qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
1,748,719 | <p>So there are a few basic formulas I'd like to start with, $W=\int_0^bFdx$, $F=ma$, and $a=\frac{d^2}{dt^2}x$.</p>
<p>In words, Work $(W)$ is defined as the area under a Force versus Displacement $(F/x)$ graph, Force is defined mass times acceleration $(m\cdot a)$, and acceleration is defined as the second derivativ... | Taiyo Terada | 152,321 | <p>The reason what you have there is tricky, is it usually results in a differential equation in $x$. so you can't in general take that integral. you might be able to find some identities, but that is about it. </p>
<p>Here is an example:
If you want acceleration not constant, a classic example is a spring where you a... |
45,911 | <p>I've been wondering for some time now about the difference between a point and a vector. In high school, it was very important to distinguish them from each other, and we used the notation $(x,y,z)$ for points and $[x,y,z]$ for vectors. We always had to translate the point $P=(a,b,c)$ to the vector $\overrightarrow{... | Benoît Kloeckner | 38,289 | <p>The simplest way to understand the importance of distinguishing points from vector is to consider subspaces. For example, imagine a plane $P$ in $\mathbb{R}^3$ that does not contain the origin. Then, if you add coordinate-wise two points of $P$, the result is not in $P$. The operation point minus point gives as a re... |
249,908 | <p>This seems like a very inefficient way of doing what I want. I generate all permutations of (for example) <code>{1, 1, 2, 2}</code>, then for each permutation I generate its rotations, select the first one in <code>Sort</code> order, and use that for <code>GatherBy</code>.</p>
<pre><code>list = {1, 1, 2, 2};
display... | Coolwater | 9,754 | <pre><code>With[{p = Permutations[list]}, Extract[p, #] & /@
ConnectedComponents[MapIndexed[UndirectedEdge,
Lookup[PositionIndex[p], RotateLeft[p, {0, 1}]]]]]
{{{1, 1, 2, 2}, {1, 2, 2, 1}, {2, 1, 1, 2}, {2, 2, 1, 1}}, {{1, 2, 1, 2}, {2, 1, 2, 1}}}
</code></pre>
|
1,865,062 | <p>I have seen some examples of inner automorphisms of Lie algebras. Can anyone please give me an example of an automorphism of Lie algebras that is not inner (with proof).
Note - An automorphism is said to be inner if it is of the form $exp(adx)$ for $adx$ nilpotent where $adx(y)$=$[x,y]$.
Thanks for any help.</p>
| Dietrich Burde | 83,966 | <p>The Lie algebra ${\mathfrak s}{\mathfrak l}_n({\mathbb C})$ has an automorphism given by $A\mapsto -A^t$. It is not inner for $n>2$, but inner for $n=2$, in which case it is given by
$$
A\mapsto -A^t=X^{-1}AX, \quad \text{with} \quad X=\begin{pmatrix} 0 & 1 \cr -1 & 0 \end{pmatrix}.
$$
This answers the qu... |
1,990,804 | <p>I know that we can define the exponential by a function $f: \mathbb{N}^2 \rightarrow \mathbb{N}$ by letting:</p>
<p>$f(m,o) = 1$ and $f(m,n+1) = f_x(m^n,m)$ where $f_x$ is the multiplication function, which we know is recursive.</p>
<p>I would then let $g = s(z)$ and hence $g(n) = 1$ for each $n \in \mathbb{N}$ , ... | Bram28 | 256,001 | <p>The $*$ works like a NAND for a binary logic, with $a$ the 'True' element, and $b$ the 'False' element.</p>
|
3,358,592 | <p>In the following quote, what does the notation <span class="math-container">$\{a_n\}$</span> mean?</p>
<blockquote>
<p>Дана последовательность Фибоначчи <span class="math-container">$\{a_n\}$</span>.</p>
</blockquote>
<p><strong>Translation:</strong> "You are given the Fibonacci sequence <span class="math-contai... | Coffee | 496,684 | <p>I would comment if I only were privileged.</p>
<p>Have in mind that the function you want to construct is only piecewise analytic. Maybe you can obtain this function as the limit of some sequence of functions, which is presumably the exercise here, given the question tags.</p>
<p>I suspect that there won't be any ... |
306,848 | <p>The question is in the title. Suppose that $X$ and $Y$ are two pointed connected CW-complexes. I was wondering if there exists a spectral sequence computing the homology of the function space $$H_{\ast}(map_{\ast}(X,Y);k) $$
where $k$ is a fixed field. Could we say something interesting in the case when $H_{\ast}(X;... | John Klein | 8,032 | <p>Let $X$ be a simplicial set and $Y$ a space (or simplicial set) . Then $F_\ast(X,Y)$ is a cosimplicial space and we can consider its homology spectral sequence. Bousfield gave conditions for when this will converge to the homology
of $F_\ast(X,Y)$ with field coefficients. See here:</p>
<p>On the Homology Spectral S... |
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>
| Stella Biderman | 45,118 | <p>Related to the answer about NP-complete problems, there are a number of theorems that state "either x is true, or P=NP." The most interesting of these in my opinion are hardness of approximation results. For example: "Given two graphs on $n$ vertices, one with max clique size $n^\alpha$ and one with max clique size ... |
3,522,752 | <p>Solve the following equation:
<span class="math-container">$$y=x+a\tan^{-1}p$$</span>
<span class="math-container">$$\text{where p}=\frac{dy}{dx}$$</span>
Differentiating both side w.r.t. x,
<span class="math-container">$$\frac{dy}{dx}=1+\frac{a}{1+p^2}\frac{dp}{dx}\\
\implies p=1+\frac{a}{1+p^2}\frac{dp}{dx}$$</spa... | Ankita Pal | 739,790 | <p><span class="math-container">$$y=x+a\tan^{-1}p\\
\begin{align}
&\implies\frac{dy}{dx}=1+\frac{a}{1+p^2}\frac{dp}{dx}\\
&\implies p=1+\frac{a}{1+p^2}\frac{dp}{dx}\\
&\implies (p-1)(1+p^2)=a\frac{dp}{dx}\\
&\implies\int{\frac{adp}{(p^2+1)(p-1)}}=\int{dx}\\
&\implies\frac a2\int{\frac{1}{p-1}-\frac{... |
125,451 | <p>If $W$ is a subspace of a finite dimensional vector space $V$ and $\{g_{1},g_{2},\cdots, g_{r}\}$ is a basis of the annihilator $W^{\circ}=\{f \in V^{\ast}| f(a)=0, \forall a \in W\}$, then $W=\cap_{i=1}^{r} N_{g_{i}}$, where for $f \in V^{\ast}$, $N_{f}=\{a \in V| f(a)=0\}$ </p>
<p>How shall I prove this?</p>
| mdp | 25,159 | <p>A hint:</p>
<p>To show $W\subset\bigcap^r_{i=1}N_{g_i}$, you just need to unpack definitions.</p>
<p>For the other inclusion, you should use the definition of the linear structure on $V^*$ to show that if $g_i(a)=0$ for all $i$, then in fact $f(a)=0$ for all $f\in W^\circ$. Then you either need to use (if you know... |
125,451 | <p>If $W$ is a subspace of a finite dimensional vector space $V$ and $\{g_{1},g_{2},\cdots, g_{r}\}$ is a basis of the annihilator $W^{\circ}=\{f \in V^{\ast}| f(a)=0, \forall a \in W\}$, then $W=\cap_{i=1}^{r} N_{g_{i}}$, where for $f \in V^{\ast}$, $N_{f}=\{a \in V| f(a)=0\}$ </p>
<p>How shall I prove this?</p>
| Elchanan Solomon | 647 | <p>We wish to prove that </p>
<p>$$W = \bigcap_{i=1}^{r} N_{g_{i}}$$</p>
<hr>
<p>Step $1$: Proving $W \subset \bigcap_{i=1}^{r} N_{g_{i}}$</p>
<p>Let $w \in W$. We know that the annihilator $W^{o}$ is the set of linear functionals that vanish on $W$. If $g_{i}$ is in the basis for $W^{o}$, it is certainly <strong>i... |
2,167,855 | <p>Let $f(t)$ be a differentiable function for $t$ $\in$ $[0,1]$ satisfying the above,</p>
<p>Does $f(t)$ have any fixed points?</p>
<p>I can easily prove there always exists fixed points without the second condition using $MVT$,</p>
<p>does $0$ $\leq$ $\frac{\partial f(t)}{\partial t}$ $\leq$ $\frac 12$ change anyt... | Reginald Dick | 421,431 | <p>Here is a hint. Let g(t)=f(t)-t.
Now use the intermediate value theorem to show that there are fixed points.
For bonus, show that the fixed point is unique!
