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 |
|---|---|---|---|---|
4,461,327 | <p>To show that they are equal, I need to show</p>
<p><span class="math-container">$\bigcap_{n=1}^{\infty}[0,1+1/n) \subset [0,1]$</span> and <span class="math-container">$[0,1] \subset \bigcap_{n=1}^{\infty}[0,1+1/n)$</span></p>
<p>My attempt is: let <span class="math-container">$x \in [0,1] \Rightarrow 0 \leq x \leq ... | Jordan Miller | 1,062,506 | <p>To simplify this, let <span class="math-container">$p$</span> be the statement <span class="math-container">$x\in [0,1]$</span> and <span class="math-container">$q$</span> be the statement <span class="math-container">$x\in \bigcap_{n=1}^{\infty}[0,1+\frac{1}{n})$</span></p>
<p><span class="math-container">$x\in [0,... |
4,461,327 | <p>To show that they are equal, I need to show</p>
<p><span class="math-container">$\bigcap_{n=1}^{\infty}[0,1+1/n) \subset [0,1]$</span> and <span class="math-container">$[0,1] \subset \bigcap_{n=1}^{\infty}[0,1+1/n)$</span></p>
<p>My attempt is: let <span class="math-container">$x \in [0,1] \Rightarrow 0 \leq x \leq ... | Eparoh | 617,769 | <p>Observe that if <span class="math-container">$x \in \bigcap_{n=1}^{\infty}[0,1+1/n)$</span> then you have that
<span class="math-container">$$x \in [0,1+1/n), \forall n \in \mathbb{N}\hspace{3mm} (1)$$</span>
That implies that <span class="math-container">$x \geq 0$</span> and it remains to prove that <span class=... |
4,186,743 | <p>Is there a trigonometric function explaining <span class="math-container">$\cos(x)+\sin(x)$</span>? if not, what is <span class="math-container">$\cos(x)+\sin(x)$</span> as a function of <span class="math-container">$\cos(x)$</span>?</p>
| Luca Ghidelli | 176,416 | <p><span class="math-container">$$\sqrt 2 \cos (x-\pi/4)$$</span></p>
<p>By the <a href="https://mathworld.wolfram.com/ProsthaphaeresisFormulas.html" rel="nofollow noreferrer">Prostaphaeresis-Simpson-formula</a>. If you need, you can convert sinuses into cosinuses via <span class="math-container">$\sin x =\cos(\pi/2-x)... |
3,946,591 | <blockquote>
<p>The sides of a triangle are on the lines <span class="math-container">$2x+3y+4=0$</span>, <span class="math-container">$ \ \ x-y+3=0$</span>, and <span class="math-container">$5x+4y-20=0$</span>. Find the equations of the altitudes of the triangle.</p>
</blockquote>
<p>Should I find the vertices first? ... | DanielV | 97,045 | <p>Transition matrix with two states, the string has an <span class="math-container">$b$</span> or it doesn't. State 0 will be no bs and state 1 will be some bs. <span class="math-container">$M_{i, j}$</span> is "how many ways can we transist from a string of type <span class="math-container">$i$</span> to a str... |
2,966,871 | <blockquote>
<p>Define the unit sphere as <span class="math-container">$S^1=\{x\in \mathbb{R}^2: \|x\|=1\}$</span></p>
<p>Also define the real projective line as <span class="math-container">$\mathbb{R}P^1=S^1/(x\sim-x)$</span></p>
</blockquote>
<p>We can consider the mapping <span class="math-container">$f:S^1\rightar... | Ashvin Swaminathan | 259,604 | <p>The map <span class="math-container">$f$</span> is surjective because in polar coordinates, it is given by <span class="math-container">$e^{i\theta} \mapsto e^{2i\theta}$</span>, and every angle <span class="math-container">$\psi \in [0, 2\pi)$</span> can be uniquely represented as <span class="math-container">$2\th... |
2,237,441 | <p>Let $n$ be a natural number.</p>
<p>I need to prove that $9 \mid 4^n-3n-1$</p>
<p>Could anyone give me some hints how to prove it without using induction.</p>
| lhf | 589 | <p>By the binomial theorem, $4^n = (1+3)^n = 1 + 3n + 9a$.</p>
|
2,403,201 | <p>How do I solve for $x$:</p>
<p>$$\log\left(\frac{1.07^x}{1050-2.5x}\right)=\log\left(\frac{1.2}{828}\right)$$</p>
<p>If I raise both sides to the power of $10$, I get:
$\dfrac{1.07^x}{1050-2.5x}=\frac{1}{690}$</p>
<p>Then I'm stuck. How do I solve this ?</p>
<p>As suggest by @Kevin, I have decided to add my take... | Barry Cipra | 86,747 | <p>It's convenient to rewrite the equation to be solved for $x$ as</p>
<p>$$690\cdot1.07^x+2.5x=1050$$</p>
<p>As others have indicated in comments and answers, equations of this form cannot, in general, be solved exactly; the best that can done is a numerical approximation. There are various ways to go about obtaini... |
3,454,514 | <blockquote>
<p>How to change the integration order in the given integral?
<span class="math-container">$$
\int\limits_0^1dx\int\limits_0^1dy\int\limits_0^{x^2+y^2}fdz\rightarrow
\int\limits_?^?dz\int\limits_?^?dy\int\limits_?^?fdx
$$</span></p>
</blockquote>
<p>I tried to make a graphic interpretation, but it see... | Andrei | 331,661 | <p>It's not a good idea to reverse the order of integrals in this case, it will make things more complicated. But it can be done. You need to first understand what is the domain of integration. If you just look at the integral over <span class="math-container">$z$</span>, and you think about <span class="math-container... |
3,917,912 | <p>I am reading an article where the author seems to use a known relationship between the sum of a finite sequence of real positive numbers <span class="math-container">$a_1 +a_2 +... +a_n = m$</span> and the sum of their reciprocals. In particular, I suspect that
<span class="math-container">\begin{equation}
\sum_{i=1... | Albus Dumbledore | 769,226 | <p>by <a href="https://en.wikipedia.org/wiki/Cauchy%E2%80%93Schwarz_inequality" rel="nofollow noreferrer">Cauchy schwarz inequality</a><span class="math-container">$$\left(\sum_{i=1}^{n}{\sqrt{a_i}}^2\right)\left(\sum_{i=1}^{n}\frac{1}{{\sqrt{a_i}}^2}\right)\ge (\sum_{i=1}^{n} 1)^2=n^2$$</span></p>
<hr />
<p>WLOG <span... |
3,981,983 | <p>I had an interesting math problem presented to me some time ago by a friend (he stated it in non-mathematical terms). At what angle would you launch a projectile from a spaceship/satellite such that it left that object and went on to hit another orbiting object? Then as a supplemental question he asked at what angle... | Henno Brandsma | 4,280 | <p><span class="math-container">$x$</span> is a limit point of <span class="math-container">$A$</span>, by definition if</p>
<p>for all (open) neighbourhoods <span class="math-container">$U$</span> of <span class="math-container">$x$</span>, <span class="math-container">$U \cap (A\setminus\{x\}) \neq \emptyset$</span> ... |
3,000,862 | <p>I can name at least 4 different ways of representing <span class="math-container">$\exp$</span> function:</p>
<ol>
<li>Taylor series: For <span class="math-container">$x \in \mathbb{R}, \exp(x) = \sum_{k=0}^{\infty} \frac{x^k}{k!}$</span>.</li>
<li>Differential equation: <span class="math-container">$f: \mathbb{R} ... | Aleksas Domarkas | 562,074 | <p>With free computer algebra system Maxima 5.42.1
<a href="https://i.stack.imgur.com/aqC6L.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/aqC6L.png" alt="enter image description here"></a></p>
|
1,816,109 | <p>Does anyone know how to deal with the integral
$$\int_0^{\pi /2}\cot^n(x)dx$$</p>
<p>with $n\in (-1,1)$?</p>
