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 |
|---|---|---|---|---|
136,453 | <p>For every $k\in\mathbb{N}$, let
$$
x_k=\sum_{n=1}^{\infty}\frac{1}{n^2}\left(1-\frac{1}{2n}+\frac{1}{4n^2}\right)^{2k}.
$$
Calculate the limit $\displaystyle\lim_{k\rightarrow\infty}x_k$.</p>
| Thomas Klimpel | 12,490 | <p>The prime numbers have connections to pseudo random numbers. They might also have connections to "true randomness", but I'm not aware if there has been much progress on the conjectures which point in this direction. I wonder whether the Riemann Hypothesis is equivalent to a statement about the relation of the prime ... |
497,015 | <p>So this is an excercise.. Does anyone have a hint? </p>
<p><strong>If A is both orthogonal and a orthogonal projector. What can you then conlcude about A?</strong></p>
<p>I know that an $n\times n$ matrix $P$ is an orthogonal projector if it is both idempotent ($P^2 = P$) and symmetric ($P = P^T$ ). Such a matrix ... | Berci | 41,488 | <p>Very good.</p>
<p>So, if $P$ is in addition orthogonal, then
$P=P^2=P^TP=I$.</p>
|
497,015 | <p>So this is an excercise.. Does anyone have a hint? </p>
<p><strong>If A is both orthogonal and a orthogonal projector. What can you then conlcude about A?</strong></p>
<p>I know that an $n\times n$ matrix $P$ is an orthogonal projector if it is both idempotent ($P^2 = P$) and symmetric ($P = P^T$ ). Such a matrix ... | user1551 | 1,551 | <p>It actually suffices to assume that $P$ is idempotent (instead of being a <em>orthogonal</em> projector) and $P$ is an orthogonal matrix: $P=P^TP^2=P^TP=I$.</p>
|
1,731,382 | <p>Notice that the parabola, defined by certain properties, is also the trajectory of a cannon ball. Does the same sort of thing hold for the catenary? That is, is the catenary, defined by certain properties, also the trajectory of something?</p>
| Oscar Lanzi | 248,217 | <p>A freely suspended chain or string forms a <a href="https://en.wikipedia.org/wiki/Catenary" rel="nofollow">catenary</a>.</p>
|
123,202 | <blockquote>
<p>Let $X,Y$ be vectors in $\mathbb{C}^n$, and assume that $X\ne0$. Prove
that there is a symmetric matrix $B$ such that $BX=Y$.</p>
</blockquote>
<p>This is an exercise from a chapter about bilinear forms. So the intended solution should be somehow related to it.</p>
<p>Pre-multiplying both sides by... | Phira | 9,325 | <p>I propose that you choose a basis containing $X$ and think about what the equation tells you about $B$ in that basis. </p>
<p>You can very easily find a symmetric $B$ in that basis. </p>
<p>Now, you have to just think about what kind of basis change does not destroy symmetry and choose your basis accordingly.</p>
... |
1,279,564 | <p>I try to be rational and keep my questions as impersonal as I can in order to comply to the community guidelines. But this one is making me <strong>mad</strong>. Here it goes.
Consider the uniform distribution on $[0, \theta]$. The likelihood function, using a random sample of size $n$ is
$\frac{1}{\theta^{n}}$.<b... | Antoni Parellada | 152,225 | <p>$\theta$ is the parameter to estimate, which corresponds to the upper bound of the $U(0,\theta)$. The observed samples are $x_1=2,\,x_2=4$, and $x_3=8$. The likelihood function to maximize is $\mathcal{L}(\theta|X) = \frac{1}{\theta^n}$ with $X$ corresponding to the observed values (a vector, really), $\theta$ the u... |
370,599 | <p>If A is an invertible $nxn$ matrix prove that:$ adj(adjA)=(A)(detA)^{n-2}$
I have done this but it somewhere went wrong:
$ adj(adjA)=adj(A^{-1} detA)=(A^{-1}detA)^{-1} det(A^{-1}detA)=AdetA det(A^{-1}detA)= Adet(AA^{-1}detA)=A (detA)^n $ </p>
| anon | 11,763 | <p>Let $X$ and $Y$ be subsets of the naturals. For any given $y\in Y$, the number of pairs $a,b\in X$ such that $a+b=y$ is at most equal to the number of $a\in X$ such that $a\le y$. Thus the number of pairs of primes adding to $m$ grows asymptotically no more than $\frac{\log m}{m}$ by the prime number theorem; at any... |
31,414 | <p>I'm experimenting with different algorithms that approximate pi via iteration and comparing the result to pi. I want to both visualise and perhaps know the function (if any) that describes the increasing trend in accuracy as the number of iterations rises. </p>
<p>For example, 1 iteration might give me 3.0, 10 ite... | Arnoud Buzing | 105 | <p>There are many examples on the Wolfram Demonstrations web site on this topic:</p>
<p><a href="http://demonstrations.wolfram.com/search.html?query=pi%20approximations">http://demonstrations.wolfram.com/search.html?query=pi%20approximations</a></p>
<p>These examples come with the code that generated them and allow y... |
1,548,667 | <p>Consider the following steady state problem</p>
<p>$$\Delta T = 0,\,\,\,\, (x,y) \in \Omega, \space \space 0 \leq x \leq 4 ,\space \space \space\space 0 \leq y \leq 2 $$</p>
<p>$$ T(0,y) = 300, \space \space T(4,y) = 600$$</p>
<p>$$ \frac{\partial T}{\partial y}(x,0) = 0, \space \space \frac{\partial T}{\partial ... | Hosein Rahnama | 267,844 | <p><strong>Hints</strong></p>
<p>1) Start from here </p>
<p>$$Y^{''} - \lambda Y = 0\\
Y^{'}(0)=Y^{'}(4)=0$$</p>
<p>find the eigenvalues $\lambda_i$ and the eigenfunctions $Y_i(y)$.</p>
<p>2) Find the $X_i(x)$ </p>
<p>$$X^{''} + \lambda X = 0$$</p>
<p>3) Form the infinite sum</p>
<p>$$T(x,y) = \sum\limits_{i =... |
1,341,385 | <p>I want to be a mathematician or computer scientist. I'm going to be a junior in high school, and I skipped precalc/trig to go straight to AP Calc since I've studied a lot of analysis and stuff on my own. My dad wants me to memorize about 30 trig identities (though some of them are very similar) since I'm missing tri... | David K | 139,123 | <p>In favor of rote memorization, unless all your exams are open-notes
you will find that it is very awkward when you have $n$ minutes left to
complete $m$ problems and you are busy re-discovering each trig identity
that you need to use as you go along.
Working out the identity to use in one part of the problem can eas... |
1,531,646 | <p>Find the following limit</p>
<p>$$
\lim_{x\to0}\left(\frac{1+x2^x}{1+x3^x}\right)^\frac1{x^2}
$$</p>
<p>I have used natural logarithm to get</p>
<p>$$
\exp\lim_{x\to0}\frac1{x^2}\ln\left(\frac{1+x2^x}{1+x3^x}\right)
$$</p>
<p>After this, I have tried l'opital's rule but I was unable to get it to a simplified for... | marty cohen | 13,079 | <p>$\lim_{x\to0} f(x)\\
\text{where } f(x)
=\left(\frac{1+xa^x}{1+xb^x}\right)^\frac1{x^2}
$</p>
<p>Let
$g(x, a)
=x a^x
$.
