qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
3,433,383 | <p>Is it possible to write, let's say :- (a^2 + 1/a^2 - 1) as (a-b)^2.</p>
<hr>
<p>So I have to solve a problem (as practice, no test or assignment) that says to show if the roots of a given equation are real.</p>
<p>The equation is :- </p>
<p><span class="math-container">$$x^2 - 2\left(m + \frac1m \right)x + 3... | Quanto | 686,284 | <p>Note that the discriminant,</p>
<p><span class="math-container">$$m^2 + \frac 1{m^2} -1 = \left(m - \frac 1m\right)^2+1>0$$</span></p>
<p>So, the roots are real.</p>
|
3,433,383 | <p>Is it possible to write, let's say :- (a^2 + 1/a^2 - 1) as (a-b)^2.</p>
<hr>
<p>So I have to solve a problem (as practice, no test or assignment) that says to show if the roots of a given equation are real.</p>
<p>The equation is :- </p>
<p><span class="math-container">$$x^2 - 2\left(m + \frac1m \right)x + 3... | Allawonder | 145,126 | <p>Your discriminant should be <span class="math-container">$$\left[-2\left(m+\frac1m\right)\right]^2-4(1)(3)=4\left(m^2+2m\frac1m+\frac{1}{m^2}-3\right)=4\left(m^2-1+\frac{1}{m^2}\right)=4\left(m^2-2+\frac{1}{m^2}+1\right)=4\left(m^2-2m\frac1m+\frac{1}{m^2}+1\right)=4\left(m-\frac1m\right)^2+4,$$</span> which is alway... |
1,357,148 | <p>Stokes' theorem generalizes the fundamental theorem of calculus (part 2) using differential forms. Is there a known generalization of part 1?</p>
<h2>edit</h2>
<p>In case anyone is unaware, The fundamental theorem of calculus part 1 states that the derivative of the map $t \mapsto \int_{a}^{t} f(s) ds$ is equal to... | Muphrid | 45,296 | <p>Consider the following equivalent formulation of the first fundamental theorem: given a function $f$ defined on an interval $I = [a,b]$. Let $G$ be the 1d Green's function for differentiation: $G(s) = \text{sgn } s/2$. Then an antiderivative $F$ for the function $f$ can be found by, for $x \in I$,</p>
<p>$$F(x)... |
581,599 | <p>Is it generally true that
$$\frac{\mathrm d}{\mathrm da} \int_{-\infty}^{a-y} f(x)\, \mathrm dx = f(a-y),$$
where $a$ and $y$ in the expression are constants?</p>
<p>To give context to the question, I am reading about the function convolution of two independent random variables $X, Y$. </p>
| Potato | 18,240 | <p>Assuming $f$ is (for example) continuous, you are correct. To prove this, use the fundamental theorem of calculus in conjunction with the chain rule.</p>
<p>Let $$F(z)=\int_{-\infty}^{z} f(x) \ dx.$$</p>
<p>By the fundamental theorem, $F'(z)=f(z)$. In your problem, you want to evaluate the derivative of $F(a-y)$ ... |
1,368,447 | <p><img src="https://i.stack.imgur.com/TL80w.jpg" alt="enter image description here"></p>
<p>So I tried the good old Calculus 1 approach and turned this into an optimization problem. The equations got REALLY hairy, but it was okay since this was the graphing calculator section of the exam. I called the longer part of ... | Claude Leibovici | 82,404 | <p>Using the long way by algebra.</p>
<p>As answered by André Nicolas, we try to maximize $$f(x_2)=\sqrt{25-x_2^2}\left(\sqrt{x_2^2-21}+x_2\right)$$ The derivative simplifies to $$f'(x_2)=-\frac{\left(\sqrt{x_2^2-21}+x_2\right) \left(x_2
\left(\sqrt{x_2^2-21}+x_2\right)-25\right)}{\sqrt{(25-x_2^2) \left(x_2^2-21\... |
865,639 | <p>I was doing some computations for research purposes, which led me to this integral:</p>
