qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
3,134,991 | <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="... | Vasili | 469,083 | <p>Your approach is good also, you probably made a mistake in calculations. The number of favorable outcomes is
<span class="math-container">$$\binom{9}{0}+\binom{9}{2}+\binom{9}{4}+\binom{9}{6}+\binom{9}{8}=1+36+126+84+9=256$$</span>
The number of all possible outcomes is <span class="math-container">$512$</span> thus... |
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="... | Sri-Amirthan Theivendran | 302,692 | <p>The probability generating function of a Binomiall random variable <span class="math-container">$X\sim \text{Bin}(n, 1/2)$</span> with probability of success <span class="math-container">$1/2$</span> is given by
<span class="math-container">$$
g_{X}(t)=Et^X=\sum_{k=0}^nP(X=k)t^k=\sum_{k=0}^n\binom{n}{k}\frac{t^k}{2^... |
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>
| mm-aops | 81,587 | <p>$200$. easy to see you can't have more cause every black ball can be a neighbour of at most $2$ white balls. to obtain such an arrangement just put a white ball on the left and on the right of each black one, put them in a row, it's evident such an arrangement satisfies your conditions.</p>
|
455,979 | <p>Suppose we have three 6-sided die that all share the same common bias:</p>
<p>For a single dice: let the probability of rolling a 2 or $P(2) = 2{\times}P(1$), let the probability of rolling a 3 or $P(3) = 3{\times}P(1)$, and so on...</p>
<p>Such that:
$P(2) = 2P(1), P(3) = 3P(1), P(4) = 4P(1), P(5)=5P(1), P(6)=6P... | Stephen Herschkorn | 27,997 | <p>I think the best way to handle this is via the probability generating function $\pi(z) = Ez^X = \frac1{21}\sum_{k=1}^6 k z^k$ for the value $X$ from a roll of a single die. The probability generating function for the sum of $n$ independent rolls is $[\pi(z)]^n$. Multiply out the polynomilas (a CAS - <em>e.g.</em>... |
64,130 | <p>This is an arithmetic follow-up to my previous question <a href="https://mathoverflow.net/questions/64112/does-there-exist-a-non-trivial-semi-stable-curve-of-genus-1-with-only-4-singular">Does there exist a non-trivial semi-stable curve of genus >1 with only 4 singular fibres</a> </p>
<p>Let $k$ be an algebraica... | Qing Liu | 3,485 | <p>There are some classical examples of such surfaces. For any prime number $p\ge 11$ different from 13, the modular curve $X_0(p)$ has good reduction away from $p$, and semi-stable reduction at $p$ (equal to the union of two projective lines intersecting at supersingular $j$'s). This is proved by Deligne-Rapoport (see... |
2,736 | <p><a href="https://mathoverflow.net/questions/18989/generating-classical-groups-over-finite-local-rings">Generating Classical Groups over Finite Local Rings</a> asks a question that, according to the poster's own 'answer' <a href="https://mathoverflow.net/a/19098/2383">https://mathoverflow.net/a/19098/2383</a>, is not... | Community | -1 | <p>Yes, you can, and likely should, try again. It could be seen as problematic to submit the exact same proposal again, yet since it is substantially different and informed by the feedback you got this seems perfectly fine.</p>
<p>Under certain circumstances, rather not in this case, I would even support submitting t... |
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>
| Macavity | 58,320 | <p>This is to show that $x=1$ always works. So we need to show that
$$f(a)=(a-1)-\log a \ge 0, \quad \forall a> 0$$</p>
<p>$$f'(a) = 1-\frac1a = \begin{cases} < 0, && a < 1 \\ > 0, && a> 1 \end{cases}$$</p>
<p>So the function is decreasing for $a < 1$ and increasing for $a> 1$, ... |
2,002,201 | <p>simplify <span class="math-container">$\sqrt{(45+4\sqrt{41})^3}-\sqrt{(45-4\sqrt{41})^3}$</span>.</p>
<blockquote>
<p>1.<span class="math-container">$90^{\frac{3}{2}}$</span></p>
<p>2.<span class="math-container">$106\sqrt{41}$</span></p>
<p>3.<span class="math-container">$4\sqrt{41}$</span></p>
<p>4.<span class="ma... | Ennar | 122,131 | <p>When I see expression where both $\alpha = a+b\sqrt{n}$ and $\beta =a-b\sqrt n$ occur, I immediately calculate $\alpha + \beta = 2a$ and $\alpha\beta = a^2-nb^2$ since they are guaranteed to be integers (more precisely, the minimal polynomial of both of them is $x^2 - (\alpha+\beta)x+\alpha\beta$ which might be help... |
2,002,201 | <p>simplify <span class="math-container">$\sqrt{(45+4\sqrt{41})^3}-\sqrt{(45-4\sqrt{41})^3}$</span>.</p>
<blockquote>
<p>1.<span class="math-container">$90^{\frac{3}{2}}$</span></p>
<p>2.<span class="math-container">$106\sqrt{41}$</span></p>
<p>3.<span class="math-container">$4\sqrt{41}$</span></p>
<p>4.<span class="ma... | Mathew Mahindaratne | 525,941 | <p>Considering all positive values, $45\pm4\sqrt{41}$ can be written as $45\pm4\sqrt{41}=(\sqrt{41}\pm2)^2$, thus, the given expression can be simplified as follows: $$\sqrt{(45+4\sqrt{41})^3}-\sqrt{(45-4\sqrt{41})^3}=\left(\sqrt{45+4\sqrt{41}}\right)^3-\left(\sqrt{45-4\sqrt{41}}\right)^3=\left(\sqrt{\left(\sqrt{41}+2\... |
4,510,384 | <p>In exercise 2.13 of page 43 of the book <a href="https://rads.stackoverflow.com/amzn/click/com/0134746759" rel="nofollow noreferrer" rel="nofollow noreferrer">Mathematical Proofs: A Transition to Advanced Mathematics</a> the reader is asked to state the logical negation of some statements. Of these, I find the autho... | Dan Christensen | 3,515 | <p>Let <span class="math-container">$x,y$</span> and <span class="math-container">$z$</span> be the sides of a triangle. We have: <span class="math-container">$x\neq y,
~ x\neq z,~ y\neq z$</span></p>
<p>Let <span class="math-container">$len(s)$</span> be the length of side <span class="math-container">$s$</span>.</p>
... |
3,032,950 | <p>I have the following Cauchy problem. I do not know where to start, so I would appreciate any help and tips.