For small $x$,
$g(x, a)
=xe^{x \ln a}
\approx x(1+x \ln a + x^2 \ln^2a/2 + O(x^3))
= x(1+x \ln a + O(x^2))
$
so</p>
<p>$\begin{array}\\
\frac{1+xa^x}{1+xb^x}
&\approx \frac{1+x(1+x \ln a + O(... |
2,895,382 | <p>Let $J_n=\{1,\dots,n\}$. How do I show that the set of all functions $J_n\to \mathbb N$ is countable? Any function is given by specifying the images of $1,\dots,n$. There are $|\mathbb N|$ options for the image of each $i=1,
\dots, n$. So intuitively, the set of such functions is the union of $n$ copies of $\mathbb ... | mvw | 86,776 | <p>Each function is given by a $n$-tuple $(f_1, \dotsc, f_n) \in \mathbb{N}^n$ of natural numbers, where $f_i = f(i)$.</p>
<p>You can use the <a href="https://en.wikipedia.org/wiki/Pairing_function" rel="nofollow noreferrer">Cantor pairing function</a>,
$$
\pi(x, y) := y + \sum_{i=0}^{x+y} i = y+\frac{1}{2} (x + y) (... |
2,895,382 | <p>Let $J_n=\{1,\dots,n\}$. How do I show that the set of all functions $J_n\to \mathbb N$ is countable? Any function is given by specifying the images of $1,\dots,n$. There are $|\mathbb N|$ options for the image of each $i=1,
\dots, n$. So intuitively, the set of such functions is the union of $n$ copies of $\mathbb ... | qualcuno | 362,866 | <p>Let $\mathcal{J_k} = \{f : J_k \to \mathbb{N} \}$ be the set of functions from $J_k$ to $\mathbb{N}$. Now, the mapping </p>
<p>$$
\begin{align}
& \mathcal{J_k} \ \longrightarrow \ \mathbb{N}^k \\
& \quad f \mapsto (f(1), \dots, f(k) )
\end{align}
$$ </p>
<p>is bijective, so it suffices to show that $\ma... |
2,660,316 | <p>$$\frac{\mathrm{d}}{\mathrm{d}y}\left(\frac 2{\sqrt{2\pi}}\int_0^{\sqrt y} \exp\left(-{\frac{x^2}{2}}\right) \,\mathrm{d}x\right).$$</p>
<p>I try to integrate first and then do the differentiation but it's not easy. I want to know other way to do it. Thank you.</p>
| Community | -1 | <blockquote>
<p><strong>Theorem (<a href="https://en.wikipedia.org/w/index.php?title=Legendre%27s_formula&oldid=813253390" rel="nofollow noreferrer">Legendre's formula</a>):</strong> For any prime $p$ and natural number $n$, the $p$-adic valuation of $n!$ is given by
$$ v_p(n!) = \sum_{i=1}^{\infty} \left\lflo... |
15,033 | <p>I have noticed in <a href="https://meta.mathoverflow.net/questions/833/who-are-the-mathoverflow-moderators">this post</a> that at MO they have e-mail address <code>moderators@mathoverflow.net</code>, which can be used to contact moderators.</p>
<p>Is there a similar address for moderators of this site? If not, woul... | Willie Wong | 1,543 | <p>There is one significant difference between MathOverflow and us which makes it much, much more useful for them to have a moderators e-mail account. And that is the fact that the MathOverflow brand is <em>not owned by StackExchange</em> and that there is a <a href="https://meta.mathoverflow.net/questions/969/who-owns... |
600,097 | <p>I am stuck on the following problem from an exercise in my analysis book: </p>
<blockquote>
<p>Show that $$\int_0^4 x \mathrm d(x-[x])=-2$$ where $[x]$ is the greatest integer not exceeding $x$. </p>
</blockquote>
<p>I think I have to partition the interval $[0,4]$ into some suitable subintervals and here I see ... | Ross Millikan | 1,827 | <p>You can define $u=x-[x]$, which will make the $d(x-[x])$ much more friendly. Then $x=u+[x]$ and it seems natural to split the integral into $[0,1],[1,2],[2,3],[3,4]$ On each interval you can see what to do with the $[x]$. That said, I am not getting $\frac 32$ for an answer.</p>
|
600,097 | <p>I am stuck on the following problem from an exercise in my analysis book: </p>
<blockquote>
<p>Show that $$\int_0^4 x \mathrm d(x-[x])=-2$$ where $[x]$ is the greatest integer not exceeding $x$. </p>
</blockquote>
<p>I think I have to partition the interval $[0,4]$ into some suitable subintervals and here I see ... | copper.hat | 27,978 | <p>If we let $f(x) = x -\lfloor x \rfloor$, then the Lebesgue Stieltjes measure corresponding to $f$ can be written as $\mu_f = m -\sum_n \delta_n$, where $m$ is the Lebesgue measure and $\delta_n$ is the Dirac measure concentrated at $n$.</p>
<p>Then $\int_0^4 x d \mu_f = \int_0^4 xdx - \int_0^4 x d(\sum_n \delta_n) ... |
4,007,987 | <p>So define a polynomial <span class="math-container">$P(x) = 4x^3 + 4x - 5 = 0$</span>, whose roots are <span class="math-container">$a, b $</span> and <span class="math-container">$c$</span>. Evaluate the value of <span class="math-container">$(b+c-3a)(a+b-3c)(c+a-3b)$</span></p>
<p>Now tried this in two ways (both... | Z Ahmed | 671,540 | <p>We have <span class="math-container">$a+b+c=0, ab+bc+ca=1, abc=5/4$</span>
Le <span class="math-container">$y=a+b-3c=a+b+c-4c=-4c \implies y=-4x$</span>
Let us transform <span class="math-container">$x^3+4x-5=0$</span> bt <span class="math-container">$x=y/-4$</span> to get a the <span class="math-container">$y$</spa... |
2,441,359 | <p>It's an example given in my book after monotone convergence theorem and dominated convergence theorem (without explanation) : </p>
<p>Find an equivalent of $$\int_0^{\pi/2}\dfrac {dx} {\sqrt{\sin^2(x)+\epsilon \cos^2(x)}}$$</p>
<p>when $\epsilon\to 0^{+}$.</p>
<p>Inspired of the theorems, I naturally think of the... | Rob Arthan | 23,171 | <p>For a somewhat different example, let $P$ and $Q$ be propositional variables. Then intuitionistic propositional logic does not prove $(P \land \lnot\lnot Q) \to (\lnot\lnot P \land Q)$ (and hence, by symmetry, it does not prove $(Q \land \lnot\lnot P) \to (\lnot\lnot Q \land P)$). To see this note that if intuitioni... |
108,200 | <p>If you are to calculate the hypotenuse of a triangle, the formula is:</p>
<p>$h = \sqrt{x^2 + y^2}$</p>
<p>If you don't have any units for the numbers, replacing x and y is pretty straightforward:
$h = \sqrt{4^2 + 6^2}$</p>
<p>But what if the numbers are in meters?<br />
$h = \sqrt{4^2m + 6^2m}$ <em>(wrong, would... | Robin Stammer | 14,037 | <p>You just have to sort this a little bit. Let's assume we're talking about an closed Riemannian manifold $(M,g)$ with its Laplace-Beltrami-Operator $\Delta_g$. Then you have the heat kernel as fundamental solution of the heat equation:</p>
<p>$$ \mathcal{K} \in C^{\infty}(M \times M \times \mathbb{R}^+)$$
Note that ... |
3,739,911 | <p><a href="https://i.stack.imgur.com/aSEt6.png" rel="nofollow noreferrer">question</a></p>
<p><a href="https://i.stack.imgur.com/G6EHM.png" rel="nofollow noreferrer">options and answers</a></p>
<p>The interval in which the function <span class="math-container">$f(x)=\sin(e^x)+\cos(e^x)$</span> is increasing is/are?</p... | zkutch | 775,801 | <p><span class="math-container">$$f^{'}(x)=e^x(\sin e^x - \cos e^x)>0$$</span> gives (first letter in alphabet)</p>
|
3,347,391 | <blockquote>
<p>Find the maximum and minimum values of <span class="math-container">$x^2 + y^2 + z^2$</span> subject to the equality constraints <span class="math-container">$x + y + z = 1$</span> and <span class="math-container">$x y z + 1 = 0$</span></p>
</blockquote>
<p>My try:</p>
<p>Let <span class="math-conta... | Dr. Sonnhard Graubner | 175,066 | <p>Hint: With <span class="math-container">$$z=1-x-y$$</span> we get
<span class="math-container">$$x^2+y^2+(1-x-y)^2$$</span> and the equation <span class="math-container">$$xy-x^2y-xy^2+1=0$$</span>
Now you can eliminate <span class="math-container">$x$</span> or <span class="math-container">$y$</span>, and you will ... |
226,346 | <p>I have the three dimensional Laplacian <span class="math-container">$\nabla^2 T(x,y,z)=0$</span> representing temperature distribution in a cuboid shaped wall which is exposed to two fluids flowing perpendicular to each other on either of the <span class="math-container">$z$</span> faces i.e. at <span class="math-co... | Bill Watts | 53,121 | <p>This is more of an extended comment than an answer, but it occurred to me that your solution is incomplete. You have a double <span class="math-container">$Cos$</span> series in <span class="math-container">$m$</span> and <span class="math-container">$n$</span>, and unlike <span class="math-container">$Sin$</span> ... |
1,595,658 | <p>$$ \text { Given the function: }f:\mathcal{N^+} \to \mathcal{N^+} where f \left(k\right) = \sum_{i=0}^k \,4^i. $$ </p>
<p>Examining the prime factorizations of f(k) for k= 1...48, many factors appear in a regular pattern. </p>
<p>QUESTION: </p>
<ol>
<li>Is there a proof that these patterns continue for larger ... | Slade | 33,433 | <p>We have $f(k) = \frac{1}{3} (4^{k+1}-1)$.</p>
<p>Let $p\neq 2,3$ be prime. By Fermat's Little Theorem, if $p \mid f(k)$, then $p\mid f(k+p-1)$. So there should be lots of patterns among divisors of the $f(k)$.</p>
<p>For example, we have $5\mid f(1+4k)$ for $k=0,1,2,\ldots$</p>
|
1,595,658 | <p>$$ \text { Given the function: }f:\mathcal{N^+} \to \mathcal{N^+} where f \left(k\right) = \sum_{i=0}^k \,4^i. $$ </p>
<p>Examining the prime factorizations of f(k) for k= 1...48, many factors appear in a regular pattern. </p>
<p>QUESTION: </p>
<ol>
<li>Is there a proof that these patterns continue for larger ... | Robert Israel | 8,508 | <p>$$f(k) = \sum_{i=0}^k 4^i = \dfrac{4^{k+1}-1}{3}$$
If $d$ is coprime to $2$ and $3$, then $d$ divides $f(k)$ if and only if
$4^{k+1} \equiv 1 \mod d$, i.e. iff $k+1$ is a multiple of the order of $4$ in the multiplicative group $U_d$ of units in $\mathbb Z/d\mathbb Z$.