<p>$$I(n) = \int_0^{\infty} (t^2+t^4)^n e^{-t^2-t^4}\,dt.$$</p>
<p>This is very suggestively written so as to employ a parametric differentiation technique as so:</p>
<p>$$\left(\frac{\partial^n}{\partial \alpha^n}\right)\int_... | mike | 75,218 | <p>I would think that the parametric differentiation technique is probably easier to carry through:</p>
<p>$$\frac{\partial}{\partial r}K_{\nu}(r)= -(1/2)(K_{\nu-1}(r) +
K_{\nu+1}(r))......(1)$$</p>
<p>EDIT:
From the answer by O.L., we know that:</p>
<p>$$K_{\nu+1}(r)-K_{\nu-1}(r)=\frac{2\nu}{r}K_{\nu}(r)....... |
865,639 | <p>I was doing some computations for research purposes, which led me to this integral:</p>
<p>$$I(n) = \int_0^{\infty} (t^2+t^4)^n e^{-t^2-t^4}\,dt.$$</p>
<p>This is very suggestively written so as to employ a parametric differentiation technique as so:</p>
<p>$$\left(\frac{\partial^n}{\partial \alpha^n}\right)\int_... | Start wearing purple | 73,025 | <p>Let us make the change of variables $t=\sinh \frac{x}{4}$. Since
$$t^2+t^4=\frac{\sinh^2\frac{x}{2}}{4}=\frac{\cosh x-1}{8}$$
and $dt=\frac14 \cosh\frac{x}{4}dx$, the initial integral can be rewritten as
\begin{align}
I(n)&=\frac{e^{\frac18}}{4^{n+1}}\int_0^{\infty}\cosh\frac{x}{4}\sinh^{2n}\frac{x}{2}e^{-\fra... |
3,566,997 | <p>When I go through examples of calculating the intersection of two planes, there seems to be a convention of choosing an arbitrary point in order to solve the linear equations in question and get a particular point on the intersection line.</p>
<p>Let's take an example from a <a href="https://math.stackexchange.com/... | Community | -1 | <p><span class="math-container">$$a^y=x$$</span> differentiates on <span class="math-container">$x$</span> as</p>
<p><span class="math-container">$$(a^y)'=1$$</span></p>
<p>and by the chain rule,</p>
<p><span class="math-container">$$(a^y)'=\frac{d\,a^y}{dx}=\log(a)\,a^y\frac{dy}{dx}=\log(a)\,x\,y'.$$</span></p>
|
3,566,997 | <p>When I go through examples of calculating the intersection of two planes, there seems to be a convention of choosing an arbitrary point in order to solve the linear equations in question and get a particular point on the intersection line.</p>
<p>Let's take an example from a <a href="https://math.stackexchange.com/... | Matthew Leingang | 2,785 | <p>An alternative derivation avoids implicit differentiation altogether. Note that
<span class="math-container">$$
y = \log_a x = \frac{\ln x}{\ln a} = \frac{1}{\ln a} \cdot \ln x
$$</span>
Since <span class="math-container">$a$</span> is constant,
<span class="math-container">$$
y' = \frac{1}{\ln a} \cdot \fr... |
173,252 | <p>Is my proof of proposition 6.2 on page 75 correct? (it's different from what they do in the book)</p>
<p>Proposition 6.2.: $M$ is a Noetherian $A$ module $\iff$ every submodule of $M$ is finitely generated.</p>
<p>My proof:</p>
<p>$\implies$ Assume $M$ has a submodule $N$ that is not finitely generated. Say, $N =... | Colin McQuillan | 3,643 | <p>The first part is not quite correct: you have assumed $N$ is countably generated.</p>
<p>The second part looks good, though you should make sure you know why the fact that $N$ is finitely generated implies that the sequence is stationary.</p>
|
173,252 | <p>Is my proof of proposition 6.2 on page 75 correct? (it's different from what they do in the book)</p>
