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 |
|---|---|---|---|---|
466,909 | <p>How many ways can you split the numbers 1 to 5 into two groups of varying size?
For example: '1 and 2,3,4,5' or '1,2 and 3,4,5' or '1,2,3 and 4,5'. How many combinations are there like this? What is the formula?</p>
| Jared | 65,034 | <p>Great question. It seems to me that there are two decisions that should be made. The first is to determine the size of the two groups. After that decision has been made, we must then decide which elements will go in which groups.</p>
<p>Suppose we split the $5$ numbers into a group of size $1$ and a group of siz... |
466,909 | <p>How many ways can you split the numbers 1 to 5 into two groups of varying size?
For example: '1 and 2,3,4,5' or '1,2 and 3,4,5' or '1,2,3 and 4,5'. How many combinations are there like this? What is the formula?</p>
| Barry Cipra | 86,747 | <p>The OP has specified that the order of the groups in the splitting doesn't matter, and also that the two groups must both be non-empty. One of the groups must have the $1$, and that group either has or doesn't have each of the other four numbers, except that it can't have all four (which would leave the other group... |
3,912,722 | <blockquote>
<p>Circle of radius <span class="math-container">$r$</span> touches the parabola <span class="math-container">$y^2+12x=0$</span> at its vertex. Centre of circle lies left of the vertex and circle lies entirely within the parabola. What is the largest possible value of <span class="math-container">$r$</span... | Narasimham | 95,860 | <p><a href="https://i.stack.imgur.com/xx8Ry.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/xx8Ry.png" alt="enter image description here" /></a></p>
<p>It is helps to learn and remember standard forms of the parabola along with the osculating circle radii at their vertices.
<span class="math-containe... |
32,111 | <p>Let p : Y -> X be an n-sheeted covering map, where X and Y are topological spaces. If X is compact, prove that Y is compact.</p>
<p>I realize that this seems like a very simple problem, but I want to stress the lack of assumptions on X and Y. For example, this is very easy to prove if we can assume that X and Y are... | Dick Palais | 7,311 | <p>Well, the obvious argument that any sequence has a convergent subsequence that your three friends used for the metrizeable case generalizes easily to show that any net has a convergent subnet in the general case.</p>
|
2,210,317 | <p>Simplify $\sum^{n}_{k=1} (-1)^k(n-k)!(n+k)!$. </p>
<p>I thought of interpreting a term $(n-k)!(n+k)!=\frac{(2n)!}{\dbinom{2n}{n-k}}$, but I do not know of any ways to sum this efficiently.</p>
| Jack D'Aurizio | 44,121 | <p>By using Euler's Beta function:
$$\sum_{k=1}^{n}(-1)^k \Gamma(n+k+1)\Gamma(n-k+1) = \Gamma(2n+2)\sum_{k=1}^{n}\int_{0}^{1}(-1)^k z^{n+k}(1-z)^{n-k}\,dz $$
equals:
$$ (2n+1)! \int_{0}^{1} z^{n+1}\left[(-z)^n-(1-z)^n\right]\,dz =(2n+1)!\left[\frac{(-1)^n}{2n+2}-B(n+1,n+2)\right]$$
or:
$$ \color{red}{(-1)^n\frac{(2n+1)... |
1,217,805 | <p>Let's say I have a set of linearly independent vectors, collected in a square matrix $\mathbf{M}$.</p>
<p>I know that I could orthogonalize these vectors with the QR decomposition,</p>
<p>$\mathbf{M} = \mathbf{QR}$</p>
<p>where $\mathbf{Q}$ is my orthogonal set of vectors.</p>
<p>I'm curious if it is possible to... | texasflood | 201,600 | <p>If $E$ is full rank (and it will be if $M$ is square and full rank), you can take the square root of it</p>
<p>$$UEV^H$$
$$=UE^{1/2}IE^{1/2}V^H$$</p>
<p>And so you can just absorb $E^{1/2}$ into $U$ and $V$.</p>
<p>It should go without saying that this is no longer a singular value decomposition and may lose all ... |
907,729 | <p>I am sorry if the numbers are not formatted, I have searched but found nothing on how. I am trying to multiply $$(2x^3 - x)\left(\sqrt{x} + \frac {2}{x}\right)$$ together and I arrive at a different answer that does not match the one given. The approach I take is to assign the numbers powers then multiply them by th... | amWhy | 9,003 | <p>We distribute just as we would distribute $$(a + b)(c + d) = a(c+d) + b(c + d) = ac + ad + bc + bd.$$</p>
<hr>
<p>$$(2x^3 - x)(\color{blue}{\sqrt x + 2x^{-1}}) = 2x^3(\color{blue}{\sqrt x + 2x^{-1}}) - x(\color{blue}{\sqrt x + 2x^{-1}})\\ =2x^3\sqrt x + 4x^3x^{-1} - x\sqrt x - 2xx^{-1}\\ = 2x^{6/2}x^{1/2} + \frac ... |
907,729 | <p>I am sorry if the numbers are not formatted, I have searched but found nothing on how. I am trying to multiply $$(2x^3 - x)\left(\sqrt{x} + \frac {2}{x}\right)$$ together and I arrive at a different answer that does not match the one given. The approach I take is to assign the numbers powers then multiply them by th... | Barkas | 170,882 | <p>Terms with negative and fractional exponents are dealt with in very much the same way as natural exponents.</p>
<p>you have</p>
<ul>
<li>$x^a*x^b=x^{a*b}$</li>
<li>$x^{-a}=\frac{1}{x^a}$</li>
</ul>
<p>Sometimes fractional exponents are written as roots. For example the exponent $\frac{1}{2}$ is more commonly know... |
1,794,673 | <p>Prove that there exist two real Gaussian distribution $X$ and $Y$ such that the vector $(X,Y)$ is not Gaussian. </p>
<p>The way that i tried to solve this problem was choosing $X=X$, and take before the linear combination $X-X$ or stuff like that; but in the definition of Gaussian distribution that i have the const... | Community | -1 | <p>At most this requires the relative smoothing theorem: if $f: X \to N$ is $C^0$, and smooth on some closed subset $M$, then it is homotopic to a smooth map, with $f|_M$ fixed by this homotopy. Apply this to $X = S^n \times M$ to see that the map is a surjection on homotopy groups. (That is, we're smoothing a "sphere'... |
3,464,383 | <p>I couldn't find any substantial list of 'strange infinite convergent series' so I wanted to ask the MSE community for some. By <em>strange</em>, I mean infinite series/limits that <strong>converge when you would not expect them to and/or converge to something you would not expect</strong>.</p>
<p>My favorite conver... | Descartes Before the Horse | 592,365 | <p>A series from user <a href="https://math.stackexchange.com/users/276986/reuns">Reuns</a>, which he proves in a <a href="https://math.stackexchange.com/questions/3464938/a-closed-form-for-re-sum-k-1-infty-dfraci-sigma-0kk/3464986#3464986">previous question</a> of mine:
<span class="math-container">$$\sum_{k=1}^\infty... |
3,848,676 | <p>Let <span class="math-container">$n$</span> and <span class="math-container">$k$</span> be nonnegative integers. The binomial coefficient <span class="math-container">$\binom{n}{k}$</span> is equal to the number of ways to choose <span class="math-container">$k$</span> items from a set with <span class="math-contain... | littleO | 40,119 | <p>Let's imagine what happens if we expand
<span class="math-container">$$
(x+y)^3 = \underbrace{(x+y)}_{\text{factor 1}} \underbrace{(x+y)}_{\text{factor 2}} \underbrace{(x+y)}_{\text{factor 3}}
$$</span>
without collecting like terms. The result is
<span class="math-container">\begin{align}
(x + y)^3 &= x\cdot x... |
2,231,899 | <p>How to prove $\text{Con}(PA + \text{Con}(PA))$ in $ZFC$?</p>
<p>I found an example of this claim <a href="https://mathoverflow.net/a/50174/2003">here</a> but I'm looking for a more detailed explanation.</p>
<p>An equivalent statement is: there is no proof in $PA$ that $PA$ is inconsistent.</p>
