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 |
|---|---|---|---|---|
4,403,081 | <p><span class="math-container">$$\int \dfrac{dx}{x\sqrt{x^4-1}}$$</span></p>
<p>I need to solve this integration.
I solved and got <span class="math-container">$\dfrac12\tan^{-1}(\sqrt{x^4-1}) + C$</span>, however the answer given in my textbook is <span class="math-container">$\dfrac12\sec^{-1}(x^2) + C$</span></p>
<... | Community | -1 | <p>I think your problem is already solved with B. Goddard's answer, Here's an easy way to solve the given integral.</p>
<hr />
<p>We have,
<span class="math-container">$$\int\dfrac{dx}{x\sqrt{x^4-1}}$$</span></p>
<p>Let <span class="math-container">$x^2 = \sec(\theta) \implies 2x \, dx = \sec(\theta) \tan(\theta) \, d\... |
815,432 | <p>I have sometimes seen notations like $a\equiv b\pmod c$. How do we define the notation? Have I understood correctly that $c$ must be an element of some ring or does the notation work in magmas in general?</p>
| Bill Dubuque | 242 | <p>In a commutative ring <span class="math-container">$\,R,\, $</span> <span class="math-container">$\, a\equiv b\pmod c\,$</span> means <span class="math-container">$\,c\mid a-b,\ $</span> i.e. <span class="math-container">$\,a-b = cr\,$</span> for some <span class="math-container">$\,r \in R,\,$</span> i.e. <span cla... |
4,474,806 | <p>I use the following method to calculate <span class="math-container">$b$</span>, which is <span class="math-container">$a$</span> <strong>increased</strong> by <span class="math-container">$x$</span> percent:</p>
<p><span class="math-container">$\begin{align}
a = 200
\end{align}$</span></p>
<p><span class="math-cont... | John Douma | 69,810 | <p>It is easier to translate these expressions from English to Math than to think in terms of multiplication and division. The latter way leads to memorizing an arbitrary rule which won't stay with you.</p>
<p>We are saying we want to increase <span class="math-container">$a$</span> by <span class="math-container">$x\%... |
4,474,806 | <p>I use the following method to calculate <span class="math-container">$b$</span>, which is <span class="math-container">$a$</span> <strong>increased</strong> by <span class="math-container">$x$</span> percent:</p>
<p><span class="math-container">$\begin{align}
a = 200
\end{align}$</span></p>
<p><span class="math-cont... | JonathanZ supports MonicaC | 275,313 | <p>I think there are two issues at play here.</p>
<p><strong>Issue #1</strong>: Increase vs. decrease. That determines whether you use <span class="math-container">$1 + \frac{p}{100}$</span> or <span class="math-container">$1-\frac{p}{100}$</span>, respectively.</p>
<p>Using <span class="math-container">$I$</span> for ... |
1,296,420 | <p>I was trying to find an example such that $G \cong G \times G$, but I am not getting anywhere. Obviously no finite group satisfies it. What is such group?</p>
| Mikko Korhonen | 17,384 | <p>Take $$G = H \times H \times H \times \cdots$$ for $H$ any nontrivial group.</p>
|
3,165,937 | <p>Are orthogonal groups are lie groups? I think parameter space points corresponds to elements with determinant -1 break analytic property of lie groups , what is the general condition to check a group is lie group or not ?</p>
| José Carlos Santos | 446,262 | <p>Yes, orthogonal groups are Lie groups. Since, <span class="math-container">$O(n,\mathbb R)$</span> consists of two copoes of <span class="math-container">$SO(n,\mathbb R)$</span>, if <span class="math-container">$SO(n,\mathbb R)$</span> is a Lie group, then so is <span class="math-container">$O(n,\mathbb R)$</span>.... |
1,011,718 | <p>$P1 : \sin(x/y)$ .</p>
<p>I tried using $y=mx$. $f$ becomes $\sin(1/m)$, so limit doesn't exist. But it is too easy. Am I right?</p>
<p>$P2 : F = x^2\log(x^2+y^2)$</p>
<p><img src="https://i.stack.imgur.com/pEaSs.jpg" alt="enter image description here"></p>
<p><img src="https://i.stack.imgur.com/CEI2K.jpg" alt=... | Przemysław Scherwentke | 72,361 | <p>P1 is easy indeed and you are correct.</p>
<p>In P2 the limit is 0, but your attempt shows only limits on lines $y=mx$. A correct version is similar to that from answers to your previous question:
$$
|x^2\log(x^2+y^2)|\leq|(x^2+y^2)\log(x^2+y^2)|.
$$
But you should know, that
$$
\lim_{x\to0^+}x\ln{x}=0,
$$
and thi... |
3,964,237 | <p>I am trying to understand the meaning of this exercise question in a logic textbook.</p>
<blockquote>
<p>For each of the following statement forms,find a statement form that is logically equivalent to its negation and in which negation signs apply only to statement letters.<br/>
i. <span class="math-container">$A \r... | Neptune | 825,557 | <p>It means you are only using <span class="math-container">$\neg$</span> in front of <span class="math-container">$A$</span> <span class="math-container">$B$</span> or <span class="math-container">$C$</span>. So instead of writing <span class="math-container">$\neg(A \lor B)$</span>, for example, you would write <span... |
2,658,195 | <p>I have the following problem with which I cannot solve. I have a very large population of birds e.g. 10 000. There are only 8 species of birds in this population. The size of each species is the same.</p>
<p>I would like to calculate how many birds I have to catch, to be sure in 80% that I caught one bird of each s... | SK19 | 509,159 | <p>If we assume that each die roll is done <strong>independent</strong> of the others and further assume that <strong>each side has the same probability</strong>, then this experiment is just the same as throwing a perfect coin until one party wins. There is no way to predict which of the eight predictions would be mor... |
546,572 | <p><img src="https://i.stack.imgur.com/aJ2t5.jpg" alt="enter image description here"></p>
<p>Could anyone tell me how to solve 9b and 10? I've been thinking for five hours, I really need help.</p>
| detnvvp | 85,818 | <p>Hint: take $A=\mathbb N$ and $B=\{n+\frac{1}{n}\left|\right.n\in\mathbb N,n\geq 2\}$.</p>
|
2,473,951 | <p>The definition I have of a convex function $f: \mathbb{R} \rightarrow \mathbb{R}$ is that for every $x, y \in \mathbb{R}$ and every $\lambda \in [0, 1]$,
$$
f(\lambda x + (1-\lambda )y) \leq \lambda f(x) + (1- \lambda )f(y).$$</p>
<p>By proving that slopes increase I mean that for $x \leq y \leq z$, we get $$\frac... | Community | -1 | <p><strong>Hint</strong>.</p>
<p>Note that if $x\leqslant y\leqslant z$, then we can write
$$
y=\lambda x+(1-\lambda)z
$$
for some $\lambda\in[0,1]$. <em>Explicitly</em>, you can find what is $\lambda$ in terms of $x,y$ and $z$. On the other hand, the definition of convexity tells you
$$
f(y)\leqslant \lambda f(x)+(1... |
2,399,842 | <p>I ran $\frac{d^n}{dx^n}[(x!)!]$ through <em><a href="https://www.wolframalpha.com/input/?i=nth%20derivative%20of%20(x!)!" rel="nofollow noreferrer">Wolfram|Alpha</a></em>, which returned</p>
<blockquote>
<p>$$\frac{\partial^n(x!)!}{\partial x^n} = \Gamma(1+x!)\,R(n,1+x!)$$
for</p>
<ul>
<li><p>$R(n,x)=\ps... | leonbloy | 312 | <p>The superscript notation means (I think) the derivative: $$R^{(0,1)}(−1+n,x)=\frac{\partial R(n,x)}{\partial x}\Bigr|_{(−1+n,x)}$$</p>
<p>$R(n,x)$ is so ugly, it seems, that it cannot be expressed in some closed form but only as a recursion in $n$:</p>
<p>$$R(n,x)=\frac{d \ln[\Gamma(x)]}{dx} \,R(-1+n,x)+\frac{\par... |
1,612,220 | <p>This is an exercise page 7 from Sutherland's book Introduction to Metric and Topological Spaces.</p>
<blockquote>
<p>Suppose that <span class="math-container">$V,X,Y$</span> are sets with <span class="math-container">$V\subseteq X\subseteq Y$</span> and suppose that <span class="math-container">$U$</span> is a subse... | DanielWainfleet | 254,665 | <p>Using $+$ to mean union, and multiplication to mean intersection,and $-$ for complement (That is $a-b=a\backslash b$) we have $$V=X-(X-V).$$ We also have $$X-V=X U \;\text { and } X=X Y.$$ Hence $$V=X-(X-V)=X-X U=( X Y) -( X Y)U=X(Y-Y U)=X(Y-U)=$$ $$=X\cap (Y\backslash U).$$ Note that the condition $U\subset Y$ wa... |
1,465,490 | <p>I have a set of target coordinates and a set of actually clicked coordinates which should be approximately the same, but not identical. The y coordinates are equal, however, the x-coordinates differ, such that negative coordinates are closer together than larger/positive coordinates.
