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 |
|---|---|---|---|---|
4,619,164 | <p>So one way to solve this ordinary differential equation is by computing the integral of both sides of <span class="math-container">$$\frac{y'}{y} = 1$$</span>
However, I did it another way and I think I made a mistake, but where is it? What I did is:
<span class="math-container">$$y' = y $$</span>
<span class="math-... | PrincessEev | 597,568 | <p>You're treating <span class="math-container">$y$</span> as a constant when you say</p>
<p><span class="math-container">$$\int y \, dx = xy+C$$</span></p>
<p>In reality, <span class="math-container">$y$</span> is a function of <span class="math-container">$x$</span>. For instance, this claims that</p>
<p><span class=... |
3,546,274 | <p>I am trying out the first probability problem at this <a href="https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-041-probabilistic-systems-analysis-and-applied-probability-fall-2010/assignments/MIT6_041F10_assn04.pdf" rel="nofollow noreferrer">link</a> which looks something like this:</p>
<p... | Yanirmr | 750,776 | <p>The probability of an event A is written as <span class="math-container">$P(A)$</span>, <span class="math-container">$p(A)$</span>, or <span class="math-container">$Pr(A)$</span>.</p>
<p>I'm recommending on this answers for your question: </p>
<ul>
<li><a href="https://stats.stackexchange.com/questions/260711/uppe... |
3,546,274 | <p>I am trying out the first probability problem at this <a href="https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-041-probabilistic-systems-analysis-and-applied-probability-fall-2010/assignments/MIT6_041F10_assn04.pdf" rel="nofollow noreferrer">link</a> which looks something like this:</p>
<p... | Graham Kemp | 135,106 | <p><span class="math-container">$\mathsf P(X=x)$</span> is the probability for the event of <span class="math-container">$X=x$</span>.</p>
<p><span class="math-container">$p_{\small X\!}(x)$</span> is the probability mass function of random variable <span class="math-container">$X$</span>, measured at <span class="mat... |
1,294,792 | <p>Can a 2-dimensional sphere be decomposed into two disjoint homeomorphic subspaces? If yes, can these subspaces be non-discrete / connected / have some other good properties?</p>
| Timbuc | 118,527 | <p>Some ideas. There's quite a few things to justify, among them taking the Principal Value of the series after the logarithmic differentiation.</p>
<p>Take the infinite product for the sine function:</p>
<p>$$\sin \pi z=\pi z\prod_{n=1}^\infty\left(1-\frac{z^2}{n^2}\right)=\pi z\prod_{0\neq n=-\infty}^\infty\left(1+... |
712 | <p>I read <a href="https://math.stackexchange.com/questions/625/why-is-the-derivative-of-a-circles-area-its-perimeter-and-similarly-for-spheres">this question</a> the other day and it got me thinking: the area of a circle is $\pi r^2$, which differentiates to $2 \pi r$, which is just the perimeter of the circle. </p>
... | Isaac | 72 | <p>If you use the formula you describe for squares, your measurements are coming from one corner of the square. Imagine those measurements growing slowly. The square will grow, but only along the two sides opposite the corner from which you measured, so the derivative of the area formula is only the perimeter on thos... |
3,841 | <p>Taking tori in symmetric products and "miraculously" proving that the Floer homology is independent of choices always seemed, well, miraculous. Some time ago Max Lipyanski explained to me the origins of this construction from gauge theory on surfaces, a la Atiyah-Floer conjecture, which I have then forgotten. What ... | Ben Webster | 66 | <p>I think the crude answer is that there is (or maybe just should be) an extended 4 dimensional TQFT that assigns the Fukaya category of a symmetric product to a surface, and the usual Heegard-Floer Lagrangian to a 3 manifold. So, the usual definition of Heegard-Floer is the gluing formula for a Heegard splitting, an... |
941,596 | <p>Assuming I'm flipping $M$ biased coins with different probability for heads $p_i, i=\{1,...,M\}$. What is the probability of having $k$ times head? Is there a distribution function known for this?</p>
| mathse | 136,490 | <p>You may want to look for "Poisson binomial distribution", e.g., <a href="http://en.wikipedia.org/wiki/Poisson_binomial_distribution" rel="nofollow">http://en.wikipedia.org/wiki/Poisson_binomial_distribution</a></p>
|
4,340,560 | <p>I have to teach limits to infinity of real functions of one variable.
I would like to start my course with a beautiful example, not simply a basic function like <span class="math-container">$1/x.$</span> For instance, I thought of using the functions linked to the propagation of covid-19 and show that, under the bas... | Chilote | 113,061 | <p><strong>Application:</strong> A limit can be interpreted as a prediction of the result of an experiment that can be held (hypothetically) for eternity without having to do it for eternity.</p>
<p>For example, if you throw a 6-side dice the probability of getting a 1 is <span class="math-container">$\frac{1}{6}.$</sp... |
2,133,769 | <p>Asymptotic notations are a little vague for me at the moment. Here I have a problem that asks me if two equations are equal to each other, and it involves the big O notation. I wrote my question in the image in magenta color.</p>
<p><a href="https://i.stack.imgur.com/hxDx3.jpg" rel="nofollow noreferrer"><img src="h... | Joffan | 206,402 | <p>Your suggested solution assumes $x_4$ and $x_5$ are zero. The given solution acknowledges that there will be two remaining unknowns and set those to be $s=x_4$, $t=x_5$ (as you can see from the last two elements of the tuples they multiply) and solves for those.</p>
<p>[You can think of this as two more equations:<... |
240,139 | <p>Does a quadratic form always come from symmetric bilinear form ?
We know when $q(x)=b(x,x)$ where $q$ is a quadratic form and $b$ is a symmetric bilinear form.
But when we just take a bilinear form and $b(x,y)$ and write $x$ instead of $y$,does it give us a quadratic form ?</p>
| Christopher A. Wong | 22,059 | <p>Yes, every quadratic form (over a finite-dimensional $\mathbb{R}$-space) can be expressed in terms of a symmetric bilinear form, because if your quadratic form $Q(x)$ (for $x \in \mathbb{R}^n$) is written as
$$ Q(x) = \sum_{i\le j} c_{ij} x_i x_j $$
then $Q(x) = x^T Ax$, where $A$ is a symmetric matrix given by $A =... |
2,167,541 | <p>Is this expression true?$$y(x) =\lim_{h \rightarrow 0}\frac{f(x+h)-f(x)}{h} = \lim_{h \rightarrow 0}\frac{f(x)-f(x-h)}{h}$$</p>
<p>I'm taking the limit of h goes to 0 for y(x) and I got the latter equation. Can I just write y(x) = df(x)/dx?</p>
| Yusuf Kanat | 669,262 | <p><span class="math-container">$=\dfrac{1}{(2k+1)!}\displaystyle\sum_{i=0}^k \frac{(2k+1)!}{(2i+1)!(2k)(2i)!}$</span></p>
<p><span class="math-container">$=\dfrac{1}{(2k+1)!} \displaystyle\sum_{i=0}^k \begin{pmatrix} 2k+1 \\ 2i+1 \end{pmatrix}$</span></p>
<p><span class="math-container">$=\dfrac{1}{(2k+1)!}\begin{pm... |
4,387,141 | <p>I have the following instance of a semidefinite program (SDP):</p>
<p><span class="math-container">$$
d + f \to \max
$$</span></p>
<p><span class="math-container">$$
a + b + c = 1
$$</span></p>
<p><span class="math-container">$$
\begin{pmatrix}
a & d & e\\
d & b & f\\
e & f & c
\end{pmatrix} ... | Ben Grossmann | 81,360 | <p>I will assume that <span class="math-container">$r \leq n$</span>. Otherwise, there's some extra "bookkeeping" to be done regarding the singular values of <span class="math-container">$A$</span>.</p>
<p>It's easy to show that taking <span class="math-container">$X$</span> to be the leading left singular ve... |
4,387,141 | <p>I have the following instance of a semidefinite program (SDP):</p>
<p><span class="math-container">$$
d + f \to \max
$$</span></p>
<p><span class="math-container">$$
a + b + c = 1
$$</span></p>
<p><span class="math-container">$$
\begin{pmatrix}
a & d & e\\
d & b & f\\
e & f & c
\end{pmatrix} ... | KBS | 532,783 | <p>I can propose an alternative solution. The objective is to find a rank <span class="math-container">$r$</span> matrix <span class="math-container">$X\in\mathbb{R}^{m\times r}$</span> that mimimizes <span class="math-container">$||X^TA||_F$</span> and that satisfies <span class="math-container">$X^TX=I$</span>.</p>
<... |
1,326,652 | <p>What would be the nature of the roots of the equation $$2x^2 - 2\sqrt{6} x + 3 = 0$$</p>
<p>My book says that as the discriminant is 0 so the roots are rational and equal.