<p>If $PA$ could pr... | Jonathan Schilhan | 265,128 | <p>Well, in ZFC there is a model of PA + Con(PA), namely the natural numbers. Also the soundness theorem (if a theory has a model, it is consistent) is a theorem in ZFC so it is also a theorem of ZFC that the above is consistent.</p>
<p>Edit: I think I understand your approach now. But I don't think it could lead to ... |
2,231,899 | <p>How to prove $\text{Con}(PA + \text{Con}(PA))$ in $ZFC$?</p>
<p>I found an example of this claim <a href="https://mathoverflow.net/a/50174/2003">here</a> but I'm looking for a more detailed explanation.</p>
<p>An equivalent statement is: there is no proof in $PA$ that $PA$ is inconsistent.</p>
<p>If $PA$ could pr... | Asaf Karagila | 622 | <p>The completeness theorem for first-order logic tells us that a theory $T$ is consistent if and only it has a model. Moreover, since $\sf ZFC$ proves the completeness theorem, in order to show that $\sf PA+\operatorname{Con}(PA)$ is consistent, we just need to show it has a model.</p>
<p>But this is not quite what y... |
818,850 | <p>I want to generate numbers $1$ to $10$ with uniform probability distribution. So I write the numbers $1$ to $10$ in the natural order. I keep writing the next $10$ number block by permuting the first $10$ numbers. Will this long string can be considered random with uniform distribution? Is there a problem?</p>
| Community | -1 | <p>Think of a meterstick. Suppose you mark a point on it.</p>
<p>The meterstick is divided up into centimeters: you can narrow down where the mark is by checking which two centimeter marks it lies between.</p>
<p>Each centimeter is split into ten millimeters. You can further narrow down where the mark is within the c... |
248,167 | <p>I want to know why, when I look at the Julia sets of the quadratic family, I see only a finite number of repeating patterns, rather than a countable infinity of them.</p>
<p>My question is specifically about the interaction of these three theorems:</p>
<p><strong>Theorem 1</strong>: Let $z_0\in\mathbb{C}$ be an re... | Mark McClure | 46,214 | <p>Julia sets are all very closely related to self-similar sets - each one can be thought of as the invariant set of something like an iterated function system. Specifically, the Julia set of $f(z)=z^2 +c$ is the closure of the repelling periodic points of $f$. Thus, it makes sense that the Julia set itself should be a... |
3,911,489 | <p>How to prove <span class="math-container">$$I=\int_{0}^{\infty} \frac{\sin x}{x(x^2+1)}dx=\frac{\pi}{2}(1-\frac{1}{e})\, ?$$</span></p>
<p>Let <span class="math-container">$$f(z)=\frac{\sin z}{z(z^2+1)}=\frac{e^{iz}-e^{-iz}}{2i\,z(z^2+1)}.$$</span></p>
<p>Then I have no idea how to deal with <span class="math-contai... | Bernkastel | 551,169 | <p>You can notice since your <span class="math-container">$f$</span> is even it is
<span class="math-container">$$\int_0^{\infty} f(x) \text{d}x=\frac{1}{2} \int_{-\infty}^{\infty} f(x) \text{d}x$$</span></p>
|
1,226,176 | <p>The following serie is obviously convergent, but I can not figure out how to calculate its sum :</p>
<p>$$\sum \frac{6^n}{(3^{n+1}-2^{n+1})(3^n - 2^n)}$$</p>
| Bumblebee | 156,886 | <p>You can use one of the following identities to telescope the series: <span class="math-container">$$\dfrac{3^{n+1}}{3^{n+1}-2^{n+1}}-\dfrac{3^n}{3^n-2^n}=\dfrac{2^n}{3^n-2^n}-\dfrac{2^{n+1}}{3^{n+1}-2^{n+1}}=\dfrac{6^n}{(3^{n+1}-2^{n+1})(3^n-2^n)}$$</span></p>
|
413,982 | <p>How can I divide this using long division?</p>
<p>$$\frac{ax^3-a^2x^2-bx^2+b^2}{ax-b}$$</p>
<p><strong>Edit</strong></p>
<p>Sorry guys I wrote it wrong... Fixed it now.</p>
| Billy | 39,970 | <p>You may have more luck computing the reciprocal, $\dfrac{ax^3 - a^2x^2 - bx^2 + b^2}{ax-b}$, and then inverting it again at the end.</p>
<p>(When you've done that, try artificially 'setting' $ax = b$ in the expression $ax^3 - a^2x^2 - bx^2 + b^2$ and see what you get. What happens, and why?)</p>
|
413,982 | <p>How can I divide this using long division?</p>
<p>$$\frac{ax^3-a^2x^2-bx^2+b^2}{ax-b}$$</p>
<p><strong>Edit</strong></p>
<p>Sorry guys I wrote it wrong... Fixed it now.</p>
| Michael Hardy | 11,667 | <p>$$
\begin{array}{rccccccccccccc}
& & x^2 & - & ax & - & b \\[12pt]
ax-b & ) & ax^3 & - & (a^2+b)x^2 & + & b^2 \\
& & ax^3 & - & bx^2 \\[12pt]
& & & & -a^2x^2 & + & b^2 \\
& & & & -a^2x^2 & + & abx \\[12pt... |
588,725 | <p>Prove the following statement:</p>
<p>$$\frac{1}{x}<\ln(x)-\ln(x-1)<\frac{1}{x-1}$$</p>
<p>Proof:</p>
<p>$$\frac{-1}{x^2}<\frac{1}{x(x-1)}<\frac{-1}{(x-1)^2}$$</p>
<p>$$e^{(\frac{-1}{x^2})}<e^{(\frac{-1}{x(x-1)})}<e^{(\frac{-1}{(x-1)^2})}$$</p>
<p>$$\lim_{x\to\infty}e^{(\frac{-1}{x^2})}<\li... | Nick Peterson | 81,839 | <p>You've made this much too difficult, and your proof is a bit shady.</p>
<p>Try a different tact:</p>
<p>Note that for any $x>1$,
$$
\int_{x-1}^{x}\frac{1}{t}\,dt=\ln(x)-\ln(x-1).
$$
However, $\frac{1}{t}$ is monotone decreasing on $[x-1,x]$; can you come up with a way to use this to bound the above integral?</p... |
2,046,773 | <p>Given a function $f$ that is differentiable at a point $x_0$, if we define (using the Riemann integral)</p>
<p>$$F(x) = \int_a^x f$$</p>
<p>Can we necessarily say that $F^{\prime}(x)$ is continuous at $x_0$? Going back and forth between $f$ and $F$ confuses me a bit. I <em>think</em> that the Fundamental Theorem o... | zhw. | 228,045 | <p>You can't say $F'$ is continuous at $x_0$ because $F'$ may not exist in a full neighborhood of $x_0.$ Take the interval $[-1,1]$ with $x_0=0$ for example. Choose any sequences $a_n,b_n$ you like such that $1\ge b_1 > a_1 > b_2 > a_2 > \cdots \to 0^+.$ Define $f(x) = x^2$ on each $[a_n,b_n],$ $f=0$ everyw... |
1,012,730 | <p>A purely imaginary number is one which contains no non-zero real component. </p>
<p>If I had a sequence of numbers, say $\{0+20i, 0-i, 0+0i\}$, could I call this purely imaginary? </p>
<p>My issue here is that because $0+0i$ belongs to multiple sets, not just purely imaginary, is there not a valid case to say that... | Vishnu | 693,070 | <p>As other answers say, zero is purely real as well as purely imaginary. Let me give an intuitive explanation based on the graphical representation of a complex number.</p>
<p>In the Argand plane, the points on the real axis represent complex numbers which are "purely real" as their imaginary part is zero. On the oth... |
2,289,754 | <p>Let $f_t: \mathbb R^2 \to \mathbb R$ define by $f_t(x,y) = t \sin(t|x^2+y^2-1|)$.</p>
<p>For all $\phi\in D(\mathbb R^2-\{(0,0)\})$, we denote by $T$ the map given by
$$T(\phi)=\lim_{t\to +\infty}\int_{\mathbb R^2} f_t(x,y) \, \phi(x,y)\, \mathrm{d}x\, \mathrm{d}y$$</p>
<p>How can prove that $T$ is a distribution ... | Michael L. | 153,693 | <p>First, we note that for $\mathrm{supp}(\phi)\subset \Omega = B_{r_2}(0)\setminus B_{r_1}(0)$ for some $0 < r_1 < r_2$, \begin{align*} I_t(\phi) &:= \int_0^{2\pi}\int_{r_1}^{r_2} t\sin(t\lvert r^2-1\rvert)\phi(r, \theta)\,r\,\mathrm{d}r\,\mathrm{d}\theta \\ &= \int_{r_1}^{r_2} t\sin(t\lvert r^2-1\rvert)... |
42,192 | <p>In search for a Machian formulation of mechanics I find the following problem. In Machian mechanics absolute space does not exists, and the only real entities are the relative distances between the particles. As a consequence, the configuration space of a N-particle system is the set of the distances on a set of N e... | Anton Petrunin | 1,441 | <p>Your question is not well stated.