Good luck!</p>
|
2,223,163 | <p>I don't have any idea on how to prove it, and I need it for one of my questions which is still unanswered: <a href="https://math.stackexchange.com/questions/2192947/what-is-the-largest-number-smaller-than-100-such-that-the-sum-of-its-divisors-is?noredirect=1#comment4521040_2192947">What is the largest number smaller... | User8128 | 307,205 | <p>We see $$10^n - 4 = 6 + \sum^{n-1}_{k=1} 9\cdot 10^k$$ and $6$ divides $9\cdot 10^k$ for any $k\ge 1$ since each of those is an even number divisible by $3$. </p>
|
2,374,282 | <p>I am trying to find all connected sets containing $z=i$ on which $f(z)=e^{2z}$ is one to one.
I have no idea how to approach.
Can someone give me some hints?
Thank you</p>
| Claude Leibovici | 82,404 | <p>There is a very simple solution (assuming that you can use programs) : convert dates to Julian days and subract the two numbers.</p>
<p>For example, in the book <a href="http://numerical.recipes/" rel="nofollow noreferrer">"Numerical Recipes"</a> you could find a subroutine named <strong>julday</strong>.</p>
<p>Y... |
604,836 | <p>Prove if <span class="math-container">$a \equiv c \pmod{n}$</span> and <span class="math-container">$b \equiv d \pmod n$</span> then <span class="math-container">$ab \equiv cd \pmod{n}$</span>.</p>
<p>I tried to use <span class="math-container">$(a-c)(b-d) = ab-ad-cb+cd$</span>, but it seem doesn't work.</p>
| Brian M. Scott | 12,042 | <p>You need to subtract and add the same thing:</p>
<p>$$ab-cd=ab-ad+ad-cd=\ldots$$</p>
|
4,002,925 | <p>I'm struggling a bit to understand some aspects of compactness in infinite sets.</p>
<p>We say that <span class="math-container">$[0,1]$</span> is compact by Heine-Borel, but that means that for all open covers of <span class="math-container">$[0,1], \exists$</span> a finite subcover of <span class="math-container">... | Son Gohan | 865,323 | <p>No, it does not mean that "no infinite collection of open sets between 0 and 1 inclusive exists". It means that for every (also infinite) cover (of open sets) that contains <span class="math-container">$[0,1]$</span>, we can find a finite number of them (open sets) that contains <span class="math-container... |
4,002,925 | <p>I'm struggling a bit to understand some aspects of compactness in infinite sets.</p>
<p>We say that <span class="math-container">$[0,1]$</span> is compact by Heine-Borel, but that means that for all open covers of <span class="math-container">$[0,1], \exists$</span> a finite subcover of <span class="math-container">... | Mark Bennet | 2,906 | <p>To take one of your questions.</p>
<p>Suppose we take as a cover of the closed unit interval the set containing every every open sub-interval.</p>
<p>Note that the endpoints have to be included in a cover so there will be sets of the form <span class="math-container">$[0, a)$</span> and <span class="math-container">... |
723,707 | <p>I'm trying to understand what the relation is between the direct product and the quotient group. </p>
<p>If we let $H$ be a normal subgroup of a group $G$, then it is not too difficult to show that the set of all cosets of $H$ in $G$ forms a quotient group $G/H$:
\begin{equation}
G/H = \{ g H \mid g \in G \}
\end{e... | mesel | 106,102 | <p>Your solution is not wrong but it has unnecassary steps. You can simply use following arguments.</p>
<p>Let $\pi:G\times H\to G$ be projection map .i.e. $\pi(g,h)=g$. It is clear that map is onto.</p>
<p><strong>Claim$1:$</strong> $\pi$ is an homomorphism;</p>
<p>$$\pi((g_1,h_1)(g_2,h_2))=\pi((g_1g_2,h_1h_2))=g_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> <... | Mr.Fry | 68,477 | <p>All squares modulo $3$ are: $\{0,1\} $.</p>
|
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> <... | Krishnaar | 155,879 | <p>IT IS CLEAR THAT ONE CAN WRITE ANY INTEGER IN THE FORM OF - 3K OR 3K+1 OR 3K+2 AS 0,1,2 ARE THE RESIDUES SET FOR 3. THUS WHEN WE SQUARE IT AND DIVIDE THE EXPRESSION BY 3- WE GET ONLY 0 AND 1 AS THE REMAINDER- IE;- FOR 3K IT IS OBVIOUSLY 0, FOR THE OTHER TWO IT IS 1. THUS WE CAN NEVER WRITE IT AS X^2 CONGRUENT TO 2 (... |
3,281,503 | <blockquote>
<p>For natural numbers <span class="math-container">$a$</span> and <span class="math-container">$b$</span>, what is the greatest value of <span class="math-container">$b$</span> so that <span class="math-container">$a^b$</span> has <span class="math-container">$b$</span> digits?</p>
</blockquote>
<p>I k... | Taha Direk | 676,723 | <p>You are trying to find the maximum <span class="math-container">$b$</span> that satisfies <span class="math-container">$10^{b-1}<9^b<10^b$</span> (it is easy to show <span class="math-container">$a=9$</span> is best possible) and as you said <span class="math-container">$b=21$</span> satisfy that. Also if a "<... |
3,281,503 | <blockquote>
<p>For natural numbers <span class="math-container">$a$</span> and <span class="math-container">$b$</span>, what is the greatest value of <span class="math-container">$b$</span> so that <span class="math-container">$a^b$</span> has <span class="math-container">$b$</span> digits?</p>
</blockquote>
<p>I k... | fleablood | 280,126 | <p>If <span class="math-container">$a^b$</span> has <span class="math-container">$b$</span> digits then </p>
<p><span class="math-container">$10^{b-1} \le a^b < 10^b$</span>.</p>
<p><span class="math-container">$\log 10^{b-1} \le \log a^b < \log 10^b$</span> (Notation: <span class="math-container">$\log$</span>... |
2,545,226 | <p>Suppose $a_n$ is a positive sequence but not necessarily monotonic. </p>
<p>For the series $\sum_{n=1}^\infty \frac{1}{a_n}$ and $\sum_{n=1}^\infty \frac{a_n}{n^2}$ I can find examples where both diverge: $a_n = n$, and where one converges and the other diverges: $a_n = n^2$.</p>
<p>Can we find example where both... | Eric Thoma | 35,667 | <p>By Cauchy-Schwarz, we have
$$
\sum_{n=1}^N \frac{1}{n}= \sum_{n=1}^N \frac{1}{\sqrt{a_n}} \cdot \sqrt{\frac{a_n}{n^2}} \leq \left(\sum_{n=1}^N \frac{1}{{a_n}} \right)^{1/2}\left(\sum_{n=1}^N \frac{a_n}{n^2}\right)^{1/2}
$$
and the left hand side goes to $\infty$ as $N \to \infty$. So the right must as well, meaning ... |
912,002 | <p>I'm solving some programming puzzle and it has come down to this:</p>
<p>I've a fraction, say 12/13, and I need to multiply it with a smallest possible natural number (say x) to get a whole number. How do I solve for x?</p>
<p>I intuitively feel I need to use LCM to solve this but haven't been able to pin down on ... | GuiguiDt | 171,756 | <p>If the fraction is allready reduce ($\implies gcd(num;den)=1$) the smallest $x$ is denominator.</p>
<p>Or to reduce the fraction you have to divide both num and den by $gcd(num;den)$ so the new denominator is $den/gcd(num;den)$.</p>
<p>So the smallest $x$ is $den/gcd(num;den)$.</p>
|
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... | Henno Brandsma | 4,280 | <p>Not true. Some examples, let C be the Cantor set :</p>
<p>$C \times C \simeq C \times \{0\}$ but $C \not\simeq \{0\}$.</p>
<p>$\mathbb{Q} \times \mathbb{Q} \simeq \mathbb{Q} \times \mathbb{Z}$ but $\mathbb{Q} \not\simeq \mathbb{Z}$ </p>
<p>$[0,1] \times [0,1]^{\mathbb{N}} \simeq [0,1]^2 \times [0,1]^{\mathbb{N}}... |
4,398,873 | <p>Given f: (0,1) <span class="math-container">$\rightarrow$</span> <span class="math-container">$\mathbb R$</span>. f is a continuous function and improper integrable.</p>
<p>If <span class="math-container">$\int_{0}^{x}f(t)dt = 0$</span> <span class="math-container">$\forall x \in [0,1] $</span>,</p>
<p>Does <span cl... | Peter Szilas | 408,605 | <p>Assume there is a <span class="math-container">$z \in (0,1)$</span> with</p>
