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 |
|---|---|---|---|---|
28,811 | <p>There are lots of statements that have been conditionally proved on the assumption that the Riemann Hypothesis is true.</p>
<p>What other conjectures have a large number of proven consequences?</p>
| Joel David Hamkins | 1,946 | <p>Set theory is of course completely saturated with this feature, since the independence phenomenon means that a huge proportion of the most interesting natural set-theoretic questions turn out to be independent of the basic ZFC axioms. Thus, most of the interesting work in set theory is about the relations beteween t... |
28,811 | <p>There are lots of statements that have been conditionally proved on the assumption that the Riemann Hypothesis is true.</p>
<p>What other conjectures have a large number of proven consequences?</p>
| Tobias Kildetoft | 4,614 | <p>In the representation theory of a reductive algebraic group $G$ in positive characteristic $p$, there is a conjecture known as the Humphreys-Verma conjecture, which states that an indecomposable injective module for a Frobenius kernel $G_r$ of $G$ should lift uniquely to a module for $G$. There is also a refinement ... |
1,879,283 | <p>It is said that greatest possible value of $380$ (2 s.f rounded) is $385$,and least one is $375$. How this is possible? if i try to round $385$ (2 s.f) it should be $390$.
As the least possible value of $390$ is $385$.</p>
<p>How can i resolve these issues?</p>
| marty cohen | 13,079 | <p>For A and C,
I would use
the point-slope form
of a line.
It states that
a line through
$(u, v)$
with slope $m$
has the form
$y-v = (x-u)m
$.</p>
<p>Note that this is true
when $x=u$ and $y=v$,
so it passes through the point.
Also note that
the equation can be written
$\dfrac{y-v}{x-u}
=m
$,
so its slope is $m$.</p>... |
1,879,283 | <p>It is said that greatest possible value of $380$ (2 s.f rounded) is $385$,and least one is $375$. How this is possible? if i try to round $385$ (2 s.f) it should be $390$.
As the least possible value of $390$ is $385$.</p>
<p>How can i resolve these issues?</p>
| Bernard | 202,857 | <p>For A), the most direct way is to use the vector form of the equation: if $(x_A,y_A)$ are the coordinates of the point $A$, and $\vec n$ is a normal vector, the equation is
$$\overrightarrow{AM}\cdot \vec n=0,\quad\text{i.e.}\quad 2x+5(y+2)=0.$$</p>
<p>For B), if the given poibts have different abscissae, there's a... |
467,609 | <blockquote>
<p>Find the value of
$$\int _0 ^ \pi \dfrac{x}{1+\sin^2(x)} dx $$</p>
</blockquote>
<p>I have tried using $\int_a ^bf(x) dx=\int_a^b f(a+b-x)dx$</p>
<p>$\displaystyle \int _0 ^ \pi \dfrac{x}{1+\sin^2(x)} dx=\int _0 ^ \pi \dfrac{\pi-x}{1+\sin^2(x)} dx=I$</p>
<p>I couldn't go any further with that!</p... | Start wearing purple | 73,025 | <p>Now add the two integrals to get
$$2I=\pi\int_0^{\pi}\frac{dx}{1+\sin^2x}=\pi\int_{0}^{2\pi}\frac{dy}{3-\cos y},\tag{1}$$
where we used $\cos2x=1-2\sin^2x$ and the change of variables $y=2x$ to get the second equality.
The last integral in (1) can be easily calculated by residues (in fact the integrals of this type ... |
2,167,855 | <p>Let $f(t)$ be a differentiable function for $t$ $\in$ $[0,1]$ satisfying the above,</p>
<p>Does $f(t)$ have any fixed points?</p>
<p>I can easily prove there always exists fixed points without the second condition using $MVT$,</p>
<p>does $0$ $\leq$ $\frac{\partial f(t)}{\partial t}$ $\leq$ $\frac 12$ change anyt... | Pete Caradonna | 164,325 | <p>You definitely have fixed points, by an Intermediate Value theorem argument: let $g(t):= f(t)-t$. Then $g(0) \ge 0$ and $g(1)< 0$ so using only continuity one necessarily obtains a fixed point, which solves $g(t^*)=0$. </p>
<p>But, $f'(t) \ge 0$ implies that your function is always increasing, and $f' \le \fra... |
2,223,163 | <p>I don't have any idea on how to prove it, and I need it for one of my questions which is still unanswered: <a href="https://math.stackexchange.com/questions/2192947/what-is-the-largest-number-smaller-than-100-such-that-the-sum-of-its-divisors-is?noredirect=1#comment4521040_2192947">What is the largest number smaller... | Sri-Amirthan Theivendran | 302,692 | <p>Note that $4^n\equiv 4\mod 2$ and $4^n\equiv 4\mod 3$ so by the Chinese Remainder theorem $4^n\equiv 4\mod 6$ i.e. $10^n\equiv 4\mod 6$ </p>
|
2,223,163 | <p>I don't have any idea on how to prove it, and I need it for one of my questions which is still unanswered: <a href="https://math.stackexchange.com/questions/2192947/what-is-the-largest-number-smaller-than-100-such-that-the-sum-of-its-divisors-is?noredirect=1#comment4521040_2192947">What is the largest number smaller... | Bernard | 202,857 | <p>Because it is even and $\;10^n-4\equiv1^n-1\equiv0\mod3$, so it's divisible by $2$ and $3$.</p>
|
2,868,172 | <p>The power rule states that for any real number $r$, </p>
<p>$$\frac{d}{dx}x^r=rx^{r-1}$$</p>
<p>Now one common way to prove this is to use the definition $x^r=e^{r\ln x}$, where $e^x$ is defined as the inverse function of $\ln x$, which is in turn defined as $\int_1^x\frac{dt}{t}$.</p>
<p>But this puts the cart b... | Community | -1 | <p><strong>Hint:</strong></p>
<p>$$x^r:=\lim_{q\to r,\\q\in\mathbb Q}x^q.$$</p>
<p>Then</p>
<p>$$(x^r)'=\lim_{h\to0}\frac{\lim_{q\to r,\\q\in\mathbb Q}((x+h)^q-x^q)}{h}=\lim_{q\to r,\\q\in\mathbb Q}\lim_{h\to0}\frac{((x+h)^q-x^q)}{h}=\lim_{q\to r,\\q\in\mathbb Q}qx^{q-1}=rx^{r-1}.$$</p>
<p>The hard part is to justi... |
2,113,062 | <blockquote>
<p>Find $x^2+y^2$ if $x^2-\frac{2}{x}=3y^2$ and $y^2-\frac{11}{y}=3x^2$.</p>
</blockquote>
<p>My try:</p>
<p>$$\frac{y^2-\frac{11}{y}}{3}-\frac{2}{x}=3y^2$$</p>
<p>then?</p>
| dxiv | 291,201 | <p>[ <em>EDIT</em> #3 ] The previously posted answer remains copied below, under <em>Alternative Solution</em>.</p>
<p>Write the equations as:</p>
<p>$$
\begin{cases}
\begin{align}
2 &= x\,\left(x^2-3y^2\right) \\
11 &= y\,\left(y^2-3x^2\right)
\end{align}
\end{cases}
$$</p>
<p>Square the two and add t... |
3,281,540 | <p>I wrote an algorithm by combining Fermat's Little Theorem and Euler's Method. However, I am experiencing a problem in Euler's method. </p>
<p>For instance, If I take <span class="math-container">$(A, B, M)$</span> such that <span class="math-container">$A^B mod(M)$</span>.</p>
<p>When the initial values are <span ... | Community | -1 | <p>Apply Fermat's little theorem and extentions, for both of <span class="math-container">$p_1,p_2$</span>, then apply Chinese remainder theorem ( with LCM extension if needed) to the exponents for each to find the intersection. Apply to B Solve mod M. </p>
|
2,374,282 | <p>I am trying to find all connected sets containing $z=i$ on which $f(z)=e^{2z}$ is one to one.
I have no idea how to approach.
Can someone give me some hints?
