qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
3,757,864 | <p>Given a diagram like this,
<a href="https://i.stack.imgur.com/Xwum0.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Xwum0.png" alt="enter image description here" /></a></p>
<p>Where <span class="math-container">$O$</span> is the center and <span class="math-container">$OA = \sqrt{50}$</span>, <spa... | farruhota | 425,072 | <p>Refer to the figure:</p>
<p><span class="math-container">$\hspace{4cm}$</span><a href="https://i.stack.imgur.com/5hjeO.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/5hjeO.png" alt="enter image description here" /></a></p>
<p>From the right triangle <span class="math-container">$ACD$</span>: <spa... |
83,167 | <p>Let $X$ be a compact complex surface of general type which a ball quotient. Is it true that $\pi_{1}(X)$ can not contain ${\mathbb{Z}}^{2}$ as a subgroup? What kind of infinite abelian groups can occur as a subgroup of $\pi_{1}(X)$?</p>
| BS. | 6,451 | <p>If I understand your question, you ask if a cocompact torsion free subgroup of $PU(2,1)$ (namely $\Gamma=\pi_{1}(X)$) can contain a ${\mathbb Z}^2$. </p>
<p>This is not the case, because $\Gamma$ is a Gromov-hyperbolic group <a href="http://en.wikipedia.org/wiki/Hyperbolic_group" rel="nofollow">http://en.wikipedia.... |
1,634,520 | <p>I <a href="https://math.stackexchange.com/questions/1632455/mathematical-meaning-of-certain-integrals-in-physics/1633320#1633320">have been told</a> that the Helmholtz decomposition theorem says that</p>
<blockquote>
<p>every <em>smooth</em> vector field $\boldsymbol{F}$ [where I am not sure what precise assumpti... | Self-teaching worker | 111,138 | <p>Thanks to a conversation I have had with user <a href="https://math.stackexchange.com/users/112478/trialanderror">TrialAndError</a>, whom I deeply thank, <a href="https://math.stackexchange.com/questions/1632455/mathematical-meaning-of-certain-integrals-in-physics/1633320#comment3523919_1633320">here</a>, I think I ... |
1,910,109 | <p>$$\int \frac{1}{\sqrt{x} (1 - 3\sqrt{x})}$$</p>
<p>I tried with the substitution $u = 1-3\sqrt{x}$</p>
<p>I am confused with how to finish this problem I know I am supposed to substitute $u$ and $\text{d}u$ in but I am not sure how to finish it.</p>
| Mike | 17,976 | <p>The substitution is a good one. If $u=1-3x^{1/2}$, then</p>
<p>$$du=\frac12(-3x^{-1/2})dx=-\frac3{2\sqrt x}dx$$</p>
<p>$$-\frac23\int\frac1{1-3\sqrt x}\left(-\frac3{2\sqrt x}dx\right)=-\frac23\int\frac{du}u$$</p>
|
1,057,819 | <p>The number $128$ can be written as $2^n$ with integer $n$, and so can its every individual digit. Is this the only number with this property, apart from the one-digit numbers $1$, $2$, $4$ and $8$? </p>
<p>I have checked a lot, but I don't know how to prove or disprove it. </p>
| phil_20686 | 150,039 | <p>Every such number is of the form of a sum of f(n,m) = 2^n*10^m where m is an integer and n is 0,1,2,3. You can gain insight into this problem by writing this problem out in binary. Every power of two is a 1 followed by zeros, so 1, 10, 100, 1000 is 1,2,4,8 etc. So multiplying a binary by 2 adds a zero to the right. ... |
1,832,812 | <blockquote>
<p>Let $M$ be a nonempty subset of $\mathbb Z^+$ such that for every element $x$ in $M,$ the numbers $4x$ and $\lfloor \sqrt x \rfloor$ also belong to $M.$ Prove that $M = \mathbb Z^+$.</p>
</blockquote>
<p>Suppose $a \in M$. Then so are $4ak$ and $\lfloor \sqrt{4ak} \rfloor$ for every positive integer ... | coffeemath | 30,316 | <p>Starting at any $m \in M$ and repeatedly applying $f(x)=[\sqrt{x}]$ (I use $[u]$ for floor of $u$) one eventually gets $1 \in M,$ then repeatedly multiplying by $4$ we have $4^k \in M,$ then applying $f$ to these we have $2^k \in M$ for all $k,$ i.e. $M$ contains the powers of $2.$
In what follows we use "integer in... |
1,201,955 | <blockquote>
<p><em>Question</em>: If $f(n)$ is $O(g(n))$ and $g(n)$ is $O(f(n))$, is $f(n) = g(n)$?</p>
</blockquote>
<p>I'm studying for a discrete mathematics test, and I'm not sure if this is true or false. Since Big-OH ignores constant multiples, wouldn't $f(n) = c\, g(n)$?</p>
| MathMajor | 113,330 | <p>Your thoughts are correct. Consider $f(n) = n$ and $g(n) = 2f(n)$ as a counter example.</p>
|
3,060,456 | <p>Can any one explain me the intuition behind this formula ? (with permutation example)</p>
<pre><code>P(n, k) = P(n-1, k) + k* P(n-1, k-1)
</code></pre>
| Arthur | 15,500 | <p>Here is an example which should give you some insight into how I believe you should think about the formula.</p>
<p>Let's say <span class="math-container">$P(n, k)$</span> counts the number of teams of <span class="math-container">$k$</span> people you can make from a roster of <span class="math-container">$n$</spa... |
2,693,143 | <p>For example, to convert $0.25$ to binary. Using this algorithm it gives the correct result $0.01$:</p>
<blockquote>
<ol>
<li>Multiply by two</li>
<li>take decimal as the digit</li>
<li>take the fraction as the starting point for the next step</li>
<li>repeat until you either get to 0 or a periodic number<... | ncmathsadist | 4,154 | <p>Long division works in any base. If you pay attention, you can spot the periodicities in the expansion. So sharpen your Miss Wormwood pencil and check it out.</p>
<p>Example: I will obtain the binary expansion of 1/3 this way.</p>
<p><code>
0.010
-----------------
11 ) 1.00000000000
... |
2,693,143 | <p>For example, to convert $0.25$ to binary. Using this algorithm it gives the correct result $0.01$:</p>
<blockquote>
<ol>
<li>Multiply by two</li>
<li>take decimal as the digit</li>
<li>take the fraction as the starting point for the next step</li>
<li>repeat until you either get to 0 or a periodic number<... | David K | 139,123 | <p>The following is the algorithm I think you described. Step 2 seemed unclear to me, so I have rephrased it below.</p>
<p>Start with $0.25_\mathrm{ten}$
(the subscript reminds us which base the number is in).