<p>Apparently it is a well known identity: it is listed in the <a href="http://mathworld.wolfram.com/Cotangent.html" rel="nofollow">page of wolfram for the cotangent</a>, where it says that it equals $2^{-1... | Jack D'Aurizio | 44,121 | <p>If $\alpha\in(-1,1)$
$$I(\alpha)=\int_{0}^{\pi/2}\cot^{\alpha}(x)\,dx = \int_{0}^{\pi/2}\tan^{\alpha}(x)\,dx = \int_{0}^{+\infty}\frac{t^{\alpha}}{1+t^2}\,dt $$
and the last integral boils down to a value of the <a href="https://en.wikipedia.org/wiki/Beta_function" rel="nofollow">Euler's beta function</a> through th... |
3,763,744 | <p>The helix is a curve <span class="math-container">$x(t) \in \mathbb{R}^3$</span> defined by:</p>
<p><span class="math-container">$$
x(t) = \begin{bmatrix}
\sin(t) \\
\cos(t) \\
t
\end{bmatrix}
$$</span></p>
<p>and it takes the classic shape:</p>
<p><a href="https://en.wikipedia.org/wiki/File:Rising_circular.gif" rel... | SZB1 | 1,045,629 | <p>I think another idea which might be a generalization of helices to <span class="math-container">$\mathbb{R}^n$</span> is that we take the parametrization of a <span class="math-container">$\mathbb{R}^{n-1}$</span> sphere and then we translate it by a function of the parameters which are used at generating the sphere... |
2,853,668 | <blockquote>
<p>Show that $$\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{x^n}$$ converges for every $x>1$.</p>
</blockquote>
<p>let $a(x)$ be the sum of the series. does $a$ continious at $x=2$? differentiable?</p>
<p>I guess the first part is with leibniz but I am not sure about it.</p>
| Nosrati | 108,128 | <p>Using <a href="https://en.wikipedia.org/wiki/Root_test" rel="nofollow noreferrer">root test</a>
$$\lim_{n\to\infty}\sqrt[n]{\left|\dfrac{(-1)^n}{x^n}\right|}=\dfrac{1}{|x|}<1$$
then the series is converge for $|x|>1$.</p>
|
2,538,305 | <p>Here is the question I am struggling with:</p>
<p>A box has 16 Balls, of which 8 are Green, 6 are Red, and 2 are Blue. If you draw 2 Balls with replacement, what is the probability of getting 1 Green Ball and 1 Blue Ball in no particular order?</p>
<p>I see three different ways to get an answer to this problem. Pl... | fleablood | 280,126 | <p>Method 1:</p>
<p>This is specifying order. The $\frac 8{16}$ applies to a specific drawing as does the $\frac 2{16}$. As blue first then green is as equally likely as green first than blue. The probability is $\frac{8}{16}*\frac{2}{16}*2$</p>
<p>Method 2:</p>
<p>This is correcct.</p>
<p>"Note: This confuses m... |
3,134,991 | <p>If nine coins are tossed, what is the probability that the number of heads is even?</p>
<p>So there can either be 0 heads, 2 heads, 4 heads, 6 heads, or 8 heads.</p>
<p>We have <span class="math-container">$n = 9$</span> trials, find the probability of each <span class="math-container">$k$</span> for <span class="... | Peter | 82,961 | <p>The easiest way to see this : Consider the number of heads we have in the first <span class="math-container">$8$</span> coins.</p>
<ul>
<li>If this number is even, we need a tail, we have probability <span class="math-container">$\frac{1}{2}$</span> </li>
<li>If this number is odd, we need a head, we have probabili... |
3,134,991 | <p>If nine coins are tossed, what is the probability that the number of heads is even?</p>
<p>So there can either be 0 heads, 2 heads, 4 heads, 6 heads, or 8 heads.</p>
<p>We have <span class="math-container">$n = 9$</span> trials, find the probability of each <span class="math-container">$k$</span> for <span class="... | Kyle Miller | 172,988 | <p>If <span class="math-container">$h$</span> denotes getting a heads and <span class="math-container">$p$</span> denotes getting tails, let's write <span class="math-container">$\frac{1}{2}h+\frac{1}{2}p$</span> for the notion of a fair coin: half the time it turns up heads, and half the time tails.</p>
<p>If we expa... |
3,134,991 | <p>If nine coins are tossed, what is the probability that the number of heads is even?</p>
<p>So there can either be 0 heads, 2 heads, 4 heads, 6 heads, or 8 heads.</p>
<p>We have <span class="math-container">$n = 9$</span> trials, find the probability of each <span class="math-container">$k$</span> for <span class="... | Brian Tung | 224,454 | <p>A useful way to think about this problem, especially for the case of generally unfair coins, is in terms of a recurrence. Let <span class="math-container">$p$</span> be the probability that the coin flips heads, and let <span class="math-container">$q_n$</span> be the probability, after <span class="math-container"... |
780,895 | <p>A collection of black and white balls are to be arranged on a straight line such that each ball has at least one neighbor of different color. If there are 100 black balls, then the maximum number of white balls that allows such an arrangement is? </p>
| talegari | 27,440 | <p>$200$</p>
<p>There can be at most two white balls between two black balls. If there were more than two white balls between some two black balls, one of the white ball ends up having two white neighbors. There can be one white ball to the left of the leftmost black ball and one white ball to the right of the rightmo... |
1,291,107 | <p>Let $X$ be random variable and $f$ it's density. How can one calculate $E(X\vert X<a)$?</p>
<p>From definition we have:</p>
<p>$$E(X\vert X<a)=\frac{E\left(X \mathbb{1}_{\{X<a\}}\right)}{P(X<a)}$$</p>
<p>Is this equal to:</p>
<p>$$\frac{\int_{\{X<a\}}xf(x)dx}{P(X<a)}$$</p>
<p>? If yes, then ho... | bnosnehpets | 224,658 | <p>There are already answers on how to get the real solutions, so I will only show you the non-real solutions.</p>
<p>You have obtained that $(x-5)^3=-1$. Expanding and simplifying we get:
$$x^3-15x^2+75x-124=0$$
However, we know that $x=4$ is a solution so we can say that $(x-4)(ax^2+bx+c)=0$. You can equate coeffici... |
1,458,405 | <p>(p ∨ q) → r ≡ (p → q) ∨ (p → r) could be valid or invalid</p>
<p>I need to prove it using logical equivalences (can't use truth table)</p>
<p>This is how far I've gotten by working with the right side:</p>
<p><-> p→(q v r)</p>
<p><-> ¬p v (q v r) </p>
<p>then commutative law</p>
<p><-> (q v r) v ¬p </... | skyking | 265,767 | <p>Well the proposition is invalid. </p>
<p>The systematic way to show this is to bring it to conjunctive normal form (this can be more easily done with carnaugh tables, but that would probably count as a truth table). We start with the statement:</p>
<p>$((p\lor q)\rightarrow r) \leftrightarrow ((p\rightarrow q)\lor... |
135,675 | <p>Let $D$ be the Dirac-Operator on $\mathbb{R}^n$ or more generally the Dirac spinor bundle $\mathcal{S}\to M$ of a (semi-)Riemannian spin manifold $M$. Then we consider $D$ as an unbouded Operator on $\mathcal{H}=L^2(\mathbb{R}^n)$ with domain $C^\infty_c(\mathbb{R}^n,\mathbb{C}^N)$. Then it is said that the operator... | Mikhail Katz | 28,128 | <p>What is it that you find mysterious exactly? On a compact manifold $M$, the eigenvalues of $\langle D\rangle$ tend to infinity, hence the eigenvalues of $\langle D\rangle^{-1}$ tend to zero, so that we have a compact operator, namely the image of unit ball is compact. Multiplication by $f$ is a bounded operator, th... |
991,878 | <p>How can it be proven that a cycle of length k is an even permutation if and only if k is odd?
