines -> Automatic, GridLinesStyle -> Directive[Red]]
</code></pre>
<p>It looks like this:</p>
<p><a href="https://i.stack.imgur.com/KcyZo.png" rel="noreferrer"><img src="https://i.stack.imgur.com/KcyZo.png" alt="enter imag... | Young | 41,016 | <p>Adding a <code>Frame</code> cleans it up nicely:</p>
<pre><code>Plot[Sin[x], {x, 0, 2 Pi}, GridLines -> Automatic,
GridLinesStyle -> Directive[Red], Frame -> True]
</code></pre>
<p><a href="https://i.stack.imgur.com/PTXfd.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/PTXfd.png" alt=... |
1,368,455 | <p>"I take a journey and, due to heavy traffic, crawl along the first half of the complete distance of my journey at an average speed of $10$ mph. How fast would I have to travel over the second half of the journey to bring my average speed to $20$ mph?"</p>
<p>At work, this has been a topic of a long debate.<br>
Prop... | orion | 137,195 | <p>Let $x$ be the whole distance. The time to complete the journey is</p>
<p>$$t=\frac{x/2}{v_1}+\frac{x/2}{v_2}$$</p>
<p>The average speed over the whole journey is then</p>
<p>$$v=\frac{x}{t}=\frac{2}{1/v_1+1/v_2}=\frac{2v_1v_2}{v_1+v_2}$$
You want $v=2v_1$, which would mean $v_2/(v_1+v_2)$ would have to be $1$. T... |
734,248 | <p>Example of two open balls such that the one with the smaller radius contains the one with the larger radius.</p>
<p>I cannot find a metric space in which this is true. Looking for hints in the right direction. </p>
| XtremeCurling | 136,072 | <p>Hint: look at $\mathbb{R}^2$. You can define some weird metrics on $\mathbb{R}^2$. It might be easiest if you define the metric to be very trivial (i.e. some constant) for any two points outside of a small subset of $\mathbb{R}^2$ (say, the nonnegative part of one of the axes). I remember answering this question for... |
734,248 | <p>Example of two open balls such that the one with the smaller radius contains the one with the larger radius.</p>
<p>I cannot find a metric space in which this is true. Looking for hints in the right direction. </p>
| Arno | 128,989 | <p>If you just want containment, the trivial metric space with a single point is the best example.</p>
<p>If you want that the ball with the larger radius is a proper subset of the one with the smaller radius they will have to have different centers, but then it is rather straight-forward.</p>
<p>Depending on your pr... |
2,771,823 | <p>The question is if the modulus of a multiplication, i.e. $a*a*a$ modulus $n$, is the same when we take the modules at each step of the multiplication.
So if </p>
<p>$(((a \text{ mod } n)* a \text{ mod } n) * a \text{ mod } n) = a*a*a \text{ mod } n$?</p>
| Peter Melech | 264,821 | <p>Yes, because $\mathbb{Z}\rightarrow\mathbb{Z}/(n),a\mapsto a+(n)$ is a homomorphism, the so called canonical homomorphism.
To see this You just have to prove that multiplication on $\mathbb{Z}/(n)$ is well defined: Indeed if $b-b'\in(n)$ then
$$ab'-ab=a(b'-b)\in (n).$$
Thus $(a+(n))(b+(n))=(a\mod(n))(b\mod(n))=ab+(n... |
3,170,871 | <p>Could anyone please give me a hint on how to compute the following integral?</p>
<p><span class="math-container">$$\int \sqrt{\frac{x-2}{x^7}} \, \mathrm d x$$</span></p>
<p>I'm not required to use hyperbolic/ inverse trigonometric functions.</p>
| Community | -1 | <p>With <span class="math-container">$y:=\dfrac1x$</span>,</p>
<p><span class="math-container">$$\int\sqrt{\frac{x-2}{x^7}}dx=-\int\sqrt{\left(\frac1y-2\right)y^7}\,\frac{dy}{y^2}=-\int y\sqrt{1-2y}\,dy.$$</span></p>
<p>Then by parts,</p>
<p><span class="math-container">$$-\int y\sqrt{1-2y}\,dy=\frac13y(1-2y)^{3/2}-... |
373,357 | <p>I've tought using split complex and complex numbers toghether for building a 3 dimensional space (related to my <a href="https://math.stackexchange.com/questions/372747/what-are-the-uses-of-split-complex-numbers?noredirect=1">previous question</a>). I then found out using both together, we can have trouble on the pr... | rschwieb | 29,335 | <p>Congratulations: the multiplication table for basis elements that you have laid out indicate that you have independently discovered the <a href="http://en.wikipedia.org/wiki/Clifford_algebra" rel="noreferrer">Clifford algebra</a> of a two dimensional vector space with metric signature $(1,-1)$, also denoted as $C\el... |
373,357 | <p>I've tought using split complex and complex numbers toghether for building a 3 dimensional space (related to my <a href="https://math.stackexchange.com/questions/372747/what-are-the-uses-of-split-complex-numbers?noredirect=1">previous question</a>). I then found out using both together, we can have trouble on the pr... | Anixx | 2,513 | <p>You have discovered <a href="https://en.wikipedia.org/wiki/Split-quaternion" rel="nofollow noreferrer">split-quaternions</a>. You can compare the multiplication table there and in your question.</p>
<p>This algebra is not commutative and has zero divisors. So, it combines the "negative" traits of both quat... |
71,952 | <p><strong>Background</strong></p>
<p>Take 2 convex sets in $\mathbb{R}^2$, or 3 convex sets in $\mathbb{R}^3$, or generally, $n$ convex sets in $\mathbb{R}^n$. "Mixed volume" assigns to such a family $A_1, \ldots, A_n$ a real number $V(A_1, \ldots, A_n)$, measured in $\mathrm{metres}^n$. </p>
<p>As I understand it... | pinaki | 1,508 | <p>I think the first three properties do indeed characterize mixed volume. For example, in two dimensions they imply that</p>