To start out with, the binary number is $0_\mathrm{two}.$</p>
<blockquote>
<ol>
<li>Multiply by two</li>... |
1,731,364 | <p>So the question asks:</p>
<blockquote>
<p>Let $X_1,X_2,X_3\sim \operatorname{Exp}(\lambda)$ be independent (exponential) random variables (with $\lambda> 0$).<br>
(a) Find the probability density function of the random variable $Z = \max \{X_1,X_2,X_3\}$.<br>
(b) Let $T = X_1+X_2/2+X_3/3$, use moment gener... | Joseph | 313,328 | <p>First you would look at the distribution of $Z$. Notice that the moment generating function for $Z$ is equal to
$$
\int_0^{\infty } 3 \lambda e^{t x} e^{-3 \lambda x} \left(e^{\lambda x}-1\right)^2 \, dx=\int_0^{\infty } 3 \lambda e^{t x} e^{-3 \lambda x} e^{2\lambda x} \, dx - \int_0^{\infty } 3 \lambda e^{t... |
2,252,090 | <p>Someone posed this question to me on a forum, and I have yet to figure it out. If $a,b,c,d$ are the zeroes of:</p>
<p>$$x^4-7x^3+2x^2+5x-1=0$$
Then what is the value of $$ \frac1a +\frac1b +\frac1c +\frac1d $$</p>
<p>I can figure out the zeroes, but they are wildly complex. I'm sure there must be an easier way. </... | dxiv | 291,201 | <p>If $P(x)=x^4-7x^3+2x^2+5x-1$ has roots $a,b,c,d$ then $\frac{1}{a}, \frac{1}{b}, \frac{1}{c}, \frac{1}{d}$ are the roots of $P(\frac{1}{x})=0$ $\iff$ $1-7x+2x^2+5x^3-x^4=0$. By Vieta's relations, the sum of the roots of the latter is $5$.</p>
|
1,608,299 | <p>I am completely stuck on a question. I've done it 4 times, each times got different result, but never correct.</p>
<p>The third term of an arithmetic progression is 71 and the seventh term is 55. Find the sum of the first 45 terms.</p>
<p>Any ideas? Thanks</p>
| SchrodingersCat | 278,967 | <p>Let the first term of the arithmetic series be $a$ and the common difference be $d$.</p>
<p>Hence the third term = $a+(3-1)d=a+2d$</p>
<p>and the seventh term = $a+(7-1)d=a+6d$</p>
<p>Now given that $$a+2d=71$$
$$a+6d=55$$
This implies that $4d=-16 \Rightarrow d=-4$</p>
<p>Also we have that $a=79$</p>
<p>So sum... |
3,586,346 | <p>Basically, I'd like to model sin x, but make it's derivative tend towards 0, so as x increases, it becomes a constant y = 0. The function begins like a typical sin x function, but slowly the fluctuation decreases until it isn't there anymore. If this works as I'm trying to have it work, I think some constant between... | mjw | 655,367 | <p>Let <span class="math-container">$$f(x) = e^{-\alpha x} \sin x \quad \alpha <<1$$</span></p>
<p>Here is a graph of this function, and also another nice choice that was suggested <span class="math-container">$g(x)=\frac{\sin x}{1+\alpha^2 x^2}$</span>, both with <span class="math-container">$\alpha = \frac{1}{... |
1,579,616 | <p>So I know it's true for $n = 5$ and assumed true for some $n = k$ where $k$ is an interger greater than or equal to $5$.</p>
<p>for $n = k + 1$ I get into a bit of a kerfuffle.</p>
<p>I get down to $(k+1)^2 + 1 < 2^k + 2^k$ or equivalently:</p>
<p>$(k + 1)^2 + 1 < 2^k * 2$.</p>
<p>A bit stuck at how to pro... | lhf | 589 | <p>$\displaystyle
2^n = (1+1)^n > \binom{n}{0}+\binom{n}{1}+\binom{n}{2}+\binom{n}{3}>n^2+1
$
iff $n>5$. (*)</p>
<p>The case $n=5$ is proved by inspection.</p>
<p>(*) This seems to be a cubic inequality but it reduces to an easy quadratic.</p>
|
160,161 | <p>Is there an unlabeled locally-finite graph which is a Cayley graph of an infinitely many non-isomorphic groups with respect to suitably chosen generating sets?</p>
| YCor | 14,094 | <p>Here I answer your additional question about finitely presented groups. The answer is then no.</p>
<p><b>Affirmation.</b> Only finitely many finitely presentable groups may have the same given Cayley graph.</p>
<p>This works as follows. Let $X_1$ be the Cayley graph of some f.p. group. By finite presentability, th... |
140,819 | <p>Everybody loves the good old quadratic Mandelbrot set. As you probably know, both it and the corresponding quadratic Julia sets are defined by the iteration $f(z) = z^2 + c$.</p>
<p>You might expect, however, that $f(z) = az^2 + bz + c$ would give you more possibilities. However, all the books on the subject assert... | MathematicalOrchid | 29,949 | <p>OK, I <em>think</em> I see what J.M. is getting at now:</p>
<p>If I write $g(z) = (z + \alpha)^2 + \beta$, then we have</p>
<p>$$(z + \alpha)^2 + \beta = z^2 + (2\alpha)z + (\alpha^2 + \beta)$$</p>
<p>In other words, taking $f(z) = z^2 + c$ and adding a linear term is like performing one addition before the squar... |
2,150,085 | <p>I've always been puzzled by the value $0^0$. I remembered that one of my professor claimed that it was truely equal to $1$. However I think that most people would say, from an analytic point of view, that it is indeterminate which I agree with. Translating $0^0$ into $e^{0 \ln(0)}$ makes the problem appear explicitl... | GEdgar | 442 | <p>The real problem with defining $0^0=1$ is not the case $z^z$. </p>
<p>We say $0^0$ is indeterminate because there are sequences $a_n, b_n$ with $$\lim a_n = 0,\quad\text{and}\quad\lim b_n = 0$$ but not $$\lim a_n^{b_n}=1.$$ These cases do not have $a_n = b_n$. They have $a_n$ much smaller than $b_n$</p>
<p>This... |
1,382,374 | <p>This is an exercise from Rudin's <em>Principles of Mathematical Analysis</em>, Chapter $6$.</p>
<blockquote>
<p>Suppose $f$ is a real, continuously differentiable function on $[a, b]$, $f(a) = f(b) = 0$, and
$$\int_a^b f^2(x)\, dx = 1.$$
Prove that
$$\int_a^b xf(x)f'(x)\,dx = -\frac{1}{2}$$
and that
$$... | Hagen von Eitzen | 39,174 | <p>Assume $f'(x)=\lambda xf(x)$. Then $f$ is identically zero.
Indeed assume $f(x_0)\ne 0$ for some $x_0\in [a,b]$. Then $x_0>a$ and we can let $a'$ be the infimum of all numbers such that $f$ has no zero in $(a',x_0)$.
Then $f(a')=0$ (this is where we use $f(a)=0$).
In the interval $(a',x_0)$ we have
$$\frac{\ma... |
3,736,580 | <p>Show that for <span class="math-container">$n>3$</span>, there is always a <span class="math-container">$2$</span>-regular graph on <span class="math-container">$n$</span> vertices. For what values of <span class="math-container">$n>4$</span> will there be a 3-regular graph on n vertices?</p>
<p>I think this q... | G Cab | 317,234 | <p>Let me propose a "chemical approach" :<br />
<em>we want a mixture of elements of atomic weight <span class="math-container">$\{ 1,2,3,4 \}$</span> such that the resulting average atomic weight is <span class="math-container">$\pi$</span></em>.</p>
<p>We should then have the following diophantine system<b... |
8,307 | <p>I have a program that gives the average distance as the output. When I tried to repeat finding the average distance 100 times using Table, It failed to generate the output. This is the program.</p>
<pre><code>Xarray = A @@@ Tuples[Range[0, 4], 3];
Table[M = RandomSample[Xarray, 7];
energies = RandomVariate[Exponent... | Matariki | 379 | <p>Try this. It is not beautiful but ...</p>
<pre><code>xarray = A @@@ Tuples[Range[0, 4], 3];
distanceBetween[{n_, m_}, list_] :=
Norm[List @@ list[[n, 1]] - List @@ list[[m, 1]]];
meanDist := Module[{m, c, f, list, energies},
m = RandomSample[xarray, 7];
energies = RandomVariate[ExponentialDistribution[1.5]... |
927,188 | <p>This question has been on my mind for a very long time, and I thought I'd finally ask it here. </p>
<p>When I was 6, my dad pulled me out of school. The classes were too easy; the professors, too dull. My father had been man of philosophy his entire life (almost got a PhD in it) and regretted not having a more q... | Andy | 175,311 | <p>I loved math in school but struggled at uni, and lost plenty of motivation when unable to connect what I was doing to any sort of career (pure "theoretical" math was my strength). It did however lead into cryptography, programming and IT which is now the career I love.</p>
<p>What I'm attempting to express is that... |
927,188 | <p>This question has been on my mind for a very long time, and I thought I'd finally ask it here. </p>
<p>When I was 6, my dad pulled me out of school. The classes were too easy; the professors, too dull. My father had been man of philosophy his entire life (almost got a PhD in it) and regretted not having a more q... | I Like to Code | 71,654 | <p>I interpret your question as how to determine
if you are good enough to be a mathematics professor
and to do academic research for a living.</p>
<p><strong>Note:</strong> You may wish to ask your question on <a href="http://academia.stackexchange.com">http://academia.stackexchange.com</a>
as there are quite a few m... |
133,370 | <p>In differential geometry of surfaces, how can one define a non-zero Torsion tensor? It seems that the connection you provide has always to be symmetric since, by definition,
$$\Gamma^{\gamma}_{\alpha\beta}\equiv\mathbf{a}^{\gamma}\cdot\mathbf{a}_{\alpha,\beta}=\mathbf{a}^{\gamma}\cdot\mathbf{r}_{,\alpha\beta}=\mathb... | Robert Bryant | 13,972 | <p>I think that the OP is asking a more specific question than whether or not a surface has a connection that is not metric or not torsion free. It seems that the OP is assuming that the surface $M$ comes equipped with an immersion $\mathbf{r}:M\to\mathbb{E}^3$ into (oriented) Euclidean $3$-space and is asking whether... |
617,275 | <p>$E$ is normed vector space.Let $f\in E^*$ in a bounded linear functional from $E$ to $C$ and fix $x\in E$. We have $$\forall y\in E;\ \ \ \ f(y-x)\leq \frac{1}{2}\|y\|^2-\frac{1}{2}\|x\|^2$$
And I have proven $f(x)=\|x\|^2$ and $\|x\|\leq \|f\|$. Prove that $\|f\|=\|x\|$.</p>