e.g.:</p>
<pre><code>target_y ... | Hagen von Eitzen | 39,174 | <p>Make a sketch! In total you have two half circles and two line segments of length distance plus twice the radius.</p>
|
40,493 | <p>I am trying to use Mathematica to solve a relatively simple ODE involving parameter(s). I would like to use a set of conditions to solve for the particular solution of the ODE. I understand how to make Mathematica find values for the constants that arise during the process of solving the ODE, but what about solving ... | Nasser | 70 | <pre><code>sol = T[t] /. First@DSolve[{T'[t] == -k*(T[t] - Ta)}, T[t], t]
</code></pre>
<p><img src="https://i.stack.imgur.com/wVWI5.png" alt="Mathematica graphics"></p>
<pre><code>sol = sol /. C[1] -> c
</code></pre>
<p><img src="https://i.stack.imgur.com/mba0H.png" alt="Mathematica graphics"></p>
<pre><code>eq... |
1,224,202 | <p>Does the following equation makes any sense at all?</p>
<p>$$
\frac{1}{|X|\cdot|Y|}\sum\limits_{x \in X}\sum\limits_{y \in Y}\begin{cases}
1 & \mathrm{if~} x > y\\
0.5 & \mathrm{if~} x = y\\
0 & \mathrm{if~} x < y
\end{cases}
$$</p>
<p>For every comparison of $x$ a... | Community | -1 | <p>$$\frac{1}{|X|\cdot|Y|}\left(|\{(x,y):x \in X,y\in Y:x>y\}|+\frac12|\{(x,y):x \in X,y\in Y:x=y\}|\right).$$</p>
<hr>
<p>Alternatively,
$$\frac1{|X|\cdot|Y|}\left(\sum_{\substack{x\in X \\ y \in Y}}(x>y)+\frac12\sum_{\substack{x\in X \\ y \in Y}}(x=y)\right)$$</p>
|
11,353 | <p>Thinking about the counterintuitive <em>Monty Hall Problem</em> (stick or switch?),
revisited in <a href="https://matheducators.stackexchange.com/a/11346/511">this ME question</a>,
I thought I would issue a challenge:</p>
<blockquote>
<p>Give in one (perhaps long) sentence a convincing explanation of why <em>swit... | quid | 143 | <p>If you "stay" then you win when the prize is behind the <em>one</em> door your originally selected, yet when you "switch" you win when the prize is behind one of the <em>two</em> doors you originally did not select.</p>
|
231,773 | <p>Let $G=(V,E)$ be an undirected graph. We form a graph $H=(V',E')$ from $G$ such that </p>
<ul>
<li>$V' = V \cup \{ w_e \mid e \in E \}$, and </li>
<li>$E' = \{ aw_e, bw_e \mid ab = e \in E \} \cup \{ w_e w_f \mid e,f \text{ are adjacent edges in }G \}$. </li>
</ul>
<p>Informally, $H$ is built from $G$ by subdividi... | assaferan | 74,819 | <p>The answer is true, using the following construction.
Let <span class="math-container">$B$</span> be the quaternion algebra of discriminant <span class="math-container">$p$</span> and let <span class="math-container">$O$</span> be a maximal order with an element <span class="math-container">$x$</span> satisfying <sp... |
2,911,187 | <p>Lines (same angle space between) radiating outward from a point and intersecting a line:</p>
<p><a href="https://i.stack.imgur.com/52HY4.png" rel="noreferrer"><img src="https://i.stack.imgur.com/52HY4.png" alt="Intersection Point Density Distribution"></a></p>
<p>This is the density distribution of the points on t... | user | 505,767 | <p>It can be shown assuming that the quantity from the ray is proportional to the angular interval that the denisty distribuiton function is in the form</p>
<p>$$y=\frac{d}{H+x^2}$$</p>
<p>where $d$ is related to the intensity and $H$ is the distance of the source point from the line.</p>
<p>Here is the <a href="htt... |
3,764,030 | <p>If <span class="math-container">$|z| = \max \{|z-1|,|z+1|\}$</span>, then:</p>
<ol>
<li><span class="math-container">$\left| z + \overline{z} \right| =1/2$</span></li>
<li><span class="math-container">$z + \overline{z} =1$</span></li>
<li><span class="math-container">$\left| z + \overline{z} \right| =1$</span></li>
... | Kavi Rama Murthy | 142,385 | <p>There is no complex number <span class="math-container">$z$</span> such that <span class="math-container">$|z|$</span> is the maximum of <span class="math-container">$|z-1|$</span> and <span class="math-container">$|z+1|$</span>.</p>
<p>One can say that all four options are vacuously true!</p>
<p>Proof: write <span ... |
2,635,635 | <p>I have searched a lot and don't really understand the answers I've come across. So apologies in advance if I'm repeating a common question.</p>
<p>The problem is as follows: Distribute $69$ identical items across
$4$ groups where each groups needs to contain at least $5$ items.</p>
<p>The way I see the problem: $... | AleksandrH | 287,621 | <p>Precisely, yes. What you did is correct. If you're interested, the "rigorous" mathematical approach would be the following:</p>
<p>$x_1 + x_2 + x_3 + x_4 = 69$, where $x_i \geq 5$ for all $i \in [1,4]$</p>
<p>Let's take $x_1$ as an example:</p>
<p>$x_1 \geq 5$</p>
<p>By simple subtraction from both sides, we see... |
3,608,097 | <p>Three strings totaling a length <span class="math-container">$U= 3 a + 4b + 2 \pi r$</span> cut into three parts together enclose minimum total area</p>
<p><span class="math-container">$$ A= \frac{\sqrt3 a^2}{4} + b^2+\pi r^2,$$</span></p>
<p>when they are made into shapes of an equilateral triangle, square and ci... | José Carlos Santos | 446,262 | <p>If <span class="math-container">$\lvert x\rvert\leqslant\frac13$</span>, then<span class="math-container">$$\left\lvert\frac{3^n}{2n^2+5}x^n\right\rvert\leqslant\frac1{2n^2+5},$$</span>and the series converges (apply the comparison test to this series and to the series <span class="math-container">$\sum_{n=1}^\infty... |
3,488,226 | <p>We have <span class="math-container">$$\ln(v-1) - \ln(v+3) = \ln(x) + C$$</span></p>
<p>Multiplying through by e gives:</p>
<p><span class="math-container">$$(v-1)/(v+3) = x + e^C$$</span></p>
<p>But the answer in the textbook is:</p>
<p><span class="math-container">$$(v-1)/(v+3) = Dx$$</span></p>
<p>Where <spa... | Community | -1 | <p><span class="math-container">$ln(\frac{v-1}{v+3})$</span> <span class="math-container">$=$</span> <span class="math-container">$ln(x)+C$</span> <span class="math-container">$\implies$</span> <span class="math-container">$\frac{v-1}{v+3} = Dx$</span>.</p>
|
2,867,521 | <p>I am interested in calculating the following double summation:</p>
<p>$\sum_{n=2}^ \infty \sum_{k =0}^{n-2}\frac{1}{4}^k \frac{1}{2}^{n-k-2}$</p>
<p>I don't really know where to start, so I was hoping someone could point me to some resource where I could learn the terminology/methodology associated with solving su... | William Elliot | 426,203 | <p>Let the directed set be (0,1) and the net assign x to x in R.<br>
This net converges to 1 and its image along with 1 is (0,1]<br>
which is not compact.</p>
|
362,250 | <p>Let <span class="math-container">$\nu$</span> be a <em>finite</em> Borel measure on <span class="math-container">$\mathbb{R}^n$</span> and define the shift operator <span class="math-container">$T_a$</span> on <span class="math-container">$L^p_{\nu}(\mathbb{R}^n)$</span> by <span class="math-container">$f\to f(x+a)$... | Yemon Choi | 763 | <p>This was intended as an extended comment but started to have too many formulas, so I thought that it would be more legible if posted as an answer.