But discriminant can be used for determining the nature of roots only when the roots are rational numbers. Is the answer in the book wrong becau... | Bernard | 202,857 | <p>If the discriminant of $ax^2+bx+c$ is $0$, there is a double root, i. e. the quadratic can be written as $\,a(x-\xi)^2\,$ for some real number $\xi$ (if the coefficients $a,b,c$ are real, of course). Furthermore, this double root is equal to:
$$\xi=-\frac b{2a}.$$
So here the double root is actually <em>irrationa... |
3,014,843 | <p>My calculusbook simply states that:
<span class="math-container">$$\sin^2\left(\frac{\sqrt{x}}{2}\right)\le\left(\frac{\sqrt{x}}{2}\right)^2$$</span>
...but I don't immediately see why this is true. </p>
<p>What (probably) simple trick am I missing? Apparently it is "obvious"...</p>
| Matthew C | 25,959 | <p>This will follow if you show that <span class="math-container">$\sin(t) \leq t$</span> for all <span class="math-container">$t\geq 0$</span>: To do that, you can try to show that if <span class="math-container">$f(0) = g(0)$</span> and <span class="math-container">$f'(t) \leq g'(t)$</span> for all <span class="math-... |
3,014,843 | <p>My calculusbook simply states that:
<span class="math-container">$$\sin^2\left(\frac{\sqrt{x}}{2}\right)\le\left(\frac{\sqrt{x}}{2}\right)^2$$</span>
...but I don't immediately see why this is true. </p>
<p>What (probably) simple trick am I missing? Apparently it is "obvious"...</p>
| hamam_Abdallah | 369,188 | <p>By MVT</p>
<p><span class="math-container">$$\sin(X)-\sin(0)=X\cos(c)$$</span></p>
<p>thus</p>
<p><span class="math-container">$$|\sin(X)|\le |X|$$</span></p>
<p>and</p>
<p><span class="math-container">$$\sin^2(X)\le X^2$$</span></p>
<p>now apply to
<span class="math-container">$$X=\frac{\sqrt{x}}{2}$$</span><... |
303,944 | <p>Interesting problem I spotted while learning:</p>
<blockquote>
<p>Let <span class="math-container">$X=\left\{1,..,n\right\}$</span>. We randomly select subset of <span class="math-container">$X$</span> and name it <span class="math-container">$A$</span>. Each subset if equally likely.</p>
<p>a) Find the expected val... | Ross Millikan | 1,827 | <p>For a) you can pair each subset with its complement to show that a given element is in half the subsets, so is in the chosen subset half the time.</p>
<p>For b) each element has the same chance to be chosen. If you want to get $k$ of them, you need that chance to be $\frac kn$.</p>
<p>Good thinking on both.</p>
|
432,223 | <p>I wanted to know, how can I factor $x^6 +5x^3 +8$, I have no idea. Is there any method to know if a polynomial is factored. Just some advice will do.</p>
<p>Help appreciated.</p>
<p>Thanks.</p>
| Eric Tressler | 26,785 | <p>Let $y = x^3$ to obtain $y^2 + 5y + 8 = 0$. This factors as $y = \frac{-5 \pm \sqrt{-7}}{2} = \frac{-5}{2} \pm \frac{\sqrt{7}}{2}i$. These have modulus $r = \sqrt{25/4 + 7/4} = 2\sqrt{2}$. Now solve $5/2 = 2\sqrt{2} \cos \theta$ to find the angle $\theta = \cos^{-1}(5/4\sqrt{2})$. Our two roots correspond to $re^{\p... |
17,885 | <p>In most education systems, Mathematics is a compulsory subject from primary school all the way to the start of university. A common reason given is that essential concepts like addition and multiplication are taught to the children. </p>
<p>But for many high school students, especially those who are keen on pursuin... | JRN | 77 | <p>Underwood Dudley answers the question "What is mathematics education for?" in <a href="https://www.ams.org/notices/201005/rtx100500608p.pdf" rel="noreferrer">this article</a> from 2010 (<em>Notices of the American Mathematical Society</em>, vol. 57, no. 5, pp. 608-613) (even though the title of the article... |
17,885 | <p>In most education systems, Mathematics is a compulsory subject from primary school all the way to the start of university. A common reason given is that essential concepts like addition and multiplication are taught to the children. </p>
<p>But for many high school students, especially those who are keen on pursuin... | WGroleau | 13,472 | <p>Because</p>
<blockquote>
<p>many high school students, especially those who are keen on pursuing
the humanities, they do not see any point in studying Mathematics for
the rest of their schooling life, or how any concept in Mathematics
could possibly be applied in their future work.</p>
</blockquote>
<p>sho... |
17,885 | <p>In most education systems, Mathematics is a compulsory subject from primary school all the way to the start of university. A common reason given is that essential concepts like addition and multiplication are taught to the children. </p>
<p>But for many high school students, especially those who are keen on pursuin... | Aaron Peterson | 13,494 | <p>Mathematic notation and numbers are fundamental to basic communication and problem solving.</p>
<p>Our existence depends on it.</p>
<p>It is also very fun. </p>
<p>I think this question would come up less often if we presented kids with problems to solve, and then they realize that they need tools. </p>
<p>My ph... |
372,181 | <p>The theory of real closed fields is decidable. The <a href="https://en.wikipedia.org/wiki/Hyperreal_number" rel="noreferrer">hyperreals</a> satisfy that theory, so we can interpret statements in the theory of real closed fields as being about hyperreals.</p>
<p>If we add a unary predicate for "is a standard rea... | Emil Jeřábek | 12,705 | <p>Yes, the theory is decidable.</p>
<p>If <span class="math-container">$F$</span> is an ordered field and <span class="math-container">$R\subseteq F$</span> a non-cofinal subfield, then
<span class="math-container">$$O=\{x\in F:\exists u\in R\:(-u\le x\le u)\}$$</span>
is a convex valuation ring of <span class="math-c... |
1,313,216 | <p>Its my first time on here and my maths is poor so please be kind. I am working on a Masters dissertation focused on document clustering methods in which I would like to apply a weight based on the time interval between two documents. </p>
<p>I am looking for some help coming up with a function to express a time int... | Hrodelbert | 133,762 | <p>The simplest approach in my opinion is to consider the function
$$
w(d) =\frac{1}{d},
$$
where $d$ is the number of days difference. It is obvious that $w(1) = 1$ and that $0<w(d)<1$ for all $d\geq1$, as you wanted, and by coincidence, this function also has the property that $w(9)=\frac{129}{9}w(129)$, or mor... |
4,349,748 | <p>I would appreciate some help with this.</p>
<p>Is:
<span class="math-container">$$
\nabla \cdot \left( \nabla \vec{v}\right)^T= \nabla \left( \nabla \cdot \vec{v}\right)
$$</span></p>
<p>How can I show this? Is the gradient of a vector mathematically defined?</p>
<p>Best regards</p>
| Son Gohan | 865,323 | <p>To me it is ok, but since you are in a metric space you could use sequences as well to characterize the closure of your set.</p>
<p>If instead you wanted to do it because you wanted to exercise with nets (whose convergence is sometimes called Moore-Smith convergence), then all you have to do is to mimick the sequenc... |
64,022 | <p><strong>Edit</strong></p>
<p>(As Robert pointed out, what I was trying to prove is incorrect. So now I ask the right question here, to avoid duplicate question)</p>
<p>For infinite independent Bernoulli trials with probability $p$ to success, define a random variable N which equals to the number of successful tri... | Srivatsan | 13,425 | <p>You want to compute the probability of $s$ successes for $s = 0, 1, 2, \ldots$. Here the crucial point is that $s$ is fixed first, and then you compute the probability that you get $s$ successes when you throw infinitely many coins (each of success probability $p$). In other words, we want
$$
\lim_{m \to \infty} b... |
64,022 | <p><strong>Edit</strong></p>
<p>(As Robert pointed out, what I was trying to prove is incorrect. So now I ask the right question here, to avoid duplicate question)</p>
<p>For infinite independent Bernoulli trials with probability $p$ to success, define a random variable N which equals to the number of successful tri... | uforoboa | 15,453 | <p>Let us call $E_{k,n}:=$ probability of winning exactly $k$ times after $n$ trials. Let now $$E_k=\lim_{n\to+\infty}E_{k,n}.$$</p>
<p>It holds
$$P(E_k)=\lim_{n\to\infty}P(E_{k,n})=\lim_{n\to+\infty}\binom{n}{k}p^k(1-p)^{n-k}$$
Because $E_{k,n}\subseteq E_{k,n+1}$ and of course one has</p>
<p>$$0\leq P(E_k)= \lim_{... |
823,641 | <p>Find and classify the critical points of this function: $f(x,y)= (x^y)-(xy)$ in the domain $x>0, y>0$.</p>