<p>Proposition 6.2.: $M$ is a Noetherian $A$ module $\iff$ every submodule of $M$ is finitely generated.</p>
<p>My proof:</p>
<p>$\implies$ Assume $M$ has a submodule $N$ that is not finitely generated. Say, $N =... | Community | -1 | <p>Your first part is strange. I would rephrase the argument as follows: Suppose that $M$ is a Noetherian $A$ - module and there exists a submodule $N$ that is not finitely generated. Now choose $f_1 \in N$. Since $N$ is not finitely generated, $\langle f_1 \rangle \subsetneqq N$ and so we can choose $f_2$ such that $f... |
3,340,723 | <p>How can we find the total number of ways in which we can divide <span class="math-container">$n$</span> elements into two subsets such that none of them are empty and the union of both sets should be equal to the whole set?</p>
<p>Eg. If <span class="math-container">$S=\{1,2,3\}$</span>, the answer can be <span cla... | mdnestor | 519,413 | <p>Use the power set of <span class="math-container">$S$</span>. For example when <span class="math-container">$S=\{1,2,3\}$</span> the power set is</p>
<p><span class="math-container">$$ \mathcal P(S) = \Big\{ \emptyset, \{1\}, \{2\}, \{3\}, \{1,2\}, \{1,3\}, \{2,3\}, \{1,2,3\} \Big\}$$</span></p>
<p>Choosing a pair... |
2,918,453 | <p>After reading <a href="https://math.stackexchange.com/questions/208317/show-sum-n-0-infty-frac1a2n2-frac1a-pi-coth-a-pi2a2/208407#208407">Show $\sum_{n=0}^\infty\frac{1}{a^2+n^2}=\frac{1+a\pi\coth a\pi}{2a^2}$</a> and noodling around on wolfram alpha, I discovered </p>
<p>$$ \begin{align}
&\coth(x \pi)=\frac{x... | Szeto | 512,032 | <p>I don’t have a list, but I can present the method to make a list systematically.</p>
<p>Let $P(x,n),Q(x,n)$ be two polynomials of $x,n$.</p>
<p>Suppose
$$Q(x,r_1(x))=Q(x,r_2(x))=\cdots=Q(x,r_k(x))=0$$ for all $x$.</p>
<p>Suppose the sum
$$S(x)=\sum^\infty_{n=-\infty}\frac{P(x,n)}{Q(x,n)}$$ converges whenever $r... |
3,054,558 | <p>I've just learned a throrem which states that : A metric space has the structure of a topological space in which the open sets are unions of balls . </p>
<p>But the theorem only told me there "exist" one topology with respect to the metric.When we refer to topology on a metric space <span class="math-container">$S$... | Community | -1 | <p>Yes. So you can say that a set <span class="math-container">$U$</span> is open iff for each <span class="math-container">$x\in U$</span>, there exists an open ball <span class="math-container">$B(x,r)$</span> centered at <span class="math-container">$x$</span> with <span class="math-container">$B(x,r)\subset U$</sp... |
10,978 | <p>So, in <a href="https://math.meta.stackexchange.com/a/10083/43351">this post</a>, I suggested to replace <a href="https://math.stackexchange.com/questions/tagged/rootsystem" class="post-tag" title="show questions tagged 'rootsystem'" rel="tag">rootsystem</a> with <a href="https://math.stackexchange.com/quest... | user642796 | 8,348 | <p>I have merged <a href="https://math.stackexchange.com/questions/tagged/rootsystem" class="post-tag" title="show questions tagged 'rootsystem'" rel="tag">rootsystem</a> into <a href="https://math.stackexchange.com/questions/tagged/root-systems" class="post-tag" title="show questions tagged 'root-systems&#... |
4,386,333 | <p>This is Exercise 1.6(iii) from Intro. to K-Theory for <span class="math-container">$C^*$</span>-Algebras by Rørdam et al.</p>