(You don't actually state if there are <span class="math-container">$\nu$</span>-null sets which are not Lebesgue null, so I'm going to build an example which is mutually... |
1,862,807 | <blockquote>
<p>Show that for <span class="math-container">$x,y,z\in\mathbb{Z}$</span>, if <span class="math-container">$x$</span> and <span class="math-container">$y$</span> are coprime and <span class="math-container">$z$</span> is nonzero, then <span class="math-container">$\exists n\in\mathbb{Z}$</span> such that... | Bill Dubuque | 242 | <p>This can be solved <em>intuitively</em> by using a slight twist on Euclid's idea for generating new primes. Euclid employed <span class="math-container">$\,1 + p_1\cdots p_n$</span> is coprime to <span class="math-container">$\,c = p_1\cdots p_n.\,$</span> Stieltjes noted the generalization that, furthermore, <span... |
2,550,365 | <p>I encountered a question regarding orthogonal projection. I only have some clues to approach this questions and am not sure whether my thoughts can be used to answer the question. Could anyone check my thoughts and correct it or figure out a better solution, if needed? Thanks in advance.</p>
<p>Suppose $W \subseteq... | amd | 265,466 | <p>You’ve found an eigenvalue with geometric multiplicity 4 and one with geometric multiplicity 2. The sum of their eigenspaces is six-dimensional, so must be all of $\mathbb R^6$. $P$ is therefore diagonalizable and in fact is similar to $\operatorname{diag}(1,1,1,1,0,0)$.</p>
|
3,074,668 | <p>Good evening,</p>
<p>Could someone please demonstrate why this property is valid?</p>
<blockquote>
<p>Given <span class="math-container">$\sigma\in S_n$</span></p>
<p><span class="math-container">$$\left|\prod_{i<j} \frac{\sigma(j)-\sigma(i)}{j-i}\right|=1$$</span></p>
</blockquote>
| Nick Peterson | 81,839 | <p>This is because <span class="math-container">$\sigma$</span> is a permutation, and therefore a one-to-one correspondence. </p>
<p>You can rewrite the product in terms of numerators and denominators by way of
<span class="math-container">$$
\begin{align*}
\left\lvert\prod_{i<j}\frac{\sigma(j)-\sigma(i)}{j-i}\rig... |
60,810 | <p>I am looking for a proof of the fact that if $f:\mathbb{R}\to \mathbb{R}$ is a group automorphism of $(\mathbb{R},+)$ that also preserves order, then there exists a positive real number $c$ s.t. $f(x)=cx$ for all $x\in \mathbb{R}$. If anyone can point to a reference, that will be great.</p>
| Dylan Moreland | 3,701 | <p>I don't have a reference, but here is an outline of how I would do this. Since we can approximate any real number from above and below by elements of $\mathbf Q$ and $f$ preserves the order, we can reduce to the case of rational $x$. Clearly we want $f(1) = c > 0$ to be our constant. Now, if $b$ is a positive int... |
1,989,950 | <p>I was doing proof of open mapping theorem from the book Walter Rudin real and complex analysis book and struck at one point. Given if $X$ and $Y$ are Banach spaces and $T$ is a bounded linear operator between them which is $\textbf{onto}$. Then to prove $$T(U) \supset \delta V$$ where $U$ is open unit ball in $X$ an... | Claude Leibovici | 82,404 | <p><strong>Hint</strong></p>
<p>Assuming $\cos(\theta)\neq 0$,
$$\sec(\theta) + \tan(\theta) = \frac{3}{2}\implies 1+\sin(\theta)= \frac{3}{2}\cos(\theta) $$ Now, use the tangent half-angle substitution $t=\tan(\frac\theta 2)$ to get $$5 t^2+4 t-1=0\implies (t+1) (5 t-1)=0$$ One of the roots cannot be used (which one ... |
894,764 | <p>I am having problems understanding how to solve this question. </p>
<p>Find a linear function that satisfies both of the given conditions.</p>
<p>$f(-1) = 5, f(1) = 6$</p>
<p>Thanks,
<i>Note: i have the answer, just need help understanding</i></p>
| nchar | 132,757 | <p>A linear function (in one variable) is of the form $f(x) = ax + b$, where $a$ and $b$ are constants. The two conditions $f(-1) = 5$ and $f(1) = 6$ determine the value of $a$ and $b$. </p>
<p>To be explicit, using these two conditions, we have
$$ -a+b = 5,$$
$$ a+b = 6.$$
Now solve the equations above to get $a$ an... |
376,484 | <p>My questions are motivated by the following exercise:</p>
<blockquote>
<p>Consider the eigenvalue problem
$$
\int_{-\infty}^{+\infty}e^{-|x|-|y|}u(y)dy=\lambda u(x), x\in{\Bbb R}.\tag{*}
$$
Show that the spectrum consists purely of eigenvalues. </p>
</blockquote>
<p>Let $A:L^2({\Bbb R})\to L^2({\Bbb R})$ be ... | Hagen von Eitzen | 39,174 | <p>The function
$$f(n)=\begin{cases}1&\text{if }n\text{ odd}\\0&\text{if }n\text{ even}\end{cases}$$
is multiplicative and we have $\lim_{m\to\infty}f(p^m)=0$ if we let $p=2$. Since $\lim_{n\to\infty}f(n)$ does not exist, I guess that we may assume that <em>for all</em> primes $p$ we have $\lim_{m\to\infty}f(p... |
254,623 | <p>For example 100 is even and 100/2= 50 is also even</p>
<p>But 30 is also even but 30/2=15 is odd</p>
<p>Now let's say I have a number as large as 10^10000000000...</p>
<p>I want to know how many steps are involved in cutting this number in half. When the number is even, I divide it by 2. When it's odd, I subtract... | N. S. | 9,176 | <p>Yes you can get the chain for $n=10$, but for large numbers it is not easy.</p>
<p>Write your number $n$ in binary. What you are doing is the following:</p>
<ul>
<li>If the last digit is 0, you erase it.</li>
<li>If the last digit is 1, you make it a zero.</li>
</ul>
<p>For $n=10$ in binary you have $n=1010$. Thu... |
1,923,149 | <p>Preimage is defined as $X = \{x \in \mathbb{R}^n:Ax \in S\}$, where $A$ is a linear mapping $A \in \mathbb{R}^{m,n}$, and $S \subseteq \mathbb{R}^m$ is a sequentially-closed set.</p>
<p>The definition of sequential-closedness is that $S$ is sequentially-closed iff every convergent sequence in $S$ has its limit in $... | David K | 139,123 | <p>Interpret $f(z) = \binom zn$ as the number of combinations of $n$ objects that can be selected from a set of $z$ distinct objects
where $n, z \in \mathbb Z$ and $z \geq n \geq 0$.</p>
<p>For $n = 0$, we have $f(z) = 1$, which is a convex function.</p>
<p>For $n \geq 1$, consider a set of $z+1$ distinct objects,
$S... |
1,285,177 | <p>I know this is very simple and I'm missing something trivial here...</p>
<p>I'm having trouble converting this set of equations to polar form:</p>
<p>$$
\dot{x_1}=x_2-x_1 (x_1^2+x_2^2-1)\\
\dot{x_2}=-x_1-x_2 (x_1^2+x_2^2-1)
$$</p>
<p>where</p>
<p>$$
r= (x_1^2+x_2^2)^{1/2}\\
\theta=\arctan\left(\frac{x_2}{x_1}\ri... | David Holden | 79,543 | <p>$$
A^\mathrm{T}=-A
$$
implies that $A$ has zeroes on the main diagonal, since they change sign on the RHS, but remain unchanged on the LHS. if you incorporate that into your representation, it should be easy to answer the question about dimension and basis.</p>
<blockquote>
as a related exercise you might like t... |
1,176,435 | <p>Consider $G(4,p)$ - the random graph on 4 vertices. What is the probability that vertex 1 and 2 lie in the same connected component?</p>
<p>So far, I have considered the event where 1 and 2 do not lie in the same component. Then vertex 1 must lie in a component of order 1, 2 or 3 that doesn't contain vertex 2. Howe... | Siméon | 51,594 | <p>Let $E_{ij} = E_{ji}$ denote the event that the edge $\{i,j\}$ is added in the graph. The event <em>"1 is connected to 2"</em> is the disjoint union of the five following events:</p>
<ul>
<li>$E_{12}$, with probability $p$</li>
<li>$\overline{E_{12}} \wedge E_{13} \wedge E_{32}$, with probability $ p^2(1-p)$</li>
<... |
2,540,007 | <p>I have the following question. Find the matrix representation of the transormation $T:\mathbb{R}^3\to\mathbb{R}^3$ that rotates any vector by $\theta=\frac{\pi}{6}$ along the vector $v=(1,1,1)$.</p>
<p>A hint is given to find the rotation matrix about the $z-axis$ by $\frac{\pi}{6}$which is $$
\begin{bmatrix}
\f... | Lee Mosher | 26,501 | <p>The Gram-Schmidt process is a rather tedious way to do such a simple geometric task.</p>
<p>First, the plane $P$ orthogoal to $v = (1,1,1)$ is $x+y+z=0$.</p>
<p>Second, pick any vector $P$ in that plane, say $w = (1,-1,0)$.</p>
<p>Third, let $u$ be the cross product of $v,w$:
$$u = v \times w = (1,1,1) \times (1... |
362,895 | <p>I have been having a lot of trouble teaching myself rings, so much so that even "simple" proofs are really difficult for me. I think I am finally starting to get it, but just to be sure could some one please check this proof that $\mathbb Z[i]/\langle 1 - i \rangle$ is a field. Thank you.</p>
<p>Proof: Notice that ... | davidlowryduda | 9,754 | <p>Your answer is great, but I'd like to give a different view as well.</p>
<p>A standard first or second example of a Euclidean Domain is the Gaussian integers $\mathbb{Z}[i]$, so that in particular the Gaussian integers form a principal ideal domain. We also know that in PIDs, nonzero prime ideals are maximal. So if... |
2,551,683 | <blockquote>
<p>A jet has a $5\%$ chance of crashing on any given test flight. Once it crashes the program will be halted. Find the probability that the program lasts less than three flights.</p>
</blockquote>
<p>The correct answer to this question is $0.1426$, but I can't figure out how to get it.</p>
<p>Here's my... | Dean | 393,411 | <p>The probability that the jet survives after three tests: $(0.95)^3$. The probability for the other cases is $1-(0.95)^3=0.1426$.</p>
|
3,632,097 | <p>Given a sheaf <span class="math-container">$F$</span> on a topological space <span class="math-container">$X$</span> and <span class="math-container">$U$</span> is an open subset of <span class="math-container">$X$</span>. Denote <span class="math-container">$F|_U$</span> be the restricted sheaf of <span class="math... | Bueggi | 683,835 | <p>Yes we have.