<p>I am having trouble treating x and y as constants when taking partial derivatives.</p>
| Claude Leibovici | 82,404 | <p><strong>Hint</strong></p>
<p>If the names make trouble to you because they are names of variables, consider first $f_1(x)=x^k-kx$ and compute its derivative with respect to $x$ as usual; this will give you $$\frac {df_1(x)}{dx}=k x^{k-1}-k$$ so, replacing $k$ by $y$, $$f'_x(x,y)=y x^{y-1}-y$$ Do a similar thing usi... |
2,534,369 | <p>I am trying to work my through the exercises in Spivak's <em>Calculus on Manifolds.</em> I am currently working on the exercises in Chapter 3 which deals with Integration. I am having trouble with the following question:</p>
<blockquote>
<p>Let:</p>
<p>\begin{equation}
f(x,y)=\begin{cases}
0, & \te... | RRL | 148,510 | <p>Hint: For any partition $P$ of $A$ the lower sum $L(P,f) = 0$ since any rectangle must contain a point $(x,y)$ where $x$ is irrational and $f(x,y) = 0.$ Next show that the upper sum $U(P,f)$ can be arbitrarily close to zero if the partition is sufficiently fine. Just extend the proof for the one-dimensional case gi... |
123,675 | <p>What are some "applications to" / "connections with" topology that one could hope to reasonably cover in a first course on topos theory (for master students)? I have an idea of what parts of the theory I would like to cover, however, I would love some more nice examples and applications. Of course, I have some ideas... | Ivan Di Liberti | 104,432 | <p>I will copy and paste the description of chapter 2 and 3 of my master thesis.</p>
<p>Chapter 2 plays an important motivational role in the thesis and is aimed at pointing out geometric characteristics of a Grothendieck topos. For a topological space $X$ one can consider the category of its open sets $\mathcal{O}(X)... |
2,008,437 | <p>Given four points in the plane, there exists a one-dimensional family of conics through these, often called a pencil of conics. The locus of the centers of symmetry for all of these conics is again a conic. What's the most elegant way of computing it?</p>
<p>I know I could choose five arbitrary elements from the pe... | MvG | 35,416 | <p>Based on the fundamental theorem of projective geometry, it must be possible to take any projectively invariant property which can be described as a polynomial in the coordinates of the involved points and turn that into an expression in determinants of these homogeneous coordinates of these points instead. Using ra... |
2,008,437 | <p>Given four points in the plane, there exists a one-dimensional family of conics through these, often called a pencil of conics. The locus of the centers of symmetry for all of these conics is again a conic. What's the most elegant way of computing it?</p>
<p>I know I could choose five arbitrary elements from the pe... | MvG | 35,416 | <p><em>While there were many good answers here, I feel that two comments were even more useful to me, so I'll combine them to a CW answer.</em></p>
<p><a href="https://math.stackexchange.com/users/305862/jeanmarie">@JeanMarie</a> quoted <a href="http://en.wikipedia.org/wiki/Nine-point_conic" rel="nofollow noreferrer">... |
1,085,164 | <p><strong>I am currently taking a course on Numerical PDE. The course covers the following topics listed below.</strong> </p>
<p><strong>Chapter 1: Solutions to Partial Dierential Equations:</strong></p>
<p><strong>Chapter 2: Introduction to Finite Elements:</strong></p>
| Zaheer kiyani | 964,430 | <p>If any one tell me the basic and easy book with examples to understand the course of Numerical methods of PDE's.
Contents are given:
Boundary and initial conditions, Polynomial approximations in higher dimensions. Finite Difference Method: Finite Difference approximation. Finite Element Method: The Galerkin method i... |
3,024,456 | <p>I am working on a problem and I am confused if exponents can be split up in the manner below. </p>
<p><span class="math-container">$$x^{k+2} = x^{k+1}+x^{k+1} = 2x^{k+1} $$</span></p>
<p>I apologize if this is a simple question I tried to look up exponent rules but I couldn't find exponents + a number. </p>
| PrincessEev | 597,568 | <p>No, they definitely cannot be split up like that. To my knowledge there isn't any general law for <span class="math-container">$x^a + x^b = \text{something}$</span>.</p>
<p>A few handy exponent laws can be found at <a href="http://mathworld.wolfram.com/ExponentLaws.html" rel="nofollow noreferrer">http://mathworld.w... |
2,315,599 | <p>I am trying to evaluate the integral<br>
$\int (\frac{1}{z(\exp(z)-1)}dz$ $\mathbb{C}:|z|=1\}$</p>
<p>using Laurent theorem. $0$ is a singularity point and both denominators becomes $0$ when $z=0$. So it's hard to find a method to evaluate this using Residue theorem. Could someone please show me how to solve thi... | Saketh Malyala | 250,220 | <p><strong>I would appreciate if someone could check this over for me:</strong></p>
<p>Consider $d$, the distance between her and the tram. </p>
<p>Also consider $r_0$ and $r_1$, the speeds of her and the tram.</p>
<p>We have (1) --> $d=(r_0+r_1)t$.</p>
<p>We also have (2) --> $d+\frac{1}{5}r_1=\frac{1}{5}r_0+(r_0+... |
118,658 | <p>I have set of files in the format 20160615-1-0.asc, 20160615-2-0.asc,... 20160615-5-0.asc etc. The following commands can List the file names at once which can be used to import later. </p>
<pre><code> files = Range[1, 5];
c = "D:\\20160615\\20160615-" <> ToString[#] <> "-0.asc" & /@files
</cod... | george2079 | 2,079 | <p>a good use for the "new in 10" <code>StringTemplate</code> : </p>
<pre><code> Table[
StringTemplate["D:\\20160615\\20160615-`1`-`2`.asc"][i, j], {i, 3}, {j, 3}]
</code></pre>
<blockquote>
<p><a href="https://i.stack.imgur.com/x637Z.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/x637Z.png"... |
3,556,109 | <p>Suppose that <span class="math-container">$f : (0, 2] \to \mathbb{R}$</span> is a continuous function and that <span class="math-container">$\lim_{x \to 0}f(x) = L$</span> for some <span class="math-container">$L \in \mathbb{R}$</span>.
Define <span class="math-container">$g : [0, 2] \to \mathbb{R}$</span> via <spa... | Rezha Adrian Tanuharja | 751,970 | <p>Actually Holder’s inequality for <span class="math-container">$p=q=2$</span> is stronger than CS inequality.</p>
<p>Triangle inequality:</p>
<p><span class="math-container">$\left|\sum{x_{i}y_{i}}\right|\leq\sum{\left|{x_{i}y{i}}\right|}$</span></p>
<p>Therefore:</p>
<p><span class="math-container">$\left|\sum{x... |
40,669 | <p><a href="http://en.wikipedia.org/wiki/Wang_tile">Wang tiles</a> are (by Wikipedia): "equal-sized squares with a color on each edge which can be arranged side by side (on a regular square grid) so that abutting edges of adjacent tiles have the same color; the tiles cannot be rotated or reflected."</p>
<p>The usual d... | Ben Standeven | 29,318 | <p>There's a newer proof given in "Two-by-Two Substitution Systems and the Undecidability of the Domino Problem", by Nicolas Ollinger. It seems to be available online at: <a href="http://hal.inria.fr/docs/00/26/01/12/PDF/sutica.pdf" rel="noreferrer">http://hal.inria.fr/docs/00/26/01/12/PDF/sutica.pdf</a>. Hopefully, it... |
2,204,834 | <p>I have to show by induction that this function is a multiple of 8. I have tried everything but I can only show that is multiple of 4, some hints? The function is
$$5^{n+1}+2\cdot 3^n+1 \hspace{1cm}\forall n\ge 0$$, because it is a multiple of 8, you can say that$$5^{n+1}+2\cdot 3^n+1=8\cdot m \hspace{1cm}\forall m\i... | User8976 | 98,414 | <p>By induction:</p>
<p>It is true for case $n=1$, Let it be true for $n=k$ then </p>
<p>$5^{k+2} +2.3^{k+1} +1 = 5.5^{k+1}+3.2.3^k+1 = 2(5^{k+1} -1) + 3(5^{k+1}+2.3^k+1)$</p>
<p>The second part is a multiple of $8$ and one can easily show that $(5^{k+1} -1)$ is a multiple of $4$.</p>
<p>Hint for showing $(5^{k+1... |
4,263,836 | <blockquote>
<p>I would like to compute <span class="math-container">$\mathrm{Ext}^i_{\mathbb Z}(\mathbb Q, \mathbb Z/2\mathbb Z)$</span>.</p>
</blockquote>
<p>So from the definition I learned in my class, I need to find a free resolution of <span class="math-container">$\mathbb Q$</span> over <span class="math-contain... | Jun Koizumi | 966,250 | <p>Taking an explicit free resolution of <span class="math-container">$\mathbb{Q}$</span> is not impossible but difficult.
Instead we can use the functoriality of <span class="math-container">$\operatorname{Ext}^i$</span> to determine <span class="math-container">$\operatorname{Ext}^i(\mathbb{Q},\mathbb{Z}/2)$</span>.