<span class="math-container">$$\frac{\partial^2 Y(t, x)}{\partial t^2} = 9\frac{\partial^2 Y(t,x)}{\partial x^2} - 2Z(t,x)$$</span>
<span class="math-container">$$\frac{\partial^2 Z(t, x)}{\partial t^2} = 6\fr... | Yuri Negometyanov | 297,350 | <p><span class="math-container">$$\begin{cases}
2Y(t,x) = 6\dfrac{\partial^2 Z(t,x)}{\partial x^2} - \dfrac{\partial^2 Z(t, x)}{\partial t^2}\\[4px]
2\dfrac{\partial^2 Y(t, x)}{\partial t^2} = 18\dfrac{\partial^2 Y(t,x)}{\partial x^2} - 4Z(t,x),
\end{cases}$$</span>
so
<span class="math-container">$$\dfrac{\partial^2}{... |
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... | lab bhattacharjee | 33,337 | <p><span class="math-container">$$u=\sec x+\sec y=\dfrac{4\cos\dfrac{x+y}2\cos\dfrac{x-y}2}{\cos(x-y)+\cos(x+y)}$$</span></p>
<p><span class="math-container">$$u=\dfrac{4\cos C\cdot t}{2t^2-1+\cos2c}$$</span></p>
<p>which is a quadratic equation in <span class="math-container">$t=\cos\dfrac{x-y}2$</span></p>
<p>As <... |
1,598,545 | <p>Maybe I am not well versed with the actual definition of mean, but I have a doubt. On most resources, people say that arithmetic mean is the sum of $n$ observations divided by n. So my first question: </p>
<blockquote>
<p>How does this formula work? Is there any derivation to it? If not,
then while creating thi... | Ahmed S. Attaalla | 229,023 | <p>I suppose how you would come up with mean:</p>
<p>Suppose we have played some soccer games, and here is the list of the goals in the soccer games we played:</p>
<p>$${1,1,1,1,1,9}$$</p>
<p>And we want to get a sense of how we did as a whole or how much we scored per game as a whole.</p>
<p>Well looking at the li... |
761,286 | <p>let $G$ be an infinite group of the form $G_1 \oplus G_2 \oplus \dots \oplus G_n$ where each $G_i$ is a <strong>non trivial</strong> group and $n>1$. Prove that $G$ is not cyclic.</p>
<p><strong>Attempt</strong> : Let $G = G_1 \oplus G_2 \oplus \dots \oplus G_n$ be cyclic.</p>
<p>then $\exists ~g =(g_1,g_2,....... | ml0105 | 135,298 | <p>It would be $\Omega(n^{2})$ time, but not $O(n^{2})$ time. The intuition for this is that you multiply the complexities of inner loops.</p>
<p>Consider as well, straight from the definition of Big-O. Suppose your algorithm was $O(n^{2})$. Then $n * n\sqrt{n} \leq C * n^{2}$, for some positive constant $C$. We then ... |
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... | Ned | 67,710 | <p>Your argument (the second one) seems to miss the FIRST ace aspect of the situation. The <span class="math-container">$3*51!$</span> orderings in the numerator have the spade ace immediately following another ace but some have yet a different ace occurring before both of them, so they shouldn't be counted.</p>
|
2,299,466 | <p>For the first-order language with vocabulary $(E)$ (the binary relation $E$ which holds if two vertices have an edge) together with a set $G$ of vertices, I've been told that the property "a symmetric graph is connected" cannot be axiomatized by any set of first-order sentences. </p>
<p>I think the proof involved t... | Eric Wofsey | 86,856 | <p>Your description of $T$ is not complete; here's what it should be. Suppose there exists a first order axiomatization $T_0$ of connected graphs. Now let $T$ be the union of $T_0$ and your sentences "there is no path of length $n$ between $x$ and $y$" for each $n$. Any finite subset of this $T$ has a model, since y... |
813,517 | <p>Deduce that the next integer greater than $(3+\sqrt 5)^n$ is divisible by $2^n$</p>
<p>I tried expanding it by binomial theorem but got nothing</p>
| lab bhattacharjee | 33,337 | <p>HINT:</p>
<p>Let $a=3+\sqrt5, b=3-\sqrt5$</p>
<p>So, $a,b$ are the roots of $x^2-6x+4=0\implies t_{n+2}=6t_{n+1}-ta_n$ where $t_m=(3+\sqrt5)^m$</p>
|
813,517 | <p>Deduce that the next integer greater than $(3+\sqrt 5)^n$ is divisible by $2^n$</p>
<p>I tried expanding it by binomial theorem but got nothing</p>
| Mark Bennet | 2,906 | <p>There is a trick to this kind of question, which comes in handy. Note that $$0\lt 3-\sqrt 5 \lt 1$$ so so that if $$a_n=(3+\sqrt5)^n+(3-\sqrt 5)^n$$ we know that $$a_n-(3+\sqrt 5)^n=(3-\sqrt 5)^n\lt 1$$</p>
<p>Also $a_0=0, a_1=6$ are integers.</p>
<p>Now $3+\sqrt 5+3-\sqrt 5=6$ and $(3+\sqrt 5)(3-\sqrt 5)=4$ so $3... |
3,930,659 | <blockquote>
<p>Evaluate: <span class="math-container">$$ \int \frac{x^2}{\sqrt{1-x^2}}\,dx$$</span></p>
</blockquote>
<p>The solution I came across does a <span class="math-container">$u$</span>-substitution by letting <span class="math-container">$x = \sin(t)$</span>. But why <span class="math-container">$\sin(t)$</s... | Community | -1 | <p>I think that the OP's question refers to how to distinguish why to make this change of variable. Now, I will write the answer to that question.</p>
<p>Suppose you want to solve an integral of the form <span class="math-container">$$\color{blue}{\boxed{\int R(x,\sqrt{a^{2}-x^{2}})dx}}$$</span>
As in your problem, tha... |
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... | Reese Johnston | 351,805 | <p>It's the same reason that you can't argue that $\lim_{n \to \infty}(1 + \frac{1}{n})^n = 1$, even though $1 + \frac{1}{n} \to 1$ and $1^n = 1$. The issue is that when we say that the <em>limit</em> of an expression is a certain value, we just mean the expression gets <em>very close</em> to that value - we make no pr... |
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... | zhw. | 228,045 | <p>Just to give you some intuition, note that since $1-x < 1/(1+x)$ for $0<x<1,$</p>
<p>$$ (1-1/n)^n < \frac{1}{(1 + 1/n)^n}$$</p>
<p>for $n>1.$ Now the right side $\to 1/e.$ Thus if the limit of the left side exists, it has to be $\le 1/e.$ </p>
|
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>$$\Sigm... | Bernard | 202,857 | <p>You can say more for 1):</p>
<p>Set $\displaystyle s_2n=\sum_{k=1}^{2n}(-1)^{k+1}\frac{k+1}k $. Grouping pairs of consecutive terms, you can write
$$s_{2n}=\sum_{i=1}^n\biggl(\frac{2i}{2i-1}-\frac{2i+1}{2i}\biggr)=\sum_{i=1}^n\frac{1}{2i(2i-1)}=\sum_{i=1}^n\biggl(\frac{1}{2i-1}-\frac{1}{2i}\biggr)=\sum_{k=1}^{2n}\f... |
337,930 | <p>Given two polynomials</p>
<p>$$
p(x) = a_0 + a_1 x + a_2 x^2 + \ldots + a_{n-1}x^{n-1} \\
q(x) = b_0 + b_1 x + b_2 x^2 + \ldots + b_{n}x^{n}
$$</p>
<p>And the series expansion from their rational polynomial</p>
<p>$$
\frac{p(x)}{q(x)} = c_0 + c_1 x + c_2 x^2 + \ldots
$$</p>
<p>is it possible to recover the the o... | Felix Marin | 85,343 | <p><span class="math-container">$\newcommand{\+}{^{\dagger}}%
\newcommand{\angles}[1]{\left\langle #1 \right\rangle}%
\newcommand{\braces}[1]{\left\lbrace #1 \right\rbrace}%
\newcommand{\bracks}[1]{\left\lbrack #1 \right\rbrack}%
\newcommand{\dd}{{\rm d}}%
\newcommand{\isdiv}{\,\left.\right\vert\,}%
\newcommand{\... |
337,930 | <p>Given two polynomials</p>
<p>$$
p(x) = a_0 + a_1 x + a_2 x^2 + \ldots + a_{n-1}x^{n-1} \\
q(x) = b_0 + b_1 x + b_2 x^2 + \ldots + b_{n}x^{n}
$$</p>
<p>And the series expansion from their rational polynomial</p>
<p>$$
\frac{p(x)}{q(x)} = c_0 + c_1 x + c_2 x^2 + \ldots
$$</p>
<p>is it possible to recover the the o... | g------ | 161,506 | <p>Consider the $K\times K$ matrix
$$B= \left[ \begin{array}{ccccccc}
1 & 0 & 0 & \cdots & 0 & 0 & 0 \\
1 & 1 & 0 & \cdots & 0 & 0 & 0 \\
0 & 1 & 1 & \cdots & 0 & 0 & 0 \\
0 & 0 & 1 & \cdots & 0 & 0 & 0 \\
& & &... |
337,930 | <p>Given two polynomials</p>
<p>$$
p(x) = a_0 + a_1 x + a_2 x^2 + \ldots + a_{n-1}x^{n-1} \\
q(x) = b_0 + b_1 x + b_2 x^2 + \ldots + b_{n}x^{n}
$$</p>
<p>And the series expansion from their rational polynomial</p>
<p>$$
\frac{p(x)}{q(x)} = c_0 + c_1 x + c_2 x^2 + \ldots
$$</p>
<p>is it possible to recover the the o... | Marko Riedel | 44,883 | <p>Using a coefficient-extractor e.g. <span class="math-container">$[z^k] (1+z)^n = {n\choose k}$</span>, we
find</p>
<p><span class="math-container">$$\sum_{k=0}^n {R\choose k} {M\choose n-k}
= \sum_{k=0}^n {R\choose k} [z^{n-k}] (1+z)^M
\\ = [z^n] (1+z)^M \sum_{k=0}^n {R\choose k} z^k.$$</span></p>
<p>Now we may ... |
622,076 | <p>Continuity $\Rightarrow$ Intermediate Value Property. Why is the opposite not true? </p>
<p>It seems to me like they are equal definitions in a way. </p>
<p>Can you give me a counter-example? </p>
<p>Thanks</p>
| user44197 | 117,158 | <p>$$ f(x) = \sin(1/x), ~~ x \gt 0$$
and $$f(0) =0$$</p>
<p>This is <em>not</em> continuous at $x=0$ but clearly satisfies the intermediate value property.</p>
|
442,950 | <p>I would like to show <span class="math-container">$\lim\limits_{r\to\infty}\int_{0}^{\pi/2}e^{-r\sin \theta}\text d\theta=0$</span>.</p>
<p>Now, of course, the integrand does not converge uniformly to <span class="math-container">$0$</span> on <span class="math-container">$\theta\in [0, \pi/2]$</span>, since it has... | cool | 79,292 | <p>It's only enough to show that</p>
<p><span class="math-container">$$ \int\limits_{0}^{\pi/2}{e^{-r\sin\theta}\text d\theta}\le \int\limits_{0}^{\pi/2}{e^{-r\frac{2}{\pi}\theta}\text d\theta}=\frac{\pi}{2r}\left(1-e^{-r}\right) \to 0 \quad (r \to +\infty)$$</span></p>
|
442,950 | <p>I would like to show <span class="math-container">$\lim\limits_{r\to\infty}\int_{0}^{\pi/2}e^{-r\sin \theta}\text d\theta=0$</span>.</p>
<p>Now, of course, the integrand does not converge uniformly to <span class="math-container">$0$</span> on <span class="math-container">$\theta\in [0, \pi/2]$</span>, since it has... | Guy Fsone | 385,707 | <h2>Very simple trick:</h2>
<p>By studying the function $[0,\frac{\pi}{2}]\ni\theta \mapsto\frac{\sin\theta}{\theta}$ that </p>
<p>$$ \color{blue}{\sin\theta \geq \frac{2}{\pi}\theta ~~ \forall \theta\in [0,\frac{\pi}{2}] } $$
therefore we get that
$$\lim_{R\to\infty}\int_0^{\frac{\pi}{2}} e^{-R\sin\theta}d\theta\le... |
422,225 | <p>The proof uses this lemma which I understand: </p>
<p>$\mathbf {Lemma}$: Suppose $x$ and $y$ are positive real numbers such that $x>y$. If we decrease $x$ and increase $y$ by some positive quantity $E$ such that $x-E \ge y+E$, then $(x-E)(y+E) \gt xy$ . $\;$Hence, by subtracting $E$ from $x$ and adding it to $y$... | chizhek | 165,778 | <p>The proof that uses the Lemma is rather hard-going.