| x tang | 97,573 | <p>Since $f(x)=\| x \|^2,$ Let $\bar{x}=\frac{x}{\| x\|},$ $E_1 = \{ \lambda \bar{x}: \lambda\in\mathbb{R} \}.$ Then $\|f\|_{E_1} = \| x \|,$ then by Hahn-Banach, $f$ can be extended into $E,$ and $\| f \|=\|f\|_{E_1} =\|x\|.$</p>
|
3,383,206 | <p><strong>Question</strong>: Can <span class="math-container">$\int_0^\infty \frac{\sqrt{x}}{(1+x)^2} dx$</span> be computed with residue calculus?</p>
<p>The integral comes from computing <span class="math-container">$\mathbb{E}(\sqrt{X})$</span> where <span class="math-container">$X=U/(1-U)$</span> and <span class=... | fleablood | 280,126 | <blockquote>
<p>Because supposedly the they are adherent if you can have a neighborhood centered on it which intersects the set. And I would think that is possible?</p>
</blockquote>
<p>They are adherent if <em>ALL</em> neighborhoods intersect; not just one.</p>
<p>And it should be clear that if <span class="math-c... |
4,036,558 | <p><span class="math-container">$f(x)=e^x(x^2+x)$</span>, derive <span class="math-container">$\dfrac{d^n\,f(x)}{dx^n}$</span></p>
<p>may use Leibniz formula but i'm not sure:(</p>
| Adib Akkari | 890,705 | <p>It seems a lit bit hard but if you try to find the pattern from n=1 to n=2 ,
so <span class="math-container">$(e^x (x^2 + x))\prime$</span> = <span class="math-container">$e^x (x^2 + 3x +1)$</span> this is for n egale to 1 you cannot get the pattern , so for <span class="math-container">$n = 2\quad , (e^x(x^2 + x))\... |
1,216,392 | <blockquote>
<p>$P,Q$ are polynomials with real coefficients and for every real $x$ satisfy $P(P(P(x)))=Q(Q(Q(x)))$. Prove that $P=Q$.</p>
</blockquote>
<p>I see only that these polynomials are same degree</p>
| Slade | 33,433 | <p>It is easy to see that $P$ and $Q$ have the same leading term. Without loss of generality, assume that both are monic. Note that $P$ and $Q$ also have the same constant term. (Note: as zhw has pointed out, this part is not so obvious. I'll try to update with an argument shortly.)</p>
<p>Suppose that $P\neq Q$. ... |
2,227,280 | <p>For every positive number there exists a corresponding negative number. Would that imply that the number of positive numbers is "equal" to the number of negative numbers? (Are they incomparable because they both approach infinity?)</p>
| étale-cohomology | 53,113 | <p>The notion of "counting" is made precise in mathematics by using functions. These functions are <a href="https://en.wikipedia.org/wiki/Bijection" rel="nofollow noreferrer">bijections</a>. To "count" the elements in a set $X$, you establish a bijection from $X$ to a subset of the natural numbers $\mathbb{N}$.</p>
<p... |
2,409,268 | <p><strong>Confirm that the identity $1+z+...+z^n=(1-z^{n+1})/(1-z)$ holds for every non-negaive integer $n$ and every complex number $z$, save for $z=1$</strong></p>
<p>I have tried to prove this by induction but I am not sure that I am doing things right, for $ n = 1 $ we have $ (1-z ^ 2) / (1-z) = (1-z) (1+ z) / (1... | fleablood | 280,126 | <p>You can also prove that </p>
<p>$(1-z)\sum_{i=0}^n z^i=$</p>
<p>$\sum_{i=0}^n (1-z)z^i=$ (technically this assummes distribution is known for all finite sums; that can be proven by induction for the precise and anal).</p>
<p>$\sum_{i=0}^n (z^i-z^{i+1})=$</p>
<p>$\sum_{i=0}^nz^i - \sum_{i=0}^nz^{i+1}= $ (technic... |
2,621 | <p>Let $A$ be a commutative Banach algebra with unit.
It is well known that if the Gelfand transform $\hat{x}$ of $x\in A$ is non-zero, then $x$ is invertible in $A$ (the so called Wiener Lemma in the case when $A$ is the Banach algebra of absolutely convergent Fourier series).</p>
<p>As a converse of the above, let ... | J. M. ain't a mathematician | 498 | <p>Tangential to John D. Cook's reply, and also somewhat related: Monte Carlo also finds application in the solution of (partial, stochastic) differential equations, of which cubature (nobody ever uses MC in the one-dimensional case practically ;) ) is but a specialized case. As already mentioned by John, the pain deal... |
2,436,336 | <p>I am a bit puzzled. Trying to solve this system of equations: </p>
<p>\begin{align*}
-x + 2y + z=0\\
x+2y+3z=0\\
\end{align*}</p>
<p>The solution should be \begin{align*}
x=-z\\
y=-z\\
\end{align*} </p>
<p>I just don't get the same solution. Please advice.</p>
| Ewan Cazar | 1,006,389 | <p>Start with the 2 equations</p>
<p>-x + 2y + z = 0</p>
<p>x + 2y + 3z = 0</p>
<p>Simplify</p>
<p>-x + 2y + z = x + 2y + 3z</p>
<p>Then solve for x</p>
<p>-x + z = x + 3z</p>
<p>-x = x + 2z</p>
<p>-2x = 2z</p>
<p>x = -z</p>
<p>Then solve for z</p>
<p>-x + z = x + 3z</p>
<p>-x = x + 2z</p>
<p>-2x = 2z</p>
<p>x = -z</p>... |
2,894,126 | <blockquote>
<p>$$\int \sin^{-1}\sqrt{ \frac{x}{a+x}} dx$$</p>
</blockquote>
<p>We can substitute it as $x=a\tan^2 (\theta)$ . Then:</p>
<p>$$2a\int \theta \tan (\theta)\sec^2 (\theta) d\theta$$</p>
<p>Using integration by parts will be enough here. But I wanted to know if this particular problem can be solved by... | Deepesh Meena | 470,829 | <p><strong>Method 1:</strong>
$$I=2a\int \theta\cdot \tan \theta\cdot \sec^2 \theta d\theta$$</p>
<p>write it as </p>
<p>$$I=2a\int \theta\cdot (\sec \theta\cdot \sec \theta \tan \theta) d\theta$$
now apply integration by parts </p>
<p><strong>Method 2:</strong></p>
<p>Directly apply integration by parts</p>
<p... |
2,678,077 | <p>I need to solve this differential equation:
$$\frac{du}{dr}=\frac{4+\sqrt{r}}{2+\sqrt{u}}$$</p>
<p>I did it and got
$$u=\frac{2r^{\left(\frac{3}{2}\right)}}{3\sqrt{u}+2}+\frac{4r}{\sqrt{u}+2}+C$$ but my homework system is marking this as wrong. Why is that?</p>
| user577215664 | 475,762 | <p><strong><em>Just a hint</em></strong> </p>
<p>$$\frac{du}{dr}=\frac{4+\sqrt{r}}{2+\sqrt{u}}$$
$$(2+\sqrt{u})du=(4+\sqrt{r})dr$$
$$\int(2+\sqrt{u})du=\int(4+\sqrt{r})dr$$
$$2u+\frac 23 u^{3/2}+K=\int(4+\sqrt{r})dr$$
$$..............$$
Do the same for the right side of the equation</p>
|
4,634,180 | <p><span class="math-container">$$\int \frac{\sin^2(x)dx}{\sin(x)+2\cos(x)}$$</span></p>
<p>I tried to use different substitutions such as <span class="math-container">$t=\cos(x)$</span>, <span class="math-container">$t=\sin(x)$</span>, <span class="math-container">$t=\tan(x)$</span>, and after expressing <span class="... | K.K.McDonald | 302,349 | <p>It is possible to solve it without rational fractions. The denominator can be written as</p>
<p><span class="math-container">$$\sin(x)+2\cos(x) = \sqrt 5\left(\frac{1}{\sqrt 5}\sin x + \frac{2}{\sqrt 5}\cos x\right) = \sqrt 5 \sin\left(x+\theta\right)$$</span></p>
<p>where <span class="math-container">$\sin\theta = ... |
640,769 | <p>I'm to prove that every proper ideal is a product of maximal ideals which are uniquely determined up to order.
I have no idea even how to start in the proof to solve this question :(
May anybody help me ? </p>
| copper.hat | 27,978 | <p>Let $$\phi(t) = e^{-ct} F(t)$$ Then $\phi(0) = 0$, and $\phi(t) \ge 0$ for all $ t \ge 0$.</p>
<p>Furthermore, $$\phi'(t) = e^{-ct}(F'(t) - c F(t)) \le 0$$hence $\phi(t) = \int_0^t \phi'(\tau) d \tau \le 0$, and so $\phi(t) =0 $ for all $t \ge 0$.</p>
<p>If $ϕ(t)=0$ for all $t≥0$ , then $F(t)=0$ for all $t≥0$ . ... |
166,460 | <p>I work with linear combinations of graphs,
$$c_1 G_1 + c_2 G_2 + \dotsc,$$
and I want to represent them in my Mathematica code. I represent graphs as adjacency matrices, e.g.</p>
<pre><code>{{0,1},{1,0}}
</code></pre>
<p>The next step is to write down linear combinations of these matrices. However, I want to imple... | Sarah Stanley | 55,480 | <p>Use a HoldAll instead of an unevaluated function, or define your AdjMtx function as HoldAll</p>
<pre><code>In[127]:= AdjMtx[x___] := HoldAll[x];
In[128]:= lin = 5*AdjMtx[{{0, 1}, {1, 0}}] + 3*AdjMtx[{{1, 0}, {0, 1}}]
Out[128]= 5 HoldAll[{{0, 1}, {1, 0}}] + 3 HoldAll[{{1, 0}, {0, 1}}]
In[129]:= lin[[1]]
Out[129]... |
4,066,942 | <p>This is a problem from Kenneth A Ross 2nd Edition Elementary Analysis:</p>
<p>Show that the infinite series,<span class="math-container">$$\sum_{n=1}^{\infty} \frac{(-1)^n}{n+x^2}$$</span> converges uniformly for all <span class="math-container">$x$</span>, and by termwise differentiation, compute <span class="math-... | Martin R | 42,969 | <p><em>Preliminary remarks:</em> <span class="math-container">$f(a) = f(b)$</span> means that the function can be extended to a continuous function on <span class="math-container">$\Bbb R$</span> with period <span class="math-container">$b-a$</span>. Then either or <span class="math-container">$x_1$</span> and <span cl... |
1,776,260 | <p>After understanding the Cardano's formula for solving the depressed cubic (of the form $x^3+mx=n$, of course), I tried to find the solution of the equation $$x^3+6x=20.$$
After plugging into the formula
$$x=(n/2+\sqrt{ \frac{n^2}{4}+ \frac{m^3}{27} })^{1/3}+(-n/2+\sqrt{ \frac{n^2}{4}+ \frac{m^3}{27} })^{1/3}$$
where... | André Nicolas | 6,312 | <p>With the benefit of hindsight we notice that $10+\sqrt{108}=10+6\sqrt{3}$ and
$$10+6\sqrt{3}=(1+\sqrt{3})^3.$$
Similarly,
$$-10+6\sqrt{3}=(-1+\sqrt{3})^3.$$
Take the (real) cube roots and subtract.</p>
|
101,191 | <p>A few years ago I <a href="http://math.sfsu.edu/federico/Articles/arrangem.pdf">computed</a> the Tutte polynomials of the matroids given by the classical Coxeter groups, and found that their generating functions are all simple variations of the series $\sum_n \frac{x^n y^{n^2}}{n!}$.