Thank you</p>
| Dave | 334,366 | <p>I wrote a small program (I am a <strong>beginner</strong> programmer, so it may not be a very efficient way of doing this) to compute the number of days passed between two dates. One would enter the first date in the form D/M/Y and then the second in the same form, and this will spit out the number of days in betwee... |
134,455 | <p>I have an expression which consists of terms with undefined function calls <code>a[n]</code>:</p>
<pre><code>example = 1 - c^2 + c a[1] a[2] + 1/2 c^2 a[1]^2 a[2]^2 + c a[1] a[3]
</code></pre>
<p>Now I want to transform each term with individual <code>a</code>s to a different function <code>v[m1,m2,m3]</code>, suc... | Edmund | 19,542 | <p>You may use <a href="http://reference.wolfram.com/language/ref/CoefficientRules.html" rel="nofollow noreferrer"><code>CoefficientRules</code></a> and may find the <a href="http://reference.wolfram.com/language/guide/PolynomialAlgebra.html" rel="nofollow noreferrer">Polynomial Algebra</a> guide useful.</p>
<pre><cod... |
604,836 | <p>Prove if <span class="math-container">$a \equiv c \pmod{n}$</span> and <span class="math-container">$b \equiv d \pmod n$</span> then <span class="math-container">$ab \equiv cd \pmod{n}$</span>.</p>
<p>I tried to use <span class="math-container">$(a-c)(b-d) = ab-ad-cb+cd$</span>, but it seem doesn't work.</p>
| jgon | 90,543 | <p>Hint:</p>
<p>Congruence relation is transitive, and if $a\equiv b \mod(n)$, then $ax\equiv bx \mod n$. Play around.</p>
|
2,435,505 | <p>It is a question from permutations and combination chapter and its ans is 48 as given in book! Please help me to do this. I am unable to figure out the solution! Please help!</p>
| TomGrubb | 223,701 | <p>Hint: you can tell whether or not a number is even by looking solely at its last digit. Try conditioning on the last digit and counting each case individually.</p>
|
723,707 | <p>I'm trying to understand what the relation is between the direct product and the quotient group. </p>
<p>If we let $H$ be a normal subgroup of a group $G$, then it is not too difficult to show that the set of all cosets of $H$ in $G$ forms a quotient group $G/H$:
\begin{equation}
G/H = \{ g H \mid g \in G \}
\end{e... | Pedro | 23,350 | <p>Suppose $G=H\times K$. We want to show $G/K\simeq H$. Consider the map $H\times K\to H\times 0\simeq H$ given by $(h,k)\to (h,0)$. What is the kernel of this? What is it's image? </p>
<p>In general, the whole point of the direct product is given groups $H$ and $K$ to form a group $G=H\times K$ with copies $\hat K=1... |
815,661 | <p>Let $m$ be the product of first n primes (n > 1) , in the following expression :</p>
<p>$$m=2⋅3…p_n$$</p>
<p>I want to prove that $(m-1)$ is not a complete square.</p>
<p>I found two ways that might prove this . My problem is with the SECOND way . </p>
<p><strong>First solution (seems to be working) :</strong> <... | lab bhattacharjee | 33,337 | <p>For any integer $\displaystyle x, x\equiv0, 1,2\pmod3$</p>
<p>$\displaystyle\implies x^2\equiv0,1^2\equiv1,2^2\equiv1\pmod3$</p>
|
900,884 | <p>If $ A = \{ m^n| \text{ } m, n \in Z \text { and } m, n \ge 2 \} $, then how find $\sum_{k \in A} \frac{1}{k-1} $?</p>
| barak manos | 131,263 | <p>Not a complete answer to your question, but if $A$ is a multi-set (where any element may appear more than once), then your series is larger than the sum of all the following series put together:</p>
<ul>
<li>$\sum\limits_{n=2}^{\infty}\frac{1}{2^n}=\frac{1}{2-1}-\frac{1}{2}=\frac{1}{2}$</li>
<li>$\sum\limits_{n=2}^... |
2,786,656 | <p>I know that if a group is generated by a single element then the group is abelian but does this mean that if a group is abelian then its conjugacy class is composed of a single element?</p>
| Aweygan | 234,668 | <p>Tsemo is correct, in that the fact that $T$ maps the unit sphere of $X$ bijectively onto the unit sphere of $Y$ implies that $T$ is an isometry. Furtermore, note that
\begin{align*}
\operatorname{Ran}(T^*)_\perp&=\{x\in X:\forall f\in Y^*,\langle T^*f,x\rangle=0\}\\
&=\{x\in X:\forall f\in Y^*, \langle f,T... |
813,715 | <p>Say I am asked to find, in expanded form without brackets, the equation of a circle with radius 6 and centre 2,3 - how would I go on about doing this?</p>
<p>I know the equation of a circle is $x^2 + y^2 = r^2$, but what do i do with this information?</p>
| please delete me | 153,520 | <p>The equation of a circle with centre $(a,b)$ and radius $r$ is $(x-a)^2+(y-b)^2=r^2$.</p>
|
813,715 | <p>Say I am asked to find, in expanded form without brackets, the equation of a circle with radius 6 and centre 2,3 - how would I go on about doing this?</p>
<p>I know the equation of a circle is $x^2 + y^2 = r^2$, but what do i do with this information?</p>
| Nicky Hekster | 9,605 | <p>The equation gets like this: $$(x-2)^2+(y-3)^2=36,$$which can bee seen as translating a circle with radius 6 and center $(0,0)$ (the equation you mentioned) to the new center $(2,3)$.</p>
|
3,469,252 | <p>At first: I am new to differential equations, so this question might seam a little bit obvious.</p>
<p>The differential eqation was <span class="math-container">$y'x^3 = 2y -5$</span>.
I rearranged it to: <span class="math-container">$\frac{dx^3}{dx} = \frac{d(2y-5)}{dy}$</span>.</p>
<p>The Problem is, if i derive... | Community | -1 | <p>Your equation is</p>
<p><span class="math-container">$$\frac{dy}{dx}x^3=2y-5.$$</span></p>
<p>You cannot rearrange it to</p>
<p><span class="math-container">$$\frac{dx^3}{dx} = \frac{d(2y-5)}{dy}$$</span></p>
<p>(why would those differentials appear at the numerators ?) but to</p>
<p><span class="math-container... |
2,704,102 | <p>Let $X,Y,Z$ be topological spaces. Is the following statement true?
$X \times Z \cong Y \times Z \implies X \cong Y$?
how would you prove it? </p>
<p>and I know that if $A \cong B$, and $a \in A$ that there is a $b \in B$, such that $A\setminus{\{a\}} \cong B\setminus{\{b\}}.$ How would you prove the same for remov... | JKEG | 217,837 | <p>In addition to the counterexamples given in Henno Brandsma's answer, there is a very pathological and very interesting counterexample: In '<a href="http://www.ams.org/journals/proc/2006-134-12/S0002-9939-06-08596-0/S0002-9939-06-08596-0.pdf" rel="nofollow noreferrer">A counterexample related to topological sums – Ya... |
525,885 | <p>Why is $$\frac{\sum_{i=1}^n a_i^{n+1}}{\sum_{i=1}^{n}a_i^n} \geq \frac{\sum_{i=1}^n a_i}{n}$$
where $n$ is some positive natural number, and all $a_i$s are assumed to be positive real number?</p>
| Noam D. Elkies | 93,983 | <p>This is the special case $(r,s) = (1,n)$ of the following inequality:</p>
<p><strong>Lemma.</strong>
<em>Suppose $r$, $s$, and $a_i$ ($1 \leq i \leq n$) are positive reals. Then</em>
$$
n \sum_{i=1}^n a_i^{r+s} \geq \sum_{i=1}^n a_i^r \sum_{i=1}^n a_i^s,
$$
<em>with equality</em> iff <em>the $a_i$ are all equal</e... |
29,670 | <p>I have $a_k=\frac1{(k+1)^\alpha}$ and $c_k=\frac1{(k+1)^\lambda}$, where $0<\alpha<1$ and $0<\lambda<1$, and we have a infinite sequence $x_k$ with the following evolution equation.
$$
x_{k+1}=\left(1-a_{k+1}\right)x_{k}+a_{k+1}c_{k+1}^{2}
$$
I have proven that $x_k$ is bounded and obviously positive. ... | Did | 6,179 | <p>Let us show that $x_k\to0$.</p>
<p>For every positive $u$, there exists a finite integer $k(u)$ such that $c_{k+1}^2\le u$ for every $k\ge k(u)$, hence $x_{k+1}\le(1-a_{k+1})x_k+a_{k+1}u$. </p>
<p>-- If there exists $k\ge k(u)$ such that $x_k\le u$, then $x_i\le u$ for every $i\ge k$, hence $\limsup x_i\le u$. </p... |
2,475,938 | <blockquote>
<p>How can I factor the polynomial <span class="math-container">$x^4-2x^3+x^2-1$</span>?</p>
</blockquote>
<p>This is an exercise in algebra. I have the solution showing that
<span class="math-container">$$
x^4-2x^3+x^2-1=(x^2-x-1)(x^2-x+1).
$$</span></p>
<p>But the solution does not show any details.... | Crescendo | 390,385 | <p>Let’s examine the first three terms of your quartic$$y=x^4-2x^3+x^2-1$$The coefficients are in the order of $1,-2,1$. Thus$$\begin{align*}x^4-2x^3+x^2-1 & =x^2(x^2-2x+1)-1\\ & =x^2(x-1)^2-1\\ & =\left[x(x-1)-1\right]\left[x(x-1)+1\right]\\ & =(x^2-x-1)(x^2-x+1)\end{align*}$$where the second to last s... |
2,233,138 | <p>Let ${x_n}$ be defined by </p>
<p>$$x_n : = \begin{cases} \frac{n+1}{n}, &\text{if } n \text{ is odd}\\
0,&\text{if } n \text{ is even}.