<p>$V(A_1, A_2) = \frac{1}{2}(V(A_1 + A_2, A_1 + A_2) - V(A_1, A_1) - V(A_2,A_2))$ <br>
$= \frac{1}{2}(Vol(A_1 + A_2) - Vol(A_1) - Vol(A_2)),$</p>
<p>which gives the formula of mixed volume i... |
887,656 | <p>Is there a closed form (complex) solution $z(t)$ to the equation</p>
<p>\begin{align}
\frac{dz}{dt}=f(t)\bar{z},
\end{align}</p>
<p>(the bar means complex conjugate) for any given complex valued function $f$ of a real variable $t$? The usual approach to deal with separable equations gives
\begin{align}
\int\frac{1... | JJacquelin | 108,514 | <p>One cannot give a general answer because a closed form $z(t)$ exists or doesn't exist depending on the kind of function $f(t)$. As shown below, one have to solve a linear system of two équations. This is equivalent to a second order linear ODE. All second order linear ODE cannot be analytically solved and solutions ... |
2,660,595 | <p>Ten people are sitting in a row, and each is thinking of a negative integer no smaller than $-15$. Each person subtracts, from his own number, the number of the person sitting to his right (the rightmost person does nothing). Because he has nothing else to do, the rightmost person observes that all the differences w... | DonAntonio | 31,254 | <p>For $\;x=2k+1\;$ an odd integer:</p>
<p>$$n=1: (2k+1)^{2^1}=4k^2+4k+1=4k(k+1)+1=1\pmod 4=2^2\;\checkmark$$</p>
<p>Suppose truth for all integers up to $\;n\;$ and we shall prove now for $\;n\;$ :</p>
<p>$$x^{2^{n+1}}=\left(x^{2^n}\right)^2\stackrel{\text{Ind. Hyp.}}=\left(1+m2^{n+1}\right)^2=1+m2^{n+2}+m2^{2n+2}\... |
1,206,195 | <p>I am trying to find the maximum of $x^{1/x}$. I don't know how to find the derivative of this. I have plugged in some numbers and found that $e^{1/e}$ seems to be the maximum at around 1.44466786. I don't know if this is the maximum, and I would like an explanation of why it is/what the maximum is. essentially, how ... | danimal | 202,026 | <p>One way to do it is to rewrite $$x^{1/x}$$ as $$e^{\ln x \over x}$$
and then differentiate using the chain rule.</p>
|
1,083,841 | <p>I have extracted the below passage from the wikipedia webpage - Point (geometry): </p>
<blockquote>
<p>In particular, the geometric points do not have any length, area, volume, or any other dimensional attribute. </p>
</blockquote>
<p>I think the above passage imply\ies that the point is zero dimensional. If... | user21820 | 21,820 | <p>It depends on your definition of "line" and "point" as Hurkyl mentioned. In pure Euclidean geometry with only the geometric axioms you can't talk about dimension at all. If you add the Cantor-Dedekind axiom, then Euclidean geometry can be embedded in $\mathbb{R}^3$, and then you can talk about dimension, which is si... |
1,083,841 | <p>I have extracted the below passage from the wikipedia webpage - Point (geometry): </p>
<blockquote>
<p>In particular, the geometric points do not have any length, area, volume, or any other dimensional attribute. </p>
</blockquote>
<p>I think the above passage imply\ies that the point is zero dimensional. If... | Agora345 | 427,668 | <p>I'm none too bright, but I like Euclid, and was of the opinion that points passing through themselves generate lines, lines through themselves areas, and areas through themselves volumes (in mathematical imagination land, where reality lives). Because of this I am starting to think the unit represents the magnitude ... |
2,762,391 | <p>Let $x,y>0$ s.t. $x^3+y^3\geq 2$.</p>
<p>Show that $x^2+y^2\geq x+y $.</p>
<p>I analyse the case when $x,y\geq 1$ but I don't know to solve the case when $x\geq 1\geq y $.</p>
| Hagen von Eitzen | 39,174 | <p>Note that
$$ \begin{align}(x+y)(x^2+y^2)&=x^3+y^3+xy(x+y)\\&\tag1\ge 2+xy(x+y)\\&=2+\frac{(x+y)^2-(x^2+y^2)}{2}(x+y)\\
&=2+\frac 12(x+y)^3-\frac12(x+y)(x^2+y^2)\end{align}$$
Therefore,
$$ \tag2(x+y)(x^2+y^2)\ge \frac 43+\frac13(x+y)^3$$
So if the first factor is larger, we first find from $(1)$
$$ x... |
3,493,151 | <p>This is a calculus problem from a high school math contest in Greece,from 2012.</p>
<p>I wish to know some solutions for this. I attempted to solve it.</p>
<blockquote>
<p>Let <span class="math-container">$f:\Bbb{R} \to \Bbb{R}$</span> differentiable such that <span class="math-container">$\lim_{x \to +\infty}f(... | Matematleta | 138,929 | <p>Here is an approach that only uses the easily proved fact that </p>
<p><span class="math-container">$\underset{x\to\infty }\lim\frac{e^x}{p(x)}=\infty\ \text{whenever}\ p \ \text{is a polynomial}. \tag1$</span> </p>
<p>Indeed, there is an <span class="math-container">$x_0\in \mathbb R^+$</span> such that <span c... |
96,110 | <p><span class="math-container">$A = \begin{pmatrix}
0 & 1 &1 \\
1 & 0 &1 \\
1& 1 &0
\end{pmatrix} $</span></p>
<p>The matrix <span class="math-container">$(A+I)$</span> has rank <span class="math-container">$1$</span> , so <span class="math-container">$-1$</span> is an eigenvalue with an al... | Patrick Da Silva | 10,704 | <p>It is well known that the algebraic multiplicity of an eigenvalue is greater or equal than the geometric multiplicity of that eigenvalue (i.e. the dimension of its eigenspace). Therefore, knowing that the rank of $(A-(-1)I)$ is $1$, you know that the algebraic multplicity of $-1$ is at least $2$ because if we denote... |
136,453 | <p>For every $k\in\mathbb{N}$, let
$$
x_k=\sum_{n=1}^{\infty}\frac{1}{n^2}\left(1-\frac{1}{2n}+\frac{1}{4n^2}\right)^{2k}.