<p><span class="math-container">$f(z) >0$</span> (or <span class="math-container">$f(z)<0$</span>).</p>
<p>Since f is continuos there is
a</p>
<p><span class="math-container">$e>0$</span> s.t. for <span class="math-container">$z-e ... |
2,354,609 | <p>I have to approximate $\sqrt2$ using Taylor expansion with an error $<10^{-2}$.</p>
<p>I noticed that I can do MacLaurin expansion of $\sqrt{x+1}$ then put $x=1$</p>
<p>So: $$\sqrt{x+1}=1 + \dfrac{x}{2} - \dfrac{x^2}{8} + \dfrac{x^3}{16} + {{\frac1{2}}\choose{n+1}}x^{n+1}(1+\xi)^{-\frac1{2}-n}$$</p>
<p>I have ... | Community | -1 | <p>The series$\sum_{k=0}^\infty\binom{1/2}{k}$ is alternating (which is the reason why it converges, by the way). For such series this estimate holds $$\left\lvert\sum_{k=0}^\infty a_k-\sum_{h=0}^n a_h\right\rvert\le \rvert a_{n+1}\rvert$$</p>
<p>See, for instance, the proof of <a href="https://en.wikipedia.org/wiki/A... |
587,878 | <p>Let $X$ be a reflexive Banach space of infinite dimension. </p>
<p>a) Prove that there exists a sequence $x_n$ such that $\| x_n \|=1$ and $x_n$ converges weakly to $0$.</p>
<p>b) Let $x_n$ be a sequence such that $\forall f \in X' \quad \exists \lim\limits_{n\to\infty} f(x_n)<\infty$ .Prove that $x_n$ converge... | Norbert | 19,538 | <p>a) Let $\{z_n\}_{n\in\mathbb{N}}\subset S_X$ be such a sequence that $\Vert z_n-z_m\Vert\geq 1/2$ provided $n\neq m$. This sequence exist thanks to <a href="https://math.stackexchange.com/questions/163500/an-application-of-riesz-lemma">Riesz lemma</a>. Since $X$ is reflexive its <a href="https://math.stackexchange.c... |
4,065,797 | <p>Just to give a simple numerical example but in general the variables <span class="math-container">$x,y,z,u,v$</span> are not equal.</p>
<p><span class="math-container">$113= 2*4^2 + 2*4^2 +2*4^2 + 4^2 +1^2$</span></p>
<p>I am looking for a general method to solve this type of equation or a piece of software to do th... | Giulio R | 807,789 | <p>We have
<span class="math-container">$$
I_n=2\int_0^{\pi/2}\sin^n(x)\log(\sin(x)) dx=
\int_0^{\pi/2}\sin^n(x)\log(1-\cos^2(x)) dx
$$</span>
<span class="math-container">$$
=-\sum_{k\geq 1}\frac 1k\int_0^{\pi/2}\sin^n(x)\cos^{2k}(x) dx.
$$</span></p>
<p>With the help of Euler's Beta function
<span class="math-contain... |
114,664 | <p>How would one evaluate $\int_0^1 {\ln(1+x)\over x}\,dx$?</p>
<p>I'd like to do this without approximations. Not quite sure where to start. What really bothers me is that I came across this while reviewing my old intro to calculus book... but I'm fairly certain I've exhausted all the basic methods they teach in th... | Ragib Zaman | 14,657 | <p>$$ \int^1_0 \frac{ \log (1+x) }{x} dx = \int^1_0 \sum_{n=1}^{\infty} (-1)^{n-1} \frac{x^{n-1}}{n} dx$$ </p>
<p>$$ =\sum_{n=1}^{\infty} (-1)^{n-1} \int^1_0 \frac{x^{n-1} }{n} dx = \sum_{n=1}^{\infty} (-1)^{n-1} \frac{1}{n^2}. $$</p>
<p>Denote $\displaystyle S = \sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}.$... |
139,105 | <p>Can a (finite) collection of disjoint circle arcs in $\mathbb{R}^3$ be interlocked in the sense in that they cannot be separated, i.e. each moved arbitrarily far from one another while remaining disjoint (or at least never crossing) throughout?
(Imagine the arcs are made of rigid steel; but infinitely thin.)
The arc... | Cristi Stoica | 10,095 | <p>There is already a beautiful correct <a href="https://mathoverflow.net/a/139114/10095">answer</a>.</p>
<p>Mine is just an illustrated comment, for the case $\mathbb R^2$, although the question was about $\mathbb R^3$.</p>
<p>Here is how three arcs can be locked, in $\mathbb R^2$:</p>
<p><img src="https://i.stack.... |
1,687,868 | <p>This might be a really obvious question so I apologize in advance, but I'm having trouble seeing when matrices are commutative for general nxn matrices. For example, when proving tr(AB)=tr(BA), I can easily prove this in a 2x2 matrix but I'm getting confused for proving it in a nxn matrix.</p>
<p>I've searched onlin... | Fryie | 282,149 | <p>Basically, you have <span class="math-container">$$(A_{11}B_{11}+A_{12}B_{21}+\ldots)+(A_{21}B_{12}+A_{22}B_{22}+\ldots)+\ldots$$</span> and by simply regrouping the terms you get <span class="math-container">$$(A_{11}B_{11}+A_{21}B_{12}+\ldots)+(A_{12}B_{21}+A_{22}B_{22}+\ldots)+\ldots.$$</span></p>
<p>The products... |
3,114,208 | <p>Say I have a biased coin that shows heads with probability <span class="math-container">$p \in ]1/3,1/2[$</span> and I initially have capital of <span class="math-container">$100 $</span>EUR. Every time heads is shown, my capital is doubled, in the other case I pay half of my capital. Let <span class="math-container... | Mike Earnest | 177,399 | <p>Let <span class="math-container">$X_n$</span> be your wealth after <span class="math-container">$n$</span> flips, and let <span class="math-container">$Q_{n}=X_n/X_{n-1}$</span>. Then
<span class="math-container">$$
X_n=X_0\times Q_1\times Q_2\times\dots \times Q_n
$$</span>
Let <span class="math-container">$Y_n=\lo... |
939,747 | <p>Denote by $a_n$ the sum of the first $n$ primes. Prove that there is a perfect square between $a_n$ and $a_{n+1}$, inclusive, for all $n$.</p>
<p>The first few sums of primes are $2$, $5$, $10$, $17$, $28$, $41$, $58$, $75$. It seems there is a perfect square between each pair of successive sums. In addition, we ca... | Robert Lewis | 67,071 | <p>Not in general, since the rank of $XX^T$ is must be either $0$ or $1$, whereas the rank of $A$ is unconstrained, i.e. we have <em>a priori</em> $1 \le \text{rank}(A) \le n$. To see that $0 \le \text{rank}(XX^T) \le 1$, note that with</p>
<p>$X = (x_1, x_2, \ldots, x_n)^T, \tag{1}$</p>
<p>a column vector, we have<... |
3,225,553 | <p>Show that <span class="math-container">$4x^2+6x+3$</span> is a unit in <span class="math-container">$\mathbb{Z}_8[x]$</span>.</p>
<p>Once you have found the inverse like <a href="https://math.stackexchange.com/questions/3172556/show-that-4x26x3-is-a-unit-in-mathbbz-8x">here</a>, the verification is trivial. But how... | Ehsaan | 78,996 | <p>If <span class="math-container">$R$</span> is a commutative ring: the units in <span class="math-container">$R[x]$</span> are the polynomials whose constant term is a unit, and whose higher order coefficients are nilpotent. You can apply this directly to your example.</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... | achille hui | 59,379 | <p>Notice $\sum_{k=1}^n k = \frac{n(n+1)}{2}$. For the identity at hand,</p>
<p>$$\sum_{k=1}^n k^5 + \sum_{k=1}^n k^7 \stackrel{?}{=} 2 \left(\sum_{k=1}^n k\right)^4$$
If one compute the difference of successive terms in RHS, we find</p>
<p>$$\begin{align}{\rm RHS}_n - {\rm RHS}_{n-1}
&= 2\left(\frac{n(n+1)}{2}\r... |
73,111 | <blockquote>
<p>A trapezoid was inscribed into a semicirle of radius R. The side of
the trapezoid is slanting alpha against the base which is the diameter
of the semicirlce. Compute the area of the trapezoid.</p>
</blockquote>
<p>So the base is 2R. The bad thing is: that's all I know. How should I move on? Don't... | Ross Millikan | 1,827 | <p>Hint: both base angles are $\alpha$ to make the top parallel to the base. Draw a radius to the upper corners and you have three isosceles triangles.</p>
|
2,303,163 | <blockquote>
<p>Let $T$ be a linear operator on the finite-dimensional space $V.$ Suppose there is a linear operator $U$ on $V$ such that $TU=I.$ Prove that $T$ is invertible and $U=T^{-1}.$</p>
</blockquote>
<p>Attempt: Let $\dim V=n$ and $\{\alpha_i\}_{i=1}^n$ a basis for $V$. We claim that $\{U(\alpha_i)_{i=1}^n\... | Sahiba Arora | 266,110 | <p>As $TU =I$, therefore $TU$ is onto. This implies $T$ is onto.
