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 |
|---|---|---|---|---|
508,550 | <p>I read in an article the Euclidean distance formula can be estimated with about 6% relative error with the following formula. Would you please <strong>why</strong> this is true and <strong>where</strong> can I find such estimations? Is it possible to <strong>extend</strong> it for higher dimensions?</p>
<p>$d(a,b) ... | Did | 6,179 | <p>Let $u= \max(|a_x-b_x|,|a_y-b_y|)$ and $v=\min(|a_x-b_x|,|a_y-b_y|)$, then algebra shows that
$$
d(a,b)=u+r(u,v)\cdot v,\qquad r(u,v)=\frac{v}{u+\sqrt{u^2+v^2}}.
$$
Why one would think that $r(u,v)\approx0.365$ always, escapes me. In fact, $r(u,v)$ can be anywhere in the interval $[0,\sqrt2-1)\approx[0,0.414)$ and $... |
1,986,247 | <p>The nth Catalan number is :
$$C_n = \frac {1} {n+1} \times {2n \choose n}$$
The problem 12-4 of CLRS asks to find :
$$C_n = \frac {4^n} { \sqrt {\pi} n^{3/2}} (1+ O(1/n)) $$
And Stirling's approximation is:
$$n! = \sqrt {2 \pi n} {\left( \frac {n}{e} \right)}^{n} {\left( 1+ \Theta \left(\frac {1} {n}\right) \right... | epi163sqrt | 132,007 | <p>We can use Stirling's approximation formula
\begin{align*}
n!=\sqrt{2\pi n}\left(\frac{n}{e}\right)^n\left(1+O\left(\frac{1}{n}\right)\right)
\end{align*}
to prove:</p>
<blockquote>
<p>The following is valid</p>
<p>\begin{align*}
C_n=\frac{1}{n+1}\binom{2n}{n}= \frac {4^n} { \sqrt {\pi} n^{3/2}} (1+ O... |
3,909,972 | <p>I used Photomath and Microsoft Math to compute an equation, but they gave me two different results (-411 and -411/38) Why did that happen and which is the correct answer?</p>
<p><a href="https://i.stack.imgur.com/egt5A.jpg" rel="nofollow noreferrer">https://i.stack.imgur.com/egt5A.jpg</a>
<a href="https://i.stack.im... | Robby the Belgian | 19,298 | <p>Note the first step in the second solution. It starts by multiplying both sides by 38.</p>
<p>Anyway, this is not a good way to do arithmetic. You are trying to find a solution that makes <span class="math-container">$\frac{411}{38}$</span> equal to <span class="math-container">$0$</span>, which cannot be done.</p>
... |
396,440 | <p>Suppose we have the function $$f(x) = \frac{x}{p} + \frac{b}{q} - x^{\frac{1}{p}}b^{\frac{1}{q}}$$ where $x,b \geq 0 \land p,q > 1 \land \frac{1}{p}+\frac{1}{q} = 1$</p>
<p>I am trying to show that $b$ is the absolute minimum of $f$. </p>
<p>I proceeded as follows:</p>
<p>$$\frac{df(x)}{dx} = \frac{1}{p} - \fr... | Community | -1 | <p>Note that $$\dfrac{p}{q(p-1)}=1$$ since $\dfrac1p + \dfrac1q = 1$. This is because, we have $\dfrac1q = 1 - \dfrac1p = \dfrac{p-1}p \implies \dfrac{p}{q(p-1)} = 1$.</p>
|
223,582 | <p>Maps $g$ maps $\left\{1,2,3,4,5\right\}$ onto $\left\{11,12,13,14\right\}$ and $g(1)\neq g(2)$. How many g are there.</p>
<p><strong>My answer</strong>:
I transformed the question to a easy-understand way and find out the solution.
Consider there are five children and four seats. Two of them are willing sitting to... | Mick | 42,351 | <ol>
<li>From AC, find its midpoint F.</li>
<li>Draw the circle using F as center and FA (or FC) as radius.</li>
<li>The point(s) of intersection of the circles is D. </li>
</ol>
|
2,788,276 | <p>Let$\ f_n (x)=n^2x(1-x)^n$ I need to prove that$\ f_n→0$ in the interval$\ [0,1]$.</p>
<hr>
<p>Let$\ f_n(x) = nx^n$ prove that$\ f_n→0$ in the interval$\ [0,1)$.</p>
<p>For both of these sequences I tried the following:</p>
<p>By taking the function$\ f(x)=0$ we can see that</p>
<p>$$\lim_{n\rightarrow\infty}f_... | Youngsu | 84,157 | <p>The existence of $f(X)$ guarantees that every element in $R$ is integral over $\mathbb{Z}/(n)$. In other words, $\mathbb{Z}/(n) \subset R$ is an integral extension. So, $\dim R = \dim \mathbb{Z}/(n) = 0$.</p>
|
3,069,684 | <p>My question goes like this</p>
<p>If 5a+4b+20c=t, then what is the value of t for which the line ax+by+c-1=0 always passes through a fixed point?</p>
<p>I tried but couldn't solve it so I looked at the solution. The solution says that the equation has 2 independent parameters. I get that. If we choose a and b, c a... | amd | 265,466 | <p>I’ll speak to your first question, which wasn’t really addressed in the other answer. </p>
<p>A line in the plane normally has two degrees of freedom. You’re probably used to specifying them via slope and <span class="math-container">$y$</span>-intercept (with an exception for vertical lines). For lines that go th... |
3,634,416 | <p>First of all, English is not my native language, but Chinses is. I tried to spilt the integration interval into 2 pieces: <span class="math-container">$ [0, 1-1/n] $</span> and <span class="math-container">$ [1-1/n, 1] $</span>. In both intervals I use the mean value theorem:
<span class="math-container">$$
\in... | Community | -1 | <p>Consider <span class="math-container">$\int_0^{1-n^{-1/2}}\frac1{1+x^n}\,dx$</span> and <span class="math-container">$\int_{1-n^{-1/2}}^1\frac1{1+x^n}\,dx$</span> instead. The second integral is still bounded above by <span class="math-container">$n^{-1/2}$</span>.</p>
<p>For the first integral, <span class="math-c... |
2,629,133 | <p>In keno, the casino picks 20 balls from a set of 80 numbered 1 to 80. Before the draw is over, you are allowed to choose 10 balls. What is the probability that 5 of the balls you choose will be in the 20 balls selected by the casino?</p>
<p>My attempt: The total number of combinations for the 20 balls is $80\choose... | bames | 464,998 | <p>Another way to think about this is to realize that the casino must choose $5$ balls from the $10$ that you chose and $15$ balls from the $70$ that you didn't choose. So:
$$P = \frac{\binom{10}{5} \binom{70}{15}}{80\choose 20} \approx 0.0514...$$</p>
|
1,146,050 | <p>given $f(x)=\frac{x^4+x^2+1}{x^2+x+1}$.</p>
<p>Need to find the min value of $f(x)$.</p>
<p>I know it can be easily done by polynomial division but my question is if there's another way</p>
<p>(more elegant maybe) to find the min? </p>
<p><strong>About my way</strong>: $f(x)=\frac{x^4+x^2+1}{x^2+x+1}=x^2-x+1$. (... | Community | -1 | <p>$$f(x)=\frac{x^4+x^2+1}{x^2+x+1}=x^2-x+1$$
$$f'(x) = 2x-1=0,x=\frac12$$
$$f''(\frac12)=2\gt0\text{ hence this is a local minimum by the second derivative test}$$</p>
|
2,573,458 | <p>Given $n$ prime numbers, $p_1, p_2, p_3,\ldots,p_n$, then $p_1p_2p_3\cdots p_n+1$ is not divisible by any of the primes $p_i, i=1,2,3,\ldots,n.$ I dont understand why. Can somebody give me a hint or an Explanation ? Thanks.</p>
| Sri-Amirthan Theivendran | 302,692 | <p>Towards a contradiction, if some prime $p_i$ ($1\leq i\leq n$) divides $p_1p_2p_3\cdots p_n+1$, then because $p_i$ also divides $p_1p_2p_3\cdots p_n$, it follows that $p_i\mid 1$ (the difference), a contradiction. </p>
|
2,877,080 | <p>Let A denote a commutative ring and let e denote an element of A such that $e^2 = e$.
How to prove that $eA \times (1 - e)A \simeq A$?