$$
Calculate the limit $\displaystyle\lim_{k\rightarrow\infty}x_k$.</p>
| tocs | 30,071 | <p>Relative primes are also useful when building gear trains. It helps to reduce wear. If one gear has a number of teeth that is a factor of the other gear the same teeth always come into contact with one another. </p>
<p>For instance if one gear has 10 teeth and the other 20, the first tooth on the 10t gear always co... |
20,771 | <p><strong>Background:</strong></p>
<p>Let $G$ be a profinite group. If $M$ is a discrete $G$-module, then $M=\varinjlim_U M^U$, where the direct limit is taken with respect to inclusions over all open normal subgroups of $G$, and one naturally has $H^n(G,M)\simeq\varinjlim H^n(G/U,M^U)$, where the cohomology groups o... | Leonid Positselski | 2,106 | <p>For example, the profinite group cohomology $H^2(\hat{\mathbb Z}, \mathbb Z_p)$, where $\mathbb Z_p$ is considered as a trivial discrete $\hat{\mathbb Z}$-module, is isomorphic to $H^1(\hat{\mathbb Z},\mathbb Q_p/\mathbb Z_p)$ (since $H^i(\hat{\mathbb Z},\mathbb Q_p)=0$ for $i>0$). Which is isomorphic to $\mathb... |
2,992,127 | <p>I know that <span class="math-container">$ad\neq bc $</span> is sufficient for <span class="math-container">$z$</span> irrational because if <span class="math-container">$ad = bc$</span> then <span class="math-container">$\frac{ax+b}{cx+d} = \frac{ax+b}{cx+d} \frac{cb}{ad} = \frac{cax+cb}{cax+da}\frac{b}{d}$</span> ... | fleablood | 280,126 | <p><span class="math-container">$\frac {ax+b}{cx + d} =z$</span></p>
<p><span class="math-container">$ax + b = z(cx + d) $</span></p>
<p><span class="math-container">$x(a-cz) = zd - b$</span>. (<span class="math-container">$cz$</span> is irrational and <span class="math-container">$a$</span> is rational so <span cla... |
4,108,926 | <p>I was reading Axler's Linear Algebra Done Right, and the following appears as exercise <span class="math-container">$3$</span> in chapter <span class="math-container">$5$</span>, section A:</p>
<blockquote>
<p>Suppose <span class="math-container">$T \in \mathcal{L}(V)$</span> and <span class="math-container">$T^2 = ... | hamam_Abdallah | 369,188 | <p><strong>hint</strong></p>
<p><span class="math-container">$ T $</span> and <span class="math-container">$ T+I $</span> are bijective.</p>
<p>For any vector <span class="math-container">$ u\ne 0$</span>,
<span class="math-container">$$(T+I)u\ne 0$$</span>
and</p>
<p><span class="math-container">$$(T-I)((T+I)u)=(T^2-... |
4,108,926 | <p>I was reading Axler's Linear Algebra Done Right, and the following appears as exercise <span class="math-container">$3$</span> in chapter <span class="math-container">$5$</span>, section A:</p>
<blockquote>
<p>Suppose <span class="math-container">$T \in \mathcal{L}(V)$</span> and <span class="math-container">$T^2 = ... | copper.hat | 27,978 | <p>Suppose <span class="math-container">$y=Tx-x$</span>, then <span class="math-container">$Ty = -y$</span> and so we must have <span class="math-container">$y=0$</span>. Hence <span class="math-container">$Tx=x$</span>.</p>
|
220,996 | <p>I was wondering if for every NFA there exists an equivalent DFA? I think the answer is yes. How would one <em>prove</em> it? Since I'm just starting up learning about Automata I'm not confused about this and especially the proof of such a statement.</p>
| greendragons | 71,982 | <p>It can be done in two steps:<br/>
1) Use subset construction to construct DFA from NFA.<br/>
2) Then show for any w $\in$$\sum$$^*$,<br/> $\hat\delta$$_D$({q},a) = $\hat\delta$$_N$(q,a). That is for any string and for any set of states they both take you to same set of states.</p>
|
88,861 | <p>If $n$ is an integer, how do you know whether $n^n$ is a perfect square, without a calculator?</p>
<p>The actual question is: "<em>how many integers between $1$ and $100$ inclusive, raised to their own power, are perfect squares?</em>".</p>
| Pete L. Clark | 299 | <p>Here are some hints:</p>
<p>1) Any number raised to any even (positive integer) power is a perfect square: $a^{2b} = (a^b)^2$.</p>
<p>2) Any perfect square raised to any (positive integer) power is a perfect square:
$(a^2)^b = a^{2b} = (a^b)^2$.</p>
<p>3) The first two hints suggest that you should consider sepa... |
88,861 | <p>If $n$ is an integer, how do you know whether $n^n$ is a perfect square, without a calculator?</p>
<p>The actual question is: "<em>how many integers between $1$ and $100$ inclusive, raised to their own power, are perfect squares?</em>".</p>
| whitehat | 20,651 | <p>It is clear that if $n$ is an even number then, $n$ to power $n$ is surely a perfect square.</p>
<p>Now consider the case that n is an odd number. Let us do <a href="http://en.wikipedia.org/wiki/Fundamental_theorem_of_arithmetic">prime power factorization</a> of n.</p>
<p>$$\text{so }n = a_1^{p_1} \times a_2^{p_2}... |
4,267,862 | <p>If k balanced n-sided dice are rolled, what is the probability that each of the n different numbers will appear at least once?</p>
<p>A case of this was discussed <a href="https://math.stackexchange.com/questions/264408">here</a>, but I’m not sure how to extend this. Specifically, I’m not sure how the to calculate t... | JMoravitz | 179,297 | <p>Without loss of generality, assume the dice are all uniquely identifiable (<em>e.g. by color or by order in which they are rolled</em>).</p>
<p>There are <span class="math-container">$n^k$</span> equally likely different ways in which the dice may all be rolled.</p>
<p>Now... consider the event that a <span class="m... |
313,712 | <p>Consider $\mathbb Q\otimes \mathbb Q$, where $\mathbb Q$ is considered as $\mathbb Z$-algebra and consider $\mathbb Q$ as a right $\mathbb Q\otimes\mathbb Q$ module. Then is it true that $\mathbb Q$ is projective $\mathbb Q\otimes\mathbb Q$-module?</p>
| Pete L. Clark | 299 | <p>Since $\mathbb{Q} \otimes_{\mathbb{Z}} \mathbb{Q} \cong \mathbb{Q}$ is a field, every module over it is projective. </p>
|
313,712 | <p>Consider $\mathbb Q\otimes \mathbb Q$, where $\mathbb Q$ is considered as $\mathbb Z$-algebra and consider $\mathbb Q$ as a right $\mathbb Q\otimes\mathbb Q$ module. Then is it true that $\mathbb Q$ is projective $\mathbb Q\otimes\mathbb Q$-module?</p>
| Martin Brandenburg | 1,650 | <p>I assume that the module structure is induced by the ring map $\mathbb{Q} \otimes_{\mathbb{Z}} \mathbb{Q} \to \mathbb{Q}$, $a \otimes b \mapsto ab$. But this is an isomorphism, so that the module is free of rank $1$. More generally, if $A \to B$ is an epimorphism in the category of commutative rings, then $B \otimes... |
878,517 | <p>Is there any more solutions to this functional equation $f(f(x))=x$?</p>
<p>I have found: $f(x)=C-x$ and $f(x)=\frac{C}{x}$.</p>
| Shabbeh | 165,678 | <p>$f(x) = c-x$</p>
<p>$f(x) = c/x$</p>
<p>$f(x) = \frac{c_{1}-x}{c_{2}x+1}$</p>
<p>$f(x) = \frac{1}{2}\left ( \sqrt[]{{c_{1}^2}+c_{2}-4x^2} \right )+ c_{1}x$</p>
<p>$f(x) = \sqrt[3]{c-x^3}$</p>
<p><a href="http://www.wolframalpha.com/input/?i=f%28f%28x%29%29%3Dx" rel="nofollow">http://www.wolframalpha.com/input/?... |
878,517 | <p>Is there any more solutions to this functional equation $f(f(x))=x$?</p>
<p>I have found: $f(x)=C-x$ and $f(x)=\frac{C}{x}$.</p>
| Anixx | 2,513 | <p>Any function whose graphic is symmetric against the line y=x.</p>
|
4,353,958 | <p><a href="https://i.stack.imgur.com/zxyBa.png" rel="nofollow noreferrer">In this example</a>, it says that the phase of the complex number <span class="math-container">$i$</span> is <span class="math-container">$\pi/2$</span> and <span class="math-container">$-1$</span> has phase <span class="math-container">$\pi$</s... | peter.petrov | 116,591 | <p>This formula does not quite work because it returns a number in <span class="math-container">$(-\pi/2, \pi/2)$</span>. That is the range of <span class="math-container">$\arctan$</span>.</p>
<p>The phase/argument of a non-zero complex number is a number (i.e. angle expressed in radians) in the interval <span class="... |
2,036,943 | <p>any riemann integrable function is a smooth piecewise function?</p>
<p>Is true for fundamental theorem calculus?</p>
| Henricus V. | 239,207 | <p>Consider <a href="https://en.wikipedia.org/wiki/Thomae's_function" rel="nofollow noreferrer">Thomae's Function</a>. This function is continuous on all irrational numbers, hence is Riemann integrable by Lebesgue's criterion of Riemann integrability, but it is not piecewise smooth.</p>
|
1,279,564 | <p>I try to be rational and keep my questions as impersonal as I can in order to comply to the community guidelines. But this one is making me <strong>mad</strong>. Here it goes.