I've wondered if there is a mor... | Abdelmalek Abdesselam | 7,410 | <p>I don't know if this might help, but Alan Sokal has extensively studied this kind of series (with $n$ choose 2 instead of the $n^2$ exponent for $y$). See the material for <a href="http://www.maths.qmul.ac.uk/~pjc/csgnotes/sokal/" rel="nofollow">these recent lectures</a>.</p>
|
929,502 | <p>Here are two succinct statements of the 'same' question:</p>
<p><strong>Statement 1:</strong>
Take $a>0$ and $S \subseteq \mathbb{R}^N; S=\{(x_1,\dots,x_N)| \frac{1}{N}\sum_i x_i = a; x_i>0\}$. Define a 'product function' $f:S\rightarrow \mathbb{R}^N; f(x_1,\dots,x_i)=\prod_ix_i$. There are many proofs that t... | Ewan Delanoy | 15,381 | <p>Your statements are false. Consider for example $\vec{c}=(2,100,498)$ and $\vec{d}=(25,25,550)$. Both $\vec{c}$ and $\vec{d}$ have mean $a=200$. And $||\vec{c}-\vec{a}||_1=596<700=||\vec{d}-\vec{a}||_1$, but $f(\vec{c})=99600<343750=f(\vec{d})$.</p>
<p>You probably forgot some additional conditions.</p>
|
139,385 | <p>Can anyone help me prove if $n \in \mathbb{N}$ and is $p$ is prime such that $p|(n!)^2+1$ then $(p-1)/2$ is even?</p>
<p>I'm attempting to use Fermats little theorem, so far I have only shown $p$ is odd.</p>
<p>I want to show that $p \equiv 1 \pmod 4$</p>
| Simon Markett | 30,357 | <p>If I am not mistaken this is just a special case of the first supplement to <a href="http://en.wikipedia.org/wiki/Quadratic_reciprocity" rel="nofollow">the quadratic reciprocity law</a>:</p>
<p>$p|x^2+1\Rightarrow x^2 \equiv -1 \pmod p$. This is solvable if and only if $p \equiv 1 \pmod 4$. And you have a given sol... |
2,899,829 | <p>$\newcommand{\d}{\mathrm{d}}$</p>
<blockquote>
<p>Evaluate the integral using the indicated substituion. $$\int \cot x \csc^2x \,\d{x}, \qquad u= \cot x .$$</p>
</blockquote>
<p>Differentiating both sides of $u$, then making the substitution: $$
\begin{align}
u &=
\phantom{-}\cot x, \\
\d u &= -\cot x\c... | egreg | 62,967 | <p>You have $du=-\csc^2x\,dx$, rather than your wrong differentiation. This implies the integral is
$$
\int\cot x\csc^2x\,dx=\int-u\,du=-\frac{1}{2}u^2+c=-\frac{1}{2}\cot^2x+c
$$
On the other hand, rewriting the integral as
$$
\int\frac{\cos x}{\sin^3x}\,dx=\int(\sin x)^{-3}d(\sin x)=-\frac{1}{2}\frac{1}{\sin^2x}+c
$$
... |
3,715,824 | <p>I proved that <span class="math-container">$$\lim_{n\to\infty}\left(1+\frac{x^2}{n^2}\right)^{\frac{n}{2}}=1$$</span>
using L'Hospital's rule. But is there a way to prove it without L'Hospital's rule? I tried splitting it as
<span class="math-container">$$\lim_{n\to\infty}n^{-n}(n^2+x^2)^{\frac{n}{2}},$$</span>
but ... | L F | 221,357 | <p>This has the form <span class="math-container">$\displaystyle\lim_{n\to\infty} (1+1/n)^{n}=e$</span>. </p>
<p><span class="math-container">$$\lim_{n\to\infty}\left(1+\frac{x^2}{n^2}\right)^{\frac{n}{2}}=\lim_{n\to\infty}\left(1+\frac{x^2}{n^2}\right)^{\frac{n}{2}{\color{red} {\frac{n}{x^2}\cdot\frac{x^2}{n}} }}=\li... |
3,715,824 | <p>I proved that <span class="math-container">$$\lim_{n\to\infty}\left(1+\frac{x^2}{n^2}\right)^{\frac{n}{2}}=1$$</span>
using L'Hospital's rule. But is there a way to prove it without L'Hospital's rule? I tried splitting it as
<span class="math-container">$$\lim_{n\to\infty}n^{-n}(n^2+x^2)^{\frac{n}{2}},$$</span>
but ... | Gosrabios | 771,183 | <p>I have an algebraic solution. Let's our limit be <span class="math-container">$L$</span>:
<span class="math-container">$$L=\lim_{n\rightarrow\infty}\left(1+\frac{x^2}{n^2}\right)^\frac{n}{2}$$</span>
Now, we make two changes of variables: <span class="math-container">$$t = \frac{n}{2} $$</span> and
<span class="math... |
44,391 | <p>The general equation of a conic is $A x^2 + B x y + C y^2 + D x + E y + F = 0$. At Wikipedia, there is an equation for the eccentricity, based on ABCDEF. </p>
<p>Is there a similar equation for getting the foci or directrix for a general ellipse, parabola, hyperbola from ABCDEF? Please assume that a non-degenera... | Jerry | 11,957 | <p>You'll first want to check either the discriminant or the eccentricity of your conic before proceeding to use any expression(s) for the foci; the central conics have two foci while the parabola only has one.</p>
<p>For the central conics, it is known that the two foci are at a distance $a\epsilon$ from the center, ... |
2,655,018 | <p>I have a quick question regarding a little issue.</p>
<p>So I'm given a problem that says "$\tan \left(\frac{9\pi}{8}\right)$" and I'm supposed to find the exact value using half angle identities. I know what these identities are $\sin, \cos, \tan$. So, I use the tangent half-angle identity and plug-in $\theta = \f... | user | 505,767 | <p>You need to use </p>
<p>$$\tan 2x=\frac{2\tan x}{1-\tan^2x}$$</p>
<p>with $$x=\frac98\pi$$</p>
<p>and since we know that $$\tan 2x=\tan \frac94\pi=\tan \frac{\pi}4=1$$</p>
<p>we have with $x=\tan \frac98\pi$</p>
<p>$$x^2+2x-1=0$$</p>
<p>which gives </p>
<p>$$y=\tan \frac98\pi=\sqrt 2 -1$$</p>
<p>as acceptabl... |
3,856,567 | <p>I’m new to number theory and I’m solving questions in the textbook one by one.