To sections define the same germ if they agree on an open neighborhood of a point p. So they will also agree on a smaller neighborhood.</p>
|
2,252,317 | <p>I have a rather challenging question on my assignment and I have put in my best effort for now. I think I just need a tiny nudge to set me in the right direction to finish this proof. If you could have a look, that would be great!</p>
<hr>
<p><strong>Background on Cosets and Operations Defined on $V/W$</strong></p... | Sarvesh Ravichandran Iyer | 316,409 | <blockquote>
<p>$w_3 + W = W$!</p>
</blockquote>
<p>Proof : $x \in w_3 + W \implies x = w_3 + w$ for some $w \in W $, which means $x \in W$ since $W$ is a subspace.</p>
<p>On the other hand, $x \in W \implies x = w_3 + (x - w_3) \in w_3 + W$, since $W$ is a subspace, so $x-W_3 \in W$.</p>
<p>Hence, continuing fro... |
902,522 | <p>How would I simplify a fraction that has a radical in it? For example:</p>
<p>$$\frac{\sqrt{2a^7b^2}}{{\sqrt{32b^3}}}$$</p>
| FWE | 170,600 | <p>See for example <a href="http://rads.stackoverflow.com/amzn/click/0521356539" rel="nofollow"><em>Lambek and Scott: Introduction to higher order categorical logic</em></a>, ch 0.1 (unfortunately not in the net but for sure in ur university library). There first is defined a <strong>graph</strong>, then a <strong>dedu... |
813,395 | <p>I have read that linear independence occurs when:</p>
<p>$$\sum_{i=1}^n a_i v_i =0$$</p>
<p>Has only $a_i=0$ as a solution, but what if all $v_i$ were $0$ then $a_i$ could vary and still yield $0$. Does that mean that such a vector set is not linearly independent?</p>
<p>What if I have:</p>
<p>Let $\{c_0,c_1,c_2... | DeepSea | 101,504 | <p>By definition of linearly independent vectors the space $X$ itself is supposed to consist a non-zero vector $v$, thus not all of the $v_i$'s are zero vectors. </p>
|
3,115,090 | <p>This was supposedly an easy limit, and it is suspiciously similar to a Riemann sum, but I can't quite figure out for what function. </p>
<p><span class="math-container">$$\lim_{n\to\infty}{\frac{1}{n} {\sum_{k=3}^{n}{\frac{3}{k^2-k-2}}}}$$</span></p>
<p>Well, even the fact that <span class="math-container">$\frac{... | Romeo | 28,746 | <p>Isn't it <span class="math-container">$k^2 -k -2 = (k+1)(k-2)$</span>? In that way,
<span class="math-container">$$
\frac{3}{k^2-k-2} = \frac{1}{k-2} - \frac{1}{k+1}
$$</span>
and this is likely to be telescopic. </p>
|
1,988,731 | <p>Can one of you math this for me?</p>
<p>You've got a deck of 88 cards. Let's call the first card "1", the next "2", and so on.</p>
<p>What are the odds of pulling "1", "2", and "3" out of the deck, in that order?</p>
| Chas Brown | 167,790 | <p>If you mean "what are the odds that the first card drawn is $1$, the second card drawn is $2$, and the third card drawn is $3$?": There are $88!$ different possible shuffles of the deck of $88$ cards. There are "only" $(88-3)!$ shuffles that have the first three cards being $1,2,3$ in that order. So the probability ... |
951,332 | <p>the integer 220, 251 304 represent three consecutive perfect squares in base b. Determine the value of b.</p>
| lab bhattacharjee | 33,337 | <p>HINT:</p>
<p>Use the identity, $$(a+2)^2+a^2-2(a+1)^2=2$$</p>
<p>Here $(a+2)^2=(304)_b=3b^2+4$ and so on</p>
|
3,290,839 | <p>I want to prove that that given <span class="math-container">$f:R^2 \rightarrow R$</span> which is continuous with compact support s.t the integral of <span class="math-container">$f$</span> for every straight line <span class="math-container">$l$</span> is zero (<span class="math-container">$\int f(l(t))\mathrm{d}t... | Daniele Tampieri | 317,063 | <p>The problem you pose is the weak version of a classical result in harmonic analysis, which does not require the continuity nor the compactness of the support of the datum <span class="math-container">$f$</span>: in order to prove this stronger result, first note that
<span class="math-container">$$
\begin{align}
\in... |
2,956,744 | <p><a href="https://i.stack.imgur.com/lhDyB.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/lhDyB.png" alt="https://i.imgur.com/vvQDuVa.png" /></a></p>
<p>I'll provide a quickly-drawn representation of what the problem is.