Consider the uniform distribution on $[0, \theta]$. The likelihood function, using a random sample of size $n$ is
$\frac{1}{\theta^{n}}$.<b... | Michael Hardy | 11,667 | <p>Suppose the observed order statistics are <span class="math-container">$1,2,3.$</span></p>
<p>You have <span class="math-container">$L(\theta) = \dfrac 1 {\theta^3}$</span> for <span class="math-container">$\theta\ge3.$</span></p>
<p>And <span class="math-container">$L(\theta)=0$</span> for <span class="math-contain... |
370,599 | <p>If A is an invertible $nxn$ matrix prove that:$ adj(adjA)=(A)(detA)^{n-2}$
I have done this but it somewhere went wrong:
$ adj(adjA)=adj(A^{-1} detA)=(A^{-1}detA)^{-1} det(A^{-1}detA)=AdetA det(A^{-1}detA)= Adet(AA^{-1}detA)=A (detA)^n $ </p>
| Inceptio | 63,477 | <p>$\dfrac{x+y}{n}=n^2 \implies x+y=2^{\alpha+2 \alpha}$</p>
<p>$x+y =2^{3 \alpha} \implies x+y=8^ {\alpha}$</p>
<p>To have $y>x>n$, you need to have $\alpha \ge 2$</p>
<p>I see $x,y \not \in$ prime number set.For instance when $\alpha=2$, $(x,y)=(49,15).(9,55)$ you get only few solutions for $\alpha=2$ becaus... |
31,414 | <p>I'm experimenting with different algorithms that approximate pi via iteration and comparing the result to pi. I want to both visualise and perhaps know the function (if any) that describes the increasing trend in accuracy as the number of iterations rises. </p>
<p>For example, 1 iteration might give me 3.0, 10 ite... | Yves Klett | 131 | <p>As pointed out by Rahul, <code>ListLogLogPlot</code> does the job nicely and points you into the right direction model-wise:</p>
<pre><code>data = {{10, 0.19809}, {50, 0.039984}, {100, 0.019998}, {500,
0.00399998}, {1000, 0.002}, {20000, 0.0001}, {100000,
0.00002}, {500000, 4.*10^-6}};
ListLogLogPlot[dat... |
1,531,646 | <p>Find the following limit</p>
<p>$$
\lim_{x\to0}\left(\frac{1+x2^x}{1+x3^x}\right)^\frac1{x^2}
$$</p>
<p>I have used natural logarithm to get</p>
<p>$$
\exp\lim_{x\to0}\frac1{x^2}\ln\left(\frac{1+x2^x}{1+x3^x}\right)
$$</p>
<p>After this, I have tried l'opital's rule but I was unable to get it to a simplified for... | zhw. | 228,045 | <p>Define $f(x) = x(2^x-3^x)/(1+x3^x).$ Then the expression equals $(1+f(x))^{1/x^2}.$ Apply $\ln$ to get</p>
<p>$$\tag 1 \frac{1}{x^2}\ln ( 1 +f(x))=\frac{f(x)}{x^2}\frac{\ln ( 1 +f(x))}{f(x)}.$$</p>
<p>Now $f(x) \to 0$ as $x\to 0.$ Because $[\ln(1+u)]/u \to 1$ as $u\to 0$ (simply because $\ln'(1) = 1),$ the second ... |
232,276 | <p>I can prove with the triangle inequality that the unit sphere in $R^n$ is convex, but how to show that it is strictly convex?</p>
| Fly by Night | 38,495 | <p>To show that the closed unit ball $B$ is strictly convex we need to show that for any two points $x$ and $y$ in the boundary of $B$, the chord joining $x$ to $y$ meets the boundary only at the points $x$ and $y$.</p>
<p>Let $x,y \in \partial B$, then $||x|| = ||y|| = 1.$ Now consider the chord joining $x$ to $y$. W... |
4,007,987 | <p>So define a polynomial <span class="math-container">$P(x) = 4x^3 + 4x - 5 = 0$</span>, whose roots are <span class="math-container">$a, b $</span> and <span class="math-container">$c$</span>. Evaluate the value of <span class="math-container">$(b+c-3a)(a+b-3c)(c+a-3b)$</span></p>
<p>Now tried this in two ways (both... | José Carlos Santos | 446,262 | <p>Note that:</p>
<ul>
<li><span class="math-container">$b+c-3a=a+b+c-4a=-4a$</span>;</li>
<li><span class="math-container">$a+b-3c=a+b+c-4c=-4c$</span>;</li>
<li><span class="math-container">$c+a-3b=a+b+c-4b=-4b$</span>.</li>
</ul>
<p>So, you're after <span class="math-container">$(-4a)\times(-4c)\times(-4b)=-64abc$</... |
738,455 | <p>Sources: <a href="https://rads.stackoverflow.com/amzn/click/0495011665" rel="nofollow noreferrer"><em>Calculus: Early Transcendentals</em> (6 edn 2007)</a>. p. 890, Section 14.3. Exercise 50b, c.. </p>
<blockquote>
<p><img src="https://i.stack.imgur.com/8ggG2.png" alt="enter image description here"></p>
</blo... | Umberto P. | 67,536 | <p>Here $f$ is a function of a single variable and $f'$ refers to the ordinary derivative. </p>
|
2,965,193 | <p>Basically the question is asking us to prove that given any integers <span class="math-container">$$x_1,x_2,x_3,x_4,x_5$$</span> Prove that 3 of the integers from the set above, suppose <span class="math-container">$$x_a,x_b,x_c$$</span> satisfy this equation: <span class="math-container">$$x_a^2 + x_b^2 + x_c^2 = 3... | Mohammad Riazi-Kermani | 514,496 | <p>The remainder of every integer in dividing by <span class="math-container">$3$</span> is either <span class="math-container">$0$</span>,<span class="math-container">$1$</span>,or <span class="math-container">$2$</span>.</p>
<p>Thus the remainder of a square in dividing by <span class="math-container">$3$</span> is ... |