I know it can be done using the fact that a permutation which exchanges two elements but leaves the rest unchanged is an odd permutation.</p>
| Dr. Sonnhard Graubner | 175,066 | <p>we have to distinguish two cases
if $0<a<1$ then we have $x\geq \frac{a-1}{\log(a)}$ in the case that $a>1$ we have
$x\le \frac{a-1}{\log(a)}$ and for $a=1$ we have $0\le 0$</p>
|
203,673 | <p>Let $X$ be a locally compact Hausdorff space. Does there exist a locally finite open covering consisting of relatively compact sets?</p>
| Joseph Van Name | 22,277 | <p>The condition that you are referring to when coupled with local compactness is precisely paracompactness. Not every locally compact space is paracompact as the example of Arthur Fischer illustrates and which one can look up <a href="http://topology.jdabbs.com/">here</a>.</p>
<p>$\mathbf{Theorem}$</p>
<p>Suppose th... |
2,275,016 | <p>I am currently going through Dummit/Foote's <em>Abstract Algebra</em>, and was asked to prove the above for a specific case but was wondering if it holds in the general case.</p>
<p>I have a feeling it might be false but I am bad at coming up with counterexamples so I tried to think of some simple contradictions if... | Vitor Borges | 303,852 | <p>Notice that $xa = ax$ iff $xax^{-1} = a$. Multiplying both sides by $x^{-1}$ shows that $x^{-1}xax^{-1} = ax^{-1} = x^{-1}a$.</p>
|
108,409 | <p>Given a commutative ring <span class="math-container">$A$</span> with unity, Grothendieck used universal polynomials to define a <em>special</em> <span class="math-container">$\lambda$</span>-ring structure on <span class="math-container">$\Lambda(A):=1+t\:A[[t]]$</span>. Suppose <span class="math-container">$A$</sp... | Baptiste Calmès | 4,763 | <p>I'm not sure what product you are thinking of on $\Lambda^0(A)$, but the one I'm thinking of, and the one that I believe is implicitly used in Fulton-Lang is the usual product on power series. So in particular, $(1+a_1 t+\cdots)\cdot (1+b_1 t+\cdots) = 1 + (a_1 + b_1)t + \cdots$.</p>
<p>There is no problem of gradi... |
1,176,615 | <p>I am invited to calculate the minimum of the following set:</p>
<p>$\big\{ \lfloor xy + \frac{1}{xy} \rfloor \,\Big|\, (x+1)(y+1)=2 ,\, 0<x,y \in \mathbb{R} \big\}$.</p>
<p>Is there any idea?</p>
<p>(The question changed because there is no maximum for the set (as proved in the following answers) and I assume ... | Lozenges | 219,277 | <p>There is a minimum of $6$ achieved at $x=y=\sqrt{2}-1$</p>
<p>to prove this, start with the identity</p>
<p>$\left(x-\frac{1}{x}\right)\left(y-\frac{1}{y}\right)+\left(\frac{y}{x}+\frac{x}{y}\right)=\left(x y+\frac{1}{x y}\right)$</p>
<p>a simple verification shows that $\left(x-\frac{1}{x}\right)\left(y-\frac{1}... |
75,880 | <p>Say $f:X\rightarrow Y$ and $g:Y\rightarrow X$ are functions where $g\circ f:X\rightarrow X$ is the identity. Which of $f$ and $g$ is onto, and which is one-to-one?</p>
| Arturo Magidin | 742 | <p>If it is just a matter of remembering what the right conclusion is, here's the picture I always use to remember:
$$\begin{array}{rcl}
&\bullet &\\
&&\searrow\\
\bullet\rightarrow& \bullet & \rightarrow\bullet\\
X\quad\quad&Y&\quad\quad Z
\end{array}$$
The compositum is one-to-on... |
2,362,790 | <p>I'm interested in the problem linked with <a href="https://math.stackexchange.com/questions/770117/determinant-of-circulant-matrix/2362754#2362754">this answer</a>. </p>
<p>Let $ f(x) = a_n + a_1 x + \dots + a_{n-1} x^{n-1} $ be polynomial with distinct $a_i$ which are <strong>primes</strong>. </p>
<p>(Po... | Jyrki Lahtonen | 11,619 | <p>This is certainly possible. The easiest way is to use pairs of twin primes and $n=4$. Such as
$$
f(x)=7+5x+11x^2+13x^3
$$
where $f(-1)=0$ and $x+1$ is a common factor of $f(x)$ and $x^4-1.$</p>
<hr>
<p>Extending the same idea to third roots of unity. Consider
$$
f(x)=7+5x+17x^2+29x^3+31x^4+19x^5.
$$
Because $7+29=... |
879,640 | <p>Does a matrix have only one inverse matrix (like the inverse of an element in a field)? If so, does this mean that</p>
<p>$A,B \text{ have the same inverse matrix} \iff A=B$?</p>
| Pauly B | 166,413 | <p>Yes, it is unique. To show this, assume a matrix $A$ has two inverses $B$ and $C$, so that $AB=I$ and $AC=I$. Therefore $AB=AC \implies BAB=BAC \implies B=C$. So the inverse is indeed unique. For the second question, note that $(A^{-1})^{-1}=A$ so that if $A$ and $B$ both have inverse $A^{-1}$, then $A^{-1}$ has a u... |
3,243,503 | <p>If <span class="math-container">$x + y = 2c$</span>, find minimum value of
<span class="math-container">$ \sec x +\sec y $</span> if <span class="math-container">$x,y\in(0,\pi/2)$</span>, in terms of <span class="math-container">$c$</span>.</p>
<p>I was able to solve by differentiating the equation and got the ans... | Key Flex | 568,718 | <p>Here is a method without differentiation.</p>
<p>Since <span class="math-container">$x,y\in(0,\pi/2)$</span></p>
<p><span class="math-container">$\sec x,\sec y>0$</span></p>
<p>By using <span class="math-container">$A.M\ge G.M$</span>, we have
<span class="math-container">$$\dfrac{\sec x+\sec y}{2}\ge\sqrt{\s... |
885,778 | <ol>
<li><p>Is there any group of order 36 with no subgroup of order 6?</p></li>
<li><p>Is there any group of order $p^2q^2$ with no subgroup of order $pq$?</p></li>
<li><p>Is there any group of order $p^{2m}q^{2m}$ with no subgroup of order $p^mq^m$?</p></li>
<li><p>Is there any group of order $p^{2m}q^{2n}$ with no s... | kabumm | 167,588 | <p>This is not really a complete answer (yet). But since it's an interesting question and since I cannot post comments due to forum restrictions, I'll hint you to an article I found: <a href="http://matwbn.icm.edu.pl/ksiazki/fm/fm92/fm9211.pdf" rel="nofollow noreferrer">http://matwbn.icm.edu.pl/ksiazki/fm/fm92/fm9211.p... |
1,450,476 | <p>I'm in number theory and I currently have these problems assigned as homework. I've looked through the sections containing these problems and I've solved/proved most of the other problems, but I can't figure these ones out.</p>
<ol>
<li>For $n>1$, show that every prime divisor of $n!+1$ is an odd integer that is... | Community | -1 | <p>Even though the cardinality of $\mathbb{Z}$ and $\mathbb{Q}$ are same, its not a bijection (its an embedding but not a surjection). $\frac{1}{2}$ is not in the image set. </p>
|
1,507,526 | <p>Let $S$ be the portion of the sphere $x^2+y^2+z^2=9$, where $1\leq x^2+y^2\leq4$ and $z\geq0$. Calculate the surface area of $S$</p>
<p>Ok i'm really confused with this one. I know i have to apply the surface area formula but and possibly spherical coordinates but i can't seem how to get the integral out.</p>
<p>T... | Narasimham | 95,860 | <p>I shall outline symbolically at first:</p>
<p>Due to possibility two axial positions and independence of $ \theta$ ( full $ 0 to 2 \pi) you can use either spherical coordinate system or cylindrical. I used the latter.</p>
<p>$$ dA= 2 \pi r ds = 2 \pi \frac {dz}{\cos \phi} =2 \pi R dz $$</p>
<p>$$ (\because \c... |
3,845,475 | <p>Here's what I'm tasked with showing:</p>
<p>Let <span class="math-container">$(a_n)$</span> be a convergent sequence with <span class="math-container">$a_n\rightarrow a$</span> as <span class="math-container">$n\rightarrow\infty$</span>. By the Algebraic Limit Theorem, we know that <span class="math-container">$(a_n... | Brian M. Scott | 12,042 | <p>If you order the numbers <span class="math-container">$1$</span> and <span class="math-container">$2$</span> and then insert <span class="math-container">$3$</span>, you can get all <span class="math-container">$6$</span> permutations of <span class="math-container">$1,2$</span>, and <span class="math-container">$3$... |
3,845,475 | <p>Here's what I'm tasked with showing:</p>
<p>Let <span class="math-container">$(a_n)$</span> be a convergent sequence with <span class="math-container">$a_n\rightarrow a$</span> as <span class="math-container">$n\rightarrow\infty$</span>. By the Algebraic Limit Theorem, we know that <span class="math-container">$(a_n... | user2661923 | 464,411 | <p>In your analysis of set <span class="math-container">$S$</span>, you didn't take it
far enough. As you indicated, there are <span class="math-container">$3!$</span> total
orderings.</p>
<p>Further, as you indicated, there are <span class="math-container">$2!$</span> total orderings of the elements besides
the <span... |
2,952,803 | <blockquote>
<p>Let <span class="math-container">$a<b$</span> and <span class="math-container">$a,b\in\Bbb R$</span>. Then there is <span class="math-container">$c\in\Bbb R\setminus\Bbb Q$</span> such that <span class="math-container">$a<c<b$</span>.</p>
</blockquote>
<hr>
<p><strong>My attempt:</strong></... | John Doe | 399,334 | <p>You have<span class="math-container">$$\int(M-P)^{-1}\,dP=k\int dt$$</span>
This gives <span class="math-container">$$M-P=e^{-k(t-t_0)}$$</span> where <span class="math-container">$t_0$</span> is a constant of integration. Then we get<span class="math-container">$$ P=M-e^{-k(t-t_0)}=M-e^{-kt}e^{kt_0}=M-\alpha e^{-kt... |
1,137,565 | <blockquote>
<p>Let <span class="math-container">$A\Delta C\subseteq A\Delta B$</span>. (<span class="math-container">$\Delta$</span> denotes symmetric difference.)</p>
<p>Prove <span class="math-container">$A\cap B \subseteq C$</span>.</p>