... |
2,561,245 | <p>Here is the question:
Let $f(x)$ be an integrable function on $[-1,1]$. $f(x)$ is continuous at $x=0$. Define $g(x)=\int_0^{\int_0^xf(n)dn}f(y)dy$ and prove $g'(0)=f(0)^2$.</p>
<p>My initial thought was to look at $g(0)$ and say when you take an integral from $a$ to $b$ and $a=b$ then the integral is $0$, but I do ... | Dylan | 135,643 | <p>Here's what I would do. Let
$$h(x) = \int_0^x f(n)\ dn $$</p>
<p>then we immediately have $h'(x) = f(x)$ by the FTOC, and $h(0) = 0$</p>
<p>By definition
$$ g(x) = \int_0^{h(x)} f(y)\ dy $$</p>
<p>Using the FTOC again we have
$$ g'(x) = f\big(h(x)\big)\cdot h'(x) = f\big(h(x)\big) \cdot f(x) $$</p>
<p>Therefore
... |
3,048,340 | <blockquote>
<p>Suppose <span class="math-container">$f$</span> is a convex function and differentiable everywhere on <span class="math-container">$(0, \infty)$</span> and satisfies
<span class="math-container">$$\lim_{x \rightarrow\infty}{f(x)}=A$$</span>
where <span class="math-container">$A$</span> is an arbit... | Kavi Rama Murthy | 142,385 | <p>For a convex differentiable function we canwrite <span class="math-container">$f(x)=f(0)+\int_0^{x} f'(t)\, dt$</span>. The hypothesis implies that <span class="math-container">$\int_0^{x} f'(t)\, dt$</span> remains bounded as <span class="math-container">$ x\to \infty$</span>. Note that <span class="math-container"... |
356,217 | <p>Find the following infinite sum : $$q\sin a+q^2\sin 2a+\ldots+q^n\sin na+\ldots$$ where $|q|<1$ .It would be good if you could find it without the help of any auxiliary sequences using only trigonometric formulas.</p>
| Community | -1 | <p><strong>Hint</strong>
We calculate the sum of geometric series
$$\sum_{n=1}^\infty q^ne^{ina}$$
then we take the imaginary part.</p>
|
2,362,485 | <blockquote>
<p>Evaluate: $$\lim_\limits{x\to \pi/4} \frac {2-\csc^2 x}{1-\cot x}$$</p>
</blockquote>
<p>My Attempt:
\begin{align}\lim_\limits{x\to \pi/4} \frac {2-\csc^2 x}{1-\cot x}&=\lim_{x\to \pi/4} \frac {2-\csc^2 x}{1-\cot x}\\\\
&=\lim_{x\to \pi/4} \frac {2-\frac {1}{\sin^2 x}}{1-\frac {\cos x}{\sin x... | Claude Leibovici | 82,404 | <p>Considering
$$ A=\dfrac {2-\csc^2( x)}{1-\cot(x)}$$ let $x=y+\frac \pi 4$ to get (after simplification) $$A=\frac{2 \cos (y)}{\sin (y)+\cos (y)}$$ and now, look for the limit when $y\to 0$.</p>
|
1,842,093 | <p><a href="https://i.stack.imgur.com/A5U3D.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/A5U3D.png" alt="enter image description here"></a></p>
<p>$ABCD$ forms a square. $CDE$ forms a triangle. Given $\measuredangle AED=15^{\circ}$ and $DE=CE$, prove $\triangle CDE$ is equilateral. </p>
<p>The q... | Amritanshu | 349,576 | <p>Ok ! Here we go ..</p>
<p>$ABCD$ is a square.
That means -> $AB=BC=CD=AD$</p>
<p>Now if you carefully notice, $CD$ is the base of the $\Delta CDE$</p>
<p>Given that $DC = DE$</p>
<p>We can conclude that $\Delta ADE$ is an isosceles triangle.
Therefore ,
$\angle AED = \angle EAD$</p>
<p>So, $\angle EAD = 15º$</p... |
7,900 | <p>This semesters I am teaching a course to 12-13 year old kids in which they are supposed to learn the basics of spreadsheet usage.</p>
<p>I am having difficult in coming up with fun / interesting exercises to teach them the basic functions, such as average, max, minimum and the like. One of the biggest problems is h... | celeriko | 3,237 | <p>Based on your unfortunate lack of time my answer is going to be a list of suggestions that I believe could be completed and meaningful given the amount of time you have with your students. Not all of them pertain directly with math (calculations, computations, formulas, etc), but I believe that 1) the real power of... |
7,900 | <p>This semesters I am teaching a course to 12-13 year old kids in which they are supposed to learn the basics of spreadsheet usage.</p>
<p>I am having difficult in coming up with fun / interesting exercises to teach them the basic functions, such as average, max, minimum and the like. One of the biggest problems is h... | Amir Asghari | 1,217 | <p>you might find this paper "<a href="http://link.springer.com/article/10.1007/s10758-005-8420-9" rel="nofollow">Designing Spreadsheet-Based Tasks for Purposeful Algebra</a>" (and a couple of some others written by the same authors and about the same year) useful. </p>
<p>PS. Originally, I meant to put this answer as... |
7,900 | <p>This semesters I am teaching a course to 12-13 year old kids in which they are supposed to learn the basics of spreadsheet usage.</p>
<p>I am having difficult in coming up with fun / interesting exercises to teach them the basic functions, such as average, max, minimum and the like. One of the biggest problems is h... | Jasper | 1,147 | <p><em><a href="http://rads.stackoverflow.com/amzn/click/0393310728" rel="nofollow">How to Lie with Statistics</a></em>. Many graphs in newspapers are misleading because the authors used Excel's default graph settings, instead of making sure to set the range of the y-axis to start at zero.</p>
<p>Are you using a rece... |
2,760,113 | <p>Suppose $f'$ is continuous on $[a,b]$ and $\epsilon >0$ is given.
Prove that $\exists\; \delta>0$ such that $$\left\lvert \frac{f(t)-f(x)}{t-x}-f'(x)\right\rvert<\epsilon$$</p>
<p>$\forall \;0<|t-x|<\delta, a\le x \le b ,a\le t\le b$.</p>
<p>How to solve this problem since using compactness argumen... | Christian Blatter | 1,303 | <p>Since $f'$ is continuous we may write
$$f(t)-f(x)=\int_x^tf'(\xi)\>d\xi=(t-x)\int_0^1f'\bigl(x+\tau(t-x) \bigr)\>d\tau\ ,$$
and therefore
$${f(t)-f(x)\over t-x}-f'(x)=\int_0^1\bigl(f'\bigl(x+\tau(t-x)\bigr) -f'(x)\bigr)\>d\tau\qquad(t\ne x)\ .\tag{1}$$
Now $f'$ is automatically uniformly continuous on $[a,b... |
1,207,134 | <p>Given the image:
<img src="https://i.stack.imgur.com/EJ3ax.jpg" alt="enter image description here"></p>
<p>and that $x_0 = 1, y_0=0$ and $\text{angles} \space θ_i
, i = 1, 2, 3, · · ·$ can be arbitrarily picked.</p>
<p>How can I derive a recurrence relationship for $x_{n+1}$ and $x_n$?</p>
<p>I actually know what... | marty cohen | 13,079 | <p>This is a generalization of the
"Spiral of Theodorus"
which is discussed in many places.
Here is the obvious one:
<a href="http://en.wikipedia.org/wiki/Spiral_of_Theodorus" rel="nofollow">http://en.wikipedia.org/wiki/Spiral_of_Theodorus</a></p>
<p>Using the methods there
as a guide.</p>
<p>(I guess this should be ... |
1,207,134 | <p>Given the image:
<img src="https://i.stack.imgur.com/EJ3ax.jpg" alt="enter image description here"></p>
<p>and that $x_0 = 1, y_0=0$ and $\text{angles} \space θ_i
, i = 1, 2, 3, · · ·$ can be arbitrarily picked.</p>
<p>How can I derive a recurrence relationship for $x_{n+1}$ and $x_n$?</p>
<p>I actually know what... | Narasimham | 95,860 | <p>The problem is not well-posed. There are several solutions to the given problem depending on discretization choice. One more condition is needed to solve the finite "spiral". I am assuming that the real aim here is to find the spiral which is described by the tip of rotating vector.</p>
<p>An extreme choice of disc... |
2,338,797 | <p>A bag contains $3$ red balls and $4$ blue balls. Find the probability of picking, at random, two balls of </p>
<p>a) the same color </p>
<p>b) different colors</p>
| Mandar Kulkarni | 458,156 | <p>I think it is important to note here that the order in which balls are chosen does not matter.</p>
<p>The total number of ways of drawing 2 balls out of 7 is 7C2 i.e. 21</p>
<p>For first case we either draw 2 red (3C2) or 2 blue balls (4C2), for total favorable cases of 9</p>
<p>Hence probability of drawing same ... |
1,455,098 | <p>I am reading the book Foundation of Machine Learning, and the author has many proofs for different theorems. Here is one part of a proof about Perceptron algorithm which I don't quite understand</p>
<p>Note: $w_t$ is a vector</p>
<p>Let
\begin{equation}
w_{t+1} = \begin{cases}
w_t + ny_t x_t & \text{if } y_t(... | Greg Martin | 16,078 | <p>This is an algebraic identity that has very little to do with the specific function: for <em>any</em> $f\colon \mathbb Z_{\ge0} \to \mathbb R_{\ge0}$ be any function satisfying $f(0)=0$, we have
$$
f(T+1) = \sqrt{\sum_{0\le t\le T} \big( f(t+1)^2-f(t)^2 \big)}.