I thought that $\phi: A \mapsto eA \times (1 - e)A, \ \phi(a) = (ea, (1-e)a)$ is an isomorphism but I don't know how to prove that $\phi$ is a bijection.</p>
| gandalf61 | 424,513 | <p>Remember that your primary objective in this model lecture is to communicate clearly and effectively to your scenario audience (first year analysis students), not to impress you actual (experienced) audience with the breadth and depth of your knowledge.</p>
<p>I imagine your biggest challenge when delivering your m... |
169,531 | <p>Let me preface this by saying that I have essentially no background in logic, an I apologize in advance if this question is unintelligent. Perhaps the correct answer to my question is "go look it up in a textbook"; the reasons I haven't done so are that I wouldn't know which textbook to look in and I wouldn't know ... | Thomas Andrews | 7,933 | <p>If you had a theory, $T$, with a recursively enumerable axiom set, and it completely resolved all arithmetic questions, then, if the set of arithmetic questions was recursively enumerable in $T$, you could recursively enumerate all the arithmetic statements in $T$ which are provable in $T$, and hence you'd have an r... |
2,304,318 | <p>Let $\sum_{n=1}^\infty a_n$ and $\sum_{n=1}^\infty b_n$ be two real series, where we have $\lim_{n\rightarrow\infty} \frac{a_n}{b_n} = M >0$. Show that either one of these options happen:</p>
<ol>
<li>Both series converge</li>
<li>Both series diverge</li>
</ol>
<p>I have no clue on how to solve this. Can someo... | Martin Argerami | 22,857 | <p>Clue: for $\varepsilon=M/2$ and large enough $n$, $$\frac M2\,b_n\leq a_n\leq \frac{3M}2\,b_n.$$</p>
|
1,513,373 | <p>Let M be a cardinal with the following properties:<br>
- M is regular<br>
- $\kappa < M \implies 2^\kappa < M$<br>
- $\kappa < M \implies s(\kappa) < M$ where $s(\kappa)$ is the smallest strongly inaccessible cardinal strictly greater than $\kappa$ </p>
<p>My question is: Is M a Mahlo cardinal ? If s... | Cosmonut | 287,070 | <p>Thanks a lot for your answers, Wojowu and Andreas.
I find I am not able to simply reply with a comment while being a guest, so I am writing it this way.</p>
<p>The mental picture I am getting of Mahlo versus inaccessibles is somewhat like $\aleph_1$ versus the Veblen hierarchy of countable ordinals - you keep recur... |
2,196,539 | <p>I'm having a complete mind blank here even though i'm pretty sure the solution is relatively easy.</p>
<p>I need to make X the subject of the following equation:</p>
<p>$$AB - AX = X $$</p>
<p>All i've done so far is:
$$A(B-X) = X$$
$$B-X = A^{-1} X$$</p>
<p>Not sure if thats right?</p>
<p>Thanks in advance.</p... | rookie | 346,911 | <p>Whatever you have written is correct if inverse of A exists.</p>
<p>Hint for another way of writing an expression for $X$: $AB = (I+A)X.$</p>
|
211,803 | <p>I ended up with a differential equation that looks like this:
$$\frac{d^2y}{dx^2} + \frac 1 x \frac{dy}{dx} - \frac{ay}{x^2} + \left(b -\frac c x - e x \right )y = 0.$$
I tried with Mathematica. But could not get the sensible answer. May you help me out how to solve it or give me some references that I can go over... | doraemonpaul | 30,938 | <p>$\dfrac{d^2y}{dx^2}+\dfrac{1}{x}\dfrac{dy}{dx}-\dfrac{ay}{x^2}+\left(b-\dfrac{c}{x}-ex\right)y=0$</p>
<p>$\dfrac{d^2y}{dx^2}+\dfrac{1}{x}\dfrac{dy}{dx}-\left(ex-b+\dfrac{c}{x}+\dfrac{a}{x^2}\right)y=0$</p>
<p>Let $y=\dfrac{u}{\sqrt{x}}$ ,</p>
<p>Then $\dfrac{dy}{dx}=\dfrac{1}{\sqrt{x}}\dfrac{du}{dx}-\dfrac{u}{2x\... |
211,803 | <p>I ended up with a differential equation that looks like this:
$$\frac{d^2y}{dx^2} + \frac 1 x \frac{dy}{dx} - \frac{ay}{x^2} + \left(b -\frac c x - e x \right )y = 0.$$
I tried with Mathematica. But could not get the sensible answer. May you help me out how to solve it or give me some references that I can go over... | Przemo | 99,778 | <p>Consider a slightly more general ODE.
<span class="math-container">\begin{equation}
\frac{d^2 y(x)}{d x^2} + \frac{1}{x} \frac{d y(x)}{d x} + \left(-\frac{a}{x^2} + b - \frac{c}{x} - e x + e_1 x^2 \right) y(x)=0
\end{equation}</span>
If we write:
<span class="math-container">\begin{equation}
y(x)=x^{\sqrt{a}}\cdot \... |
1,460,488 | <p>There are $4$ girls and $3$ boys but there are only $5$ seats. How many ways can you seat the $3$ boys together?</p>
<p>The order of the seat matters, for example:
there's the order
$B_1$ $B_2$ $B_3$ $G_2$ $G_4$
and there's
$B_2$ $B_3$ $B_1$ $G_2$ $G_4$</p>
<p>Here's my answer:
There are $3!$ ways to seat the $3$ ... | N. F. Taussig | 173,070 | <p>You are correct. Here is another approach: </p>
<p>We can select the two girls in $\binom{4}{2}$ ways. We treat the three boys as a unit, so we have three objects to permute (the two girls we select and the unit of three boys). We can permute the three objects in $3!$ ways. We can also permute the unit consist... |
63,052 | <p>Suppose I have a square matrix $M$, which you can think of as the weighted adjacency matrix of a graph $G$. I want to order the vertices of $G$ in such a way that the entries of the matrix $M$ are clustered. By this I mean that the weights that are close in value should appear close in $M$.</p>
<p>I know Mathematic... | kh40tika | 6,395 | <p><a href="https://en.wikipedia.org/wiki/Spectral_clustering" rel="nofollow noreferrer">Spectral clustering</a> might be a good candidate here. Generally, spectral clustering works as following:</p>
<ol>
<li><p>Find a few largest eigenvectors of the adjacency matrix by magnitude, let's say we choose largest M vectors.... |
944,840 | <p>For vectors u, w, and v in a vector space V, I am trying to prove:</p>
<p>If $u + w = v + w$ then $u = v$</p>
<p><strong>without</strong> using the additive inverse and only using the 8 axioms which define a vector space. I am coming up short. I don't see how to do this without assuming that if $u + w = v + w$ the... | rschwieb | 29,335 | <p>Apparently the only approach for a vector space $V$ that avoids additive inverses <em>in $V$</em> is to use $w'=(-1)w$, which would leverage additive inverses in <em>the field</em> and not in the abelian group. Presumably your axioms (whatever they are) include that $0w=0$, $1w=w$ and that scalars distribute, allowi... |
3,595,451 | <p><strong>Question:</strong></p>
<p>Let <span class="math-container">$P_{3}(\mathbb{R})$</span> have the standard inner product and <span class="math-container">$U$</span> be the subset spanned by the two vectors (which are polynomials) <span class="math-container">$u_{1}=1+2x-3x^2$</span> and <span class="math-conta... | Graham Kemp | 135,106 | <blockquote>
<p>How would I represent the distance R from the point (X,Y) and would this be the cdf or pdf?</p>
</blockquote>
<p>By definition of Euclidean Distance, <span class="math-container">$R=\sqrt{X^2+Y^2}$</span>. </p>
<p>So immediately we know the support is <span class="math-container">$\{r: 1\leqslant r... |
3,920,469 | <p>The topic of <a href="https://en.wikipedia.org/wiki/Perfect_number#Odd_perfect_numbers" rel="nofollow noreferrer">odd perfect numbers</a> likely needs no introduction.</p>
<p>The question is as is in the title:</p>
<blockquote>
<p>If <span class="math-container">$p^k m^2$</span> is an odd perfect number with special... | mathlove | 78,967 | <p>This is a partial answer.</p>
<blockquote>
<p>Does the inequality still hold when the constant <span class="math-container">$C > 1$</span>?</p>
</blockquote>
<p>Yes, one can get
<span class="math-container">$$\frac{\sigma(m^2)}{p^k} \le \frac{m^2 - p^k}{\color{red}{4/3}}\tag1$$</span></p>
<p>I don't know if this ... |
164,137 | <p>I want to animate a Point moving on an ellipse, but the angle I need to use is a numerical solution from an epression. How can I get mathematica to just take the value from the animated nummerical solution and use it as a variable for my pointfunction?
This is my Code and this way it does not work. I think I have to... | Coolwater | 9,754 | <pre><code>FindRoot[t, {t, 0}]
</code></pre>
<p>Returns <code>{t -> 0.}</code>. You need to extract the numeric value, e.g.</p>
<pre><code>t /. FindRoot[t, {t, 0}]
</code></pre>
<p>which returns <code>0.</code>. Inserting the needed <code>ψ /.</code> in your code makes it work:</p>
<pre><code>Animate[Show[curvE,... |
164,137 | <p>I want to animate a Point moving on an ellipse, but the angle I need to use is a numerical solution from an epression. How can I get mathematica to just take the value from the animated nummerical solution and use it as a variable for my pointfunction?