</blockquote>
<p>I am getting ready for a test and I could really use proof ver... | avz2611 | 142,634 | <p>for $(x,y)\rightarrow t$
$$(x,-y)\rightarrow -t$$
$$(-x,-y)\rightarrow \frac{1}{t}$$
$$(-x,y)\rightarrow \frac{1}{-t}$$
now you can get every point so you can see to get $x=1$ the parameter should tend to infinity</p>
|
1,137,565 | <blockquote>
<p>Let <span class="math-container">$A\Delta C\subseteq A\Delta B$</span>. (<span class="math-container">$\Delta$</span> denotes symmetric difference.)</p>
<p>Prove <span class="math-container">$A\cap B \subseteq C$</span>.</p>
</blockquote>
<p>I am getting ready for a test and I could really use proof ver... | bubba | 31,744 | <p>A hyperbola has two pieces (often called "branches"). The implicit equation $x^2 - y^2=1$ describes both of these pieces -- the points on both pieces satisfy the equation. </p>
<p>But the parametric equation represents only a portion of the curve, with the point $(1,0)$ missing.</p>
<p>To understand the situation,... |
30,653 | <p>I have a data file which contains thousands of lines and each line has eight elements. Here is a small sample of the data file</p>
<pre><code>-4.00 -0.80 0.1886024468848907E+01 0.1467147621657460E+01 -.1217067274319363E+01 0.7206100000000000E+03 0.7693457688734395E-12 5
-4.00 -0.70 0.143035... | Mike Honeychurch | 77 | <p>You could write functions to generalize this but for your specific problem:</p>
<pre><code>data = Import["list.txt", "Data"];
ntot = Length[data];
</code></pre>
<p>(a) to get percentages of elements in the last column (you could optionally <code>Sort</code> this list too)</p>
<pre><code>Inner[Times, Tally[data[[A... |
4,341,172 | <p><span class="math-container">$(x+1)^2y''+(x+1)y'+y=x^2+2\sin(\ln(x+1)), y(0)=\frac{1}{5},y'(0)=2$</span></p>
<p>My solution:</p>
<p><span class="math-container">$y''+\frac{y'}{(x+1)}+\frac{y}{(x+1)^2}=\frac{x^2+2\sin(\ln(x+1))}{(x+1)^2}$</span></p>
<p>First of all , I found the equation solution of <span class="math... | Chee Han | 242,589 | <p>The two equations you have using variation of parameters is incorrect. Given a second-order linear inhomogeneous DE <span class="math-container">$y'' + p(x)y' + q(x)y = f(x)$</span> with homogeneous solution <span class="math-container">$y_c(x) = c_1y_1(x) + c_2y_2(x)$</span>, a particular solution <span class="math... |
2,547,933 | <p>Consider the integral $I=\displaystyle\int_{R}\int f(x,y)dx dy$ over the region $R$, given by the triangle with vertices $(0,0),(1,1)$ and $(2,0)$. </p>
<p>This is an isosceles triangle with one side lying along the $x-$axis. So, our domain is not "nice" to find the bounds for integral I assume, since even if we w... | Martin Argerami | 22,857 | <p>If the limits were the same, they would have to be $1$, since $$\left(1+\frac1n\right)^n>1$$ for all $n$, while $$\left(1-\frac1n\right)^n<1$$ for all $n$.</p>
<p>So if you expect the first limit to be $e$ and you know that $e>1$, you cannot expect the second one to be equal to it. </p>
<p><strong>Edit:... |
2,757,562 | <p>I'm work through some questions relating to connection coefficents. My question is more about the summation notation being used. Why is there no starting point (or end point) for the summation here? For the i, j, k, I take these to range from 1 to 2 for a 2 dimensional surface. Does this just imply that l does likew... | pureundergrad | 517,979 | <p>Another example of use of a similar notation which might make more sense to you is </p>
<p>$$\sum_{n\in \mathbb{N}}n=1+2+3+\dots$$</p>
<p>If the set is not specified directly under your sum, then $l$ probably runs over some set of values which make sense, maybe try reading back, that set was probably stated either... |
1,552,055 | <p>Why is it true that all irrational numbers are non-terminating/non-repeating decimals?</p>
<p>By definition, an irrational number is one that can't be expressed as a ratio of integers.</p>
| user236182 | 236,182 | <p>The definition: a number is irrational if and only if it's not rational, i.e. it can't be expressed as a ratio of two integers. This answers one part of your question.</p>
<p>The other part: I'll prove the contrapositive. If <span class="math-container">$x$</span> has a repeating decimal expansion (this includes te... |
1,527,197 | <p>So in the case where data points have the same variance $\sigma^2$, the estimator (in normal equation form) can be written as </p>
<p>$$\theta=(X^TX)^{-1}X^TY$$</p>
<p>I'm not sure how to derive a similar formula when the data points have different variances, and thus the covariance matrix would be</p>
<p>$$\Sigm... | Henno Brandsma | 4,280 | <p>As to (2), you are right that the alternating series test says it converges.</p>
<p>The $n$-th term does not converge to $0$ (it doesn't even converge, period.) in 1, so, no convergence. In any series (alternating or not), if it converges , then $a_n$ tends to $0$.</p>
<p>In (3) the AST also works. </p>
|
3,762,624 | <p>This is a pretty common question in probability and there are already a few answers on the site. HOWEVER, my Professor has put a little twist on it and I can't piece it together anymore.</p>
<p><strong>QUESTION:</strong></p>
<p>Let <span class="math-container">$(X_n)$</span> be a sequence of geometric random variabl... | Robert Israel | 8,508 | <p>Let <span class="math-container">$f = g + h$</span>. So your assumptions are <span class="math-container">$g(0)=h(0) = 0$</span>, <span class="math-container">$g'(0) \ge 0$</span>, <span class="math-container">$h'(0) \ge 0$</span>, <span class="math-container">$g''(s) > 0$</span> and <span class="math-container... |
622,076 | <p>Continuity $\Rightarrow$ Intermediate Value Property. Why is the opposite not true? </p>
<p>It seems to me like they are equal definitions in a way. </p>
<p>Can you give me a counter-example? </p>
<p>Thanks</p>
| hot_queen | 72,316 | <p>A very strong counterexample would be a function whose range is all of R on every interval.</p>
|
442,950 | <p>I would like to show <span class="math-container">$\lim\limits_{r\to\infty}\int_{0}^{\pi/2}e^{-r\sin \theta}\text d\theta=0$</span>.</p>
<p>Now, of course, the integrand does not converge uniformly to <span class="math-container">$0$</span> on <span class="math-container">$\theta\in [0, \pi/2]$</span>, since it has... | nullUser | 17,459 | <p>On <span class="math-container">$[0,\pi/2]$</span> the sine is nonnegative and so <span class="math-container">$|e^{-r\sin \theta}| \leq 1$</span> for <span class="math-container">$r \geq 0$</span>. It follows by dominated convergence that
<span class="math-container">$$
\lim_{r \to \infty} \int_0^{{\pi/2}} e^{-r \... |
442,950 | <p>I would like to show <span class="math-container">$\lim\limits_{r\to\infty}\int_{0}^{\pi/2}e^{-r\sin \theta}\text d\theta=0$</span>.</p>
<p>Now, of course, the integrand does not converge uniformly to <span class="math-container">$0$</span> on <span class="math-container">$\theta\in [0, \pi/2]$</span>, since it has... | robjohn | 13,854 | <p>This can also be handled by <a href="http://en.wikipedia.org/wiki/Dominated_convergence_theorem" rel="nofollow">Monotone Convergence</a>.</p>
<p>The functions $f_r(\theta)=1-e^{-r\sin(\theta)}$ monotonically converge to
$$
\lim_{r\to\infty}f_r(\theta)=f(\theta)=\left\{\begin{array}{}
1&\text{if }0\lt\theta\le\f... |
3,889,175 | <p>Is there a way to calculate:<span class="math-container">$\sum_{i=0}^{k} {2k+1\choose k-i}$</span> using only:</p>
<ul>
<li>symmetry;</li>
<li>pascal's triangle;</li>
<li>one of these sums: <span class="math-container">$$\sum_{i=0}^{k} {n+i\choose i}={n+k+1\choose k}$$</span> and <span class="math-container">$${p\ch... | Community | -1 | <p>At worse, multiplying an equation by a common expression can introduce alien solutions (when you multiply by zero), but not lose any: if <span class="math-container">$a=b$</span>, <span class="math-container">$ac=bc$</span> remains true.</p>
<p>Your equation has meaning only when <span class="math-container">$x\ne1$... |
3,889,175 | <p>Is there a way to calculate:<span class="math-container">$\sum_{i=0}^{k} {2k+1\choose k-i}$</span> using only:</p>
<ul>
<li>symmetry;</li>
<li>pascal's triangle;</li>
<li>one of these sums: <span class="math-container">$$\sum_{i=0}^{k} {n+i\choose i}={n+k+1\choose k}$$</span> and <span class="math-container">$${p\ch... | Rhys Hughes | 487,658 | <p>Losing or gaining solutions to an equation happens when you perform an operation which is either not uniquely invertible, or an inverse of such an operation. For example, with squaring we see:
<span class="math-container">$$y=x\to y^2=x^2\to y=\pm x$$</span></p>
<p>The function <span class="math-container">$x^2$</sp... |
2,836,539 | <p>Recently I run into this integral</p>
<p>$$\mathcal{J} = \int_{0}^{\pi/2} \frac{x \log \left ( 1-\sin x \right )}{\sin x} \, \mathrm{d}x$$</p>
<p>I don't know to what it evaluates. I tried several approaches.</p>
<p><strong>1st:</strong> Differentiation under the integral sign</p>
<p>Consider the function $\disp... | Jack D'Aurizio | 44,121 | <p>The given problem is equivalent to the evaluation of
$$ \int_{0}^{1}\frac{\arcsin(x)}{\sqrt{1-x^2}}\cdot\frac{\log(1-x)}{x}\,dx =\sum_{n\geq 1}\frac{4^n}{2n\binom{2n}{n}}\int_{0}^{1}x^{2n-2}\log(1-x)\,dx=\sum_{n\geq 1}\frac{4^n H_{2n-1}}{2n\binom{2n}{n}(1-2n)}$$