For example, the order of $4$ in $U_7$ is $3$ ... |
2,406,107 | <p><a href="https://i.stack.imgur.com/ccIr4.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/ccIr4.png" alt="enter image description here"></a></p>
<p><a href="https://i.stack.imgur.com/zCPUz.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/zCPUz.png" alt="enter image description he... | Berci | 41,488 | <p>There is an unnamed set $X$ here, and $R,S,T$ are all binary relations on this set, i.e. $R,S,T\subseteq X\times X$.</p>
<p>The notation $\ x\,R\,y\ $ for a relation $R$ with this notation means nothing else but $(x,y)\in R$.</p>
|
2,306,570 | <p>I'm new to this topic and trying to solve system of equations over the field $Z_{3}$:
$$\begin{array}{rcr} x+2z & = & 1 \\ y+2z & = & 2 \\ 2x+z & = & 1 \end{array}$$</p>
<p>I solved the system but I have roots:
$$x=1/3,
y=4/3, z=1/3$$ and it's probably not right. Can you help with this one?<... | Ofek Gillon | 230,501 | <p>$$\begin{pmatrix} -4& 3\\ 7& -5 \end{pmatrix} X = \begin{pmatrix} 2&2\\-2&2 \end{pmatrix} $$
The inverse matrix of $\begin{pmatrix} a&b\\c&d \end{pmatrix}$ is $\frac{1}{ad-bc} \begin{pmatrix} d& -b\\ -c& a \end{pmatrix}$ meaning the inverse of $\begin{pmatrix} -4& 3\\ 7& -5 \e... |
440,844 | <p>Suppose we have a linear map $A \colon V \to V$ on a finite- dimensional vector space, and $W \leq V$ it's invariant subspace. Then we have obviously $\operatorname{Ker} A + W \subseteq A^{-1}(W)$.</p>
<p>Is it then necessary $\operatorname{Ker} A + W = A^{-1}(W)$ ?</p>
<p>I can prove it in case $A$ is a projector... | Brian Rushton | 51,970 | <p>What would an inverse look like? For every element $y$ of $Y$, it would send $y$ to an element of $X$ in the preimage of $Y$. You can just make up such a map by choosing $g(y)$ to be any point in $f^{-1}(y).$ </p>
|
53,698 | <p>A point moves from point A to point B. Both points are known, and so is the distance between them.</p>
<p>It starts with a known speed of V<sub>A</sub>, accelerate (with known constant acceleration a) reaching V<sub>X</sub> (unknown), then start decelerating (with known constant acceleration -a) until it reach the ... | Henry | 6,460 | <p><strong>Hint:</strong> You can calculate the time to reach a speed of $V_x$ and a further time to reach $V_B$, so you can find the total distance traveled as a function of $V_x$. </p>
<p>Set this distance equal to $|B-A|$ and solve for $V_x$.</p>
|
53,698 | <p>A point moves from point A to point B. Both points are known, and so is the distance between them.</p>
<p>It starts with a known speed of V<sub>A</sub>, accelerate (with known constant acceleration a) reaching V<sub>X</sub> (unknown), then start decelerating (with known constant acceleration -a) until it reach the ... | André Nicolas | 6,312 | <p>We write down some equations, in a semi-mechanical way, and then solve them.<br>
Let $D$ be the (known) total distance travelled. </p>
<p>It is natural to introduce some additional variables. The intuition probably works best if we use time. So let $s$ be the length of time that we accelerated, and $t$ the lengt... |
3,911,221 | <p>I am working on a probability exercice and I am trying to calculate E(Y) which comes down to this expression :</p>
<p><span class="math-container">$$ E(Y) = \int_{-∞}^{+∞} y\frac{e^{-y}}{(1+e^{-y})^{2}} \, \mathrm{d}y $$</span></p>
<p>I tried to use integrals by part but it diverges and I can't find a good change of... | PierreCarre | 639,238 | <p>This density is an even function, and the first moment integrand is an odd function. The expected value is then zero.</p>
<p>If we denote <span class="math-container">$f(y)=\dfrac{e^{-y}}{(1+e^{-y})^2}$</span>, you can see that
<span class="math-container">$$
f(-y)=\frac{e^y}{(1+e^y)^2} = \frac{e^{2y} \cdot e^{-y}}{... |
1,782,558 | <p><strong>Problem</strong></p>
<p>I have two differential equations</p>
<p>$ \frac{dx}{dt} + \frac{dy}{dt} + x + y = 0$</p>
<p>$ 2 \frac{dx}{dt} + \frac{dy}{dt} + x = 0 $</p>
<p>initial conditions: $y(0) = 1$ and $x(0) = 0$</p>
<p><strong>Attempt</strong></p>
<p>I've solved the system via the Matrix method of s... | jdods | 212,426 | <p>As noted in the other solution, the matrix method is overkill, but since you're interested, here it is.</p>
<p>Your matrix system is $\mathbf{x}'=A\mathbf{x}$ where
$$A=\left(\begin{matrix}
0 & 1 \\
-1 & -2
\end{matrix}\right)$$</p>
<p>The eigenvalue is $-1$ with a multiplicity of $2$. The first eigenvect... |
1,801,946 | <p>I need to find the equation of tangent line passing $(2,3)$ and perpendicular to $3x+4y=8$. Need help in this and also show me how you got the answer. I will be very thankful.</p>
| peter.petrov | 116,591 | <p>The equation is: $-4x + 3y = a$ </p>
<p>By taking $(-4,3)$ (in front of $x$ and $y$) you make the line<br>
perpendicular to the given one which has $(3,4)$. </p>
<p>You determine the $a$ by putting $(2,3)$ in there.<br>
You get: $-4x + 3y = 1$ </p>
|
2,300,049 | <p>these are my toughs:</p>
<p>$$z^2 = 1 + 2i \Longrightarrow (x+yi)(x+yi) = 1 + 2i$$</p>
<p>so: $x^2-y^2 = 1$ and $2xy = 2$</p>
<p>then i got that $x = 1/y$ but i cant continue to find the real- and imaginary part of z anymore. Appriciated any help</p>
| Rodrigo Dias | 375,952 | <p>In general, we have this</p>
<p><strong>Lemma:</strong> If $z=a+ib \in \mathbb{C}$, with $a,b\in\mathbb{R}$, then
$$ w = \sqrt{\frac{|z|+a}{2}} + i\epsilon\sqrt{\frac{|z|-a}{2}},$$
where $\epsilon =\pm 1$ according to $b=\epsilon|b|$, satisfies $w^2 = z$.</p>
<p><strong>Proof:</strong> Let $w=x+iy$ satisfying $w^2... |
4,385,676 | <blockquote>
<p>Let <span class="math-container">$Y_n$</span> be a sequence of non-negative i.i.d random variables with <span class="math-container">$EY_n = 1$</span> and <span class="math-container">$P(Y_n = 1) < 1$</span>. Consider the martingale process formed by <span class="math-container">$X_n = \prod_{k=1}^n ... | Snoop | 915,356 | <p>Since <span class="math-container">$X_n$</span> is a positive martingale, it is also a supermartingale bounded below by <span class="math-container">$0$</span>, therefore <span class="math-container">$X_n\to X_\infty$</span> a.s. by supermartingale convergence. Now consider that <span class="math-container">$P(|Y_n-... |
1,809,017 | <p>Let $U$ be an open set containing $0$ and $f:U \rightarrow C$ a holomorphic function such that $f(0)=0$ and $f^{'}(0)=2$.Prove that there exists an open neighbourhood $0 \in V \subset U $ and a holomorphic injective function $h:V \rightarrow V$ such that $h(f(z))=2h(z)$. Since I don't have any idea where to start, I... | Doug M | 317,162 | <p>Should the minus sign be there. Probably. If it is.</p>
<p>$x^2 + 4y^2 - (2-z)^2 \le 0$ is a double cone, with an elliptical cross section.</p>
<p>$z^2\ge 0$ is a trivial statement.</p>
<p>$z^2\le 2$ is the space between 2 planes.