<p>If <span class="math-container">$A$</span> is a unital <span class="math-container">$C^*$</span>-algebra, and <span class="math-container">$a \in A$</span> is invertible, then I want to sh... | Josse van Dobben de Bruyn | 246,783 | <p>Since <span class="math-container">$a^*a$</span> is normal, it follows from the <a href="https://math.stackexchange.com/a/2405957">linked answer</a> that <span class="math-container">$C^*(a^*a) = C^*(a^*a,1)$</span>. Therefore we have <span class="math-container">$1,(a^*a)^{-1} \in C^*(a^*a) \subseteq C^*(a)$</span>... |
2,675,758 | <p>For example:</p>
<p>There are two green marbles and two red marbles in a bag. Two marbles are chosen at random, what is the probability that the two marbles which have been chosen are of the same colour?</p>
<p>Ordered + distinct: <br>
Marbles = $G1, G2, R1, R2$ <br>
Event space = $\{|G1~G2|, |G2~G1|, |R1~R2|, |R2... | Andrew Li | 344,419 | <p>I'm not sure how you multiplied by $10$ on one side, while performed exponentiation on the other in the first step. Did you mean: </p>
<p>$$10^{\log\left((x+2)^2\right)} = 10^2$$
$$(x+2)^2 = 100 \rightarrow x^2+4x-96 =0\rightarrow (x+12)(x-8)=0$$</p>
<p>To get $x=8$ as the solution since $-12<0$. Regardless, yo... |
270,875 | <p>When I run</p>
<pre><code>FullSimplify[Sqrt[(1 - Sqrt[1 - 2 x^2 y^2])^2], {1 > 2 x^2 y^2 > 0}]
</code></pre>
<p>I obtain <code>1 - Sqrt[1 - 2 x^2 y^2]</code>. However, when I run</p>
<pre><code>FullSimplify[ExpandAll[Sqrt[(1 - Sqrt[1 - 2 x^2 y^2])^2]], {1 > 2 x^2 y^2 > 0}]
</code></pre>
<p>I am never abl... | Void | 46,818 | <p>As kindly pointed out by JimB in the comments, it turns out that the function I needed was given by Carl Woll in <a href="https://mathematica.stackexchange.com/a/230700/46818">this answer</a>:</p>
<pre><code>denestSqrt[e_, domain_, X] :=
Replace[
z /. Solve[
Simplify[Reduce[Reduce[
z == e &&... |
95,477 | <p>I am seeking a function $f: \mathbb{R}^3 \to \mathbb{R}^3$
that has these properties:</p>
<p>(1) When iterated $n$ times starting from some $p$,
connecting the points in order
with segments and closing last to first,</p>
<p>$$(p, f(p), f^2(p), \ldots, f^n(p), p)$$</p>
<p>results in a simple (non-self-intersectin... | Lee Mosher | 20,787 | <p>There is a body of work starting with Birman and Williams, and continued by Ghrist, Holmes, and Sullivan, concerning knottedness of closed orbits of flows. The flows they consider are certain Axiom A flows on $S^3$ with interesting basic sets, for example the Lorentz attractor. Williams' paper "Lorentz knots are pri... |
95,477 | <p>I am seeking a function $f: \mathbb{R}^3 \to \mathbb{R}^3$
that has these properties:</p>
<p>(1) When iterated $n$ times starting from some $p$,
connecting the points in order
with segments and closing last to first,</p>
<p>$$(p, f(p), f^2(p), \ldots, f^n(p), p)$$</p>
<p>results in a simple (non-self-intersectin... | Yoav Kallus | 20,186 | <p>If you parametrize a torus in $\mathbb{R}^3$ as $(x(u,v),y(u,v),z(u,v))$, $0\le u,v<1$, you can easily generate the torus knot $(3,q)$ (with crossing number $2q$) for $q$ large enough and not divisible by three by letting $f(u,v) = (u+3/q^2 \mod 1,v+1/q \mod 1)$ and $n=q^2$. So you just have to embed a continuum ... |
638,971 | <p>What is the definition of $R_{ijkl}$ in terms of metrics on a manifold?</p>
<p>I know what the definition of the riemann tensor, $R^l_{ink}$, is. But what exactly is meant by $R_{ijlk}$?</p>