which is a <em>twisted</em> hypergeometric series. On ... |
1,930,743 | <p>We have a map <span class="math-container">$f:P(X)\to P(X)$</span>, where <span class="math-container">$P(X)$</span> means the part of <span class="math-container">$X$</span> and the function is monotone (by considering inclusion "<span class="math-container">$\subseteq$</span>"). So <span class="math-container">$\... | Brian M. Scott | 12,042 | <p>HINT: Consider the set $\bigcup\{A\subseteq X:A\subseteq f(A)\}$. (Be sure to show that there is at least one $A\subseteq X$ such that $A\subseteq f(A)$.)</p>
|
3,832,383 | <p>Below is a problem I found, however, after many attemps I can not seem to get a solution.</p>
<p><strong>Problem:</strong> Let <span class="math-container">$E \subset [0,1]$</span>. Show that if <span class="math-container">$m^*(E) + m^*([0,1] \setminus E) = 1$</span>, then <span class="math-container">$E$</span> is... | Sumanta | 591,889 | <p>Let <span class="math-container">$A$</span> be any Lebesgue measurable subset of <span class="math-container">$[0,1]$</span>. So, Lebesgue measurability of <span class="math-container">$A$</span> gives <span class="math-container">$$m^*(E)=m^*(A\cap E)+m^*(A^c\cap E)$$</span> and <span class="math-container">$$m^*(E... |
3,515,649 | <blockquote>
<p><a href="https://i.stack.imgur.com/qqIay.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/qqIay.jpg" alt="enter image description here" /></a></p>
</blockquote>
<p><strong>My try</strong> :</p>
<p>we can deal three different cases of removing two biased coin and 1 unbiased coin.</p>
<p... | Robert Shore | 640,080 | <p>Here's how I think about these problems. There are <span class="math-container">$50$</span> coins with <span class="math-container">$2$</span> sides each, or <span class="math-container">$100$</span> possible outcomes, each equally likely. Of those possible outcomes, <span class="math-container">$53$</span> result... |
1,254,896 | <p>I am trying to solve identify the expected value of a statistic that involves a fraction. I have simplified the expression to:</p>
<p>$E[\frac{1}{1+ \sum_i x_i}] = E[\frac{1}{1+ T}]$</p>
<p>However, I am not sure how to proceed. Is there anyway to simplify this through algebra, i.e. simplify the expression to the ... | Paolo Leonetti | 45,736 | <p>Notice that by the Legendre symbol properties we have
$$
\left(\frac{a}{p}\right) \equiv a^{\frac{p-1}{2}} \equiv x \pmod p
$$
where $x$ can be only $-1$,$0$ or $1$. That's why if you have a $n$-th power in a diophantine equation and $2n+1$ is prime, then you're really lucky, although it is not a strange guess xd</p... |
1,254,896 | <p>I am trying to solve identify the expected value of a statistic that involves a fraction. I have simplified the expression to:</p>
<p>$E[\frac{1}{1+ \sum_i x_i}] = E[\frac{1}{1+ T}]$</p>
<p>However, I am not sure how to proceed. Is there anyway to simplify this through algebra, i.e. simplify the expression to the ... | user26486 | 107,671 | <p>You should check modulo $p=\text{lcm}(a_1,a_2,\ldots,a_n)+1$, if it is prime (here $a_1,a_2,\ldots,a_n$ are the degrees of the variables). The reason this may easily work is: by FLT (if $p\nmid x$) $$x^{\text{lcm}(a_1,a_2,\ldots,a_n)}\equiv 1\pmod {\!p}$$ </p>
<p>and $a_ik_i=\text{lcm}(a_1,a_2,\ldots,a_n)$ for so... |
2,780,731 | <p>In school, I have recently been learning about simple differential equations. We know that the solution of $y'=y$ is $y=Ae^x$, where $A$ is a constant. But how can we know that it is the <strong>only</strong> solution? The only thing I can figure out is that $y$ is continuously differentiable. Help me, please.</p>
| Jan | 254,447 | <p>Suppose there is another solution $y=f(x)$. Then we must have that $y=f(x)-Ae^x$ is a solution. Now working this out gives us:
$$y'=\frac{\partial }{\partial x}(f(x) - A e^x) = f'(x) - A e^x$$
Note that we must have $y=y'$ so $f(x)=f'(x)$ clearly this only holds for $f(x) = A e^x$.</p>
|
1,598,006 | <p>(Here, $B$ is relatively compact means the closure of $B$ is compact.)</p>
<blockquote>
<ol>
<li><p>$\hat A$ is compact.</p></li>
<li><p>$\hat A=\hat {\hat A}$.</p></li>
<li><p>$\hat A$ is connected.</p></li>
<li><p>$\hat A=X$.</p></li>
</ol>
</blockquote>
<p>I try to eliminate the options by using an ... | qinxs | 448,972 | <p>For (2), <span class="math-container">$X-\hat{A}$</span> is the union of non-relatively compact connected components of <span class="math-container">$X-A$</span> by definition. Let <span class="math-container">$O$</span> be any connected component of <span class="math-container">$X-A$</span> which is not relatively ... |
1,341,486 | <p>Problem:
Find the sum to $n$ terms of
\begin{eqnarray*}
\frac{1}{1\cdot 2\cdot 3} + \frac{3}{2\cdot 3\cdot 4} + \frac{5}{3\cdot 4\cdot 5} +
\frac{7}{4\cdot 5\cdot 6}+\cdots \\
\end{eqnarray*}
Answer:
The way I see it, the problem is asking me to find this series:
\begin{eqnarray*}
S_n &=& \sum_{i=1}^... | A.A. | 240,951 | <p>see that $a_n = (2n-1)/((n*(n+1)*(n+2)=-1/2*(1/n)+3*(1/(n+1))-5/2*(1/(n+2)) = 1/2* (-1/n+1/(n+1)+5*(1/(n+1)-1/(n+2))$, so $S_n = 1/2*(-1+5+1/(n+1)-5/(n+2)) = 1/2*(4n^2+8n+5)/((n+1)*(n+2))$.</p>
|
1,341,486 | <p>Problem:
Find the sum to $n$ terms of
\begin{eqnarray*}
\frac{1}{1\cdot 2\cdot 3} + \frac{3}{2\cdot 3\cdot 4} + \frac{5}{3\cdot 4\cdot 5} +
\frac{7}{4\cdot 5\cdot 6}+\cdots \\
\end{eqnarray*}
Answer:
The way I see it, the problem is asking me to find this series:
\begin{eqnarray*}
S_n &=& \sum_{i=1}^... | Community | -1 | <hr>
<p>The good old way: compute a few terms and find a pattern.</p>
<p>$$\frac2{12},\frac7{24},\frac{15}{40},\frac{26}{60},\frac{40}{84}\cdots$$</p>
<p>By finite differences, we find that the numerators are $\dfrac{n(3n+1)}2$ and the denominators $2(n+1)(n+2)$.</p>
<hr>
<p>Check:</p>
<p>$$S_1=\frac2{12}$$and
$$... |
3,752,771 | <p>I wanted to get the full probability of 2 attempts made at 60% chance of success.</p>
<p>I was looking at a different chain of math and found my probability to hit an enemy is 60% per each attack but I was wondering how it would look at all the outcomes and the probability of it.</p>
<blockquote>
<p>6/10 * 6/10 = 36... | Cardinal | 525,594 | <p>Scenario 1: Both Hit</p>
<p>6/10 * 6/10 = 36%</p>
<p>Scenario 2: Hit first, miss second</p>
<p>6/10 * 4/10 = 24%</p>
<p>Scenario 3: Miss first, hit second</p>
<p>4/10 * 6/10 = 24%</p>
<p>Scenario 4: Miss Both</p>
<p>4/10 * 4/10 = 16%</p>
<p>Notice how this total adds to 100%. If you define success as at least one hi... |
858,576 | <p>Prove that the union of three subspaces of V is a subspace iff one of the subspaces contains the other two.</p>
<p>I can do this problem when I am working in only two subspaces of $V$ but I don't know how to do it with three. </p>
<p>What I tried is:
If one of the subspaces contains the other two, Then their union... | pre-kidney | 34,662 | <p>Suppose $W=V_1\cup V_2\cup V_3$ is a subspace. Since $W$ contains $V_1,V_2,V_3$ and is closed under linear combinations, we have $V_1+V_2+V_3\subset W$ as well. The reverse inclusion also holds, since each $V_1,V_2,V_3$ lie in the sum. Thus $$W=V_1+V_2+V_3.$$</p>
<p>Observe the inclusions
$$
W=V_1\cup V_2\cup V_3\s... |
4,276,974 | <p>I have to prove that sentence, but I'm not sure how to do that. Help!</p>
| Aaa Lol_dude | 979,418 | <p>The only thing we have to prove now is that the prime cannot be 3 mod 6.</p>
<p>If it is 3 mod 6, then it is divisible by 3. Only leaves you with 1 and 5 mod 6.</p>
<p>Note: 1 and 5 mod 6 are the p-1 and p+1, respectively.</p>
<p>The fact is relied upon that all primes greater than 3 are odd.</p>
|
2,611,676 | <p>Or consider the general problem-
Find the value of n for which x^n is just greater than x!</p>
<p>I dont know even if it is possible to find the solution or not...</p>
| Ross Millikan | 1,827 | <p>Use Stirling's approximation and take the log. $x! \approx (\frac xe)^x \sqrt {2 \pi x}$. Take the base $x$ log and round up.</p>
|
446,499 | <p>I have just learned the definition of connectedness and wikipedia gives an example of a disconnected set: <span class="math-container">$(0,1)\cup \left\{ 3 \right\}$</span> (<a href="https://en.wikipedia.org/wiki/Connected_space#Examples" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Connected_space#Exampl... | Seirios | 36,434 | <p><strong>Hint:</strong> Show that $\{3\}$ is a <a href="https://en.wikipedia.org/wiki/Clopen_set" rel="nofollow">clopen set</a>.</p>
|
3,128,352 | <p>I want to prove that when <span class="math-container">$F:K\rightarrow K[X]/\langle f\rangle $</span> is a map such that <span class="math-container">$F(a)=a+\langle f \rangle$</span>, then <span class="math-container">$F$</span> is an embedding from <span class="math-container">$K$</span> to <span class="math-cont... | Community | -1 | <p>By long division, the expression is reworked as the sum of a polynomial and a remainder term</p>
<p><span class="math-container">$$\frac c{x+7}.$$</span></p>
<p>The polynomial plays no role, and the derivatives of <span class="math-container">$(x+7)^{-1}$</span> at <span class="math-container">$x=5$</span> are <sp... |
2,867,404 | <p>I have a problem how to get the area from the picture.