As $V$ is finite dimensional therefore this also shows that $T$ is one-one. Thus, $T$ is invertible and inverse has to be $U$.</p>
|
117,933 | <p>I couldn't find similar question being asked here. The closest one I can find is <a href="https://mathoverflow.net/questions/11366/when-to-split-merge-papers">When to split/merge papers?</a>. Here is my situation: I proved a theorem. When I try to type it, I found that it's very long. Since it's long, I splitted it ... | Andreas Blass | 6,794 | <p>Since the two papers together prove one main theorem (if I correctly understand the first few lines of the question), it seems reasonable to submit them to the same journal. I can imagine a referee or editor being unhappy about being asked to publish part 1, which builds up to a big theorem that will appear in a di... |
117,933 | <p>I couldn't find similar question being asked here. The closest one I can find is <a href="https://mathoverflow.net/questions/11366/when-to-split-merge-papers">When to split/merge papers?</a>. Here is my situation: I proved a theorem. When I try to type it, I found that it's very long. Since it's long, I splitted it ... | Kevin R. Vixie | 30,269 | <p><strong>What I would do</strong>: Put them both on arXiv.org and then submit the second one to the same journal. </p>
<p>Readers will thank you (for the arXiv versions <strong>and</strong> for only having to go to one journal to read the whole result).</p>
|
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 }$$</p>
<p>Then I tried to integrate it,I got $2\ln(x+1)+\frac{2}{x+1}+C$ as my answer. Am I right? please correct me if I'm wrong.</p>
| Mathematics | 290,994 | <p>Your answer is almost correct, but you should put modulus sign in your log's argument as it must be positive i.e your answer should be;
$$2\ln|x+1|+\frac{2}{x+1}+C$$</p>
<p>You can try one more method too.</p>
<p>$$\int \frac{2x}{x^2+2x+1}dx$$
Try to create derivative of denominator in numerator
$$\int \frac{2x+2-... |
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>
| Itakura | 229,346 | <p>$$\int\frac{2x}{x^2+2x+1}\text{d}x$$</p>
<p>let $u=x^2+2x+1$ and $\frac{du}{dx}=2x+2$ and now $dx=\frac{du}{2(x+1)}$</p>
<p>$$\int\frac{2x}{u}\cdot \frac{du}{2(x+1)}$$</p>
<p>Now since $x^2+2x+1=(x+1)^2$, therefore $u^\frac{1}{2}=x+1$ then $u^\frac{1}{2}-1=x$</p>
<p>$$\int\frac{u^\frac{1}{2}-1}{u^\frac{3}{2}}du$... |
3,896,709 | <ul>
<li>Any idea to evaluate the sum
<span class="math-container">$$
\sum_{j=m}^{k}\frac{\binom{m}{2m - j\,\,}}{\binom{k}{j}}
\quad\mbox{with}\quad m \leq k < 2m - 1.
$$</span></li>
<li>I have found the sum for <span class="math-container">$k=2m-1$</span>. In fact, it is verified that
<span class="math-container">$... | vonbrand | 43,946 | <p>Here <a href="https://maxima.sourceforge.net" rel="nofollow noreferrer">maxima</a>'s Zeilberger package (take a peek at <a href="https://www.math.upenn.edu/%7Ewilf/AeqB.html" rel="nofollow noreferrer">Petkovsek, Wilf, Zeilberger "A = B"</a> for the gory details) says your sum isn't Gosper summable. That me... |
2,930,003 | <p>Can there be a relation which is reflexive, symmetric, transitive, and antisymmetric at the same time? I tried to find so.</p>
<p>If <span class="math-container">$A = \{ a,b,c \}$</span>. Let <span class="math-container">$R$</span> be a relation which is reflexive, symmetric, transitive, and antisymmetric.</p>
<p>... | Mohammad Riazi-Kermani | 514,496 | <p>Your answer is correct and you can easily generalize it to a set with more elements </p>
<p>Apparently the only solution to your question is the diagonal relation, <span class="math-container">$$R=\{(x,x)|x\in A \}$$</span> for any set A.</p>
|
3,884,891 | <p>I am reading an article about "Longest Paths in Digraphs". In the proof there is a step that they considered as a trivial step, it seems easy but I am not being able to write or concieve an exact proof for it.</p>
<p>We have a strong digraph <span class="math-container">$D$</span>, that has minimum in degr... | Fareed Abi Farraj | 584,389 | <p>I will use @Mike idea in his answer, but write a much shorter version of the answer.</p>
<p>Let <span class="math-container">$P=v_p\cdots v_1$</span> be the longest path in <span class="math-container">$D$</span>, (starting from <span class="math-container">$v_p$</span> to <span class="math-container">$v_1$</span>).... |
88,511 | <p>In version <a href="http://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn102.html">10.2</a> there is a new experimental function: <a href="http://reference.wolfram.com/language/ref/FindFormula.html"><code>FindFormula[]</code></a>.</p>
<p>I suspect that a <a href="https://en.wikipedia.org/wiki/Symbolic_... | Giorgia | 30,942 | <p>The Experimental function <code>FindFormula[]</code> at the moment is using a combination of different methods: it combines non linear regression with Markov chain Monte Carlo methods (e.g. Metropolis–Hastings algorithm). In the future (possibly in V$10.3$) there will be an option allowing the user to choose which m... |
1,991,950 | <p>Let $m$ and $n$ be relatively prime integers, with $n \ne 0,1$, so that $a=m/n$ is a non-integer rational fraction. Let $p$ be an odd prime.</p>
<blockquote>
<p><strong>QUESTION 1</strong>: Can Fermat's Little Theorem be applied, <em>i.e.</em>, can one say $a^p \equiv a\!\pmod{p}$, without any further considerati... | Robert Israel | 8,508 | <p>If $p$ divides $n$, $m/n \mod p$ makes no sense. If $p$ divides $m$, $a^p \equiv a \equiv 0 \mod p$. If $p$ divides neither, $a^{p-1} \equiv m^{p-1}/n^{p-1} \equiv 1 \mod p$.</p>
|
197,877 | <p>According to answer of Denis Serre to <a href="https://mathoverflow.net/questions/197773/a-geometric-property-of-singular-matrices">this question</a>, the manifold of singular matrices in $M_{n}(\mathbb{R})$ is defined as follows:
$$M=\{A\in M_{n}(\mathbb{R})\mid \text{rank}(A)=n-1\}$$</p>
<p>So we define a (line b... | Daniel Valenzuela | 52,936 | <p>Consider the natural map $M \to\mathbb RP^n$ assigning $x\mapsto ker x$. Then obviously your line bundle arises as the pullback of the tautological bundle. But now you can write down a homotopically non-trivial map $S^1 \to M \to \mathbb RP^n$ (try to restrict your attention to hit $\mathbb RP^1\subset \mathbb RP^n$... |
90,112 | <p>When reading "Chebyshev centers and uniform convexity" by Dan Amir I encountered the following result which is apparently "known and easy to prove". I'm sure it is, but I can't find a proof and am failing to prove it myself.</p>
<p>The result (slightly simplified) is</p>
<p>If $X$ is a uniformly convex space (i.e.... | Stephan Fackler | 21,704 | <p>In addition to S. Ivanov's proof, I give you a reference to the literature. A proof can be found in Classical Banach Spaces II by J. Lindenstrauss & L. Tzafriri in Section 1.e directly after the definition of uniform convexity on p. 60.</p>
|
3,979,686 | <ul>
<li><a href="https://www.britannica.com/science/derivative-mathematics" rel="nofollow noreferrer">Derivative</a></li>
</ul>
<p>This article says the following:</p>
<blockquote>
<p>To find the slope at the desired point, the choice of the second point needed to calculate the ratio represents a difficulty because, i... | Leonard Neon | 818,617 | <p><span class="math-container">$$
\displaystyle I=\frac{1}{\sqrt{a}}\ln\left({2\sqrt{a}\sqrt{ax^2+bx+c}+2ax+b}\right)+C
\label{eq1} \tag{eq1}\\
$$</span></p>
<p><span class="math-container">$$
\displaystyle I=-\frac{1}{\sqrt{-a}}\arcsin{\left(\frac{2ax+b}{\sqrt{b^2-4ac}}\right)}+C