\end{cases}$$</p>
<p>I am pretty sure about $\lim_{n\to\infty}\inf x_n = 0$ </p>
<p>because if ${x_1} = 2$, $x_2 = 0$, $x_3 = 4/3$, $x_4 = 0$ so $\lim_{n->\infty}\... | Chappers | 221,811 | <p>Continuing on from your first equality,
$$ \sum_{i=1}^n \sum_{j=1}^n a_i a_j \leq \sum_{i=1}^n \sum_{j=1}^n | a_i a_j | \leq \sum_{i=1}^n \sum_{j=1}^n \frac{a_i^2 + a_j^2}{2} = \frac{n}{2}\sum_{i=1} a_i^2 + \frac{n}{2}\sum_{j=1} a_j^2 = n \sum_{i=1}^n a_i^2 $$
by the AM–GM inequality.</p>
|
4,327,537 | <p>I have a question which states that
"In a group of 23 people what is the probability that there are two people with the same birthday? Assume there are 365 days in a year. Ignore leap years and such complications. Assume there is an equal probability of a person being born on each day of the year.". I solv... | Thomas | 128,832 | <p>If you think of the graph of <span class="math-container">$f$</span> as a a mountain range then the condition <span class="math-container">$\frac{\partial f}{\partial y}(x_0,y_0) \neq 0$</span> means that, along the curve <span class="math-container">$f(x,y) =0$</span>, the mountain range has a nonzero slope in <spa... |
2,081,641 | <p>I figured out that $\lim_{n \to \infty}\frac{(3n^2−2n+1)\sqrt{5n−2}}{(\sqrt{n}−1)(1−n)(3n+2)} $ is $-\sqrt{5}$ but I don't know how to prove it.</p>
| 8hantanu | 395,310 | <p>Multiplying numerator and denominator with $\frac{1}{n^{5/2}}$, we get</p>
<p>$$\lim_{n\to\infty}\frac{(3-\frac{2}{n}+\frac{1}{n^2})(\sqrt{5-\frac{2}{n}})}{(1-\frac{1}{n^{1/2}})(\frac{1}{n}-1)(3+\frac{2}{n})}=\frac{3\times \sqrt 5}{-1 \times 3}=-\sqrt5$$</p>
<p>Since $\lim_{n\to\infty}\frac{1}{n}=0.$</p>
|
1,923,808 | <p>Suppose we have a function $g:\mathbb{R}\rightarrow\mathbb{R}$ defined
as:
$$
g(x)=x^{2}
$$
Now, we know that this function is not onto because it is not defined
for negative values of $g.$ However, that is because we have <em>defined</em> the mapping from the set of real numbers to the set of real numbers.
If in fa... | G Tony Jacobs | 92,129 | <p>Sometimes the codomain is specified for reasons outside of the function you're looking at. You might not know the range of your function, but you know some suitable codomain.</p>
<p>Here is a modest example from linear algebra. Let $A$ be an $m\times n$ matrix, then we can ask whether the equation $Ax=b$ has a solu... |
1,923,808 | <p>Suppose we have a function $g:\mathbb{R}\rightarrow\mathbb{R}$ defined
as:
$$
g(x)=x^{2}
$$
Now, we know that this function is not onto because it is not defined
for negative values of $g.$ However, that is because we have <em>defined</em> the mapping from the set of real numbers to the set of real numbers.
If in fa... | 5xum | 112,884 | <p>We don't <em>have</em> to allow for such cases, but it's easier. It's easier to look at real valued functions as all having the same domain, because then it's easier to look at compositions of functions.</p>
<p>If We would always limit functions to their domains, then a function like</p>
<p>$$f(x)=e^{\sin x}$$</p>... |
1,923,808 | <p>Suppose we have a function $g:\mathbb{R}\rightarrow\mathbb{R}$ defined
as:
$$
g(x)=x^{2}
$$
Now, we know that this function is not onto because it is not defined
for negative values of $g.$ However, that is because we have <em>defined</em> the mapping from the set of real numbers to the set of real numbers.
If in fa... | E. Joseph | 288,138 | <p>I agree with the current answers, and I want to give you another important example.</p>
<p>If you consider the mapping</p>
<p>$$\exp\colon \mathbb C\to \mathbb C,$$</p>
<p>it is not trivial but it is an important theorem that $\exp(\mathbb C)=\mathbb C\setminus \{0\}$.</p>
<p>Once you have proved that you can sa... |
3,712,256 | <p>I am trying to prove that: </p>
<blockquote>
<p>For nonempty subsets of the positive reals <span class="math-container">$A,B$</span>, both of which are bounded above, define
<span class="math-container">$$A \cdot B = \{ab \mid a \in A, \; b \in B\}.$$</span>
Prove that <span class="math-container">$\sup(A \cd... | Calum Gilhooley | 213,690 | <p>Here's a completely different idea. (I've left only the straightforward proof of Lemma 1 to be filled in.)</p>
<p><strong>Lemma 1.</strong> If <span class="math-container">$E$</span> is a nonempty subset of <span class="math-container">$\mathbb{R}$</span> that is bounded above, and <span class="math-container">$c &... |
25,707 | <p>[Edit: to explain some things that were not so clear in the original post]</p>
<p>I believe in being straightforward. Without linking to the specific question, people will just treat this as another vague gripe with nothing to discuss. But to talk about the specific issues with a specific post is equivalent to poin... | Pedro | 23,350 | <p>If you identify incorrect answers to a post, you can <em>downvote</em> and <em>comment</em>, and you can also <em>post an answer</em> you deem better and (hopefully) correct. The site is not immune to errors, and it is the duty of all of us to make sure we preserve a database of correct and useful answers. </p>
<p>... |
28,751 | <p>$X \sim \mathcal{N}(0,1)$, then to show that for $x > 0$,
$$
\mathbb{P}(X>x) \leq \frac{\exp(-x^2/2)}{x \sqrt{2 \pi}} \>.
$$</p>
| cardinal | 7,003 | <p>Since for $t \geq x > 0$ we have that $1 \leq \frac{t}{x}$,
$$
\mathbb{P}(X > x) = \frac{1}{\sqrt{2\pi}} \int_x^\infty 1 \cdot e^{-t^2/2} \,\mathrm{d}t \leq \frac{1}{\sqrt{2\pi}} \int_x^\infty \frac{t}{x} e^{-t^2/2} \,\mathrm{d}t = \frac{e^{-x^2/2}}{x \sqrt{2\pi}} .
