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 |
|---|---|---|---|---|
8,658 | <p>$f(x) = \frac{1}{\cos x}$</p>
<p>$f'(x) = \frac{\sin(x)}{\cos^2(x)}$</p>
<p>$f''(x) = \frac{2\sin^2(x)+\cos^2(x)}{\cos^3(x)}$</p>
<p>$f^{(3)}(x) = \frac{6\sin^3(x)+5\cos^2(x)\sin(x)}{cos^4(x)}$</p>
<p>$\vdots$</p>
<p>$f^{(n)}(x) = \frac{ ?}{cos^{n+1}(x)}$</p>
<p>Some of these are easy: <a href="http://darkwing... | J. M. ain't a mathematician | 498 | <p>For completeness: the Wolfram Functions site gives <a href="http://functions.wolfram.com/ElementaryFunctions/Sec/20/02/0001/" rel="nofollow">a series representation</a> for the $n$-th derivative of the secant:</p>
<p>$$\frac{\mathrm d^n}{\mathrm dx^n}\sec\,x=\sum_{j=0}^\infty \frac{(-1)^j}{(2j-n)!} E_{2j} x^{2j-n}$... |
168,020 | <p>Let $R$ be an local Artinian ring, with maximal ideal $\mathfrak{m}$.</p>
<p>Let $e$ be the smallest positive integer for which $\mathfrak{m}^e=(0)$.</p>
<p>Let $t$ be the smallest positive integer for which $x^t=0$ for all $x \in \mathfrak{m}$.</p>
<p>We know $t \leq e$, with equality holding whenever $\mathfrak... | Mohan | 9,502 | <p>If $R$ contains a field of characteristic zero, then $e=t$. This follows from the fact that if $V$ is a finite dimensional vector space over a field of characteristic zero, the image of the map $V\to S^dV$, $v\mapsto v^d$ generates $S^dV$ as a vector space for any $d$. In your case, suffices to prove that $\mathfrak... |
889,111 | <p>This is a problem from Apostol's Real Analysis book.
$$\text{Find if }\sum_{n=1}^{\infty}\dfrac{1}{n^{1+\frac{1}{n}}}\text{ converges or diverges. }$$
I tried to compare with $\displaystyle \sum_{n=1}^{\infty}\dfrac{1}{n^p}$ for suitable $p$, but $p>1$ fails always. I tried to show $\displaystyle \sum_{k=1}^{\in... | Ted Shifrin | 71,348 | <p>An alternative (and, conceptually, a powerful) way to think about such problems is to use the limit comparison test. Note that $n^{1+1/n} = n\cdot n^{1/n}$. What is $\lim\limits_{n\to\infty}n^{1/n}$?</p>
|
1,250,258 | <p>It's been 10 years since my last math class so I'm very rusty. How would I go about proving
$$3^n < n!$$
where $n \geq 7$?</p>
<p>I understand that factorials grow faster than set values with a variable exponent. Just not sure how to start proving it mathematically.</p>
| Daniel W. Farlow | 191,378 | <p>We can prove it by induction. For $n\geq 7$, let $S(n)$ denote the statement
$$
S(n) : 3^n < n!.
$$
<strong>Base case ($n=7$):</strong> $S(7)$ says that $3^7 = 2187<5040=7!$, and this is true. </p>
<p><strong>Induction step:</strong> Fix some $k\geq 7$, and assume that $S(k)$ is true where
$$
S(k) : 3^k < ... |
1,238,210 | <p>How we can solve that $\lim _{_{x\rightarrow \infty }}\int _0^x\:e^{t^2}dt$ ?</p>
<p>P.S: This is my method as I thought:
$\int _0^x\:\:e^{t^2}dt>\int _1^x\:e^tdt=e^x-e$ which is divergent, so all your answers, helped me to think otherwise, maybe my method help something else :D</p>
| robjohn | 13,854 | <p>$$
\begin{align}
\int_0^xe^{t^2}\,\mathrm{d}t
&\ge\int_0^x\frac tx\,e^{t^2}\,\mathrm{d}t\\
&=\frac1{2x}\left(e^{x^2}-1\right)
\end{align}
$$
As $x\to\infty$, the function on the right goes to $\infty$ extremely fast.</p>
|
695,648 | <p>How can I count the numbers of $5$ digits such that at least one of the digits appears more than one time? </p>
<p>My thoughts are:<br>
I count all the possible numbers of $5$ digits: $10^5 = 100000$. Then, I subtract the numbers that don't have repeated digits, which I calculate this way: $10*9*8*7*6$ $= 30240 $. ... | Priyanshu Kumar | 406,522 | <p>You're doing it wrong.</p>
<p>Total no. of $5$ digit numbers that are possible are $9⋅10⋅10⋅10⋅10$ because $0$ can't be kept on the first place, else it will end up being a $4$ digit number. $= 90000$</p>
<p>Similarly, when any digit is not repeated, it would be $9⋅9⋅8⋅7⋅6$ as one option in less for the first digi... |
208,615 | <p>Maybe I'm not using the best programming practices in Mathematica, but my notebooks usually contain a mixture of "definitions" and "computations" with these definitions. Say,</p>
<pre><code>f[x_] := x^2 (* Definition *)
f[10] (* Computation *)
g[x_] := f[x] - 1/f[x] (* Definition *)
g[100] (* Computation *)
</code>... | Alexey Popkov | 280 | <blockquote>
<p>Then each time I open the notebook anew, to ensure that all definitions are in place the easiest way is to evaluate everything. But some computations can be costly, and I do not need their results.</p>
</blockquote>
<p>One way is to wrap every evaluation by <code>TimeConstrained</code> using <code>$P... |
308,329 | <p>I need help with writing $\sin^4 \theta$ in terms of $\cos \theta, \cos 2\theta,\cos3\theta, \cos4\theta$.</p>
<p>My attempts so far has been unsuccessful and I constantly get developments that are way to cumbersome and not elegant at all. What is the best way to approach this problem?</p>
<p>I know that the answe... | Community | -1 | <p>By repeatedly applying the formulas
\begin{eqnarray*}
\sin(x)\sin(y)&=&{1\over2}\left[\cos(x-y)-\cos(x+y)\right]\\[5pt]
\sin(x)\cos(y)&=&{1\over2}\left[\sin(x-y)+\sin(x+y)\right]
\end{eqnarray*}
you will see how to write odd powers of sine as a linear combination of sines,
and even powers of sine as... |
966,278 | <p>Given a recursive relation
$$a_n = \begin{cases}
(1 - 2b_n)a_{n-1} + b_n, & n > 1 \\
\frac{1}{2}, & n =1
\end{cases}
$$, how can I expression $a_n$ in term of $b_i, i \in \{1, 2, \dots n\}$?</p>
| Claude Leibovici | 82,404 | <p>Consider the case of $n=2$ with $a_1=\frac{1}{2}$. $$a_2=(1 - 2b_2)a_1 + b_2=(1 - 2b_2)\frac{1}{2}+b_2=\frac{1}{2}$$ You can repeat that for ever and, whatever $b_n$ could be, all $a_n=\frac{1}{2}$.</p>
|
2,464,890 | <p>Here is link to some limit questions:</p>
<p><a href="https://i.stack.imgur.com/2rM9f.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/2rM9f.png" alt="Example" /></a>
Can anyone explain how has answers were derived? In (a), how can we cancel out <span class="math-container">$(x-2)$</span>? And how ... | Fimpellizzeri | 173,410 | <p>There's a factor of $(x-2)$ in the denominator, and two factors of $(x-2)$ in the numerator. The cancelling removes one factor of each. What remains is</p>
<p>$$\frac{x-2}{x+2}$$</p>
<p>As $x$ approaches $2$, the numerator approaches $0$ and the denominator approaches $4$. This is a well defined quotient: $0/4=0$,... |
819,054 | <p>Having the following inequality, for a real-valued function $f$ which is twice differentiable:</p>
<p>$f(a+h)-f(a)\geq f(a)-f(a-h)$ for any $a \in\mathbf{R}$, $h > 0$.</p>
<p>and assuming that $f$ is bounded, I see on a graph that $f$ is constant but I can't prove it properly.
I tried to prove that $f'(x)=0$ fo... | Siminore | 29,672 | <p>$$
\frac{f(a+h)-2f(a)+f(a-h)}{h^2} \xrightarrow{h \to 0}{\, f''(a)} \quad \text{(by L’Hôpital’s rule)}.
$$
Actually, your condition looks like a convexity condition at any point, and convex functions are either unbounded or constant on the whole $\mathbb{R}$.</p>
|
819,054 | <p>Having the following inequality, for a real-valued function $f$ which is twice differentiable:</p>
<p>$f(a+h)-f(a)\geq f(a)-f(a-h)$ for any $a \in\mathbf{R}$, $h > 0$.</p>
<p>and assuming that $f$ is bounded, I see on a graph that $f$ is constant but I can't prove it properly.