\label{eq2} \tag{eq2}\\
$$</span></p>... |
1,999,194 | <p>Determine the following system of equations has 'a unique solution', 'many solutions' or 'no solution':
$$\begin{cases}
& x + 2y + z &= 1\\
&2x + 2y - 2z &= 4\\
&-x + 2y - 3z &= 5
\end{cases}
$$</p>
<p>Answer = A unique solution</p>
<p>How is it a unique solution? C... | Casper Thalen | 385,675 | <p>The easiest way is to show that the corresponding coefficient matrix is non-singular. </p>
<p>Define A = $\begin{bmatrix}1&2&1\\2&2&-2\\-1&2&-3\end{bmatrix}$</p>
<p>It is a property that if this matrix is non-singular, that the system of linear equations corresponding to this matrix has exa... |
400,838 | <p>I need to find $$\lim_{x\to 1} \frac{2-\sqrt{3+x}}{x-1}$$</p>
<p>I tried and tried... friends of mine tried as well and we don't know how to get out of:</p>
<p>$$\lim_{x\to 1} \frac{x+1}{(x-1)(2+\sqrt{3+x})}$$</p>
<p>(this is what we get after multiplying by the conjugate of $2 + \sqrt{3+x}$)</p>
<p>How to proce... | Alex | 38,873 | <p>Multiply both numerator and denominator by $2+\sqrt{3+x}$, simplify, cancel out. I get $-\frac{1}{4}$</p>
|
2,497,875 | <p>Define $\sigma: [0,1]\rightarrow [a,b]$ by $\sigma(t)=a+t(b-a)$ for $0\leq t \leq 1$. </p>
<p>Define a transformation $T_\sigma:C[a,b]\rightarrow C[0,1]$ by $(T_\sigma(f))(t)=f(\sigma(t))$ </p>
<p>Prove that $T_\sigma$ satisfies the following:</p>
<p>a) $T_\sigma(f+g)=T_\sigma(f)+T_\sigma(g)$</p>
<p>b) $T_\sigma... | Siong Thye Goh | 306,553 | <p>Hint:$$\lim_{x\to 1} \frac{\sin(1-\sqrt{x})}{(\sqrt{x}-1)(\sqrt{x}+1)}$$</p>
|
2,497,875 | <p>Define $\sigma: [0,1]\rightarrow [a,b]$ by $\sigma(t)=a+t(b-a)$ for $0\leq t \leq 1$. </p>
<p>Define a transformation $T_\sigma:C[a,b]\rightarrow C[0,1]$ by $(T_\sigma(f))(t)=f(\sigma(t))$ </p>
<p>Prove that $T_\sigma$ satisfies the following:</p>
<p>a) $T_\sigma(f+g)=T_\sigma(f)+T_\sigma(g)$</p>
<p>b) $T_\sigma... | Rebellos | 335,894 | <p>Let's try to work around our <em>limit</em> :</p>
<p>$$\lim_{x\to 1} \frac{\sin(1-\sqrt{x})}{x-1} = \lim_{x\to 1} \frac{\sin(1-\sqrt{x})}{(\sqrt{x}-1)(\sqrt{x}+1)}$$</p>
<p>since $x-1 = (\sqrt{x}-1)(\sqrt{x}+1). $</p>
<p>A known standar limit tells us that : </p>
<p>$$\lim_{x \to 0} \frac{\sin x}{x} = 1$$</p>
<... |
3,936,102 | <p>Can you have a function <span class="math-container">$f \notin L^1$</span> but its Fourier transform <span class="math-container">$\hat{f} \in L^1$</span>? Ive been playing around with examples and I cant find one, but I also cant prove one doesn't exist.</p>
| GEdgar | 442 | <p>More on the example of @md2pepe.
Line 18 on this page below from a textbook says that the Fourier transform of
<span class="math-container">$\frac{W}{\pi}\operatorname{sinc}(Wt)$</span> is <span class="math-container">$\operatorname{rect}(\frac{\omega}{2W})$</span></p>
<p><a href="https://i.stack.imgur.com/WHOul.png... |
3,657,026 | <p>I need to prove expression using mathematical induction <span class="math-container">$P(1)$</span> and <span class="math-container">$P(k+1)$</span>, that:</p>
<p><span class="math-container">$$
1^2 + 2^2 + \dots + n^2 = \frac{1}{6}n(n + 1)(2n + 1)
$$</span></p>
<p>Proving <span class="math-container">$P(1)$</span>... | JCH | 467,371 | <p>True for <span class="math-container">$n=1.$</span></p>
<p>Let <span class="math-container">$n \geq 1$</span> and suppose true for <span class="math-container">$ n. $</span></p>
<p>Then </p>
<p><span class="math-container">$1^2 + 2^2 + ...+ n^2 + (n+1)^2 = \frac{1}{6}n(n+1)(2n+1)+ (n+1)^2 = \frac{1}{6}(n+1)(2n^2 ... |
571,955 | <p>I've tried solving this problem every way I know how and I just can't get it. I've looked at similar problems of this type, and I still cannot get an answer that seems right.</p>
<p>Parametric Equations:</p>
<p><strong>a) Write the distance between the line and the point as a function of s</strong></p>
<p><strong>b... | Cameron Buie | 28,900 | <p>In the first part, we are taking $s$ to be some <em>fixed</em> (but unspecified) value. The distance from <em>a point</em> on the parametric line to the given point is indeed $$\sqrt{(1+t)^2+\left(2-\frac12t\right)^2+(s-2t-1)^2}.\tag{$\star$}$$ To find the distance from <em>the parametric line, itself</em> to the gi... |
129,287 | <p>Suppose $p(x_1, x_2, \cdots, x_n)$ is a symmetric polynomial. Given any univariate polynomial $u$, we can define a new polynomial $q(x_1, x_2, \cdots, x_{n+1})$ as</p>
<p>$q(x_1, x_2, \cdots, x_{n+1}) = u(x_1)p(x_2, x_3, \cdots, x_{n+1}) + u(x_2)p(x_1, x_3, \cdots, x_{n+1}) + \cdots \\ \phantom{q(x_1, x_2, \cdots, ... | David Wehlau | 16,684 | <p>Your sum is (up to a scalar) the transfer or trace of $u(x_{n+1})p(x_1,x_2,\dots,x_n)$. Given any
polynomial $h(x_1,x_2,\dots,x_n,x_{n+1})$ its transfer or trace (with respect to the symmetric group, $S_{n+1}$ is the symmetric polynomial $\sum_{\sigma\in S_{n+1}} \sigma \cdot h(x_1,x_2,\dots,x_n,x_{n+1})$. This ... |
3,718,347 | <p>If <span class="math-container">$x$</span> and <span class="math-container">$y$</span> are real numbers and <span class="math-container">$$x^2+y^2-4x-6y-1=0$$</span>what is the minimum value of <span class="math-container">$x + y$</span>?</p>
<p>I tried to factor it but I couldn't. Then I tried to make squares so al... | J.G. | 56,861 | <p>The circle <span class="math-container">$(x-2)^2+(y-3)^2=14$</span> minimizes <span class="math-container">$x+y$</span> when meeting its leftmost tangent of the form <span class="math-container">$x+y=c$</span>, hence of gradient <span class="math-container">$-1$</span>, with a radius of gradient <span class="math-co... |
1,743,542 | <p>I'm looking for a continuous random variable with the following properties</p>
<ul>
<li>It is not bounded towards $+\infty$.</li>
<li>The expected value of the <em>maximum</em> of x-many draws out of that random variable has a closed-form solution.</li>
</ul>
<p>The more standard and well-known it is, the better. ... | FooBar | 112,643 | <p>Something else I was hinted at is to use the Frechet distribution, as it preserves maxima. </p>
<p>More precisely, if $X_i \sim Frechet(\alpha, s, m)$, then $\max \{X_1, \cdot X_n\} = Frechet(\alpha, n^\frac{1}{\alpha}s, m)$.</p>
|
4,028,534 | <p>I have three points with coordinates: <span class="math-container">$A (5,-1,0),B(2,4,10)$</span>, and <span class="math-container">$C(6,-1,4)$</span>.</p>
<p>I have the following vectors <span class="math-container">$\overrightarrow {CA} = (-1, 0, -4)$</span> and <span class="math-container">$\overrightarrow{CB} = (... | Andrei | 331,661 | <p>If you calculate the angle, you introduce numerical errors. The most elegant way to calculate the area is to use the cross product. You will not need any trigonometric functions:
<span class="math-container">$$A=\frac12|\vec{CA}\times\vec{CB}|$$</span>
So all you need will be some multiplications, additions(subtract... |
2,333,847 | <p>A function $f(x) = k$ and the domain is $\{-2,-1,\dotsc,3\}$. Would I say
$$x = \{-2,-1,\dotsc,3\}\quad\text{or}\quad x \in \{-2,-1,\dotsc,3\} \ ?$$
Thanks. </p>
| MPW | 113,214 | <p>Your second alternative is the correct one.</p>
<p>The value of $x$ is not a <em>set</em> of numbers, rather it is <em>in a set</em> of numbers.</p>
|
756,662 | <p>I am trying to figure out what random variables are measurable with respect to sigma algebra given by $[1-4^{-n}, 1]$ where $n= 0, 1, 2, ....$ if $[0,1]$ is the sample space. I believe I can do with with indicator functions but I'm not sure how to write this.