This is my Code and this way it does not work. I think I have to... | Bob Hanlon | 9,362 | <pre><code>ρ = 1; ε = 4/5; T = 10 π;
curvE = ParametricPlot[{ρ/(1 - ε^2) Cos[ψ], ρ/Sqrt[1 - ε^2] Sin[ψ]}, {ψ, 0, 2 π}];
sol[t_?NumericQ] := FindRoot[{ψ - ε Sin[ψ] == 2 π t/T}, {ψ, 0}]
Animate[
Show[
curvE,
Graphics[{PointSize[Large], Red,
Point[{ρ/(1 - ε^2) Cos[ψ], ρ/Sqrt[1 - ε^2] Sin[ψ]}]} /. sol[t]]],
... |
4,029,249 | <blockquote>
<p>Prove that the function <span class="math-container">$f(x)=e^x-(ax^2+bx+c)$</span> has 3 solutions at most .</p>
<p><span class="math-container">$a$</span>,<span class="math-container">$b$</span> and <span class="math-container">$c$</span> are constants.</p>
</blockquote>
<p>This is the information give... | Bob Dobbs | 221,315 | <p>I like this question but you must correct the statement of the question as we want to solve the equation <span class="math-container">$f(x)=0$</span> where <span class="math-container">$f(x)=e^x-ax^2-bx-c$</span>. And the claim is that it has at most <span class="math-container">$3$</span> roots or in other words <s... |
84,204 | <p>Say I have some object or quantity and an instance or special case of it, how to formally write this down? </p>
<p>I don't (just) mean that $X$ is a set and $x$ an element, i.e. $x\in X$ is not it. I'm dealing with things as general like "<em>the specific group $g$ is a group/is a case of a group</em>". Or "<em>the... | hmakholm left over Monica | 14,366 | <p>If I understand the question correctly, you want to know how to write down a formal formula that says that $g$ is a group?</p>
<p>Usually one would just introduce a predicate specifically for saying this, so your formula would be
$$\mathrm{Group}(g)$$
The definition of this predicate would be as an abbreviation of<... |
3,453,483 | <p>I'm reading Serre's <span class="math-container">$\textit{A course in Arithmetic}$</span> where he defines a Dirichlet series to be an infinite sum of the form
<span class="math-container">$$f(z) = \sum\limits_{n=1}^{\infty} a_ne^{-\lambda_nz}
$$</span>
where <span class="math-container">$\lambda_n$</span> is an in... | Henry Crawford | 274,113 | <p>Let <span class="math-container">$d(z) = \sum\limits_{n} a_n e^{-\lambda_n z}$</span> be a Dirichlet series converging in some non-empty half plane <span class="math-container">$H$</span>.
Proposition <span class="math-container">$6$</span> on page <span class="math-container">$66$</span> of the book mentioned impli... |
3,258,372 | <p>I'm doing a practice exam questions and am stuck at this question:</p>
<blockquote>
<p>Are there topological spaces X,Y (each with more than one point), such that [0,1] is homeomorphic to X×Y? What if we replace [0,1] with R?</p>
</blockquote>
<p>I'm not even sure how to start tackle it, any help and clues will ... | José Carlos Santos | 446,262 | <p><strong>Hints:</strong> Prove that:</p>
<ul>
<li>If <span class="math-container">$X\times Y$</span> is homeomorphic to <span class="math-container">$[0,1]$</span>, then both <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> are connected.</li>
<li>If <span class="math-container">$x... |
3,863,495 | <p>This is my solution to an old exam problem that I'd appreciate some feedback on. The problem:</p>
<blockquote>
<p>Let <span class="math-container">$f:[0,\infty)\to\mathbb{R}$</span>, <span class="math-container">$f\geq 0$</span> and <span class="math-container">$\int _0^{\infty} f(x) dx=L<\infty;$</span> that is,... | RRL | 148,510 | <p>For a very general approach, we have <span class="math-container">$f \in L^1([0,\infty))$</span> since <span class="math-container">$f$</span> is nonnegative and improperly Riemann integrable.</p>
<p>Thus, by the dominated convergence theorem,</p>
<p><span class="math-container">$$\lim_{R \to \infty} \frac{1}{R}\int... |
31,480 | <p>I'm having difficulty with my math, fractions and up. I used to understand it all, but it's been so long since I've touched the book (I finished it a couple of months ago, picked it up to review everything), I seem to have forgotten it. </p>
<p>The explanations inside of the individual chapters do no good. They nev... | JonnyQuiznos | 9,968 | <p>Agreed w/ Khan Academy, but there are videos out there that teach you essential Calculus topics in less than several minutes. </p>
|
116,634 | <p>Let $G$ be a graph and $e$ be an edge of the graph $G$ such that the subgraph $G\setminus e$
is connected. The subgraph $G\setminus e$ is the subgraph of $G$ obtained by deletion of the edge $e$ of $G$.
Assume that $G$ has $n$ vertices.
Is it true that $\lambda(G)-\lambda(G\setminus e)\geq \frac{1}{n}$?
Here $\lamb... | Chris Godsil | 1,266 | <p>It's always useful to test these questions on actual examples. The largest eigenvalue of the cycle $C_n$ is 2 and the largest eigenvalue of the path $P_n$ on $n$ vertices is $2\cos(\pi/(n+1))$. When $n=8$, this is 1.879385 and $2-1.8793852=0.120615 <1/8$.</p>
<p>In fact it is not hard to see that for large $n$ t... |
116,634 | <p>Let $G$ be a graph and $e$ be an edge of the graph $G$ such that the subgraph $G\setminus e$
is connected. The subgraph $G\setminus e$ is the subgraph of $G$ obtained by deletion of the edge $e$ of $G$.
Assume that $G$ has $n$ vertices.
Is it true that $\lambda(G)-\lambda(G\setminus e)\geq \frac{1}{n}$?
Here $\lamb... | Shahrooz | 19,885 | <p>This conjecture is not true in general. For example, let $G$ be a graph that obtained from joining the end vertex of $P_3$, $P_3$ and $P_4$, where it is an star-like tree. This graph has largest eigenvalue equal to $2.02852$. Now join the end vertex of $P_4$ to the end vertex of $P_3$. Therefore we added an edge to ... |
80,456 | <p>Given an array <code>sel</code> and an index position <code>i0</code>, how can I find the position of the nearest (left or right) nonzero element?