$$
To see this, it's equivalent to show that
$$
f(T+1)^... |
1,472,901 | <h2>Code first :</h2>
<pre><code>sum = 0
for(i=1; i<=n; i++){
for(j=1; j<=i; j++){
if(j%i == 0){
for(k=1; k<=n; k++){
sum = sum + k;
}
}
}
}
</code></pre>
<hr>
<p>Total no. of iterations in $j = 1+2+3+4+ \dot... | Sujith Sizon | 262,120 | <h2>Critical points</h2>
<p>refers to the set of all points which satisfies at least one of the following conditions :</p>
<p>(A) $f'(x)=0$ ie: turning points.</p>
<p>(B) $f'(x)$ does not exist.</p>
<p>(C) sign changes of $f'(x)$ in nbd of $x$ (not a completely necessary condition as these will be included in (B))<... |
4,164,405 | <p>Grothendieck has proven that whenever <span class="math-container">$X\longrightarrow\operatorname{Spec}(A)$</span> is a proper morphism of Noetherian schemes, <span class="math-container">$F$</span> is coherent over <span class="math-container">$X$</span> and flat over <span class="math-container">$\operatorname{Spe... | Aphelli | 556,825 | <p>The answer is that the groups of the Cech complex aren’t finitely generated except when <span class="math-container">$X$</span> is finite over <span class="math-container">$A$</span>. I’m studying the cohomology of <span class="math-container">$\mathcal{O}_X$</span>.</p>
<p>Indeed, let <span class="math-container">$... |
650,291 | <p>I'm not so good at combinatorics, but I want to know if my answer for this question is right. Originally this question is written in spanish and it says:</p>
<blockquote>
<p>Se dispone de una colección de 30 pelotas divididas en 5 tamaños distintos y 6 colores diferentes de tal manera que en cada tamaño hay los s... | Woria | 119,309 | <p>In this kind of problems, the key idea is to transform it into steps and cases in a way that you can use "Addition law" and "Multiplication law". Here you need tow different pairs of balls of the same size, so</p>
<p><strong>Step 1:</strong> Select 2 different sizes from those existing 5 sizes;</p>
<p><strong>Step... |
1,712,933 | <p>I am slightly ashamed to be asking this, but I have been recently reflecting on changing variables in very simple problems. If I missed a question that already discusses this please point it out to me and I will delete this one.
Anyhow writing this will probably be a learning experience.</p>
<p>Directly from the Wi... | Jasper | 316,202 | <p>This answer will give an idea on why change of variables works/is allowed when <em>integrating</em>.</p>
<p>I will give you an idea (heuristic) on why these requirements for $u$ are needed. The basic idea behind substition of variables is that you choose a different basis over which you know the solution. </p>
<p>... |
13,230 | <p>Introduction:</p>
<p>Let A be a subset of the naturals such that <span class="math-container">$\sum_{n\in A}\frac{1}{n}=\infty$</span>. The <a href="https://en.wikipedia.org/wiki/Erd%C5%91s_conjecture_on_arithmetic_progressions" rel="nofollow noreferrer">Erdos Conjecture</a> states that A must have arithmetic progr... | Thomas | 20,974 | <p>An idea that I came up with is that if the conjecture is true, then there would be a finite upper bound on how big the sum could be if the set had no arithmetic progression of length k, lets call this upper bound f(k). Obviously f(1)=0, as if there are any members then there is an arithmetic progression of that leng... |
2,226,307 | <p>Let <span class="math-container">$c\in \mathbb{R}$</span> and <span class="math-container">$f:\mathbb{R}\to \mathbb{R}$</span> is continuous at <span class="math-container">$c$</span>. If for every positive <span class="math-container">$\delta$</span>, there is a point <span class="math-container">$y\in (c-\delta, c... | L.F. Cavenaghi | 248,387 | <p>Hint: $\delta_n := 1/n$. Then there exists $y_n$ such that $f(y_n) = 0.$ But $0 = \lim f(y_n) = f(\lim y_n)$ when $n\to \infty.$ But $\lim y_n = c.$</p>
|
322,598 | <p><a href="https://en.wikipedia.org/wiki/Partially_ordered_set" rel="noreferrer">Partially ordered sets</a> (<em>posets</em>) are important objects in combinatorics (with <a href="https://gilkalai.wordpress.com/2019/02/05/extremal-combinatorics-v-posets/" rel="noreferrer">basic connections to extremal combinatorics</a... | Tri | 51,389 | <p>Let <span class="math-container">$P,Q$</span> be posets. Let <span class="math-container">$Q^P$</span> denote the poset of order-preserving maps from <span class="math-container">$P$</span> to <span class="math-container">$Q$</span>, where <span class="math-container">$f\le g$</span> if <span class="math-container"... |
244,875 | <p>\begin{align}
\min_{x} c^Tx \\
s.t.~Ax=b
\end{align}
Note that here $x$ is unrestricted. I need to prove that the dual of this program is given by
\begin{align}
\max_{\lambda} \lambda^Tb \\
s.t.~\lambda^TA\leq c^T
\end{align}</p>
<p>But in the constraint, I always get an equality (using what I learnt)
\begin{align}... | littleO | 40,119 | <p>You wrote the dual problem correctly. Perhaps whoever wrote your assignment forgot to include the constraint $x \geq 0$ in the primal problem.</p>
<p>Edit: here's how I derived the dual problem. The Lagrangian is
\begin{align}
L(x,\nu) &= \langle c, x \rangle + \langle \nu, b - Ax \rangle \\
&= \langle c,... |
2,944 | <p>Let $f$ be a diffeomorphism, say from $\mathbb R^n$ to $\mathbb R^n$ , such as the transition map between two coordinate charts on a differentiable manifold.</p>
<p>A differential $n$-form (or rather its coefficient function which is obtained by using the canonical one-chart atlas on $\mathbb R^n$) then transforms ... | J. M. ain't a mathematician | 498 | <p>Briefly, $p\rightarrow q$ can be read as "if $p$, then $q$"; "if not p then not q" would be the "inverse", and "if not q then not p" is the "contrapositive", which has the same truth value as the original proposal.</p>
|
1,363,407 | <p>If $x,y,z$ are elements of a group such that $xyz=1,$ then which of the following are true?</p>
<ol>
<li>$yzx=1$</li>
<li>$yxz=1$</li>
<li>$zxy=1$</li>
<li>$zyx=1$</li>
</ol>
<p>I have found options 1 and 3 to be correct, but how to prove that options 2 and 4 are wrong (that is, what the given an... | egreg | 62,967 | <p>You have
$$
xyzz^{-1}=1z^{-1}
$$
so
$$
xy=z^{-1}
$$
hence
$$
zxy=zz^{-1}=1
$$</p>
<p>Similarly, you can show that $yzx=1$. So you're right in saying that 1 and 3 hold *without any other assumption on the group.</p>
<p>Since 2 and 4 obviously hold when the group is abelian, you can find a counterexample only in a n... |
1,363,407 | <p>If $x,y,z$ are elements of a group such that $xyz=1,$ then which of the following are true?</p>
<ol>
<li>$yzx=1$</li>
<li>$yxz=1$</li>
<li>$zxy=1$</li>
<li>$zyx=1$</li>
</ol>
<p>I have found options 1 and 3 to be correct, but how to prove that options 2 and 4 are wrong (that is, what the given an... | lisyarus | 135,314 | <p>1 and 3 are true, which follows from a little more general result: if $x_1 x_2 \dots x_n = 1$ and $\pi$ is a cyclic permutation, then $x_{\pi(1)}x_{\pi(2)}\dots x_{\pi(n)} = 1$.</p>
<p>To prove that 2 and 4 are false, it is sufficient to provide counterexamples.</p>
<p>2) Let $z = 1$, then $xyz$ = $xy$ and $yxz$ =... |
367,638 | <p>Reading over a book on computability, it asserts that in P.C., if A is a theorem, then A has arbitrarily many proofs. I can't see how that would work, would you do an infinite loop in the sequence of well-formed-formulae?</p>
| xyzzyz | 23,439 | <p>A proof is a sequence of steps that leads to desired conclusion at the end. Nobody says that all these steps have to be relevant. For instance, just before concluding the proof you can put lots of logical tautologies. For a human reader, they will obviously be irrelevant to the proof, but nevertheless, the new proof... |