Some ideas I got are not good enough to get the correct value of the whole element.</p>
<p><a href="https://i.stack.imgur.com/PeQ5O.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/PeQ5O.jpg" alt="enter image description here"></a></p>
| mathcounterexamples.net | 187,663 | <p>You don’t have enough data to compute the area (if 13.62 denotes the lenght of the basis as it seems on the OP figure). The area depends on the « balance » between the left and the right sides of the house. And this seems unknown from your picture.</p>
|
345,735 | <p>If <strong>two planes</strong> are <strong>intersected</strong> <em>by making a straight line, like <span class="math-container">$AB$</span></em> then</p>
<blockquote>
<p>Does the angle between two planes (see figure) <strong>always</strong> given by the
angle between normal vectors (<span class="math-container">$n_... | SSumner | 25,240 | <p>Yes. See <a href="http://www.netcomuk.co.uk/%7Ejenolive/vect14.html" rel="nofollow noreferrer">here</a></p>
<blockquote>
<p>It[The angle between two planes] is defined as the angle between 2 lines, one in each plane, so that they are at right angles to the line of intersection of the 2 planes (like the angle between... |
3,512,521 | <p>Let <span class="math-container">$x_1, x_2,\dots, x_n$</span> be positive real numbers. Let <span class="math-container">$A$</span> be the <span class="math-container">$n\times n$</span> matrix whose <span class="math-container">$i,j^\text{th}$</span> entry is <span class="math-container">$$a_{ij}=\frac{1}{x_i+x_j}.... | Ben Grossmann | 81,360 | <p>Regarding the hint: just as a sum of positive semidefinite operators is positive semidefinite, so is an integral of positive semidefinite operators positive semidefinite.
So, since <span class="math-container">$T(t)$</span> is positive semidefinite for all <span class="math-container">$t \geq 0$</span>, it follows t... |
3,512,521 | <p>Let <span class="math-container">$x_1, x_2,\dots, x_n$</span> be positive real numbers. Let <span class="math-container">$A$</span> be the <span class="math-container">$n\times n$</span> matrix whose <span class="math-container">$i,j^\text{th}$</span> entry is <span class="math-container">$$a_{ij}=\frac{1}{x_i+x_j}.... | A.Γ. | 253,273 | <p>Hint (without integrals): Let <span class="math-container">$X=\operatorname{diag}(x_1,x_2,\ldots,x_n)$</span> and <span class="math-container">$e$</span> be the vector of all ones. </p>
<ol>
<li>Prove that the Cauchy matrix <span class="math-container">$C$</span> satisfies the equation
<span class="math-container">... |
2,477,302 | <p><a href="https://i.stack.imgur.com/AtXyT.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/AtXyT.png" alt="enter image description here"></a></p>
<p>I solved the problem. I am getting the final answer as 8/15, however, the book says the answer is 7/15.Seems like they haven't subtracted 7/15 from 1 ... | Nick Peterson | 81,839 | <p><strong>Hint:</strong></p>
<p>If $x=0$, this is pretty straight-forward; I'll give you a hint for the $x>0$ case, though.</p>
<p>Let $\epsilon>0$. As you said, you can write
$$
\lvert\sqrt{x_n}-\sqrt{x}\rvert=\left\lvert\frac{x_n-x}{\sqrt{x_n}+\sqrt{x}}\right\rvert=\frac{\lvert x_n-x\rvert}{\sqrt{x_n}+\sqrt... |
2,477,302 | <p><a href="https://i.stack.imgur.com/AtXyT.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/AtXyT.png" alt="enter image description here"></a></p>
<p>I solved the problem. I am getting the final answer as 8/15, however, the book says the answer is 7/15.Seems like they haven't subtracted 7/15 from 1 ... | MrYouMath | 262,304 | <p>Hint: First look at the case that the limit is equal to $x=0$. The assume $x\neq 0$ and find an upper bound for the fraction.</p>
<p>$$\frac{1}{\sqrt{x_n}+\sqrt{x}}$$</p>
|
1,311,367 | <p>Recently, I am considering a question, as is well known, Cauchy's inequality is a famous and useful inequality. $$\left|\int_{a}^{b}f(x)dx\right|^2\leq|b-a| \int_{a}^{b}f^2(x)dx.$$ My question is: can we obtain a inequality such that
$$\left|\int_{a}^{b}f(x)dx\right|^2\ge |A|\times \left|\int_{a}^{b}f(x)^2dx\right... | marty cohen | 13,079 | <p>No you can't,
because you can choose
$f(x)$ so that
$\int_a^b f(x) dx = 0$.</p>
<p>If you restrict
$f$ so that
it is always positive,
then I think you need
a bound on $f$.</p>
<p>(added later)</p>
<p>There <em>is</em> a reversed-inequality
of the discrete C-S inequality:</p>
<p>If $0 < m < a_i/b_i < M$,... |
1,419,315 | <p>I have a particular scenario.</p>
<p>In this scenario, we have the standard cubic equation,</p>
<pre><code>ax^3 + bx^2 + cx + d = y
</code></pre>
<p>as well as 3 points that are graphed, <a href="https://i.imgur.com/VCZKuGW.png" rel="nofollow noreferrer">as can be seen in this graph</a>. (The line is irrelevant ... | m0_as | 266,546 | <p>To say about positive (negative) (semi-) definite, you need to find eigenvalues of A. Then,
1) If all eigenvalues are positive, A is positive definite
2) If all eigenvalues are non-negative, A is positive semi-definite
3) If all eigenvalues are negative, A is negative definite
4) If all eigenvalues are non-positive,... |
509,635 | <p>If every chain in a lattice is complete (we take the empty set to be a chain), does that mean that the lattice is complete? If yes, why? </p>
<p>My intuition says yes, and the reasoning is that we should somehow be able to define a supremum of any subset of the lattice to be the same as the supremum of some chain r... | Brian M. Scott | 12,042 | <p>If $L$ is not complete, it has a subset with no join; among such subsets let $A$ be one of minimal cardinality, say $A=\{a_\xi:\xi<\kappa\}$ for some cardinal $\kappa$. For each $\eta<\kappa$ let $A_\eta=\{a_\xi:\xi\le\eta\}$; $|A_\eta|<\kappa$, so $A_\eta$ has a join $b_\eta$. Clearly $b_\xi\le b_\eta$ whe... |
4,500,928 | <p>How many ten-digit positive integers are there such that all of the
following conditions are satisfied:</p>
<p>(a) each of the digits 0, 1, ... , 9 appears exactly once;</p>
<p>(b) the first digit is odd;</p>
<p>(c) five even digits appear in five consecutive positions?</p>
<p>From Combinatorics by Pavle Mladenovic<... | MathFail | 978,020 | <p>The first digit must be odd, I use X to mark it. I use <span class="math-container">$[-----]$</span> to mark the five consecutive positions for even integers. I use <span class="math-container">$[~~]$</span> to mark other seat for odd integers. So we have <span class="math-container">$5$</span> ways to locate the fi... |
393,250 | <p>Let <span class="math-container">$G$</span> be a finitely generated residually finite group and let <span class="math-container">$M$</span> be a finitely generated <span class="math-container">$\mathbb{Z}[G]$</span>-module.</p>
<p><strong>Question</strong>: Must <span class="math-container">$M$</span> be residually ... | YCor | 14,094 | <p>It's true, and due to Ph. Hall, when <span class="math-container">$G$</span> is virtually nilpotent, and more generally (Roseblade) when <span class="math-container">$G$</span> is virtually polycyclic.</p>
<p>When <span class="math-container">$G=\mathbf{Z}\wr\mathbf{Z}$</span> there exists an infinite simple <span c... |
4,363,409 | <blockquote>
<p>Define <span class="math-container">$X_0=\alpha\in(0,1)$</span> the initial capital and <span class="math-container">$X_n$</span> as the remaining capital after each game.