I'm able to do it with a loop and a couple of awful If's, but I was looking for a functional way...</p>
<pre><code> lr=Length[sel];
For[i = 0, i <= lr, i++,
... | kglr | 125 | <pre><code>nrstNZP[l_] := With[{nF = Nearest[Flatten@SparseArray[l]["NonzeroPositions"]]},
With[{nrst = nF[#, 2]}, DeleteCases[nrst, #][[1]]] & /@ #] &
</code></pre>
<p>Example:</p>
<pre><code>SeedRandom[1]
sel = RandomInteger[{0, 2}, 20]
(* {1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 1, 2, 0, 0, 1, 1} ... |
1,212,254 | <p>I am trying to find all pairs of side integers (a, b) for a given hypothenuse number n so
that (a, b, n) is a Pythagorean triple, i.e.,$ a^2 + b^2 = n^2$</p>
<p>The approach i am using is </p>
<ol>
<li>Sorting the array in ascending order </li>
<li>Finding the square of each element in array</li>
</ol>
<p>for(j->... | Tim Raczkowski | 192,581 | <p>Hint: Let $N=\{n\in\Bbb N: x^n\notin\alpha\}$. This is non-empty, because $x>1$, so let $n+1=\min N$.</p>
|
1,212,254 | <p>I am trying to find all pairs of side integers (a, b) for a given hypothenuse number n so
that (a, b, n) is a Pythagorean triple, i.e.,$ a^2 + b^2 = n^2$</p>
<p>The approach i am using is </p>
<ol>
<li>Sorting the array in ascending order </li>
<li>Finding the square of each element in array</li>
</ol>
<p>for(j->... | CIJ | 159,421 | <p>Even though I asked (and solved) this question a lot of time ago, here is a solution for future references.</p>
<p>Case I. $\alpha=1^*.$ This case is trivial since $x^{-1}\in\alpha$ and $x^0\notin\alpha.$</p>
<p>Case II. $\alpha\subsetneq1^*.$ Let $a\in\alpha\cap\mathbb Q_{>0}$ and let $m$ and $n$ be positive i... |
198,521 | <p>I am trying to overlap two <code>Graphics</code> objects <code>g1</code> and <code>g2</code> with <code>Show</code>. However, I found that when the coordinates of each object is defined to quite different ranges, I need to "scale" and "shift" the coordinates of one object to get the desired look. </p>
<p>For exampl... | C. E. | 731 | <p>I'm not saying that this is what you need for your problem, but sometimes when you want to superimpose graphics like this, you're looking for <code>Inset</code>:</p>
<pre><code>Show[
g1,
Graphics@Inset[
Show[g2, PlotRangePadding -> 0],
{0, 0}, {Center, Center}, {2, 2}
],
Axes -> True, ImageSize -&... |
495,622 | <p>Well, it may seem trivial, but I cannot find it on google. Is a constant function continuously differentiable, of all orders?</p>
<p>Thank you.</p>
| Umberto P. | 67,536 | <p>Yes. $f'$ and all higher derivatives are identically equal to zero.</p>
|
229,558 | <p>When we say that a set $S$ is denumerable, that is, there is a bijection $S \to \omega$, do we mean that there <em>exists</em> such a bijection or do we mean that we have one and are talking about a pair $(S,f)$?</p>
<p>I'm asking because it makes a difference to whether I need choice in some proofs or whether I do... | Ittay Weiss | 30,953 | <p>Denumerable means there exists a bijection between the given set and the set $\mathbb {N}$. This indeed, as you point out, creates subtleties for certain proofs. Further to your comment about the countable union of countable sets being countable, it can be shown that without countable choice this result is false. Th... |
2,586,618 | <p>I'm trying to study for myself a little of Convex Geometry and I have some doubts with respect the proof of the Theorem 1.8.5 of the book Convex Bodies: The Brunn-Minkowski Theory. Before I presented the proof and my doubts, I will put the definitions used in the theorem below.</p>
<p><span class="math-container">$\... | Community | -1 | <p>Your parametrization is correct. You missed the jacobian factor during the change of variable. $\int_{-2}^{2} \int_{0}^{3\sqrt{1-x^2/4}}-2dxdy=-2.\frac{1}{2}$area of allipse=$-6\pi$
($\because \frac{x^2}{a^2}+\frac{y^2}{b^2}$, area enclosed by the closed ellipse=$\pi a b$)</p>
|
1,428,905 | <p>I have two functions:</p>
<p>$n!$</p>
<p>$2^{n^{2}}$</p>
<p>What is the difference between the growth of these two? My thought is that $2^{n^2}$ grows much faster than $n!$. </p>
| bof | 111,012 | <p>$$\frac{2^{n^2}}{n!}=\frac{2^n}1\cdot\frac{2^n}2\cdot\frac{2^n}3\cdot\cdots\cdot\frac{2^n}n\rightarrow\infty$$</p>
|
10,949 | <p>Is it known whether every finite abelian group is isomorphic to the ideal class group of the ring of integers in some number field? If so, is it still true if we consider only imaginary quadratic fields?</p>
| Arturo Magidin | 742 | <p>At least as of 1999, this was still an open question, according to the MathReview of a paper by Marc Perret (<em>On the ideal class group problem for global fields</em>, J. Number Theory <strong>77</strong> (1999), no. 1, pages 27-35; MR1695698 (2000d:11135), review by Bruno Anglès). </p>
<p>Luther Claborn proved i... |
1,673,771 | <p>I was wondering if this proof is valid. </p>
<p>I use $[x]$ to denote the floor of $x$.</p>
<p><strong>Problem</strong> </p>
<p>Prove that</p>
<p>$$[mx] = \sum_{k=0}^{m-1} \, \bigg[x+\frac{k}{m} \bigg]$$</p>
<p>where $m \in \mathbb{N}$ and $x \in \mathbb{R}$.</p>
<p><strong>Proof</strong></p>
<p>Let $m \in \m... | heropup | 118,193 | <p>I think it's a good effort but your proof is difficult to follow due to some issues with presentation and definition. For example, the partition of $[0,1) = \bigcup_{k=1}^m [k/m, (k+1)/m)$ is problematic because it obviously fails to contain $0$. Also, if you say "let $p = 1, 2, \ldots, m$ and consider the $p^{\rm... |
379,893 | <p>Define the sequence <span class="math-container">$b_1=1$</span> and
<span class="math-container">$$b_n=\sum_{k=1}^{n-1}\binom{n-1}k\binom{n-1}{k-1}b_kb_{n-k}.$$</span></p>
<p>By now, there is enough in the literature that <span class="math-container">$C_n$</span> is odd iff <span class="math-container">$n=2^k-1$</sp... | David Kern | 165,619 | <p>Multicategories and bicategories, to me, are first of all completely
orthogonal generalisations of <em>monoidal</em> categories, with virtual
double categories as a common generalisation of multicategories and
(strict) <span class="math-container">$2$</span>-categories (they are to multicategories as categories are
... |
1,269,447 | <p>I consider the space $C^1[a, b]$ of (complex) functions that are at least once differentiable on $[a, b]$. I want to show that</p>
<p>$$||f||_{C^1} := ||f||_\infty + ||f'||_\infty$$</p>
<p>defines a norm on $C^1[a, b]$.</p>
<p>Now it's easy to see that $||f||_{C^1}$ is non-negative, and that it's zero iff f = 0, ... | minthao_2011 | 40,146 | <p>Solution.</p>
<p><strong><em>First way</em></strong></p>
<p>From the first equation, we have
$$\begin{cases}
2x^2\leqslant 1,\\
y^2 \leqslant 1
\end{cases}
\Leftrightarrow
\begin{cases}
-\dfrac{1}{\sqrt{2}} \leqslant x \leqslant \dfrac{1}{\sqrt{2}},\\
- 1 \leqslant y \leqslant 1.
\end{cases}$$
Then, the conditio... |
4,616,048 | <p>How can I solve this system of coupled differential equations?</p>
<p><span class="math-container">$\frac{d^2\rho}{d\lambda^2}=\frac{5\rho}{(5\rho^2+4t^2)^2}$</span> <span class="math-container">$\frac{d^2t}{d\lambda^2}=\frac{4t}{(5\rho^2+4t^2)^2}$</span></p>
<p>Is it something I could input in the Wolfram calculato... | Hans Lundmark | 1,242 | <p>I hope you'll excuse me if I write <span class="math-container">$x(t)$</span> and <span class="math-container">$y(t)$</span> instead of <span class="math-container">$\rho(\lambda)$</span> and <span class="math-container">$t(\lambda)$</span>.</p>
<p>The system has the form
<span class="math-container">$$
\begin{pmatr... |
160,801 | <p>Here is a vector </p>
<p>$$\begin{pmatrix}i\\7i\\-2\end{pmatrix}$$</p>
<p>Here is a matrix</p>
<p>$$\begin{pmatrix}2& i&0\\-i&1&1\\0 &1&0\end{pmatrix}$$</p>
<p>Is there a simple way to determine whether the vector is an eigenvector of this matrix?</p>
<p>Here is some code for your conven... | Henrik Schumacher | 38,178 | <p>Yes! We just check whether $h.y = (u + I v) y$ holds for some real $u, v \in \mathbb{R}$.</p>
<pre><code>h = {{2, I, 0}, {-I, 1, 1}, {0, 1, 0}};
y = {I, 7 I, -2};
expr = Norm[h.y - (u + I v) y, 2]^2 // ComplexExpand;
Minimize[expr, {u, v}]
</code></pre>
<blockquote>
<p>{623/6, {u -> 17/18, v -> 0}}</p>
</blockqu... |
2,080,042 | <blockquote>
<p>I am interested of finding examples of non-zero homomorphisms $f:R\to S$ of rings with unity such that $f(1_R)\neq 1_S$.</p>
</blockquote>
<p>I will provide one example and I will be glad if others can also give examples. </p>
| Mustafa | 400,050 | <p>Let be $f:\mathbb Z \to M_2(\mathbb Z); f(a)= \left( \array{a&0\\0&0}\right) $ then $f$ is ring homomorphism and $f(1)=\left(\array{1&0\\0&0}\right) \ne \left(\array{1&0\\0&1}\right)$ . </p>
|
725,602 | <p>I am trying to prove the 'second' triangle inequality:
$$||x|-|y|| \leq |x-y|$$</p>
<p>My attempt:
$$----------------$$
Proof:
$|x-y|^2 = (x-y)^2 = x^2 - 2xy + y^2 \geq |x|^2 - 2|x||y| + |y|^2 = (||x|-|y||)^2$</p>
<p>Therefore $\rightarrow |x-y| \geq ||x|-|y||$</p>
<p>$$----------------$$</p>
<p>My questions are... | user137500 | 137,500 | <p>Induction will work fine. Assuming it works for $n$, we have $$F_{n+2}F_n-F_{n+1}^2=(F_{n+1}+F_n)F_n-F_{n+1}^2=F_{n+1}(F_n-F_{n+1})+F_n^2=F_{n+1}(-F_{n-1})+F_n^2$$ $$=-\left(F_{n+1}F_{n-1}-F_n^2\right)=-(-1)^n=(-1)^{n+1}$$</p>
|
1,591,371 | <p>If we start with $n$ elements and at each step split them into $2$ parts randomnly and repeat with both sub-parts until parts of only $1$ element are left, in how many different ways can these elements be separated?