$$</p>
|
1,473,318 | <blockquote>
<p>How many numbers can by formed by using the digits $1,2,3,4$ and $5$ without repetition which are divisible by $6$?</p>
</blockquote>
<p><strong>My Approach:</strong></p>
<p>$3$ digit numbers formed using $1,2,3,4,5$ divisible by $6$ </p>
<p>unit digit should be $2/4$ </p>
<p>No. can be $XY2$ &... | N. F. Taussig | 173,070 | <p>For a number to be divisible by $6$, it must be divisible by both $2$ and $3$. If it is divisible by $2$, it must be even, so the units digit must be $2$ or $4$. If it is divisible by $3$, the sum of its digits must be divisible by $3$.</p>
<p>The only one-digit positive integer that is divisible by $6$ is $6$ it... |
3,577,249 | <p>Let be <span class="math-container">$X$</span> a topological space and let be <span class="math-container">$Y\subseteq X$</span> such that <span class="math-container">$\mathscr{der}(Y)=\varnothing$</span>: so for any neighborhood <span class="math-container">$I_y$</span> of <span class="math-container">$y\in Y $</s... | drhab | 75,923 | <p>Not true in general.</p>
<p>Let <span class="math-container">$a\neq b$</span> and <span class="math-container">$\tau=\{\varnothing,\{a\},\{a,b\}\}$</span>.</p>
<p>Then <span class="math-container">$\tau$</span> is topology on <span class="math-container">$X=\{a,b\}$</span>.</p>
<p><span class="math-container">$Y... |
3,536,135 | <blockquote>
<p>There are five multiple choice questions on a test, with four choices per question. A student was given 10 questions to study for the test and the teacher picked 5 out of 10 questions to put on the test. The student memorizes 7 of the 10 answers of the questions given. If the student encounters the th... | WW1 | 88,679 | <p>Total ways of choosing the 5 questions is <span class="math-container">$\binom {10}5 $</span></p>
<p>To get 100% split into 4 cases</p>
<p>case 1: all five memorized , no lucky guesses</p>
<p><span class="math-container">$$ P_1 = \frac {\binom {7}5}{\binom {10}5} $$</span></p>
<p>case 2: 4 memorized , 1 lucky g... |
75,005 | <p>Let's imagine a guy who claims to possess a machine that can each time produce a completely random series of 0/1 digits (e.g. $1,0,0,1,1,0,1,1,1,...$). And each time after he generates one, you can keep asking him for the $n$-th digit and he will tell you accordingly.</p>
<p>Then how do you check if his series is <... | GM2001 | 17,808 | <p>All the sequences you mentioned have a really low Kolmogorov complexity, because you can easily describe them in really short space. A random sequence (as per the usual definition) has a high Kolmogorov complexity, which means there is no instructions shorter then the string itself that can describe or reproduce the... |
2,568,157 | <p>Consider the following:</p>
<p>$$(1^5+2^5)+(1^7+2^7)=2(1+2)^4$$</p>
<p>$$(1^5+2^5+3^5)+(1^7+2^7+3^7)=2(1+2+3)^4$$</p>
<p>$$(1^5+2^5+3^5+4^5)+(1^7+2^7+3^7+4^7)=2(1+2+3+4)^4$$</p>
<p>In General is it true for further increase i.e.,</p>
<p>Is</p>
<p>$$\sum_{i=1}^n i^5+i^7=2\left( \sum_{i=1}^ni\right)^4$$ true $\f... | lhf | 589 | <p>Both sides are polynomials in $n$ of degree $8$. Since they coincide for $n=0,\dots,8$, they are equal.</p>
<p>Any $9$ points will do. Taking $n=-4,\dots,4$ is probably easier to do by hand.</p>
|
2,568,157 | <p>Consider the following:</p>
<p>$$(1^5+2^5)+(1^7+2^7)=2(1+2)^4$$</p>
<p>$$(1^5+2^5+3^5)+(1^7+2^7+3^7)=2(1+2+3)^4$$</p>
<p>$$(1^5+2^5+3^5+4^5)+(1^7+2^7+3^7+4^7)=2(1+2+3+4)^4$$</p>
<p>In General is it true for further increase i.e.,</p>
<p>Is</p>
<p>$$\sum_{i=1}^n i^5+i^7=2\left( \sum_{i=1}^ni\right)^4$$ true $\f... | Hypergeometricx | 168,053 | <p>Here are some interesting observations, which are too long to be included as a comment. </p>
<p>A useful reference can be found <a href="https://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Beardon201-213.pdf" rel="nofollow noreferrer">here</a>.</p>
<p>Denoting $\displaystyle\sum_{r=1}^n r^m=\sigma_... |
3,482,441 | <p>Very confused on how to deal with these direct sum problems.</p>
<p>Problem:Suppose <span class="math-container">$U=\{(x,y,x+y,x-y,2x) \in \mathbb{F}^{5}:x,y \in \mathbb{F}\}$</span></p>
<p>Find a subspace <span class="math-container">$W$</span> of <span class="math-container">$\mathbb{F}^{5}$</span> such that <sp... | Community | -1 | <p>I don't see your answer panning out.</p>
<p>Here's another method:</p>
<p><span class="math-container">$U$</span> is the span of <span class="math-container">$\{(1,0,1,1,2), (0,1,1,-1,0)\}$</span>. (To see this, plug in <span class="math-container">$x=1,y=0$</span>, and <span class="math-container">$x=0, y=1$</span>... |
3,482,441 | <p>Very confused on how to deal with these direct sum problems.</p>
<p>Problem:Suppose <span class="math-container">$U=\{(x,y,x+y,x-y,2x) \in \mathbb{F}^{5}:x,y \in \mathbb{F}\}$</span></p>
<p>Find a subspace <span class="math-container">$W$</span> of <span class="math-container">$\mathbb{F}^{5}$</span> such that <sp... | Anurag A | 68,092 | <p>Since <span class="math-container">$U=\text{span}\left(\left\{\right(1,0,1,1,2), \, (0,1,1,-1,0)\}\right)$</span> so to find a <span class="math-container">$W$</span>, one can choose <span class="math-container">$W=U^{\perp}$</span> (orthogonal complement of <span class="math-container">$U$</span>), i.e.
<span clas... |
281,735 | <blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="https://math.stackexchange.com/questions/202452/why-is-predicate-all-as-in-allset-true-if-the-set-is-empty">Why is predicate “all” as in all(SET) true if the SET is empty?</a> </p>
</blockquote>
<p>In don't quite understand this quantification ov... | André Nicolas | 6,312 | <p>A peculiar explanation. </p>
<p>Whatever $Q(y)$ may be, it is true that for all $y\in \emptyset$, the sentence $Q(y)$ is true. For the empty set is $\dots$ empty. Every unicorn likes wine. </p>
|
4,554,831 | <blockquote>
<p>Let <span class="math-container">$(X,d)$</span> be a metric space. Prove that if the point <span class="math-container">$x$</span> is on the boundary of the open ball <span class="math-container">$B(x_0,r)$</span> then <span class="math-container">$d(x_0,x)=r$</span>.</p>
</blockquote>
<p>I find this di... | Sam | 530,289 | <p>We start by defining the boundary of a set.
Let <span class="math-container">$(X, \mathcal{T})$</span> be a topological space. Let <span class="math-container">$A$</span> be a set in <span class="math-container">$X$</span>, the boundary of <span class="math-container">$A$</span> is defines as
<span class="math-conta... |
2,005,555 | <p>When I was solving a DE problem I was able to reduce it to </p>
<p>$$e^x \sin(2x)=a\cdot e^{(1+2i)x}+b\cdot e^{(1−2i)x}.$$ </p>
<p>For complex $a,b$. Getting one solution is easy $(\frac{1}{2i},-\frac{1}{2i})$ but I was wondering what are all the values for complex $a,b$ that satisfy the equation. </p>
| DanielWainfleet | 254,665 | <p>When $x=0$ we have $0=a+b.$ So $b=-a.$ So $e^x\sin 2x=ae^{(1+2i)x}-ae^{(1-2i)x}=2iae^x\sin 2x.$ When $\sin 2x\ne 0$ this reduces to $1=2ia.$</p>
|
4,627,821 | <p>Consider the difference equations</p>
<p><span class="math-container">$$x(k+1) = f(x(k)) \qquad (1)$$</span></p>
<p>and</p>
<p><span class="math-container">$$y(k+1) = g(y(k)) \qquad (2)$$</span></p>
<p>where <span class="math-container">$g = f \circ f$</span>.</p>
<p>In <em>An Introduction to Difference Equations (3... | Adam Rubinson | 29,156 | <p><span class="math-container">$f\ $</span> is bounded. Suppose <span class="math-container">$\ y_1\ $</span> is a lower bound and <span class="math-container">$\ y_2\ $</span> is an upper bound of <span class="math-container">$\ f.\ $</span></p>
<p>Let <span class="math-container">$\ \varepsilon > 0.\ $</span> Sin... |
1,684,124 | <p>Here is my attempt:</p>
<p>$$ \frac{2x}{x^2 +2x+1}= \frac{2x}{(x+1)^2 } = \frac{2}{x+1}-\frac{2}{(x+1)^2 }$$</p>
<p>Then I tried to integrate it,I got $2\ln(x+1)+\frac{2}{x+1}+C$ as my answer. Am I right? please correct me if I'm wrong.</p>
| 3SAT | 203,577 | <p>$$=\int\frac{2x+2}{x^2+2x+1}dx-2\int\frac{dx}{x^2+2x+1}$$</p>
<p>set $t=x^2+2x+1$ and $dt=(2x+2)dx$</p>
<p>$$=\int\frac{1}{t}-2\int\frac{dx}{(x+1)^2}dx$$</p>
<p>Set $\nu=x+1$ and $d\nu=dx$</p>
<p>$$=\int\frac{1}{t}-2\int\frac{1}{\nu ^2}d\nu=\ln|t|+\frac{2}{\nu}+\mathcal C=\color{red}{\ln|x^2+2x+1|+\frac{2}{x+1}+... |
1,501,940 | <p>This is related to a <a href="https://math.stackexchange.com/questions/1501852/why-does-this-statement-not-hold-when-me-0/1501925#1501925">question</a> I just asked, that I now think was based on wrong assumptions.</p>
<p>It is true that if <span class="math-container">$f=a$</span> a.e. on the interval <span class="... | saz | 36,150 | <blockquote>
<p>Now $E_0 = h^{-1}(\mathbb{R} \backslash \{0\})$ [...] Since $E_0$ is open and $m(E_0)=0$, it must be empty.</p>
</blockquote>
<p>No, that's not correct. The continuity of $h$ gives that</p>
<p>$$U := h^{-1}(\mathbb{R} \backslash \{0\}) = \{x \in \mathbb{R}; g(x) \neq f(x)\}$$</p>
<p>is open. This s... |
1,858,095 | <p>Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a function with:</p>
<p>$$f(x) = x - \arctan{x}$$</p>
<p>We consider the sequence $(x_{n})$ with $x_{0} > 0$ and $x_{n + 1} = f(x_{n})$, for any $n \in \mathbb{N}$.</p>
<p>Prove that $(x_{n})$ is convergent and find its limit.</p>
<p>So far, I've proved that $f(x... | amcalde | 168,694 | <p>Look at the graph: (from Wolfram Alpha)</p>
<p><a href="https://i.stack.imgur.com/5krNw.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/5krNw.png" alt="enter image description here"></a></p>
<p>The way to answer this question is to prove that, say for $x > 0$, $f(x) < x \le 0$. It should b... |
218,020 | <p>I came across the following statement: Let $R$ be a complete local Noetherian commutative ring. If $A$ is a commutative $R$-algebra that is finitely generated and free as a module over $R$, then $A$ is a semi-local ring that is the direct product of local rings. (I'm unsure if completeness or the Noetherian conditio... | Gustavo Hospital | 219,590 | <p>If you have any sum s and want to know the smallest number which digits add up to that sum, you just have to use modulo 9 arithmetic. </p>
<p>You calculate s mod 9, that will be your first cypher to the left, then you solve for q and that will be how many 9s you add to the right.</p>
<p>For example:</p>
<p>16 mo... |
110,896 | <p>Now I have a function, say $f(k,z)=e^{-kz}(1+kz)$</p>
<p>I want to find the $n$th $\log$ derivative with respect to z.