2,965,193 | <p>Basically the question is asking us to prove that given any integers <span class="math-container">$$x_1,x_2,x_3,x_4,x_5$$</span> Prove that 3 of the integers from the set above, suppose <span class="math-container">$$x_a,x_b,x_c$$</span> satisfy this equation: <span class="math-container">$$x_a^2 + x_b^2 + x_c^2 = 3... | Rosie F | 344,044 | <p>Any integer is in one of 3 sets:</p>
<ul>
<li>multiples of 3</li>
<li>integers of the form <span class="math-container">$3k+1$</span> where <span class="math-container">$k$</span> is an integer</li>
<li>integers of the form <span class="math-container">$3k+2$</span> where <span class="math-container">$k$</span> is ... |
3,274,766 | <p>I have been trying to do this problem:</p>
<p>Solve <span class="math-container">$$\sec(x)=\tan(x),\quad 0≤x<2π$$</span></p>
<p>I started by rewriting <span class="math-container">$\sec(x)$</span> as <span class="math-container">$\frac{1}{\cos(x)}$</span>.</p>
<p>I then rewrote <span class="math-container">$\t... | Martund | 609,343 | <p><span class="math-container">$$\sec^2x-\tan^2x=1$$</span>
It fails at the solution of this problem. Hence there is no solution, because this identity never fails.</p>
<p>Hope it helps:)</p>
|
3,739,911 | <p><a href="https://i.stack.imgur.com/aSEt6.png" rel="nofollow noreferrer">question</a></p>
<p><a href="https://i.stack.imgur.com/G6EHM.png" rel="nofollow noreferrer">options and answers</a></p>
<p>The interval in which the function <span class="math-container">$f(x)=\sin(e^x)+\cos(e^x)$</span> is increasing is/are?</p... | Community | -1 | <p><strong>Hint:</strong></p>
<p><span class="math-container">$$\sin y+\cos y=\sqrt2\sin\left(y+\frac\pi4\right)$$</span> is growing in <span class="math-container">$$\left[-\frac{3\pi}4,\frac{\pi}4\right]+2k\pi$$</span></p>
<p>and the transformation</p>
<p><span class="math-container">$$y=e^x$$</span> is invertible.</... |
2,409,580 | <p>Recall that </p>
<blockquote>
<p><strong>Theorem (Bessel inequality).</strong> Let $(e_k)$ be an orthonormal sequence in an inner product space $X$. Then for every $x \in X $, $$\sum_{k=1}^{\infty} |\langle x,e_k \rangle|^2 \le \|x\|^2 .$$</p>
</blockquote>
<p>The proof results in $\le$ and not just $=$. Can $\l... | Gribouillis | 398,505 | <p>In a Hilbert space, $\sum_0^\infty |\langle x, e_k\rangle|^2 = \|P(x)\|^2$ where $P$ is the orthogonal projection on the closed linear span of the $e_k$ and one has
$$P(x) = \sum_0^\infty \langle x, e_k\rangle e_k$$
This closed linear span can be smaller than the whole space. Take any orthonormal sequence and remove... |
242,203 | <p>What's the derivative of the integral $$\int_1^x\sin(t) dt$$</p>
<p>Any ideas? I'm getting a little confused.</p>
| Joe | 24,942 | <p>You can use a nice theorem called the <a href="http://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus" rel="nofollow">Fundamental Theorem of Calculus </a>. Here, we're mainly worried about FTC part 1. Below is a summary of what FTC part 1 says.</p>
<p>Let $f$ be a continuous, function defined on $[a,b]$ and
$... |
488,983 | <p>I'm trying to prove the group isomorphism $(\Bbb Z[x]/(x^{n+1}))^\times\cong\Bbb Z/2\Bbb Z \times \prod _{i=1}^n \Bbb Z$.</p>
<p>Obviously I tried to establish a ring isomorphism from $\Bbb Z[x]/(x^{n+1})$ to some ring $R$, a direct product of easier rings, and prove that $R^\times$ equals the RHS of the original i... | Bruno Joyal | 12,507 | <p>In any ring $R$, if $a \in R$ is nilpotent and $u\in R$ is a unit, then $a+u$ is a unit. Indeed, if $a^n=0$ then since $u^n = u^n - (-a)^n = (u+a)(u^{n-1} + u^{n-1}(-a)+\dots + (-a)^{n-1})$ is a unit, so is $u+a$. Nilpotent elements act on the units as infiniteseimal translations.</p>
<p>Equipped with this result, ... |
182,316 | <p>I am trying to find an example of a separable Hausdorff space which has a non-separable subspace. This led me to ask the question in the title: is the set of irrationals, regarded as a subspace of the real line, separable or non-separable?</p>
<p>A space is separable if it contains a countable dense subset. A su... | Brian M. Scott | 12,042 | <p>$\Bbb R$ is second countable (i.e., has a countable base), so it’s hereditarily separable. Specifically, let $\mathscr{B}$ be the set of all open intervals with rational endpoints; $\Bbb Q$ is countable, so $\mathscr{B}$ is countable. Enumerate $\mathscr{B}=\{B_n:n\in\Bbb N\}$, and for $n\in\Bbb N$ let $x_n$ be any ... |
2,924,165 | <p>Assume that $\mathbb R$ is an ordered field (i.e. $\mathbb R$ is a model of real numbers). We define the set of natural numbers $\mathbb N$ as the smallest inductive set containing $1_\mathbb R$ (multiplicative identity of the field $\mathbb R$), where by definition a set $X\subset \mathbb R$ is inductive if $x\in X... | David K | 139,123 | <p>You can change the order of integration of your volume so that it is
<span class="math-container">$$
V = \int_0^2 \int_0^{2\pi} \int_0^{2-r\cos \theta - r \sin \theta}
r \,dz\,d\theta\,dr.