I tried to prove that $f'(x)=0$ fo... | Community | -1 | <p>Show the contrapositive, that if $f$ is not constant but satisfies the given condition, then $f$ is not bounded.</p>
<p>If $f$ is not constant then there exist $p,q$ with, say, $p<q$ and $f(p)\ne f(q)$. I'll consider only the case $f(p)<f(q)$, leaving the case $f(p)>f(q)$ to you.</p>
<p>Set $x_k=p+k(q-p)... |
1,807,456 | <p>I don't know asymptotic behaviour of the integral $$\int_{0}^{\infty}\frac{du}{\sqrt{4\pi u^{3}}}\left(1-\frac{e^{-\Omega u}}{\sqrt{\frac{1-\exp\left(-2u\right)}{2u}}}\right),$$
when I read a physics paper. It says that the integral have asymptotic behaviour $\log\left(\pi\Omega B\right)/\sqrt{2\pi}$, when $\Omega\t... | Jack D'Aurizio | 44,121 | <p>We may replace $\sqrt{\frac{2u}{1-e^{-2u}}}$ with $\sqrt{2u+1}$ and compute the resulting integral in terms of the <a href="https://en.wikipedia.org/wiki/Confluent_hypergeometric_function" rel="nofollow">Tricomi $U$ function</a>
$$ \sqrt{2}\cdot U\left(-\frac{1}{2},1,\frac{\Omega}{2}\right)\approx \frac{1}{\sqrt{2\p... |
3,425,369 | <p>I'm taking a linear algebra course and I'm having trouble proving linear (in)dependence of functions. I understand that I have to prove that the <span class="math-container">$a_1f(x) + a_2g(x) = 0$</span> but I don't know how to actually do that. For example given a pair of functions 1 and t, how do you prove linear... | Brian Moehring | 694,754 | <p>Note that we don't <em>prove</em> <span class="math-container">$a_1f(x) + a_2g(x) = 0.$</span> We <em>assume</em> that and then prove that <span class="math-container">$a_1=a_2=0.$</span></p>
<p>Since functions are defined by their values, one way you can do this is to choose certain <span class="math-container">$... |
3,839,129 | <p>I think I was able to find the answer from some guesswork, but I'm unsure of how to prove the result.</p>
<p>If <span class="math-container">$n=5$</span>, then there are <span class="math-container">$2^5=32$</span> total subsets by the power set rule.</p>
<p>The subsets that contain <span class="math-container">$1$<... | Brian M. Scott | 12,042 | <p>There is a bijection <span class="math-container">$\varphi$</span> between subsets of <span class="math-container">$A$</span> containing <span class="math-container">$1$</span> and <span class="math-container">$2$</span> and subsets of <span class="math-container">$B=\{3,4,\ldots,n\}$</span>: if <span class="math-co... |
3,839,129 | <p>I think I was able to find the answer from some guesswork, but I'm unsure of how to prove the result.</p>
<p>If <span class="math-container">$n=5$</span>, then there are <span class="math-container">$2^5=32$</span> total subsets by the power set rule.</p>
<p>The subsets that contain <span class="math-container">$1$<... | nonuser | 463,553 | <p>You just have to ''add'' some subset of <span class="math-container">$\{3,4,5...,n\}$</span> to the set <span class="math-container">$\{1,2\}$</span>. But the number of such subsets is exactly <span class="math-container">$2^{n-2}$</span> and you are done.</p>
|
2,342,124 | <p><a href="https://i.stack.imgur.com/QdbFG.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/QdbFG.png" alt="enter image description here"></a></p>
<p>Well this seems like <span class="math-container">$1-|t|$</span> for <span class="math-container">$|t|<1$</span> and <span class="math-container">... | Robert Z | 299,698 | <p>Yes, you are correct. Since the integrand depends only on $r$, it is constant on each spherical layer and
$$\iiint_C \sqrt{x^2+y^2+z^2}\, dx \, dy \, dz=\int_{r=0}^1 r\cdot (\mbox{surface area of the sphere of radius $r$})\, dr\\=
\int_{r=0}^1 r\cdot (4\pi r^2)\, dr=4\pi\cdot\left[\frac{r^4}{4}\right]_0^1=\pi.$$... |
1,156,738 | <p>Is it fine to say "Groups $A$ and $B$ are isomorphic." or should one say "Groups $A$ and $B$ are isomorphic to each other."?</p>
| kolonel | 104,564 | <p>It is the same thing. An isomorphism between $A$ and $B$ implies that your found a bijection between $A$ and $B$ that is also a homomorphism, which remains clear in both statements.</p>
|
3,186,627 | <p>Proposition: Let A be a subset of R which is bounded below. Let B be a subset of R which is bounded above. If <span class="math-container">$\inf(A) < \sup (B) $</span> then there is some <span class="math-container">$a \in A$</span> and <span class="math-container">$b \in B$</span> such that <span class="math-con... | Marco Vergamini | 661,708 | <p>Your proof works, but there is a little flaw: you're not considering the case <span class="math-container">$I \in A, S \not\in B$</span> or viceversa; by the way, dividing in cases is useless cause what you do in the last case can be modified to work in every case. You can simply change <span class="math-container"... |
4,041,641 | <p>This is the answer given:</p>
<p><a href="https://i.stack.imgur.com/6dyU4.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/6dyU4.png" alt="enter image description here" /></a></p>
<p>I think this answer is not correct because the line given by y=-2x/3 makes an angle that is below the x axis, so the... | Sandipan Dey | 370,330 | <p>Here the slope of the line = <span class="math-container">$m = tan \gamma = -\frac{2}{3}$</span>.</p>
<p><span class="math-container">$\therefore sin \gamma = \frac{m}{\sqrt{1+m^2}},\; cos \gamma = \frac{1}{\sqrt{1+m^2}}$</span>.</p>
<p>The transformation matrix for reflection around the line is given by the product... |
285,034 | <p>I currently don't see how to solve the following integral:</p>
<p>$$\int_{-1/2}^{1/2} \cos(x)\ln\left(\frac{1+x}{1-x}\right) \,dx$$</p>
<p>I tried to solve it with integration by parts and with a Taylor series, but nothing did help me so far.</p>
| nbubis | 28,743 | <p>You can't solve the indefinite integral using elementary functions. This integral can be solved by parts however if you use $Ci(x)$ - the <a href="http://mathworld.wolfram.com/CosineIntegral.html" rel="nofollow">Cosine Integral</a>. The actual value as @Psx mentioned is of course zero since the integrand is odd.</p>... |
658,078 | <p>I'm embarrassed to ask this question, but my child has the following homework question:</p>
<p>"Use absolute value to describe the relationship between a negative credit card balance and the amount owed."</p>
<p>I'm not sure for what it is they're looking. Clearly a <code>-$25</code> balance means you have <code>... | bsdshell | 54,105 | <p>$$
\text{Given e = -5, only show right inverse}\\
\text{let b is the inverse of a}\\
\Rightarrow\text{a*b=e}\\
\Rightarrow a + b + 5 = e = -5\\
\Rightarrow a + b = -5 - 5\\
\Rightarrow a + b + (-a) = -5-5+(-a)\quad(\because \text{I always can add an Integer in both side})\\
\Rightarrow b = -a -10\quad(\because \text... |
658,078 | <p>I'm embarrassed to ask this question, but my child has the following homework question:</p>
<p>"Use absolute value to describe the relationship between a negative credit card balance and the amount owed."</p>
<p>I'm not sure for what it is they're looking. Clearly a <code>-$25</code> balance means you have <code>... | Kevin Arlin | 31,228 | <p>Here's a restatement of what's going on here: we have a function $\varphi(n)=n+5$ from $\mathbb{Z}$ to itself and want to check that $n*m=\varphi(n+m)$ is also a group structure. So, it may leave the waters less muddy if we state the problem more generally: given a set $S$ with a group structure $(*,e)$ and a functi... |
2,130,141 | <p>I am having troubles with the following excercise:</p>
<p>$P(A\times B) = Q$ and $Q = \lbrace V\times W \ \vert \ V\in P(A), W\in P(B)\rbrace$ </p>
<p>So I have to prove or disprove. I know that $P(A\times B) \neq Q$ and being specific $P(A\times B) \not\subset Q$ and $Q \subset P(A\times B)$. In addition; </p>
<... | Martin Argerami | 22,857 | <p>Any topology arising from a metric is Hausdorff. But with $\{\emptyset,M\} $ you cannot separate $a $ and $b $.</p>
|
3,878,723 | <blockquote>
<p>Find the value of <span class="math-container">$k$</span> if the curve <span class="math-container">$y = x^2 - 2x$</span> is tangent to the line <span class="math-container">$y = 4x + k$</span></p>
</blockquote>
<p>I have looked at the solution to this question and the first step is the "equate the... | JC12 | 736,604 | <p>If you want two lines to be tangent, you only want them to touch at one point, meaning there is only one solution to:</p>
<p><span class="math-container">$$x^2-2x=4x+k$$</span></p>
<p>Which simplifies to:</p>
<p><span class="math-container">$$x^2-6x-k=0$$</span></p>
<p>Note that the above only has one solution if th... |
317,547 | <p>Let $X$ and $Y$ have joint mass function</p>
<p>$f(j,k)=\frac {c(j+k)a^{j+k}}{j!k!}$, $j,k\geq 0$</p>
<p>where $a$ is a constant. Find $c$</p>
<p>This sum seems hard to to. How to complete this sum?</p>
| Coiacy | 62,460 | <p>This is nothing but some calculation. Fix $k$ first,</p>
<p>When $k=0,\ \sum_{j=0}^{\infty} \frac{c(j+k)\cdot a^{j+k}}{j!\cdot k!}=\sum_{j=0}^{\infty} \frac{c\cdot j\cdot a^{j}}{j!}=ca\cdot e^a$</p>
<p>When $k\ge 1$,
$$ \sum_{j=0}^{\infty} \frac{c(j+k)\cdot a^{j+k}}{j!\cdot k!}=\sum_{j=0}^{\infty} \frac{c\cdot j\c... |
3,438,048 | <p>I've recently obtained my University entrance papers from 1967 (yes,52 years ago!) and I found the question below difficult. I presume the answer is a symmetric expression in the differences between alpha,beta and gamma.Am I missing some obvious trick? Any help would be appreciated.</p>
<p>Simplify and evaluate the... | Root | 466,361 | <p>Your proof of a) is not right. Note that interval size tending to zero is different from it being zero.