like $(z\partial_z)^{(n)}f(k,z)$ (or $(\partial_{\ln z})^{(n)}f(k,z)$
if you like), where the $(n)$ denotes that we take the derivative $n$
times. </p>
<p>I found the answer in <a href="https://... | J. M.'s persistent exhaustion | 50 | <p>There is a nice closed form in terms of the Stirling subset numbers for these:</p>
<pre><code>lDerivative[n_Integer?Positive][f_] := Composition[Function, Evaluate] @
(StirlingS2[n, Range[n]].Table[#^\[FormalK] Derivative[\[FormalK]][f][#],
{\[FormalK], 1, n}])
</code></pre>
<p>For i... |
3,409,068 | <p>I have to find (and prove) the infimum and supremum of the following set:</p>
<p><span class="math-container">$M_1:=\{x\in\mathbb{Q} \mid x^2 < 9\}$</span></p>
<p>On first glance, I would say:</p>
<p><span class="math-container">$\inf M_1=-3 $</span><br>
<span class="math-container">$\sup M_1=3$</span></p>
<p... | Mark Viola | 218,419 | <p>In order to be analytic at <span class="math-container">$x=0$</span>, the function and all of its derivatives must exist in a neighborhood of <span class="math-container">$x=0$</span>. However, if <span class="math-container">$f(x)=x\log(|x|)$</span> and <span class="math-container">$f(0)=0$</span>, we see that <sp... |
69,711 | <blockquote>
<p>Find an equation of the tangent line to the graph of $y= \sqrt{x-3}$ that is perpendicular to $6x+3y-4=0$. </p>
</blockquote>
<p>I don't understand what it's asking. Is this the normal line? How do I solve this?</p>
| Altar Ego | 11,020 | <p>First, determine the slope of the line $6x + 3y - 4 = 0$. Here, $m = -2$.<br><br>
Then we calculate the perpendicular slope to $-2$ as $1/2$ (why?).<br><br></p>
<p>Then we want to find where the slope of the tangent to $y = \sqrt{x - 3}$ is equal to $1/2$.<br>
In other words, where $y' = \frac{1}{2\sqrt{x - 3}}... |
1,811,443 | <p>Let $(a_n)$ be a sequence of rational numbers, where <strong>all rational numbers are terms</strong>. (<em>i.e. enumeration of rational numbers</em>)</p>
<p>Then, is there any convergent sub-sequence of $(a_n)$?</p>
| E. Joseph | 288,138 | <p>Yes there is. </p>
<p>Consider the interval $I:=[0,1]$. Since it contains infinitely many rationals, your sequence will have value in $I$ forever.</p>
<p>So you can extract the sub-sequece such that all the values $(a_{\varphi(n)})$ of the new sequence are in $I$ :</p>
<p>$$\forall n \quad a_{\varphi(n)}\in I.$$<... |
2,497,875 | <p>Define $\sigma: [0,1]\rightarrow [a,b]$ by $\sigma(t)=a+t(b-a)$ for $0\leq t \leq 1$. </p>
<p>Define a transformation $T_\sigma:C[a,b]\rightarrow C[0,1]$ by $(T_\sigma(f))(t)=f(\sigma(t))$ </p>
<p>Prove that $T_\sigma$ satisfies the following:</p>
<p>a) $T_\sigma(f+g)=T_\sigma(f)+T_\sigma(g)$</p>
<p>b) $T_\sigma... | lab bhattacharjee | 33,337 | <p>Let $1-\sqrt x=y\implies x=(1-y)^2$</p>
<p>$$\lim_{x\to 1}\frac{\sin(1-\sqrt{x})}{x-1}=\lim_{y\to0}\dfrac{\sin y}y\cdot\lim_{y\to0}\dfrac1{y-2}=?$$</p>
|
129,287 | <p>Suppose $p(x_1, x_2, \cdots, x_n)$ is a symmetric polynomial. Given any univariate polynomial $u$, we can define a new polynomial $q(x_1, x_2, \cdots, x_{n+1})$ as</p>
<p>$q(x_1, x_2, \cdots, x_{n+1}) = u(x_1)p(x_2, x_3, \cdots, x_{n+1}) + u(x_2)p(x_1, x_3, \cdots, x_{n+1}) + \cdots \\ \phantom{q(x_1, x_2, \cdots, ... | P Vanchinathan | 22,878 | <p>Don't know if there is a name. Possibly this is known to Newton; the inductive proof of Newton's theorem on elementary symmetric polynomials goes along similar lines.</p>
<p>When we start with some polynomial and take the sum over its orbit under $S_{n+1}$ it will be invariant under $S_{n+1}$. You are starting wit... |
3,210,295 | <p>I wondered if anybody knew how to calculate a percentage loss/gain of a process over time?</p>
<p>Suppose for example Factory A conducted activity over 6 periods.</p>
<p>In t-5, utilisation of resources was: 80%
t-4: 70%
t-3: 80%
t-2: 100%
t-1: 90%
t: 75%</p>
<p>Therefore, but for the exception of two periods ago... | Community | -1 | <p>For this example, just over 1 day production has been lost if you think of all days having similar normal production, as Ross Millikan's answer shows. Some processes compound though. If this were percentage left of a given stock value on each day without reset, then it works as follows:<span class="math-container">$... |
1,977,031 | <p>How would I parameterise this curve in 3D?
I am confused since the diagrams deal with three variables in total – should I use complex numbers? I'm only used to two diagrams and haven't encountered a problem with three like this.</p>
| hamam_Abdallah | 369,188 | <p>Hint:</p>
<p>in $xy$ plane, you use polar coordinates as follows.</p>
<p>$x=r\cos(t)$ and $y=r\sin(t)$ with</p>
<p>$r=ae^{-bt}$ and you choose the right parameters $a,b.$</p>
<p>to get $z$ , you replace $x$.</p>
|
2,333,847 | <p>A function $f(x) = k$ and the domain is $\{-2,-1,\dotsc,3\}$. Would I say
$$x = \{-2,-1,\dotsc,3\}\quad\text{or}\quad x \in \{-2,-1,\dotsc,3\} \ ?$$
Thanks. </p>
| DMcMor | 155,622 | <p>Here's one way to look at it:</p>
<p>When you write $x = \lbrace -2, -1, \dotsc ,3\rbrace$ you are saying "$x$ is equal to the set consisting of the integers $-2$ through $3$". If $x$ were really a set then you'd be fine, but if you want to say that the set consists of the possible values for $x$, that is it's the... |
80,966 | <p>I wonder if it there exists a topological compact group $G$ (by compact, I mean Hausdorff and quasi-compact) and a non-zero group morphism
$\phi : G \to \mathbb{Z}$ (without assuming any topological condition on this morphism).</p>
<p>For compact Lie groups, using the exponential map, the answers is no, but in ge... | Sean Eberhard | 20,598 | <p>Sorry for resurrecting such an old question, but I think we can give a much simpler proof here. We'll reduce the problem from $G$ to the Bohr compactification $B\mathbf{Z}$ of $\mathbf{Z}$, then from $B\mathbf{Z}$ to the profinite completion $\hat{\mathbf{Z}}=\prod_p\mathbf{Z}_p$ of $\mathbf{Z}$, and then we'll argu... |
1,642,029 | <p>I'm looking at my textbooks steps for calculating the complexity of bubble sort...and it jumps a step where I don't know what exactly they did. </p>
<p><a href="https://i.stack.imgur.com/XaztP.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/XaztP.png" alt="enter image description here"></a></p>
... | Svetoslav | 254,733 | <p>You can take as many subsequences of $\{a_n\}$ as you want. </p>
<p>The subsequence $\{b_n\}$ is already <strong>convergent</strong>, so any further subsequence of $\{b_n\}$ will be convergent and will have the same limit as $\{b_n\}$.</p>
<p>If you have omitted only a finite number of terms from $\{a_n\}$ to get ... |
660,315 | <p>Let $A,B,C$ be sets. Identify a condition such that $A \cap C = B \cap C$ together with your condition implies $A=B$. Prove this implication. Show that your condition is necessary by finding an example where $A \cap C = B \cap C$, but $ A \neq B$</p>
<p>Edit: I've read the wrong proposition/definition. UGH! The que... | Mauro ALLEGRANZA | 108,274 | <p>The condition for having </p>
<blockquote>
<p>$A \cap C = B \cap C$ but at the same time $A \neq B$</p>
</blockquote>
<p>impose you that</p>
<blockquote>
<p>$A \cap C = \emptyset = B \cap C$</p>
</blockquote>
<p>This does not imply that $A$ and $B$ are both <em>empty</em>.</p>
<p>Let $\quad C = \{ x \in \ma... |
1,385,936 | <p><em>I was wondering how to approximate $\sqrt{1+\frac{1}{n}}$ by $1+\frac{1}{2n}$ without using Laurent Series.</em></p>
<p>The reason why I ask was because using this approximation, we can show that the sequence $(\cos(\pi{\sqrt{n^{2}-n}})_{n=1}^{\infty}$ converges to $0$. This done using a mean-value theorem or L... | Mark Bennet | 2,906 | <p>This is not so different from other approaches (exactly equivalent to some), but has a different perspective based on pure computation. </p>
<p>Suppose $a$ is an estimate for the square root of $b$ so that $$a^2=b+h$$where $h$ is taken to be small (see below).</p>