$$</span></p>
<p>Notice that this is like the shell method of integrating a volume of revolution, except that (because it ... |
34,294 | <p>Let $M$ be a 2-dimensional Riemannian manifold of non-positive curvature everywhere, of genus > 1. Let $\textbf{D} \subset \textbf{C}$ be the open unit disc in the complex plane, the universal cover of $M$. Let $\gamma \subset \textbf{D}$ be a curve representing a geodesic in $M$ which is entirely in a region of z... | Sam Nead | 1,650 | <p>Suppose that $S$ is your Riemannian surface and $X \subset S$ is a flat subsurface (that is, locally isometric to $\mathbb{R}$ with the usual metric). Let's suppose that $X$ has some nontrivial topology. For example, $X$ is a unit disk minus one-half of a unit disk, and the core curve of $X$ is essential in $S$. <... |
3,059,676 | <blockquote>
<p>Why is the sum of all external angles in a convex polygon <span class="math-container">$360^\circ$</span>? </p>
</blockquote>
<p>From my understanding, for each vertex in a convex polygon, there exist exactly <span class="math-container">$2$</span> exterior angles corresponding to it, which are both ... | MJD | 25,554 | <p>As the comment says, there are two equal exterior angles at each vertex, one on the left of the vertex and one on the right. When we say that "the sum of the exterior angles is 360°", we mean that the sum of the left-side angles is 360° and that the sum of the right-side angles is 360°, not that the sum of the two s... |
362,801 | <blockquote>
<p>$f:[0,1]\to\mathbb{R}^2$ is continuous, $f(0) \in B_{1}(0,0)$ and $f(1) \in B_{1}(10,10)$. Prove there exists $t \in [0,1]$ such that $f(t) \in \{(x,y): x+y=5\}$. </p>
</blockquote>
<p>I am thinking we need to use extreme value theorem or intermediate value theorem. Which one and how? </p>
<p>Just ... | yohBS | 21,295 | <p>Here's my hint: consider the change of coordinates</p>
<p>$$u=x+y$$
$$v=x-y$$</p>
<p>Rephrasing the question in these coordinates:</p>
<blockquote>
<p>$f:[0,1]\to\mathbb{R}^2$ is continuous, $f(0) \in B_{1}(0,0)$ and $f(1) \in B_{1}(20,0)$. Prove there exists $t \in [0,1]$ such that the first coordinate of $f(t... |
2,164,994 | <p>Is the ratio test for convergence applicable to the below series:</p>
<p>$$\sum_{n=1}^\infty \frac{n^3+1}{\sqrt[3]{n^{10} + n}}$$</p>
<p>I already know that the series diverge. I want to confirm if the ratio test is applicable or not?</p>
| Ángel Mario Gallegos | 67,622 | <p>What about of the integral test?</p>
<p>We have $$\frac{n^3+1}{\sqrt[3]{n^{10}+n}}> \frac{n^3+1}{\sqrt[3]{n^{10}+n^{10}}}= \frac{n^3+1}{2n^{10/3}}$$
Now, the integral
$$\int_1^{\infty}\frac{x^3+1}{2x^{10/3}}dx$$
diverges. Then, the given series diverges too.</p>
|
2,164,994 | <p>Is the ratio test for convergence applicable to the below series:</p>
<p>$$\sum_{n=1}^\infty \frac{n^3+1}{\sqrt[3]{n^{10} + n}}$$</p>
<p>I already know that the series diverge. I want to confirm if the ratio test is applicable or not?</p>
| Praneet Srivastava | 233,186 | <p>If the limit of the ratio $$\lim_{n \to \infty} \frac{a_{n+1}}{a_n} = 1$$ Then the Ratio Test is <strong>Inconclusive</strong>. The test does not tell you anything about the series. The series may diverge or converge conditionally or absolutely.</p>
<p><strong>As such, it would not be correct to say that the series... |
880,928 | <p>$A,B,C,D,E,F,G$</p>
<p>A list consists of all possible three-letter arrangements formed by using the letters above such that the first letter is $D$ and one of the remaining letters is $A$. If no letter is used more than once in an arrangement in the list and one three-letter arrangement is randomly selected from t... | Bridgeburners | 166,757 | <p>Your attempt would be right... if the actual question was "what are the chances of choosing DCA out of any random assortment of letters?" But you have to realize that the probability of your question is much higher, since the possibilities have been significantly reduced. </p>
<p>First, since the first letter is a ... |
2,762,323 | <p>I need to find the asymptotic behavior of $$\sum_{j=1}^N \frac{1}{1 - \cos\frac{\pi j}{N}}$$
as $N\to\infty$.</p>
<p>I found (using a computer) that this asymptotically will be equivalent to $\frac{1}{3}N^2$, but don't know how to prove it mathematically.</p>
| José Carlos Santos | 446,262 | <ol>
<li>Yes, the first assertion is false, but you should provide an example. For instance, you can take $X=\mathbb R$ with the usual topology and $Y=[-1,1]$ with the discrete topology.</li>
<li>$i(U)=U=V\cap Y$, which is open in $X$, since both $V$ and $Y$ are open in $X$.</li>
</ol>
|
3,278,761 | <p>Suppose a student says : "if 17 is even, then 2 is not a divisor of 17". </p>
<p>Surely his teacher would tell him he is wrong, saying that when a number is even, this number has 2 as divisor. The teacher would correct with " if 17 were even, then 2 would be a divisor of 17". In other words, the student's claim con... | Bram28 | 256,001 | <p>First of all, the fact that a material implication is considered true as soon as its antecedent is false, is not the same as <a href="https://en.wikipedia.org/wiki/Principle_of_explosion" rel="nofollow noreferrer">ex falso sequitur quodlibet</a>, which says that any statement <em>follows from</em> a contradiction.<... |
134,796 | <p>Example list below. All elements are in the form {1 or 0, 1 or 0, 1 or 0}, with a least one of the numbers 0 and 1 in the element (so excluding {1,1,1} and {0,0,0}) </p>
<pre><code>ListA = {{1, 1, 0}, {1, 1, 0}, {1, 1, 0}, **{0, 1, 1}**, {1, 0, 1}, {1, 0, 1}, {1,
0, 1}}
</code></pre>
<p>I want a command to rep... | Edmund | 19,542 | <p>You may construct a pattern with <a href="http://reference.wolfram.com/language/ref/Longest.html" rel="nofollow noreferrer"><code>Longest</code></a> and utilise it with <a href="http://reference.wolfram.com/language/ref/ReplaceAll.html" rel="nofollow noreferrer"><code>ReplaceAll</code></a>.</p>
<p>With</p>
<pre><c... |
4,385,676 | <blockquote>
<p>Let <span class="math-container">$Y_n$</span> be a sequence of non-negative i.i.d random variables with <span class="math-container">$EY_n = 1$</span> and <span class="math-container">$P(Y_n = 1) < 1$</span>. Consider the martingale process formed by <span class="math-container">$X_n = \prod_{k=1}^n ... | John Dawkins | 189,130 | <p>By Jensen's inequality, <span class="math-container">$b:=E\left(\sqrt{Y_1}\right)<\sqrt{E(Y_1)}=\sqrt{1}=1$</span>. Therefore <span class="math-container">$E\left(\sqrt{X_n}\right)=b^n\to 0$</span> as <span class="math-container">$n\to\infty$</span>. By Fatou's lemma, <span class="math-container">$E\left(\sqrt{X}... |
264,594 | <p>I need to make a proof but I can't come to the solution:
<p>For every vertex of oriented graph with vertices $U_{1},U_{2},\ldots,U_{n}$ we've got $s_{+}(U)$ the number of edges, which come to the vertex $U$, and $s_{-}(U)$ the number of edges which leave from the vertex.