In particular I did not understand why embedding into a plane is "preferred embedding".</p>
<p>Here are some associations...</p>
<p><strong>4-point case.</strong></p>
<p>A generic 4-point metric space can be isometrically embedded into two different model planes (i.e. simply conn... |
382,295 | <p>Please help. I haven't found any text on how to prove by induction this sort of problem: </p>
<p>$$
\lim_{n\to +\infty}1 + \frac{1}{4} + \frac{1}{4^2} + \cdots+ \frac{1}{4^n} = \frac{4}{3}
$$</p>
<p>I can't quite get how one can prove such. I can prove basic divisible "inductions" but not this. Thanks.</p>
| Michael Hardy | 11,667 | <p>Archimedes did this (Thomas Heath's translation of <em>Quadrature of the Parabola</em>, Proposition 23). He showed that (expressed in modern notation)
$$
\underbrace{1+\frac14+\frac1{4^2}+\cdots+\frac1{4^n}}+\underbrace{\frac13\left(\frac1{4^n}\right)}
$$
<b>does not depend on the number of terms.</b> All you ... |
2,608,455 | <p>Could someone please help me with how do I calculate the sum of the $$\sum_{n=1}^{\infty}\frac{1}{4n^{2}-1}$$ infinite series? I see that $$\lim_{n\rightarrow\infty}\frac{1}{4n^{2}-1}=0$$ so the series is convergent based on the Cauchy's convergence test. But how do I calculate the sum? Thank you.</p>
| Dave | 334,366 | <p>$$\frac{1}{4n^2-1}=\frac{1}{2}\left(\frac{1}{2n-1}-\frac{1}{2n+1}\right)$$</p>
|
2,160,671 | <p>I got stuck with this problem.</p>
<p>\begin{matrix}
1 & 1 & 1 & 1\\
0 & 1 & 1 & 0\\
0 & 0 & 1 & 1\\
\end{matrix}</p>
<p>Consider the $3\times 4$ matrix $\bf A$ (above). Do the columns of $\bf A$ span $\mathbb R^3$? </p>
<p>Prove your answer. Als... | mathreadler | 213,607 | <ol>
<li>One way to do this is by <strong>Kronecker products</strong>, where you write matrix multiplication as a matrix and then solve an equation system.</li>
<li>Another is by using some Pseudo-inverse, for example <strong>Moore-Penrose pseudoinverse</strong>. For this example you can calculate it like:</li>
</ol>
... |
3,043,319 | <p>I just begin with the 3- dimension function. I dont really understand how to begin with this problem.
Function <span class="math-container">$$f(x,y)= (x^2+4y^2-5)(x-1)$$</span>
Find where <span class="math-container">$$f(x,y)=0; f(x,y)>0; f(x,y)<0.$$</span></p>
<p>To find where <span class="math-container">$... | Sambo | 454,855 | <p>Showing that these properties hold is a straightforward application of the definitions, with some elementary properties of bijections. So, I suspect what you are having trouble with is formulating a proper proof. I will show reflexivity as an example.</p>
<p>We want to show that for any set <span class="math-contai... |
7,725 | <p>Basically the textbooks in my country are awful, so I searched on the web for a precalculus book and found this one: <a href="http://www.stitz-zeager.com/szprecalculus07042013.pdf">http://www.stitz-zeager.com/szprecalculus07042013.pdf</a></p>
<p>However, it does not cover convergence,limits etc. and those topics we... | Chris C | 201 | <p>It depends on how Calculus is treated. </p>
<p>At my university and others I've attended (US), the concept of the limit is usually treated early in the first Calculus class in order to talk about continuity and derivatives. The core idea of sequences is brushed over in the introductory courses (except in the advanc... |
190,392 | <p>How to prove that the Torus and a Ball have the same Cardinality ?
The proof is using the Cantor Berenstein Theorem.
I now that they are subsets of $\mathbb{R}^{3}$ so I can write $\leq \aleph$ but I do not know how to prove $\geqslant \aleph$.
Thanks</p>
| Ilmari Karonen | 9,602 | <p>I'm assuming that you're talking about solid balls and tori in $\mathbb R^3$.</p>
<p>To prove that these sets are equipollent using the <a href="http://en.wikipedia.org/wiki/Cantor%E2%80%93Bernstein%E2%80%93Schroeder_theorem" rel="nofollow">Cantor–Bernstein–Schröder theorem</a>, all you need to do is to show that:<... |
4,094,333 | <blockquote>
<p>Suppose <span class="math-container">$a_1,a_2>0$</span> and
<span class="math-container">$a_{n+2}=2+\dfrac{1}{a_{n+1}^2}+\dfrac{1}{a_n^2}(n\ge 1)$</span>. Prove
<span class="math-container">$\{a_n\}$</span> converges.</p>
</blockquote>
<p>First, we may show <span class="math-container">$\{a_n\}$</spa... | Anon | 307,340 | <p>Let's use <strong>fixed point iteration</strong>s.</p>
<p><strong>Observe</strong>:
Substituting <span class="math-container">$x_n > 2$</span> gives us a upper bound on <span class="math-container">$x_{n}$</span> (<span class="math-container">$n \geq 3$</span>). And substituting this upper bound back, gives us a ... |
4,094,333 | <blockquote>
<p>Suppose <span class="math-container">$a_1,a_2>0$</span> and
<span class="math-container">$a_{n+2}=2+\dfrac{1}{a_{n+1}^2}+\dfrac{1}{a_n^2}(n\ge 1)$</span>. Prove
<span class="math-container">$\{a_n\}$</span> converges.</p>
</blockquote>
<p>First, we may show <span class="math-container">$\{a_n\}$</spa... | Martin R | 42,969 | <p>Another approach: Let <span class="math-container">$L \approx 2.3593$</span> be the unique solution of <span class="math-container">$L = 2 + 2/L^2$</span> in the interval <span class="math-container">$[2, 2.5]$</span>, and <span class="math-container">$b_n = a_n - L$</span>. We want to show that <span class="math-co... |
3,200,431 | <p>We can say, that any field <span class="math-container">$\mathbb{K}$</span> -- <span class="math-container">$1$</span>-dim vector space on itself: <span class="math-container">$\mathbb{K}_{\mathbb{K}}$</span>. So any vector of one another finite-dimensional vector space <span class="math-container">$V_{\mathbb{K}}$<... | elidiot | 574,565 | <p>Because if <span class="math-container">$\mathbb K$</span> is a field, <span class="math-container">$\mathbb K^n$</span> remains a <span class="math-container">$\mathbb K$</span>-vector space but may not be a field (with the same addition). For example, <span class="math-container">$\mathbb R$</span> is a field, but... |
3,044,390 | <p>As we know, each random variable is responsible for associating some random events to the probability values. These random events belong to the specific population, and that random variable represent that population. In other words, random variables are representatives of their populations. By virtue of this, they p... | J.G. | 56,861 | <p>The classical central limit theorem states that, given a large sample of independent values <span class="math-container">$X_n$</span> from the same finite-<span class="math-container">$\mu$</span>-and<span class="math-container">$\sigma$</span> distribution, <span class="math-container">$\frac{1}{\sqrt{n}}\sum_{i=1}... |
1,357,148 | <p>Stokes' theorem generalizes the fundamental theorem of calculus (part 2) using differential forms. Is there a known generalization of part 1?</p>
<h2>edit</h2>
<p>In case anyone is unaware, The fundamental theorem of calculus part 1 states that the derivative of the map $t \mapsto \int_{a}^{t} f(s) ds$ is equal to... | Bob Terrell | 15,643 | <p>Sorry to have just seen this question 7 years late. An answer is in the Poincar'e Lemma. See Spivak's Calculus on Manifolds for example. I wrote an exposition for the vector field version in 3 dimensions that is on my web page bterrell.net.</p>
|
3,566,997 | <p>When I go through examples of calculating the intersection of two planes, there seems to be a convention of choosing an arbitrary point in order to solve the linear equations in question and get a particular point on the intersection line.</p>
<p>Let's take an example from a <a href="https://math.stackexchange.com/... | Peter Szilas | 408,605 | <p>Option:</p>
<p><span class="math-container">$a^y=x$</span>; </p>
<p>Tale <span class="math-container">$\log_e$</span> of both sides: </p>
<p><span class="math-container">$y \log a=\log x$</span>;</p>
<p>Differentiate with respect to <span class="math-container">$x$</span>:</p>
<p><span class="math-container">$y... |
951,104 | <p>Problem :
If $$T_r =\frac{r}{4r^4+1}$$ then the value of $$\sum^{\infty}_{r=1} T_r$$ is ? </p>
<p>How to start such problem I am not getting any clue on this please suggest thanks .</p>
| Mathronaut | 53,265 | <p>Hint:$$4r^4+1=(2r^2+1+2r)\cdot(2r^2+1-2r)$$</p>
|
2,450,838 | <p>So I am hoping to show that $f(x,y)=\dfrac{x}{y}$ is continuous when $x>0$ and $y>0$.</p>
<p>I am not sure how to approach this problem. My idea was that taking $$\dfrac{\partial f}{\partial x}=\dfrac{1}{y}$$ and $$\dfrac{\partial f}{\partial y}=\dfrac{-x}{y^2}$$</p>
<p>My logic was that since both partials ... | zwim | 399,263 | <p>For $f(x)$ and $g(y)$ continuous then in $(x_0,y_0)$ where $g(y_0)\neq0$ we have:</p>
<ul>