1,511,753 | <p>Using the concept of self-similarity, it's possible to encode the decimal expansion of a number as a sort of 'fractal' object. For instance, consider the sequence,</p>
<p>$$(1) \quad C_0=0.1, \ C_1=0.101, \ C_2=0.101000101, 0.101000101000000000101000101,...,C_n$$</p>
<p>The astute reader will notice this is analog... | BrianO | 277,043 | <p>It depends on the definition of the real in question. Some, like $\sqrt 2$ and $\pi$, are defined precisely enough that their decimal expansions can be computed to arbitrary precision. For the two reals just mentioned, the methods used aren't the same, and each takes advantage of aspects of the defined real peculiar... |
2,957,440 | <p>Knowing <span class="math-container">$f(1) = 2, f(4)=7,f'(1) = 5,f'(4)=3$</span></p>
<p>Find
<span class="math-container">$\int_1^4xf''(x)dx$</span></p>
<p>It is obvious that if there was no <span class="math-container">$x$</span> the answer would be <span class="math-container">$f'(4) - f'(1)$</span> from the fun... | Robert Lewis | 67,071 | <p>To evaluate, according to the given data, the integral</p>
<p><span class="math-container">$\displaystyle \int_1^4 xf''(x) \; dx, \tag 1$</span></p>
<p>set</p>
<p><span class="math-container">$u(x) = x, \; v(x) = f'(x); \tag 2$</span></p>
<p>then</p>
<p><span class="math-container">$du = dx, \; dv = f''(x) dx; ... |
1,350,733 | <p>The following (multiple choice) problem is from a test review.</p>
<blockquote>
<p>For the given matrix $A$, find a basis for the corresponding eigenspace for the given eigenvalue.</p>
<p>$$A = \begin{bmatrix}1 & 6 & 6 \\ 6 & 1 & -6 \\ -6 & 6 & 13\end{bmatrix};\quad \lambda = 7.$$</p>... | Rory Daulton | 161,807 | <p>As @GTonyJacobs correctly said in a comment, your answer is correct, but there are infinitely many answers to the question and the one you found is not the one given as one of the multiple choice answers.</p>
<p>You asked @GTonyJacobs in a comment of your own what to do in such a case. The first thing is for your t... |
3,336,592 | <p>I could prove the following result from my Real Analysis course:</p>
<blockquote>
<p>Let <span class="math-container">$f:[0,1] \rightarrow [0,1]$</span> be an increasing mapping. Then it has a fixed point.</p>
</blockquote>
<p>I understand that this is a very baby version of Tarski’s Fixed Point Theorem. Now, I ... | Noah Schweber | 28,111 | <p>As Chris Eagle said, your example for (1) is wrong. Removing the characteristic specification does the trick (as they observe), but there are also far simpler examples. For instance, take the empty language <span class="math-container">$\{\}$</span> (so only "<span class="math-container">$=$</span>" allowed, besides... |
103,358 | <p>I am trying to illustrate some simple ideas with exponents. I can manually express something like $5^4$ as $5 \cdot 5 \cdot 5 \cdot 5$, but wondered how to get <em>Mathematica</em> to do that for me.</p>
<p>I found the example below in the documentation, but can't figure out how to "massage" it to work with a num... | Searke | 144 | <p>First, try resetting Mathematica by following this article:</p>
<p><a href="http://support.wolfram.com/kb/12464" rel="nofollow">http://support.wolfram.com/kb/12464</a></p>
<p>Even if there's no reason why it should work, please try it. </p>
<hr>
<p>If that doesn't work, then try running the Mathematica kernel:</... |
3,619,835 | <blockquote>
<p>Find the maximum of the function <span class="math-container">$f(x,y) = (a + x)(b + y)$</span> under the constraint <span class="math-container">$d = x + y$</span>, where <span class="math-container">$a$</span>, <span class="math-container">$b$</span>, <span class="math-container">$d$</span> are known... | jobseeker_68141 | 771,246 | <p>It's labeled as pre-calculus problem so I'll solve it without calculus.</p>
<p><span class="math-container">$f(x,y)=(a+x)(b+y)=(a+x)(b+d-x)=-x^2+(b+d-a)x+(ab+ad)=-(x-(b+d-a)/2)^2+(ab+ad+(b+d-a)^2/4)$</span></p>
<p>The maximum can be obtained at <span class="math-container">$x=(b+d-a)/2$</span> and the maximum valu... |
547,784 | <p>Given random variable $F_{x}(n)=P(X=n)=\frac{c}{n(n+1)}$ calculate $c$ and $P(X>m)$ for $m=1,2,3...$.</p>
<p>First of all $\lim\sum\limits_{n \in \mathbb{N}} F_{x}(n)=1 $, so $$\lim\sum\limits_{n \in \mathbb{N}} F_{x}(n)=\frac{c}{n(n+1)} + \frac{c}{(n+1)(n+2)} ... \leq c(\frac{1}{2}+\frac{1}{4} ...)\leq c$$ </p>... | Community | -1 | <p>$2^{n+1}=2^n.2=2.\sum_{v=0}^{n} \binom{n}{v}= \binom{n}{0}+\big\{\binom{n}{0}+ \binom{n}{1}\big\} +\big\{ \binom{n}{1}+\binom{n}{2}\big\}+\dots + \binom{n}{n}+ \binom{n}{n}$</p>
<p>It would be necessary to use :$$\binom{n}{r}+\binom{n}{r-1}=\binom{n+1}{r}$$</p>
|
547,784 | <p>Given random variable $F_{x}(n)=P(X=n)=\frac{c}{n(n+1)}$ calculate $c$ and $P(X>m)$ for $m=1,2,3...$.</p>
<p>First of all $\lim\sum\limits_{n \in \mathbb{N}} F_{x}(n)=1 $, so $$\lim\sum\limits_{n \in \mathbb{N}} F_{x}(n)=\frac{c}{n(n+1)} + \frac{c}{(n+1)(n+2)} ... \leq c(\frac{1}{2}+\frac{1}{4} ...)\leq c$$ </p>... | Haha | 94,689 | <p>You can use the Newton's binomial that states that $(a+b)^n=\sum_{k=0}^n \binom{n}{k}a^{k}b^{n-k}$. Thus you have that $2^n=(1+1)^n=\sum_{k=0}^n \binom{n}{k}1^{k}1^{n-k}=\sum_{k=0}^n \binom{n}{k}$</p>
|
483,602 | <p>$$
\large\sqrt{8+\sqrt{5}-\sqrt{6-2\sqrt{5}}}
$$</p>
<p>I am asked to write the following as a ratio of two integers (i.e.,) show that this value is rational
(Do not use calculator)</p>
<p>What I tried doing was putting everything to the exponent of $.5$ and then was unsure how to proceed </p>
| Josephine Moeller | 950 | <p>Andre's hint is great, but more generally, if you need to show that a quantity containing a square root is rational, try completing the square on the quantity under the root. </p>
<p>The most inside root is best to start with because it's easier to analyze and things might cancel.</p>
<p>For example, to do that on... |
2,654,702 | <p>I'm working on generating a counter example that shows projections from $X \times Y$ do not always map closed sets to closed sets. I'm working backwards, that is, from the decompositions to the product space. My attempt is to see if I might generate a closed product space whose projection maps to clopen sets. </p>
... | fleablood | 280,126 | <p>You are missing the point of the exercise.</p>
<p>The exercise is to <em>extend</em> the definition of $b^n; n \in \mathbb N$ to $b^x; x \in \mathbb R$.</p>
<p>At this point $b^n$ is defined to mean "$b$ multiplied by itself $n$ times". This fine but it's not a very useful definition. What if $x \not \in \mathbb... |
2,790,910 | <blockquote>
<p>Let <span class="math-container">$X \sim N (0, 1)$</span> and <span class="math-container">$Y ∼ N (0, 1)$</span> be two independent random variables, and define <span class="math-container">$Z = \min(X, Y )$</span>. Prove that <span class="math-container">$Z^2\sim\chi^2(1),$</span> i.e. Chi-Squared with... | honeybadger | 471,078 | <p>Since $X$ and $Y$ are independent, we can see that $f(x,y) = f(x)f(y)$. </p>
<p>Now, if we transform our axis from $X,Y$ to $X_1,Y_1$, where $X_1 = \frac{1}{\sqrt{2}}(X+Y)$ and $X_2 = \frac{1}{\sqrt{2}}(X-Y)$ - essentially, we are rotating the $XY$ space by a +45 degrees. </p>
<p>Since the distribution $f(x,y)$ h... |
4,115,308 | <p>Hi I have a problem where I need to solve the following set of equations:</p>
<p><span class="math-container">$$
v = U u
$$</span></p>
<p><span class="math-container">$$
u = 1 -Uv
$$</span></p>
<p><span class="math-container">$$