I made a mistake we don't split them in half we split them in a random place.</p>
| true blue anil | 22,388 | <p>Imagine a log of wood which you want to cut into $n$ parts.</p>
<p>A little thought will show that you need to make $(n-1)$ cuts.</p>
<p>These cuts can be in any order, thus # of ways to do it = $(n-1)!$</p>
<p>e.g. if $n = 4,$ you need $(4-1)! = 3\times2\times1 = 6$</p>
|
2,128,380 | <p>Okay, I must admit that I am lost on how to do this. I have looked up videos and tutorials about this, and they helped a little. The main thing is that my professor asked for us to solve this without using the "determinant method." I have just started linear algebra, so I am still trying to understand determinants a... | Reinhard Meier | 407,833 | <p>The proof can be given using the distributive property of the cross product and the fact that $c(v\times w) = (cv)\times w = v\times (cw)$ for vectors $v$ and $w$ and a scalar $c$:
$$
A\times B = (A_x \hat i + A_y \hat j + A_z \hat k)\times (B_x \hat i + B_y \hat j + B_z \hat k)\\
= A_xB_x(\hat i\times\hat i)+A_xB_y... |
125,165 | <p>Hi friends,</p>
<p>I have some questions concerning the critical values of motives, in the sense of Deligne. I will only look at motives of the form $h^i(X)$ where $X$ is a smooth projective algebraic variety over $\mathbb{Q}$. If I understand correctly, the notion of critical value depends only on the Hodge number... | David Loeffler | 2,481 | <p>In your example (with Hodge structure of weight 3 concentrated in bidegrees (2, 1) and (1, 2)) there is only one critical value, at s = 2. At all other integer values of $s$, either $L_\infty(M, s)$ or $L_\infty(M^\vee, 1-s)$ will have a pole.</p>
|
365,986 | <p>If $A$ is an $n \times n$ matrix with $\DeclareMathOperator{\rank}{rank}$ $\rank(A) < n$, then I need to show that $\det(A) = 0$.</p>
<p>Now I understand why this is - if $\rank(A) < n$ then when converted to reduced row echelon form, there will be a row/column of zeroes, thus $\det(A) = 0$</p>
<p>However, I... | Abel | 71,157 | <p>Here goes the sketch of a proof.</p>
<p>Let $A$ be your matrix. Do Gaussian elimination on $A$ so that you end up with an upper triangular matrix $U$.
During this process, you either</p>
<p>1) Switch rows, which switches the sign of the determinant.</p>
<p>2) Add a multiple of a row to another row, which preserve... |
1,129,712 | <p>So I'm complete stuck with something. I know it the following statements are true (or at least the seem to be from the results that I got from messing around with it a bit on MATLAB), but I don't understand why they are true or how to show so.
Let $A$ be and $m$X$n$ matrix. Show that:</p>
<p>a) if $x \in N(A^TA)$ t... | Pablo Repetto | 79,624 | <p>Let us show that (c) and (d) are both consecuences of (b):</p>
<p>Let $A\in R^{nxm}$ and $nul(A) = nul(A^TA)$</p>
<blockquote>
<p>$ \dim nul(A)
= \dim nul(A^TA) $</p>
<p>$ n-rank(A) = n-rank(A^TA)$</p>
<p>$rank(A)=rank(A^TA)$</p>
</blockquote>
<p>and, if $A$ has linearly independent columns, $nul(A)=\... |
1,293,207 | <p>A ray of light travels from the point $A$ to the point $B$ across the border between two materials. At the first material the speed is $v_1$ and at the second it is $v_2$. Show that the journey is achieved at the least possible time when Snell's law: $$\frac{\sin \theta_1}{\sin \theta_2}=\frac{v_1}{v_2}$$ holds. </p... | Hagen von Eitzen | 39,174 | <p>The total travel time, which is the sum of travel times from $A$ to the surface and fromthe surface to $B$. The two part times are obtained as length divided by speed, the two lengths depend on the angles.</p>
|
1,556,645 | <p>I am new to the axiom of choice, and currently working my way through some exercises. I am struggling with the following exercise:</p>
<p><strong>Exercise -</strong>
Prove the Axiom of Choice (every surjective $f: X \to Y$ has a section) in the following two special cases:</p>
<ol>
<li>Y is finite</li>
<li>X is co... | Clément Guérin | 224,918 | <p>The countable axiom of choice refers to the case where $Y$ is countable, so it is kind of different.</p>
<p>Let us see how we could do this. Since $X$ is countable there is a bijection $\phi$ with $\mathbb{N}$. Now you have a well defined order on $X$ :</p>
<p>$$x\leq y \text{ by definition if } \phi(x)\leq \phi(y... |
148,185 | <p>Let $ X = \mathbb R^3 \setminus A$, where $A$ is a circle. I'd like to calculate $\pi_1(X)$, using van Kampen. I don't know how to approach this at all - I can't see an open/NDR pair $C,D$ such that $X = C \cup D$ and $C \cap D$ is path connected on which to use van Kampen. </p>
<p>Any help would be appreciated. Th... | Lilith | 114,997 | <p>You should definitely check out the Hatcher's Algebraic Topology book page 46. </p>
<p>It was very hard for me to imagine at first but $\mathbb{R}^3 - S^1$ deformation retracts onto $S^1 \vee S^2$ so just choose $S^1$ and $S^2$ for $C$ and $D$ respectively, since the space is formed as wedge product of two spaces, ... |
23,471 | <p>I'm trying to find an explanation for the different sizes I'm seeing for fonts added to graphics in different ways, and haven't yet located an easy to understand explanation. Here's a minimal example:</p>
<pre><code>Graphics[
{LightGray,
Rectangle[{0, 0}, {72, 72}],
Red,
Style[
Text["Hig", {0, 0... | geordie | 4,626 | <p>The <code>FontSize</code> refers to <a href="http://en.wikipedia.org/wiki/Point_%28typography%29" rel="noreferrer">printers points</a>, which by convention are 1/72 of an inch. So the red 'Hig' is fixed to an external font size and will not scale with the graphic. However, you are right to say that this font size do... |
2,604,093 | <p>I would like to study the convergence of the series:</p>
<p>$$\sum_{n=1}^\infty \frac{\log n}{n^2}$$</p>
<p>I could compare the generic element $\frac{\log n}{n^2}$ with $\frac{1}{n^2}$ and say that
$$\frac{1}{n^2}<\frac{\log n}{n^2}$$ and $\frac{1}{n^2}$ converges but nothing more about.</p>
| N. S. | 9,176 | <p><strong>Hint</strong> Apply the integral test, and integrate by parts.</p>
|
216,031 | <p>Using image analysis, I have found the positions of a circular ring and imported them as <code>xx</code> and <code>yy</code> coordinates. I am using <code>ListInterpolation</code> to interpolate the data:</p>
<pre><code>xi = ListInterpolation[xx, {0, 1}, InterpolationOrder -> 4, PeriodicInterpolation -> True,... | Cesareo | 62,129 | <p>After normalizing the data scaling conveniently we can proceed approximating the data points by a conic (ellipse). This can be done as</p>
<pre><code>f[p_List] := c1 p[[1]]^2 + c2 p[[2]]^2 + c3 p[[1]] p[[2]] + c4 p[[1]] + c5 p[[2]] + c6
data0 = Import["testShape.csv"];
factor = 0.01;
data = data0*factor;
xmax = Ma... |
1,811,028 | <p>I want to know which of the following methods is right for graphing $f(ax-b)$ from $f(x)$ and why:</p>
<p>Method 1. First Horizontally Translate it ($f(x)$) by $b$, then Horizontally Stretch/Compress it by $a$.</p>
<p>Method 2. First Horizontally Stretch/Compress it by $a$ then Horizontally Translate it by $b/a$.<... | Dietrich Burde | 83,966 | <p>A connected compact Lie group $G$ has a reductive Lie algebra $L=[L,L]\oplus Z(L)$, where $[L,L]$ is semisimple. Furthermore it has finite center $Z(G)$ if and only if $[G,G]=G$, i.e., if and only if $[L,L]=L$. It follows that a connected compact Lie group $G$ satisfies $[L,L]=L$ if and only if $Z(L)=0$, so that $L=... |
2,278,018 | <p>Let $\;\;\displaystyle \sum_{n=1}^\infty U_n\;$ be a divergent series of positive real numbers.</p>
<p>Then, show that the series $\;\displaystyle\sum_{n=1}^\infty \dfrac{U_n}{1+U_n}\;$ is divergent.</p>
<p>Is there is any easy method to prove it? </p>
| Rigel | 11,776 | <p>Assume, for the sake of contradiction, that $\sum_n \frac{U_n}{1+U_n}$ is convergent.</p>
<p>Then $\dfrac{U_n}{1+U_n} \to 0$, hence
$$U_n = \frac{\frac{U_n}{1+U_n}}{1-\frac{U_n}{1+U_n}}\to 0$$
so that there exists $N\in\mathbb{N}$ such that $0\leq U_n \leq 1$ for every $n\geq N$.</p>
<p>Since
$$
\frac{U_n}{1+U_n... |
552,474 | <p>If there are,
Are there unity <strong>(but not division)</strong> rings of this kind?