<li>$\forall \varepsilon>0,\exists \delta_1>0\mid |x-x_0|<\delta_1\implies |f(x)-f(x_0)|<\varepsilon$</li>
<li>$\forall \varepsilon>0,\exists \delta_2>0\mid |y-y_0|<\delta_2\implies |g(y)-g(y_0)|<\vare... |
833,353 | <p>Let $a,b \geq 0$ and $0<p,q < 1$ s.t. $p + q = 1$.</p>
<p>Is it true that $a^p b^q \leq a+b$?</p>
| Sandeep Silwal | 138,892 | <p>The comments given are counter examples. A famous inequality that resembles yours is</p>
<p>$a,b,p,q$ are positive real numbers and $\frac{1}p+\frac{1}q = 1$ then,</p>
<p>$$\frac{a^{p}}{p} + \frac{b^{q}}{q} \ge ab.$$</p>
|
3,489,896 | <p>It is clear to me that <span class="math-container">$\liminf X_n \le \limsup X_n$</span> but every time that I see that <span class="math-container">$\liminf X_n \le \sup X_n$</span> it sound to me not so obvious awkward. Could someone help me to understand this inequality?<span class="math-container">$ $</span> <sp... | Bernard | 202,857 | <p>By definition,
<span class="math-container">$$\limsup X_n=\lim_k\Bigl(\sup_{n\ge k} X_n\Bigr), $$</span>
and the sequence <span class="math-container">$(S_k)=\bigl(\sup_{n\ge k} X_n\bigr)_k$</span> is <em>non-increasing</em>, so
<span class="math-container">$$\liminf X_n\le\limsup X_n\le S_1=\sup X_n.$$</span></p>... |
2,600,062 | <p>I'm stuck at doing this problem. It's actually a problem related to circuit analysis. I'm given this equation:
$$\frac{(R_3+\frac{1}{\omega C}I)*(\omega Li)}{(R_3+\frac{1}{\omega Ci})+(\omega Li)}$$
Now to rationalize it I will multiply by
$$\frac{R_3+\frac{1}{\omega C}i-\omega Li}{R_3+\frac{1}{\omega C}i-\omega Li}... | Dr. Sonnhard Graubner | 175,066 | <p>use that $$\frac{R_3-\left(\frac{1}{\omega C}\omega L\right)}{R_3+i\left(\frac{1}{\omega C}-\omega L\right)}$$ and multiply numerator and denominator by $$R_3-i\left(\frac{1}{\omega C}-\omega L\right)$$</p>
|
509,487 | <p>Prove $1+{n \choose 1}2+{n \choose 2}4+...+{n \choose n-1}2^{n-1}+{n \choose n}2^n=3^n$ using combinatorial arguments. I have no idea how to begin solving this, a nudge in the right direction would be appreciated. </p>
| anon | 11,763 | <p>Hint: $\sum\binom{n}{k}2^k$ counts the pairs $(A,f)$ with $A\subseteq\{1,\cdots,n\}$ and $f:A\to\{1,2\}$.</p>
|
3,682,514 | <blockquote>
<p>The hyperbola is given with the following equation: <span class="math-container">$$3x^2+2xy-y^2+8x+10y+14=0$$</span>
Find the asymptotes of this hyperbola. (<span class="math-container">$\textit{Answer: }$</span> <span class="math-container">$6x-2y+5=0$</span> and <span class="math-container">$2x+2y... | amd | 265,466 | <p>You say that you got stuck when trying <span class="math-container">$k=\frac13$</span>. That’s likely because that’s not the slope of either asymptote. If a line has direction vector <span class="math-container">$(l,m)$</span>, then its slope is equal to <span class="math-container">$\frac ml$</span>, not <span clas... |
680,660 | <p>Recently I came across this question:</p>
<p>Given a random permutation of integers 1, 2, 3, …, n with a discrete, uniform distribution, find the expected number of local maxima. (A number is a local maxima if it is greater than the number before and after it.) For example, if n=4 and our permutation was 1, 4, 2, 3... | André Nicolas | 6,312 | <p>Let $n\ge 3$. Suppose that the permutation $\alpha$ takes $k$ to $a_k$ ($k=1$ to $n$). Let $\alpha'$ be the permutation that takes $k$ to $(n+1)-a_k$. Then the number of local maxima (minima) of $\alpha$ is the same as the number of local minima (maxima) of $\alpha'$. Thus by symmetry the expected number of local ma... |
3,923 | <p>I propose the following lemma and its proof. It is related to row-null and column-null matrices - i.e. matrices whose rows and columns both sum to zero. Could you please give your opinion on the plausibility of the lemma, and the validity of the proof?</p>
<p><strong>Lemma:</strong> Let $Z\in\text{GL}(n,\mathbb{R})... | Laurent Lessard | 1,356 | <p>Here is an alternate way of proving your Lemma. I'm not sure if its any simpler than your proof -- but it's different, and hopefully interesting to some.</p>
<p>Let $S$ be the set of $n\times n$ matrices which are row-null and column-null. We can write this set as:
$$
S = \left\{ Y\in \mathbb{R}^{n\times n} \,\mid\... |
631,054 | <p>Let $A$ is in $M_3(\mathbb R^3)$ which is not a diagonal matrix. Pick out the cases when $A $ is diagonalizable over $\mathbb R$:</p>
<p>a. when $A^2=A$;</p>
<p>b. when $(A-3I)^2=0$;</p>
<p>c. when $A^2+I=0$.</p>
<p>My attempt is if $A$ is diagonalizable then there is some invertible $P$ s.t. $PAP^{-1}$=$D$. </p... | Robert Lewis | 67,071 | <p>Here's how I would dispose of these issues:</p>
<p>For (c.), note that a matrix $A \in M_3(R)$ has at least one real eigenvalue, since its characteristic polynomial is of degree 3, and real polynomials of odd degree have at least one real root. But if $A^2 + I = 0$, the eigenvalues $\lambda$ of $A$ must satisfy $\... |
3,556,490 | <p>Can someone explain me how to solve the equation <span class="math-container">$\frac{x-n}{x+n} = e^{-x}$</span>, where <span class="math-container">$n$</span> is a non-zero natural number ? Unfortunately, I have not even an idea how to start. Any hint is much appreciated. </p>
<p>Many thanks in advance. </p>
| Claude Leibovici | 82,404 | <p>The solution of <span class="math-container">$$ e^{-x}=\frac{x-n}{x+n} $$</span> is given in terms of the <a href="https://arxiv.org/abs/1408.3999" rel="nofollow noreferrer">generalized Lambert function</a> (have a look at equation <span class="math-container">$(4)$</span> in the linked paper). It is far to be eleme... |
1,457,797 | <p>I'm told that a circle intersects the y-axis at $(0,0)$ and $(0, -4)$.</p>
<p>I have a tangent that starts at $(0, -6)$. I want to find the point of intersection.</p>
<p>I am taking the midpoint of the circle to be $(0, -2)$ and the radius to be $2$ from the points above.</p>
<p>The equation of the circle then is... | Tushant Mittal | 272,305 | <p>Your assumption that the tangent at (0,-6) is y=-6 is incorrect as this line doesn't touch the circle and therefore you have no point on the real plane that satisfies both the circle and the line.</p>
<p>You will have to find the equation of tangent using its property that the perpendicular distance of line from ce... |
1,157,528 | <p>How to find the corresponding probability distribution function for the following distribution function ?</p>
<p>$$F (x)=
\left\{
\begin{array}{ll}
0 & \text{if } x<0 \\
x^2 & \text{if } 0\leq x\leq \frac{1}{2} \\
\frac{1}{25} \left(1-3 (3-x)^2\right) & \text{if }\frac{1}{2}<x\leq 3 \\
1 ... | Chinny84 | 92,628 | <p>$$
F(x) =
\begin{cases}
0, & \text{if $\,x$<0} \\
x^2, & \text{if 0$\,\leq x \leq \frac{1}{2}$}\\
\frac{1}{25}\left(1-3(3-x)^2\right), & \text{if $\,\frac{1}{2} < x < 3$}\\
1, & \text{if $\,3 \leq x $}
\end{cases}
$$
thus
$$
f(x) =
\begin{cases}
2x, & \text{if 0$\,\leq x \leq \frac{1}{... |
3,879,201 | <p>Let <span class="math-container">$E$</span> be a normed <span class="math-container">$\mathbb R$</span>-vector space, <span class="math-container">$\mu$</span> be a finite signed measure on <span class="math-container">$(E,\mathcal B(E))$</span> and <span class="math-container">$$\hat\mu:E'\to\mathbb C\;,\;\;\;\varp... | Marc | 503,755 | <p>Your answer is correct.</p>
<p>Note that it might be easier to <em>start</em> dividing by <span class="math-container">$X$</span> since you will have one factor out right away. Dividing by <span class="math-container">$X$</span> is fine; it's just dividing by a linear term <span class="math-container">$X-a$</span> w... |
1,927,650 | <p>In the diagram below, DE is a chord of the circle that goes through c,d and e. A is the center of the circle. The perpendicular line from centre A intersects DE at B and the circle at C, </p>
<p>DE=100cm
BC= 10cm
AB=X</p>
<p>I need to calculate the length of AB / x.</p>
<p>Basically i need to calculate the radius... | N.S.JOHN | 302,172 | <p>You have $DB= 50$ and $AD=AC=x+10$</p>
<p>Now use Pythagoream theorem and find $x$</p>
|
2,993,299 | <p>If anyone can help me with how to go about solving these kind of equations i would really appreciate it. :-)</p>
<p><span class="math-container">$$\sqrt{36-2x^2} = 4$$</span></p>
<p>Solve for X</p>
| user | 505,767 | <p>I often use </p>
<p><span class="math-container">$$\ldots=\lim_ {x \to \infty}\frac{x+\frac1x}{1-\frac1x}=\left(\frac{\infty+0}{1-0}\right)=\infty$$</span></p>
<p>or directly as <span class="math-container">$x\to\infty$</span></p>
<p><span class="math-container">$$\ldots=\frac{x+\frac1x}{1-\frac1x}\to \infty$$</s... |
3,219,260 | <p>Trying to find the easiest and most general method in finding last <span class="math-container">$n$</span> digits of a number.