Here is one :
If <span class="math-container">$m\geq 1$</span> and <span class="math-container">$n\geq2$</span> , which both of them are natural numbers , prove this statement:</p>
<p><span class="math-container">$$(n-1)^2 | (n^m-1) \iff ... | Servaes | 30,382 | <p><strong>Hint 1:</strong> Use your factorization of <span class="math-container">$a^n-b^n$</span> to show that
<span class="math-container">$$(n-1)^2\mid(n^m-1)\qquad\Leftrightarrow\qquad (n-1)\mid(n^{m-1}+n^{m-2}+\ldots+n+1).$$</span></p>
<p><strong>Hint 2:</strong></p>
<blockquote class="spoiler">
<p> Divide <span ... |
41,174 | <p>I am trying to find the precise statement of the correspondence between stable Higgs bundles on a Riemann surface $\Sigma$, (irreducible) solutions to Hitchin's self-duality equations on $\Sigma$, and (irreducible) representations of the fundamental group of $\Sigma$. I am finding it a bit difficult to find a refere... | Andrei Halanay | 1,220 | <p>Stable Higgs bundles $(E,\theta)$ with vanishing Chern class
over a compact smooth Riemann surface (modulo the action of $\mathbb{C}^*: \theta \mapsto t\cdot \theta$) are in bijection with irreducible ($GL(n))-$representations of $\pi_1(X)$. This result in its full generality is due to C.Simpson (for any smooth proj... |
2,088,229 | <p>Is there a neat way to find the largest integer that divides another integer fully, within a range. As an example, I would like to find the largest integer smaller than 1131 that divides 3500 completely. </p>
<p>So far I have just tried by breaking up 3500 into its prime components and guessing, coming to 875, but ... | Dan Brumleve | 1,284 | <p>There is a <a href="https://cstheory.stackexchange.com/questions/4769/an-np-complete-variant-of-factoring/4785">randomized reduction from SUBSET-SUM</a> to this problem, and it is NP-complete assuming a weak version of Cramér's conjecture. So there is no neat general solution, but of course there are many special c... |
2,203,988 | <p>I'm reading the book <i>Heat Transfer</i> by J.P. Holman. On the chapter of unsteady-state conduction, page 140, the author remarks:</p>
<blockquote>
<p>The final series solution is therefore:
$${\theta(x,t) \over \theta_i} =
{4\over \pi} \sum^{\infty}_{n=1} {1\over n} e^{-\left({n\pi/L}\right)^2\alpha \,t}\si... | N. F. Taussig | 173,070 | <blockquote>
<p>In how many ways can eight rings be placed on three fingers if the order in which the rings are placed on the fingers does not matter?</p>
</blockquote>
<p>All that matters is which finger receives which ring. There are three choices of finger for each of the eight rings, so there are $3^8$ ways of ... |
3,975,832 | <p>I think the following claim is clearly correct, but I cannot prove it.</p>
<blockquote>
<p>Let <span class="math-container">$A$</span> and <span class="math-container">$B$</span> be sets. If <span class="math-container">$f:A \times B \to \mathbb{R}$</span> satisfies <span class="math-container">$f(a, b) \leq C_a$</s... | Calvin Lin | 54,563 | <p>The claim is not correct, so find a counter example.</p>
<p>Hint:</p>
<blockquote class="spoiler">
<p> Let <span class="math-container">$f(x, x ) = x $</span></p>
</blockquote>
|
110,373 | <p>Are there classes of infinite groups that admit Sylow subgroups and where the Sylow theorems are valid?</p>
<p>More precisely, I'm looking for classes of groups <span class="math-container">$\mathcal{C}$</span> with the following properties:</p>
<ul>
<li><span class="math-container">$\mathcal{C}$</span> includes th... | Jim Humphreys | 4,231 | <p>A number of older papers by V.P. Platonov (in Russian, often followed by English translations) deal with periodic linear groups or linear algebraic groups in which the notions of Sylow theory make sense and where some results from the finite case actually generalize. One of the more substantial papers deals especi... |
110,373 | <p>Are there classes of infinite groups that admit Sylow subgroups and where the Sylow theorems are valid?</p>
<p>More precisely, I'm looking for classes of groups <span class="math-container">$\mathcal{C}$</span> with the following properties:</p>
<ul>
<li><span class="math-container">$\mathcal{C}$</span> includes th... | Geoff Robinson | 14,450 | <p>Amalgams of finite groups provide another example. Let <span class="math-container">$A$</span> and <span class="math-container">$B$</span> be finite groups and let <span class="math-container">$C = A \cap B.$</span> Suppose that <span class="math-container">$P$</span> is a Sylow <span class="math-container">$p$</spa... |
1,275,437 | <p>My question is simply: for which values of $n$ is it possible to divide any given angle into $n$ equal parts using only a compass and a straight edge? I know that it is possible for $2$ and not possible for $3$, but is it possible for any integers that are not of the form $2^k$?</p>
| André Nicolas | 6,312 | <p>The only possibility is indeed numbers of the form $2^k$.</p>
<p>We use the famous characterization of constructible regular polygons. The $\frac{360^\circ}{N}$ angle is straight edge and compass constructible if and only if $N$ is of the shape
$$N=2^k p_1\cdots p_s,\tag{1}$$
where the $p_i$ are <strong>distinct</s... |
1,275,437 | <p>My question is simply: for which values of $n$ is it possible to divide any given angle into $n$ equal parts using only a compass and a straight edge? I know that it is possible for $2$ and not possible for $3$, but is it possible for any integers that are not of the form $2^k$?</p>
| Piquito | 219,998 | <p>It is not possible.You know how to construct the square root, so successively you can do with the half of any of 1/2 , 1/4, 1/8,.....(see the formule of sin ($\alpha$/2)). But you can not get 1/3 of the angle which is, as you know, a famous classic problem of impossibility (see the formule of sin ($\alpha$/3)) and n... |
4,124,777 | <p>I'm trying to find out whether <span class="math-container">$\sum _{n=0}^{\infty }\left(\cos^n\left(\frac{1}{\sqrt{n}}\right)-\frac{1}{\sqrt{e}}\right)$</span> converges or not. I've tried with taylor series but it doesn't lead me anywhere except with the fact that <span class="math-container">$\lim_{n \to \infty}\c... | epi163sqrt | 132,007 | <p>Here is a somewhat different perspective.</p>
<p>It is quite common when we talk about <span class="math-container">$\mathbb{R}$</span> to think of the complete ordered field of <span class="math-container">$\left(\mathbb{R},+,\cdot,<\right)$</span>. But we also have to consider the <em>context</em> in which the... |
656,560 | <p>I'm trying to get a solution for:</p>
<p>$4^{2x+1}-3^{3x+1}=4^{2x+3}-3^{3x+2}$</p>
<p>My main problem is that I don't know how to combine this potencys!</p>
<p>Ive also thought about another function that would bring me same difficulties:</p>
<p>$6^x=36*9.75^{x-2}$</p>
<p>What am I supposed to do? </p>
| Ben Grossmann | 81,360 | <p>For the first one, we have
$$
4^{2x+1}-3^{3x+1}=4^{2x+3}-3^{3x+2} \implies\\
3^{3x+2}-3^{3x+1} = 4^{2x+3}-4^{2x+1} \implies\\
(3-1)3^{3x+1} = (4^2 - 1)4^{2x+1} \implies\\
2\cdot 3^{3x+1} = 5\cdot 3\cdot4^{2x+1} \implies\\
3^{3x} = 5 \cdot 2^{4x+1}
$$
I think that's the simplest we can get it. From there, I suppose... |
656,185 | <blockquote>
<p>let sequence $\{G_{n}\}$ such
$$G_{1}=1,G_{3}=3,G_{2n}=G_{n}$$
$$G_{4n+1}=2G_{2n+1}-G_{n},G_{4n+3}=3G_{2n+1}-2G_{n}$$</p>
</blockquote>
<p>If such $G_{n}=n$, then we said $n$ is 'good'.
How many 'good' numbers $n$, such that $n<2^{100}?$</p>
<p><strong>My try:</strong></p>
<p>since
$$\begin{... | Neil W | 119,166 | <p>For every $n \in \mathbb{N}$, $n$ can be written uniquely in the form $\sum_{j=0}^{m} a_j 2^j$ with $a_j \in \{0, 1\}$ for $j=0,1,...,m-1$ and $a_m = 1$</p>
<p>Now for $n = \sum_{j=0}^{m} a_j 2^j$ with $a_j \in \{0, 1\}$ for $j=0,1,...,m-1$ and $a_m = 1$,</p>
<p>define</p>
<p>$f(n) = \sum_{j=0}^{m} a_{m-j} 2^j$</... |
1,575,397 | <p>I need help calculating
$$\lim_{n\to\infty}\left(\frac{1}{n^{2}}+\frac{2}{n^{2}}+...+\frac{n}{n^{2}}\right) = ?$$</p>
| Clement C. | 75,808 | <p><strong>Hint:</strong></p>
<ul>
<li><p>First possibility: Rewrite this
$$
\frac{1}{n}\sum_{k=1}^n \frac{k}{n}
$$
and apply results you know (?) on Riemann sums with the function $f\colon x\in [0,1]\mapsto x$.</p></li>
<li><p>Second possibility:
Explicitly compute $\sum_{k=1}^n k$. Now, you can divide by $n^2$ and t... |
4,044,953 | <p>I would like some help to prove the following equality :
<span class="math-container">$$\sum_{i=0}^n \binom{n}i^2=\binom{2n}n$$</span>
I wanted to do a proof by induction :
<span class="math-container">$$\sum_{i=0}^{n+1} \binom{n+1}i^2=1+\sum_{i=1}^{n+1} \binom{n+1}i^2=1 + \sum_{i=0}^{n} \binom{n+1}{i+1}^2=1+\sum_{i... | qfwfq | 894,108 | <p>The binomial theorem says <span class="math-container">$$(1+x)^n=\sum^n_{i=0}\binom{n}{i}x^i,$$</span> and we know that
<span class="math-container">$$(1+x)^n(1+x)^n=(1+x)^{2n}.$$</span> Comparing the coefficient of <span class="math-container">$x^n$</span>, we get <span class="math-container">$$\sum^n_{i=0}\binom{n... |
4,044,953 | <p>I would like some help to prove the following equality :
<span class="math-container">$$\sum_{i=0}^n \binom{n}i^2=\binom{2n}n$$</span>
I wanted to do a proof by induction :
<span class="math-container">$$\sum_{i=0}^{n+1} \binom{n+1}i^2=1+\sum_{i=1}^{n+1} \binom{n+1}i^2=1 + \sum_{i=0}^{n} \binom{n+1}{i+1}^2=1+\sum_{i... | awkward | 76,172 | <p>One way to proceed is to prove a more general identity by induction, and then deduce the identity in the problem statement as a corollary.</p>
<p>The more general identity is called "Vandermonde's Identity":
<span class="math-container">$$\sum_{i=0}^k \binom{m}{i} \binom{n}{k-i} = \binom{m+n}{k} \tag{1}$$<... |
1,526,474 | <p>Find the natural number $k <117$ such that $2^{117}\equiv k \pmod {117}$.</p>
<p>I know $117$ is the product of $3$ and $37$.</p>
<p>$2^{117}\equiv 2 \pmod 3$
$2^{117}\equiv 31 \pmod {37}$.