$-\sqrt{2} \le z \le \sqrt{2}$
and $-\sqrt{2}, \sqrt{2}$ are both below the verte... |
1,345,364 | <p>I am struggling with this question: </p>
<blockquote>
<p>Let $\{a_n\}$ be defined recursively by $a_1=\sqrt2$, $a_{n+1}=\sqrt{2+a_n}$. Find $\lim\limits_{n\to\infty}a_n$. HINT: Let $L=\lim\limits_{n\to\infty}a_n$. Note that $\lim\limits_{n\to\infty}a_{n+1}=\lim\limits_{n\to\infty}a_n$, so $\lim\limits_{n\to\infty... | Chiranjeev_Kumar | 171,345 | <p>$$L=\sqrt{2+L}$$, squaring both side we have,</p>
<p>$$L^2-L-2=0$$</p>
<p>$$(L-2)(L+1)=0$$</p>
<p>which gives $L=2,-1$</p>
<p>since $a_n\gt 0$, for all $n\in \Bbb N$, Hence $L=2$</p>
|
24,704 | <p>It seems that often in using counting arguments to show that a group of a given order cannot be simple, it is shown that the group must have at least <span class="math-container">$n_p(p^n-1)$</span> elements, where <span class="math-container">$n_p$</span> is the number of Sylow <span class="math-container">$p$</spa... | Plop | 2,660 | <p>That's because it is not true in general. Look at $2$-Sylows in $S_5$: they have nontrivial intersection.</p>
|
24,704 | <p>It seems that often in using counting arguments to show that a group of a given order cannot be simple, it is shown that the group must have at least <span class="math-container">$n_p(p^n-1)$</span> elements, where <span class="math-container">$n_p$</span> is the number of Sylow <span class="math-container">$p$</spa... | Derek Holt | 2,820 | <p>In some situations, to prove that groups of order $n$ cannot be simple, you can use the counting argument if all Sylow subgroups have trivial intersection, and a different argument otherwise.</p>
<p>For example let $G$ be a simple group of order $n=144 = 16 \times 9$. The number $n_3$ of Sylow 3-subgroups is 1, 4 o... |
4,112,771 | <p>This is <strong>Exercise 2.3.4</strong> of Robinson's <em>"A Course in the Theory of Groups (Second Edition)"</em>. My universal algebra is a little rusty, so <a href="https://math.stackexchange.com/q/3590724/104041">this question</a> is not what I'm looking for; besides, I ought to be able to use tools gi... | Sean Clark | 1,306 | <p>I can't tell if I am misunderstanding the question, so my apologies if the following answer is totally missing the point. It seems like the question is essentially: given a set of words <span class="math-container">$W$</span>, show that given <span class="math-container">$G,H\in \mathfrak B(W)$</span>, any subgroup ... |
3,293,383 | <p><span class="math-container">$$ \frac{ln{x}}{(x^3-1)} <\frac{x}{x^3} , \forall x \in[2,\infty) $$</span></p>
<p>This is specifically for an improper integral question, where the left term needs to be proven convergent or divergent for the interval <span class="math-container">$$ [2,\infty) $$</span></p>
| Z Ahmed | 671,540 | <p>Let <span class="math-container">$$f(x)=\ln x-x+\frac{1}{x^2} \Rightarrow f'(x)=\frac{1}{x}-1-\frac{2}{x^3}<0, ~\mbox{for}~x\ge 1. $$</span> So <span class="math-container">$f(x)$</span> is decreasing function on <span class="math-container">$[1,\infty].$</span> This means that
<span class="math-container">$f(x) ... |
2,811,155 | <p>I learned recently that there are mathematical objects that can be proven to exist, but also that can be proven to be impossible to "construct". For example see this answer on MSE:<br>
<a href="https://math.stackexchange.com/questions/2808804/does-the-existence-of-a-mathematical-object-imply-that-it-is-possible-to-c... | Noah Schweber | 28,111 | <p>Arguably, your question cannot be answered in a satisfying way (unless you're a formalist).</p>
<p>Ultimately most mathematicians don't spend too much time thinking about ontology - a sort of "naive Platonism" may be adopted, although when pressed I think we generally retreat from that stance - but I think the "sta... |
458,922 | <p>Recently I've stumbled across this claim:</p>
<blockquote>
<p>Peano axioms can be deduced in ZFC</p>
</blockquote>
<p>I found a lot of info regarding this claim (e.g. what would (one version of) the natural numbers look like within the universe of sets: $0 = \emptyset$, $n + 1 = n \cup \{n\}$), but not what the ... | hmakholm left over Monica | 14,366 | <p>Do you mean the original Peano Axioms (with an unrestricted second-order induction axiom), or Peano Arithmetic (known as PA, with a first-order axiom <em>scheme</em>)? Though it's not much different for the purpose of this question.</p>
<p>Before you can start deriving the PA axioms, of course, you run into the pro... |
3,808,077 | <p>I'm trying to show that <span class="math-container">$P(A\triangle B)=P(A)+P(B)–2P(A\cap B)$</span>. Knowing that <span class="math-container">$A\triangle B=(A\cap B^{c})\cup(A^{c} \cap B)$</span>.</p>
<p>So, what I did was this:</p>
<p><span class="math-container">\begin{equation*}
\begin{aligned}
P(A\triangle B)&a... | CSch of x | 566,601 | <p>Using the fact that: <span class="math-container">$$P(K \cup R) = P(K) + P(R) - P( K \cap R)$$</span></p>
<p>and substituting : <span class="math-container">$K = A \cap B^c$</span> and <span class="math-container">$R = A^c \cap B$</span> we get:</p>
<p><span class="math-container">$$P ( A \triangle B) = P(A \cap B^c... |
289,708 | <p>The <a href="https://en.wikipedia.org/wiki/Catalan_number" rel="noreferrer">Catalan numbers</a> <span class="math-container">$C_n$</span> count both </p>
<ol>
<li>the Dyck paths of length <span class="math-container">$2n$</span>, and </li>
<li>the ways to associate <span class="math-container">$n$</span> repeated a... | Timothy Chow | 3,106 | <p><b>EDIT:</b> I can complete half of the proof, showing that the magma order refines the Dyck order.</p>
<p><hr>
Following Martin Rubey's comment, there is a standard bijection between association orders and Dyck paths that uses <a href="https://en.wikipedia.org/wiki/Reverse_Polish_notation" rel="noreferrer">reverse... |
1,120,816 | <p>$$ f(x) =
\begin{cases}
x^{-1} & \text{for $x<-1$} \\
ax+b & \text{for $-1\le x\le \frac 12$} \\
x^{-1} & \text{for $x>\frac 12$} \\
\end{cases}$$</p>
<p>I don't understand how I am supposed to find the value of the constants. It seems as if there is not enough information to determine that. I ... | Tim Raczkowski | 192,581 | <p>$\lim_{x\to-1^-}f(x)=-1$, So we need $ax+b=-1$ for $x=-1$. Hence $b-a=-1$. On the other hand, $\lim_{x\to1/2^+}f(x)=2$. Hence $\frac12a+b=2$.</p>
|
2,533,834 | <p>For a complex number $z$, I came across a statement that $\ln(e^{z})$ is not always equal to $z$. Why is this true?</p>
<p>Thanks for the help.</p>
| blat | 382,972 | <p>Let $C \in \Bbb R$. Choose $n \ge e^C $. Then we have (note that $\ln$ is increasing)</p>
<p>$$\ln(n) \ge ln(e^C) = C.$$</p>
<p>That means $\ln$ is not bounded and hence it diverges.</p>
|
195,790 | <blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="https://math.stackexchange.com/questions/19796/name-of-this-identity-int-e-alpha-x-cos-beta-x-space-dx-frace-al">Name of this identity? $\int e^{\alpha x}\cos(\beta x) \space dx = \frac{e^{\alpha x} (\alpha \cos(\beta x)+\beta \sin(\beta x))}{\alpha... | Cameron Buie | 28,900 | <p>The answers above are good. So far, in each of them, and in your question itself, the difference quotient is taken to be $$\frac{f(x+h)-f(x)}h,\quad h\neq 0.\tag{1}$$ (The $h\neq 0$ part is important, since the expression is meaningless if $h=0$.) Now, there's no real problem with using this--and in fact, it is equi... |
4,099,804 | <p>I need to characterize every finitely generated abelian group G that has the following property:
<span class="math-container">$$\frac{G}{S} \text{ is cyclic for every } \lbrace0\rbrace \lneq S\leq G$$</span>
Given the problems before this one, I believe I am supposed to use the structure theorem figure out the under... | Community | -1 | <p>Let <span class="math-container">$G=\langle x_1,\dots, x_n\rangle $</span>. Then <span class="math-container">$G/\langle x_1\rangle\cong \langle x_2,\dots,x_n\rangle $</span> is cyclic. It follows that <span class="math-container">$G $</span> is generated by two elements.</p>
<p>More can be said: it's easy to se... |
608,909 | <p>Is it possible to solve analytically the following equation?