Thanks!</p>
| Davide Giraudo | 9,849 | <p>The collection of intervals $[1-4^{-n}]$, $n\geqslant 0$ does not form a $\sigma$-algebra, but if we define $I_n:=[1-4^{-n},1-4^{-n-1})$, then the collection of sets of the form $\bigcup_{n\in J}I_n$, $J\subset\mathbb N$, plus singleton $\{1\}$, is a $\sigma$-algebra. </p>
<p>If a $\sigma$-algebra is generated by ... |
2,709,832 | <p><a href="https://en.wikipedia.org/wiki/Cantor_set#Construction_and_formula_of_the_ternary_set" rel="noreferrer">Cantor set</a>
See the link, I am referring to cantor set on the real line. I wish to show that it is compact. I am doing this by pointing following arguments. I am not sure if this is enough.</p>
<ol>
<li... | Xander Henderson | 468,350 | <p>This question is tagged <a href="/questions/tagged/solution-verification" class="post-tag" title="show questions tagged 'solution-verification'" rel="tag">solution-verification</a>, which indicates that the asker is seeking a critique of their argument. As neither of the previous answers have addressed this... |
208,883 | <p>Let $\bar{\rho}: G_K\to PGL_n(\mathbb{C})$ be projective representation of the absolute Galois group of a number field $K$ and $\varphi\in Aut(G_K)$.</p>
<p>A theorem of Tate tells us that we can always lift $\bar{\rho}$ to some $\rho: G_K \to GL_n(\mathbb{C})$. I am wondering if there is a lift $\rho$ whose kerne... | Lasse Rempe | 3,651 | <p><strong>EDIT.</strong> I am returning to this many years after the fact. The essence of the answer remains unchanged, but I am writing it now as a more detailed proof.</p>
<p>Suppose that we are given a surface <span class="math-container">$S$</span> with a countable "<span class="math-container">$K$</span>-qc ... |
2,747,509 | <p>How would you show that if <span class="math-container">$d\mid n$</span> then <span class="math-container">$x^d-1\mid x^n-1$</span> ?</p>
<p>My attempt :</p>
<blockquote>
<p><span class="math-container">$dq=n$</span> for some <span class="math-container">$q$</span>. <span class="math-container">$$ 1+x+\cdots+x^{d-1}... | Mark Bennet | 2,906 | <p>You can always do:</p>
<p>$f(y)=1+y+ \dots +y^{r-1}$ so that $yf(y)=y+y^2+\dots +y^r$ and $yf(y)-f(y)=(y-1)f(y)=y^r-1$</p>
<p>Then put $y=x^d$ with $dr=n$ and obtain $(x^d-1)f(x^d)=x^n-1$ and by construction $f(x^d)$ is a polynomial.</p>
|
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... | kglr | 125 | <p>Remove <code>{t, 0, 2 Pi}</code> from your <code>SpC1</code> to get a 3D parametrized line </p>
<pre><code>SpC1b = ParametricPlot3D[{r[th] Cos[th], z[th], r[th] Sin[th]}, {th, 0, thmax},
PlotStyle -> {Thick, Magenta}, PlotLabel -> "SPACE_CURVE"];
Show[SpC1b, SpC2]
</code></pre>
<p><img src="https:... |
1,253,778 | <p>When we say that an inequality is sharp, does it mean that it is "the best" inequality we can get between the two quantities involved?</p>
<p>For example, I read that we would say that the inequality
$$
\frac{a^2+b^2}{2}\geq ab
$$
is sharp, but wouldn't $\frac{a^2+b^2}{2}$ on the RHS be sharper than $ab$?</p>
<p>D... | Nate 8 | 226,768 | <p>As was said, many times bounds are not as strong as they theoretically could be. In math, this comes because a specific quality is hard to compute. A basic example is computing world population-- you could say "The population of the world is greater than 3 billion." but this inequality might not be very good.</p>
<... |
733,908 | <p>How do i start off with integrating the below function? i tried applying trig substitution and U substitution. how do i go about solving this function? should i split them up further into 2 separate functions ? need some help in this as i can't seem to figure out how to continue on with it </p>
<p>$$\int\fra... | lab bhattacharjee | 33,337 | <p>$$I=\int\frac{x^3}{\sqrt{x^2-1}}dx=\int\frac{x^2}{\sqrt{x^2-1}}x\ dx$$</p>
<p>Setting $\displaystyle\sqrt{x^2-1}=y,\frac{2x}{2\sqrt{x^2-1}}dx=dy$ and $\displaystyle x^2=y^2+1$</p>
<p>$$I=\int(y^2+1)dy=\cdots $$</p>
<hr>
<p>Alternatively,</p>
<p>Using Trigonometric substitution (<a href="http://www.sosmath.com/c... |
657,047 | <p>So I have $a^n = b$. When I know $a$ and $b$, how can I find $n$?</p>
<p>Thanks in advance!</p>
| Community | -1 | <p>Exponent problems like finding $n$ if we know the value of both $a$ and $b$ in the equation $a^n = b$ can be solved using <a href="http://en.wikipedia.org/wiki/Logarithm" rel="nofollow">logarithms</a>:</p>
<p>$$
\begin{align}
a^n=b&\Rightarrow \log \left(a^n\right)=\log (b) \\
&\Rightarrow n\log(a)=\log(b) ... |
271,259 | <p>My goal is to have NumericQ[h[j]]=True for any j regardless of whether j may be symbolic with no defined value.</p>
<p>Setting NumericQ[h[j_]]=True does not work and as I understand the NumericFunction attribute only works when the input is also numeric.</p>
<p>One solution might be to unprotect NumericQ and then se... | user293787 | 85,954 | <p>Instead of unprotecting <code>NumericQ</code>, one can define an <a href="https://reference.wolfram.com/language/ref/UpValues.html" rel="noreferrer">upvalue</a> for <code>h</code>:</p>
<pre><code>h/:HoldPattern[NumericQ[h[_]]]=True;
</code></pre>
<p>This means that the definition is stored with <code>h</code> rather... |
482,003 | <p>I need help with the following limit $$\lim_{n\to\infty}\sum_{k=1}^n \frac{1}{\sqrt{kn}}$$</p>
<p>Thanks.</p>
| Mikasa | 8,581 | <p>Think about $$\int_0^1 f(x)dx,~~~f(x)=\frac{1}{\sqrt{x}}
$$</p>
<p>Indeed: $$\int_a^b f(x)dx=\lim_{n\to\infty}\sum_{i=1}^nf\left(a+\frac{b-a}{n}i\right)\left(\frac{b-a}{n}\right)$$</p>
|
482,003 | <p>I need help with the following limit $$\lim_{n\to\infty}\sum_{k=1}^n \frac{1}{\sqrt{kn}}$$</p>
<p>Thanks.</p>
| mrf | 19,440 | <p>Your sum can be interpreted as a Riemann sum:</p>
<p>$$\sum_{k=1}^n \frac{1}{\sqrt{kn}} = \frac1n \sum_{k=1}^n \sqrt{\frac{n}{k}}.