For my perspective on the proof see my answer to the
<a href="https://math.stackexchange.com/questions/919140">related question</a>;
it also answers your question, I think.</p>
<p>I gather that the AM-GM inequality you are talking about is
$\,(a_1\cdots a_n)^{1/n}... |
1,930,743 | <p>We have a map <span class="math-container">$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$</span>"). So <span class="math-container">$\... | Aloizio Macedo | 59,234 | <p>I'll generalize the nice answer by @Brian and give a curiosity (that you don't need to show that there exists at least one $A \subset X$ such that $A \subset f(A)$!).</p>
<p><em>Definition:</em> Given a partially ordered set $X$ and a subset $A$, $\sup(A)$ is defined as (if it exists) the element $s$ such that:</p>... |
3,785,982 | <p>Given the following ODE,</p>
<p><span class="math-container">$$\frac{{dy}}{{dx}}=\cos ({x})-\sin ({y})+{x}^{2}; \quad {y}\left({x}_{0}=-1\right)=y_0=3$$</span></p>
<p>I have to use the Taylor Series Method to compute the value of <span class="math-container">$y(x)$</span> at <span class="math-container">$x=-0.8$</sp... | Bernard | 202,857 | <p><strong>Hint</strong>:</p>
<p>It is simpler here to use <em>Hadamard's formula</em>:
<span class="math-container">$$\frac 1R=\limsup |a_n|^{1/n}=\limsup\Bigl(\frac{n+2}n\Bigr)^{\!n}. $$</span></p>
|
4,112,308 | <p>I was just exploring a little bit on Desmos, and was trying to figure out something somewhat interesting. I'm familiar that this is an elliptic curve, but ALL I know about them is that they are of the form <span class="math-container">$y^2=x^3+ax+b$</span>. Nothing else, really....</p>
<p>So, here's what I'm thinkin... | Math Lover | 801,574 | <p>For the curve to be self intersecting, we take the form</p>
<p><span class="math-container">$y^2 = (x-p)^2 (x-q) = x^3 - (2p+q)x^2 + (2pq+p^2)x-p^2q \ $</span>. Please note that the curve forms only for <span class="math-container">$x \geq q$</span></p>
<p>As <span class="math-container">$x^2$</span> term is zero, <... |
2,385,369 | <p>Explain why $K_{2,3}$ cannot have a Hamilton cycle.</p>
<p>I can visibly see and show why this is the case, but is there a mathematical proof or specific way of explaining how this Hamilton cycle cannot exist? Thanks a ton for all the help!</p>
| Bob Krueger | 228,620 | <p>A more general solution, and possibly the reason why you can visibly see that there is no Hamiltonian cycle, is as follows:</p>
<p>Let $S$ be the larger part of this bipartite graph. Then $|S| = 3$, $|N(S)| = 2$, and $S$ is an independent set. Whenever there is a set of vertices $S$ of a graph with $S$ an independe... |
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>
| Phil H | 554,494 | <p>I've heard this kind of question before. An anti-derivative will yield a definitive delta area under a graph of a function between any 2 limits. Seeing there is only one delta area, any different expressions defining it would essentially be the same. </p>
<p>The area of a right triangle $1/2xy$ or $1/2x^2 \tan \the... |
1,876,287 | <p><strong>Question:</strong></p>
<p>Let P be a point where the normal (in the point where the x-coordinate is h) to the curve</p>
<p>$$y = e^{2x} - 2x$$</p>
<p>cuts the y-axis. Determine the y-coordinates of P when h goes to 0.</p>
<p><strong>Attempted solution:</strong></p>
<p>I first decided to draw the followi... | Claude Leibovici | 82,404 | <p>In the same spirit as MathInferno's answer, not using L'Hôpital's rule, consider Taylor series around $h=0$ $$e^{2h}= 1+2 h+2 h^2+O\left(h^3\right)$$ So, $$e^{2h}-2h=1+2 h^2+O\left(h^3\right)$$ $$2e^{2h}-2=4 h+4 h^2+O\left(h^3\right)$$ then $$e^{2h}-2h+\frac h{2e^{2h}-2}=1+2 h^2+O\left(h^3\right)+\frac h{4 h+4 h^2+O... |
2,481,046 | <p>I have a question that asks to show that $S^2 = \{(x,y,z) \in \mathbb{R}^3|x^2+y^2+z^2=1\}$ is a differentiable manifold. My professor says that one way to do this is to define the following 6 parametrizations of the sphere, which cover the entire sphere.</p>
<p>$\vec{\phi_{i}}:V \to \mathbb{R}^3$ where $V = \{(u,v... | choco_addicted | 310,026 | <p>Since $\displaystyle \int_1^{\infty} \frac{1}{y^2}dy$ converges and $|\frac{1}{y^2}\sin(y+\frac{1}{y})| \le \frac{1}{y^2}$ for $y\in [1,\infty)$, $\displaystyle \int_1^\infty \frac{1}{y^2}\sin\left(y+\frac{1}{y}\right)dy$ converges absolutely by the comparison test. According to <a href="http://www.wolframalpha.com/... |
2,481,046 | <p>I have a question that asks to show that $S^2 = \{(x,y,z) \in \mathbb{R}^3|x^2+y^2+z^2=1\}$ is a differentiable manifold. My professor says that one way to do this is to define the following 6 parametrizations of the sphere, which cover the entire sphere.</p>
<p>$\vec{\phi_{i}}:V \to \mathbb{R}^3$ where $V = \{(u,v... | J.G. | 56,861 | <p>Or you could just use the fact that the integrand has modulus $\le 1$. The behaviour at one point, $x=0$, doesn't change the integral.</p>
|
4,276,974 | <p>I have to prove that sentence, but I'm not sure how to do that. Help!</p>
| Anonmath101 | 306,753 | <p><span class="math-container">$p+1 $</span> and <span class="math-container">$p-1$</span> are both even and one of <span class="math-container">$p-1, p, p+1$</span> is a multiple of <span class="math-container">$3$</span> but of course it cannot be <span class="math-container">$p$</span> itself. So <span class="math-... |
76,853 | <p>I have a list of stock symbols and related information containing some entries <code>Missing["NotAvailable"]</code>. I would like to delete all nested lists which contain a NotAvaiable entry, as <em>Mathematica</em> obviously does not support these instruments anymore (see also <a href="http://reference.wolfram.com/... | Karsten 7. | 18,476 | <p><code>Missing["NotAvailable"]</code> is not a string. Its <code>Head</code> is <code>Missing</code>, therefore you can use</p>
<pre><code>instruments = DeleteCases[indexMaster, _Missing]
</code></pre>
<blockquote>
<pre><code>{"^RDM-SO", "AMEX"}
</code></pre>
</blockquote>
|
76,853 | <p>I have a list of stock symbols and related information containing some entries <code>Missing["NotAvailable"]</code>. I would like to delete all nested lists which contain a NotAvaiable entry, as <em>Mathematica</em> obviously does not support these instruments anymore (see also <a href="http://reference.wolfram.com/... | Jinxed | 24,763 | <p>Just use the same pattern as what you don't want to see:</p>
<pre><code>DeleteCases[indexMaster,Missing["NotAvailable"]]
(* {"^RDM-SO", "AMEX"} *)
</code></pre>
|
891,575 | <p>The circumference of a circle has length 90 centimeters, Three points on the circle divide the circle into three equal lengths. Three ants A, B, and C start to crawl clockwise on the circle, with starting from one of the three points. Initially A is ahead of B and B is ahead of C. Ant A crawls 3 centimeters per seco... | Haukur Þorgeirsson | 167,969 | <p>At the start, C is at location 0, B is at location 30 and A is at location 60. It is easy to see that C will catch up to B in 6 seconds at location 60. And then they will meet again at location 60 every 18 seconds after that. When will A be at location 60? Well, he starts out there and he gets back there every 30 se... |