Basically, there is a line <span class="math-container">$$l: y=-x+b$$</span> a... | Eleven-Eleven | 61,030 | <p>Generally, the alternate definition of the derivative is </p>
<p><span class="math-container">$$f'(c)=\lim_{x\rightarrow c}\frac{f(x)-f(c)}{x-c}$$</span></p>
<p>Note that this definition requires a limit. Thus we must have that </p>
<p><span class="math-container">$$\lim_{x\rightarrow c^{\color{red}{-}}}\frac{f(... |
2,561,125 | <p>Hey having trouble finishing this question.</p>
<p>Prove by induction that $n^3 \le 2^n$ for all natural numbers $n\ge 10$.</p>
<p>This is what I have so far:</p>
<p>Base step: For $n = 10$ </p>
<p>$1000 \le 1024$</p>
<p>Assumption Step: For $n = k$</p>
<p>Assume $k^3 \le 2^k$</p>
<p>Induction step: For $n =... | Community | -1 | <p><strong>Hint:</strong></p>
<p>We have to show that $(k+1)^3<2^{k+1}$. Note that:</p>
<p>$$(k+1)^3=k^3\cdot\frac{(k+1)^3}{k^3}=k^3\left(\frac{k+1}k\right)^3\;.$$</p>
<p>Since $k\ge 10$, $\frac{k+1}k=1+\frac1k\le 1+\frac1{10}=\frac{11}{10}$, and therefore </p>
<p>$$\left(\frac{k+1}k\right)^3\le \,\, ??$$</p>
|
4,132,998 | <p>I figure it has something to do with minimal polynomials and the Jordan canonical form, I just can't piece it together. I would really appreciate a hint on how to start it.</p>
| lhf | 589 | <p>Your hunch is correct: The minimal polynomial must be divide <span class="math-container">$t^m$</span> and have degree at most <span class="math-container">$n$</span>.</p>
|
274,249 | <p>I'd like to write the current data, but the Monitor and PutAppend are being written in such a way that the function <strong>pot</strong> does not match the current variables <strong>(a and b)</strong>. How can I fix this?</p>
<pre><code>f[a_, b_] = Exp[-Sqrt[b*x + Cos[c]]] + a*y;
PP[a_, b_] := NIntegrate[f[a, b], {... | Ulrich Neumann | 53,677 | <p>Start with</p>
<pre><code>NonlinearModelFit[data, a Exp[b (x - 1990)], {a, b }, x]
</code></pre>
|
4,263,784 | <p>I'm trying to find the multiplicity of <span class="math-container">$z=0$</span> on <span class="math-container">$f(z)=z\cos(z)-\sin(z)$</span> using complex analysis.</p>
<p>I'm new to complex analysis and the argument principle/Rouché's theorem so I'm not quite sure where to start. I can prove how many zero's this... | kelalaka | 338,051 | <p>One Time Pad (OTP) requires uniform random key bits to be information-theoretically secure. Once the key is transmitted securely with a trusted carrier, than the messages transmitted between the two sides are secure as long as the keystream is never used again ( keywords: two-time pad, many-time pad, and crib-draggi... |
3,322,492 | <p><strong>Prove:</strong> <span class="math-container">$A \cap (B - C) = (A \cap B) − (A \cap C)$</span></p>
<p>I can understand this using Venn Diagrams, however I am struggling to translate this into a formal proof. </p>
| Community | -1 | <p>The most common strategy for proving that sets <span class="math-container">$S=T$</span> is to show that <span class="math-container">$S\subset T$</span> and <span class="math-container">$T\subset S$</span>. And the most common strategy for proving <span class="math-container">$S\subset T$</span> is to prove that <... |
3,322,492 | <p><strong>Prove:</strong> <span class="math-container">$A \cap (B - C) = (A \cap B) − (A \cap C)$</span></p>
<p>I can understand this using Venn Diagrams, however I am struggling to translate this into a formal proof. </p>
| Cornman | 439,383 | <p>To show that two sets are equal you normaly show the two relations <span class="math-container">$\subseteq$</span> and <span class="math-container">$\supseteq$</span>.</p>
<p>This is how it is done, where I write it down explanatory and not how I would do so in a mathematical proof.</p>
<p>So we first want to show... |
3,322,492 | <p><strong>Prove:</strong> <span class="math-container">$A \cap (B - C) = (A \cap B) − (A \cap C)$</span></p>
<p>I can understand this using Venn Diagrams, however I am struggling to translate this into a formal proof. </p>
| user0102 | 322,814 | <p>Here it is an approach based on the definition of difference and the De Morgan Laws:</p>
<p><span class="math-container">\begin{align*}
(A\cap B) - (A\cap C) & = (A\cap B)\cap(\overline{A\cap C}) = (A\cap B)\cap(\overline{A}\cup\overline{C})\\
& = (A\cap B\cap\overline{A})\cup(A\cap B\cap\overline{C})\\
&am... |
4,463 | <p>It seems that most authors use the phrase "elementary number theory" to mean "number theory that doesn't use complex variable techniques in proofs." </p>
<p>I have two closely related questions.</p>
<ol>
<li>Is my understanding of the usage of "elementary" correct?</li>
<li>It appears that advanced techniques fro... | Kevin O'Bryant | 935 | <p>Elementary means that I know how to reduce it to "counting on fingers", college algebra, and linear algebra. That doesn't mean that other areas are to be avoided at all, though. </p>
<p>This definition works differently for different people, as that "I know how" expression is vague. In my book, probability is eleme... |
4,463 | <p>It seems that most authors use the phrase "elementary number theory" to mean "number theory that doesn't use complex variable techniques in proofs." </p>
<p>I have two closely related questions.</p>
<ol>
<li>Is my understanding of the usage of "elementary" correct?</li>
<li>It appears that advanced techniques fro... | user717 | 717 | <p>I refine my answer: I think that problems in elementary number theory can be characterized as problems in number theory for which both the problem and its solution can be understood in a fair amount of time by someone with undergraduate mathematical knowledge. The condition that also the solution is elementary and d... |
4,463 | <p>It seems that most authors use the phrase "elementary number theory" to mean "number theory that doesn't use complex variable techniques in proofs." </p>
<p>I have two closely related questions.</p>
<ol>
<li>Is my understanding of the usage of "elementary" correct?</li>
<li>It appears that advanced techniques fro... | Thomas Riepe | 451 | <p>I would regard as 'elementary number theory' that what needs no previous mathematical knowledge, esp. no abstract algebra, no Galois theory and no (complex) analysis. 'Elementary number theory' could include what Euler and Gauss did and of course "elementary" means not "simple". I would regard the reducability of st... |
4,463 | <p>It seems that most authors use the phrase "elementary number theory" to mean "number theory that doesn't use complex variable techniques in proofs." </p>
<p>I have two closely related questions.</p>
<ol>
<li>Is my understanding of the usage of "elementary" correct?</li>
<li>It appears that advanced techniques fro... | Ilya Nikokoshev | 65 | <p>To me, "elementary" = <strong>in principle can be understood by a person who only knows high school math</strong>. </p>
<p>Notes:</p>
<ul>
<li>The person can be assumed to be extremely smart or a genius.</li>
<li>What to mean by <em>high school math</em> can vary from person to person, but derivatives are certainl... |
4,463 | <p>It seems that most authors use the phrase "elementary number theory" to mean "number theory that doesn't use complex variable techniques in proofs." </p>
<p>I have two closely related questions.</p>
<ol>
<li>Is my understanding of the usage of "elementary" correct?</li>
<li>It appears that advanced techniques fro... | Konstantinos Gaitanas | 38,851 | <p>In a book i saw a nice answer:<br>
''Elementary Number Theory'' is Number Theory which is based on mathematics you can find in Euclid's ''Elements''</p>
|
194,421 | <p>This is homework. The problem was also stated this way: </p>
<p>Let A be a dense subset of $\mathbb{R}$ and let x$\in\mathbb{R}$. Prove that there exists a decreasing sequence $(a_k)$ in A that converges to x.</p>
<p>I know:</p>
<p>A dense in $\mathbb{R}$ $\Rightarrow$ every point in $\mathbb{R}$ is either in A o... | Shankara Pailoor | 39,210 | <p>Hint: Recursively define $a_n \in A$ such that $a_n \in (x, a_{n-1})$. </p>
|
3,516,754 | <p>I got stuck while doing exercise of the Apostol's Calculus, the exercise 28 of Section 5.5.</p>
<p>Here's the question</p>
<hr>
<p>Given a function <span class="math-container">$f$</span> such that the integral <span class="math-container">$A(x) = \int_a^xf(t)dt$</span> exists for each <span class="math-container... | symplectomorphic | 23,611 | <p>Paramanand gives an explicit counterexample, which proves that your <em>claim</em> is false. But I want to add this further answer to expose the flaw in your <em>reasoning</em>.</p>
<p>Your argument appears to rely on the following missing assumption: if <span class="math-container">$g$</span> and <span class="math-... |
189,266 | <p><strong>Q1:</strong> If a Morse function on a smooth closed $n$-manifold $X$ has critical points of only index $0$ and $n$, does it follow that $X\approx \mathbb{S}^n\coprod\ldots\coprod\mathbb{S}^n$?</p>
<p>I think the following question is essential in regard to the one above:</p>
<p><strong>Q2:</strong> If $f$ ... | André Nicolas | 6,312 | <p>There are many choices, particularly if you do not ask that the "distance" function satisfy the triangle inequality.</p>
<p>For example, if the vectors are $(x_1,x_2,\dots,x_n)$ and $(y_1,y_2,\dots,y_n)$ we can use $\sum (x_i-y_i)^2$ (the square of the Euclidean distance). This has some very nice properties that ha... |
2,846,114 | <p>$$\int\frac{2}{x(3x-8)}dx=P\cdot \ln\left|x\right|+Q\cdot \ln\left|3x-8\right|$$</p>
<p>Find out what P and Q are equal to.</p>
<p>This is what I worked out:</p>
<p>$$\frac{A}{x}+\frac{B}{3x-8}=\frac{2}{x(3x-8)}$$
$$-\frac{1}{4}=A,\ \ \ \frac{3}{4}=B$$
$$P=A, Q=B$$</p>
<p>why is the answer $P=-\frac{1}{4}, Q=\fr... | Key Flex | 568,718 | <p>You have the integral $\int\dfrac{2}{x(3x-8)}=2\int\dfrac{1}{x(3x-8)}$</p>
<p>Now if you solve $\dfrac{1}{x(3x-8)}$ using partial fractions then you will get
$$\dfrac{1}{x(3x-8)}=\frac{3}{8(3x-8)}-\frac{1}{8x}$$
Now,
$$2\int\dfrac{1}{x(3x-8)}=2\int\frac{3}{8(3x-8)}-\frac{1}{8x}$$
$$=2\left(\frac18\ln|3x-8|-\frac18\... |
261,031 | <p>i hope some of you can support to solve my problem, i need to work on data in the following way, where the length of each of the lists or sublists is equal. As an example i want to share the data-pattern with you:</p>
<pre><code>list1={a,b,c};
list2={{d,e,f},{g,h,i},......} (in reality the number of sublists in list... | Syed | 81,355 | <p>A minor variation based on @kglr's answer.</p>
<pre><code>list1 = {a, b, c};
list2 = {{d, e, f}, {g, h, i}};
MapThread[List, {list1, #}] & /@ list2
</code></pre>
<hr />
<p>Using Transpose:</p>
<pre><code>Transpose[{list1, #}] & /@ list2
</code></pre>
<hr />
<p>Result:</p>
<blockquote>
<p>{{{a, d}, {b, e}, {c... |
4,422,824 | <p><strong>Edit: This question involves derivatives, please read my prior work!</strong></p>
<p>This question has me stumped.</p>
<blockquote>
<p>A car company wants to ensure its newest model can stop in less than 450 ft when traveling at 60 mph. If we assume constant deceleration, find the value of deceleration th... | VERNON HEZRON | 1,002,019 | <p>Well, I believe the solution is rather simple. We have however been given not a fixed distance but an upper constraint on the distance to be travelled before it stops. Therefore, I will derive a simple equation of motion and use it to compute the dece;eration required. First of all, we are farmiliar with the fact th... |
180,296 | <p>I need an algorithm to decide quickly in the worst case if a 20 digit integer is prime or composite.</p>
<p>I do not need the factors.</p>
<p>Is the fastest way still a prime factorization algorithm? Or is there a faster way given the above relaxation?</p>
<p>In any case which algorithm gives the best worst case... | M Turgeon | 19,379 | <p>Write down what it means for an integer to have an inverse modulo $n$, and recall <a href="http://en.wikipedia.org/wiki/B%C3%A9zout%27s_identity" rel="nofollow">Bézout's identity</a>.</p>
|
157,074 | <blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="https://math.stackexchange.com/questions/155685/prove-binomp-1k-equiv-1k-pmod-p">Prove $\binom{p-1}{k} \equiv (-1)^k\pmod p$</a> </p>
</blockquote>
<p>The question is as follows:</p>
<blockquote>
<p>Let $p$ be prime. Show that ${p \choose k}\b... | André Nicolas | 6,312 | <p>It turns out that we do not even need Wilson's Theorem. Note the identity
$$\binom{p-1}{k+1}(k+1)=\binom{p-1}{k}(p-k-1).$$
This is easily obtained from the fact that $\binom{n}{m}=\frac{n!}{m!(n-m)!}$.