<p>Then $$\left(a-\frac h{2a}\right)^2=a^2-h+\left... |
65,192 | <p>Multivariate parameters appear to present a jagged appearance of integrands (using default Runge-Kutta ODE integration intervals?) in ParametricPlot3D plotting on a single argument. </p>
<p>Higher Mesh.. ing to 200 improves sectors' jagging (large step secants appearing instead of tangent) somewhat, but still colo... | mgamer | 19,726 | <p>I´m not quite sure whether I understood your problem right, but I increased the Mesh, set the PlotPoints higher (I just choose a higher number) and MaxRecursion to 1, i.e.:</p>
<pre><code>SpC1 = ParametricPlot3D[{r[th] Cos[th], z[th], r[th] Sin[th]}, {t, 0,
2 Pi}, {th, 0, thmax}, PlotStyle -> {Thick, Magenta... |
4,144,083 | <p>I am currently doing an internship in a research laboratory ( I am in my third year of Bachelor ) and I'm really struggling with the things I have to do.</p>
<p>For instance, here's something I'm having trouble with.</p>
<p>Let <span class="math-container">$L$</span> be a finite Galois extension of <span class="math... | hm2020 | 858,083 | <p><strong>Question:</strong> "Please forgive my mistakes, I'm still learning, and English isn't my first language. Thank you for your help!"</p>
<p><strong>Answer:</strong> It seems you may reduce to the case of <span class="math-container">$I$</span> being a power of a prime ideal. Since <span class="math-c... |
2,908,361 | <p>I tried to solve this inequality by taking the square outside the floor function $[y]$ (greatest integer less than $y$)but it was wrong since if $x=2.5$ then $[x]= 2$ and $x^2=4$ while $[x^2]=[6.25]=6$.</p>
| dmtri | 482,116 | <p>I think the answer should be: $x\le-\sqrt{27}\qquad $ or $\qquad -5\le{x}\le-\sqrt{22}\qquad $
or
$\qquad 0\le{x}\qquad $
Here is a sketch:</p>
<p>For $$-4\le{x}\le-1$$ it is $$x^2+5x+4\le0$$ or $$x^2+5x\le-4$$, but also
$$ [x^2]+5[x]\le{x^2+5x}$$
so there is no solution in $[-4, -1]$.</p>
<p>For $$-5\le{x}\... |
336,943 | <p>Given $P(A) = 0.5$ and $P(A \cup (B^c \cap C^c)^c)=0.8$.</p>
<p>Determine $P(A^c \cap (B \cup C))$.</p>
<p>I know from DeMorgans law that: $(B^c \cap C^c)^c = (B \cup C)$.</p>
<p>Edit:</p>
<p>Also how can I "prove" that P(X)=P(Y) if and only if $P(X \cap Y^c) = P(X^c \cap Y)$? </p>
| Berci | 41,488 | <ol>
<li>Let $X:=B\cup C$, as you noted, we have $P(X\cup A)=0.8$ and $P(X\cap A^c)=P(X\setminus A)$ is the question. For this, use that $X\setminus A$ and $A$ are disjoint and that $(X\setminus A)\cup A=X\cup A$. (All gets clear once you draftly draw them..)</li>
<li>As $X\cap Y$ is disjoint to $X\setminus Y=X\cap Y^c... |
1,316,008 | <p><img src="https://i.stack.imgur.com/xNGPi.png" alt="enter image description here"></p>
<p>The problem is shown in the image. I'm not able to post images yet.. What are the next steps to to find how tall the triangle is? So far i see, that the 3 triangles are similar; however, even by these similarities and by the ... | Narasimham | 95,860 | <p>The angles in the figure are not drawn to proper angular proportion. BY is not convincing as a bisector of $XBA$. </p>
<p>Anyway, we can even verbally show its error.</p>
<p>If $ PAB + PBA$ is 90 degrees, due to bisections given the double angles sum </p>
<p>$YAB + XBA$ must be 180 degrees.</p>
<p>It makes AY p... |
336,827 | <p>A covering map $p:C\to X$ is finite when for each $x\in X$ we have $|p^{-1}(x)|<\infty.$ I have to prove that such a covering map has to be closed. I'm having trouble with it. </p>
<p>When $p$ is a covering map, we can take open neighborhoods $U_x$ of every point $x\in X$ such that $p^{-1}(U_x)$ is a disjoint un... | Georges Elencwajg | 3,217 | <p>Let $(U_i)_{i\in I}$ be an arbitrary open cover of a a topological space $X$.<br>
The crucial remark is that a subset $F\subset X$ is closed in $X$ if and only if for each $i\in I$ the intersection $F\cap U_i$ is closed in $U_i$. </p>
<p>In your situation, if you have a closed subset $A\subset C$ you should apply ... |
1,324,062 | <p>Evaluate: </p>
<blockquote>
<p>$$\lim_{h \rightarrow 0} \frac{e^{2h}-1}{h}$$</p>
</blockquote>
<p>Now one way would be using the Maclaurin expansion for $e^{2x}$</p>
<p>However, can we solve it using the definition of the derivative (perhaps considering $f(x)=e^x$)? Many thanks for your help! $$$$
EDIT: I forgo... | Vincenzo Oliva | 170,489 | <p>Noting $e^{2h}-1=(e^h-1)(e^h+1)$ and recalling a notable limit does the job.</p>
|
1,324,062 | <p>Evaluate: </p>
<blockquote>
<p>$$\lim_{h \rightarrow 0} \frac{e^{2h}-1}{h}$$</p>
</blockquote>
<p>Now one way would be using the Maclaurin expansion for $e^{2x}$</p>
<p>However, can we solve it using the definition of the derivative (perhaps considering $f(x)=e^x$)? Many thanks for your help! $$$$
EDIT: I forgo... | MCT | 92,774 | <p>Note that, with $f(x) = e^x$, we have $\lim \limits_{h \to 0} \dfrac{e^h - 1}{h} = f'(0) = 1$.</p>
<p>Therefore, $\lim \limits_{h \to 0} \frac{e^{2h} - 1}{h} = \lim \limits_{h \to 0} \frac{(e^h - 1)}{h}(e^h + 1) = 1 \cdot (1 + 1) = \boxed 2$.</p>
|
2,231,391 | <p>I'd like a low-discrepancy sequence of points over a 3D-hypercube <span class="math-container">$[-1,1]^3$</span>, but don't want to have to commit to a fixed number <span class="math-container">$n$</span> of points beforehand, that is just see how the numerical integration estimates develop with increasing numbers ... | Martin Roberts | 575,045 | <p><em>As the OP cross-posted this <a href="https://math.stackexchange.com/questions/2231391/how-can-one-generate-an-open-ended-sequence-of-low-discrepancy-points-in-3d/2845473#2845473">question from Math stackexchange</a>, I have also cross-posted the answer that I wrote there.</em></p>
<h1>#</h1>
<p>The simplest tr... |
2,231,391 | <p>I'd like a low-discrepancy sequence of points over a 3D-hypercube <span class="math-container">$[-1,1]^3$</span>, but don't want to have to commit to a fixed number <span class="math-container">$n$</span> of points beforehand, that is just see how the numerical integration estimates develop with increasing numbers ... | Paul B. Slater | 217,460 | <p>The original question was posed in April, 2017. Now, a few days ago, I extended the question </p>
<p><a href="https://mathematica.stackexchange.com/questions/143457/how-can-one-generate-an-open-ended-sequence-of-low-discrepancy-points-in-3d">https://mathematica.stackexchange.com/questions/143457/how-can-one-generat... |
994,620 | <p>I need to solve $f(2x)=(e^x+1)f(x)$. I am thinking about Frobenius type method:
$$\sum_{k=0}^{\infty}2^ka_kx^k=\left(1+\sum_{m=0}^{\infty}\frac{x^m}{m!}\right)\sum_{n=0}^{\infty}a_nx^n\\
\sum_{k=0}^{\infty}(2^k-1)a_kx^k=\left(\sum_{m=0}^{\infty}\frac{x^m}{m!}\right)\left(\sum_{n=0}^{\infty}a_nx^n\right)=\sum_{m=0}^{... | RE60K | 67,609 | <p>As suggested by @Shivang in comments, substituting $g(x)=f(x)/(e^x-1)$ gives:
$$f(2x)=(e^x+1)f(x)\implies g(2x)=g(x)$$
From a known process one can then prove that g is constant and thus f is $a(e^x-1)$ where a is a constant.</p>
|
994,620 | <p>I need to solve $f(2x)=(e^x+1)f(x)$. I am thinking about Frobenius type method:
$$\sum_{k=0}^{\infty}2^ka_kx^k=\left(1+\sum_{m=0}^{\infty}\frac{x^m}{m!}\right)\sum_{n=0}^{\infty}a_nx^n\\
\sum_{k=0}^{\infty}(2^k-1)a_kx^k=\left(\sum_{m=0}^{\infty}\frac{x^m}{m!}\right)\left(\sum_{n=0}^{\infty}a_nx^n\right)=\sum_{m=0}^{... | bot | 92,881 | <p>What about $f(x)=D(x)(e^x-1)?$</p>
<p>But the additional condition:</p>
<p>There exist $\lim\limits_{x\to 0} \frac{f(x)}{x}$</p>
<p>will delete such counterexamples </p>
|
2,323,223 | <p>I am having a hard time with this question for some reason. </p>
<p>You and a friend play a game where you each toss a balanced coin. If the upper faces on
the coins are both tails, you win \$1; if the faces are both heads, you win \$2; if the coins
do not match (one shows head and the other tail), you lose \$1.