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>
| user326210 | 326,210 | <p>Well, if </p>
<p>$$x^n > x!$$</p>
<p>then we can take the logarithm of both sides without affecting the order:</p>
<p>$$\log(x^n) > \log(x!)$$</p>
<p>We get:</p>
<p>$$n \log(x) > \log(1) + \log(2) + \log(3) + \ldots + \log(x)$$</p>
<p>Let's divide by $\log(x)$ on both sides (if $x=1$ then $\log(x) = 0... |
1,656,145 | <p>Let the real function of two real variables$$u(x,y) =
\begin{cases}
x, & \quad \text{if } |y|>|x| \\
-x, & \quad \text{if } otherwise
\\ \end{cases} $$</p>
<p>Is there a sequence $\{(x_n,y_n)\}_{n \geq 0}$ which converge to $(0,0)$ such that $\lim_{n \to \infty} u(x_n,y_n) \not= u(0,0)$?</p... | Robert Israel | 8,508 | <p>Hint: $|u(x,y) - u(0,0)| = |x|$. </p>
|
3,128,352 | <p>I want to prove that when <span class="math-container">$F:K\rightarrow K[X]/\langle f\rangle $</span> is a map such that <span class="math-container">$F(a)=a+\langle f \rangle$</span>, then <span class="math-container">$F$</span> is an embedding from <span class="math-container">$K$</span> to <span class="math-cont... | Claude Leibovici | 82,404 | <p>For simplicity, I let <span class="math-container">$x=y+5$</span> to make
<span class="math-container">$$\frac{x^3 -2x+1}{x+7}=\frac{y^3+15 y^2+73 y+116}{y+12}$$</span> and using the long division
<span class="math-container">$$\frac{y^3+15 y^2+73 y+116}{y+12}=\frac{29}{3}+\frac{95 y}{18}+\frac{175 y^2}{216}+\frac{4... |
1,521,779 | <p>I have a homework question that I want to make sure I'm getting it right.</p>
<p>This is a joint probability table for the proportions of survey respondents who smoke and who have had heart attacks.</p>
<p><kbd> &n... | poetasis | 546,655 | <p>It would be easy if <span class="math-container">$\space 2017\space $</span> were a perfect square but there are no better approaches that i know of except for limiting the search.
<span class="math-container">$$x^2+y^2=2017\implies y=\sqrt{2017-x^2}\\
\implies\bigg\lceil\sqrt{2017-\big(\big\lfloor\sqrt{2017}\;\big\... |
441,792 | <p>There are many objects in mathematics that have the term "chiral" in their name, for instance, chiral algebra by Beilinson and Drinfeld, chiral de Rham complex, chiral Koszul duality etc. Some people told me that chiral algebras are <span class="math-container">$2$</span>-dimensional analogue of associativ... | AXidenT | 88,421 | <p>A vertex operator algebra describes the algebra of local operators in the chiral part of a 2d CFT. Typically one sees a VOA described depending on a complex coordinate <span class="math-container">$z$</span>. To describe a full 2d CFT, you would typically need to also include an "anti-chiral" VOA depending... |
3,151,452 | <p>The context of the question is that a bakery bakes cakes and the mass of cake is demoted by <span class="math-container">$X$</span> such that <span class="math-container">$X \sim N(300, 40^2)$</span>. A sample of 12 cakes is taken and the mean of the sample is 292g. The question wants me to find the <span class="mat... | farruhota | 425,072 | <p>The hypothesis testing:
<span class="math-container">$$H_0: \mu =300\\
H_1:\mu \ne 300 \\
z=\frac{\bar{x}-300}{40/\sqrt{12}}=-0.6928\\
p\text{-value}=P(z<-0.6928)=0.244 \ \\
\text{Reject $H_0$ if $p<\frac{\alpha}{2}$}: \ 0.244\not < 0.05 \Rightarrow \text{Fail to Reject} \ H_0.$$</span>
Note: <span class="m... |
1,222,909 | <p>I was thinking to convert to cartesian coordinates and then find when the slope of the tangent line is $1$, but I get a messy equation $2\cos^2\theta -2\sin^2\theta=4\sin^2\theta\cos\theta$
I was wondering if there was an easy way as it is hard to get values from this.</p>
<p>Edit: The equation ends up simplifying ... | goldenratio | 226,628 | <p>This is the equation of the circle of center $(0,1)$ and radius $1$. </p>
<p>$$r = 2\sin(\theta) \iff r = 2(\frac{y}{r}) \iff r^2 = 2y \iff x^2 + y^2 = 2y \iff x^2 + (y - 1)^2 = 1$$</p>
<p>Differentiate $x^2 + y^2 = 2y$ with respect to $x$..</p>
|
272,057 | <p>Let $\mu$ and $\nu$ be two probability measures on $\mathbb R^n$ with finite first moment. Denote by $d:=W_1(\mu,\nu)$, where $W_1(\cdot,\cdot)$ stands for the Wasserstein distance of order $1$. </p>
<p>My question is the following: Let $X$ be a random variable defined on some probability space (rich enough) with l... | Iosif Pinelis | 36,721 | <p>$\newcommand{\R}{\mathbb R}
\newcommand{\B}{\mathcal B}
\newcommand{\la}{\lambda}
\newcommand{\Si}{\Sigma}
\renewcommand{\c}{\circ}
\newcommand{\tr}{\operatorname{tr}}$</p>
<p>The desired function $f$ and random variable (r.v.) $G$ can be built recursively, by induction, using the increasing rearrangement/<a href="... |
3,518,221 | <p>So I had this complex integral </p>
<blockquote>
<p>If <span class="math-container">$0 \leq y \leq 1$</span>, find the maximum value of the integral
<span class="math-container">$$
\int_0^y \left(x^4 + (y-y^2) \right)^{1/2}\, dx
$$</span></p>
</blockquote>
<p>I differentiated the integral using the leibniz rul... | Claude Leibovici | 82,404 | <p>If there is no typo, I have the feeling that you face a difficult problem.</p>
<p>If
<span class="math-container">$$f(y)=\int_0^y g(x,y)\,dx$$</span> then
<span class="math-container">$$f'(y)=g(y,y)+\int_0^y \frac{\partial g(x,y)}{\partial y}\,dx$$</span> So, for your case
<span class="math-container">$$f'(y)=\sqrt... |
736,684 | <p>I'm trying to figure the probability that <span class="math-container">$X < Y$</span> with:</p>
<p><span class="math-container">$$X, Y \in \mathbb R^+;\ X\in [0,5] ; \ Y \in [0,2]$$</span>
What is the law to use?</p>
| Did | 6,179 | <p><strong>IF</strong> the random variables are independent and <strong>IF</strong> they are uniformly distributed on the range you indicate then $E(Y)=1$ and $P(X\lt y)=\frac15y$ for every $y$ in $(0,2)$ hence $P(X\lt Y)=\frac15E(Y)=\frac15$.</p>
<p>More generally, if $X$ and $Y$ are independent and uniform on the in... |
3,371,964 | <p>Let be <span class="math-container">$O_{2}$</span> the orthogonal group, that is, the group of reflections and rotations of <span class="math-container">$\mathbb{R}^{2}$</span>. His center is <span class="math-container">$\{ \pm I\} \simeq \mathbb{Z}_{2}$</span>. I'm having problems to study the center of the quotie... | erFuricksen | 479,710 | <p><span class="math-container">$O_2$</span> is generated by rotations and symmetries, which means that all the elements of <span class="math-container">$O_2$</span> can be written as <span class="math-container">$R_\theta S^\epsilon$</span>, where <span class="math-container">$R_\theta $</span> is a rotation of an ang... |
1,263,865 | <p>So I have that $700=7\cdot2^2\cdot5^2$ and I got that $3^2\equiv1\pmod2$ so then $3^{1442}\equiv1\pmod2$ also $3^2\equiv1\pmod{2^2}$ so $3^{1442}\equiv1\pmod{2^2}$ which covers one of the divisors of $700$. Im not sure if I'm supposed to use $2$ or $2^2$ and I was able to find that $3^2\equiv-1\pmod5$ so $3^{1442}\e... | Anurag A | 68,092 | <p>Since $\phi(700)=240$, therefore from Euler's theorem