I know the trick lies in finding the remainder when the number is divided by <span class="math-container">$10^n$</span> but still not able to perform the rest of the steps to reach the answe... | J. W. Tanner | 615,567 | <p>Repeated squaring:</p>
<p><span class="math-container">$$7^4 = 2401 \mod 10000$$</span>
<span class="math-container">$$7^8 = 2401^2\equiv 4801 \mod 10000$$</span>
<span class="math-container">$$7^{16}\equiv 4801^2\equiv 9601 \mod 10000$$</span>
<span class="math-container">$$7^{32}\equiv 9201 \mod 10000$$</span>
<s... |
3,219,260 | <p>Trying to find the easiest and most general method in finding last <span class="math-container">$n$</span> digits of a number.
I know the trick lies in finding the remainder when the number is divided by <span class="math-container">$10^n$</span> but still not able to perform the rest of the steps to reach the answe... | lab bhattacharjee | 33,337 | <p><span class="math-container">$7^{128}\equiv(1-50)^{64}\equiv1-64\cdot50+\binom{64}250^2-\binom{64}350^3+\cdots+50^{64}$</span></p>
<p><span class="math-container">$\equiv1-64\cdot50\pmod{10^4}$</span></p>
<p>as <span class="math-container">$10^4$</span> divides <span class="math-container">$50^n,n\ge4,$</span></p>... |
3,030,317 | <p>I have four points on a rectangular grid <span class="math-container">$(x_1,y_1)$</span>, <span class="math-container">$(x_1,y_2)$</span>, <span class="math-container">$(x_2,y_1)$</span> and <span class="math-container">$(x_2,y_2)$</span>. I also have the value of a third variable <span class="math-container">$z$</... | Community | -1 | <p>Interpolate along the grid lines using Hermite cubic splines. <a href="https://en.wikipedia.org/wiki/Cubic_Hermite_spline" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Cubic_Hermite_spline</a></p>
<p>Then inside a tile, use the Coons method. <a href="https://en.wikipedia.org/wiki/Coons_patch" rel="nofoll... |
1,580,736 | <p>I know that $\arcsin(x) + \arccos(x) = \frac{\pi}2$,</p>
<p>but how to use that to solve the following question?</p>
<p>$$2\arcsin(x)-3\arccos(x)=\frac{\pi}6 $$</p>
| Mythomorphic | 152,277 | <p><strong>Hint:</strong></p>
<p>Rearrange your known identity into:
$$\arcsin(x) =\frac{\pi}2-\arccos(x)$$</p>
<p>and substitute it into the equation. Then the equation should be in terms of $\arccos x$. Solve for $x$.</p>
|
4,412,371 | <p>Would appreciate some help with proving the following statement:</p>
<p>Let <span class="math-container">$a_{n}$</span> be a sequence, and <span class="math-container">$\lim _{n\rightarrow \infty }a_{n}=1 $</span>.</p>
<p>Prove that <span class="math-container">$\lim _{n\rightarrow \infty }a_{1}+a_{2}+\ldots +a_{n}=... | Kavi Rama Murthy | 142,385 | <p>There exists <span class="math-container">$m$</span> such that <span class="math-container">$a_n >\frac 1 2$</span> for <span class="math-container">$n \geq m$</span>. So, for <span class="math-container">$n >m$</span> we have <span class="math-container">$$a_1+a_2+\cdots+a_{m-1}+a_m+a_{m+1}+\cdots+a_n$$</sp... |
3,570,084 | <blockquote>
<p>find the value of <span class="math-container">$f(0)$</span>, given <span class="math-container">$f(f(x))=0$</span> has roots as 1 & 2, <span class="math-container">$f(x) = x^2 + \alpha x + \beta$</span></p>
</blockquote>
<p><strong>My attempt:</strong></p>
<p>If <span class="math-container">$k_... | Vishu | 751,311 | <p>Note that <span class="math-container">$f(f(x))= (x^2+ax+b)^2 + a(x^2+ax+b) + b$</span></p>
<p>Put <span class="math-container">$x=1,x=2$</span> and set to 0:</p>
<p><span class="math-container">$(1+a+b)^2 + a(1+a+b) + b = 0$</span> <strong>_(i)
<span class="math-container">$(4+2a+b)^2 + a(4+2a+b) + b =0$</span> _... |
128,009 | <p>Hi,
Is there any known sequence such that the sum of a combination of one subsequence never equals another subsequence sum. The subsequences should have elements only from the parent sequence.</p>
<p>Thanks
Sundi</p>
| Community | -1 | <p>There is the following description of $G$ invariant pseudodifferential operators on a Riemannian homogeneous space $G/H$: The Schwartz kernels are smooth outside the diagonal and conormal with respect to the diagonal; this is equivalent to Beals' commutator characterization mentioned in Pedro Lauridsen Ribeiro's ans... |
1,851 | <p>Is it possible to modify closing reasons menu? If so I would love to see a small addition, maybe others have other wishes too.</p>
<p>So, the last <a href="https://mathematica.meta.stackexchange.com/q/1010/5478">Granular off-topic close reasons</a> lead to creating</p>
<blockquote>
<p>This question arises due to... | Jason B. | 9,490 | <p>No it shouldn't because then it will be used more often and I don't think that being "too localized and unlikely to help future visitors" is, by itself, a valid reason to close a question. </p>
<p>If it's localized and unlikely to help future visitors and not of interest to you, then leave it unanswered. If the q... |
3,293,233 | <p>My robot has a laser time-of-flight distance sensor. Holding the distance and target constant: When I take single set of readings, three standard deviations usually is 3% to 6% of the mean.</p>
<p>If I take multiple sets of readings, each set will show three standard deviations to be 3-6% of the mean, BUT, averagi... | eyeballfrog | 395,748 | <p>If the subsets are all the same size, the averages are necessarily the same. Suppose you divide your <span class="math-container">$N$</span> measurements <span class="math-container">$a$</span> into <span class="math-container">$n$</span> groups of <span class="math-container">$m$</span>. Then
<span class="math-cont... |
4,232,539 | <p><a href="https://en.wikipedia.org/wiki/Braikenridge%E2%80%93Maclaurin_theorem" rel="nofollow noreferrer">Braikenridge–Maclaurin theorem</a> is the converse to <a href="https://en.wikipedia.org/wiki/Pascal%27s_theorem" rel="nofollow noreferrer">Pascal's theorem</a>.</p>
<p><a href="https://i.stack.imgur.com/g70d9.jpg... | brainjam | 1,257 | <p>If you search for information like "sympy determinant slow" you get lots of hits. The gist of the problem is that the det() function seems to be trying to simplify as it goes, and can get bogged down once the matrices are <span class="math-container">$4\times 4$</span> or larger.</p>
<p>One problem with y... |
4,232,539 | <p><a href="https://en.wikipedia.org/wiki/Braikenridge%E2%80%93Maclaurin_theorem" rel="nofollow noreferrer">Braikenridge–Maclaurin theorem</a> is the converse to <a href="https://en.wikipedia.org/wiki/Pascal%27s_theorem" rel="nofollow noreferrer">Pascal's theorem</a>.</p>
<p><a href="https://i.stack.imgur.com/g70d9.jpg... | auntyellow | 919,440 | <p>I found simply <code>Matrix(mat).det(method='domain-ge')</code> works. See <a href="https://docs.sympy.org/latest/modules/matrices/matrices.html#sympy.matrices.matrices.MatrixDeterminant.det" rel="nofollow noreferrer">here</a> for details.</p>
<p>Full code:</p>
<pre><code>from sympy import *
def L(a, b):
return... |
114,909 | <p>Consider this simple code:</p>
<pre><code>Grid[RandomInteger[{0, 1}, {10, 10}], Background -> LightBlue]
</code></pre>
<p><a href="https://i.stack.imgur.com/PfWwb.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/PfWwb.png" alt="output"></a></p>
<p>Is it possible to modify the code so that the... | ubpdqn | 1,997 | <pre><code>r = RandomInteger[{0, 1}, {10, 10}];
bg = Thread[SparseArray[r]["NonzeroPositions"] -> LightBlue];
Grid[r, Background -> {None, None, bg}]
</code></pre>
<p><a href="https://i.stack.imgur.com/jmdPS.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/jmdPS.png" alt="enter image descriptio... |
1,406,462 | <p>I've been studying an introductory book on set theory that uses the <a href="https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory">ZFC</a> set of axioms, and it's been really exciting to construct and define numbers and operations in terms of sets. But now I've seen that there are other alternatives to ... | Paul Sinclair | 258,282 | <p>Yes. Category theory is one example of a theory than transcends ZFC. Some other examples include Hyperreals and Surreals. There are many more.</p>
|
1,391,014 | <p>Let $f,g:[0,1] \rightarrow [0,1]$ be continuous functions such that $f\circ g =g\circ f$. Prove that there exists $x \in [0,1]$ such that $f(x)=g(x)$</p>
| Paul Sinclair | 258,282 | <p>I haven't worked through it all, but I would start by assuming wlog that $f(0) < g(0)$, then use the composition reversal to show that $f < g$ cannot hold everywhere. Then the result will follow from the intermediate value theorem.</p>
|
2,495,789 | <p>\begin{pmatrix}4&-6&2&-6\\ \:-2&3&-1&3\end{pmatrix}</p>
<p>Am I supposed to move the $0 $ to the other side of the equation and find the inverse of the matrix above or is there another way?</p>
| Tancredi | 487,564 | <p>A $4\times 2$ matrix <strong>can not have</strong> an inverse. You are searching for the kernel of $A$. With a few <a href="https://en.wikipedia.org/wiki/Gaussian_elimination" rel="nofollow noreferrer">Gauss moves</a> on the rows of $A$ you can simplify the sistem $A·x = 0$ getting the solutions. In this case the so... |
1,046,582 | <p>Question : </p>
<p>$$\lim_{x \to 0} \frac{e^{\tan^3x}-e^{x^3}}{2\ln (1+x^3\sin^2x)}$$</p>
<p>Here I tried $\tan x$ and $\sin x$ expansion in numerator and denominator which are as follows : </p>
<p>$$\tan x =x +\frac{x^3}{3}+\frac{2x^5}{15} \cdots$$ </p>
<p>$$\sin x = x-\frac{x^3}{3!}+\frac{x^5}{5!}\cdots$$ </p>... | Dr. Sonnhard Graubner | 175,066 | <p>the series of your term is $${\frac {1}{2}}+{\frac {8}{15}}{x}^{2}+{\frac {1}{2}}{x}^{3}+{\frac {
367}{945}}{x}^{4}+O \left( {x}^{5} \right)
$$ thus the limit for $x$ tends to zero exists and is $$\frac{1}{2}$$.