But $2^{117}\equiv 44 \pmod {117}$.</p>
<p>I can't seem to understand how to get $44$. Can anyone help me understand?</p... | DanielWainfleet | 254,665 | <p>Using the totient function, we have $\Phi (13)=12$ and $\Phi (9)=6$. Since $\gcd (2,13)=\gcd (2,9)=1$ we have $1\equiv 2^{12}\pmod {13}$ and $2^{12}\equiv (2^6)^2\equiv 1 \pmod 9$. Since $\gcd (9,13)=1$ this requires $2^{12}\equiv 1\pmod {117}$. Hence $2^{117}=(2^{12})^9 2^9\equiv 2^9 \equiv 44\pmod {117}$.</p>
|
1,071,040 | <p>I found <a href="https://math.stackexchange.com/questions/549065/how-exactly-do-you-measure-circumference-or-diameter">How exactly do you measure circumference or diameter?</a> but it was more related to how people measured circumference and diameter in old days.</p>
<p><strong>BUT</strong> now we have a formula, b... | Chootar Laal | 444,555 | <p>We can pretend to measure the circumference of a circle by saying
circumference = $\pi$ * diameter</p>
<p>Since $\pi$ itself is an approximation, a "measurement" of the circumference will always and forever be just an approximation and NEVER an exact number. It is quite interesting because one can clearly see a c... |
1,177,721 | <p>A fair $6$-sided die is rolled $6$ times independently. For any outcome, this is the set of numbers that showed up at least once in the different rolls. For example, the outcome is $(2,3,3,3,5,5)$, the element set is $\{2,3,5\}$. What is the probability the element set has exactly $2$ elements? how about $3$ element... | Dale M | 55,635 | <p>For two groups:</p>
<p>There are $6\choose 2$ ways of selecting the 2 groups. There are ${6\choose 2}-2$ ways to for 6 dice to roll those numbers.</p>
<p>Can you work it out for 3 groups?</p>
|
11,994 | <p>Now that we get to see the SE-network wide list of "hot" questions, I am just shaking my head in disbelief. At the time I am writing this, the two hot questions from Math.SE are titled (get a barf-bag, quick)</p>
<ul>
<li><a href="https://math.stackexchange.com/q/599520/8348">https://math.stackexchange.com/q/599520... | Logan M | 8,473 | <p>I could say a lot about this, but I'll do my best to be as brief as possible. The hot questions list is something that I've been collecting data on for a few months now. Barring a few strange cases, I have a formula which seems to be approximately correct for most questions. Note that the formula is different from t... |
1,584,080 | <p>I've started to learn probability, and the first thing I saw was the question about the probability of getting at least one six in $4$ rolls of a dice.
I understand that it's easier to do $1-(\frac{5}{6})^4$ because it says "at least", but what if it said the probability of getting exactly one six in those $4$ rolls... | Dr. Sonnhard Graubner | 175,066 | <p>$$e^{ab}+e^{ac}=e^{ac}\left(\frac{e^{ab}}{e^{ac}}+1\right)=e^{ac}\left(e^{ab-ac}+1\right)=e^{ac}\left(e^{a(b-c)}+1\right)$$</p>
|
1,584,080 | <p>I've started to learn probability, and the first thing I saw was the question about the probability of getting at least one six in $4$ rolls of a dice.
I understand that it's easier to do $1-(\frac{5}{6})^4$ because it says "at least", but what if it said the probability of getting exactly one six in those $4$ rolls... | Simply Beautiful Art | 272,831 | <p>For the complex extension of $\cos(x)$, we have $$\cos(x)=\frac{e^{ix}+e^{-ix}}2$$Which is how we evaluate $\cos(i)$.</p>
<p>From here, we note your expression:$$e^{ab}+e^{ac}$$If we allow $b=-c$, then we have $$e^{ab}+e^{-ab}=2\cos(\frac{ab}i)$$If not, then we can't really do much.</p>
<p>We can also factor some ... |
2,305,689 | <blockquote>
<p>If $x^6-12x^5+ax^4+bx^3+cx^2+dx+64=0$ has positive roots then find $a,b,c,d$.</p>
</blockquote>
<p>I did something but that don't deserve to be added here, but what I thought before doing that is following:</p>
<ol>
<li>For us, Product and Sum of roots are given.</li>
<li>Roots are positive.</li>
<l... | Arpan1729 | 444,208 | <p>Use AM-GM on the roots.</p>
<p>Say $a_1,a_2,a_3,a_4,a_5,a_6$ are the roots of the equation.</p>
<p>Then $a_1+a_2+a_3+a_4+a_5+a_6\geq 6(a_1\times a_2\times a_3\times a_4\times a_5\times a_6)^{1/6} $</p>
<p>Now $a_1+a_2+a_3+a_4+a_5+a_6=12$</p>
<p>And also $6(a_1\times a_2\times a_3\times a_4\times a_5\times a_6)^{... |
2,672,908 | <p>Hey I was given this question in my discrete math class, and I'm unsure of what I should do!</p>
<blockquote>
<p>Prove that if $x$ is coprime with $6$ and $x$ is coprime with $8$, then $x$ is coprime with 24.</p>
</blockquote>
<p>I think I have to use the GCD theorem or co-primality theorem but I don't think wha... | David | 119,775 | <p><strong>Hint</strong>: no need to use Bezout, use the contrapositive.</p>
<ul>
<li>Suppose that $x$ is <strong>not</strong> coprime with $24$.</li>
<li>What can you say about prime factors of $x$?</li>
<li>Is it possible for $x$ to be coprime with $6$?</li>
</ul>
<p>Good luck!</p>
|
374,380 | <p>I am having trouble understanding the factor group, $\mathbb{R}$/$\mathbb{Z}$, or maybe i'm not. Here's what I am thinking.</p>
<p>Okay, so i have a group $G=(\mathbb{R},+)$, and I have a subgroup $N=(\mathbb{Z},+)$. Then I form $G/N$. So this thing identifies any real number $x$ with the integers that are exact... | zarathustra | 73,997 | <p>One proves that $\mathbb R/\mathbb Z$ is isomorphic to the group of unit-modulus complex numbers (let's call it $G$), which is a circle, isn't it?</p>
<p>Let's prove the isomorphism. Take $\varphi : \mathbb R \rightarrow G$ defined by $\varphi(\theta) = e^{2\pi i\theta}$. We have $\varphi(\theta + \theta') = e^{2\p... |
2,258,697 | <p>I recently encountered this question and have been stuck for a while. Any help would be appreciated!</p>
<p>Q: Given that
$$\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{5} \tag{1} \label{eq:1}$$
$$abc = 5 \tag{2} \label{eq:2}$$
Find $a^3 + b^3 + c^3$. It wasn't specified in the question but I think it can be... | Kenny Lau | 328,173 | <p>Denote $e_1 = \begin{bmatrix}1\\0\end{bmatrix}$ and $e_2 = \begin{bmatrix}0\\1\end{bmatrix}$.</p>
<p>Let the matrix in question be $A$.</p>
<p>Observe that $A = \begin{bmatrix}I&I\\I&I\end{bmatrix}$, where $I = \begin{bmatrix}1&0\\0&1\end{bmatrix}$.</p>
<p>Then, $A$ is similar to $B = \begin{bmatr... |
2,076,908 | <blockquote>
<p><strong>Question:</strong> Prove that $e^x, xe^x,$ and $x^2e^x$ are linearly independent over $\mathbb{R}$.</p>
</blockquote>
<p>Generally we proceed by setting up the equation
$$a_1e^x + a_2xe^x+a_3x^2e^x=0_f,$$
which simplifies to $$e^x(a_1+a_2x+a_3x^2)=0_f,$$ and furthermore to
$$a_1+a_2x+a_3x^2=... | kobe | 190,421 | <p>Setting $x = 0$ in the equation $a_1 + a_2x + a_3x^2 = 0$ results in $a_1 = 0$. Then $a_2x + a_3x^2 = 0$ for all $x\in \Bbb R$. Setting $x = 1$ gives $a_2 + a_3 = 0$, and setting $x = -1$ gives $-a_2 + a_3 = 0$. Solving the system of equations will yield $a_2 = a_3 = 0$.</p>
|
2,763,735 | <p>Is it true that $$\mathbb{Z/4Z\subseteq Z/2Z}$$
Why precisely? Or the reverse $$\mathbb{Z/2Z \subseteq Z/4Z}$$ holds? I'm a beginner. How do I justify the true inclusion?