$$\left(x+\frac{1}{x}\right)^{\frac{1}{x}}=A$$
with $A\gt 1$? I tried to transform it in the following:
$\frac{1}{x}\ln\left(x+\frac{1}{x}\right)=B$ with $B=\ln(A)$, but it seems to be still unsolvable. Is there some trick to solve it? Thanks.</p>
| Stefan Hamcke | 41,672 | <p><span class="math-container">$\require{AMScd}$</span>
There is a retraction of <span class="math-container">$D^n\times I\twoheadrightarrow D^n×\{0\}\cup S^{n-1}×I$</span> defined via
<span class="math-container">$$r(x,t)=\begin{cases}
\left(\frac{2x}{2-t},\ 0\right) &\text{, if }t\le2(1-||x||) \\
\le... |
4,187,498 | <p>I am studying the proof of the Prime Number Theorem and I want to show that the function <span class="math-container">$\frac{\zeta'(s)}{\zeta(s)}$</span> has a simple pole at <span class="math-container">$s=1$</span>.</p>
<p>I think that if I can find the Laurent series expansion of <span class="math-container">$\ze... | Joshua Stucky | 749,752 | <p>If all one cares about is knowing that there is a simple pole at <span class="math-container">$s=1$</span> (and perhaps what its residue is), this can actually be done quite quickly using some standard complex-analytic results. For a reference, see <a href="https://terrytao.wordpress.com/2014/12/05/245a-supplement-2... |
523,932 | <p>I've got a system of equations which is:<br></p>
<p>$\begin{cases} x=2y+1\\xy=10\end{cases}$</p>
<p>I have gone into this: $x=\dfrac {10}y$.
<br>
How can I find the $x$ and $y$?</p>
| Shravan40 | 40,230 | <p><strong>Hint :</strong> </p>
<p>This kind of equation can be solved by substituting the value of $ x $ or $ y $ in the first equation.And the above equation will become quadratic, solve for it</p>
<p>$ x = 2y +1 \dots (1)$</p>
<p>$xy = 10 $
$ \implies x = \frac{10}{y}$</p>
<p>Put the value of x in equation (1)</... |
1,242,317 | <p>If I have a unitary square matrix $U$ ie. $U^{\dagger}U=I$ ( $^\dagger$ stands for complex conjugate and transpose ), then for what cases is
$U^{T}$ also unitary. One simple case I can think of is $U=U^{T}$ ( all entries of $U$ are real, where $^T$ stands for transpose ). Are there any other cases ?</p>
| Ben Grossmann | 81,360 | <p>It's going to be true in <em>all</em> cases.</p>
<p>In particular, if $U$ is unitary, then
$$
(U^T)^\dagger U^T = [UU^\dagger]^T = I^T = I
$$</p>
|
1,705,453 | <p>I have a list of prime numbers which can be expressed in the form of $3x+1$. One such prime of form $3x+1$ satisfies the expression: $a^2+b^2-ab$.</p>
<p>Now I am having list of prime numbers of form $3x+1$ (i.e., $7,19 \ldots$). But I am unable to find the $a$ and $b$ which satisfy the above expression.</p>
<p>Th... | Lutz Lehmann | 115,115 | <p>For $f(x)=\sin(x)$ you get the nice formula
$$
f^{(k)}(x)=\sin\Bigl(x+k\frac\pi2\Bigr)
$$
Thus $f^{(4)}(x)=\sin(x)$.</p>
<p>In the error formula, the argument of the 4th derivative is some not known point inside the interval $(x-h,x+h)$. As a first estimate one can take the value at $x$ if $h$ is small.</p>
<p>Apa... |
3,403,272 | <p><p> I'm currently taking abstract algebra and I'm very lost.</p>
<blockquote>
<p>Let <span class="math-container">$G = (\Bbb Z/18\Bbb Z, +)$</span> be a cyclic group of order <span class="math-container">$18$</span>.</p>
<p>(1) Find a subgroup <span class="math-container">$H$</span> of <span class="math-cont... | Shaun | 104,041 | <p>Here <span class="math-container">$\lvert G/H\rvert=\lvert G\rvert/\lvert H\rvert=18/3=6$</span>.</p>
<p>Also, <span class="math-container">$G\cong\langle a\mid a^{18}\rangle,$</span> so
<span class="math-container">$$\begin{align}
G/H&\cong\langle b\mid b^{18/3}\rangle \\
&\cong \langle b\mid b^6\rangle ... |
2,722,609 | <p>In a past thread it was mentioned that $x \in A$ is a predicate. I know $\exists x$ and $\forall x$ are quantifiers but are they also predicates themselves? What about when combined with "in" itself (or whatever this operator is called)? e.g. $\exists x \in A$ or $\forall x \in A$</p>
| goblin GONE | 42,339 | <p>Yes and no.</p>
<p>Let $X$ denote a set. A predicate on $X$ is basically a subset of $X$. A quantifier on $X$ is basically a collection of subsets of $X$. In particular, we can think of the existential quantifier as the collection of all non-empty subsets of $X$, and we can think of the universal quantifier as the ... |
1,618,699 | <p>Let A be a non-empty subset of $\mathbb{R}$ that is bounded above and put $s=\sup A$<br>
Show that if $s\notin A$ the the set $A\cap (s-ε,s)$ is infinite for any $ε>0$ </p>
<p>This has to be solved using contradiction, by supposing $A\cap (s-ε,s)$ is an finite set. But I am not sure how to proceed after this.</... | Christian Blatter | 1,303 | <p>Formulas like
$$1+2+3+4+\ldots=-{1\over12}\tag{1}$$
are peddled even in the $21^{\rm st}$ century to impress simple minded people, and have per se no mathematical content whatsoever. It is true that one could replace the sum $\sum_{k=1}^\infty k$ on the left hand side of $(1)$ by a sum of the form
$$\sum_{k=1}^\inft... |
4,146,629 | <p>I'm reading D. E. Knuth's book "Surreal Numbers". And I'm completely stuck in chap. 6 (The Third Day) because there is a proof I don't understand. Alice says</p>
<blockquote>
<p>Suppose at the end of <span class="math-container">$n$</span> days, the numbers are <span class="math-container">$$x_1<x_2<... | mjqxxxx | 5,546 | <p>Conway's second rule says that</p>
<blockquote>
<p>One number is less than or equal to another number if and only if no member of the first number's left set is greater than or equal to the second number, and [other stuff].</p>
</blockquote>
<p>So <span class="math-container">$x_i=x_i$</span> implies that no member ... |
50,113 | <p>What are some good books on field and Galois theory?</p>
| Chris Godsil | 1,266 | <p>David Cox "Galois Theory" Wiley 2004 is my current favorite. Lots of interesting material and very nicely written.</p>
|
50,113 | <p>What are some good books on field and Galois theory?</p>
| Ian Agol | 1,345 | <p>Chapter 1.5 The Absolute Galois Group of a Finite Field
Might be useful from <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2445111" rel="nofollow noreferrer">Field Arithmetic</a> by Fried and Jarden.</p>
|
2,946,379 | <p>The question posed is the following: Let <span class="math-container">$X$</span> be a Banach Space and let <span class="math-container">$T:X\to X$</span> be a Lipschitz-Continuous map. Show that, for <span class="math-container">$\mu$</span> sufficiently large, the equation
<span class="math-container">\begin{equat... | Bernard | 202,857 | <p>If you make the substitution <span class="math-container">$\;t=\mathrm e^x\iff x=\ln t$</span>, so that <span class="math-container">$\;\mathrm dx=\dfrac{\mathrm d t}t$</span>, we obtain
<span class="math-container">$$\int_{1}^{\infty}\frac{\mathrm e^{x}+\mathrm e^{3x}}{\mathrm e^{x}-\mathrm e^{5x}}\,\mathrm dx=\int... |
4,188,020 | <p>I know that there exists a connection on a principal bundle and via parallel transport it is possible to define a a covariant derivative on the associated bundle.</p>
<p>However, can we also define a covariant derivative on the principal bundle. I.e. something that can differentiate a section along a vector field? O... | Mozibur Ullah | 26,254 | <blockquote>
<p>I know that there exists a connection on a principal bundle and via parallel transport it is possible to define a a covariant derivative on the associated bundle.</p>
</blockquote>
<p>Ehresmann connections are the geometric version of connections. They are generally available on all fibre bundles and no... |
359,742 | <p>I have a mathematical problem that leads me to a particular necessity. I need to calculate the convolution of a function for itself for a certain amount of times. </p>
<p>So consider a generic function $f : \mathbb{R} \mapsto \mathbb{R}$ and consider these hypothesis:</p>
<ul>
<li>$f$ is continuos in $\mathbb{R}$.... | SSF | 131,617 | <p>Well for your trials with Gaussians you're only rescaling them. My guess would be that it will tend to an infinitely stretched Gaussian regardless of the initial function. It least it is equivalent to having a sum of $n \to \infty$ random variables with the same probability density function $f$, which should tend to... |
37,013 | <p><strong>Question:</strong> What are some interesting or useful applications of the Hahn-Banach theorem(s)?</p>