Now note that $p-k-1\equiv -(k+1)\pmod p$. Thus
$$\binom{p-1}{k+1}(k+1)\equiv -\binom{p-1}{k}(k+1)\pmod p.$$
If $0... |
2,579,572 | <blockquote>
<p>In an election, $10\%$ voters did not participate and $1200$ votes are found invalid. The winner gets $68\%$ of total voting list and he won by $56400$ votes. Find the votes polled in favor of losing candidate.</p>
</blockquote>
<p>I can't understand what should I do with the number of invalid votes.... | N. F. Taussig | 173,070 | <p>Let $x$ be the number of voters on the list. The total number of votes is equal to the number of votes received by the winner plus the number of votes received by the loser plus the number of voters who did participate in the election plus the number of voters whose votes were invalidated. </p>
<p>The winner rece... |
1,127,596 | <p>I am trying to show that the value of $\int^\infty_0$$\int^\infty_0$ sin($x^2$+$y^2$) dxdy is $\frac{\pi}{4}$ using Fresnel integrals. I'm having trouble splitting apart the integrand in order to actually be able to use the Fresnel integrals. Any help is appreciated. </p>
<p>Answer:
$\int^\infty_0$ $\int^\infty_0$... | user 1591719 | 32,016 | <p>Does it work like that? $$\int_{0}^{\infty}\int_{0}^{\infty} \sin(x^{2}+y^{2}) \,dx\ dy=\Im\left\{\int_{0}^{\infty}\int_{0}^{\infty} e^{i(x^2+y^2)} \ dx \ dy\right\}=\Im\left\{\int_{0}^{\infty}e^{i x^2} \ dx\int_{0}^{\infty} e^{i y^2} \ dy\right\}=\frac{\pi}{4}$$</p>
|
3,015,596 | <p>A short introduction: The independence number <span class="math-container">$\alpha(G)$</span> of a graph <span class="math-container">$G$</span> is the cardinality of the largest independent vertex set. Independent vertex set is made only of vertices with no edges between them. </p>
<p><a href="https://i.stack.imgu... | Yair Caro | 1,067,865 | <p>Induction on the order of <span class="math-container">$G$</span>. It's true for <span class="math-container">$|G| = 1$</span>. Now assume it's true for <span class="math-container">$|G| \leq n$</span> and let's prove it for <span class="math-container">$|G| = n+1$</span>.</p>
<p>Choose a vertex <span class="math-co... |
3,206,138 | <p>I have the matrix <span class="math-container">$A=\begin{pmatrix}
1 & -1\\
1 &1
\end{pmatrix}$</span> and <span class="math-container">$A^{n}=\begin{pmatrix}
x_{n} & -y_{n}\\
y_{n} &x_{n}
\end{pmatrix}$</span></p>
<p>At first exercise I found that <span class="math-container">$2A-A^{2}=2I_{2}$<... | E.H.E | 187,799 | <p><span class="math-container">$$x=\frac{i}{1+i}=\frac{i(1-i)}{(1+i)(1-i)}=\frac{i+1}{2}$$</span></p>
|
3,206,138 | <p>I have the matrix <span class="math-container">$A=\begin{pmatrix}
1 & -1\\
1 &1
\end{pmatrix}$</span> and <span class="math-container">$A^{n}=\begin{pmatrix}
x_{n} & -y_{n}\\
y_{n} &x_{n}
\end{pmatrix}$</span></p>
<p>At first exercise I found that <span class="math-container">$2A-A^{2}=2I_{2}$<... | DanielWainfleet | 254,665 | <p><span class="math-container">$[1].$</span> Suppose <span class="math-container">$A\implies B\implies C \implies x\in \{0,1\}.$</span> You cannot conclude, from this, that <span class="math-container">$x\in \{0,1\}\implies A.$</span> This does not mean that it isn't useful to use one-way implications. It's very usefu... |
3,197,683 | <p>Here is the theorem that I need to prove</p>
<blockquote>
<p>For <span class="math-container">$K = \mathbb{Q}[\sqrt{D}]$</span> we have</p>
<p><span class="math-container">$$\begin{align}O_K = \begin{cases}
\mathbb{Z}[\sqrt{D}] & D \equiv 2, 3 \mod 4\\
\mathbb{Z}\left[\frac{1 + \sqrt{D}}{2}\... | J. W. Tanner | 615,567 | <p>Let <span class="math-container">$\alpha=p+q\sqrt D$</span> with <span class="math-container">$p,q\in\mathbb Q$</span>. The other root of its minimal polynomial is <span class="math-container">$\bar \alpha=p-q\sqrt D.$</span> </p>
<p>The minimal polynomial is <span class="math-container">$(x-\alpha)(x-\bar\alpha)... |
1,375,958 | <p>I am looking for a bounded funtion $f$ on $\mathbb{R}_+$ satisfying $f(0)=0$, $f'(0)=0$ and with bounded first and second derivatives. My intitial idea has been to consider trigonometric functions or compositions of them, but I still haven't found an adequate one. Any ideas would be greatly appreciated.</p>
| David C. Ullrich | 248,223 | <p>f=0.</p>
<p>Now it turns out a reply must be at least thirty characters. Hmm. Humdee hum...</p>
|
1,888,187 | <p>I am working on algebraic functions and I am stuck on this problem:</p>
<p>$f(x) = a * r^x$<br>
$(2,1),(3,1.5)$<br></p>
<p>This would be a simple problem if it weren't for that $1.5$ -</p>
<p>So, I have plugged in $2$ and $1$ into the function and this is what I got:
<a href="https://i.stack.imgur.com/MVbxv.png" ... | Salech Alhasov | 25,654 | <p>So if we have the function f(x)=ar^x, then by substituting the first point given, namely $(2,1)$, we'll get:</p>
<p>$$1=ar^2$$</p>
<p>Similarly, substituting $(3, \frac{3}{2})$, we end up with</p>
<p>$$\frac{3}{2}=ar^3$$</p>
<p>Now, to find $a\neq 0$ and $r\neq 0 $, you should divide this two equations, what you... |
1,914,686 | <p>A fleet of nine taxis is dispatched to three airports, in such a way that three go to airport A, five go to airport B and one goes to airport C.</p>
<p>If exactly three taxis are in need of repair. What is the probability that every airport receives one of the taxis requiring repairs.</p>
<p>My method was total nu... | Henry | 6,460 | <p>A probability is usually between $0$ and $1$. </p>
<p>Try a simpler question: two airports, namely D with two taxis and E with two. Suppose the taxis are $T_1$, $T_2$, $T_3$, $T_4$ and the even numbers are broken.</p>
<p>The four equally probably cases with a broken taxi at each airport are </p>
<ul>
<li>$\{T_... |
82,765 | <p><strong>Bug introduced in 9.0 and persisting through 12.2</strong></p>
<hr />
<p>I get the following output with a fresh Mathematica (ver 10.0.2.0 on Mac) session</p>
<pre><code>FullSimplify[Exp[-100*(i-0.5)^2]]
(* 0. *)
Simplify[Exp[-100*(i-0.5)^2]]
(* E^(-100. (-0.5+i)^2) *)
</code></pre>
<p><code>FullSimplif... | Silvia | 17 | <p>Although confirmed by Wolfram officially, I will still hesitate to call this a bug.</p>
<p>The expression DOES very close to zero (though not a constant) everywhere except in a small neighborhood around $i=0.5$:</p>
<pre><code>LogPlot[E^(-25 (1 - 2 i)^2), {i, -1, 2}, PlotRange -> All]
</code></pre>
<p><img src... |
3,830,231 | <p>I'm trying to prove the following 'covariance inequality'
<span class="math-container">$$
|\text{Cov}(x,y)|\le\sqrt{\text{Var}(x)}\sqrt{\text{Var}(y)}\,,
$$</span>
where covariance and variance are defined using discrete values,
<span class="math-container">$$
\text{Cov}(x,y) = \frac{1}{n-1}\sum_{i=1}^n \big[(x_i-\b... | tommik | 791,458 | <p>Without considering continuous or discrete, the following is a very elementary proof.</p>
<p>Let's consider the rv <span class="math-container">$Z=Y+aX$</span> and let's calculate its variance.</p>
<p><span class="math-container">$$V(Z)=V(Y)+a^2V(X)+2a cov(X,Y)\geq0$$</span></p>
<p>As per the fact that variance cann... |
3,882,214 | <p>I have a question that involves an Argand diagram. The complex number <strong>u = 1 + 1i</strong> is the center of that circle, and the radius is one. In other words, <span class="math-container">$$|z - (1 + 1i)| = 1$$</span></p>