C... | Satish Ramanathan | 99,745 | <p>$Variance= 50 Var(X) = 50.[{.25(1-.25)^2 + .25(2-.25)^2 +.5*(-1-.25)^2}]=84.375$</p>
|
3,516,189 | <p>I've been struggling with the following exercise for quite some time already:</p>
<blockquote>
<p>Consider a linear space <span class="math-container">$\mathbb{V} = \mathcal{C}\left(\left[a, b\right]\right)$</span> and let <span class="math-container">$f_{1},\ldots, f_{n}$</span> be linearly independent functions... | user1551 | 1,551 | <p><em>(I'll call this a recursive algorithm rather than mathematical induction, but one may disagree.)</em></p>
<p>Let <span class="math-container">$\mathbf f=(f_1,f_2,\ldots,f_n)^T$</span>.</p>
<ul>
<li>Pick any nonzero vector <span class="math-container">$v_1$</span>.</li>
<li>Since <span class="math-container">$f... |
326,094 | <p>Suppose <span class="math-container">$(f_n)_n$</span> is a countable family of entire, surjective functions, each <span class="math-container">$f_n:\mathbb{C}\to\mathbb{C}$</span>. Can one always find complex scalars <span class="math-container">$(a_n)_n$</span>, not all zero, such that <span class="math-container... | Noam D. Elkies | 14,830 | <p>One expects there to be no such <span class="math-container">$a_n$</span> in general, because the
"typical" entire functions is surjective (those that aren't are of the
special form <span class="math-container">$z \mapsto c + \exp g(z)$</span>). An explicit example is
<span class="math-container">$f_n(z) = \cos z/... |
2,856,180 | <p>I've been learning some about counting and basic combinatorics. But some scenarios were not explained in my class...</p>
<p><strong>Example problem:</strong> You are given 6 tiles. 1 is labeled "1", 2 are labeled "2", and 3 are labeled "3".</p>
<p><strong>Problem 1:</strong> How many different ways can you arrange... | jgon | 90,543 | <p>Not sure this is the best method, but the way I would actually solve such a set of questions would be the following:</p>
<p>Use generating function methods for the second question.
The explicit collection of submultisets of the six tiles is given by the terms of $(1+x)(1+y+y^2)(1+z+z^2+z^3)$ (the $x$ terms correspo... |
271,343 | <p>I need help finding the integral of $\sin(\sqrt{x})dx$. I have the answer here but would like to know how to get there. </p>
| Hanul Jeon | 53,976 | <p><strong>Hint</strong>: $$\sin(\sqrt{x}) = \frac{\sqrt{x}\sin(\sqrt{x})}{\sqrt{x}}$$
and take $u=\sqrt{x}$, $2 du=\frac{dx}{\sqrt{x}}$.</p>
|
271,343 | <p>I need help finding the integral of $\sin(\sqrt{x})dx$. I have the answer here but would like to know how to get there. </p>
| Community | -1 | <p>$$J = \underbrace{\int \sin(\sqrt{x})dx = \int \sin(t) 2t dt }_{\sqrt{x} = t \implies x = t^2 \implies dx = 2t dt}$$
$$I = \int t \sin(t) dt = -\int t d(\cos(t)) = - \left(t \cos(t) - \int \cos(t) dt \right) = - t \cos(t) + \sin(t) + c$$
Hence, $$J = - 2t \cos(t) + 2\sin(t) + k = -2 \sqrt{x} \cos(\sqrt{x}) + 2 \sin(... |
3,710,804 | <p><span class="math-container">$$f(x) = \int \frac{\cos{x}(1+4\cos{2x})}{\sin{x}(1+4\cos^2{x})}dx$$</span></p>
<p>I have been up on this problem for an hour, but without any clues. </p>
<p>Can someone please help me solving this?</p>
| trancelocation | 467,003 | <p>It is quite straight forward after rewriting the integrand:</p>
<p><span class="math-container">$$\frac{\cos x (1+4\cos 2x )}{\sin x (1+4\cos^2 x)}= \frac{\cos x}{\sin x}\left(1 - \frac{4\sin^2 x}{1+4\cos^2 x}\right)$$</span>
<span class="math-container">$$= \frac{\cos x}{\sin x} - \frac{2\sin 2x}{3+2\cos 2x}$$</sp... |
2,141,182 | <p>In the case of $$\sqrt{(x_n-\ell_1)+(y_n-\ell_2)}\leq \sqrt{(x_n-\ell_1)^2} + \sqrt{(y_n-\ell_2)^2} = |x_n-\ell_1|+|y_n-\ell_2|$$</p>
<p>it is true, if we take the rise the two sides in the power of $2$ we get:</p>
<p>\begin{align}
& (x_n-\ell_1)+(y_n-\ell_2)\leq \left( \sqrt{(x_n-\ell_1)^2}+\sqrt{(y_n-\ell_2)... | Matija Sreckovic | 367,539 | <p>Quite simply, the answer to your question from the title is "yes", if we assume that $a$ and $b$ are non-negative real numbers.</p>
<p>Since both of the sides are positive, we can square the entire equation, and we get a trivial inequality: $$ \sqrt{a+b} \leq \sqrt{a} + \sqrt{b} \iff a+b \leq a+b+2\sqrt{ab} \iff 0 ... |
2,342,051 | <p>I am totally new to statistics. I'm learning the basics.</p>
<p>I came upon this question while solving Erwin Kreyszig's exercise on statistics.
The problem is simple. It asks to calculate standard deviation after removing outliers from the dataset.</p>
<p>The dataset is as follows: 1, 2, 3, 4, 10.