a) is vacuuously true for null set to put it differently there is no x the 'for all x'statement is always true. For <span class="math-container">$\mathbb{R}$</span> it clear that as for any real number say interv... |
145,785 | <p>Let $V$ be a $\mathbb{C}$-vector space of finite dimension. Denote its $d$-th symmetric power by $V^{\odot d}$. I am looking for a proof that $V^{\odot d}$ is generated by the elements $v^{\odot d}$ for $v\in V$. </p>
<p>A different way to look at it is the following: Consider the polynomial ring $R=\mathbb{C}[x_1,... | user8268 | 8,268 | <p>Let $W$ be the (finite-dimensional) vector space generated by $d$-th powers of linear functions in $x_1,\dots,x_n$. Let $h_1,\dots,h_d$ be such linear functions. Consider the polynomial map $f:\mathbb{C}^d\to W$ given by $f(t_1,\dots,t_d)=(t_1h_1+\dots+t_d h_d)^d$. As
$$\frac{\partial}{\partial t_1}\cdots\frac{\par... |
1,772,562 | <p>Let $f:\mathbb{C}\rightarrow\mathbb{C}$ be holomorphic. If we have $|f(z)|\leq|z|^n$ for some $n\in\mathbb{N}$ and all $z\in\mathbb{C}$, then $f$ is a polynomial.</p>
<p>I tried to apply Liouville's theorem but it does not help.</p>
<p>Thanks for your help.</p>
| lhf | 589 | <p>$|f(z)|\leq|z|^n$ implies $f(0)=0$.</p>
<p>Writing $f(z)=z^m g(z)$ with $g(0)\ne0$ implies $m \ge n$ and so $|z^{m-n}g(z)| \le 1$.</p>
<p>Now apply Liouville's theorem.</p>
|
3,075,869 | <p>The following is quoted from <a href="https://en.wikipedia.org/wiki/Quotient_space_(topology)" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Quotient_space_(topology)</a></p>
<p>Quotient maps q : X → Y are characterized among surjective maps by the following property: if Z is any topological space and f :... | drhab | 75,923 | <p>Let <span class="math-container">$\tau $</span> denote the quotient topology on <span class="math-container">$Y $</span> induced by <span class="math-container">$q $</span>.</p>
<p>Let <span class="math-container">$\rho $</span> denote a topology on <span class="math-container">$Y $</span> such that the character... |
491,748 | <p>I got stuck on this problem and can't find any hint to solve this. Hope some one can help me. I really appreciate.</p>
<blockquote>
<blockquote>
<p>Give an example of a collection of sets $A$ that is not locally finite, such that the collection $B = \{\bar{X} | X \in A\}$ is locally finite.</p>
</blockquote... | user87690 | 87,690 | <p>Just take some non-empty open set $U$ such that $\overline{U} \setminus U$ is infinite. Actually $U$ does not have to be open, just non-empty and such that $\overline{U} \setminus U$ is infinite. This includes the correct example of dense subset of $\mathbb{R}$ other people mentioned.</p>
|
2,360,819 | <p>Probability question that I don't understand? It is on our assignment for probability and I can't seem to figure out how it has to do with probability or how to solve it.</p>
| Cameron Buie | 28,900 | <p>Well, a diagonal connects two non-adjacent vertices, yes? Well, how many segments can be drawn which connect two <em>distinct</em> vertices of an octagon? How many segments can be drawn which connect two <em>adjacent</em> vertices of an octagon? Now what?</p>
|
545,003 | <p>I have a proof that I am trying to prove and I am getting stuck at the inductive hypothesis. This is my theorem:</p>
<blockquote>
<p>For all real numbers $n>3$, the following is true: $n + 3 < n!$.</p>
</blockquote>
<p>I have proven true for $n = 4$, and will assume true for some arbitrary value $k$, i.e.,... | Patrick | 50,809 | <p>Induction is overkill here.</p>
<p>For $n \gt 3$ we have that $n+3 \lt n + n = 2\cdot n \lt 1\cdot 2 \cdot3\cdots (n-1) \cdot n =n!$</p>
|
1,213,344 | <p>A solid half-ball $H$ of radius $a$ with density given by $k(2a-\rho)$, where $k$ is a constant. Find its mass.</p>
<p>You of course use spherical coordinates so $dV=\rho ^2 \sin\phi d\rho d\phi d\theta$. It is clear to see that the limits are $\rho \in [0,a]$ and $\theta \in [0,2\pi]$. The limits for $\phi$ are no... | Timbuc | 118,527 | <p>$$I_n:=\int_0^1\frac{x^n}{x^n+1}dx=\int_0^1\left(1-\frac1{x^n+1}\right)dx=1-\int_0^1\frac{dx}{x^n+1}$$</p>
<p>$$I_{n+1}=\int_0^1\frac{x^{n+1}}{x^{n+1}+1}dx=\int_0^1\left(1-\frac1{x^{n+1}+1}\right)dx=1-\int_0^1\frac{dx}{x^{n+1}+1}$$</p>
<p>Now, it is trivial that for $\;x\in[0,1]\;$ we have</p>
<p>$$\frac1{x^{n+1}... |
533,534 | <p>Let X = {x(i)} be a group of n data with mean = μ(x) and variance $= σ(x)^2$.</p>
<p>We use the symbol S(x(i)) to represent the sum of all the x's.</p>
<p>Similar notations will be used for the group Y.</p>
<p>Supposed that Y is formed by adding an extra element x(n+1) to X and the value of that element is greate... | Juan Sebastian Lozano | 87,284 | <p>I think it is useful too look at this graphically first. </p>
<p><img src="https://i.stack.imgur.com/1nT7q.jpg" alt="graph"></p>
<p>Now that you can see this graphic, lets consider the geometric interpretation of the limit. If I have the limit $\lim\limits_{x \to 0} f(x)$, them I am essentially asking: What is the... |
2,848,891 | <p>Find the number of solutions of $$\left\{x\right\}+\left\{\frac{1}{x}\right\}=1,$$ where $\left\{\cdot\right\}$ denotes Fractional part of real number $x$.</p>
<h2>My try:</h2>
<p>When $x \gt 1$ we get</p>
<p>$$\left\{x\right\}+\frac{1}{x}=1$$ $\implies$</p>
<p>$$\left\{x\right\}=1-\frac{1}{x}.$$</p>
<p>Letting... | J.G. | 56,861 | <p>Now multiply by $n+f$; solve a quadratic to express $f$ in terms of $n$. Don't forget to check negative solutions too.</p>
|
2,848,891 | <p>Find the number of solutions of $$\left\{x\right\}+\left\{\frac{1}{x}\right\}=1,$$ where $\left\{\cdot\right\}$ denotes Fractional part of real number $x$.</p>
<h2>My try:</h2>
<p>When $x \gt 1$ we get</p>
<p>$$\left\{x\right\}+\frac{1}{x}=1$$ $\implies$</p>
<p>$$\left\{x\right\}=1-\frac{1}{x}.$$</p>
<p>Letting... | robjohn | 13,854 | <p><strong>Observations</strong></p>
<p>I am assuming that $\{x\}=x-\lfloor x\rfloor$, which is in $[0,1)$.</p>
<p>There are no integer solutions; if $x\in\mathbb{Z}$, then $\{x\}+\left\{\frac1x\right\}=0+\left\{\frac1x\right\}\lt1$.</p>
<p>Since $\{x\}+\left\{\frac1x\right\}=1$ is stable under $x\leftrightarrow\fra... |
23,566 | <p>I love math, and I used to be very good at it. The correct answers came fast and intuitively. I never studied, and redid the demonstration live for the tests (sometimes inventing new ones). I was the one who answered the tricky questions in class (8 hours of math/week in high school)... You get the idea.</p>
<p>As ... | Phil | 1,088 | <p>I highly reccommend this book <a href="http://rads.stackoverflow.com/amzn/click/0201558025" rel="nofollow">Concrete Mathematics: A Foundation for Computer Science</a>. While the entire book is not that relevant, I think a few chapters (on Binomial Coefficients and Discrete Probability) can be helpful.</p>
<p>The bo... |
23,566 | <p>I love math, and I used to be very good at it. The correct answers came fast and intuitively. I never studied, and redid the demonstration live for the tests (sometimes inventing new ones). I was the one who answered the tricky questions in class (8 hours of math/week in high school)... You get the idea.</p>
<p>As ... | Alex Greene | 9,303 | <p>Interesting question. Not asking about mathematics as such, but about building up one's skills in a global environment suffering from a dearth of post-educational mathematical tuition for adults.</p>
<p>Mathematics taught in educational establishments tends, at least at the early stages of childhood, to focus on es... |
1,988,191 | <p>Today I coded the multiplication of quaternions and vectors in Java. This is less of a coding question and more of a math question though:</p>
<pre><code>Quaternion a = Quaternion.create(0, 1, 0, Spatium.radians(90));
Vector p = Vector.fromXYZ(1, 0, 0);