A player bets <span class="math-container">$1-X_n$</span> if <span class="math-container">$X_n>1/2$</span> and <span class="math-... | Josiah | 1,017,665 | <p>The basic reasoning in Andrew's answer is what you'd get for a perfect mirror. There are a few more physical effects which may be worth thinking about to get it just right.</p>
<p>One is that water is much <a href="https://en.m.wikipedia.org/wiki/Reflectance#Water_reflectance%20less%20reflective" rel="nofollow noref... |
4,413,093 | <p>Determine the radius of convergence of the series <span class="math-container">$\sum_{n=1}^{\infty}a_nz^n$</span> where <span class="math-container">$a_n=\frac{n^2}{4^n+3n}$</span></p>
<p>Now <span class="math-container">$\alpha=\limsup_{n\to \infty}(\vert a_n\vert)^\frac{1}{n}$</span> and so radius of convergence <... | Dr. Sundar | 1,040,807 | <p>When we define any function <span class="math-container">$f: A \rightarrow B$</span>, its domain <span class="math-container">$A$</span> must be carefully chosen so that <span class="math-container">$f(x)$</span> is a unique, well-defined, output for each input variable <span class="math-container">$x$</span> in the... |
225,128 | <p>I'm trying to define a function accepting only real values like this:</p>
<pre><code>f[x_Real] := x^2
f[0]
</code></pre>
<p>But it outputs</p>
<blockquote>
<p><code>f[0] </code></p>
</blockquote>
<p>and doesn't output 0.</p>
<p>Is there any reason why <code>f[x_Real]</code> doesn't work? I tested <code>f[x_Integer]<... | mgamer | 19,726 | <p>0 without the "." is an Integer, see</p>
<pre><code>Head @ 0
</code></pre>
<p>(* Integer *)</p>
<p>but</p>
<pre><code>f[0.]
</code></pre>
<p>delivers</p>
<pre><code>0.
</code></pre>
|
2,204,944 | <p>A line is a collection of infinitely many points. By definition, a point has no dimensions. But, how can infinitely many dimensionless points give rise to a line with a dimension. This is the same case with planes, solids and higher dimensions too...</p>
<p>Thanks in advance for any help..!!</p>
| user1551 | 1,551 | <p>$X_1$ and $X_2$ follow the same discrete uniform distribution (since $X_1$ and $X_2|X_1$ are uniform, so is $X_2$). Hence they have identical expectations.</p>
|
2,208,755 | <p>I got stuck on this question: find all solutions $x$ for $a\in R$:</p>
<p>$$\frac{(x^2-x+1)^3}{x^2(x-1)^2}=\frac{(a^2-a+1)^3}{a^2(a-1)^2}$$</p>
<p>I see that if we simplify we get:</p>
<p>$$\frac{(x^2-x+1)^3}{x^2(x-1)^2}=\frac{[(x-{\frac 12})^2+{\frac 34}]^3}{[(x-{\frac 12})^2-{\frac 14}]^2}$$</p>
<p>From the ex... | dxiv | 291,201 | <p>Hint: write the LHS in terms of $z = x+\cfrac{1}{x}$ as follows:</p>
<p>$$
\require{cancel}
\frac{(x^2-x+1)^3}{x^2(x-1)^2} = \frac{(x^2-x+1)^3}{x^2(x^2-2x+1)}= \frac{\bcancel{x^3}\left(x+\cfrac{1}{x}-1\right)^3}{\bcancel{x^3}\left(x+\cfrac{1}{x}-2\right)}=\frac{(z-1)^3}{z-2}
$$</p>
<p>Let $b=a+\cfrac{1}{a}$ ... |
2,677,823 | <p>How can I precisely prove the existence of a continuous function $\rho(x)$ such that $0 \leq \rho(x) \leq 1 \forall x \in R^d $ such that $g(x) \rho(x)$ is bounded and continuous for $g(x)$ continuous?Both $g(x)$ and $\rho(x)$ are defined on $R^d$.</p>
<p>My idea was that we can choose $\rho(x)$ such that $\rho(x)g... | Surb | 154,545 | <p>If $f:\mathbb C\to \mathbb R$ it's obvious since the integral of a real function is real. Suppose $f:\mathbb C\to\mathbb C$ and write $$f(z)=u(x,y)+iv(x,y)$$
with $x,y\in\mathbb R$, $u=\Re(f)$ and $v=\Im(f)$. Since $u(x,y),v(x,y)\in\mathbb R$ for all $x,y$,
$$\overline{\int f}=\overline{\int u+i\int v}=\overline{\in... |
99,799 | <p>I have a <code>Solve</code> similar to the following:</p>
<pre><code>Solve[e^2 - c^2 == -15, {e, c}, Integers]
(* {{e -> -7, c -> -8}, {e -> -7, c -> 8}, {e -> -1, c -> -4},
{e -> -1, c -> 4}, {e -> 1, c -> -4}, {e -> 1, c -> 4},
{e -> 7, c -> -8}, {e -> 7, c -... | bbgodfrey | 1,063 | <p>The unwanted <code>List</code> also can be replaced after completing <code>Solve</code>.</p>
<pre><code>Solve[e^2 - c^2 == -15 ∧ {e, c} ∈ Interval[{0, 4}], {e, c}, Integers]
/. Rule[z1_, {z2_}] -> Rule[z1, z2]
</code></pre>
<p>or</p>
<pre><code>Replace[Solve[e^2 - c^2 == -15 ∧ {e, c} ∈ Interval[{0, 4}], {... |
250,454 | <p>Is there a <code>ReplaceOnce</code> which does only one replacement if possible by trying the rules sequentially in order. Consider the following as an example:</p>
<pre><code>ReplaceOnce[{"May","5","May","5"},{"May"->1,"5"->2}]
</code></pre>
<p>shoul... | Ben Izd | 77,079 | <p>I believe a better solution exists.</p>
<pre><code>ClearAll[replaceLimitedBack, replaceLimited];
replaceLimitedBack[expr_, rulesRaw_, n_Integer, levelSpec_] :=
Block[{rules = Association[rulesRaw], one = 0},
{Replace[expr,
a_?(If[one < n && KeyExistsQ[rules, #], one += 1; True,
False] ... |
78,641 | <p>I am interested in the relation between the property of countable chain condition (ccc) and the property of separable. Could someone recommend some papers or books about this to me? thanks in advance.</p>
| Joel David Hamkins | 1,946 | <p>You will want to look at <a href="http://en.wikipedia.org/wiki/Suslin%27s_problem" rel="nofollow">Suslin lines</a>, which are examples of ccc non-separable orders. The existence of Suslin lines is independent of the axioms of ZFC, and this material is covered in any of the usual graduate set theory texts, such as Je... |
2,073,923 | <p>It seems the number of nonnegative integer solutions to the equation $xyz=n$ is given by
$$\sum\limits_{d \mid n} \tau(d)$$</p>
<p>$\tau$ is the number of divisors function. I'm wondering if there is a way to simplify this sum. Really appreciate any kind of help. Thank you.</p>
<hr>
<p>Here is my attempt so far
... | ajotatxe | 132,456 | <p>The function
$$f(n)=\sum_{d\mid n}\tau(d)$$
is multiplicative. That is, $f(mn)=f(m)f(n)$ whenever $\gcd(m,n)=1$.</p>
<p>Let's try to find a formula for powers of primes:</p>
<p>$$f(p^r)=\sum_{d\mid p^r}\tau(d)=\sum_{k=0}^r\tau(p^k)=\sum_{k=0}^r(k+1)=\frac{(r+1)(r+2)}2$$</p>
<p>Then, if the prime factorization of ... |
3,097,672 | <p>I have to find the definite integral of this:</p>
<p><span class="math-container">$$\int_2^3 \frac{dx}{(x^2-1)^{\frac{3}{2}}}$$</span></p>
<p>So let's start with the indefinite integral:</p>
<p>so <span class="math-container">$x = \sec \theta$</span> so <span class="math-container">$ dx = \sec \theta \tan \theta ... | J.G. | 56,861 | <p>You made a slight mistake: since <span class="math-container">$\sec^2\theta-1=\tan^2\theta$</span> you should have <span class="math-container">$\int\frac{\sec\theta\tan\theta d\theta}{\tan^3\theta}=\int\frac{\cos\theta d\theta}{\sin^2\theta}$</span>, now use <span class="math-container">$u=\sin\theta$</span>.</p>
|
2,624,498 | <p>Evaluate $$\lim_{n \rightarrow\infty} \sqrt[n]{3^{n} +5^{n}}$$</p>
<p>Attempt:</p>
<p>The only sort of manipulation that has come to mind is: $$e^{\frac{1}{n}ln(e^{n\ln(3)} + e^{n\ln(5)})}$$</p>