<p><strong>Motivation:</strong> Most of the time, I dislike most of Analysis. During a final examination, a question sparked my interest in the Hahn-Banach theorem(s). One of my favorite things to do is... | ncmathsadist | 4,154 | <p>One I know of is the hyperplane separation theorem for convex sets. Another is the existence of Banach generalized limits.</p>
|
37,013 | <p><strong>Question:</strong> What are some interesting or useful applications of the Hahn-Banach theorem(s)?</p>
<p><strong>Motivation:</strong> Most of the time, I dislike most of Analysis. During a final examination, a question sparked my interest in the Hahn-Banach theorem(s). One of my favorite things to do is... | user248793 | 248,793 | <p>What i know about Hann Banach Theorem is the existence of enough functionals of a dual space on a given space, and these functionals sepearate points of the space. The sufficiency of these functional guaranteed enough maps in a dual space to work with. </p>
|
2,080,716 | <p>I have the quadratic form
$$Q(x)=x_1^2+2x_1x_4+x_2^2 +2x_2x_3+2x_3^2+2x_3x_4+2x_4^2$$</p>
<p>I want to diagonalize the matrix of Q. I know I need to find the matrix of the associated bilinear form but I am unsure on how to do this.</p>
| deshu | 403,660 | <p>If you can use programmatic approach to answer your questions, which I personally find way easier than hand-written-math, you can check out that simple <strong><a href="https://jsfiddle.net/deshu/kzx4rxa3/6/" rel="nofollow noreferrer">JavaScript fiddle</a></strong> I wrote for you.</p>
<p><strong>In the right botto... |
4,359,372 | <p>My question is: Does there exist <span class="math-container">$x_n$</span> (<span class="math-container">$n\geq 0$</span>) such that <span class="math-container">$x_n$</span> is a bounded and divergent sequence with <span class="math-container">$$x_{n+m}\leq (x_n+x_m)/2$$</span> for all <span class="math-container">... | bjcolby15 | 122,251 | <p>Using base two (binary) as the weight, i.e. <span class="math-container">$1 = 2^0, 2 = 2^1, 4 = 2^2, 8 = 2^3, 16 = 2^4, 32 = 2^5$</span>, we can measure any weights we want. With <span class="math-container">$5$</span> weights, you can only weigh up to <span class="math-container">$31$</span>g, but with <span class... |
1,545,929 | <p>Assume $0<\alpha\leq 1$ and $x>0$. Does the following inequality hold?
$$(1-e^{-x})^{\alpha}\leq (1-\alpha e^{-x})$$
I know that the reverse inequality holds if $\alpha\ge 1$.</p>
| Justpassingby | 293,332 | <p>The concept that you are thinking of is <em>sequence continuity</em> (which is equivalent to continuity in metric spaces such as R^n but that requires proof).</p>
<p>If you check the condition "for every given open set containing the image there exists an open set (containing the original point) that is mapped into... |
308,520 | <p>The DE is $y' = -y + ty^{\frac{1}{2}}$. </p>
<p>$2 \le t \le 3$</p>
<p>$y(2) = 2$</p>
<p>I tried to see if it was in the <a href="http://www.sosmath.com/diffeq/first/lineareq/lineareq.html" rel="nofollow">linear form</a>. I got:</p>
<p>$$\frac{dy}{dt} + y = ty^{\frac{1}{2}}$$</p>
<p>The RHS was not a function o... | Mike | 17,976 | <p>If that $y^{\frac12}$ weren't there, you might solve the problem by multiplying by an integrating factor of $e^t$ to yield $(e^ty)'$ on the left side. Try making the substitution</p>
<p>$$z=e^ty$$
$$y^{\frac12}=z^{\frac12}e^{-\frac t2}$$
$$e^ty'+e^ty=(e^ty)'=te^ty^{\frac12}$$
$$z'=te^t(z^{\frac12}e^{-\frac t2})=te... |
490,064 | <p>Solve the Cauchy problem, $\forall t \in \mathbb{R}$,
$$ \begin{cases}
u''(t) + u(t) = |t|\\
u(0)=1, \quad u'(0) = -1
\end{cases} $$</p>
<p>The solution to the homogeneous equation is $A\cos(t) + B \sin(t)$. Empirically, $|t|$ is "more or less" a particular solution, however it is not differentiable in $0$... What ... | Felix Marin | 85,343 | <p>$\displaystyle{\xi = {\rm u}' + {\rm iu}
\Longrightarrow
\xi' = {\rm u}'' + {\rm iu}'
\Longrightarrow
\xi' - {\rm i}\xi = \left\vert t\right\vert\,;
\qquad
{\rm u} = \Im\xi}$</p>
<p>$$
{{\rm d}\left({\rm e}^{-{\rm i}t}\xi\right) \over {\rm d}t}
=
{\rm e}^{-{\rm i}t}\,\left\vert t\right\vert
\Longrightarrow
{\rm e}^... |
490,064 | <p>Solve the Cauchy problem, $\forall t \in \mathbb{R}$,
$$ \begin{cases}
u''(t) + u(t) = |t|\\
u(0)=1, \quad u'(0) = -1
\end{cases} $$</p>
<p>The solution to the homogeneous equation is $A\cos(t) + B \sin(t)$. Empirically, $|t|$ is "more or less" a particular solution, however it is not differentiable in $0$... What ... | Mikasa | 8,581 | <p>If $t\in 0^+\cup\{0\}$ then we have $u''+u=t$. Here, you can use any appropriate methods to get the general solution for this ODE. For example by using undetermined coefficients you get: $$u(t)=C_1\sin t+C_2\cos t+t$$ Remember the part $C_1\sin t+C_2\cos t$ is really the solution of the associated homogenous OE, $u... |
3,536,671 | <p>I have the following mathematical operations to use: Add, Divide, Minimum, Minus, Modulo, Multiply and Round.</p>
<p>With these I need to get a number, run it through a combination of these and return 0 if the number is negative or equal to 0 and the number itself if the number is greater than 0.</p>
<p>Is that po... | MPW | 113,214 | <p>You really want <span class="math-container">$\max\{x,0\}$</span>, which can be realized as <span class="math-container">$\boxed{-\min\{-x,0\}}$</span>.</p>
<p>In your precise language, <span class="math-container">$\operatorname{Minus}(\operatorname{Min}(\operatorname{Minus}(x),0))$</span>.</p>
<p>(I assume that ... |
4,280,426 | <blockquote>
<p>We have a bag with <span class="math-container">$3$</span> black balls and <span class="math-container">$5$</span> white balls. What is the probability of picking out two white balls if at least one of them is white?</p>
</blockquote>
<p>If <span class="math-container">$A$</span> is the event of first b... | Mark | 470,733 | <p>The closure has to contain <span class="math-container">$A$</span>, so there are only two options here: <span class="math-container">$\mathbb{R}$</span> or <span class="math-container">$\mathbb{R}\setminus\{p\}$</span>. Since <span class="math-container">$\mathbb{R}\setminus\{p\}$</span> is not a closed set (its com... |
2,511,095 | <p>Let $p$ be an odd prime. We know that the polynomial $x^{p-1}-1$ splits into linear factors modulo $p$. If $p$ is of the form $4k+1$ then we can write
$$x^{p-1}-1=x^{4k}-1=(x^{2k}+1)(x^{2k}-1).$$
The theorem of Lagrange tells us that any polynomial congruence of degree $n$ mod $p$ has at most $n$ solutions. Hence we... | gt6989b | 16,192 | <p>Here is one approach. Note from the second equation you have
$$
3a^2 = 5/b - 13b^2
$$
which we can substitute into the first to get
$$
18 = a\left(a^2+39b^2\right)
= a\left(\frac{5}{3b} - \frac{13b^2}{3}+39b^2\right)
$$
so
$$
3a^2 = \left(\frac{18\cdot 3}{\frac{5}{3b} - \frac{13b^2}{3}+39b^2} \right)^2
$$
which i... |
3,008,162 | <p>Let <span class="math-container">$A$</span> and <span class="math-container">$B$</span> be well-ordered sets, and suppose <span class="math-container">$f:A\to B$</span> is an
order-reversing function. Prove that the image of <span class="math-container">$f$</span> is finite.</p>
<p>I started by supposing not. Then... | Arthur | 15,500 | <p>Hint: The image of <span class="math-container">$g$</span> is a non-empty subset of a well-ordered set. Therefore it has a minimal element.</p>
|
300,163 | <p>I need to integrate the $z/\bar z$ (where $\bar z$ is the conjugate of $z$) counterclockwise in the upper half ($y>0$) of a donut-shaped ring. The outer circle is $|z|=4$ and the inner circle is $|z|=2$. </p>
<p><strong>My method:</strong></p>
<p>$z/\bar z = e^{2i\theta}$ - which is entire over the complex plan... | Sapph | 61,818 | <p>@rlgordonma Thank you for your help! Just a quick question: I do the same integration as you but I seem to end up integrating rie^3itheta rather than just re^3itheta (as dz=d(re^i3theta)=rie^i3theta dtheta)</p>
<p>Which is right?</p>
|
688,430 | <blockquote>
<p>How can I show the following
$$a^n|b^n \Rightarrow a|b$$ with $a,b$ integers.</p>
</blockquote>
<p>$$a^n|b^n \Rightarrow b^n=m \cdot a^n \Rightarrow b^n=(m\cdot a^{n-1}) \cdot a\qquad(1)$$
How can I continue? Do I maybe have to suppose the opposite and arrive at contradiction?