$$</p>
<p>Let $f(x) = 1/\sqrt{x}$ and let $x_k = k/n$. Then
$$\sum_{k=1}^n \frac{1}{\sqrt{kn}} = \frac1n \sum_{k=1}^n \sqrt{\frac{n}{k}} = \frac1n \sum_{k=1}^n f(x_k) \to \int_0^1 f(x... |
2,419,529 | <p>I am asked to state whether the following is true or if false to give a counterexample:</p>
<blockquote>
<p>If $A_1 \supseteq A_2 \supseteq A_3 \supseteq \ldots $ are all sets containing an infinite number of elements, then the intersection $$\bigcap_{k=1}^\infty A_k$$ is infinite as well.</p>
</blockquote>
<p>I... | gt6989b | 16,192 | <p>Not quite, because then you have a finite intersection, i.e. each $A_n$ has a finite amount of elements. How about trying $A_n = [n,\infty) \cap \mathbb{N}$? What is the intersection then?</p>
|
2,816,965 | <p>Hey so I was wondering how to differentiate $(e^2)^x$ without using the chain rule.</p>
<p>I tried but I always end up using the chain rule in this case.</p>
<p>Would appreciate some help! (No hints please).</p>
| Community | -1 | <p>If you know the rule</p>
<p>$$(a^x)'=\ln a\,a^x$$ then</p>
<p>$$((e^2)^x)'=\ln e^2\,(e^2)^x=2(e^2)^x.$$</p>
<hr>
<p>Alternatively,</p>
<p>$$(e^{2x})'=\lim_{h\to0}\frac{e^{2(x+h)}-e^{2x}}h=e^{2x}\lim_{h\to0}\frac{e^{2h}-1}h=2e^x\lim_{2h\to0}\frac{e^{2h}-1}{2h}=2e^{2x}.$$</p>
|
868,943 | <p>Can you please tell me the sum of the seires</p>
<p>$ \frac {1}{10} + \frac {3}{100} + \frac {6}{1000} + \frac {10}{10000} + \frac {15}{100000} + \cdots $ </p>
<p>where the numerator is the series of triangular numbers?</p>
<p>Is there a simple way to find the sum?</p>
<p>Thank you.</p>
| André Nicolas | 6,312 | <p>Your expression is equal to $g(1/10)$, where
$$g(x)=\frac{x}{2}\left((2)(1)+(3)(2)x+(4)(3)x^2+(5)(4)x^3+\cdots\right)$$</p>
<p>Take the power series $1+x+x^2+x^3+\cdots$ for $\frac{1}{1-x}$ and differentiate twice. We get $(2)(1)+(3)(2)x+(4)(3)x^2+\cdots$ if we do it term by term, and $\frac{2!}{(1-x)^3}$ if we do... |
2,908,361 | <p>I tried to solve this inequality by taking the square outside the floor function $[y]$ (greatest integer less than $y$)but it was wrong since if $x=2.5$ then $[x]= 2$ and $x^2=4$ while $[x^2]=[6.25]=6$.</p>
| Iti Shree | 433,761 | <p>We can you Calvin's suggested method here (see his comment) by solving the equation as we would with the quadratic equation.
You solve this equation as following, let us take $[x]$ another variable say $n$, and $[x^2]$ as $n^2 + c$. Now your equation becomes :</p>
<p>$$n^2 + c + 5n + 6 = 2$$
$$n (n + 5) = -4 - c$$<... |
1,774,084 | <p>I think it is convergent to $1$ because as $n$ tends to $\infty$ , $1/\sqrt(n)$ tends to $0$. Is it true?</p>
<p>Thanks!</p>
| Bernard | 202,857 | <p>$$n^{\frac1{\sqrt n}}\stackrel{\text{def}}{=}\mathrm e^{\frac{\log n}{\sqrt n}}.$$
Now a basic limit is $\;\lim_{n\to\infty}\dfrac{\log n}{n}=0$, from which we deduce, for any $\alpha>0$: $$\;\lim_{n\to\infty}\dfrac{\log(n^\alpha)}{n^\alpha}=\alpha\lim_{n\to\infty}\dfrac{\log n}{n^\alpha}=0\;$$
whence $\;\lim_{n... |
1,774,084 | <p>I think it is convergent to $1$ because as $n$ tends to $\infty$ , $1/\sqrt(n)$ tends to $0$. Is it true?</p>
<p>Thanks!</p>
| David Holden | 79,543 | <p>for $x \in \mathbb{R^+} $let
$$
A_x=x^{1/\sqrt{x}}
$$
giving
$$
\log A_x = \frac2{\sqrt{x}}\log \sqrt{x}
$$
differentiating $\log A_x$ wrt $x$
$$
\frac{A'_x}{A_x} = x^{-\frac32}(1-\log \sqrt{x})
$$
so $A_x$ is a decreasing function of $x$ for $x \gt e^2$ and is bounded below by 1. This implies that a limit exists a... |
106,396 | <p>An Indian mathematician, Bhaskara I, gave the following amazing approximation of the sine (I checked the graph and some values, and the approximation is truly impressive.)</p>
<p>$$\sin x \approx \frac{{16x\left( {\pi - x} \right)}}{{5{\pi ^2} - 4x\left( {\pi - x} \right)}}$$</p>
<p>for $(0,\pi)$</p>
<p>Here's ... | Steven Stadnicki | 785 | <p>This is very close to a Padé approximant, and in this case the formula is simple enough that it's easy to derive. Firstly, we know that $\sin(x)$ is $0$ at $x=0, x=\pi$; this suggests recasting in terms of the variable $y=x(\pi-x)$. What we're after is a first-order rational approximation $\sin(x) = f(y) = ... |
106,396 | <p>An Indian mathematician, Bhaskara I, gave the following amazing approximation of the sine (I checked the graph and some values, and the approximation is truly impressive.)</p>
<p>$$\sin x \approx \frac{{16x\left( {\pi - x} \right)}}{{5{\pi ^2} - 4x\left( {\pi - x} \right)}}$$</p>
<p>for $(0,\pi)$</p>
<p>Here's ... | Tatar Elemér | 82,923 | <p>That's a pretty neat visualization of how can you intiutively build, proove and understand the approximation.
<a href="https://i.stack.imgur.com/ACoat.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/ACoat.jpg" alt="enter image description here" /></a></p>
<p>The full <a href="https://www.youtube.c... |
3,048,702 | <p>Let <span class="math-container">$S$</span> be a orientable closed surface with genus <span class="math-container">$g \geq 1$</span> and let <span class="math-container">$\gamma \subset S$</span> be an immersed curve. Does there exist a finite cover of <span class="math-container">$S$</span> where <span class="math... | Moishe Kohan | 84,907 | <p>The links in Lee's answer give you a part of the story but not the whole story. There is a missing step for going from residual finiteness to the "lifting property". To explain the missing part I will need two definitions. </p>
<p><strong>Definition 1.</strong> A subgroup <span class="math-container">$H$</span> of ... |
3,632,431 | <blockquote>
<p>Consider the function <span class="math-container">$f: \mathbb{N} \to \mathbb{N}$</span> defined by <span class="math-container">$f(x)=\frac{x(x+1)}{2}$</span>. Show that <span class="math-container">$f$</span> is injective but not surjective.</p>
</blockquote>
<p>So I started by assuming that <span ... | Tuvasbien | 702,179 | <p>First notice that <span class="math-container">$\sum_{n=1}^x n=\frac{x(x+1)}{2}=f(x)$</span> for all <span class="math-container">$x\in\mathbb{N}$</span>, thus if <span class="math-container">$f(a)=f(b)$</span> with <span class="math-container">$a\neq b$</span>, we can suppose without loss of generality that <span c... |
3,392,171 | <p>We have a partial fraction equation:
<span class="math-container">$$\frac{1}{x-5} +\frac{1}{x+5}=\frac{2x+1}{x^2-25}$$</span></p>
<p>I multiplied the equation by the common denominator <span class="math-container">$(x+5)(x-5)$</span> and got <span class="math-container">$0=1$</span>. Is this correct?</p>
| Quanto | 686,284 | <p>Rearrange the equation as </p>
<p><span class="math-container">$$\frac{2x}{x^2-25}-\frac{1}{x-5}-\frac{1}{x+5}
=\frac{2x+1}{x^2-25}-\frac{2x}{x^2-25}= \frac{1}{x^2-25}=0$$</span></p>
<p>which leads to the solutions </p>
<p><span class="math-container">$$x=\pm \infty$$</span></p>
|
223,955 | <p>How can we convert a list to an integer correctly? </p>
<p><strong>{5, 22, 4, 5} -> 52245?</strong></p>
<p>When I use the command <code>FromDigits</code> in Mathematica </p>
<pre><code>FromDigits[{5, 22, 4, 5}]
</code></pre>
<p>The result is incorrect, namely <strong>7245</strong></p>
| flinty | 72,682 | <p>Here are three ways to do it. The first is the best I think:</p>
<pre><code>FromDigits@Flatten[IntegerDigits /@ {5, 22, 4, 5}]
ToExpression@StringJoin[ToString /@ {5, 22, 4, 5}]
ToExpression@StringJoin@StringCases[Characters@ToString@{5, 22, 4, 5}, DigitCharacter]
</code></pre>
|
223,955 | <p>How can we convert a list to an integer correctly? </p>
<p><strong>{5, 22, 4, 5} -> 52245?</strong></p>
<p>When I use the command <code>FromDigits</code> in Mathematica </p>
<pre><code>FromDigits[{5, 22, 4, 5}]
</code></pre>
<p>The result is incorrect, namely <strong>7245</strong></p>
| user1066 | 106 | <pre><code>ToExpression@StringJoin@IntegerString[{5, 22, 4, 5}]
</code></pre>
<blockquote>
<p>52245</p>
</blockquote>
<pre><code>ToExpression@StringJoin@IntegerString[{5, 22, 4, 5},#]&/@{2,10,16}
</code></pre>
<blockquote>
<p>{10110110100101, 52245, 51645}</p>
</blockquote>
|
223,955 | <p>How can we convert a list to an integer correctly? </p>
<p><strong>{5, 22, 4, 5} -> 52245?</strong></p>
<p>When I use the command <code>FromDigits</code> in Mathematica </p>
<pre><code>FromDigits[{5, 22, 4, 5}]
</code></pre>
<p>The result is incorrect, namely <strong>7245</strong></p>
| kcr | 49,048 | <pre><code>list = {5, 22, 4, 5};
FromDigits[StringJoin[IntegerString[list]]]
</code></pre>
<p>which gives</p>
<pre><code>52245
</code></pre>
|
1,215,273 | <blockquote>
<p>Describe the cosets of the subgroup $\langle 3\rangle$ of $\mathbb{Z}$</p>
</blockquote>
<p>The problem I have is $\mathbb{Z}$ is infinite.</p>
<p>So we know that $\langle 3\rangle=\{0,3,6,9,12,\ldots\}$ and I know the definition of cosets (in this case right cosets) is the set of all products of ha... | pjs36 | 120,540 | <p>Since $\Bbb Z$ is an <em>additive</em>, your cosets are $a + \Bbb \langle 3 \rangle$, not $a\langle 3 \rangle$, like you're calculating.</p>
<p>So, to given an example, one would be the coset $1 + \langle 3 \rangle = \{1 + n: n \in \langle 3 \rangle\} = \{\ldots, -5, -2, 1, 4, \ldots\}$.</p>
<p>So that was your on... |
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}^{... | doraemonpaul | 30,938 | <p>$\because$ the trivial solution is $f(x)=e^x-1$</p>
<p>$\therefore$ the general solution is $f(x)=\Theta(\log_2x)(e^x-1)$ , where $\Theta(x)$ is an arbitrary periodic function with unit period</p>
|
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... | Alexandre Eremenko | 25,510 | <p>The answer is no. If something does not hold for polynomials, don't expect that it will hold for entire functions:-)</p>
<p>For example, all non-constant functions of order less than <span class="math-container">$1/2$</span> are surjective.