Are there non-unity rings of this kind?</p>
<p>Sorry, I forgot writting the non division condition.</p>
| André Nicolas | 6,312 | <p>Take the "polynomials" with integer coefficients in two non-commuting variables $x$ and $y$. If you don't want a unit, use even integers only.</p>
<p>A related example replaces integer coefficients by coefficients in $\mathbb{Z}_2$. </p>
|
4,429,162 | <p><a href="https://i.stack.imgur.com/9nrUn.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/9nrUn.png" alt="enter image description here" /></a></p>
<p>This is from Rambo's Math subject GRE book.</p>
<p>One solution to this problem is to note that the equation of the circle is <span class="math-conta... | Hussain-Alqatari | 609,371 | <p>This solution is not by me, but available in Charles Rambo book,</p>
<p><a href="https://i.stack.imgur.com/DxvRJ.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/DxvRJ.png" alt="enter image description here" /></a>
<a href="https://i.stack.imgur.com/hjqH0.png" rel="nofollow noreferrer"><img src="ht... |
2,023,130 | <p>Suppose I have directed graph $G=(V,E)$ s.t at least one of the following statements is always true:</p>
<ol>
<li>for every $v$ in $V$, v doesn't have any incoming edges.</li>
<li>for every $v$ in $V$, v doesn't have any outgoing edges.</li>
</ol>
<p>How can I prove that G is bipartite? I tried to think of a proof... | Brian M. Scott | 12,042 | <p>HINT: Let $V_0$ be the set of vertices with no incoming edge and $V_1$ the set of vertices with no outgoing edge.</p>
|
813,825 | <p>In strong Induction for the induction hypothesis you assume for all K, p(k) for k
<p>If for example I am working with trees and not natural numbers can I still use this style of proof?</p>
<p>For example if I want my induction hypothesis to be that p(k) for k < n where n is a node in the tree and everything sma... | Basil | 36,042 | <p>Sure you can. As André Nicolas says, this is called <em>structural induction</em>, and it applies to data types that are inductively defined, that is, the structure, or rather, the <em>construction</em> of their elements is given (a) by some <em>base cases</em>, and (b) by some <em>step cases</em> involving already... |
1,473,418 | <p>I have difficulties with this question : </p>
<p>Given the ODE named (1) : $$x'=y+\sin (x^2y)$$ $$y'=x+\sin(xy^2)$$</p>
<p>and the :</p>
<p><strong>Definition.</strong> A <em>Petal</em> is a solution $(x(t),y(t))$ that verifies $\displaystyle \lim _{t \to \pm \infty} (x(t),y(t)) =(0,0)$.</p>
<p>How can I show th... | Archis Welankar | 275,884 | <p>))U take two case for divisible by 5 case I last digit is 0 so last place. Fixed there 3 places can be filled in 6p3 ways then case 2 last place is 5 so 5 .5.4 solutions and more 29 solutions for at least 1 2 sixes</p>
|
2,983,877 | <p>I once saw a function for generating successively more-precise square root approximations, <span class="math-container">$f(x) = \frac{1}{2} ({x + \frac{S}{x}})$</span> where S is the square for which we are trying to calculate <span class="math-container">$\sqrt S$</span>. And the function works really well, generat... | Hussain-Alqatari | 609,371 | <p>Let me tell you how to differentiate a polynomial, multiply the exponent by the coefficint and reduce <span class="math-container">$1$</span> from the exponent.</p>
<p>For example, if <span class="math-container">$f(x)=5x^3+7$</span>, then <span class="math-container">$f'(x)=15x^2$</span> [we multiplied <span class... |
2,078,796 | <p>In a contest problem book, I found a reference to Newton's little formula that may be used to find the <em>nth</em> term of a numeric sequence. Specifically, it is a formula that is based on the differences between consecutive terms that is computed at each level until the differences match. </p>
<p>An example appl... | Daniel McLaury | 3,296 | <p>Starting with the second differences we have</p>
<ul>
<li>$28 = 28$,</li>
<li>$34 = 28 + 6$,</li>
<li>$40 = 28 + 6 + 6$</li>
<li>$46 = 28 + 6 + 6 + 6$$</li>
</ul>
<p>Now for the first differences we have</p>
<ul>
<li>$40 = 40$</li>
<li>$68 = 40 + 28$</li>
<li>$102 = 40 + 28 + 34 = 40 + 28 + (28 + 6)$</li>
<li>$14... |
1,472,314 | <p>How many even numbers less than 600 can be made from the digits: 3,3,4,8,9 with each only being used once. I can't figure out what to do for the 3rd case where 3 digits are needed</p>
| user247327 | 247,327 | <p>That looks straight forward. In order to be even, the number must end in 4 or 8. If it ends in 4 then the first 4 digits must be 3, 3, 8, 9. If there were 4 distinct digits there would be 4!= 24 different orders but because two of those digits are "3", each of those orders is exactly the same as another with just... |
1,472,314 | <p>How many even numbers less than 600 can be made from the digits: 3,3,4,8,9 with each only being used once. I can't figure out what to do for the 3rd case where 3 digits are needed</p>
| N. F. Taussig | 173,070 | <p>We wish to find how many even numbers less than $600$ can be formed from the digits $3, 3, 4, 8, 9$ if each digit is used at most once. </p>
<p>Since the number is even, the units digit of each number must be $4$ or $8$.</p>
<p><strong>One-digit numbers:</strong> The only possibilities are $4$ or $8$, giving us ... |
488,258 | <p>What are the last two digits of $11^{25}$ to be solved by binomial theorem like $(1+10)^{25}$?
If there is any other way to solve this it would help if that is shown too.</p>
| Harish Kayarohanam | 30,423 | <p>A short cut for finding last two digits of any power of a number that ends in 1 .</p>
<blockquote>
<p>Last two digits of</p>
<p><span class="math-container">$$(\ldots a1)^{\displaystyle\ldots x}$$</span></p>
<p><strong>TENS DIGIT</strong>: unit's digit of (<span class="math-container">$x$</span> <span class="math-co... |
3,196,797 | <p>Suppose <span class="math-container">$gcd(m,n)=1$</span>, and let <span class="math-container">$F :Z_n→Z_n$</span> be defined by <span class="math-container">$F([a])=m[a]$</span>. Prove that <span class="math-container">$F$</span> is an automorphism of the additive group <span class="math-container">$Z_n$</span>. I ... | Andreas Caranti | 58,401 | <p>To show that the map is injective and surjective is equivalent to showing that the map has a two-sided inverse.</p>
<p>The extended Euclidean algorithm yields that there are numbers <span class="math-container">$m', n'$</span> such that
<span class="math-container">$$
m m' + n n '= 1.