<p>So what is the trick to successfully evaluate this?</p>
| Rene Schipperus | 149,912 | <p>Squeeze </p>
<p>$$5\leq \sqrt[n]{3^n+5^n}\leq 5\sqrt[n]{2}$$</p>
|
2,624,498 | <p>Evaluate $$\lim_{n \rightarrow\infty} \sqrt[n]{3^{n} +5^{n}}$$</p>
<p>Attempt:</p>
<p>The only sort of manipulation that has come to mind is: $$e^{\frac{1}{n}ln(e^{n\ln(3)} + e^{n\ln(5)})}$$</p>
<p>So what is the trick to successfully evaluate this?</p>
| user284331 | 284,331 | <p>\begin{align*}
\dfrac{1}{n}\cdot\log\left(\left(\dfrac{3}{5}\right)^{n}+1\right)\rightarrow 0,
\end{align*}
so
\begin{align*}
\left(\left(\dfrac{3}{5}\right)^{n}+1\right)^{1/n}\rightarrow e^{0}=1,
\end{align*}
and hence
\begin{align*}
\sqrt[n]{3^{n}+5^{n}}=5\left(\left(\dfrac{3}{5}\right)^{n}+1\right)^{1/n}\rightarr... |
2,746,637 | <p>We have $*$ the Hodge operator, and $d $ the exterior derivative. We define $\delta=\pm *d*$ and $\triangle=d\delta+\delta d $. <a href="https://rads.stackoverflow.com/amzn/click/0387908943" rel="nofollow noreferrer">Warner</a> (pp. 223) says that we have
$$
\triangle (E^p (M))=d\delta (E^p (M))\oplus \delta d (E^p... | Ricanry | 451,947 | <p><strong>Original answer</strong>:</p>
<p>Since you know that "the first space is a subspace of the second, and why the second is of the third", it's sufficient to show that the first space is equal to the third. As in Hodge theory, we assume manifold M is compact and oriented.</p>
<p>Set $\mathbb{H}^k(M)=\{\omega ... |
3,442,862 | <blockquote>
<p>Let <span class="math-container">$F $</span> be a subset family of the set {<span class="math-container">$ 1, 2, ..., 2017 $</span>} such that
for any <span class="math-container">$ A, B \in F $</span>, worth that <span class="math-container">$A \cap B$</span> has exactly one
element. Determine a... | Community | -1 | <p>We can rewrite the argument as follows.</p>
<p>(1). Each set in <span class="math-container">$F$</span> can be represented as a vector. For example, <span class="math-container">$\{2,3,6\}$</span> is represented by <span class="math-container">$$(0,1,1,0,0,1,0,0,...)$$</span> where a "<span class="math-containe... |
1,028,371 | <p>I have been trying to prove this, but I am having trouble understanding how to prove the following mapping I found is injective and surjective. Just as a side note, I am trying to show the complex ring is isomorphic to special $2\times2$ matrices in regard to matrix multiplication and addition. Showing these hold is... | anomaly | 156,999 | <p>The map is bijective by the argument you give; the trick is showing that it respects addition and (especially) multiplciation. Here's another way of doing so:
Consider $\mathbb{C}$ as a $2$-dimensional real vector space with basis $\{1, i\}$. You then have a map $f:\mathbb{C} \to M_2(\mathbb{R})$ (i.e., $2$-dimensio... |
1,028,371 | <p>I have been trying to prove this, but I am having trouble understanding how to prove the following mapping I found is injective and surjective. Just as a side note, I am trying to show the complex ring is isomorphic to special $2\times2$ matrices in regard to matrix multiplication and addition. Showing these hold is... | sunspots | 110,953 | <p>Consider the ordered basis <span class="math-container">$\beta=\{1,i\},$</span> as in the answer by anomaly. Then, the range <span class="math-container">$$R(\phi) = span(\phi(\beta)) = span(\{\phi(1, 0), \phi(0, 1)\}) = span(\{
\left (\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}\right ),
\left (\begin{array... |
3,419,276 | <p>I'm reading about the directional derivative:</p>
<blockquote>
<p>Let <span class="math-container">$(E,\|\cdot\|)$</span> and <span class="math-container">$(F,\|\cdot\|)$</span> be Banach spaces over the field <span class="math-container">$\mathbb{K}$</span>, and <span class="math-container">$X$</span> an open su... | user | 505,767 | <p>Recall that by <a href="https://en.wikipedia.org/wiki/Logarithm" rel="nofollow noreferrer">definition of logarithm</a> with <span class="math-container">$u>0$</span> and <span class="math-container">$b>0,\neq 1$</span> we have</p>
<p><span class="math-container">$$v=\log_b u \iff b^v=u$$</span></p>
<p>theref... |
1,984,178 | <p>I have a problem with the following exercise:</p>
<p>We have the operator $T: l^1 \to l^1$ given by</p>
<p>$$T(x_1,x_2,x_3,\dots)=\left(\left(1-\frac11\right)x_1, \left(1-\frac12\right)x_2, \dots\right)$$ for $(x_1,x_2,x_3,\dots)$ in $l^1$. Showing that this operator is bounded is easy, but I am really desperate... | Community | -1 | <p>$$\left|\left| T((x_n)_{n\in\mathbb{N}})\right|\right|_{\ell_1}=\left|\left| \left(\left(1-\frac{1}{n}\right)x_n\right)_{n\in\mathbb{N}}\right|\right|_{\ell_1}=\sum_j \left|\left(1-\frac{1}{j}\right)x_j\right|\leqslant \sum_j |x_j |=||(x_n )_{n\in\mathbb{N}} ||_{\ell_1}$$</p>
<p>hence </p>
<p>$$||T||\leqslant 1$$<... |
2,007,173 | <p>I've managed to severely confuse myself in my attempts to simplify this seemingly straightforward expression:
$$
\arctan(\cot(\alpha)),\quad\text{with $0<\alpha\leq\pi$.}
$$
It seems like maybe there are some issues with domain, as $\cot$ and $\tan$ are defined on $(0,\pi)$ and $(-\frac{\pi}{2},\frac{\pi}{2})$, r... | Jed | 281,031 | <p>Draw a right triangle with one of the angles being $\alpha$. Label the leg that contains angle $\alpha$ to be $x$ and the opposite leg to be of length 1. This way, $$\cot(\alpha)=x$$ Now let's investigate $\arctan(x)$. Since the third angle of this right triangle measures $\frac{\pi}{2}-\alpha$, you know that $\tan(... |
1,662,226 | <p>Find a sufficient statistic for $σ^2$ with $μ$ known, where $X_i$ is a random sample from $N(μ,σ^2)$</p>
<p>I was able to find a sufficient statistic for $μ$ with $σ^2$ known, but I'm stuck on finding one for $σ^2$ when $μ$ is known. Can anyone give me some help? </p>
<p>I was using the factorization method before... | parsiad | 64,601 | <p>Though squaring each and checking works equally as well, here's another way to figure it out:
$$
\sqrt{0.016}=\frac{\sqrt{1000}}{\sqrt{1000}}\sqrt{0.016}=\frac{\sqrt{16}}{\sqrt{1000}}=\frac{4}{\sqrt{1000}}\approx\frac{4}{\sqrt{1024}}=\frac{4}{\sqrt{2^{10}}}=\frac{4}{2^{5}}=\frac{4}{32}=\frac{1}{8}=0.125
$$</p>
|
13,951 | <p>let $f$ and $g $ be two real valued function , I have asked many students what is the derivative of $(fg)'$ they answered me :it is $f' \cdot g'$, then I seek why most people (students) guess that ?</p>
| Benoît Kloeckner | 187 | <p>As mentioned, a probable cause is an implicit reasoning as if every operation were a homomorphism (similar to the implicit reasoning by linearity). Similar errors include $\ln(x+y) = \ln(x)+\ln(y)$, $e^{xy}=e^x e^y$, $\int f(x)g(x) dx = \int f(x) dx \int g(x) dx$, etc.</p>
<p>Such reasoning by (very loose) analogy ... |
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