System.out.println(a + " * " + p + " = " + Quaternion.product(... | amd | 265,466 | <p>The short answer is that you have to conjugate by a half-angle quaterion instead of simply multiplying to effect a rotation. See <a href="https://math.stackexchange.com/q/40164/265466">this question</a> or <a href="https://en.m.wikipedia.org/wiki/Quaternions_and_spatial_rotation" rel="nofollow noreferrer">this Wikip... |
184,682 | <p>I have difficulties with a rather trivial topological question: </p>
<p>A is a discrete subset of $\mathbb{C}$ (complex numbers) and B a compact subset of $\mathbb{C}$. Why is $A \cap B$ finite? I can see that it's true if $A \cap B$ is compact, i.e. closed and bounded, but is it obvious that $A \cap B$ is closed?<... | user642796 | 8,348 | <p>If $A \cap B$ were infinite then by the compactness of $B$ there would be a limit point $z$. Next note that every neighbourhood of $z$ would have infinite intersection with $A \cap B$ and hence with $A$, contradicting that $A$ was discrete.</p>
<p><strong>Edit:</strong> I quickly deleted this after I started to th... |
184,682 | <p>I have difficulties with a rather trivial topological question: </p>
<p>A is a discrete subset of $\mathbb{C}$ (complex numbers) and B a compact subset of $\mathbb{C}$. Why is $A \cap B$ finite? I can see that it's true if $A \cap B$ is compact, i.e. closed and bounded, but is it obvious that $A \cap B$ is closed?<... | Stefan Geschke | 16,330 | <p>There seems to be a contradiction between the answers of Andre Nicolas and of Arthur Fischer, yet both are correct. This depends on your definition of discrete. Andre's notion of discrete is this: </p>
<p>A set $S$ in a topological space $X$ is discrete if it is discrete with respect to the subspace topology. ... |
1,012,236 | <p>A continuous time process it's nule for t < 0. In which conditions is it stationary (WSS)?</p>
<p>I know that E[x(t)] must be a constant and the autocorrelation function must depend only on the time difference t2-t1. Are there any other conditions?</p>
| N. S. | 9,176 | <p>The sequence $\left(1+\frac{1}{n}\right)^n$ is strictly increasing and converging to $e$. This implies the first inequality.</p>
<p>$\left(1+\frac{1}{n}\right)^{n+1}$ is strictly decreasing and converging to $e$. This implies the first inequality.</p>
|
674,310 | <p>I am having trouble with a proof for linear algebra. Could somebody explain to me how to prove that if $A$ and $B$ are both $n\times n$ non singular matrices, that their product $AB$ is also non singular. </p>
<p>A place to start would be helpful. Thank you for your time. </p>
| Community | -1 | <p>There's different manners to prove this result for example:</p>
<ul>
<li>Using the determinant:
$$\det(AB)=\det A\det B$$
and the fact that $C$ is singular iff $\det C=0$.</li>
<li>Using the fact that $AB$ is invertible then $A$ is surjective and $B$ is injective and that in finite dimensional space: $C$ is injecti... |
1,715,358 | <p>Being fascinated by the approximation $$\sin(x) \simeq \frac{16 (\pi -x) x}{5 \pi ^2-4 (\pi -x) x}\qquad (0\leq x\leq\pi)$$ proposed, more than 1400 years ago by Mahabhaskariya of Bhaskara I (a seventh-century Indian mathematician) (see <a href="https://math.stackexchange.com/questions/976462/a-1-400-years-old-appr... | Clerk | 550,918 | <p>@Claude Leibivici use the following two point Taylor series in x=-Pi, Pi
$$\frac{z (z-\pi )^3 (z+\pi )^3}{48 \pi ^4}-\frac{5 z (z-\pi )^3 (z+\pi )^3}{16 \pi ^6}+\frac{3 z (z-\pi )^2 (z+\pi )^2}{8 \pi ^4}-\frac{z (z-\pi ) (z+\pi )}{2 \pi ^2}$$ the cuadratic error is superior to any formula above at the same grade <... |
1,037,898 | <p>Does given integral</p>
<blockquote>
<p>$$\int_1^\infty \frac{\log(x-1)}{x(x-1)}\,dx$$</p>
</blockquote>
<p>converge? If it is convergent can we evaluate it's value?</p>
| Paul | 17,980 | <p>Hints:</p>
<p>Let $t=\log(x-1)$. Then $x=1+e^t$</p>
|
1,037,898 | <p>Does given integral</p>
<blockquote>
<p>$$\int_1^\infty \frac{\log(x-1)}{x(x-1)}\,dx$$</p>
</blockquote>
<p>converge? If it is convergent can we evaluate it's value?</p>
| Siminore | 29,672 | <p>Consider
$$
\int_1^2 \frac{\log (x-1)}{x(x-1)}dx = \int_0^1 \frac{\log u}{(1+u)u}du \sim \int_0^1 \frac{\log u}{u}du,
$$
which is divergent, since $$\frac{d}{du} \frac12 \log^2 u = \frac{\log u}{u}.$$ Hence the whole integral is divergent. By the way, the integral "at infinity" converges.</p>
|
1,037,898 | <p>Does given integral</p>
<blockquote>
<p>$$\int_1^\infty \frac{\log(x-1)}{x(x-1)}\,dx$$</p>
</blockquote>
<p>converge? If it is convergent can we evaluate it's value?</p>
| Zaid Alyafeai | 87,813 | <p>$$I=\int_0^1\frac{\log(u)-\log(1-u)}{u}\,dx = \int^1_0\frac{\log(u)}{u}\,du+\zeta(2)$$</p>
|
99,572 | <p>One of the most useful tools in the study of convex polytopes is to move from polytopes (through their fans) to toric varieties and see how properties of the associated toric variety reflects back on the combinatorics of the polytopes. This construction requires that the polytope is rational which is a real restrict... | Community | -1 | <p>The construction of Davis and Januskiewicz can be realized as an equivariant colimit. </p>
<p>Let $P$ be a simple polytope of dimension $n$ and let $G$ be either the mod 2 torus ${\mathbb{Z}}_2^n$ or the usual torus ${\mathbb{T}}^n$. A characteristic function on $P$ corresponds to a order-reserving map $\chi:{\math... |
99,572 | <p>One of the most useful tools in the study of convex polytopes is to move from polytopes (through their fans) to toric varieties and see how properties of the associated toric variety reflects back on the combinatorics of the polytopes. This construction requires that the polytope is rational which is a real restrict... | Li Yu | 471,224 | <p>In the definition of moment-angle manifold in Davis-Januszkiewicz's 1991 paper, the orbit space is a simple convex polytope <span class="math-container">$P$</span> which is contractible. We can replace <span class="math-container">$P$</span> by an arbitrary smooth nice manifold with corners <span class="math-contain... |
2,601,380 | <p>I'm trying to figure out the impedance of a capacitor. My textbook tells me the answer is $\frac{-i}{\omega C}$ and plugging that into the equation does work but I wanted to come up with that answer myself. So I wrote out the equation with what I know:</p>
<p>$$-V_0\omega C\sin\omega t = Re\left( \frac{V_0(\cos\ome... | Acccumulation | 476,070 | <p>Re() is a projective map; Re(a+bi) = a. Thus Re(z) = z-iIm(z). So given RHS = Re(z), we have that RHS+bi = z for some real b. Note that Re(z) = a does not yield a single value of z as a solution, but instead gives a vertical line in the complex plane. Each point on that line will give a different value for x.</p>
<... |
3,362,654 | <p>Let's say <span class="math-container">$C$</span> is a category, and <span class="math-container">$\mathscr{C}$</span> is a collection of morphisms in <span class="math-container">$C$</span>. I have come across the following sentence </p>
<p>"<span class="math-container">$C$</span> admits pullbacks along morphisms ... | Brian Moehring | 694,754 | <p>Let <span class="math-container">$N$</span> denote the winning position. Then for <span class="math-container">$n=0,1,\ldots, 12,$</span> <span class="math-container">$$\begin{align*}P(N > n) &= \text{probability that none of the first } n+1 \text{ birthday months match} \\ &= \frac{12}{12}\cdot\frac{11}... |
2,310,109 | <p>I'm an undergraduate with little to no background in functional analysis and topology. The whole concept of function spaces is quite fuzzy to me, and I'm having a difficult time conceptualizing it. (Things like there being different notions of compactness in general topological spaces is one of many things confusing... | Henno Brandsma | 4,280 | <p>If you are considering functions from $A \subseteq \mathbb{R}^n$ to $[a,b]$ in the pointwise topology, then for a condition like you describe in the comments: $f(v_1) +f(v_2) + f(v_3) \in C$ where $C$ is closed in $[a,b]$ (like a singleton or a finite set maybe), then the functions that satisfy it is indeed closed. ... |
322,302 | <p>Conjectures play important role in development of mathematics.