$$3^{240} \equiv 1 \pmod{700}$$
Now
$$1442 =240(6)+2$$
Therefore
$$3^{1442} \equiv 3^{240(6)} \cdot 3^{2} \equiv 9 \pmod{700}$$</p>
|
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-... | Andrew D. Hwang | 86,418 | <p>In a simple mathematical model, the surface of the pond is a plane <span class="math-container">$P$</span>; the viewer's eye is a point <span class="math-container">$E$</span> "above" <span class="math-container">$P$</span>. A point <span class="math-container">$S$</span> in the scene is visible to <span c... |
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 <... | Lorago | 883,088 | <p>The key thing there is that you <strong>don't</strong> have <span class="math-container">$f:\mathbb{R}\to\mathbb{R}$</span> for <span class="math-container">$f(x)=\sqrt{x}$</span>. Instead we usually either define it as a function <span class="math-container">$f:\mathbb{R}^+_0\to\mathbb{R}$</span>, or a function <sp... |
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... | Jaideep Khare | 421,580 | <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>Now multiply both sides by $x^2(x-1)^2$ :</p>
<p>$$(x^2-x+1)^3=x^2(x-1)^2\frac{(a^2-a+1)^3}{a^2(a-1)^2}
\\ \implies (x^2-x+1)^3-x^2(x-1)^2\frac{(a^2-a+1)^3}{a^2(a-1)^2}=0$$</p>
<p>Without expanding, it can be written as :</p>
<p>$$x^6+a_1x^... |
2,677,823 | <p>How can I precisely prove the existence of a continuous function $\rho(x)$ such that $0 \leq \rho(x) \leq 1 \forall x \in R^d $ such that $g(x) \rho(x)$ is bounded and continuous for $g(x)$ continuous?Both $g(x)$ and $\rho(x)$ are defined on $R^d$.</p>
<p>My idea was that we can choose $\rho(x)$ such that $\rho(x)g... | user284331 | 284,331 | <p>If $\displaystyle\int f$ exists, then by writing $f=u+iv$ for real $u,v$, then $\displaystyle\int u$ and $\displaystyle\int v$ exist and $\displaystyle\int f=\int u+i\int v$. Note that both $\displaystyle\int u,\int v$ are real, so $\overline{\displaystyle\int f}=\displaystyle\int u-i\int v=\int(u-iv)=\int\overline{... |
2,947,953 | <p>Given the two functions <span class="math-container">$$f(x) = \ln\left(\frac{x+1}{x-1}\right)$$</span> and <span class="math-container">$$g(x) = \ln(x+1)-\ln(x-1)$$</span>
I can justify independently why <span class="math-container">$\text{dom}(f) = (-\infty, -1) \cup (1,\infty)$</span>, and <span class="math-contai... | Mohammad Riazi-Kermani | 514,496 | <p>For this function <span class="math-container">$$f(x) = \ln\left(\frac{x+1}{x-1}\right)$$</span></p>
<p>you want the fraction to be positive so both top and bottom could be positive or both could be negative.</p>
<p>On the other hand for <span class="math-container">$$g(x) = \ln(x+1)-\ln(x-1)$$</span></p>
<p>you ... |
2,947,953 | <p>Given the two functions <span class="math-container">$$f(x) = \ln\left(\frac{x+1}{x-1}\right)$$</span> and <span class="math-container">$$g(x) = \ln(x+1)-\ln(x-1)$$</span>
I can justify independently why <span class="math-container">$\text{dom}(f) = (-\infty, -1) \cup (1,\infty)$</span>, and <span class="math-contai... | fleablood | 280,126 | <p>Here's a thought experiment:</p>
<p>Let <span class="math-container">$f: \mathbb N \to \mathbb N$</span> via <span class="math-container">$f(n) = n$</span>.</p>
<p>Let <span class="math-container">$g: \mathbb R \to \mathbb R$</span> via <span class="math-container">$g(x) = |x|$</span>.</p>
<p>Let <span class="mat... |
2,129,764 | <p>Hey guys I have a problem that I'm having trouble solving. Here is the question:</p>
<p><strong>Consider events $A, B, C$ such that $P(A\mid B) > P(A)$ and $P(B\mid C) > P(B)$. Does it follow that $P(A\mid C) > P(A)$? Either prove it to be so or provide a counterexample.</strong></p>
<p>And here is what I... | user404961 | 404,961 | <p>Your partial solution must work for an appropriate choice of $p_i$'s. However, it's probably best to think about the problem this way. </p>
<p>The problem asks you to assume that knowing $C$ makes $B$ more likely, and knowing $B$ makes $A$ more likely. Then it asks if it follows that $C$ makes $A$ more likely. </p>... |
4,080,385 | <p>Would you please compute the behavior of the following composed generalized function?</p>
<p><span class="math-container">$g(t) = $</span> <span class="math-container">$\delta(e^t)$</span></p>
<p><strong>Is it even a valid generalized function?</strong></p>
<p>Thank you very much for your time.</p>
| Calvin Khor | 80,734 | <p>If you mean the action of <span class="math-container">$\delta$</span> on <span class="math-container">$\exp$</span>, this is of course <span class="math-container">$1$</span>. If you mean something like the generalisation of change of variables <span class="math-container">$$\int\delta(e^t)f(t)dt := \int\delta(r) f... |
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... | Nasser | 70 | <p>There is probably a build in way. Too many commands and too little time :)</p>
<p>But you could always code one yourself.</p>
<pre><code>ClearAll[replaceOnce]
replaceOnce[lis_List, rules_List] := Module[{lisin = lis, n, pos},
Do[
pos = FirstPosition[lis, rules[[n, 1]]] ;
If[Not[Head[pos] === Missing],
li... |
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>
| Santi Spadaro | 11,647 | <p>If you're interested in the relationship between the ccc and separability, you should read Stevo Todorcevic's survey "Chain condition methods in topology".</p>
<p><a href="http://www.sciencedirect.com/science/article/pii/S0166864198001126" rel="nofollow">http://www.sciencedirect.com/science/article/pii/S01668641980... |
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 ... | lab bhattacharjee | 33,337 | <p><span class="math-container">$$F=\dfrac{\sec x\tan x}{(\tan^2x)^{3/2}}=\dfrac{\sec x\tan x}{|\tan^3x|}$$</span></p>
<p>For <span class="math-container">$\tan x>0,$</span></p>
<p><span class="math-container">$$F=\dfrac{\cos x}{\sin^2x}=\csc x\cot x=-\dfrac{d(\csc x)}{dx}$$</span></p>
<p>What if <span class="mat... |
781,776 | <blockquote>
<p>A red die, a blue die, and a yellow die (all six sided) are rolled. Given that no two of the dice land on the same number, what is the conditional probability that blue is less than yellow which is less than red?</p>
</blockquote>
<p>The Answer is a sixth. I have absolutely no idea how to do this tho... | nature1729 | 29,257 | <p>Number of ways will be coefficient of $x^{15}$ in </p>
<p>$$f(x)=(1+x+x^2+x^3+x^4)(1+x+x^2+\cdots+x^{15})^2=(1-x^5)(1-x^{16})^2(1-x)^{-3}=(1-x^5-2x^{16}+2x^{21}
+x^{32}-x^{37})(1+\binom{3}{1}x+\binom{4}{2}x^2+\cdots)$$</p>
<p>Thus number of ways is $\binom{17}{2}-\binom{12}{2}=70$</p>
|
235,945 | <p>Hello please help me with these trig identities and double angles as I am not sure where I am going wrong but I keep getting the wrong answer </p>
<p>This is the problem
$$
\sin(\theta+30) = 2\cos(\theta)
$$
This is my one of my incorrect solutions</p>
<p>$$\sin(\theta +30) = 2\cos(\theta)$$