| Anthony Carapetis | 28,513 | <p>It's just the Riemann tensor with an index lowered, i.e. contracted with the metric:</p>
<p>$$ R^{i}{}_{jkl} = g^{im} R_{mjkl}.$$</p>
<p>If the indices are not spaced out like this, it is ambiguous which index is being raised; so it's possible that it could also be the last index (or very rarely one of the inner o... |
638,971 | <p>What is the definition of $R_{ijkl}$ in terms of metrics on a manifold?</p>
<p>I know what the definition of the riemann tensor, $R^l_{ink}$, is. But what exactly is meant by $R_{ijlk}$?</p>
| jimbo | 115,363 | <p>The operacion raising and lowering indices is a contracción of the tensorial product of metric tensor $g_ {ij}$ or co-metrico tensor $g^{ij}$, with other arbitrary tensor. </p>
<p>This tensors allow define a isomorphism, call musical isomorphism, between the tangent space $T^1(M)$ and the cotangent space $T_1(M)$.<... |
2,068,249 | <p>I am considering block matrices $$\begin{pmatrix} A & v \\ v^T & x \end{pmatrix}$$ with $A \in \mathbb{R}^{(n-1) \times (n-1)}$, $v \in \mathbb{R}^{n-1}$, $x \in \mathbb{R}.$ Is there a rational $(n-1) \times (n-1)$ expression $p(A,v,x)$ I can form in the variables $A,v,x$ such that $$\mathrm{det}\begin{pmat... | Emilio Novati | 187,568 | <p>For blok matrices
$$
\left[
\begin{array}{c|c}
A_{11}& A_{12} \\
\hline
A_{21} & A_{22}
\end{array}
\right]
$$
if $A_{11}$ is invertible, we have the general formula ( you can see <a href="http://www.ee.iisc.ac.in/people/faculty/prasantg/downloads/blocks.pdf" rel="nofollow noreferrer">here</a> or <a href="... |
3,923 | <p>I propose the following lemma and its proof. It is related to row-null and column-null matrices - i.e. matrices whose rows and columns both sum to zero. Could you please give your opinion on the plausibility of the lemma, and the validity of the proof?</p>
<p><strong>Lemma:</strong> Let $Z\in\text{GL}(n,\mathbb{R})... | Fiktor | 695 | <p>I've checked your proof. It is correct.</p>
<p>I want to say obvious thing: I know two methods for checking the proof is correct.<br>
1. Check the answer, if you can get it from something else.<br>
1'. If you can't get the answer independently, check some of it's properties.<br>
2. Check the proof line by line.<br>... |
631,054 | <p>Let $A$ is in $M_3(\mathbb R^3)$ which is not a diagonal matrix. Pick out the cases when $A $ is diagonalizable over $\mathbb R$:</p>
<p>a. when $A^2=A$;</p>
<p>b. when $(A-3I)^2=0$;</p>
<p>c. when $A^2+I=0$.</p>
<p>My attempt is if $A$ is diagonalizable then there is some invertible $P$ s.t. $PAP^{-1}$=$D$. </p... | Community | -1 | <p>Result :</p>
<p>A matrix is diagonalizable if its minimal polynomial has no repeated roots.</p>
<p>For first case $A^2=A$ minimal polynomial could be :</p>
<ul>
<li>$x$ What does this mean if $x$ is minimal polynomial for $A$.. does something goes wrong?</li>
<li>$x-1$ What does this mean if $x$ is minimal polyno... |
3,315,558 | <p>I am having a very frustrating time with the back book that says my answer is way off but to me everything looks fine:</p>
<p><span class="math-container">\begin{align*}
(xy)y'&= x^2+3y^2\\
y' &= \frac{x^2}{xy} + \frac{3y^2}{xy}\\
y' &= \frac{x}{y} + \frac{3y}{x}\\
y' &= \frac{1}{v} + 3v\\
y' &=... | Peter Foreman | 631,494 | <p>Where did the <span class="math-container">$+1$</span> go?