Mathoverflow gives an interaction platform for mathematicians from various fields, while in general it is not always easy to get in touch with what happens in the other fields.</p>
<p><strong>Question</strong> What are the conjectures in your field prove... | Sam Hopkins | 25,028 | <p>Karim Adiprasito proved the g-conjecture for spheres in a preprint that was posted in December of last year: <a href="https://arxiv.org/abs/1812.10454" rel="noreferrer">https://arxiv.org/abs/1812.10454</a>.</p>
<p>This was probably considered the biggest open problem in the combinatorics of simplicial complexes. Se... |
322,302 | <p>Conjectures play important role in development of mathematics.
Mathoverflow gives an interaction platform for mathematicians from various fields, while in general it is not always easy to get in touch with what happens in the other fields.</p>
<p><strong>Question</strong> What are the conjectures in your field prove... | T_M | 122,587 | <p>S. T. Yau conjectured in the 80's that every compact Riemannian 3-manifold should contain infinitely many different minimal surfaces (smooth, closed). This was <a href="https://arxiv.org/abs/1806.08816" rel="noreferrer">proved last year</a> by <a href="https://web.math.princeton.edu/~aysong/" rel="noreferrer">Antoin... |
322,302 | <p>Conjectures play important role in development of mathematics.
Mathoverflow gives an interaction platform for mathematicians from various fields, while in general it is not always easy to get in touch with what happens in the other fields.</p>
<p><strong>Question</strong> What are the conjectures in your field prove... | Denis Nardin | 43,054 | <p>The <a href="https://en.wikipedia.org/wiki/Weibel%27s_conjecture" rel="noreferrer">Weibel conjecture</a> about negative K-groups was proven in 2018 by Moritz Kerz, Florian Strunk, Georg Tamme.</p>
<p>The conjecture states that if <span class="math-container">$X$</span> is a Noetherian scheme of Krull dimension <spa... |
322,302 | <p>Conjectures play important role in development of mathematics.
Mathoverflow gives an interaction platform for mathematicians from various fields, while in general it is not always easy to get in touch with what happens in the other fields.</p>
<p><strong>Question</strong> What are the conjectures in your field prove... | KConrad | 3,272 | <p>In number theory, the Sato-Tate conjecture about elliptic curves over <span class="math-container">$
\mathbf Q$</span> was a problem from the 1960s and Serre's conjecture on modularity of odd 2-dimensional Galois representation was a conjecture from the 1970s-1980s. Both were settled around 2008. (For ST conj., the ... |
322,302 | <p>Conjectures play important role in development of mathematics.
Mathoverflow gives an interaction platform for mathematicians from various fields, while in general it is not always easy to get in touch with what happens in the other fields.</p>
<p><strong>Question</strong> What are the conjectures in your field prove... | David White | 11,540 | <p><a href="https://en.wikipedia.org/wiki/Vop%C4%9Bnka%27s_principle" rel="nofollow noreferrer">Vopenka's Principle</a> is a large cardinal axiom that has several equivalent formulations. Arguably the simplest is the statement</p>
<p><em>For every proper class of graphs there exists a non-identity homomorphism between ... |
3,882,566 | <p>We have <span class="math-container">$0<b≤ a$</span>, and:</p>
<p><span class="math-container">$$\underbrace{\dfrac{1+⋯+a^7+a^8}{1+⋯+a^8+a^9}}_{A} \quad \text{and} \quad \underbrace{\dfrac{1+⋯+b^7+b^8}{1+⋯+b^8+b^9}}_{B}$$</span></p>
<p>Source: Lumbreras Editors</p>
<hr />
<p>It was my strategy:</p>
<p><span clas... | xpaul | 66,420 | <p>Let
<span class="math-container">$$ f(x)=\dfrac{1+⋯+x^7+x^8}{1+⋯+x^8+x^9}. $$</span>
Then
<span class="math-container">$$ f'(x)=-\frac{x^8(x^8+2x^7+3x^6+4x^5+5x^4+6x^3+7x^2+8x+9)}{(1+⋯+x^8+x^9)^2}\le0 $$</span>and hence <span class="math-container">$f(x)$</span> is decreasing. So if <span class="math-container">$a\g... |
25,100 | <p>Suppose one has a set $S$ of positive real numbers, such that the usual numerical ordering on $S$ is a well-ordering. Is it possible for $S$ to have any countable ordinal as its order type, or are the order types that can be formed in this way more restricted than that?</p>
| Robin Chapman | 4,213 | <p>Yes, one can have any countable ordering. Indeed any countable totally
ordered set can be embedded in $\mathbb{Q}$. Write your ordered set as
$ \lbrace a_1,a_2,\ldots \rbrace $
and define the embedding recursively: once you have placed $a_1,\ldots,a_{n-1}$
there will always be an interval to slot $a_n$ into.</p>
|
2,747,896 | <p>I am a bit confused with cardinality at the moment. I know that the cardinality of $\mathbb{R}$ is equal to $\lvert(0,1)\rvert$, but does that mean they are equal to infinity, if not what are they equal to ?</p>
| Siong Thye Goh | 306,553 | <p>The cardinaility of $\mathbb{R}$ is infinite, but note that there are various forms of infinity. </p>
<p>We use the symbol $\aleph_0$ to denote $|\mathbb{N}|$, the cardinality of the set of natural numbers.</p>
<p>The symbol $2^{\aleph_0}$ denotes $|\mathbb{R}|$. </p>
<p>It is a famous theorem of Cantor that $2^{... |
1,321,233 | <p>German Wikipedia states that the Ramsey`s theorem is a generalization of the Pigeonhole principle <a href="http://de.wikipedia.org/wiki/Satz_von_Ramsey" rel="noreferrer" title="source">source</a></p>
<p>But does not say why this is true. I am doing a presentation about the Ramsey theory and also wanna explain why t... | MJD | 25,554 | <p>Ramsey's theorem, in general, says that for given $n, t, $ and $k$, there is a number $R$ (depending on $(n,t,k)$) such that for every $R$-element set $S$ and every $k$-coloring of the $t$-element subsets of $S$ $$f : S^{\{1,2,\ldots, t\}} \to \{1,2,\ldots, k\}$$ there is an $n$-element subset $S'\subset S$ such tha... |
1,802,515 | <blockquote>
<p>Say you have a bank account in which your invested money yields 3% every year, continuously compounded. Also, you have estimated that you spend $1000 every month to pay your bills, that are withdrawn from this account.</p>
<p>Create a differential model for that, find its equilibriums and determine its ... | Rodrigo de Azevedo | 339,790 | <p>Let $x (t)$ be the amount of money in the account at time $t$ (years). Hence, if no money is spent,</p>
<p>$$\dot x = r x$$</p>
<p>where $r = \ln (1.03)$. If $\$1000$ is spent <strong>continuously</strong> every month, then we have the ODE</p>
<p>$$\dot x = r x - 12000$$</p>
<p>We have an equilibrium point when ... |
354,250 | <p><strong>Remark:</strong> All the answers so far have been very insightful and on point but after receiving public and private feedback from other mathematicians on the MathOverflow I decided to clarify a few notions and add contextual information. 08/03/2020.</p>
<h2>Motivation:</h2>