<p>Prove that: $\sum_{i=1}^{n} |(s_{+}(U_{i})... | Karolis Juodelė | 30,701 | <p>Say $S$ is any set of integers and $\sum _{s\in S} s = 0$. You can then divide $S$ into $S_+ = \{s\in S : s\geq 0\}$ and $S_- = \{s \in S : s < 0\}$. We have $\sum _{s\in S} s = \sum _{s\in S_+} s + \sum _{s\in S_-} s = \sum _{s\in S_+} |s| - \sum _{s\in S_-} |s| = 0$ and thus $\sum _{s\in S_+} |s| = \sum _{s\in ... |
1,111,854 | <p>For example:</p>
<p><img src="https://i.stack.imgur.com/xEFpG.jpg" alt="enter image description here"></p>
<p>The last three lines have a |t=ti, what does that mean?</p>
| xanthousphoenix | 209,166 | <p>The $\mid_{t=t_i}$ means evaluate the stuff before with $t_i$ substituted for $t$. </p>
|
2,572,032 | <p>I'm looking for help with <strong>(b)</strong> and <strong>(c)</strong> specifically. I'm posting <strong>(a)</strong> for completeness.</p>
<p><strong>(a)</strong> Show convergence for $a_n=\sqrt{n+1}-\sqrt{n}$ towards $0$ and test $\sqrt{n}a_n$ for convergence.</p>
<p><strong>(b)</strong> Show $b_n=\sqrt[k]{n+1}... | vadim123 | 73,324 | <p>Write $$b_n=\frac{\sqrt[k]{1+\frac{1}{n}}-\sqrt[k]{1}}{\frac{1}{\sqrt[k]{n}}}=\frac{\sqrt[k]{1+\frac{1}{n}}-\sqrt[k]{1}}{\frac{1}{{n}}}\frac{1}{n^{1-1/k}}$$
The first part is a difference quotient, hence $$\lim_{n\to \infty} b_n= f'(1)\lim_{n\to \infty}n^{1/k-1}$$
where $f(x)=\sqrt[k]{x}$. We can use calculus to co... |
1,345,364 | <p>I am struggling with this question: </p>
<blockquote>
<p>Let $\{a_n\}$ be defined recursively by $a_1=\sqrt2$, $a_{n+1}=\sqrt{2+a_n}$. Find $\lim\limits_{n\to\infty}a_n$. HINT: Let $L=\lim\limits_{n\to\infty}a_n$. Note that $\lim\limits_{n\to\infty}a_{n+1}=\lim\limits_{n\to\infty}a_n$, so $\lim\limits_{n\to\infty... | marty cohen | 13,079 | <p>This expands a comment of mine
to prove that the
sequence is monotonic
and bounded
and, therefore,
converges.</p>
<p>Suppose $a < 2$.
Then I want to show that
$a < \sqrt{2+a}
< 2$.</p>
<p>Let
$2-a = d > 0$.
Then $a = 2-d$.
$\sqrt{2+a}
= \sqrt{2+2-d}
= \sqrt{4-d}
< 2
$.
Also,
since
$(1-z)^2
= 1-2z + ... |
4,244,966 | <p>I have the ODE <span class="math-container">$y^2(1+y'^2)=4$</span> to solve this I used the substitution <span class="math-container">$y'=p$</span>
<span class="math-container">$$y^2(1+p^2)=4$$</span>
<span class="math-container">$$2y(1+p^2)dy+2py^2dp=0$$</span>
<span class="math-container">$$(p^2+1)dy+py\;dp=0$$</s... | Lutz Lehmann | 115,115 | <p><em>Alternative solution strategy:</em> Expanding the left side, you get a circle equation
<span class="math-container">$$
y^2+(yy')^2=4
$$</span>
that can be parametrized as <span class="math-container">$y=2\sin u$</span>, <span class="math-container">$yy'=2\cos u$</span> which then leads to
<span class="math-conta... |
3,293,383 | <p><span class="math-container">$$ \frac{ln{x}}{(x^3-1)} <\frac{x}{x^3} , \forall x \in[2,\infty) $$</span></p>
<p>This is specifically for an improper integral question, where the left term needs to be proven convergent or divergent for the interval <span class="math-container">$$ [2,\infty) $$</span></p>
| J. W. Tanner | 615,567 | <p><strong>Hints:</strong></p>
<p><span class="math-container">$$x^3-1=(x-1)(x^2+x+1)$$</span></p>
<p><span class="math-container">$$\ln(x)<x-1$$</span></p>
<p><span class="math-container">$$x^2+x+1>x^2$$</span> </p>
<p>(the last one for <span class="math-container">$x\ge2$</span>)</p>
|
567,204 | <p>I'm currently studying CS, and as i didn't do maths A level i'm finding the module particularly difficult. We've now changed topics and lecturer, going onto discrete maths; and i'm refusing to fall behind :P. So, i'm going to post regularly/daily questions, just to make sure i have an understanding.</p>
<p>Hopefull... | Brian M. Scott | 12,042 | <p>By definition the <em>domain</em> of $f$ is the set of all inputs for which $f$ is defined; in this case that’s $\{a,b,c,d\}=X$. This is actually implicit in the notation $f:X\to Y$, which almost always implies that that the domain of $f$ is $X$. (I say <em>almost</em> because in some areas of mathematics one deals ... |
3,045,491 | <p>The set <span class="math-container">$\{(x,y,z) \in R^3: x^8+y^4+z^8-16=0\}$</span> is a bounded set?
I guess it isn't a bounded set because from <span class="math-container">$x,y,z \geq 0$</span> i suppose it's only inferiorly bounded.
Is it correct? Please tell me the correct answer.</p>
| Logan S. | 394,984 | <p>For this question, it really matters what you mean by bounded. You originally tagged your post as with topology, so I'll assume that's what you're interested in seeing. </p>
<p>In topology, we say that a set is bounded if it can be contained in a ball of finite size. You can formally show that this set is bounded b... |
2,741,229 | <p>I have searched a lot, but i haven't found any proof about that statement. I have checked the proof of</p>
<blockquote>
<p>If <span class="math-container">$f$</span> is differentiable, then <span class="math-container">$f$</span> is continuous</p>
</blockquote>
<p>but it's not the same argument I think. Also, I ... | Community | -1 | <p>Your problem seems to be the logical relationships between the statements</p>
<ol>
<li><em>If f is differentiable, then it is continuous</em> </li>
<li><em>If the derivative of <span class="math-container">$f$</span> is continuous, then <span class="math-container">$f$</span> is continuous</em></li>
<li><em>If the ... |
129,132 | <p>Both the ratio test and the root test define a number (via a limit).</p>
<p>If both limits exist (and shows that the series is convergent), what (if any) is the relation between the 2 numbers ? are they equal ?