<span class="math-container">$$2v^2+1=Cx^4$$</span>
<span class="math-container">$$\frac{2y^2}{x^2}+1=Cx^4$$</span>
<span class="math-container">$$2y^2+x^2=Cx^6$$</span>
<span class="math-container">$$x^2+2y^2=Cx^6$$</span></p>
|
3,259,475 | <p>I am looking for a nice slick way to show</p>
<blockquote>
<p><span class="math-container">$$\int^1_0 \frac{\tanh^{-1} x}{x} \ln [(1 + x)^3 (1 - x)] \, dx = 0.$$</span></p>
</blockquote>
<p>So far I can only show the result using brute force as follows.
Let
<span class="math-container">$$I = \int^1_0 \frac{\tan... | Zacky | 515,527 | <p><span class="math-container">$$\sf I = \frac12 \int^1_0 \frac{\ln\left(\frac{1+x}{1-x}\right)}{x} \ln ((1 + x)^3 (1 - x)) dx=\frac12 \int_0^1 \frac{(a-b)(3a+b)}{x}dx$$</span>
Where we denoted <span class="math-container">$\sf a=\ln(1+x)$</span> and <span class="math-container">$\sf b=\ln(1-x)$</span>.
Now we're goin... |
2,859,276 | <p>Given a measurable space $(X,\scr{A})$ where $\scr{A}$ is a sigma algebra on $X$, I would like to know if $f:X \to \mathbb{R}$ is measurable then $cf$ where $c \in \mathbb{R}$ is also measurable.</p>
<p>A function $f:X \to \mathbb{R}$ is measurable if $[f > \alpha] = \{x \in X \:{:}\: f(x) > \alpha \} \in \ma... | Epiousios | 451,957 | <p>Note that for $b=\alpha /c$ we have $[f<b] = [f\ge b]^C$ and
$$[f\ge b] = \cap_{n=1}^\infty [f >b - 1/n]\in \mathscr A.$$</p>
<p>($A^C$ is the complement of the set $A$)</p>
|
2,590,551 | <p>Suppose we have $A,B$ which are $R$-submodules of the $R$-module M. Then , there exists an exact sequence :</p>
<p>$0\rightarrow \frac{M}{A\cap B}\rightarrow \frac{M}{A}\times \frac{M}{B}\rightarrow \frac{M}{A+B} \rightarrow 0$</p>
<p>Where $A+B=\langle A\cup B\rangle$</p>
<p>I believe that that the first functio... | OmG | 356,329 | <p>You can compute it recursively. Also, start from a node with minimum degree. For example, pick node 1. As it has 2 edges, you can compute different minimum spanning trees by 3 states.</p>
<p>First, remove $e_{12}$, and then compute MST for the other nodes ($k$).
Then, you can multiply $k$ by 2, as there is a simila... |
2,384,444 | <p>Determine the equation of the line the portion of which, intercepted by the axes, is divided by the point $(-5,4)$ in the ratio of $1:2$.</p>
<p>My Attempt:
Let the equation of straight line be
$$ax+by+c=0$$
It passes through the point $(-5,4)$.
$$-5a+4b+c=0$$</p>
| Hasek | 395,670 | <p>This kind of ODE should be solved as follows:</p>
<ol>
<li>Solve the corresponding <strong>homogeneous</strong> equation.</li>
</ol>
<p>In your case it is $y'+y\cos(x)=0$ which has solution $y=c\cdot e^{-\sin(x)}$.</p>
<ol start="2">
<li><strong>Consider constant</strong> in previous solution <strong>as a functio... |
2,384,444 | <p>Determine the equation of the line the portion of which, intercepted by the axes, is divided by the point $(-5,4)$ in the ratio of $1:2$.</p>
<p>My Attempt:
Let the equation of straight line be
$$ax+by+c=0$$
It passes through the point $(-5,4)$.
$$-5a+4b+c=0$$</p>
| Frieder Jäckel | 440,045 | <p>I always like to think of these type of ODE's in terms of the product rule.
\begin{equation}x=y'e^{\sin(x)}+y\cos(x)e^{\sin(x)}=\left(ye^{\sin(x)}\right)'
\end{equation}
So integrating both sides and dividing by $e^{\sin(x)}$ yields\begin{equation}y=e^{-\sin(x)}\left(\frac{1}{2}x^2+c\right).