$$\sin(\theta)\cos(30)... | Fly by Night | 38,495 | <p>What you have written cannot be an identity. If it were then $\sin(\theta + 30)$ must equal $2\cos\theta$ for all values of $\theta$. However, while $\sin(\theta+30)$ oscillates between $-1$ and $1$, we see that $2\cos\theta$ oscillates between $-2$ and $2$. They have different ranges and so cannot possibly be ident... |
2,677,584 | <p>I have the following question:</p>
<blockquote>
<p>Find the real values of $a$ for which the equation
$$(1+\tan^2\theta)^2 + 4a\tan\theta(\tan^2\theta + 1) + 16\tan^2\theta = 0$$
has four distinct real roots in $\left(0, \dfrac{\pi}{2}\right)$.</p>
</blockquote>
<p>I tried to solve the above equation by div... | lab bhattacharjee | 33,337 | <p>$$\sec^4t+4a\frac{\sin t}{\cos^3t}+\dfrac{16\sin^2t}{\cos^2t}=0$$</p>
<p>$$0=1+4a\sin t\cos t+16(\sin^2t\cos^2t)=1+2a\sin2t+4\sin^22t$$</p>
<p>$$-a=\dfrac{1+4\sin^22t}{2\sin2t}$$</p>
<p>As $0<2t<\pi,\sin2t>0$ $$\dfrac{1+4\sin^22t}{2\sin2t}=\dfrac1{2\sin2t}+2\sin2t\ge2\sqrt{\dfrac1{2\sin2t}\cdot2\sin2t}=... |
3,005,329 | <blockquote>
<p>Suppose that <span class="math-container">$f:[0,1]\to\mathbb{R}$</span> is continuous on <span class="math-container">$[0,1]$</span>. Show that <span class="math-container">$\{\int_0^1f(x^n)dx\}_{n=1}^\infty$</span> converges to <span class="math-container">$f(0)$</span>.</p>
</blockquote>
<p>I'm not... | hamam_Abdallah | 369,188 | <p><strong>Hint</strong></p>
<p>Let <span class="math-container">$\epsilon>0$</span> small enough.</p>
<p><span class="math-container">$$\int_0^{1-\epsilon}(f(x^n)-f(0))dx=(1-\epsilon)(f(c^n)-f(0))$$</span>
with <span class="math-container">$0\le c\le 1-\epsilon<1$</span>.
now use sequential charactersation of ... |
3,005,329 | <blockquote>
<p>Suppose that <span class="math-container">$f:[0,1]\to\mathbb{R}$</span> is continuous on <span class="math-container">$[0,1]$</span>. Show that <span class="math-container">$\{\int_0^1f(x^n)dx\}_{n=1}^\infty$</span> converges to <span class="math-container">$f(0)$</span>.</p>
</blockquote>
<p>I'm not... | Jack D'Aurizio | 44,121 | <p>Since <span class="math-container">$f(x)$</span> is continuous, for any <span class="math-container">$x\in(0,1)$</span> the sequence <span class="math-container">$f(x^n)$</span> is convergent to <span class="math-container">$f(0)$</span> as <span class="math-container">$n\to +\infty$</span>.
On the other hand the c... |
356,306 | <p>If $f:X_1 \rightarrow X_2$ and $g:X_2 \rightarrow X_3$ are homomorphisms.
If $g \circ f =0$ does it imply that $Im f \subseteq ker g$? and how to show that? do you have an example?
thanks :)</p>
| rschwieb | 29,335 | <p>Things in $Im(f)$ look like $f(x)$, any $y$ such that $g(y)=0$ is in the kernel of $g$, and $g(f(x))=0$. You have everything you need.</p>
|
89,621 | <p>All geometry in computer graphics are transformed by position * transform matrix; The issue is the fact that position is a vector with 3 components (x,y,z); And transform matrix is a 4 by 4 with one column that can be dumped(at least in my case). So my transform matrix is now a 3 by 4 matrix:<br>
axis x { x... | hardmath | 3,111 | <p>Here's the "math" way of looking at this. In three dimensions there are various "isometries", mappings that preserve distances between points. Some of these are linear transformations, and these can be represented in the usual way as multiplication by <a href="http://en.wikipedia.org/wiki/Orthogonal_matrix" rel="n... |
1,821,927 | <p>Let $V = \big\{z: |z|<5,\text{Im}(z)>0 \big\}$. Let $f$ analytic in $V$, continuous in $\overline{V}$ and suppose $$\forall x \in \left[ -5,5\right]:\ f\left( x\right) \in \mathbb{R}$$
Show that $$\limsup_{n \rightarrow \infty} \root{n}\of{\frac{f^{(n)}(1)}{n!}} \le \frac{1}{4}$$
<br><br><br>
I tried expanding... | Uria Mor | 180,241 | <p>This is a slightly different approach to solve it using Morera's Theorem. </p>
<p>Of course the point in this exercise was to find an analytic continuation of $f$ to the disk, and once that is done, one can simply expand $f$ around $z=1$, to a power series $\sum \frac{f^{(n)}(1)}{n!}(z-1)^n$ and since $f\in Hol(D(... |
99,961 | <p>I have a list with elements</p>
<pre><code>{a -> -1, b -> -2, c -> -3}
</code></pre>
<p>If I now wanted to apply a tranformation to <code>b</code> and <code>c</code> so that they would give the tranformation <code>b -> 1-10^val</code> and <code>c -> 1-10^val</code>, yielding</p>
<pre><code>{a ->... | halirutan | 187 | <p>I'm not sure whether you mean something different by <code>1-b^-2</code> or you just miscalculated, because your result is not the correct result. In general, you can transform transformation-rules like this:</p>
<pre><code>{a -> -1, b -> -2, c -> -3} /.
{
(b -> val_) :> (b -> 1 - val^-2),
... |
99,961 | <p>I have a list with elements</p>
<pre><code>{a -> -1, b -> -2, c -> -3}
</code></pre>
<p>If I now wanted to apply a tranformation to <code>b</code> and <code>c</code> so that they would give the tranformation <code>b -> 1-10^val</code> and <code>c -> 1-10^val</code>, yielding</p>
<pre><code>{a ->... | Kuba | 5,478 | <pre><code>list = {a -> -1, b -> -2, c -> -3}
</code></pre>
<p>The quesiton is unclear but let's say I know what you want :)</p>
<pre><code>MapAt[
1. - 10^# &,
Association[list],
List@*Key /@ {b, c}] // Normal
</code></pre>
<blockquote>
<p>{a -> -1, b -> 0.99, c -> 0.999}</p>
</blockquote>
<p>Y... |
3,688,680 | <p>I know cantor set and rational numbers in <span class="math-container">$\mathbb{R}$</span> are meagre. But they are all zero measure.</p>
<p>So is there any meagre set that is non-zero measure?</p>
| Integrand | 207,050 | <ol>
<li>Comparison</li>
</ol>
<p><span class="math-container">$$\underbrace{\int_{0}^{\infty} \sin^2(x) x^{-5/2}\,dx}_{I} = \underbrace{\int_{0}^{1} \sin^2(x) x^{-5/2}\,dx}_{I_1} +\underbrace{\int_{1}^{\infty} \sin^2(x) x^{-5/2}\,dx}_{I_2} $$</span>
<span class="math-container">$I_2$</span> converges by directly comp... |
632,891 | <p>I'm trying to solve this limit, for which I already know the solution thanks to Wolfram|Alpha to be $\sqrt[3]{abc}$:</p>
<p>$$\lim_{n\rightarrow\infty}\left(\frac{a^\frac{1}{n}+b^\frac{1}{n}+c^\frac{1}{n}}{3}\right)^n:\forall a,b,c\in\mathbb{R}^+$$</p>
<p>As this limit is an indeterminate form of the type $1^\inft... | Community | -1 | <p>By Taylor series we have:
$$\frac{a^\frac{1}{n}+b^\frac{1}{n}+c^\frac{1}{n}}{3}=\frac 1 3\left(3+\frac1 n(\log a +\log b+\log c)++o\left(\frac 1 n\right)\right)=1+\frac 1 n \log\sqrt[3]{abc}+o\left(\frac 1 n\right)$$
so
$$\left(\frac{a^\frac{1}{n}+b^\frac{1}{n}+c^\frac{1}{n}}{3}\right)^n=\exp\left(n\log\left(1+\fr... |
63,723 | <p>How to find $f'(a)$ where $f(x) = \sqrt{1-2x}$ ?</p>
<p>I am not too sure what to do, no matter what I do I can't get the correct answer. I know it is simple algebra but I can't figure it out.</p>
| Arturo Magidin | 742 | <p>As you surmise, you need to multiply by the conjugate; the problem is that you forgot to distribute the negative sign correctly, and you forgot to <em>divide</em> by the conjugate as well as multiply by it.