What is the relation (if any) between them and the original series (other than the fact that they say th... | Peter LeFanu Lumsdaine | 2,439 | <p>For a non-negative real series <span class="math-container">$(a_n)_{n \in \mathbb{N}}$</span>, the tests give two (possibly undefined) numbers: let’s call them <span class="math-container">$L_\textit{root} := \lim_n (a_n)^{\frac{1}{n}}$</span>, and <span class="math-container">$L_\textit{ratio} := \lim_n \frac{a_{n+... |
2,829,990 | <p>I want to calcurate</p>
<p><span class="math-container">$$ \lim_{n \to \infty} \int_{(0,1)^n} \frac{n}{x_1 + \cdots + x_n} \, dx_1 \cdots dx_n $$</span></p>
<p>I met this in studying Lebesgue integral. But, I don't know how to do at all. I would really appreciate if you could help me!</p>
<p>[Add]</p>
<p>Thanks to e... | Shashi | 349,501 | <p>You say that we might as well find the following limit:
\begin{align}
\lim_{n\to\infty} n \int^\infty_0 \left( \frac {1-e^{-t}}{t}\right)^n\,dt
\end{align}
Set $u=e^{-t}$ to get:
\begin{align}
\int^\infty_0 \left( \frac {1-e^{-t}}{t}\right)^n\,dt = \int^1_0 \frac{1}{u} \left(\frac{u-1}{\log(u)}\right)^n\,du = \int^1... |
2,829,990 | <p>I want to calcurate</p>
<p><span class="math-container">$$ \lim_{n \to \infty} \int_{(0,1)^n} \frac{n}{x_1 + \cdots + x_n} \, dx_1 \cdots dx_n $$</span></p>
<p>I met this in studying Lebesgue integral. But, I don't know how to do at all. I would really appreciate if you could help me!</p>
<p>[Add]</p>
<p>Thanks to e... | metamorphy | 543,769 | <p>A somewhat more elementary solution. <span class="math-container">$f(t)=(1-e^{-t})/t$</span> is decreasing for <span class="math-container">$t>0$</span> since <span class="math-container">$$f'(t)=\frac{(1+t)e^{-t}-1}{t^2}<0\impliedby e^t>1+t.$$</span> Thus it has an inverse: <span class="math-container">$x=... |
1,120,816 | <p>$$ f(x) =
\begin{cases}
x^{-1} & \text{for $x<-1$} \\
ax+b & \text{for $-1\le x\le \frac 12$} \\
x^{-1} & \text{for $x>\frac 12$} \\
\end{cases}$$</p>
<p>I don't understand how I am supposed to find the value of the constants. It seems as if there is not enough information to determine that. I ... | Jake O | 607,327 | <p>To make sure <span class="math-container">$f$</span> is continuous at <span class="math-container">$x=-1$</span> you want to use the definition of what it means to be continuous by solving <span class="math-container">$\lim_{x \to -1} f(x) = f(-1).$</span> Since <span class="math-container">$f$</span> is different ... |
3,577,021 | <p>Let inner product space V be defined over F or C and linear operators T on V, evaluate <span class="math-container">$T^{*}$</span> at the given vector in V.</p>
<p><span class="math-container">$V=R^2, T(a,b)=(2a+b,a-3b), x=(3,5)$</span></p>
<p>I know <span class="math-container">$T^{*}$</span> is the conjugate tra... | Brian Moehring | 694,754 | <p>I spent quite a while trying to figure out what formula you were attempting to use. In the end, I just had to rephrase what the formulas seemed to say.</p>
<ul>
<li><span class="math-container">$E[Y \mid X+Y > 1]$</span> is finding the mean <span class="math-container">$y$</span> coordinate in the triangle <spa... |
3,061,575 | <p>It is a principle and proof from Introduction to Set Theory, Hrbacek and Jech. </p>
<p>In the proof, line 1 and 2, I couldn't understand why <span class="math-container">$Q(0)$</span> is true. </p>
<p><span class="math-container">$Q(0)$</span> means that "<span class="math-container">$P(k)$</span> holds for all <s... | jmerry | 619,637 | <p>The region matters. The region is everything. It may seem odd to approximate the odd function <span class="math-container">$\sin \pi x$</span> with sums of even functions <span class="math-container">$\cos n\pi x$</span> - but that's not what we're really doing.</p>
<p>Continue that <span class="math-container">$\s... |
4,099,804 | <p>I need to characterize every finitely generated abelian group G that has the following property:
<span class="math-container">$$\frac{G}{S} \text{ is cyclic for every } \lbrace0\rbrace \lneq S\leq G$$</span>
Given the problems before this one, I believe I am supposed to use the structure theorem figure out the under... | Asinomás | 33,907 | <p>Suppose that <span class="math-container">$G$</span> is not cyclic. This means that in the factorization of <span class="math-container">$G$</span> there is a factor <span class="math-container">$\mathbb Z_{p^a}\times \mathbb Z_{p^b}$</span>. So let's write the group as <span class="math-container">$\mathbb Z_{p^a}\... |
939,110 | <p>This is a different but related question to one I asked earlier. I link to it here:</p>
<p><a href="https://math.stackexchange.com/questions/938953/to-show-that-f-is-injective-i-dont-get-this-statement">"To show that f is injective" - I don't get this statement</a></p>
<p>I am pretty new to "function... | Moishe Kohan | 84,907 | <p>Take infinite direct sum of finite abelian groups. For finitely generated examples google "Burnside problem".</p>
<p>If you do not know what is the direct sum of groups, take the group of roots of unity:
$$
\{ e^{i\pi r}: r\in {\mathbb Q}\}.
$$</p>
|
253,921 | <p>I am trying to come up with a measurable function on $[0,1]^2$ which is not integrable, but such that the iterated integrals are defined and unequal.</p>
<p>Any help would be appreciated.</p>
| Christian Blatter | 1,303 | <p>Consider the double integrals
$$I:=\int_0^1\int_0^1{y-x\over(2-x-y)^3}\ dy\ dx\ ,\qquad
J:=\int_0^1\int_0^1{y-x\over(2-x-y)^3}\ dx\ dy\ .$$
Then
$$\int_0^1{y-x\over(2-x-y)^3}\ dy={y-1\over(2-x-y)^2}\Biggr|_{y=0}^1={1\over(2-x)^2}\ .$$
It follows that
$$I=\int_0^1 {dx\over(2-x)^2}={1\over 2-x}\Biggr|_0^1={1\over2}\ .... |
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