Sorry for my mistake i corected it.</p>
|
2,536,877 | <p>I have to evaluate this integral:
$$\int \frac{\tan^5 x}{\cos^3{x}} dx.$$</p>
<p>I guess I should use substitution, but I don't know what to substitute =(</p>
| Community | -1 | <p><strong>Hint</strong></p>
<p>Notice that
$$\frac{\tan^5x}{\cos^3x}=\frac{\sin^5x}{\cos^8x}$$
Then if you substitute $t=\cos x$ you get
$$-\frac{\left(1-t^2\right)^2}{t^8}$$
and then just integrate each term.</p>
|
1,287,236 | <p>Need to calculate $\sin 15-\cos 15$? In degrees. I got zero, but it is wrong. Though, it seems to me that I was solving correct, I was doing this was:
$\sin (45-30) - \cos (45-30) = ((\sqrt(6)-\sqrt(2)/4) - ((\sqrt(6)+\sqrt(2)/4) =0$</p>
| Mythomorphic | 152,277 | <p>The second last step</p>
<p>$$\frac{(\sqrt(6)-\sqrt(2)}4 - \frac{(\sqrt(6)+\sqrt(2)}4 = \frac{-2\sqrt2}4=\frac{-\sqrt{2}}2$$</p>
|
2,636,929 | <p>I know that it can be obtained by simply differentiating the equation and finding the roots of the derivative, but it is a lengthy and tricky process. I am looking for a faster and more straightforward way.</p>
<p>A more effective and quick way to find the answer via simple differentiation will also be appreciated.... | Community | -1 | <p><strong>Hint</strong>: Use the fact that $\arcsin x+\arccos x=\frac{\pi}{2}$ on $x\in[-1,1]$: let $t=\arccos x\in[0,\pi]$, $\arcsin x=\frac{\pi}{2}-t\in\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$ and you need to minimise/maximise $(\frac{\pi}{2}-t)^4+t^4$. Now it is a lot easier to find the derivatives etc.</p>
|
4,236,875 | <p>Today, I was solving a problem which is described below.</p>
<blockquote>
<p>Show that the equation of any circle passing through the points of intersection of the ellipse <span class="math-container">$(x + 2)^2 + 2y^2 = 18$</span> and the ellipse <span class="math-container">$9(x - 1)^2 + 16y^2= 25$</span> can be w... | orangeskid | 168,051 | <p>HINT:</p>
<p>You have an equation of one circle <span class="math-container">$x^2 + y^2 = 5$</span> that passes through both points, and now add to that a multiple of the equation of the line through both points.</p>
|
1,506,719 | <p>Let us have measurable spaces $(S_1, \Sigma_1)$ and $(S_2, \Sigma_2)$.
Idea of measurable function $f$ with respect to $\Sigma_1,\Sigma_2$ is the following. $f:$ $S_1 \to S_2$ has to be such that:
$$\forall E_2: f^{-1}(E_2) \in \Sigma_1, $$
where
$ E_2 \subset \Sigma_2, E_1 \subset \Sigma_1 $.</p>
<hr>
<p>Anoth... | Alp Uzman | 169,085 | <p>John Dawkins gives a nice explanation in his answer, but I'd like to mention some further details. I believe you are confused mainly due to notation, or a so-called "notational abuse" we like to do, namely we use the same notation for a function that maps "points" to "points" and its associated image function that m... |
29,114 | <p>In <a href="https://puzzling.stackexchange.com/questions/72494/use-2-0-1-and-9-to-make-34">this</a> question on the <a href="https://puzzling.stackexchange.com/"><em>Puzzling Stack Exchange</em></a>, I wrote <a href="https://puzzling.stackexchange.com/questions/72494/use-2-0-1-and-9-to-make-34/72504#72504">this</a> ... | Davide Cervone | 7,798 | <p>MathJax handles stretchy characters like the over-and under-braces as follows: for smaller versions, it has several sizes to choose from in its fonts (just as TeX does), and will use one of these single-character versions when possible. For larger versions, the stretchy character is made from several glyphs pieced... |
241,494 | <p>Who has hint how to prove:
$\sum_{n=0}^N \sum_{k=1}^N g_k y_n = \sum_{n=1}^N \sum_{k=1}^n g_k y_{n-k}+\sum_{n=1}^N\sum_{k=1}^n g_n y_{N-k+1}$</p>
<p>Thank in advance!</p>
| Elias Costa | 19,266 | <p>We have
$$
\begin{align}
\sum_{k=1}^N \sum_{n^\prime=0}^N g_k y_n
& = \sum_{\substack{1\leq n\leq N\\ 1\leq k \leq N}} g_k y_n
\\
& = \sum_{(k,n)\in G} g_k y_n
\\
\end{align}
$$</p>
<p>Here $G=\{ (k,n) \in\mathbb{N}\times\mathbb{N} : 1\leq n\leq N, 1\leq k \leq N\}$. Set
$$
\begin{align}
G_1 &=\{ (k... |
108,781 | <p>I want to add arrowheads and labels to the following plot. </p>
<pre><code>P1 = Plot[x, {x, 0, 2}, PlotStyle -> {Dashed, Red}, Filling -> Bottom];
P2 = Plot[6 x, {x, 0, 2}, PlotStyle -> {Dashed, Blue}];
Show[P1, P2, AspectRatio -> Automatic, Frame -> True,
PlotRangePadding -> None, AxesOrigin -... | george2079 | 2,079 | <p>Draw with graphics primitives using <code>Epilog</code></p>
<pre><code>P1 = Plot[x, {x, 0, 2}, PlotStyle -> {Dashed, Red}, Filling -> Bottom];
P2 = Plot[6 x, {x, 0, 2}, PlotStyle -> {Dashed, Blue}];
Show[P1, P2, AspectRatio -> Automatic, Frame -> True,
PlotRangePadding -> None, AxesOrigin -> ... |
1,001,913 | <p>If we want to graph a horizontal line, we will do the following:</p>
<pre><code>y = 0x + 3
</code></pre>
<p>No matter the domain for x, the range for y will always be 3. Therefore, we have a horizontal line.</p>
<pre><code>y = 0(0) + 3 = (0,3)
y = 0(1) + 3 = (1,3)
y = 0(2) + 3 = (2,3)
</code></pre>
<p>Now the fo... | Hypergeometricx | 168,053 | <p>This is an interesting question.</p>
<p>Using upper negation, applying the Vandermonde identity, using upper negation again and applying symmetry gives the following:</p>
<p>$$\begin{align}
\sum_{t=0}^{n-m}{\color{red}t+k\choose \color{red}t}{n-k-\color{red}t\choose n-m-\color{red}t}&=\sum_{t=0}^{n-m}{-k-1\ch... |
3,986,812 | <p>Given <span class="math-container">$U_n = (1 + 1/n)^n$</span> , <span class="math-container">$n = 1,2,3,\ldots\;.$</span></p>
<blockquote>
<p>Show that <span class="math-container">$2 \le U_n \le 3$</span> for all n</p>
</blockquote>
<p>This is what I've done. Can anyone help?</p>
<p><span class="math-container">$$\... | N. S. | 9,176 | <p><strong>Hint</strong> Let
<span class="math-container">$$U_n = (1 + 1/n)^n \\
V_n=U_n = (1 + 1/n)^{n+1}$$</span></p>
<p>It is clear that <span class="math-container">$U_n < V_n$</span> for all <span class="math-container">$n$</span>. Use Bernoulli inequality to show that</p>
<p><span class="math-container">$$\fra... |
1,136,347 | <p>I'm trying to compute</p>
<p>$$
\omega^{3n/2 + 1} + \omega
$$</p>
<p>where $\omega$ is one of the $n^{th}$ roots of unity where $n$ is a multiple of $4$. Could anyone demonstrate how to do this?</p>
| Timbuc | 118,527 | <p>$$\omega^{\frac{3\cdot4k}2+1}=\omega^{6k+1}=\omega^{4k}\omega^{2k+1}=\omega^{2k+1}$$</p>