<p>I recently had an interesting... | Piyush Grover | 30,684 | <p>One way for reformulating all of (classical) mechanics is Peridynamics, which does away with derivatives. It is essentially a non-local reformulation.</p>
<p>Javili, Ali, et al. "Peridynamics review." Mathematics and Mechanics of Solids 24.11 (2019): 3714-3739.</p>
|
229,161 | <p>A sequence of positive integer is defined as follows</p>
<blockquote>
<ul>
<li>The first term is $1$.</li>
<li>The next two terms are the next two even numbers $2$, $4$.</li>
<li>The next three terms are the next three odd numbers $5$, $7$, $9$.</li>
<li>The next $n$ terms are the next $n$ even numbers if... | pedrosorio | 47,869 | <p>Your sequence is the following one:</p>
<p><a href="http://oeis.org/A001614" rel="nofollow">http://oeis.org/A001614</a></p>
|
2,618,856 | <p>I am trying to solve the problem:</p>
<blockquote>
<p><em>Question:</em> In a ring $R$ with identity, if every idempotent is central, then prove that for $a, b \in R$, $$ab =1 \implies ba=1$$</p>
</blockquote>
<p>I have done in the following manner:
$$ab=1\\
\implies b(ab)=b\\
\implies (ba)b-b=0\\
\implies (ba-1... | Alex Zorn | 62,875 | <p>Here's kind of a slick solution (don't read further if you want to figure it out for yourself!)</p>
<p>$$ab = 1$$
$$\Longrightarrow (ba)^{2} = b(ab)a = ba$$</p>
<p>So $ba$ is an idempotent and therefore central. So we have:</p>
<p>$$b(ba) = (ba)b$$
$$\Rightarrow ab(ba) = a(ba)b$$
$$\Rightarrow (ab)(ba) = (ab)(ab)... |
2,618,856 | <p>I am trying to solve the problem:</p>
<blockquote>
<p><em>Question:</em> In a ring $R$ with identity, if every idempotent is central, then prove that for $a, b \in R$, $$ab =1 \implies ba=1$$</p>
</blockquote>
<p>I have done in the following manner:
$$ab=1\\
\implies b(ab)=b\\
\implies (ba)b-b=0\\
\implies (ba-1... | rschwieb | 29,335 | <p>As mentioned in the comments, $ba\neq 1$ does not imply $aba\neq a$. In rings for which there exists $ab=1$ and $ba\neq 1$ (<a href="https://math.stackexchange.com/q/1917154/29335">see here for examples</a>) you have obviously that $aba=(ab)a=a$.</p>
<p>But clearly, if $ab=1$, $ba$ is at least idempotent, hence cen... |
207,865 | <p>It is known that all $B$, $C$ and $D$ are $3 \times 3$ matrices. And the eigenvalues of $B$ are $1, 2, 3$; $C$ are $4, 5, 6$; and $D$ are $7, 8, 9$. What are the eigenvalues of the $6 \times 6$ matrix
$$\begin{pmatrix}
B & C\\0 & D
\end{pmatrix}$$
where $0$ is the $3 \times 3$ matrix whose entries are all $... | Robert Israel | 8,508 | <p>Hint:
$$ \pmatrix{B & C\cr 0 & D\cr} \pmatrix{B^{-1} & E\cr 0 & D^{-1}\cr} = \pmatrix{I & BE + CD^{-1}\cr 0 & I\cr} $$
What $E$ will make $BE + CD^{-1} = 0$?</p>
|
207,865 | <p>It is known that all $B$, $C$ and $D$ are $3 \times 3$ matrices. And the eigenvalues of $B$ are $1, 2, 3$; $C$ are $4, 5, 6$; and $D$ are $7, 8, 9$. What are the eigenvalues of the $6 \times 6$ matrix
$$\begin{pmatrix}
B & C\\0 & D
\end{pmatrix}$$
where $0$ is the $3 \times 3$ matrix whose entries are all $... | PAD | 27,304 | <p>If $\lambda$ is an eigenvalue of $A$ with eigenvector $(x_1, x_2, x_3)^t$ then $(x_1,x_2,x_3,0,0,0)^t$ is an eigenvector of the block matrix. Similarly, for $D$ but you put three zeros at the beggining. </p>
|
1,497,898 | <p>Consider the polynomial $$f(x)=x^4-x^3+14x^2+5x+16$$ and $\mathbb{F}_p$ be the field with $p$ elements, where $p$ is prime. Then</p>
<ol>
<li>Considering $f$ as a polynomial with coefficients in $\mathbb{F_3}$, it has no roots in $\mathbb{F_3}$.</li>
<li><p>Considering $f$ as a polynomial with coefficients in $\mat... | IrbidMath | 255,977 | <p>You can write $f(x) = x^4 - x^3 -2x + 2 ~(\mod 7)$ and you can check that $f(1) = 0$. Hence $(x-1)$ is a factor of $f(x) ~(\mod 7)$.</p>
<p>After factoring we get $f(x) = (x-1)(x^3-2) ~(\mod 7)$. So If we proved that $x^3 - 2$ is irreducible $\mod 7$ that means point $3$ is true and point $4$ is false since if $f(x... |
3,325,114 | <blockquote>
<p>You are on an island inhabited only by knights, who always tell the truth, and knaves, who always lie. You meet two women who live there and ask the older one,</p>
<blockquote>
<p>"Is at least one of you a knave?"</p>
</blockquote>
<p>She responds yes or no, but! you do not yet have enough inf... | hunter | 108,129 | <p>They are both knights. Here's one approach.</p>
<p>The answer to question one can't be "yes." If it's "yes," the only logical possibility is that the older one is a knight and the younger is a knave. But then you already have enough information to know which is which, and the riddle says you don't. So the answer to... |
1,557,165 | <p>Prove that
$$\int_1^\infty\frac{e^x}{x (e^x+1)}dx$$
does not converge.</p>
<p>How can I do that? I thought about turning it into the form of $\int_b^\infty\frac{dx}{x^a}$, but I find no easy way to get rid of the $e^x$.</p>
| mrf | 19,440 | <p>Note that the integrand is positive. Since
$$
\lim_{x\to\infty} \frac{e^x}{e^x+1} = 1,
$$
it follows that there is an $X$ such that $\dfrac{e^x}{e^x+1} \ge \dfrac12$ for $x \ge X$. Hence
\begin{align*}
\int_1^\infty \frac{e^x}{x(e^x+1)}\,dx &= \int_1^X \frac{e^x}{x(e^x+1)}\,dx + \int_X^\infty \frac{e^x}{x(e^x+1... |
898,151 | <p>I have encountered an statement several times while proving determinant of a block matrix. </p>
<blockquote>
<p>$$\det\pmatrix{A&0\\0&D}\; = \det(A)\det(D)$$</p>
</blockquote>
<p>where $A$ is $k\times k$ and $D$ is $n\times n$ matrix. How to prove this?</p>
<p>Thanks in advance.</p>
| Patrick Da Silva | 10,704 | <p>If your matrices have coefficients in an integral domain, you can pass to the field of fractions and take an algebraic closure to use the Jordan canonical form, in which case this equation becomes trivial since both sides of the equation are the product of the products of the eigenvalues of $A$ and $D$. </p>
<p>Oth... |
1,029,485 | <p>I wish to show the following statement:</p>
<p>$
\forall x,y \in \mathbb{R}
$</p>
<p>$$
(x+y)^4 \leq 8(x^4 + y^4)
$$</p>
<p>What is the scope for generalisaion?</p>
<p><strong>Edit:</strong></p>
<p>Apparently the above inequality can be shown using the Cauchy-Schwarz inequality. Could someone please elaborate,... | GEO | 75,928 | <p>Regarding your edit and the question in the comment under OC-Sansoo's answer: (If I understand your issue right, you want reasoning for the choice of vectors?)</p>
<p>Start with the RHS of the inequality we want to show. </p>
<p>$$ 8\left(x^4+y^4\right) = \left(x^4+y^4\right)\left(2^2+2^2\right)$$
On the RHS we ... |
2,941,456 | <blockquote>
<p>Given <span class="math-container">$K$</span> elements between <span class="math-container">$1$</span> and <span class="math-container">$7$</span> (inclusive), how many ways can you arrange the elements s.t. their sum adds to <span class="math-container">$N$</span>? </p>
</blockquote>
<p>I can brute-... | G Cab | 317,234 | <p>You are looking for
<span class="math-container">$$
\eqalign{
& N_{\,b} (s,r,m) = \cr
& = {\rm No}\,{\rm of}\,{\rm integer}\;{\rm solutions}\,{\rm to}\left\{ \matrix{