How do I visualize $$\mathbb{Z/2Z \subseteq Z/4Z}$$
Thank you very much.</p>
| Yanko | 426,577 | <p>The objects $\mathbb{Z}/2\mathbb{Z}$ and $\mathbb{Z}/4\mathbb{Z}$ are <strong>sets</strong> but also they're <strong>groups</strong>.</p>
<p>Thinking about them as sets we have $\mathbb{Z}/2\mathbb{Z}=\{0,1\}$ and $\mathbb{Z}/4\mathbb{Z} = \{0,1,2,3\}$ and as <strong>sets</strong> $\{0,1\}\subseteq \{0,1,2,3\}$.</p... |
1,665,064 | <p>I have a vector quadratic equation of the form
$\boldsymbol{x}^{T} \boldsymbol{A} \boldsymbol{x} + \boldsymbol{x}^{T} \boldsymbol{b} + c = 0$<br>
where $\boldsymbol{A}$ is symmetric and for my particular case, $\boldsymbol{x} \in \mathbb{R}^{2}$. I know that the solution for this system (if it exists) can be found ... | Robert Israel | 8,508 | <p>Let's do this for $x \in \mathbb R^n$, where $A$ is a real symmetric $n \times n$ matrix.</p>
<p>If $A$ is positive definite, the function $f(x) = x^T A x + x^T b + c$ is bounded below, with minimum value $x^T b/2 + c$ occurring at the solution of $A x = -b/2$, but unbounded above. Thus a solution of $f(x) = 0$ ex... |
1,549,138 | <p>I have a problem with this exercise:</p>
<p>Proove that if $R$ is a reflexive and transitive relation then $R^n=R$ for each $n \ge 1$ (where $R^n \equiv \underbrace {R \times R \times R \times \cdots \times R} _{n \ \text{times}}$).</p>
<p>This exercise comes from my logic excercise book. The problem is that I've ... | hmakholm left over Monica | 14,366 | <p>The problem makes more sense if we assume that the $\times$ that appear in it is not the Cartesian product, but an unusual notation for <em>composition</em> of relations, which is more commonly notated with $\circ$:</p>
<p>$$ R\circ S = \{ \langle a,c\rangle \mid \exists b: \langle a,b\rangle\in S \land \langle b,c... |
53,188 | <p>Recently I read the chapter "Doctrines in Categorical Logic" by Kock, and Reyes in the Handbook of Mathematical Logic. And I was quite impressed with the entire chapter. However it is very short, and considering that this copy was published in 1977, possibly a bit out of date. </p>
<p>My curiosity has been sparked ... | David Roberts | 4,177 | <p>Might I recommend <em>Sheaves in Geometry and Logic</em> by MacLane and Moerdijk. To quote bits from the blurb:</p>
<blockquote>
<p>Sheaves also appear in logic as carriers for models of set theory as as for the semantics of other types of logic.</p>
<p>The applications to axiomatic set theory and the use in forcing... |
53,188 | <p>Recently I read the chapter "Doctrines in Categorical Logic" by Kock, and Reyes in the Handbook of Mathematical Logic. And I was quite impressed with the entire chapter. However it is very short, and considering that this copy was published in 1977, possibly a bit out of date. </p>
<p>My curiosity has been sparked ... | Justin Hilburn | 333 | <p>My answer here has a number of good references:
<a href="https://mathoverflow.net/questions/903/resources-for-learning-practical-category-theory/1954#1954">Resources for learning practical category theory</a></p>
<p>I don't recommend Goldblatt. Here is an article that elaborates on why:</p>
<blockquote>
<p>Colin... |
984,915 | <blockquote>
<p>If $A=\{a_1,...,a_n\}$ and $B=\{b_1,...,b_n\}$ are two bases of a vector space
$V$, there exists a unique matrix $M$ such that for any $f\in V$,
$[f]_A=M[f]_B$.</p>
</blockquote>
<p>My textbook uses this theorem without a proof, so I'm trying to show that it's true myself. Consider $[f]_A = (c_1,... | Brian M. Scott | 12,042 | <p>HINT: Note that</p>
<p>$$\left|\frac{(-1)^nn}{n^2+1}\right|=\frac{n}{n^2+1}<\frac{n}{n^2}\;.$$</p>
<p>There are some ideas that come up often, and with practice you’ll come to recognize them, but in general you have to attack such problems individually.</p>
|
2,668,447 | <p>Let <span class="math-container">$F$</span> be a subfield of a field <span class="math-container">$K$</span> and let <span class="math-container">$n$</span> be a positive integer. Show that a nonempty linearly-independent subset <span class="math-container">$D$</span> of <span class="math-container">$F^n$</span> rem... | Gerry Myerson | 8,269 | <p>One way to expand on Mariano's hint: </p>
<p>Suppose <span class="math-container">$v_1,\dots,v_r$</span> are <span class="math-container">$n$</span>-vectors with entries in a field <span class="math-container">$F$</span> and are linearly independent over <span class="math-container">$F$</span>. Extend to a basis <s... |
383,037 | <p>I was going through "Convergence of Probability Measures" by Patrick Billingsley. In Section 1: I encountered the following problem:</p>
<p><strong>Show that inequivalent metrics can give rise to the same class of Borel sets.</strong></p>
<p>My idea is that the 2 metrics generate different topologies but the Sigma... | user24367 | 17,523 | <p>"Show that inequivalent metrics can give rise to the same class of Borel sets."
any in-finite metric $d$ can be replaced by the metric $\frac{d}{d+1}$, which is finite and both these metrics generate the same topology and hence the same (Borel) sigma algebra</p>
<p>However if you are referring to problem $2$ in Bil... |
3,141,618 | <p>The exercise is:</p>
<blockquote>
<p>Show that if <span class="math-container">$A \subset \mathbb{R} $</span> is bounded and <span class="math-container">$ A \neq \varnothing $</span> then <span class="math-container">$sup(A)=max(\overline{A} ).$</span></p>
</blockquote>
<p>Now, I wanted to ask you <strong>wheth... | SmileyCraft | 439,467 | <p>"Further define <span class="math-container">$\overline{x}=max(\overline{A})$</span>." This is not allowed, as you need to prove that this is well-defined. Not every set has a maximum. Since the standard proof of this is done by showing that it equals the supremum, you end up in a circular reasoning.</p>
<p>Some hi... |
3,392,749 | <p>I'm trying to draw a dfa for this description</p>
<p>The set of strings over {a, b, c} that do not contain the substring aa,</p>
<p>current issue i'm facing is how many states to start with, any help how to approach this problem?</p>
| Math1000 | 38,584 | <p>A regular expression for this language is <span class="math-container">$(b\cup c\cup ab\cup ac)^*(a\cup\lambda)$</span>, there <span class="math-container">$\lambda$</span> is the empty string. To define a DFA recognizing this language, let the state space be <span class="math-container">$Q=\{q_0,q_a,q_{aa}\}$</span... |
977,446 | <p>Prove that $A\cap B = \emptyset$ iff $A\subset B^C$. I figured I could start by letting $x$ be an element of the universe and that $x$ is an element of $A$ and not an element of $B$. </p>
| MAM | 177,202 | <p>Hint: If $A \cap B = \emptyset $ then what can you say about all elements of $A$? Use this with the definition of $B^C$. $B^C$ is the set of all elements in the universe that are not in $B$, $(U-B)$.</p>
<p>$$ A \cap B = \emptyset \iff \forall x\in A, x\notin B \iff \forall x\in A,x \in B^C \iff A\subset B^C $$</p... |
3,489,347 | <p><strong>Is there a simple way to characterize the functions in <span class="math-container">$C^\infty((0,1])\cap L^2((0,1])$</span>?</strong></p>
<p>That is, given a function <span class="math-container">$f(t)\in C^\infty((0,1])$</span>, is there a necessary/sufficient condition I can check to see if it's square in... | Claude Leibovici | 82,404 | <p>In the real domain, consider the function
<span class="math-container">$$f(x)=5\log(x)-x$$</span> The first derivative cancels at <span class="math-container">$x=5$</span> and by the second derivative test, this is a maximum. So, there is a limited range of <span class="math-container">$x$</span> where <span class="... |
614,749 | <p><strong>The game:</strong></p>
<p>Given $S = \{ a_1,..., a_n \}$ of positive integers ($n \ge 2$). The game is played by two people. At each of their turns, the player chooses two <strong>different</strong> non-zero numbers and subtracts $1$ from each of them. The winner is the one, for the last time, being able to... | Paul Sinclair | 258,282 | <p>An old thread, but I noticed it in the "related"s for another question, and found it interesting.</p>
<p>Let me introduce the game of "raindrops": You have a vertical track of some finite number of "levels" and at each level is a finite number of raindrops. During each player's turn, they either move one raindrop d... |