<p>If $\;\omega\;$ is a <em>primitive</em> roots of order $\;n=4k\;$ , then $\;\omega^{2k}=-1\;$ (why?) , so</p>
<p>$$\omega^{3n/2 +1}+\omega=0$$</p>
|
1,371,921 | <p>The intersection defined by the two planes $v \bullet \begin{pmatrix} 8 \\ 1 \\ -12 \end{pmatrix} = 35$
and
$v \bullet \begin{pmatrix} 6 \\ 7 \\ -9 \end{pmatrix} = 70$
is a line. What is the equation of this line?</p>
<p>This is what I have so far: I set $v = \begin{pmatrix} a \\ b \\ c \end{pmatrix}$. I was able ... | Tucker | 256,305 | <p>With two equations and three unknowns your solution space will be a one-dimensional set (a line), exactly what you are looking for. You will need to write $(a,b,c)$ in terms of one of the parameters. For instance $a=f_{1}(c),b=f_{2}(c),c=c$.</p>
<p>It will not always be the case that a solution space will be one-di... |
2,354,094 | <p>I've been stumped by this problem:</p>
<blockquote>
<p>Find three non-constant, pairwise unequal functions $f,g,h:\mathbb R\to \mathbb R$ such that
$$f\circ g=h$$
$$g\circ h=f$$
$$h\circ f=g$$
or prove that no three such functions exist.</p>
</blockquote>
<p>I highly suspect, by now, that no non-trivial ... | Kamil Maciorowski | 331,040 | <p>(This is by no means a comprehensive and strict answer, just few observations I made).</p>
<hr>
<p>If you removed $0$ from the domain, the functions might be:</p>
<p>$$f_0 (x) \equiv -x$$
$$g_0(x) \equiv \frac 1 x$$
$$h(x) \equiv -\frac 1 x$$</p>
<p>Or better don't remove $0$ but allow $\infty$ into the domain a... |
2,354,094 | <p>I've been stumped by this problem:</p>
<blockquote>
<p>Find three non-constant, pairwise unequal functions $f,g,h:\mathbb R\to \mathbb R$ such that
$$f\circ g=h$$
$$g\circ h=f$$
$$h\circ f=g$$
or prove that no three such functions exist.</p>
</blockquote>
<p>I highly suspect, by now, that no non-trivial ... | tomasz | 30,222 | <p>For the non-continuous version, you can just consider the three imaginary quaternion units, acting by left multiplication on the space of quaternions (or just the unit quaternions) -- it has the same cardinality as the reals, so it resolves the problem in the non-continuous case.</p>
|
2,829,121 | <p>I get how to derive the ellipse equation, but I'm struggling to understand what it means intuitively. </p>
<p>You see, a circle equation can be understood very intuitively. The circle equation models how the radius of the circle can be represented using the Pythagorean theorem. But I don't understand what the ellip... | Roman Odaisky | 271,020 | <p>The equation $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ tells one which points belong to the ellipse and which don’t. Take a point with coordinates $(x, y)$, calculate the above. Is it 1? Excellent, it’s on the ellipse. Not 1? Then it’s some other point.</p>
<p>If you want to transform the equation into something that act... |
1,945,966 | <p>It is asked to prove that $\lim\limits_{n\to \infty} \sqrt{x_n} = \sqrt{\lim\limits_{n\to \infty} x_n}$, and suggested to use the following two inequalities:</p>
<p>$$a+b\leq a+ 2\sqrt{a}\sqrt{b}+b$$
$$\sqrt{b}\sqrt{b}\leq \sqrt{a}\sqrt{b}$$</p>
<p>The second inequality holds iff $a\geq b \geq 0$.</p>
<p>I've tri... | Michael Rozenberg | 190,319 | <p>It's wrong of course. </p>
<p>For $x_n\geq0$ it's true because $f$ is a continuous function, where $f(x)=\sqrt{x}$. </p>
|
1,945,966 | <p>It is asked to prove that $\lim\limits_{n\to \infty} \sqrt{x_n} = \sqrt{\lim\limits_{n\to \infty} x_n}$, and suggested to use the following two inequalities:</p>
<p>$$a+b\leq a+ 2\sqrt{a}\sqrt{b}+b$$
$$\sqrt{b}\sqrt{b}\leq \sqrt{a}\sqrt{b}$$</p>
<p>The second inequality holds iff $a\geq b \geq 0$.</p>
<p>I've tri... | Ginger88895 | 370,007 | <p>You have to make sure that <span class="math-container">$\forall n\in\mathbb{N}^*$</span>, <span class="math-container">$x_n\geq 0$</span>. Otherwise the limit doesn't exist because the sequence <span class="math-container">$\{\sqrt{x_n}\}_{n=1}^{\infty}$</span> isn't even defined(well, in <span class="math-containe... |
2,873,636 | <p>I have been self studying mathematical logic through A Concise Introduction to Mathematical Logic by Wolfgang Rautenberg and got stuck. I am unable to understand how the value matrix corresponds with the truth table of a given operation for instance the AND operation. I know that in a matrix the place an element is ... | coffeemath | 483,139 | <p>Looks like the rows,1,2 go with the truth values, T,F of p, and columns 1,2 go with the truth values T,F of q in the connective p (connecvtive) q.</p>
<p>Thus row 1 column 2 corresponds to p true (T) and q false (F), etc.</p>
|
716,856 | <p>I have the following equation:</p>
<p>\begin{align*}
\frac{d^2\theta}{dt}=\alpha(\theta-1)+\beta(\theta-1)^3-\gamma\frac{d\theta}{dt} \tag{1}
\end{align*}</p>
<p>Where $\alpha, \beta, \gamma \in \mathbb{R}$.</p>
<p>This is my solution in attempting to convert (1) into a system of first order ODE's:</p>
<p>\begin... | Sandeep Thilakan | 124,957 | <p>Yes it's fine. Here, you have taken the states of the system as $\{x_1 = \theta, x_2 = \phi\}$ and expressed </p>
<p>$\dot{x_1} = f(x_1,x_2)$, $\dot{x_2} = g(x_1, x_2)$</p>
|
1,273,789 | <p>Problem: X has mean and variance of 20.<br>
What can be said about $P(0<X<40)$? </p>
<p>Chebyshev formula = $P(|X-\mu|\geq k)\leq\frac{\sigma^2}{k^2}$ </p>
<p>The first step has $P(|X-20|\geq 20)\leq\frac{\sigma^2}{20\times20}$ </p>
<p>My question is: How did they get this part $P(|X-20|\geq 20)$? ... | Mehdi Jafarnia Jahromi | 231,513 | <p>Chebyshev formula is $P(|X-\mu|\geq k)\leq\frac{\sigma^2}{k^2}$. Just replace $\mu$ with mean and $k$ (that can be an arbitrary number) with $20$.</p>
<p><strong>Insight on this formula:</strong> It says that it is more probable to be around the mean. As much as you get far from the mean, the probability falls.</p>... |
1,332,433 | <blockquote>
<p>Let <span class="math-container">$f:\mathbb R\to \mathbb R$</span> be such that for every sequence <span class="math-container">$(a_n)$</span> of real numbers,</p>
<p><span class="math-container">$\sum a_n$</span> converges <span class="math-container">$\implies \sum f(a_n)$</span> converges</p>
<p>Prov... | PhoemueX | 151,552 | <p>Here comes a proof in the general case. We first start with the following</p>
<p><strong>Lemma</strong>: Let $-\varepsilon<y<0<x<\varepsilon$ and $N\in\mathbb{N}$.