1 \le x_{\,j} \le r + 1 = 7 \hfill \cr
x_{\,1} + x_{\,2} + \cdots + x_{\,m = K} = s + K = N \hfill \cr} \right.\quad = ... |
232,540 | <p>I'm trying to prove this conclusion but have some problems with one of the steps.</p>
<p>Assume $X_1,\ldots,X_n,\ldots$ is a sequence of Gaussian random variables, converging almost surely to $X$, prove that $X$ is Gaussian.</p>
<p>We use characteristics function here. Since $|\phi_{X_n}(t)|\leq 1$, by dominated c... | Davide Giraudo | 9,849 | <ul>
<li>First, we note that the sequence $\{\sigma_n\}$ and $\{\mu_n\}$ has to be bounded. It's a consequence of what was done in <a href="https://math.stackexchange.com/questions/116613/tightness-condition-in-the-case-of-normally-distributed-random-variables/116651#116651">this thread</a>, as we have in particular c... |
340,886 | <p>Suppose $x=(x_1,x_2),y = (y_1,y_2) \in \mathbb{R}^2$. I noticed that
\begin{align*}
\|x\|^2 \|y\|^2 - \langle x,y \rangle^2 &=
x_1^2y_1^2 + x_1^2 y_2^2 + x_2^2 y_1^2 + x_2^2 y_2 ^2 - (x_1^2 y_1^2 + 2 x_1 y_1 x_2 y_2 + x_2^2 y_2^2) \\
&=(x_1 y_2)^2 - 2x_1 y_2 x_2 y_1 + (x_2 y_2)^2 \\
&=(x_1 y_2 - x_2 y_1)... | ShreevatsaR | 205 | <p>No (it is not a square of a polynomial for $n \ge 3$), but the right generalization, proving that it is nonnegative, is that it is a <em>sum of squares</em>.</p>
<p>For instance, for $n = 3$,
$$
\begin{align*}
\|x\|^2 \|y\|^2 - \langle x,y \rangle^2 &= (x_1y_2 - x_2y_1)^2 + (x_2y_3-x_3y_2)^2 + (x_3y_1 - x_1y_3... |
1,894,199 | <p>Evaluate definite integral: $$\int_{-\pi/2}^{\pi/2} \cos \left[\frac{\pi n}{2} +\left(a \sin t+b \cos t \right) \right] dt$$</p>
<p>$n$ is an integer. $a,b$ real numbers.</p>
<p>The purpose of the integral - computing matrix elements of an electron Hamiltonian in an elliptic ring in the quantum box basis.</p>
<p>... | Robert Israel | 8,508 | <p>Write $a \sin(t) + b \cos(t) = c \cos(t-\delta)$ where $a = c \sin(\delta)$ and $b = c \cos(\delta)$. So now (depending on $n$ mod $4$) we want to look at
$\pm \int_{-\pi/2}^{\pi/2} \cos(c \cos(t-\delta))\; dt$ or $\pm \int_{-\pi/2}^{\pi/2} \sin(c \cos(t-\delta))\; dt$. </p>
<p>The integral with $\cos$ turns out ... |
1,894,199 | <p>Evaluate definite integral: $$\int_{-\pi/2}^{\pi/2} \cos \left[\frac{\pi n}{2} +\left(a \sin t+b \cos t \right) \right] dt$$</p>
<p>$n$ is an integer. $a,b$ real numbers.</p>
<p>The purpose of the integral - computing matrix elements of an electron Hamiltonian in an elliptic ring in the quantum box basis.</p>
<p>... | Kostiantyn Lapchevskyi | 361,449 | <p>Little addition on $\sin$ case:</p>
<p>$$
I=\int^{\pi/2}_{-\pi/2}\sin(c\cos(t-\delta))dt=\int^{\pi}_{0}\sin(c\sin(t-\delta))dt
$$</p>
<p>Make substitution: $g=\sin(t-\delta)\quad dt=\frac{dg}{\pm \sqrt{1-g^2}}$</p>
<p>Sign in last expression depends on range. We know that $-\frac{\pi}{2}<\delta<\frac{\pi}{... |
160,542 | <p>I suspect the following integration to be wrong. My answer is coming out to be $3/5$, but the solution says $1$.</p>
<p>$$\int_0^1\frac{2(x+2)}{5}\,dx=\left.\frac{(x+2)^2}{5}\;\right|_0^1=1.$$</p>
<p>Please help out. Thanks.</p>
| Ayman Hourieh | 4,583 | <p>$$
\left.\frac{(x+2)^2}{5}\right|_0^1 = \frac{(1+2)^2}{5} - \frac{(0+2)^2}{5} = \frac{9}{5} - \frac{4}{5} = \frac{5}{5} = 1
$$</p>
|
160,542 | <p>I suspect the following integration to be wrong. My answer is coming out to be $3/5$, but the solution says $1$.</p>
<p>$$\int_0^1\frac{2(x+2)}{5}\,dx=\left.\frac{(x+2)^2}{5}\;\right|_0^1=1.$$</p>
<p>Please help out. Thanks.</p>
| Community | -1 | <p>$$\left. \dfrac{(x+2)^2}5 \right \vert_0^1 = \dfrac{(1+2)^2}5 - \dfrac{(0+2)^2}5 = \dfrac{3^2}5 - \dfrac{2^2}5 = \dfrac{9}5 - \dfrac45 = \dfrac{9-4}5 = \dfrac55 = 1$$</p>
<p>Note that we can integrate $\displaystyle \int_{x_1}^{x_2} (x+a) dx$ in seemingly two different ways.</p>
<p>The first method is to treat $x... |
2,952,014 | <p>So I was doing some self study and came across a proposition in one of my chemical engineering course's prescribed textbooks. I can't quite get the proof out. It's to do with a particle moving through a medium such that when it makes contact with to either of two plates <span class="math-container">$L$</span> units ... | Kavi Rama Murthy | 142,385 | <p><span class="math-container">$(1,0,0,...,0)$</span> and <span class="math-container">$(0,0,...0,1)$</span> are two invariant distributions so uniqueness fails.</p>
|
327,990 | <p>So i was working on this:</p>
<p>$$
\lim\limits_{x\to1} \frac{x + \sqrt{x} - 2}{x - 1}
$$</p>
<p>and I thought to simpify my top by multiplying by a conjugate, taking everything other than the $x$ to be the $b$ from $a+b$ so that my conjugate looked like $x - \sqrt{x} + 2$.</p>
<p>The multiplication, if correct, ... | DonAntonio | 31,254 | <p>$$\frac{x+\sqrt x-2}{x-1}=\frac{(\sqrt x-1)(\sqrt x+2)}{(\sqrt x-1)(\sqrt x+1)}=\frac{\sqrt x+2}{\sqrt x+1}\xrightarrow[x\to 1]{} \frac{3}{2}$$</p>
|
1,358,270 | <p>If we have a function $f=f(r, \theta, \phi)$, where $(r, \theta, \phi)$ are spherical coordinates on $\mathbb{R}^3$, how do we compute the gradient $\nabla f$ by using the formula
$$\nabla f \cdot d\vec{r} = df ?$$
Here $\vec{r}$ is the position vector and $df=\frac{\partial f}{\partial r}dr +\frac{\partial f}{\par... | Christian Blatter | 1,303 | <p>In order to make $f$ well defined we have to assume $a_0>0$ and $B_2<\sqrt{a_0}$. Denote the domain ${\rm Re}(z)>a_0$ by $\Omega$. Let $z=a+it\in\Omega$. Then by a standard formula for square roots of complex numbers one has
$${\rm Re}\bigl(\sqrt{z}\bigr)=\sqrt{{1\over2}\bigl(\sqrt{a^2+t^2}+a\bigr)}\geq \s... |
27,904 | <blockquote>
<p>If $f(z) = (g(z),h(z))$ is continuous then $g$ and $h$ are as well.</p>
</blockquote>
<p>The converse is easy for me to prove, but I'm not seeing how to prove it using the terminology of open sets and not metric spaces.</p>
| lhf | 589 | <p>Compose $f$ with a projection. (First prove that a projection is continuous.)</p>
|
469,947 | <blockquote>
<p>Show that the presentation $G=\langle a,b,c\mid a^2 = b^2 = c^3 = 1, ab = ba, cac^{-1} = b, cbc^{-1} =ab\rangle$ defines a group of order $12$.</p>
</blockquote>
<p>I tried to let $d=ab\Rightarrow G=\langle d,c\mid d^2 =c^3 = 1, c^2d=dcdc\rangle$. But I don't know how to find the order of the new pre... | Mark Bennet | 2,906 | <p>If you note in the original presentation that the exponents of $a,b,c$ are $2,2,3$ whose product is $12$ a natural strategy would be to try to put a general element in the form $a^pb^qc^r$ and then show that these elements are distinct.</p>
|
469,947 | <blockquote>
<p>Show that the presentation $G=\langle a,b,c\mid a^2 = b^2 = c^3 = 1, ab = ba, cac^{-1} = b, cbc^{-1} =ab\rangle$ defines a group of order $12$.</p>
</blockquote>
<p>I tried to let $d=ab\Rightarrow G=\langle d,c\mid d^2 =c^3 = 1, c^2d=dcdc\rangle$. But I don't know how to find the order of the new pre... | i. m. soloveichik | 32,940 | <p>The presentation can be rewritten as $\langle d,c\mid d^2 =c^3 = (cd)^3=1 \rangle$.