$$\begin{align*}
\lim_{h\to 0}\frac{f(a+h)- f(a)}{h} &= \lim_{h\to 0}\frac{\sqrt{1-2(a+h)}-\sqrt{1-2a}}{h... |
3,069,244 | <p>Consider the following functional <span class="math-container">$\Phi:\mathbb R^n\to\mathbb R $</span>:
<span class="math-container">$$
\Phi(x)=\sum_{i=1}^{n-1}(1+x_i)(x_i-x_n)^2(2(1+x_i+x_n)+x_i x_n-x_1).
$$</span>
The computer experiments show that it is non-negative for all <span class="math-container">$x_i\geq 0$... | symchdmath | 626,816 | <p>This integral is more complicated than it looks and only requires comfort with hyperbolic trigonometric definitions. My initial instinct is to look at the expression inside the <span class="math-container">$\cosh$</span> to see if I can simplify it. In fact we have by the definition of the hyperbolic trigonometric f... |
153,409 | <p>Would you please tell me whether there is any wrong on this problem? given that $g$ is continuous on $[0,\infty)\rightarrow \mathbb{R}$ satisfying $\int_{0}^{x^2(1+x)}g(t)dt=x \forall x\in [0,\infty)$ then I need to find what is $g(2)$?</p>
| A.S | 24,829 | <p>Let $G(x) = \int _{0} ^{x} g(t) dt$. Since $g(t)$ is continuous, we can deduce that $G(x)$ exists and will be differentiable for $x \ge 0$.</p>
<p>Then, by your condition, $G(x^2(1+x))=x$.</p>
<p>Differentiating both sides with respect to $x$, we get $(2x(1+x)+x^2) \times g(x^2(1+x))=1$.</p>
<p>Simplifying, we g... |
1,087,874 | <p>I want to understand how I can count the terms of the expression $x^{m-1} + x^{m-2} +\ldots+ x^0$ when $x=1$.</p>
<p>The result is $m$, I dont know how to count them formally, any advice would be helpful. I'm desperated, not because it is required to do the above, but how can be done, I need to understand the subje... | GFauxPas | 173,170 | <p>There isn't a need to calculate anything or to use limits of any sort. As soon as you write "$x^{m-1} + \cdots + x^0$", you have <em>defined</em> the expression to have $m$ terms. There's nothing to prove.</p>
<p>To evaluate your limit, I recommend a proof by induction on $m$.</p>
|
550,188 | <p>Okay so I have an equation in my book which is as follows..
$$
\frac {a}{s(s+a)}
$$
it says "using partial fractions this can be expanded to
$$
\frac {1}{s} + \frac {-1}{s+a}
$$</p>
<p>My usual method would be to cross multiply and do something like this
$$
\frac {a}{s(s+a)} = \frac {A(s+a)}{s(s+a)} + \frac {B(s)}... | Empy2 | 81,790 | <p>Let $s=0$, so $a=Aa$. Let $s=1$, so $a=A(a+1)+B$<br>
Solve for $A$ and $B$.</p>
|
2,352,721 | <h2>Question</h2>
<blockquote>
<p>Four fair six-sided dice are rolled. The probability that the sum of the results being <span class="math-container">$22$</span> is <span class="math-container">$$\frac{X}{1296}.$$</span> What is the value of <span class="math-container">$X$</span>?</p>
</blockquote>
<h2>My Approach</h2... | G Tony Jacobs | 92,129 | <p>There aren't too many to just count.</p>
<p>Permutations of $6+6+6+4$: $\binom41=4$</p>
<p>Permutations of $6+6+5+5$: $\binom42=6$</p>
<p>These are the only options, so your numerator must be $4+6=10$</p>
|
2,352,721 | <h2>Question</h2>
<blockquote>
<p>Four fair six-sided dice are rolled. The probability that the sum of the results being <span class="math-container">$22$</span> is <span class="math-container">$$\frac{X}{1296}.$$</span> What is the value of <span class="math-container">$X$</span>?</p>
</blockquote>
<h2>My Approach</h2... | jvdhooft | 437,988 | <p>In order for the sum to equal 22, either three dice equal $6$ and one equals $4$, or two dice equal $6$ and two dice equal $5$. The number of valid outcomes thus equals:</p>
<p>$${4 \choose 1} + {4 \choose 2} = 4 + 6 = 10$$</p>
<p>As such, the probability of the four dice having a sum of $22$ equals:</p>
<p>$$\fr... |
2,352,721 | <h2>Question</h2>
<blockquote>
<p>Four fair six-sided dice are rolled. The probability that the sum of the results being <span class="math-container">$22$</span> is <span class="math-container">$$\frac{X}{1296}.$$</span> What is the value of <span class="math-container">$X$</span>?</p>
</blockquote>
<h2>My Approach</h2... | richard1941 | 133,895 | <p>Divide the dice into two pairs. The way you can get 22 is by 10 and 12, 11 and 11, and 12 and 10. The ways are 3, 4, and 3, totaling 10. Or you have looked at the dice individually and listed the winning combinations (in lexicographic order to be sure you don't miss anything).</p>
|
4,041,842 | <p>I want to solve for <span class="math-container">$t \in \mathbb{R}, u'(t)=-u(t)\ln \lvert u(t) \rvert$</span>.</p>
<p>I defined two cases: <span class="math-container">$\mathbb{R^*_+}$</span> and <span class="math-container">$\mathbb{R^*_-}$</span>.</p>
<p>For <span class="math-container">$\mathbb{R^*_+}$</span>:</p... | Kieran Mullen | 510,314 | <p>The above solution is absolutely correct. A slightly quicker way is to substitute
<span class="math-container">$$
u(t) = e^{f(t)}
$$</span>
producing the differential equation:
<span class="math-container">$$
f'=-f
$$</span>
eventually leading to the same answer.</p>
|
643,918 | <blockquote>
<p>Let $G$ be a group and $a, b \in G$. Show that $(a*b)' = a' * b'$ if and only if $a*b = b*a$.</p>
</blockquote>
<p>While this is simple to see by intuition, I am having a hard time expressing this formally. It seems as if I want to show that $(a*b)' = a' * b'$ strictly implies $a*b = b*a$, but I'm no... | Ulrik | 53,012 | <p>Hint: Remember that $(a*b)' = b' * a'$ is an identity that always holds.</p>
|
4,047,601 | <p>I did a question <span class="math-container">$\int_{0}^{1}\frac{1}{x^{\frac{1}{2}}}\,dx$</span>, and evaluating this is divergent integral yes? Then as a general form <span class="math-container">$\int_{0}^{1} \frac{1}{x^p}\,dx$</span>, <span class="math-container">$p \in \mathbb{R}$</span>, what values of <span cl... | Alan | 175,602 | <p>It's divergent for <span class="math-container">$p\leq 1$</span>. Otherwise by the fundamental theorem of calculus, you get <span class="math-container">$\frac {x^{-p+1}} {-p+1}$</span>. Plug in your 1 and 0 and then set equal to 4/3 should let you solve for p.</p>
|
418,647 | <p>Sorry if the question is dumb. I am trying to learn representation theory of finite groups from J.P.Serre's book by myself. In section 2.6 on canonical decomposition, he says that let V be a representation of a finite group G, $W_1,...,W_h$ be the distinct irreducible representations of G, and let V = $U_1 \oplus ..... | Douglas S. Stones | 139 | <p>The following graph is a simple 7-vertex graph with an isolated vertex. It contains every possible edge subject to the constraint that it has an isolated vertex.</p>
<p><img src="https://i.stack.imgur.com/V2tk0.png" alt="A 7 vertex graph with an isolated vertex"></p>
<p>Any other 7-vertex graph with an isolated v... |
3,005,100 | <p>Given the following formula
<span class="math-container">$$
\sum^n_{k=0}\frac{(-1)^k}{k+x}\binom{n}{k}\,.