Then there is a some $M\in\mathbb{N}$ and $z_{1},\dots,z_{M}\in\left\{ x,y\right\} $
with
$$
\left|\left\{ i\in\left\{ 1,\dots,M\right\} \,\... |
33,778 | <p>Let $X$ and $Y$ be independent exponential variables with rates $\alpha$ and $\beta$, respectively. Find the CDF of $X/Y$.</p>
<p>I tried out the problem, and wanted to check to see if my answer of: $\frac{\alpha}{ \beta/t + \alpha}$ is correct, where $t$ is the time, which we need in our final answer since we need... | Michael Hardy | 11,667 | <p>Here's a slightly different point of view.
\begin{align}
\Pr\left( \frac X Y \ge t \right) & = \iint\limits_{\{\,(x,y)\,:\, x\,\ge\,ty\,\ge\,0 \,\}} e^{-\alpha x} e^{-\beta y} (\alpha\beta\,d(x,y)) \\[10pt]
& = \int_0^\infty \left( \int_{ty}^\infty e^{-\alpha x} (\alpha\,dx) \right) e^{-\beta y} (\beta\,dy... |
2,661,221 | <p><strong>$\frac{1}{n}=O(\frac{1}{\ln n})$ or $\frac{1}{n}=o(\frac{1}{\ln n})$?</strong></p>
<p>I know that $n\geq\ln n, (n> 0)$, with which $\frac{1}{n}\leq \frac{1}{\ln n}$ and so $\frac{1}{n}=O(\frac{1}{\ln n})$.</p>
<p>To know if $\frac{1}{n}=o(\frac{1}{\ln n})$, I need to know if $\lim_{n\to \infty}\frac{\ln... | Sangchul Lee | 9,340 | <p>Let $U = \{\mathrm{w} \in \mathbb{R}_+^n : \langle \mathrm{w}, \mathrm{b} \rangle \neq 0 \}$ and define $f : U \to \mathbb{R}$ by</p>
<p>$$ f(\mathrm{w}) = \frac{\langle \mathrm{w}, \mathrm{a} \rangle}{\langle \mathrm{w}, \mathrm{b} \rangle}. $$</p>
<p>We know that $U$ is open and $\mathbf{1} = (1,\cdots,1) \in U$... |
2,041,849 | <p>I'm stuck on the following questions "In the alternating group $A_4$, let $H$ be the cyclic subgroup generated by $(123)$. Find all the right cosets of $H$ and all the left cosets of $H$. Is every right coset also a left coset."</p>
<p>The answer is as follows "$H = [idx_4 , (123), (132)]$ so each coset will have o... | Bernard | 202,857 | <p>This is probably because $3$-cycles generate the alternating group. In the case of $A_4$ there are only $8$ $3$-cycles, $(123),(124), (134), (234)$ and their inverses. So taking $3$ of them (different from $(123)$ and its inverse) is enough.</p>
|
158,630 | <p>Below I've quoted Wikipedia's entry that relates the Z-Transform to the Laplace Transform. The part I don't understand is $z \ \stackrel{\mathrm{def}}{=}\ e^{s T}$; I thought $z$ was actually an element of $\mathbb{C}$ and thus would be $z \ \stackrel{\mathrm{def}}{=}\ Ae^{s T}$ (but then it would be different to t... | Unknown123 | 366,054 | <h1>Why <span class="math-container">$z = e^{sT}$</span> chosen?</h1>
<p>If we define <span class="math-container">$z = e^{sT}$</span>, then the Z-transform becomes proportional to the Laplace transform of a sampled continuous-time signal.</p>
<hr>
<p>Taken from this slide
<a href="https://ccrma.stanford.edu/%7Ejos/Lap... |
102,234 | <p>Our assignment was to enter in a series and check to see if it converges by graphing. </p>
<p>However, when I attempt to graph it, I get a bunch of errors that I'm not sure what they mean. Best way to show it is a picture <a href="https://i.stack.imgur.com/ZWMds.png" rel="nofollow noreferrer"><img src="https://i.st... | eldo | 14,254 | <p>The series does not converge</p>
<pre><code>Limit[Sum[1/CubeRoot[k], {k, 1, n}], n -> Infinity]
</code></pre>
<p><a href="https://i.stack.imgur.com/otAGZ.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/otAGZ.jpg" alt="enter image description here"></a></p>
<pre><code>ListLinePlot@Table[Sum[1... |
3,723,183 | <p>I am working through a text book by <a href="http://www.hds.bme.hu/%7Efhegedus/Strogatz%20-%20Nonlinear%20Dynamics%20and%20Chaos.pdf" rel="nofollow noreferrer">Strogatz Nonlinear dynamics and chaos</a> . In chapter 2 question 2.2.1 , I am looking for an analytical solution. I have the question's answer but would lik... | Trebor | 584,396 | <p>Generally <span class="math-container">$\lim_{x\to x_0}f(x)$</span> is not to be taken as a number. This thing has no meaning by itself. The only meaningful expression is <span class="math-container">$\lim_{x\to x_0}f(x)=y$</span>, as a whole. Here <span class="math-container">$y\in\mathbb R \cup\{\pm\infty, \text{D... |
3,962,427 | <p>I have proven that the sequence is convergent.</p>
<p>Let <span class="math-container">$x_n=\frac{(2n)!!}{(2n+1)!!}$</span></p>
<p><span class="math-container">$\frac{x_{n+1}}{x_n}$</span> <span class="math-container">$=\frac{2n+2}{2n+3}<1$</span> Therefore the sequence decreases. On the other hand
<span class="m... | CHAMSI | 758,100 | <p>Observe that, for any <span class="math-container">$ n\in\mathbb{N} $</span> : <span class="math-container">$$\frac{\left(2n\right)!!}{\left(2n+1\right)!!}=\int_{0}^{1}{\left(1-x^{2}\right)^{n}\,\mathrm{d}x} $$</span></p>
<p>Since : <span class="math-container">\begin{aligned} \int_{0}^{1}{\left(1-x^{2}\right)^{n}\,... |
2,696,998 | <blockquote>
<p>Show that for any integer <span class="math-container">$k ≥ 0$</span> we have</p>
<p><span class="math-container">$$\frac{1}{(1-z)^{k+1}} = \sum\limits_{n=0}^{\infty}\binom {n+k}{n}z^{n}$$</span></p>
<p>and that it converges absolutely for any <span class="math-container">$|z| < 1$</span>.</p>
</bloc... | D_S | 28,556 | <p>Let $\mu \in F[t]$ be the minimal polynomial of $T$. It satisfies $\mu(T) = 0$, and divides any polynomial with the same property. and To say that $T$ is diagonalizable is the same thing as saying that all the roots of $\mu$ are in $F$, and that all these roots are distinct.</p>
<p>If $W$ is a subspace of $T$ whi... |
123,605 | <p>I have an equality:
$$\frac {d_L}{\phi t_1^2}=\frac {d_0}{\phi (t_1 - \Delta t)^2}$$
I can manually extract $d_L - d_0$ from this equation by expanding and re-arranging. Is there any way to do this in Mathematica? Ideally, I'd like <strong>Solve</strong> to work like this:</p>
<p>Solve$[\frac {d_L}{\phi t_1^2}==\... | Feyre | 7,312 | <pre><code>eq = dl/(ϕ t1^2) == d0/(ϕ (t1 - Δt)^2);
b = Solve[eq, dl][[1]]
a = Solve[eq, d0][[1]]
(dl /. b) - d0 /. a // Expand
</code></pre>
<blockquote>
<pre><code>(2 dl Δt)/t1 - (dl Δt^2)/t1^2
</code></pre>
</blockquote>
|
717,991 | <p>I've been stuck on this problem for a few weeks now. Any help?</p>
<p>Prove:
$\sum_{i=1}^{n}\prod_{j=0,j\neq i}^{n}\frac{x-x_j}{x_i-x_j}=1$</p>
<p>The sum of lagrange polynomials should be one, otherwise affine combinations of with these make no sense.</p>
<p>EDIT:
Can anybody prove this by actually working out ... | Community | -1 | <p>Let $P$ the polynomial with degree $n$ defined by
$$P(x)=\left(\sum_{i=0}^{n}\prod_{j=0,j\neq i}^{n}\frac{x-x_j}{x_i-x_j}\right)-1$$
then we see easily that
$$P(x_i)=0,\;\quad\forall i=0,\ldots,n$$
so $P$ has $n+1$ distinct roots hence it's the zero polynomial due to the D'Alembert's theorem. Conclude.</p>
|
3,600,139 | <blockquote>
<p>Solve <span class="math-container">$y''-y'-y=\cos x$</span>.</p>
</blockquote>
<p>After first solving the homogeneous equation we know that the solution to it is <span class="math-container">$$y(x)=a\sin(x)+b\cos(x).$$</span></p>
| gt6989b | 16,192 | <p><strong>HINT</strong>
Propose a particular solution <span class="math-container">$y_p(x) = Ax\cos x + Bx\sin x$</span> and compute <span class="math-container">$A,B$</span> by plugging into the ODE.</p>
|
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