This is the standard presentation for the symmetry group (rotations) of the regular tetrahedron where $d$ represents a rotation about an edge, $c$ represents a rotation about a face and $cd$ represents a rotation about a vertex.</p... |
288,974 | <p>Alright this maybe really funny but I want to know why is this wrong. We often come across identities which we prove by multiplying both the sides of the identity by a certain entity but why don't we multiply it by $0$. That way every identity will be proved in one single line. That is so stupid. I mean, by that way... | Tunococ | 12,594 | <p>$a = b$ implies $ac = bc$, but $ac = bc$ doesn't imply $a = b$. (Not immediately. Read below.)</p>
<p>The way you usually get $a = b$ from $ac=bc$ is by multiplying both sides with $1/c$, which is only available when $c \ne 0$.</p>
|
3,231,869 | <p>I am a little confused what this actually means: </p>
<p><span class="math-container">$e^{x+e^x}$</span></p>
<p>It is obviously not the same if I for example
<span class="math-container">$$e^{x}:= \lambda \\
e^{x+e^x} \neq \lambda^\lambda
$$</span></p>
| Vizag | 566,333 | <p>You're going wrong slightly in the last step: </p>
<p><span class="math-container">$$e^x = \lambda$$</span></p>
<p><span class="math-container">$$\implies e^{x+e^x} = e^x e^{e^x} = \lambda \times e^{\lambda}$$</span></p>
<p>Your expression
<span class="math-container">$$\lambda^{\lambda} = (e^{x})^{e^x}$$</span><... |
1,507,181 | <p><a href="https://i.stack.imgur.com/nuhUB.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/nuhUB.png" alt="enter image description here"></a></p>
<p><em>We know $G(0) = 0$</em></p>
<p>Okay, so I have the above graph but I'm having a difficult time translating it into the graph of $G(x)$.</p>
<p>W... | tomi | 215,986 | <p>A common application of this kind of question is when you have the speed $v=\frac {ds}{dt}$ which corresponds to $G'(x)$ and you want to find the displacement $s $ which corresponds to $G(x)$. In that situation it helps to remember that the area under the velocity-time curve is equal to the displacement. For you tha... |
1,675,329 | <p>What's the value of $\sum_{i=1}^\infty \frac{1}{i^2 i!}(= S)$?</p>
<p>I try to calculate the value by the following.</p>
<p>$$\frac{e^x - 1}{x} = \sum_{i=1}^\infty \frac{x^{i-1}}{i!}.$$
Taking the integral gives
$$ \int_{0}^x \frac{e^t-1}{t}dt = \sum_{i=1}^\infty \frac{x^{i}}{i i!}. $$</p>
<p>In the same, we gets... | Marco Cantarini | 171,547 | <p>Maybe it's interesting to see how to get the “closed form” in terms of hypergeometric function. Recalling the definition of generalized hypergeometric function $$_{q}F_{p}\left(a_{1},\dots,a_{q};b_{1},\dots,b_{p};z\right)=\sum_{k\geq0}\frac{\left(a_{1}\right)_{k}\cdots\left(a_{q}\right)_{k}}{\left(b_{1}\right)_{k}\c... |
934,660 | <p>Prove that for $ n \geq 2$, n has at least one prime factor.</p>
<p>I'm trying to use induction. For n = 2, 2 = 1 x 2. For n > 2, n = n x 1, where 1 is a prime factor. Is this sufficient to prove the result? I feel like I may be mistaken here.</p>
| Sylvain Biehler | 132,773 | <p>You can use a proof by contradiction. If <span class="math-container">$n>1$</span> has no prime divisor, you can build an inifinite sequence of decreasing numbers.</p>
|
1,319,288 | <p>There is a <a href="https://math.stackexchange.com/questions/1103723">similar question</a> in this site but I am not satisfied with the answer, which is basically the same as the proof in the mentioned textbook.</p>
<p>The book(Karel Hrbacek&Thomas Jech, <em>Introduction to Set Theory 3e</em>, p165) states a le... | bof | 111,012 | <p>All you can say about $\operatorname{cf}(2^{\aleph_\omega})$ is that it's some regular cardinal $\kappa$ such that
$$\aleph_{\omega+1}\le\kappa\le2^{\aleph_\omega}.$$
I think you need the axiom of choice to say even that much.</p>
<p>What's wrong with your reasoning is the unwarranted assumption that
$$2^{\aleph_\o... |
173,387 | <p>How can I indent properly long code in <em>Mathematica</em>?
Are there some best practices?</p>
| kglr | 125 | <h3>GeneralUtilities`HoldPrettyForm</h3>
<pre><code>Needs["GeneralUtilities`"]
HoldPrettyForm[Row[Table[Table[Plot[Sin[i x] Cos[j x], {x, 0, Pi}],
{i, 1, 5}], {j, 1, 3}]]]
</code></pre>
<p><a href="https://i.stack.imgur.com/rUdOh.png" rel="noreferrer"><img src="https://i.stack.imgur.com/rUdOh.png" alt="ent... |
661,182 | <p>I'm taking a discrete structures class and I would appreciate some help with a homework problem. The problem is</p>
<blockquote>
<p>Attempt to find a closed form for the sum $\displaystyle \sum_{k=1}^n k^3$
by perturbation, only to find a closed form for the following sum $\displaystyle \sum_{k=1}^n k^2$.</p>
<... | tabstop | 117,788 | <p>You've got it started out:
$$S_n+(n+1)^3 = S_0 + \sum_{k=0}^n (k+1)^3.$$
We know that $S_0=0$, and we can expand $(k+1)^3$ by binomial coefficients (or by polynomial multiplication if you're bored), so we have that the right hand side is
$$\sum_{k=0}^n (k^3+3k^2+3k+3) = S_n + 3\sum_{k=0}^n k^2 + 3\sum_{k=0}^n k + \s... |
4,280,328 | <p>I think the substitution <span class="math-container">$x=\xi+\eta,$</span> <span class="math-container">$y=\xi-\eta$</span> can be done. Then the equation takes the form <span class="math-container">$$ \begin{gathered} 38(\xi^{2}+\eta^{2})=221+33(\xi^{2}-\eta^{2}) \\ 5 \xi^{2}+71 \eta^{2}=221 \end{gathered} $$</span... | Michael Rozenberg | 190,319 | <p>We have <span class="math-container">$$19\left(x-\frac{33y}{38}\right)^2+\left(19-19\cdot\left(\frac{33}{38}\right)^2\right)y^2=221,$$</span> which gives
<span class="math-container">$$\left(19-19\cdot\left(\frac{33}{38}\right)^2\right)y^2\leq221$$</span> or
<span class="math-container">$$1\leq y\leq6.$$</span>
The ... |
549,347 | <p>How would I solve the following question. And determine if its true or false.</p>
<p>1.$\forall x \in R , \exists y\in R, x^2+y^2=-1$</p>
<p>2: $\exists x\in R,\forall y \in R, x^2+y^2=-1$</p>
<p>For the first one I think I can justify it is false.</p>
<p>As for any arbitrary x must y must be </p>
<p>$y=\sqrt{-... | amWhy | 9,003 | <p>Yes, both statements are false because the sum of two squared real numbers, whatever those numbers are, <strong><em>will never be negative</em></strong>.</p>
|
444,486 | <p>I am teaching myself real analysis, and in this particular set of lecture notes, the <a href="http://www.math.louisville.edu/~lee/RealAnalysis/IntroRealAnal-ch01.pdf" rel="nofollow">introductory chapter on set theory</a> when explaining that not all sets are countable, states as follows:</p>
<blockquote>
<p>If $S... | Y.H. Chan | 71,563 | <p>There are plenty of differences between $\mathbb{R}^2$ plane and $\mathbb{C}$ plane. Here I give you two interesting differences.</p>
<p>First, about branch points and branch lines. Suppose that we are given the function $w=z^{1/2}$. Suppose further that we allow $z$ to make a complete circuit around the origin cou... |
12,359 | <p>There are, IMO, quite a lot of badly tagged questions and... not very good tags. Some of them were discussed on meta recently; some of these discussions show, IMO, that users who created these tags don't always understand tagging system of Math.SE well enough.</p>
<p>On Meta.SO one needs 500 rep to create a new tag... | Post No Bulls | 111,742 | <p>I agree; there is much less need for new tags on an established math Q&A site than on a technology-oriented Q&A site. On a tech Q&A, users are likely to bring up questions about a new gadget or a new version of some software that was just released. Here we don't normally get questions about mathem... |
106,219 | <blockquote>
<p>Define a sequence of functions $f_n: (0,1)\rightarrow\mathbb{R}$ by<br>
$\
f(x) =
\begin{cases}
1/q^n
& \text{if } x =p/q \space(\space\mathrm{nonzero})\\
0 & \text{otherwise}
\end{cases}
$<br>
Find the pointwise limit $f$ of $\{f_n\}$ and show $\{f_n\}$ converges ... | azarel | 20,998 | <p>Uniform convergence means that $\forall \epsilon>0$ there is an $N$ such that $|f_n(x)-f(x)|<\epsilon$ for all $n\geq N$ and for all $x$(the point is that $N$ does not depend on $x$). So uniform convergence imply point wise limit convergence. </p>
|
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