U^2 = u^2 + v^2
$$</span></p>
<p>I have tried subbing <span class="math-container">$u$<... | David | 919,460 | <p>Use that rank = row rank = column rank. Let <span class="math-container">$r$</span> be the rank of the original matrix <span class="math-container">$A.$</span> Form <span class="math-container">$B,$</span> by choosing <span class="math-container">$r$</span> linearly independent columns of <span class="math-container... |
2,808,445 | <p>I should mention this up front: this is a question about semantics. It is extremely formal, and a bit pedantic. Most people won't care.</p>
<p>Is det, i.e the determinant function, a single one? Or is it multiple functions, one for each field?</p>
<p>Let $\mbox{Mat}(K)$ denote the set of square matrices over the f... | Leandro | 633 | <p>What about define it as a function with following domain and codomain
$$
\mathrm{det}: \bigcup_{n=1}^{\infty}\ \bigcup_{K:\ K\ \text{is a field}} M_{n\times n}(K)\to \bigcup_{K:\ K\ \text{is a field}} K
$$</p>
|
2,426,450 | <p>Today when I was solving problems from GRE Manhattan I ran into a strange word problem.</p>
<p><a href="https://i.stack.imgur.com/nqzes.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/nqzes.png" alt="enter image description here"></a></p>
<p>The first line of the problem already seems weird sinc... | Ken Draco | 431,886 | <p>Yes, you are absolutely right. A typo has crept in or something like that. It happens both in actual tests and in various textbooks.</p>
<p>Students who took French: $3/8\cdot420=157.5$</p>
<p>Students who took geography: $2/5\cdot 420=168$</p>
<p>Students who took both: $1/4\cdot 420=105$</p>
<p>Students wh... |
370,151 | <p>Let $f: \Bbb R → \Bbb R$ be a continuous function such that $f(x)=x$ has no real solution . Then is it true that $f(f(x))=x$ also has no real solution ? </p>
| Marc van Leeuwen | 18,880 | <p>Just a variation of the proof. Suppose $f(f(x))=x$. Define $g:x\mapsto f(x)-x$, which is continuous, and has $g(f(x))=f(f(x))-f(x)=-(f(x)-x)=-g(x)$. Then by the intermediate value theorem $g(y)=0$ for some $y$ in the closed interval bounded by $x$ and $f(x)$, and then of course $f(y)=y$.</p>
|
1,789,033 | <p>Let $X$ be an irreducible affine variety. Let $U \subset X$ be a nonempty open subset. Show that dim $U=$ dim $X$.</p>
<p>Since $U \subset X$, dim $U \leq$ dim $X$ is immediate. I also know that the result is not true if $X$ is any irreducible topological space, so somehow the properties of an affine variety have t... | Slade | 33,433 | <p>Let $X=\operatorname{Spec} A$, and pick some nonzero $f\in A$ with $\operatorname{Spec} A_f = D(f) \subset U$. Then we can lift chains of closed sets to see that $\dim D(f) \leq \dim U
\leq \dim X$.</p>
<p>But as shown <a href="https://math.stackexchange.com/questions/1784246/a-be-an-affine-k-algebra-and-f-be-a-n... |
3,720,856 | <p><strong>Question:</strong>
Prove that <span class="math-container">$\sin(nx) \cos((n+1)x)-\sin((n-1)x)\cos(nx) = \sin(x) \cos(2nx)$</span> for <span class="math-container">$n \in \mathbb{R}$</span>.</p>
<p><strong>My attempts:</strong></p>
<blockquote>
<p>I initially began messing around with the product to sum iden... | robjohn | 13,854 | <p><span class="math-container">$$
\begin{align}
\sin(nx)\cos((n+1)x)
&=\frac{\sin(nx+(n+1)x)+\sin(nx-(n+1)x)}2\tag1\\
&=\frac{\sin((2n+1)x)-\sin(x)}2\tag2\\
\sin((n-1)x)\cos(nx)
&=\frac{\sin((2n-1)x)-\sin(x)}2\tag3
\end{align}
$$</span>
Explanation:<br />
<span class="math-container">$(1)$</span>: identity... |
1,526,653 | <p>I am having troubles isolating and solving for $\frac{dy}{dx}$,</p>
<p>$\sin(x+y)=xy$</p>
<p>$\cos(x+y)(1+\frac{dy}{dx})= x\frac{dy}{dx} + y$</p>
<p>What would be the next few step to solve for $\frac{dy}{dx}$?</p>
| Michael Hardy | 11,667 | <p>$$
\cos(x+y)\left(1+ \frac{dy}{dx}\right)= x\frac{dy}{dx} + y \qquad \longleftarrow \text{ error: You need $y$ here, not $x$.}
$$
Expand:
$$
\cos(x+y)\cdot1 + \cos(x+y)\frac{dy}{dx} = x\frac{dy}{dx} + y
$$
Transpose:
$$
\cos(x+y)\frac{dy}{dx} - x\frac{dy}{dx} = \cos(x+y)\cdot1 + y
$$
Factor
$$
(\cos(x+y) - x) \frac{... |
1,526,653 | <p>I am having troubles isolating and solving for $\frac{dy}{dx}$,</p>
<p>$\sin(x+y)=xy$</p>
<p>$\cos(x+y)(1+\frac{dy}{dx})= x\frac{dy}{dx} + y$</p>
<p>What would be the next few step to solve for $\frac{dy}{dx}$?</p>
| Claude Leibovici | 82,404 | <p>Hoping that you already know it, the <a href="https://en.wikipedia.org/wiki/Implicit_function_theorem" rel="nofollow">implicit function theorem</a> is very useful. </p>
<p>In you case, consider the function $$F=\sin(x+y)-xy=0$$ Computing each derivative, considering that the other variable is a constant, gives $$F'... |
2,744,172 | <p>(Note: This question is tangentially related to this <a href="https://math.stackexchange.com/q/3609304">later one</a>.)</p>
<p>Let <span class="math-container">$$\sigma(x) = \sum_{d \mid x}{d}$$</span> denote the <em>sum of divisors</em> of <span class="math-container">$x \in \mathbb{N}$</span>, where <span class=... | mathlove | 78,967 | <p>This is a partial answer.</p>
<p><em>(II)</em> : I think that we can prove <span class="math-container">$m\lt k$</span> without assuming that <span class="math-container">$\gcd(k,m)=1$</span>. </p>
<hr>
<p>(1) Your proof for Lemma 1 is correct.</p>
<p>(2) We can prove <span class="math-container">$\frac{\sigma(m... |
4,005,381 | <p>Why the probability of a single element event is "zero" on the continuous model?, the explanation given is based on probability additivity axioms. but how? some more explanation with the source would be helpful.</p>
<p>cool thanks!</p>
| YiFan | 496,634 | <p>Suppose otherwise, so that the chance of picking any element is positive, say <span class="math-container">$p$</span>. Assuming uniform distribution, this probability is equal for all values in the interval by definition. Then for each of the infinitely many values in the interval, it is chosen with probability <spa... |
1,363,967 | <p>A group $G$ acts on a set $X$ transitively and a normal subgroup $H$ fixes a point $x_{0} \in X$, i.e. $h \cdot x_{0}=x_{0}$ for all $h \in H$. Show that $h \cdot x = x$ for all $h \in H$ and $x \in X$.</p>
<p>Since the action is transitive $\mathcal{O}_{x}=\{g\cdot x : g \in G,~ x \in X\}=X$ for any $x \in G$. </p... | P Vanchinathan | 28,915 | <p>This follows from the following: Let $G$ act on a set $X$,(need not be transitive) and let $Y$ be the subset consisting of all the fixed points for a normal subgroup $H\subset G$. Then $g.y\in Y$ for all $y\in Y\ g\in G$.</p>
|
299,263 | <p>What is the set of polynomials with integer coefficients look like? Help!</p>
<p>Thanks!</p>
| Brian M. Scott | 12,042 | <p>Here, as an example of the sort of thing that’s wanted, is a recursive definition of the set $T$ of positive integers that multiples of $3$:</p>
<blockquote>
<ol>
<li>$3\in T$. </li>
<li>If $m,n\in T$, then $m+n\in T$. </li>
<li>$T$ contains only those integers required by the first two clauses.</li>
</... |
3,827,650 | <p>Does the generalised integral</p>
<p><span class="math-container">$\int_{0}^{\pi}\frac{\sqrt x}{\sin x}dx$</span></p>
<p>converge or diverge?</p>
<p>The first thing I would do here is split it into two integrals</p>
<p><span class="math-container">$$\int_0^\pi \frac{\sqrt x}{\sin{x}}dx=\int_0^{\frac{\pi}{2}} \frac{\... | Oliver Díaz | 121,671 | <p>The first integral in the split is convergent; the second intergral in the split howeverdiverges since</p>
<p><span class="math-container">$$\int^{\pi}_{\pi/2}\frac{\sqrt{x}}{\sin x}\,dx\geq \sqrt{\pi/2}\int^\pi_{\pi/2}\frac{dx}{\sin x}$$</span>