<p>Now my issue is the following: I need to <strong>calculate the least value of |z| fo... | Karan Elangovan | 497,101 | <p>The closest point is the point on the line through the origin and <span class="math-container">$(1 + i)$</span>.</p>
<p>The distance between the origin and the centre is <span class="math-container">$|1+i| = \sqrt{2}$</span></p>
<p>The distance between the centre and the point is 1 (radius is 1)</p>
<p>So the distan... |
76,505 | <p>In the eighties, Grothendieck devoted a great amount of time to work on the foundations of homotopical algebra. </p>
<p>He wrote in "Esquisse d'un programme": "[D]epuis près d'un an, la plus grande partie de mon énergie a été consacrée à un travail de réflexion sur les <em>fondements de l'algèbre (co)homologique no... | Omar Antolín-Camarena | 644 | <p>The smash product of pointed topological spaces is not associative, i.e., $(X \wedge Y)\wedge Z$ need not be homeomorphic to $X \wedge (Y \wedge Z)$. (It fails, for example, for $X = Y = \mathbb{Q}$ and $Z = \mathbb{N}$.)</p>
|
2,550,568 | <p>Suppose we have an alphabet of $a$ letters and a word $w$ of length $r$. What is the probablity that $w$ will appear in a sequence of $n$ letters drawn at random from the given alphabet?</p>
<p>I have posted a general question since there seem to be a few of these questions appearing, and this is intended as a gene... | Francisco José Letterio | 482,896 | <p>Well, the possible number of words of length n using a characters is $a^n$</p>
<p>On the other side, the odds of w showing up is the different ways you can choose r consecutive positions in a string of n positions, which is (if I'm not mistaken) $n-r+1$, </p>
<p>So the odds are $$\frac {n-r+1}{a^n}$$</p>
<p>Edit:... |
3,902,418 | <p>Ok, so I know that the mean value of a function, <span class="math-container">$f(x)$</span>, on the interval <span class="math-container">$[a,b]$</span> is given by (or defined by?)
<span class="math-container">$$\frac{1}{b-a}\int_a^bf(x)~dx$$</span>
but I have <span class="math-container">$2$</span> basic questions... | Lee Mosher | 26,501 | <p>You wrote:</p>
<blockquote>
<p>For a finite set of values I divide the sum of the values by however many values I have to obtain the mean...</p>
</blockquote>
<p>Let's look at a Riemann sum for the expression <span class="math-container">$\frac{1}{b-a} \int_a^b f(x) \, dx$</span>, namely:
<span class="math-container... |
4,522,097 | <p><span class="math-container">$$
X = \begin{pmatrix}
1+b_1 & 1 & 0 & 0 & 0 & \frac{1}{a_{6}} \\
1+b_2 & 1 & 1 & 0 & 0 & -\frac{a_1}{a_6} \\
b_3 & 1 & 1 & 1 & 0 & -\frac{a_2}{a_6} \\
b_4 & 0 & 1 & 1 & 1 & -\frac{a_3}{a_6} \\
b_5 & 0 ... | quarague | 169,704 | <p>They key difference between your two examples is the dimension of the space. The manifold is a 2-dimensional surface in both cases. In the first case this surface is embedded into the 3-dimensional space <span class="math-container">$\mathbb{R}^3$</span>.</p>
<p>In the second case you consider the parametrizations w... |
692,582 | <p>When I was reading a paper, I found an strange derivation like
$$\int^{+\infty}_{-\infty}\mathrm{ln}(1+e^w)f(w)dw\\=\int^0_{-\infty}\ln(1+e^w)f(w)+\int^\infty_0[\ln(1+e^{-w})+w]f(w)dw$$
when $w$ is the normal random variable and $f(w)$ is the normal density.</p>
<p>Why is that natural log integration broken up into... | Gerry Myerson | 8,269 | <p>$$x^2+y(y+1)^2=0$$ gives something like what you want. </p>
<p>See also <a href="https://math.stackexchange.com/questions/276933/graph-of-an-infinitely-extending-rollercoaster-loop">Graph of an infinitely extending rollercoaster loop</a></p>
|
3,840,253 | <blockquote>
<p>How to show that <span class="math-container">$\csc x - \csc\left(\frac{\pi}{3} + x \right) + \csc\left(\frac{\pi}{3} - x\right) = 3 \csc 3x$</span>?</p>
</blockquote>
<p>My attempt:<br />
<span class="math-container">\begin{align}
LHS &= \csc x - \csc\left(\frac{\pi}{3} + x\right) + \csc\left(\frac... | lab bhattacharjee | 33,337 | <p>If <span class="math-container">$\csc3x=\csc3y\iff \sin3y=\sin3x$</span></p>
<p><span class="math-container">$$3y=3x+(-1)^nn\pi$$</span> where <span class="math-container">$n$</span> is any integer</p>
<p><span class="math-container">$y=x+\dfrac{2n\pi}3; n=-1,0,1$</span></p>
<p>Now as <span class="math-container">$\... |
2,388,738 | <blockquote>
<p>I'm messing around with doing a visualization that has nothing to do with the primes and in order to execute it correctly I need an ordered list of all point in the order that the Ulam Spiral crosses them. I've tried some of my work but have only run in to abundantly complicated paths to solution. Als... | sOvr9000 | 186,910 | <p>A piecewise, explicit function is provided by the accepted answer of <a href="https://math.stackexchange.com/questions/3157030/parametrizing-the-square-spiral?rq=1">a similar question</a>.</p>
|
3,566,469 | <p>I am confused by a discussion with a colleague. The discussion is about the period of a periodic function.</p>
<p>For example, the periodic function <span class="math-container">$$f(x)=\sin(x), \quad x\in (0,\infty)$$</span> has period <span class="math-container">$2\pi$</span>. If I change the scale and build the ... | J.G. | 56,861 | <p>Functions such as <span class="math-container">$\sin(\ln x)$</span> are called <a href="https://arxiv.org/abs/1002.1010" rel="nofollow noreferrer"><em>log-periodic</em></a> rather than periodic.</p>
|
373,068 | <p>For a real number $a$ and a positive integer $k$, denote by $(a)^{(k)}$ the number $a(a+1)\cdots (a+k-1)$ and $(a)_k$ the number
$a(a-1)\cdots (a-k+1)$. Let $m$ be a positive integer $\ge k$. Can anyone show me, or point me to a reference, why the number
$$ \frac{(m)^{(k)}(m)_k}{(1/2)^{(k)} k!}= \frac{2^{2k}(m)^{(k... | vonbrand | 43,946 | <p>Using Knuth's notation:
$$
x^{\underline{k}} = x (x - 1) \ldots (x - k + 1) = (x)_{(k)} \qquad
x^{\overline{k}} = x (x + 1) \ldots (x + k - 1) = (x)^{(k)}
$$
What you are looking for is:
$$
\frac{m^{\overline{k}} m^{\underline{k}}}{(1/2)^{\overline{k}} k!}
$$
As if $k > m$ then $m^{\underline{k}} = 0$, while all ... |
2,036,301 | <p>Can someone please help me prove that this series is convergent? <br></p>
<p>The problem is I don't know what to do with sin.<br> </p>
<p>$$\sum_{n=1}^{\infty} 2^n \sin{\frac{\pi}{3^n}} $$</p>
| user389056 | 389,056 | <p>Consider the root test. </p>
<p>Let $L = \lim_{n \to \infty} |2^n\sin(\frac{\pi}{3^n})|^{\frac{1}{n}}.$
Then $L = \lim_{n \to \infty} (2^n\sin(\frac{\pi}{3^n}))^{\frac{1}{n}} = \lim_{n \to \infty} 2\sin(\frac{\pi}{3^n})^{\frac{1}{n}} = 0$, so the series converges by root test. </p>
|
139,934 | <p>Suppose I want to solve an equation for the matrix elements of $\bar{W}$:
$$\alpha W_{ba}+\beta W_{bb}=x; \alpha W_{aa}+\beta W_{ab}=y$$</p>
<p>Using the syntax <code>Subscript[W, ij]</code> for my matrix element (on the $i$th row and $j$ th column), I get the following message:</p>
<p>Set::write: Tag Times in 2 x... | Alexey Popkov | 280 | <pre><code>file = "--------------
SCF ITERATIONS
--------------
ITER Energy Delta-E Max-DP RMS-DP [F,P] Damp
*** Starting incremental Fock matrix formation ***
0 -8693.9185205626 0.000000000000 0.02365877 0.00022580 0.2453968 0.8500
1 -8694.4485310565 -0.53... |
3,442,173 | <p>Give a counter example to each of the following:<br>
(a) G is a connected graph with a cut-vertex, then G contains a bridge.