What I did is, ... | Mundron Schmidt | 448,151 | <p>It is differentiable on $\mathbb{R}\setminus C$ using the same argument why it is continuous on $\mathbb{R}\setminus C$.<p>
By construction of $C$, you know that for each point $x\in\mathbb{R}\setminus C$ exists an open neighbourhood $U$ of $x$ such that $U\subset\mathbb{R}\setminus C$. On this neighbourhood is $f$ ... |
9,168 | <p>I'm having a doubt about how should we users encourage the participation of new members. So far I have only presented MSE to three of my fellow colleagues in grad school. In an overall way I feel like if MSE becomes too open and wide known, some of the high-rank researchers and top-class grad and undergrads users wi... | Ryan Budney | 642 | <p>I've always viewed MSE as an "anything goes" forum, as long as it's about actual mathematics. Grade-school homework problems to research problems. All of it is to be encouraged. I use filters to avoid looking at most of the things I don't want to see. </p>
|
4,026,149 | <p>If f is continuous on <span class="math-container">$[a,b]$</span> and <span class="math-container">$f(a)=f(b)$</span> then show that there exists <span class="math-container">$x,y \in (a,b)$</span> such that <span class="math-container">$f(x)=f(y)$</span></p>
<p>It looks obvious if I imagine the graph. But I am not ... | absolute0 | 883,281 | <p>Without using IVP: from the statement it's clear that <span class="math-container">$f$</span> cannot be strictly monotone in <span class="math-container">$(a,b)$</span>. If for all <span class="math-container">$x,y \in (a,b)$</span> we have <span class="math-container">$x\neq y \Rightarrow f(x) \neq f(y)$</span> (ot... |
863,846 | <p>Steven Strogatz has a great informal textbook on Nonlinear Dynamics and Chaos. I have found it to be incredibly helpful to get an intuitive sense of what is going on and has been a great supplement with my much more formal text from Perko.</p>
<p>Anyways I was wondering if anyone knew of any similar informal, intui... | Ian | 83,396 | <p>For linear algebra, I like Numerical Linear Algebra by Trefethen and Bau.</p>
|
1,823,556 | <p>Let be $X \subset F_1 \cup F_2$, where $F_1$ and $F_2$ are closed. If the function $f\colon X \longrightarrow \mathbb{R}$ is such that $f|_{X \cap F_1}$ and $f|_{X \cap F_2}$ are continuous, so prove that $f$ is continuous. </p>
<p>My attempt:</p>
<p>Suppose that $f$ is discontinuously, so exists $x \in X$ such th... | Michael Bil | 347,353 | <p>The proof falls when you pick a point on an edge. If the intervals don't overlap then the proof works fine. But when you are on the edge of the interval then you know that there is only one sided continuity. This is (I guess) where the closed part should be. For example: $f:X \rightarrow \mathbb{R}$, such that $X = ... |
1,823,556 | <p>Let be $X \subset F_1 \cup F_2$, where $F_1$ and $F_2$ are closed. If the function $f\colon X \longrightarrow \mathbb{R}$ is such that $f|_{X \cap F_1}$ and $f|_{X \cap F_2}$ are continuous, so prove that $f$ is continuous. </p>
<p>My attempt:</p>
<p>Suppose that $f$ is discontinuously, so exists $x \in X$ such th... | paul garrett | 12,291 | <p>Complementing @BrianMScott's answer, one could also avoid the proof by contrapositive/contradiction/whatever by claiming that any $x\in X$ has a sufficiently small neighborhood (if you like, the intersection of $X$ with a $\delta$-ball around it) lying entirely inside either $X\cap F_1$ or $X\cap F_2$. Then continui... |
1,289,994 | <p>If you fold a rectangular piece of paper in half and the resulting
rectangles have the same aspect ratio as the original rectangle,
then what is the aspect ratio of the rectangles?</p>
| Sean Henderson | 212,476 | <p>Assume the original rectangle is $x$ wide and $a \cdot x$ long, where a is the requested aspect ratio. If we fold the paper in half (lengthwise), the resulting rectangles will be $\frac{a \cdot x}{2}$ wide and $x$ long. Equating these ratios, we find that</p>
<blockquote class="spoiler">
<p>$$a = \frac{1}{a / 2} ... |
3,106,574 | <p>Let <span class="math-container">$(a_n) _{n\ge 0}$</span> <span class="math-container">$a_{n+2}^3+a_{n+2}=a_{n+1}+a_n$</span>,<span class="math-container">$\forall n\ge 1$</span>, <span class="math-container">$a_0,a_1 \ge 1$</span>. Prove that <span class="math-container">$(a_n) _{n\ge 0}$</span> is convergent.<br>
... | maxmilgram | 615,636 | <p>There might be an easier and more elegant solution, but this should work.</p>
<p>First observe that:
<span class="math-container">$$
a_{n+1}+a_{n}=a_{n+2}^3+a_{n+2}\geq2\sqrt{a_{n+2}^4}=2a_{n+2}^2\geq4a_{n+2}-2
$$</span>
Here I used the AM-GM inequality and the simply fact that <span class="math-container">$(a_{n+2... |
3,910,345 | <p>Recently a lecturer used this notation, which I assume is a sort of twisted form of Leibniz notation:</p>
<p><span class="math-container">$$y\,\mathrm{d}x - x\,\mathrm{d}y \equiv -x^2\,\mathrm{d}\left(\frac{y}{x}\right)$$</span></p>
<p>The logic here was that this could be used as:</p>
<p><span class="math-container... | md2perpe | 168,433 | <p>Very often it works to just think of <span class="math-container">$\mathrm{d}f$</span> as an alternative notation to <span class="math-container">$f'$</span> or the derivative of <span class="math-container">$f$</span> with respect to some unnamed variable.</p>
<p>For example, assume that <span class="math-container... |
508,790 | <p>I always see this word $\mathcal{F}$-measurable, but really don't understand the meaning. I am not able to visualize the meaning of it.</p>
<p>Need some guidance on this.</p>
<p>Don't really understand $\sigma(Y)$-measurable as well. What is the difference?</p>
| Stefan Hansen | 25,632 | <p>Let $(\Omega,\mathcal{F},P)$ be a probability space, i.e. $\Omega$ is a non-empty set, $\mathcal{F}$ is a sigma-algebra of subsets of $\Omega$ and $P:\mathcal{F}\to [0,1]$ is a probability measure on $\mathcal{F}$. Now, suppose we have a function $X:\Omega\to\mathbb{R}$ and we want to "measure" the probability of $X... |
642,324 | <p>According to <a href="http://en.wikipedia.org/wiki/Regression_analysis" rel="nofollow">wikipedia</a>, regression analysis is a statistical process for estimating the relationships among variables. Regression analysis is widely used for prediction and forecasting. So why is regression analysis also used as statistica... | Anatoly | 90,997 | <p>Regression analysis is surely an important tool for prediction and forecasting, but it is also commonly used as a statistical test for many purposes, e.g. to investigate whether a given variable is associated with another one independently of confounders, or whether a given multivariable model significantly predict... |
3,546,773 | <p>what are the real/complex zeros for:</p>
<p><span class="math-container">$t^9 - 1$</span></p>
<p>I also need to use the exponential form of complex numbers</p>
| J. W. Tanner | 615,567 | <p><span class="math-container">$\exp\left(\dfrac{2\pi i k}9\right)$</span> for <span class="math-container">$k=1$</span> to <span class="math-container">$9$</span>.</p>
<p>That's real only for <span class="math-container">$k=9$</span>.</p>
|
306,461 | <p>Let $A = \{(x,y) \in\mathbb{R}^2: a \leq (x-c)^2+(y-d)^2 \leq b\}$ for given $a,b,c, d$ real numbers. I want to show that $A$ is path-connected.</p>
<p>How can I do that?</p>
<p>I know that every open subset of $\mathbb R^2$ that is connected is path connected. But this is obviously not open so I cannot use that. ... | Ben Millwood | 29,966 | <p>Consider the following set:</p>
<p>$A^\prime=\{(x,y) \in\mathbb{R}^2: a< (x-c)^2+(y-d)^2< b\}$</p>
<p>i.e. exactly your set but with $<$ instead of $\le$. This set is open and connected, so by your comments it's path connected.</p>
<p>Show that the places in your set but not in mine can easily reach mine... |
29,255 | <p>sorry! am not clear with these questions</p>
<ol>
<li><p>why an empty set is open as well as closed?</p></li>
<li><p>why the set of all real numbers is open as well as closed?</p></li>
</ol>
| Jens Renders | 131,972 | <p>In my opinion, the other answers to this question are quite poor, as they just cite the definition of a topology which indeed states that the whole space and the empty set are open and closed. The natural question that then follows is: why define it like that?</p>
<p>The idea of topology comes from metric spaces (of... |
1,029,650 | <p>In Four-dimensional space, the Levi-Civita symbol is defined as:</p>
<p>$$ \varepsilon_{ijkl } =$$
\begin{cases}
+1 & \text{if }(i,j,k,l) \text{ is an even permutation of } (1,2,3,4) \\
-1 & \text{if }(i,j,k,l) \text{ is an odd permutation of } (1,2,3,4) \\
0 & \text{otherwise}
\end{cases}
</p>
<p>Let'... | Semiclassical | 137,524 | <p>Note that the permutations corresponding to $(i,j,k,4)$ and $(i,j,4,k)$ differ only by a transposition of the last two indices. Consequently, they have different parity and so their Levi-Civita symbols have opposite signs (assuming they do not vanish, of course). Hence the correct statement is $\epsilon_{ijk4}A^{jk}... |
1,032,535 | <p>I know $n \in \mathbb{N}$ and...</p>
<p>$$
a_n = \begin{cases}
0 & \text{ if } n = 0 \\
a_{n-1}^{2} + \frac{1}{4} & \text{ if } n > 0
\end{cases}
$$</p>
<ol>
<li><strong>Base Case:</strong></li>
</ol>
<p>$$a_1 = a^2_0 + \frac{1}{4}$$</p>
<p>$$a_1 = 0^2 + \frac{1}{4} = \frac{1}{... | Community | -1 | <p><strong>Note:</strong> <em>Base on the mvggz's answer, I will try to give also my complete with a lot of explanations answer. If something is wrong, please write in the comment section below ;)</em></p>
<hr>
<p>Let $$
a_n = \begin{cases}
0 & \text{ if } n = 0 \\
a_{n-1}^{2} + \frac{1}{4... |
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