This follows from an old theorem of Wiman that for such function <span clas... |
3,532,033 | <p>Let <span class="math-container">$(M_1,+,\times)$</span> be an algebraic structure, lets say, for example, a ring. If we have another structure <span class="math-container">$(M_2,+,\times)$</span> isomorphic to the first one does that mean that <span class="math-container">$(M_2,+,\times)$</span> is also a ring ?</p... | Shaun | 104,041 | <p>Sometimes.</p>
<p>It depends on what aspects of the two algebras are isomorphic; for example, <span class="math-container">$(M_1, +_1)$</span> could be isomorphic to <span class="math-container">$(M_2, +_2)$</span> <em>as groups</em>, but not necessarily as, say, <em>rings</em> <span class="math-container">$(M_1, +... |
3,532,033 | <p>Let <span class="math-container">$(M_1,+,\times)$</span> be an algebraic structure, lets say, for example, a ring. If we have another structure <span class="math-container">$(M_2,+,\times)$</span> isomorphic to the first one does that mean that <span class="math-container">$(M_2,+,\times)$</span> is also a ring ?</p... | mrtaurho | 537,079 | <p>I think what you are looking for is the idea of <a href="https://en.wikipedia.org/wiki/Transport_of_structure" rel="nofollow noreferrer"><em>transport of structure</em></a>. Lets stick with your example for now. Suppose, we know that <span class="math-container">$(M_1,+,\times)$</span> is a ring and we have a set <s... |
160,165 | <p>For an elliptic curve $E$ over $\Bbb{Q}$, we know from the proof of the Mordell-Weil theorem that the weak Mordell-Weil group of $E$ is $E(\Bbb{Q})/2E(\Bbb{Q})$. It is well known that
$$
0 \rightarrow E(\Bbb{Q})/2E(\Bbb{Q}) \rightarrow S^{(2)}(E/\Bbb{Q}) \rightarrow Ш(E/\Bbb{Q})[2] \rightarrow 0
$$
is an exact seque... | Matt E | 221 | <p>An $n$-descent will compute the $n$-Selmer group, which sits in a s.e.s.
$$0 \to E(\mathbb Q)/n E(\mathbb Q) \to S^{(n)}(E/\mathbb Q) \to Ш(E/\mathbb Q)[n] \to 0.$$</p>
<p>If you do a $2$-descent, it will give an upper bound on the size of $E(\mathbb Q)/2E(\mathbb Q).$ If you do a $3$-descent, it will give you an... |
105,071 | <p>As one may know, a <b>dynamical system</b> can be defined with a monoid or a group action on a set, usually a manifold or similar kind of space with extra structure, which is called the <i>phase space</i> or <i>state space</i> of the dynamical system. The monoid or group doing the acting is what I call the <i>time s... | Pablo Shmerkin | 11,009 | <p>This is more a comment than an answer but it's too long for a comment.</p>
<p>In ergodic theory (as opposed to dynamical systems), systems in which the acting group is not the integers or the reals have been widely studied, both for their own intrinsic interest and because of deep and striking applications to numbe... |
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>
| Mhenni Benghorbal | 35,472 | <p><strong>Hint:</strong> Use the substitution $\sqrt{x}=u$ or $x=u^2$ and then integration by parts.</p>
<p>$$\int \sin(\sqrt{x})dx = 2\int u\sin(u)du $$</p>
|
2,466,947 | <p>In class we are beginning discrete probability and we are being introduced to counting. He said in an example today that the word "anagram" (which has $7$ letters) can be rearranged to have $7!/3!$ possible words when we don't count the order of the a's. To me this seemed like a combination but this result reduces t... | N. F. Taussig | 173,070 | <p>Your instructor is counting distinguishable arrangements of the word <em>anagram</em>.</p>
<p>The word <em>anagram</em> has seven letters, so we have seven positions to fill with $3$ <em>a</em>s, $1$ <em>g</em>, $1$ <em>m</em>, and $1$ <em>r</em>. We can fill three of these seven positions with <em>a</em>s in $\bi... |
2,466,947 | <p>In class we are beginning discrete probability and we are being introduced to counting. He said in an example today that the word "anagram" (which has $7$ letters) can be rearranged to have $7!/3!$ possible words when we don't count the order of the a's. To me this seemed like a combination but this result reduces t... | Allawonder | 145,126 | <p>I think what you notice is that the process reminds you of the number of permutations (not selections, as you wrote) of <span class="math-container">$4$</span> out of <span class="math-container">$7$</span> objects. This is indeed the case.</p>
<p>Now, first note that to count the number of arrangements of <span cl... |
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... | Eric Naslund | 6,075 | <p>To answer your main concern, questions of any level are welcome on Math Stack Exchange. </p>
<p>Having said that, try to actively look to see if a question has been answered before, as this is far more likely with lower level questions. Questions about solving basic limits, or testing the convergence of certain s... |
1,434,420 | <p>Why is $\bigcap\limits_{n=1}^{\infty} \left( \bigcup\limits_{i=1}^{n} G_i \right)^c = \left( \bigcup\limits_{n=1}^{\infty} \left( \bigcup\limits_{i=1}^{n} G_i \right) \right)^c$? What set properties are being applied here? (The $^c$ is set complement)</p>
| Brian M. Scott | 12,042 | <p>For each $n\in\Bbb Z^+$ let $A_n=\bigcup_{i=1}^nG_i$, and let $C_n=A_n^c$; then $A_1\subseteq A_2\subseteq A_3\subseteq\ldots$, and therefore $C_1\supseteq C_2\supseteq C_3\supseteq\ldots\;$, a fact that may help you visualize the situation. You’re being asked to show that </p>
<p>$$\bigcap_{n\ge 1}C_n=\left(\bigcu... |
1,295,453 | <p>In my assignment I have to calculate to following limit. I wanted to know if my solution is correct. Your help is appreciated:</p>
<p>$$\lim_{n \to \infty}n\cos\frac{\pi n} {n+1} $$</p>
<p>Here's my solution:</p>
<p>$$=\lim_{n \to \infty}n\cos \pi \frac{n} {n+1} $$</p>
<p>Since $\frac {n} {n+1}\to 1 $ and $\cos ... | Paul | 17,980 | <p>You are right. However, we usually say its limit does not exist.</p>
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.