(b) G is a tree if and only if it contains no cycle.</p>
| egreg | 62,967 | <p>Consider <span class="math-container">$g(x)=f(x)-cx$</span>. Then <span class="math-container">$g’(x)=f’(x)-c=c-c=0$</span>.</p>
<p>Therefore <span class="math-container">$g$</span> is constant, say <span class="math-container">$g(x)=d$</span>.</p>
<p>Then <span class="math-container">$f(x)=cx+d$</span>.</p>
|
3,059,857 | <p>We are supposed to use this formula for which I can't find any explaination anywhere and our teacher didn't explain anything so if anyone could help me I would appreciate it. </p>
<p><span class="math-container">$ x = A + k \times 2\pi$</span></p>
<p>and</p>
<p><span class="math-container">$x = \pi - A + k \times... | KM101 | 596,598 | <p>Note that you have the arcsine function has a range of <span class="math-container">$\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$</span>, but sine is negative in both quadrants <span class="math-container">$3$</span> and <span class="math-container">$4$</span>, so only the quadrant <span class="math-container">$4$</s... |
2,054,175 | <p>This problem is giving me loads of confusion. I just need someone to walk through it because I have the answer and I can't get to it to save my life. I have been on it for days. Please help.</p>
<p>$$\frac{x + 3}{x - 4}\le 0$$ </p>
| jameselmore | 86,570 | <p>Hint:<br>
Let "$+$" be any positive real number, and "$-$" be any negative real number. It should be clear that:</p>
<p>$$\frac{+}{+} > 0;\ \ \frac{-}{-} > 0$$
$$\frac{+}{-} < 0;\ \ \frac{-}{+} < 0$$</p>
<p>If you can figure out for which values of $x$ has $(x+3)$ positive\negative, and which values of... |
93,274 | <p>Wielandt wrote a paper titled "Remarks on diagonable matrices".</p>
<p>According to Mathematische Werke - Mathematical Works : Linear Algebra and Analysis
by Helmut Wielandt, Hans Schneider, Bertram Huppert (Editor) page 260 this paper from Wielandt remained unpublished (at least from the 1950s to the 1980s).</p>
... | Federico Poloni | 1,898 | <p>Now that the server is back up, I am posting this as a real answer.</p>
<p>With some work, you might be able to find the proof in Wielandt's notebooks, which were TeXxed and put online <a href="http://www3.math.tu-berlin.de/numerik/Wielandt/index_en.html" rel="nofollow">here</a>.</p>
<p>The TeX source files are al... |
3,340,686 | <p>The <span class="math-container">$7$</span>th floor of a building is <span class="math-container">$23$</span>m above street level and <span class="math-container">$13$</span>th floor is <span class="math-container">$41$</span>m above street level. What is the height (above street level) of the first floor and what i... | Siddhant | 687,664 | <p>Yes it is perfectly correct. The first floor is <strong><span class="math-container">$5m$</span></strong> above the street level above which every floor with a height of <strong><span class="math-container">$3m$</span></strong>.</p>
|
1,893,168 | <p>$$\lim_{x\to 0} {\ln(\cos x)\over \sin^2x} = ?$$</p>
<p>I can solve this by using L'Hopital's rule but how would I do this without this?</p>
| Marco Cantarini | 171,547 | <p>$$\frac{\log\left(\cos\left(x\right)\right)}{\sin^{2}\left(x\right)}=\frac{1}{2}\frac{\log\left(1-\sin^{2}\left(x\right)\right)}{\sin^{2}\left(x\right)}=-\frac{\sin^{2}\left(x\right)+O\left(\sin^{4}\left(x\right)\right)
}{2\sin^{2}\left(x\right)}\stackrel{x\rightarrow0}{\rightarrow}-\frac{1}{2}.$$</p>
|
1,893,168 | <p>$$\lim_{x\to 0} {\ln(\cos x)\over \sin^2x} = ?$$</p>
<p>I can solve this by using L'Hopital's rule but how would I do this without this?</p>
| paf | 333,517 | <p>Note that $$\lim_{x\to 0}\frac{1-\cos x}{x^2}=\lim_{x\to 0}\frac{2\sin^2(\frac x2)}{x^2}=\lim_{x\to 0}\frac 14 \frac{2\sin^2(\frac x2)}{\frac{x^2}{4}}=\frac{1}{2}$$
Now,
$$\lim_{u\to 1}\frac{\ln u}{u-1}=\ln'(1)=1$$
Hence, since $u=\cos x$ tends to 1 when $x$ tends to 0:
$$\lim_{x\to 0}\frac{\ln(\cos x)}{\cos x-1}=1$... |
2,668,839 | <blockquote>
<p>Finding range of $$f(x)=\frac{\sin^2 x+4\sin x+5}{2\sin^2 x+8\sin x+8}$$</p>
</blockquote>
<p>Try: put $\sin x=t$ and $-1\leq t\leq 1$</p>
<p>So $$y=\frac{t^2+4t+5}{2t^2+8t+8}$$</p>
<p>$$2yt^2+8yt+8y=t^2+4t+5$$</p>
<p>$$(2y-1)t^2+4(2y-1)t+(8y-5)=0$$</p>
<p>For real roots $D\geq 0$</p>
<p>So $$16... | Community | -1 | <p>You can simplify the expression as</p>
<p>$$\frac12+\frac1{2(\sin x+2)^2}$$ and the extreme values are</p>
<p>$$\frac12+\frac1{2\cdot3^2},\\\frac12+\frac1{2\cdot1^2}.$$</p>
|
3,894,437 | <p>Let <span class="math-container">$\mathcal{A}$</span> be a finite set and consider the set of all sequences <span class="math-container">$\mathcal{A}^{\mathbb{Z}}$</span> on <span class="math-container">$\mathbb{Z}$</span> with values in <span class="math-container">$\mathcal{A}$</span>. This set has a cardinality o... | JunderscoreH | 632,927 | <p>For the side question, as long as <span class="math-container">$|\mathcal{A}|\ge 2$</span> we have <span class="math-container">$2^{\aleph_0}\le|\mathcal{A}^{\mathbb{Z}}|$</span> so that it's uncountable.</p>
<p>For the actual question, I'll consider <span class="math-container">$\mathcal{A}^{\mathbb{N}}$</span>, bu... |
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