$$\text{So } a \nmid ... | Arthur | 15,500 | <p>Assuming, for contradiction, that $a \nmid b$, there must be <em>some</em> prime $p$ to some power $m$ such that $p^m | a$, but $p^m \nmid b$. Then $p^{nm} | a^n$, but $p^{nm}\nmid b$ by simple counting of prime factors.</p>
|
1,325,432 | <p>$f(x) = x$ , $f(x+2\pi) = f(x) $ on $ [-\pi , \pi] $ </p>
<p>How do I know that this function is even or odd? My book says odd, but I don't understand how to work this out? </p>
<p>also why does $a_0 = 0$ and $a_n = 0$? </p>
<p>since its an odd function I thought we use the even extension? </p>
<p>i.e $$ a... | Mark Viola | 218,419 | <p>The integral of an odd function over symmetric limits is zero. To see this, we observe</p>
<p>$$\begin{align}
\int_{-a}^af(x)dx&=\int_{-a}^0f(x)dx+\int_{0}^af(x)dx\tag1\\\\
&=-\int_{0}^af(x)dx+\int_{0}^af(x)dx\tag2\\\\
&=0
\end{align}$$</p>
<p>where we used the substitution $x \to -x$ in going from $(... |
2,106,003 | <p>I was just reading about the <a href="https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox">Banach–Tarski paradox</a>, and after trying to wrap my head around it for a while, it occurred to me that it is basically saying that for any set A of infinite size, it is possible to divide it into two sets B and C su... | Mathematician 42 | 155,917 | <p>That's not what the paradox says. It says that you can take the unit ball in $\mathbb{R}^3$, divide it in certain disjoint subsets, then you can rotate and translate these subsets to obtain two unit balls. You need at least $5$ weird subsets if you want to do this 'explicitly'. The weird thing about this constructio... |
2,106,003 | <p>I was just reading about the <a href="https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox">Banach–Tarski paradox</a>, and after trying to wrap my head around it for a while, it occurred to me that it is basically saying that for any set A of infinite size, it is possible to divide it into two sets B and C su... | Timothy | 137,739 | <p>The Banach-Tarski paradox is the theorem in ZFC that a sphere can be partitioned into finitely many subets, then those subsets can be rearranged into 2 copies of the original sphere using only translation and rotation. Actually according to ZFC, it can be partitioned into only 5 parts.</p>
<p>Some people don't thin... |
413,165 | <p>I am a graduate student and I've been thinking about this fun but frustrating problem for some time. Let <span class="math-container">$d = \frac{d}{dx}$</span>, and let <span class="math-container">$f \in C^{\infty}(\mathbb{R})$</span> be such that for every real <span class="math-container">$x$</span>, <span class=... | username | 40,120 | <p>(Edit) A simplified shorter version of this answer is in Fedor Petrov's <a href="https://mathoverflow.net/questions/413165/does-iterating-the-derivative-infinitely-many-times-give-a-smooth-function-whene#comment1059300_413165">comments</a> (as Iosif Pinellis <a href="https://mathoverflow.net/questions/413165/does-it... |
413,165 | <p>I am a graduate student and I've been thinking about this fun but frustrating problem for some time. Let <span class="math-container">$d = \frac{d}{dx}$</span>, and let <span class="math-container">$f \in C^{\infty}(\mathbb{R})$</span> be such that for every real <span class="math-container">$x$</span>, <span class=... | Scot Adams | 103,568 | <p>We consider the simplest case of an elliptic operator, namely:</p>
<p>The second derivative, acting on real-valued functions defined on an interval.</p>
<p>The last theorem, at the very end of</p>
<p><a href="https://www-users.cse.umn.edu/%7Eadams005/PaulCussonQ/ptbddevenderivs.pdf" rel="nofollow noreferrer">https:/... |
830,977 | <p>I'm having some real trouble with lebesgue integration this evening and help is very much appreciated.</p>
<p>I'm trying to show that $f(x) = \dfrac{e^x + e^{-x}}{e^{2x} + e^{-2x}}$ is integrable over $(0,\infty)$.</p>
<p>My first thought was to write the integral as $f(x) = \frac{\cosh(x)}{\cosh(2x)}$ and then no... | Community | -1 | <p>The given function $f$ is continuous on $(0,\infty)$ and has a finite limit at $x=0$ and
$$f(x)\sim_\infty e^{-x}\in L^1(0,\infty)$$
so $f$ is integrable on $(0,\infty)$.</p>
|
2,553,284 | <p>I know that
$$\ln e^2=2$$
But what about this?
$$(\ln e)^2$$
A calculator gave 1. I'm really confused.</p>
| Community | -1 | <p>Consider the equality (assuming the operations are actually <em>defined</em> for <em><code>m</code></em> and <em><code>n</code></em>):</p>
<p>$$ x =\log _nm$$</p>
<p>What this means is that <em><code>x</code></em> is the number to you need to raise <em><code>n</code></em> to the power of, to get <em><code>m</code>... |
244,214 | <p>One major approach to the theory of forcing is to assume that ZFC has a countable <em>transitive</em> model $M \in V$ (where $V$ is the "real" universe). In this approach, one takes a poset $\mathbb{P} \in M$, uses the fact that $M$ is <em>countable</em> to prove that there exists a generic set $G \in V$, then defin... | Bill Mitchell | 109,444 | <p>This is really a comment on Hamkins answer, but I'm not permitted to make comments so I'll write it as an answer.</p>
<p>Using the standard Schoenfield machinery for forcing, there is no need for a well founded model. The theory of forcing is developed entirely inside the model $M$, including the definition of the... |
288,001 | <p>Points A and B are given in Poincare disc model. Construct equilateral triangle ABC.
Any kind of help is welcome.</p>
| DrBaxter | 66,881 | <p>There is indeed a shorter approach. Euclid's Proposition I holds in Hyperbolic Geometry just as well as it holds in Euclidean Geometry.</p>
<p><a href="http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI1.html" rel="nofollow">http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI1.html</a></p>
|
1,284,039 | <p>What function satisfies $f(x)+f(−x)=f(x^2)$?</p>
<p>$f(x)=0$ is obviously a solution to the above functional equation.</p>
<p>We can assume f is continuous or differentiable or similar (if needed).</p>
| Akiva Weinberger | 166,353 | <p>I'm going to put my comment as an answer.</p>
<p>$\ln|1-x|$ seems to work:
$$\ln|1-x|+\ln|1+x|=\ln|1-x^2|$$
Similarly, so does $\ln|1-x^3|$ (or any odd exponent).</p>
<p>Also, any linear combination of these works, as you can check. Thus:
$$\ln(1+x+x^2)$$
works because it's equal to $\ln|1-x^3|-\ln|1-x|$. The exam... |
3,492,435 | <p>I am reading <em><strong>Foundations of Constructive Analysis</strong></em> by Errett Bishop. In the first chapter he describes a particular construction of the real numbers. There is a intermediate definition before his primary introduction of the Real numbers:</p>
<blockquote>
<p>A sequence <span class="math-con... | Mohammad Riazi-Kermani | 514,496 | <p>There is no need for induction.</p>
<p>Integration by parts will do.