$$</span>
Consider the map <s... |
120,667 | <p>Let $V, W$ be two finite-dimensional vector spaces, $f: V\rightarrow W$ a linear map, and $U \subseteq W$ a vector subspace. I'm trying to show that $(f^{-1}(U))^0 = f^*(U^0)$, i.e. that the annihilator of the inverse image of $U$ is the image of the annihilator under the the dual $f^*$ of $f$. $(f^{-1}(U))^0 \supse... | Arturo Magidin | 742 | <p>I claim that $(f^*(U^0))^0\subseteq f^{-1}(U)$.</p>
<p>Let $\mathbf{v}\in (f^*(U^0))^0$. Then for every $\mathbf{x}\in f^*(U^0)$, we have $\langle \mathbf{v},\mathbf{x}\rangle = 0$. Therefore, for every $\mathbf{w}\in U^0$,
$$
\langle f(\mathbf{v}),\mathbf{w}\rangle = \langle \mathbf{v},f^*(\mathbf{w})\rangle
= 0... |
935,331 | <p>Previously, to integrate functions like $x(x^2+1)^7$ I used integration by parts. Today we were introduced to a new formula in class: $$\int f'(x)f(x)^n dx = \frac{1}{n+1} {f(x)}^{n+1} +c$$
I was wondering how and why this works. Any help would be appreciated. </p>
| Brightsun | 118,300 | <p>The reason the formula holds is that for the chain rule:
$$
\left(\frac{1}{n+1}f^{n+1}(x)\right)' =\frac{1}{n+1}(n+1)f^n(x) f'(x) = f^n(x) f'(x)
$$
and this shows your identity by definition of indefinite integral as anti-derivative.</p>
<p>However, applying integration by parts to the same problem we have:
$$
\int... |
119,506 | <p>Let $\kappa$ be a singular cardinal, and let $\langle \kappa_i \mid i<\mathrm{cf}(\kappa) \rangle$ be an increasing sequence of regular cardinals cofinal in $\kappa$. Recall that a scale on $\Pi_{i<\mathrm{cf}(\kappa)} \kappa_i$ is a sequence $\langle f_\alpha \mid \alpha < \kappa^+ \rangle$ such that:</p>
... | Eran | 10,708 | <p>Shelah's Dichotmoy theorem (see <a href="http://math.cmu.edu/~sunger/PCFtalk1.pdf" rel="nofollow">link text</a>) says more or less that both options are valid. Every increasing sequence can either:</p>
<p>a) Have an exact upper bound (in which case your condition fails.)</p>
<p>b) Or have an "interleaved cofinal s... |
987,620 | <p>$P$ and $Q$ are two distinct prime numbers. How can I prove that $\sqrt{PQ}$ is an irrational number?</p>
| Alexander Berliner | 634,134 | <p><strong>Proof:</strong> Assume, to the contrary, that <span class="math-container">$\sqrt{pq}$</span> is rational. Then <span class="math-container">$\sqrt{pq}=\frac{x}{y}$</span> for two integers <span class="math-container">$x$</span> and <span class="math-container">$y$</span> and we further assume that <span cla... |
33,330 | <p>Let $p$ be a complex number. Let $ z_0 = p $ and, for $ n \geq 1 $, define $z_{n+1} = \frac{1}{2} ( z_n - \frac{1}{z_n}) $ if $z_n \neq 0 $. Prove the following:</p>
<p>i) If $ \{ z_n \} $ converges to a limit $a$, then $a^2 + 1 = 0 $</p>
<p>ii) If $ p $ is real, then $ \{ z_n \} $, if defined, does not converge</... | Samrat Mukhopadhyay | 83,973 | <p>From the recursion relation we get, $$\left(\frac{z_{n+1}+i}{z_{n+1}-i}\right)=\left(\frac{z_{n}+i}{z_{n}-i}\right)^2, \forall n\geq1 $$ So iteratively, we get, $$\left(\frac{z_{n+1}+i}{z_{n+1}-i}\right)=\left(\frac{p+i}{p-i}\right)^{2^{n+1}}$$ Rearranging, we get, $$z_{n+1}=i\frac{1+w^{2^{n+1}}}{1-w^{2^{n+1}}}$$ wh... |
302,243 | <p>Let $f:[0,1]\to\mathbb{R}$ be a Lipschitz function, and $\pi f$ be its piecewise linear interpolant on an equispaced grid with $n$ points.</p>
<p>It should be true (if I am not making mistakes with the constant) that
$$
\int_0^1 |f - \pi f| \leq \frac{1}{4n} \operatorname{Lip}(f).
$$</p>
<p>Do you have a reference... | Will Sawin | 18,060 | <p>Let $E$ be the elliptic curve. Let $E_1$ and $E_2$ be two different double covers of $E$, with $E = E_1 /x_1$ and $E=E_2/x_2$ for two-torsion points $x_1,x_2$. </p>
<p>Let $M$ be the minimal resolution of singularities of $ E_1 \times E_2 /\langle (a,b) \to (-a,-b), (a,b) \to (a+ x_1, x_2-b) \rangle $.</p>
<p>The... |
615,093 | <p>How to prove the following sequence converges to $0.5$ ?
$$a_n=\int_0^1{nx^{n-1}\over 1+x}dx$$
What I have tried:
I calculated the integral $$a_n=1-n\left(-1\right)^n\left[\ln2-\sum_{i=1}^n {\left(-1\right)^{i+1}\over i}\right]$$
I also noticed ${1\over2}<a_n<1$ $\forall n \in \mathbb{N}$.</p>
<p>Then I wrote... | Tim Ratigan | 79,602 | <p>EDIT: I feel kind of stupid for not thinking of the easier ways in other posts, but I think this method is kind of cool.</p>
<p>I apologize in advance, this is a lot of math and few words.</p>
<p>$$\begin{align} \int_0^1\frac{nx^{n-1}}{1+x}\text dx&=\int_0^1nx^{n-1}\sum_{k=0}^\infty(-x)^k\text dx\\
&=\sum_... |
615,093 | <p>How to prove the following sequence converges to $0.5$ ?
$$a_n=\int_0^1{nx^{n-1}\over 1+x}dx$$
What I have tried:
I calculated the integral $$a_n=1-n\left(-1\right)^n\left[\ln2-\sum_{i=1}^n {\left(-1\right)^{i+1}\over i}\right]$$
I also noticed ${1\over2}<a_n<1$ $\forall n \in \mathbb{N}$.</p>
<p>Then I wrote... | Avi Steiner | 13,487 | <p>Male the u-substitution $u=x^n$, them apply the dominated convergence theorem. </p>
|
615,093 | <p>How to prove the following sequence converges to $0.5$ ?
$$a_n=\int_0^1{nx^{n-1}\over 1+x}dx$$
What I have tried:
I calculated the integral $$a_n=1-n\left(-1\right)^n\left[\ln2-\sum_{i=1}^n {\left(-1\right)^{i+1}\over i}\right]$$
I also noticed ${1\over2}<a_n<1$ $\forall n \in \mathbb{N}$.</p>
<p>Then I wrote... | iballa | 116,491 | <p>Thinking about the graph of $x^n$ on $[0,1]$ we observe that it stays near $0$ and then sharply jumps to $1$. As such, it makes sense to break up the integral into $[0,c)$ and $[c,1]$ (for some $c$ to be chosen later).</p>
<p>$$
a_n = \int_0^c{\frac{n x^{n-1}}{x+1}dx} + \int_c^1{\frac{n x^{n-1}}{x+1}dx} \leq \int_... |
834,508 | <p>Show that for any natural number $n$, between $n^2$ and $(n+1)^2$ one can find three distinct natural numbers $a,b,c$ such that $a^2+b^2$ is divisible by $c$.</p>
<p>A friend and I found a general case that always work with a computer problem, I would like to see a different solution, or a solution that tells the m... | Simply Beautiful Art | 272,831 | <p>A simple straightforward algorithm starts from <span class="math-container">$0$</span> and repeatedly adds numbers onto pre-made numbers until a streak is found. For example, consider <span class="math-container">$3$</span> and <span class="math-container">$5$</span>.</p>
<p>We start off with <span class="math-cont... |
1,762,522 | <p>This question was asked by one of my Professor during the class. I'm getting intuition that these functions should be one-one (I'm wrong maybe). But, I'm unable to classify all such functions.</p>
<p>Please help in this!!</p>
| Siddharth Bhat | 261,373 | <p>What have you tried so far? </p>
<p>And, as for being one-to-one, can you think of examples where a function maps, say, every open interval to $R$? what would such a function look like? is it one-one? onto?</p>
|
1,762,522 | <p>This question was asked by one of my Professor during the class. I'm getting intuition that these functions should be one-one (I'm wrong maybe). But, I'm unable to classify all such functions.</p>
<p>Please help in this!!</p>
| Brian M. Scott | 12,042 | <p>HINT: Not all bijections work, but you should be able to prove that the monotone (increasing or decreasing) bijections do. You should also be able to prove that these are precisely the continuous bijections. Going a bit further, try to prove that if $f:\Bbb R\to\Bbb R$ has an absolute maximum or minimum on some open... |
1,117,458 | <p>How do I integrate an expression of the form
$$
\frac{f'(x)}{[f(x)]^n}
$$
with respect to $x$?</p>
<p>Could I use some kind of recognition method, thus avoiding partial fractions?</p>
<p>For example:
$$
\frac{(2x+1)}{(x^2+x-1)^2}
$$</p>
| lab bhattacharjee | 33,337 | <p>$$S=\int\frac{f'(x)}{f^n(x)}dx=\int\dfrac{d[f(x)]}{f^n(x)\cdot dx}dx=\int f^{-n}(x)\ d[f(x)$$</p>
<p>For $-n+1\ne0\iff n\ne1$</p>
<p>$$S=\frac{f^{-n+1}(x)}{-n+1}+K$$</p>
<p>For $n=1,$</p>
<p>$$S=\int\frac{d[f(x)}{f(x)}=\ln|f(x)|+C$$</p>
|
1,117,458 | <p>How do I integrate an expression of the form
$$
\frac{f'(x)}{[f(x)]^n}
$$
with respect to $x$?</p>
<p>Could I use some kind of recognition method, thus avoiding partial fractions?</p>
<p>For example:
$$
\frac{(2x+1)}{(x^2+x-1)^2}
$$</p>
| Martingalo | 127,445 | <p>Chain rule. You know that
$$\frac{d}{dx}[f(x)]^n = n [f(x)]^{n-1} f'(x).$$</p>
<p>Then $$\int [f(x)]^m f'(x) dx = \frac{[f(x)]^{m+1}}{m+1} + C$$