Since
<span class="math-container">$$\frac{2}{\pi}\leq \frac{\cos x}{\p... |
232,387 | <p>A discrete random variable $X$ of values in $\mathbb N$ verifies the property that
$$P(X=k)=\cfrac 23 (k+1)P(X=k+1)$$
What is the distribution of $X$?</p>
<p>I found that
$$P(X\ge 0)=\sum_{k=0}^\infty P(X=k)=\sum_{k=0}^\infty\cfrac 23 (k+1)P(X=k+1)=\cfrac 23\sum_{k=1}^\infty kP(X=k)=\cfrac 23\text E(X)=1$$
$\ \ \ ... | André Nicolas | 6,312 | <p>For a counterexample, use $(1,0)$ and $(0,1)$.${}{}{}{}{}{}{}{}$</p>
|
530 | <p>Tau ($\tau = 2 \pi$) has more merits in its application, but pi is the established standard in industry and education. Is the trade-off of teach-ability of circle concepts worth the subsequent confusion due to pi's omnipresence?</p>
<p>How can both be most effectively taught in order to maximize student's understan... | quid | 143 | <p>One should teach $\pi$. One might discuss that there is a choice that is made that is somewhat arbitrary, and there are also reasons for a different choice but I do not see this as that relevant to make much ado about this. </p>
<p>It should perhaps also be noticed that there are two conflicting proposals for $\tau... |
530 | <p>Tau ($\tau = 2 \pi$) has more merits in its application, but pi is the established standard in industry and education. Is the trade-off of teach-ability of circle concepts worth the subsequent confusion due to pi's omnipresence?</p>
<p>How can both be most effectively taught in order to maximize student's understan... | P. Williams | 221 | <p>I cannot imagine a context where it would benefit a student to know about $\tau$ over and above knowing about $\pi$. Every text book a student will ever encounter will exclusively talk about $\pi$. Every calculator a student will ever use will have the constant $\pi$ pre-programmed in.</p>
<p>Although there are pe... |
530 | <p>Tau ($\tau = 2 \pi$) has more merits in its application, but pi is the established standard in industry and education. Is the trade-off of teach-ability of circle concepts worth the subsequent confusion due to pi's omnipresence?</p>
<p>How can both be most effectively taught in order to maximize student's understan... | vonbrand | 123 | <p>There is a simple solution to this debate:</p>
<p><a href="http://xkcd.com/1292/" rel="noreferrer" title="Conveniently approximated as e+2, Pau is commonly known as the Devil's Ratio (because in the octal expansion, '666' appears four times in the first 200 digits while no other run of 3+ digits appears... |
530 | <p>Tau ($\tau = 2 \pi$) has more merits in its application, but pi is the established standard in industry and education. Is the trade-off of teach-ability of circle concepts worth the subsequent confusion due to pi's omnipresence?</p>
<p>How can both be most effectively taught in order to maximize student's understan... | Garrett | 933 | <h1>$\tau$ should be taught in schools</h1>
<p>There's plenty of material arguing why $\tau$ is a much more intuitive and easier to teach concept (some of my favorites: <a href="http://www.maa.org/sites/default/files/pdf/Mathhorizons/apr12_aftermath.pdf">1</a>,<a href="https://www.youtube.com/watch?v=jG7vhMMXagQ">2</a... |
2,122,149 | <p>Using the standard topology on $\mathbb{R}^d$, interior of a subset $S$ of topological space $M$ is defined as the set of all elements of $S$ except the boundary of $S$. </p>
<p>However, I only understand the boundary in this sense to be defined with the standard topology on $\mathbb{R}^d$, so how would we define b... | TheGeekGreek | 359,887 | <p>Let $(X,\mathcal{T})$ be a topological space. For $A \subseteq X$ we define the <strong>interior of A</strong> as $$\operatorname{Int}A:=\bigcup\{C \subseteq X : C \subseteq A, C \in \mathcal{T}\}$$ Furthermore, the <strong>exterior of A</strong> is $$\operatorname{Ext}A := X \setminus \overline{A}$$ where $\overlin... |
451,326 | <p>I am having trouble understanding a certain part of the proof on why a function cannot approach two different limits near $a$, so I will just list the relevant parts. If this is not enough/ambiguous then please tell me and I will type out the whole proof.</p>
<p>So, suppose we now have:</p>
<p>$$ \text{if } 0<|... | Ben Grossmann | 81,360 | <p>Hint:</p>
<p>Let's begin with the expression
$$
\frac{f(t) - f(t_0)}{t - t_0} = \frac{1}{t-t_0}\left( \frac{{t_0}^n \displaystyle\int_{B(y,t)} u(x) \ dx}{|B(0,1)| t^n {t_0}^n} - \frac{{t}^n \displaystyle\int_{B(y,t_0)} u(x) \ dx}{|B(0,1)| t^n {t_0}^n}\right)
$$
And rewrite that, saying $t=t_0+\Delta t$, as
$$
\fra... |
14,167 | <p>I randomly pick a natural number <em>n</em>. Assuming that I would have picked each number with the same probability, what was the probability for me to pick <em>n</em> before I did it?</p>
| Gregory Fenn | 389,331 | <p>Hi :) I've always had a nagging suspicion about drawing (uniform) random numbers from an infinite set. I'm not convinced it's possible: here's an intuition about why I'm sceptical: it would be nice if someone could explain where I go wrong. My counterargument is structured as follows: if it's possible to draw random... |
978,985 | <p>Why must every vector in V belongs to one of the generalised eigenspaces of $T: V \to V?$ Is there a simple proof for this? Can someone provide me with an intuition behind it?</p>
<p>Note that V is an algebraically closed field.</p>
| Yiorgos S. Smyrlis | 57,021 | <p>We have that
$$
S+T=T(I+T^{-1}S).
$$
Define
$$
K=\left(\sum_{n=0}^\infty (-1)^n (T^{-1}S)^n\right)T^{-1}.
$$
Then, the series above converges as $\|T^{-1}S\|<1$, and hence
$K$ is a bounded linear operator, and is readily shown that
$$
K(S+T)=I.$$</p>
|
978,985 | <p>Why must every vector in V belongs to one of the generalised eigenspaces of $T: V \to V?$ Is there a simple proof for this? Can someone provide me with an intuition behind it?</p>
<p>Note that V is an algebraically closed field.</p>
| BBVM | 291,058 | <p>If $X=Y$ in your question, we have
\begin{equation*}
S+T: X \rightarrow X.
\end{equation*}</p>
<p>$S+T$ is a injective map, because $(S+T)(x) = 0 \Rightarrow x= 0 $.</p>
<p>Now, we want to show that for all $y \in X$, exists $x \in X$ such that $(S+T)(x)=y$. See that,
\begin{equation*}
(S+T)(x) = y \Leftrightarrow... |
1,496,676 | <p>let's $F(x)= x'Ax + a'x $</p>
<p>where $x'Ax$ is a quadratic form and $a'$ is defined as a vector. </p>
<p>$$A:= \left[ \begin{matrix} 6 &1&1 \\ 1&2&0 \\ 1&0&4\end{matrix}\right] $$</p>
<p>Does there exist a global minimum point (absolute min) for this function $F(x)$?</p>
<p>note: I know... | xjtein | 241,759 | <p>A is symmetric and we can make it diagonal using an orthogonal matrix. Then we convert our problem into the form:
$$f(x)=\lambda_1 x^2+\lambda_2 y^2+\lambda_3 z^2+ a x+by +c z$$, which is high school math.
We find the eigenvalues of A are about $\{6.6,3.6,1.7\}$, therefore there exists minimum value of f.</p>
|
1,857,630 | <p>Let $\mathcal{A}$ be a non (necessarily) unital commutative Banach algebra, and let
$$ M_{\mathcal{A}} = \{ \phi:\mathcal{A} \to \mathbb{C} : \phi \mbox{ is multiplicative and not trivial}\} $$
and
$$ \mathrm{Max}(\mathcal{A})=\{ I \lhd \mathcal{A} : I \mbox{ maximal} \}.$$
If $\mathcal{A}$ is unital, it is well kn... | Jonas Meyer | 1,424 | <p>The kernels of nonzero homomorphisms to $\mathbb C$ are <em>modular</em> ideals, terminology that might help you find more references.</p>
<p>Without any further restriction on the algebras, using the zero product is a way to provide trivial counterexamples. E.g., take $\mathbb C$ with the $0$ product, which has m... |
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