$$</span>
How can I show that this is equal to
<span class="math-container">$$
\frac{n!}{x(x+1)\cdots(x+n)}\,?
$$</span></p>
| Batominovski | 72,152 | <p>Consider the (unique) polynomial <span class="math-container">$p(x)\in\mathbb{Q}[x]$</span> of degree at most <span class="math-container">$n$</span> such that <span class="math-container">$p(-k)=1$</span> for all <span class="math-container">$k=0,1,2,\ldots,n$</span>. Clearly, <span class="math-container">$p(x)$</... |
99,378 | <p>The following equation in $\mathbb{C}$:</p>
<p>$4z^2+8|z|^2-3=0$</p>
<p>is not algebraic and has 4 solutions : $\pm\frac{1}{2}$ and $\pm i\frac{\sqrt{3}}{2}$.
The Solve function in Mathematica only returns the 2 real values :</p>
<pre><code>Solve[4 z^2 + 8 Abs[z]^2 - 3 == 0, Complexes]
(* {{z -> -(1/2)}, {z -... | ubpdqn | 1,997 | <p>Specifying <code>Complexes</code> for <code>Solve</code>or <code>Reduce</code> suffices as does just doing it yourself (as alluded to by Daniel:Lichtblau):</p>
<pre><code>x + I y /.Solve[{4 (x^2 - y^2) + 8 (x^2 + y^2) - 3 == 0, 8 x y == 0}, {x, y}]
</code></pre>
<p>yield:</p>
<pre><code> {-((I Sqrt[3])/2), (I Sqr... |
99,378 | <p>The following equation in $\mathbb{C}$:</p>
<p>$4z^2+8|z|^2-3=0$</p>
<p>is not algebraic and has 4 solutions : $\pm\frac{1}{2}$ and $\pm i\frac{\sqrt{3}}{2}$.
The Solve function in Mathematica only returns the 2 real values :</p>
<pre><code>Solve[4 z^2 + 8 Abs[z]^2 - 3 == 0, Complexes]
(* {{z -> -(1/2)}, {z -... | murray | 148 | <pre><code> Solve[4 z^2 + 8 z Conjugate[z] - 3 == 0, z]
(* {z -> -1/2}, {z -> 1/2}, {z -> (-I/2)*Sqrt[3]}, {z -> (I/2)*Sqrt[3]}} *)
</code></pre>
|
2,498,359 | <p>This is a basic probability question. </p>
<p>Persons A and B decide to arrive and meet sometime between 7 and 8 pm. Whoever arrives first will wait for ten minutes for the other person. If the other person doesn't turn up inside ten minutes then the person waiting will leave. What is the probability that they will... | Bime | 424,324 | <p>I hope your are doing well, I tried a new method to resolve the meeting problem but I didn't find the same result.</p>
<p>We have to calculate <span class="math-container">$P(|X-Y| < 1/6)$</span> where X and Y are independents uniformly distributed between 0 and 1.</p>
<p>So : <span class="math-container">$P(|X... |
3,516,921 | <p>Let <span class="math-container">$f : [−1, 0] → \mathbb{R}, x → x − x^2, n ∈ \mathbb{N}$</span> and let <span class="math-container">$P_n : x_0, . . . , x_n$</span> be an equal partition of <span class="math-container">$[−1, 0]$</span>.</p>
<ul>
<li>Compute the Riemann sum <span class="math-container">$S_{P_n} (f, ... | Paramanand Singh | 72,031 | <p>You should understand the definition of a Riemann sum. It is based on notion of partition.</p>
<p>A <em>partition</em> of a closed interval <span class="math-container">$[a, b] $</span> is a finite set <span class="math-container">$P$</span> of points from the interval <span class="math-container">$[a, b] $</span> ... |
3,366,781 | <blockquote>
<p>Let <span class="math-container">$(S, +, \cdot, 0)$</span> and <span class="math-container">$(S', \oplus, \otimes, 0')$</span> be two semirings. Then <span class="math-container">$f: S\rightarrow S'$</span> is said to be a homomorphism if for all <span class="math-container">$a, b\in S,$</span> <span... | Arthur | 15,500 | <p>I don't know whether this addresses your concern in general, but here is what happens in this specific case.</p>
<p>In this argument we are not applying (the negation of) uniform continuity to a sequence. We're applying it to a whole lot of different numbers separately. These numbers just also happen to form a sequ... |
3,908,955 | <p>Is the given series convergent or divergent? Give a reason. Show details.</p>
<p><span class="math-container">$$\sum_{n=2}^{\infty} \frac{(-i)^n}{ln \ n}$$</span></p>
<p>So maybe I'll try using the ratio test?</p>
<p>So the series converges if <span class="math-container">$$\left| \frac{z_{n+1}}{z_n} \right| < 1$... | Derek Luna | 567,882 | <p>It really is as simple as you were making it, although there are mistakes and it is badly written. You correctly realized that it is important to bound one of <span class="math-container">$|x-2|, |x+3|$</span>, namely <span class="math-container">$|x-2|$</span>.</p>
<p>It is standard to start with <span class="math-... |
1,314,219 | <p>Is there any formula for finding the last digit of the factorials?
How to approach these type of questions?
Thanks in advance.</p>
| alkabary | 96,332 | <p>well as the comments stated, you should observe that for $n \geq 5$ we have $n!$ is a multiple of $5$. For example $$5! = 5 \times 4 \times 3 \times 2 \times 1 = 24 \times 5$$</p>
<p>and $$6 ! = 6 \times 5! = 6 \times 24 \times 5$$</p>
<p>and so on. Now you should know that any multiple of $5$ is either $0$ or $5$... |
947,770 | <p>Here I have a diophantine equation featuring a homogeneous polynomial:</p>
<p>$$x^2+5y^2+34z^2+2xy-10xz-22yz=0; x, y, z\in\mathbb{Z}$$</p>
<p>I have no idea how to approach this, I've tried various substitutions like $x=py, x=qz$ but then I get a non-homogeneous polynomial of 2 variables which is no better than th... | Sandeep Silwal | 138,892 | <p>Hint: Factor as $$(2y-3z)^2+(x+y-5z)^2 = 0$$</p>
|
3,679,386 | <p>defining matrix exponentiation for natural numbers by repeated multiplication and defining it for <span class="math-container">$\frac{1}{n}$</span> by: <span class="math-container">$A^{\frac{1}{n}}$</span> is the matrix s.t. <span class="math-container">$(A^{\frac{1}{n}})^n=A$</span>.
for a rational number <span cla... | fleablood | 280,126 | <p>Suppose <span class="math-container">$\sum\limits_{k=1}^{\infty} z_k$</span> converges. That means <span class="math-container">$\{\sum\limits_{k=1}^{n} z_k\}$</span> is a cauchy sequence. So for any <span class="math-container">$\epsilon >0$</span> there is an <span class="math-container">$N$</span> so that if <... |
7,130 | <p>I'm looking for an explanation on how reducing the Hamiltonian cycle problem to the Hamiltonian path's one (to proof that also the latter is NP-complete). I couldn't find any on the web, can someone help me here? (linking a source is also good).</p>
<p>Thank you.</p>
| Aryabhata | 1,102 | <p>Note: The below is a Cook reduction and not a Karp reduction. The modern definitions of NP-Completeness use the Karp reduction.</p>
<p>For a reduction from Hamiltonian Cycle to Path.</p>
<p>Given a graph $G$ of which we need to find Hamiltonian Cycle, for a single edge $e = \{u,v\}$ add new vertices $u'$ and $v'$ ... |
258,332 | <blockquote>
<p>Prove that if $a^x=b^y=(ab)^{xy}$, then $x+y=1$.</p>
</blockquote>
<p>How do I use logarithms to approach this problem?</p>
| Hugo | 53,041 | <p>How about $x=y=0$ ? Am I missing something?</p>
|
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