Just set $m =-n$. For the case $n=1$ you get the logarithm, that is
$$\int \frac{f'(x)}{f(x)} dx = \log f(x) + C$$</p>
<p>In Your example the numerator is the derivative ... |
2,530,820 | <p>Let $1\leq p<\infty$ and $q$ be the conjugate exponent of $p$. Suppose that $\lbrace a^n \rbrace_{n=1}^{\infty} \subset \ell^q$ in a sequence in $\ell^q$ such that $f_{a^n}(x) \mapsto 0~( n \mapsto \infty)$ for all $x \in \ell^p$ where $f_{a^n}(x)=\sum_{i=1}^{\infty} a_i^{(n)}x_i$. Show that the sequence $\lbrace... | Matematleta | 138,929 | <p>If $\lbrace a^n \rbrace_{n=1}^{\infty} \subset l^q$ and $x\in l^p$, consider $J(a^n)(x)=x(a^n)=\sum_{i=1}^{\infty} a_i^{(n)}x_i.$ Since the series converges to $0,$ each element of $\left \{ J(a^n) \right \}$ is pointwise bounded, so by Banach-Steinhaus, there is an $M<\infty $ such that $\|J(a^n)\|<M$ for ... |
15,480 | <p>Say I have two lists,</p>
<pre><code>list1 = {a, b, c}
list2 = {x, y, z}
</code></pre>
<p>and I want to map a function f over them to produce</p>
<pre><code>{f[a,x], f[a,y], f[a,z], f[b,x], f[b,y], f[b,z], f[c,x], f[c,y], f[c,d]}
</code></pre>
<p>I would assume I map the function over the first list to produce a... | MinHsuan Peng | 1,376 | <p><code>Distribute</code> is also handy.</p>
<p>Assuming <code>f</code> is not <code>Listable</code>:</p>
<pre><code>In[39]:= Distribute[f[{a, b, c}, {x, y, z}], List]
Out[39]= {f[a, x], f[a, y], f[a, z], f[b, x], f[b, y], f[b, z],
f[c, x], f[c, y], f[c, z]}
</code></pre>
|
2,831,199 | <p>What is the probability of getting $6$ $K$ times in a row when rolling a dice N times?</p>
<p>I thought it's $(1/6)^k*(5/6)^{n-k}$ and that times $N-K+1$ since there are $N-K+1$ ways to place an array of consecutive elements to $N$ places.</p>
| Michael Burr | 86,421 | <p>This is really just a long comment, but I thought that it would be a mess to write it as a comment:</p>
<p>Please clarify your question with, at least, the following data (some of the answers are in the comments):</p>
<ol>
<li><p>Does the run of $6$'s need to be the only $6$'s in the string? If $k=2$, would $6656... |
3,523,205 | <p>The given series of function is as follow</p>
<blockquote>
<p><span class="math-container">$$\sum_{n=1}^\infty x^{n-1}(1-x)^{2}$$</span>
prove that given series is uniformaly convergent on <span class="math-container">$[0,1]$</span></p>
</blockquote>
<p><strong>The solution i tried</strong>-The given series fo... | Peter Szilas | 408,605 | <p>Option:</p>
<p>Weierstrass M test .</p>
<p><span class="math-container">$f_n (x)=x^{n-1}(1-x)^2$</span>;</p>
<p><span class="math-container">$f_n'(x)=$</span></p>
<p><span class="math-container">$(n-1)x^{n-2}(1-x)^2-2x^{n-1}(1-x)=0;$</span></p>
<p><span class="math-container">$(n-1)(1-x)-2x=0$</span>;</p>
<p><... |
1,439,850 | <p>So the problem states that the centre of the circle is in the first quadrant and that circle passes through $x$ axis, $y$ axis and the following line: $3x-4y=12$. I have only one question. The answer denotes $r$ as the radius of the circle and then assumes that centre is at $(r,r)$ because of the fact that the circl... | MrYouMath | 262,304 | <p>Rewrite </p>
<p>$$y'=-3\frac{y}{x}+\frac{1}{x^2}$$</p>
<p>Solving the homogenous equation is pretty eays (e.g. using method of separation) $$y_h=c_1x^{-3}$$</p>
<p>Then you guess (is faster for easy ODEs) a particular solution of the form $y_p=\frac{k}{x}$. Plug this into the equation and find k. </p>
<p><strong... |
1,439,850 | <p>So the problem states that the centre of the circle is in the first quadrant and that circle passes through $x$ axis, $y$ axis and the following line: $3x-4y=12$. I have only one question. The answer denotes $r$ as the radius of the circle and then assumes that centre is at $(r,r)$ because of the fact that the circl... | Kwin van der Veen | 76,466 | <p>When you have a first order non-linear and/or non-autonomous differential equation you could always try testing if it is an <a href="http://mathworld.wolfram.com/ExactFirst-OrderOrdinaryDifferentialEquation.html" rel="nofollow">exact differential equation</a></p>
<p>$$
p(x,y) + q(x,y) \frac{dy}{dx} = 0,
$$</p>
<p>... |
1,821,186 | <p>Why is the solution of $|1+3x|<6x$ only $x>1/3$? After applying the properties of modulus, I get $-6x<1+3x<6x$. And after solving each inequality, I get $x>-1/9$ and $x>1/3$, but why is $x>-1/9$ rejected? </p>
| cQQkie | 70,348 | <p>Well, take a dense subset of $D\subset X$ and then the metric guarantees you a countable neighbourhood basis at every point of $D$. A countable union of countable sets is again countable. Check that this is enough.</p>
|
744,034 | <p>How do I show that for all integers $n$, $n^3+(n+1)^3+(n+2)^3$ is a multiple of $9$?
Do I use induction for showing this? If not what do I use and how? And is this question asking me to prove it or show it? How do I show it? </p>
| Community | -1 | <p>$$n^3+(n+1)^3+(n+2)^3=n^3+n^3+3n^2+3n+1+n^3+6n^2+12n+8=\\
3n^3+9n^2+15n+9=3(n^3+3n^2+5n+3)$$
Thus, this amounts to showing that $n^3+3n^2+5n+3$ is divisible by $3$ for all $n$. Obviousl, the terms $3n^2,3$ are divisible by $3$. Thus, we need to check that $\forall n,n^3+5n$ is divisible by $3$. Let's proceed by indu... |
2,343,958 | <p>I am interested in a mathematical approach to quantum information theory. I have observed that several probabilists have been working in this area. What can be a suitable background and good book for this subject?</p>
| user 1987 | 243,227 | <p>I myself started learning quantum information theory with Quantum Computing by J. Gruska and Quantum Computation and Information Theory by Nielsen-Chuang. I think the former is more mathematical. It covers very well Hilbert spaces and related notions. I like very much the treatment of observalbles and measurements. ... |
2,717,821 | <p>Since we have 4 digits there is a total of 10000 Password combinations possible.</p>
<p>Now after each trial the chance for a successful guess increases by a slight percentage because we just tried one password and now we remove that password from the "guessing set". That being said I am struggling with the actual ... | Bram28 | 256,001 | <p>Since you're trying $3$ different PINS, the chances of one of them being correct is $$\frac{3}{10000}$$</p>
|
2,717,821 | <p>Since we have 4 digits there is a total of 10000 Password combinations possible.</p>
<p>Now after each trial the chance for a successful guess increases by a slight percentage because we just tried one password and now we remove that password from the "guessing set". That being said I am struggling with the actual ... | Ross Millikan | 1,827 | <p>The simple approach is that there are $10000$ possible PINs and you have tried $3$ of them, so your chance of finding the right one is $\frac 3{10000}$. In your calculation, the denominators should decrease $10000,9999,9998$, so you will get the same result.$$1-\frac {9999}{10000}\cdot \frac {9998}{9999}\cdot\frac{... |
58,870 | <p>I am teaching a introductory course on differentiable manifolds next term. The course is aimed at fourth year US undergraduate students and first year US graduate students who have done basic coursework in
point-set topology and multivariable calculus, but may not know the definition of differentiable manifold. I ... | R. Andrew Hicks | 13,754 | <p>I don't believe either of those books covers distributions and the theorem of Frobenius. Connections to partial differential equations in general I think are good topics. </p>
<p>Guillemin and Pollack is a book I like a lot, but chapters 2 & 3 (transversality and intersection) always seemed a bit specialized fo... |
58,870 | <p>I am teaching a introductory course on differentiable manifolds next term. The course is aimed at fourth year US undergraduate students and first year US graduate students who have done basic coursework in
point-set topology and multivariable calculus, but may not know the definition of differentiable manifold. I ... | Anton Petrunin | 1,441 | <p>If this is the first course in Differential geometry,
you should not go further than Gauss--Bonnet for surfaces.
I would not even consider anything with dimension >2.
By the way here is our <a href="https://arxiv.org/abs/2012.11814" rel="nofollow noreferrer">textbook</a> on the subject.
If they like Differential ... |
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