126,739 | <p><strong>I changed the title and added revisions and left the original untouched</strong> </p>
<p>For this post, $k$ is defined to be the square root of some $n\geq k^{2}$. Out of curiousity, I took the sum of one of the factorials in the denominator of the binomial theorem;
$$\sum _{k=1}^{\infty } \frac{1}{k!} \... | John Jiang | 4,923 | <p>Another way is to write $1/(k+2)$ as $1/(k+1) - 1/((k+1)(k+2))$.</p>
|
3,511,445 | <p>Well it is the problem from jmo odisha. It is of 5 marks
I tried a lot of ways but I can't get the answer. Only elementary mathematics is allowed.</p>
| Mark Bennet | 2,906 | <p>Note that <span class="math-container">$$x-4=\frac {16}{y-4}; z-6=\frac {36}{y-6}$$</span> so that <span class="math-container">$$(x-8)(z-8)=\left(\frac {16}{y-4}-4\right)\left(\frac {36}{y-6}-2\right)=64$$</span> and clearing fractions this gives <span class="math-container">$$(32-4y)(48-2y)=64(y-4)(y-6)$$</span></... |
446,456 | <p>Educators and Professors: when you teach first year calculus students that infinity isn't a number, how would you logically present to them $-\infty < x < +\infty$, where $x$ is a real number?</p>
| kjetil b halvorsen | 32,967 | <p>To accept $\infty$ and $-\infty$ as numbers, they would have to satisfy the usual rules of arithmetic of numbers. In particular, for all numbers $a$, $a-a=0$, so if infinity $\infty$ is a number we shoild have $\infty - \infty=0$. But that is not necessarily true, as we can let $\infty$ be represented by any sequenc... |
738,743 | <p>The following equation,
$$(\partial_x + i\partial_y)u - c(\partial_x+i\partial_y)au=0$$</p>
<p>($a=a(x,y)$ and $\partial_x=\frac{\partial}{\partial x}$)</p>
<p>with solution,
$$u=\exp(ca)f(x+iy)$$</p>
<p>where $f$ and $g$ are arbitrary entire functions, a is some scalar function and $c$ is a scalar.</p>
<p>How c... | Winther | 147,873 | <p>Hint: Your equation can be written</p>
<p>$$\partial_x (\log u-ca) + i\partial_y(\log u - ca) = 0$$</p>
<p>This equation has the solution $\log u - ca = g$ for any $g$ that satisfy</p>
<p>$$\partial_x g + i\partial_yg = 0$$</p>
|
755,571 | <p>$$a_n=3a_{n-1}+1; a_0=1$$</p>
<p>The book has the answer as: $$\frac{3^{n+1}-1}{2}$$</p>
<p>However, I have the answer as: $$\frac{3^{n}-1}{2}$$</p>
<p>Based on:</p>
<p><img src="https://i.stack.imgur.com/4vJrQ.png" alt="enter image description here"></p>
<p>Which one is correct?</p>
<p>Using backwards substit... | user141421 | 141,421 | <p>Your answer is incorrect, since it fails for $n=0$.</p>
|
755,571 | <p>$$a_n=3a_{n-1}+1; a_0=1$$</p>
<p>The book has the answer as: $$\frac{3^{n+1}-1}{2}$$</p>
<p>However, I have the answer as: $$\frac{3^{n}-1}{2}$$</p>
<p>Based on:</p>
<p><img src="https://i.stack.imgur.com/4vJrQ.png" alt="enter image description here"></p>
<p>Which one is correct?</p>
<p>Using backwards substit... | Community | -1 | <p>According to yours $$a_0=0$$ so the book's is correct.</p>
|
2,887,880 | <p>I read this <a href="https://www.reddit.com/r/math/comments/8frbe2/what_is_a_natural_way_to_represent_nonlinear/" rel="nofollow noreferrer">reddit</a> post and this <a href="https://math.stackexchange.com/q/1388566/553404">SE thread</a> discussing how to represent nonlinear/linear transforms in matrix notations but ... | saulspatz | 235,128 | <p>I'm reluctant to say it cannot be done, but I would think any natural use of matrices to represent functions would entail linear functions. For example, let $S$ be the set of convergent sequences of real numbers and let $A$ be an infinite matrix such that $X\in S\implies AX\in S$. That is $$(AX)_i=\sum_{j=1}^{\inf... |
218,479 | <p>I am trying to evaluate the following integral with Mathematica:</p>
<p><span class="math-container">\begin{align}
I = \int_{0}^{\infty} da \, \frac{e^{-\frac{a ^2}{4s^2}} }{a^2} \mbox{sinc}\left(\tfrac{w}{2} a \right) \delta' \left( \frac{D^2}{a}- a \right),
\end{align}</span>
where the prime on the delta ... | Ulrich Neumann | 53,677 | <p><strong>speculation:
Mathematica cannot handle <code>Derivative[1][DiracDelta][1/x-x]</code>in a right way?</strong></p>
<p>Here I'll give a simplified example which perhaps shows that Mathematica gives a wrong result, when applied to <code>Derivative[1][DiracDelta][1/x-x]</code>!</p>
<p>Let's consider the integr... |
1,190,759 | <p>I was trying to show the following
$\int_{-\infty}^{\infty} x^{2n}e^{-x^2}dx = (2n)!{\sqrt{\pi}}/4^nn!$ by using $\int_{-\infty}^{\infty} e^{-tx^2}dx = \sqrt{\pi/t}$
thus</p>
<p>I differentiated this exponential integral n times to get the following. </p>
<p>$\int_{-\infty}^{\infty} \frac{d^ne^{-tx^2}}{dt^n}dx ... | Jack D'Aurizio | 44,121 | <p>By exploiting a change of variable and integration by parts:
$$I=\int_{\mathbb{R}}x^{2n}e^{-x^2}\,dx = \int_{0}^{+\infty}z^{n-\frac{1}{2}}e^{-z}\,dz = \int_{0}^{+\infty}\frac{d^n}{dz^n}\left(z^{n-\frac{1}{2}}\right)e^{-z}\,dz$$
hence:
$$ I = \frac{(2n-1)!!}{2^n}\int_{0}^{+\infty}z^{-1/2}e^{-z}\,dz = \sqrt{\pi}\,\fra... |
1,217,175 | <p><strong>Here's the question:</strong></p>
<p>Is the following true or false?</p>
<p>There is a function $f: \mathbb R \to \mathbb R$ that satisfies the following condition:</p>
<p>For every $a \in \mathbb R $ and $ \epsilon \gt 0 $ there is $\delta \gt 0$ such that $\left| f(x)-f(a) \right| \lt \epsilon \implies ... | Andrew D. Hwang | 86,418 | <p>Mathematicians are often pedantic--"What do you <em>mean</em> by $\sqrt{\phantom{x}}$?" "What are the domain and target of this function?"--but the question "How to express (solutions of) $\sqrt{x} = -1$?" is a prime example of the <em>need</em> to be grindingly specific about the meaning of symbols.</p>
<p>As user... |
1,955,393 | <p>I have been trying to evaluate this limit:</p>
<p>$$\lim_{n\to\infty}{\sqrt[n]{4^n + 5^n}}$$</p>
<p>What methods should I try in order to proceed?</p>
<p>I was advised to use "Limit Chain Rule", but I believe there is a different approach.</p>
| StubbornAtom | 321,264 | <p><strong>HINT:</strong> </p>
<p>Courtesy of the <a href="https://en.wikipedia.org/wiki/Squeeze_theorem" rel="nofollow">Sandwich theorem</a>,
$\displaystyle\lim_{n\to\infty}(a^n+b^n)^{1/n}=\max (a,b)\quad$ where $a,b>0$.</p>
|
3,454,146 | <p>I'm trying to wrap my head around this new subject. I have to determine the validity of this argument (using a truth table): </p>
<p>"If Steve went to the movies then Maria's sister would not have stayed home. Either Steve went to the movies or Maria or both. If Maria went, then Maria’s sister would have st... | Community | -1 | <p>A reasoning with premises P1, P2, P3, etc. and conclusion C is valid iff its corresponding conditional is valid ( = is aa tautology). </p>
<p>By " corresponding conditional" I mean : " P1 & P2 & P3... --> C" </p>
<p>( For this kind of problem, this definition is perfectly OK). </p>
<p>So build the corres... |
4,200,602 | <p>Let <span class="math-container">$\alpha$</span> be a class <span class="math-container">$\mathcal{K}$</span> function defined on <span class="math-container">$[0,a)$</span>. Then
<span class="math-container">\begin{equation}
\alpha(r_1+r_2) \leq \alpha(2r_1) + \alpha(2r_2), \quad \forall r_1,\,r_2 \in [0,\,a/2).
\e... | Jens Schwaiger | 532,419 | <p>Since <span class="math-container">$\alpha$</span> is strictly increasing and since <span class="math-container">$r_1+r_2\leq2\max(r_1,r_2)\in\{2r_1,2r_2\}$</span> you get
<span class="math-container">$\alpha(r_1+r_2)\leq\alpha(2\max(r_1,r_2))\leq\alpha(2r_1)+\alpha(2r_2)$</span>.</p>
|
200,876 | <p>Is there a topological space $(C,\tau_C)$ and two points $c_0\neq c_1\in C$ such that the following holds?</p>
<blockquote>
<blockquote>
<p>A space $(X,\tau)$ is connected if and only if for all $x,y\in X$ there is a continuous map $f:C\to X$ such that $f(c_0) = x$ and $f(c_1) = y$.</p>
</blockquote>
</bloc... | Joseph Van Name | 22,277 | <p>I claim that for each cardinal $\lambda$, there is a connected space $C$ and $c_{0},c_{1}\in C$ such that whenever $|X|<\lambda$, then $X$ is connected if and only if for all $x,y\in X$ there is some continuous map $f:C\rightarrow X$ with $f(c_{0})=x$ and $f(c_{1})=y$. </p>
<p>Let $I$ be an index set and let $(X... |
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