<span class="math-container">$$\int _0^1 x^m(1-x)^ndx =$$</span></p>
<p><span class="math-container">$$(1-x)^n \frac {x^{m+1}}{m+1}|_0^1 -\int _0^1\frac {x^{m+1}}{m+1}n(1-x)^{n-1}(-1)dx=$$</span></p>
<p><span class="math-container">$$\frac{n}{... |
803,335 | <p>Note: this is particularly aimed at high-school/entry level college problems </p>
<p>When I'm learning a new topic:</p>
<p>1) I read the theory given in the textbook at the start of each topic</p>
<p>2) proceed to read the solved example problems which the textbook provides (usually 3-5 with full solutions)</p>
... | user21820 | 21,820 | <p>Usually the things that we remember are those things that we both understand completely and have found useful.</p>
<p>Understanding requires an intuitive grasp of why in the first place you want to consider certain mathematical objects, why a theorem should be true, and how intermediate objects were conceived. If y... |
277,594 | <p><a href="https://i.stack.imgur.com/yX9my.gif" rel="noreferrer"><img src="https://i.stack.imgur.com/yX9my.gif" alt="enter image description here" /></a></p>
<pre><code>Manipulate[
ParametricPlot[{Sec[t], Tan[t]}, {t, 0, u}, PlotStyle -> Dashed,
PerformanceGoal -> "Quality", Exclusions -> All,
... | Alexei Boulbitch | 788 | <p>Try the following. Here is your polynomial:</p>
<pre><code>expr = Sum[
Subscript[A, i, j]*Subscript[x, i]*Subscript[x, j], {i, 1, 5}, {j,
1, 5}]
</code></pre>
<p><a href="https://i.stack.imgur.com/PF3TZ.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/PF3TZ.png" alt="enter image description h... |
2,384,538 | <p>I am studying Linear Algebra Done Right, chapter 2 problem 6 states:</p>
<blockquote>
<p>Prove that the real vector space consisting of all continuous real valued functions on the interval $[0,1]$ is infinite dimensional.</p>
</blockquote>
<p><strong>My solution:</strong></p>
<p>Consider the sequence of functio... | José Carlos Santos | 446,262 | <p>Then the polynomial $\displaystyle x^n-\sum_{k=0}^{n-1}a_kx^k$ would have infinitely many roots, but it can have $n$, at most.</p>
<hr />
<p>Another way of dealing with this problem is based upon <em>defining</em> polynomials (in one variable $x$) as expressions of the type $a_0+a_1x+a_2x^2+\cdots+a_nx^n$, where $... |
159,446 | <p>The ordinary Thom isomorphism says $H^{*+n}(E,E_{0}) \simeq H^{*}(X)$, where $E$ is a vector bundle over $X$ and $E_{0}$ is $E$ minus the zero section. Now assume that $S$ is a non vanishing section for the vector bundle $E$. In each fiber $E_{x}$ we remove two points $0_{x}$ and $S(x)$. Then we put $E_{0,1}$for the... | Alex Degtyarev | 44,953 | <p>You get two copies of $H^*(X)$. In general ($k$ pairwise disjoint sections), by excision it's just like disjoint union of $k$ copies of the original bundle, hence $k$ copies of $H^*(X)$. (An extra observation is that, for one section, the result does not depend on its choice, as any section is homotopic to $0$.)</p>... |
2,280,052 | <p>Wolfram Alpha says:
$$i\lim_{x \to \infty} x = i\infty$$</p>
<p>I'm having a bit of trouble understanding what $i\infty$ means. In the long run, it seems that whatever gets multiplied by $\infty$ doesn't really matter. $\infty$ sort of takes over, and the magnitude of whatever is being multiplied is irrelevant. I.e... | Joonas Ilmavirta | 166,535 | <p>Perhaps this broader view can help.
(Perhaps it confuses instead!)
There are several different ways to describe ways to go to infinity.
I would identify these ways with different <a href="https://en.wikipedia.org/wiki/Compactification_(mathematics)" rel="nofollow noreferrer">compactifications</a>.
Compactification i... |
1,317,610 | <p>Let $u = u(t,x)$ satisfy the PDE
$$
\frac{\partial u}{\partial t} = \frac{1}{2}c^2\frac{\partial^2 u}{\partial x^2} + (a + bx)\frac{\partial u}{\partial x} + f u,
$$
where $a,b,c,f \in \mathbb{R}$ are constant.</p>
<p>I'm aware of solution methods for when $c \propto x^2$ (so not constant) and $a = 0$, for which I ... | Joffan | 206,402 | <p>Your calculation has a lot of overcounting, whenever there is more than one vowel present.</p>
<p>If you really want to avoid (or, say, cross-check) the "negative space" method of simply excluding options with no vowels, you could perhaps sum through the possibilities of where the first vowel is:</p>
<ul>
<li>Vowe... |
779,696 | <p>So my problem is:</p>
<p>$$\arcsin (x) = \arccos (5/13)$$ </p>
<p><strong>^ Solve for $x$.</strong></p>
<p>How would I begin this problem? Do I draw a triangle and find the $\sin(x)$ or is there a more algebraic way of doing this? Thanks in advance for any help.</p>
| Community | -1 | <p>I believe that you can without loss of generality check the independence of $A_j$ and $A_{j+1}$, since, if these are independent then $A_j$, $A_k$ are independent for all $j \not= k$.</p>
<p>So, since $P(A_j \cap A_{j+1})$ is the same as $(j+1)$th draw yielding a consecutive number and $(j+2)$th draw again yielding... |
4,114,180 | <p>The Theorem is as follows:</p>
<p>For any numbers x and y, the following statements are true:</p>
<ol>
<li><span class="math-container">$|x|<y$</span> if and only if <span class="math-container">$-y<x<y$</span></li>
<li><span class="math-container">$|x|\leq{y}$</span> if and only if <span class="math-contai... | fleablood | 280,126 | <p>Hmm... well, to take a mallet and force things to fit.</p>
<p><span class="math-container">$2|x|-3 \ge |x-1| \iff$</span></p>
<p><span class="math-container">$-2(|x| - 3) \le x-1 \le 2|x| - 3\iff$</span></p>
<p><span class="math-container">$-2|x|+3 \le x-1 \le 2|x| - 3 \iff$</span></p>
<p>(<span class="math-containe... |
200,920 | <p>I would like to create an array f containing n indices. The label of those indices is stored in a liste of length n, let's call it "list".</p>
<p>So I would like to have something like :</p>
<blockquote>
<p>{f[list[[0]]], f[list[[1]],...}</p>
</blockquote>
<p>The point is to affect the f[liste[[i]]] to some val... | Alex Trounev | 58,388 | <p>We can increase <code>n</code>, reducing the accuracy of calculations, for example</p>
<pre><code>f[m_, p_] :=
Block[{n = m, <span class="math-container">$MinPrecision = p, $</span>MaxPrecision = p},
intVars =
Table[{Subscript[t, i], -\[Infinity], \[Infinity]}, {i, 1, n}];
poly = Sum[
Subscript[t, i... |
2,655,518 | <p>$2ac=bc$
find the ratio ( $K$ )
what is the ratio of their area?
<a href="https://i.stack.imgur.com/9NPRi.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/9NPRi.png" alt="enter image description here"></a>I found out it is $2$ or $1/2$
is it true? </p>
<p>if the question isn't clear, make sure to... | Patrick Stevens | 259,262 | <p>We say that $y$ <strong>is a square root of $x$</strong> if $y^2 = x$.</p>
<p>We define a function $\sqrt{\cdot} : \mathbb{R}^+ \to \mathbb{R}$ ("the square root function") by $$\sqrt{x} := \text{the nonnegative number $y$ such that $y^2 = x$}$$</p>
<p>So you can see that $\sqrt{x}$ is a square root of $x$.</p>
<... |
1,081,447 | <p>I'm talking about a Roulette wheel with $38$ equally probable outcomes. Someone mentioned that he guessed the correct number five times in a row, and said that this was surprising because the probability of this happening was $$\left(\frac{1}{38}\right)^5$$</p>
<p>This is true if you only play the game $5$ times. H... | Empy2 | 81,790 | <p>You have a six-state system.<br>
State 1: Not on a run. Either you haven't started, or the last guess was wrong.<br>
State 2: The last guess was correct.<br>
State 3: The last two guesses were correct.<br>
State 4: The last three guesses were correct.<br>
State 5: The last four guesses were correct.<br>
State 6: Yo... |
832,710 | <p>Does there exist an algebraic structure $(\mathbb{K},+)$ such that equations of the form $x+a=x+b$, $a\neq b$ have solutions for all $a,b\in \mathbb{K}$?</p>
| Ragavendar Nannuri | 427,504 | <p>Let $L$ be the width of the lane. Consider the segment of straight line passing through one corner of the lane and the road, going from the opposite side of the road to the opposite side of the lane. Suppose segment makes an angle $\theta\in (0,\frac{\pi}{2})$ with the road. The lenght of such segment is a function ... |
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