\end{equation}</p>
|
1,300,391 | <p>For example, if I wanted to get two of a kind, I would need seven dice. This is because even if the first six were 1, 2, 3, 4, 5, 6, the next one would have to make a pair out of the previous dice.</p>
<p>So is there a formula for getting n of a kind 100% of the time?</p>
| FundThmCalculus | 153,550 | <p>I think you mean $>=n$ of <em>any</em> kind. A trivial upper bound on the number of $k$-sided dice needed is $(n-1)*k+1$. This is the worst-case distribution. Without loss of generality, we can assume the dice land $1,2,3,...,k$ for the first $k$ dice, and then repeat the pattern. This reduces the count of a give... |
2,661,113 | <p>Consider $n$ player numbered $1,2,\ldots,n$. If player $i$ fights against $j$ then $i$ wins with probability $i/(i+j)$. There are no ties.</p>
<p>A player $i_1$ is extracted at random. Then, a second different player $i_2$ is extracted at random. They fight against each other. </p>
<p>Then, we extract another play... | Steven Stadnicki | 785 | <p>To flesh out my comment a bit: you can extend your calculations out further without much difficulty, at least to $n=10$ or $n=12$ish. Iterate over all $n!$ permutations of $1..n$, and for each permutation $\rho=p_1p_2p_3\ldots p_n$ (note that this is 'mapping' notation, not cycle notation) you can compute the proba... |
2,661,113 | <p>Consider $n$ player numbered $1,2,\ldots,n$. If player $i$ fights against $j$ then $i$ wins with probability $i/(i+j)$. There are no ties.</p>
<p>A player $i_1$ is extracted at random. Then, a second different player $i_2$ is extracted at random. They fight against each other. </p>
<p>Then, we extract another play... | Paolo Leonetti | 45,736 | <p>This does also not answer the original question, but it provides a more efficient way to calculate the probabilities $p(i)$ of winning in $O(n^2)$ operations (in particular, less than exponential time).</p>
<p>For each $a \in \{1,\ldots,n\}$ and set $I$ such that $\{a\}\subseteq I\subseteq \{1,\ldots,n\}$, let $p(a... |
1,882,509 | <p>As we know that, according to multiplicative identity 2 to the power of 3 means,
1*2*2*2=8;but why 2 to the power -3 is equal to the 1/8,if i think it is the reverse of multiplicative identity then from where 1 is coming?
i don't want only mathematical proof,but also the intuitive story behind this?
Why it is turnin... | User8128 | 307,205 | <p>Say $x > 0$ and $a \in \mathbb R$. For me the easiest intution is this: $$x^a x^{-a} = x^{a+(-a)} = x^0 = 1$$ but also $$\frac{x^a}{x^a} =1.$$ Setting the two equal, we see $$x^{a}x^{-a} = \frac{x^a}{x^a}$$ and then we can divide by $x^a$ on both sides to see $$x^{-a} = \frac 1 {x^a}.$$ Of course, arguements like... |
2,643,486 | <p>I have a problem in my maths book which says </p>
<blockquote>
<p>Find the arithmetic sequence in which $T_8 = 11$ and $T_{10}$ is the additive inverse of $T_{17}$</p>
</blockquote>
<p>I don't have a first term of common difference to solve it, so I managed to make two equations to find the first term and common... | Matthew Leingang | 2,785 | <p>The answer given is correct. You are right that the relation
$$
\{(a,a),(b,b),(c,c),(d,d)\}
$$
is also an equivalence relation—it's a <em>different</em> equivalence relation. And it yields a different partition.</p>
|
3,430,744 | <p>I'm new to fourier transform but I somewhat know fourier series. Since the fourier series of <span class="math-container">$\sin(2\pi t)$</span> is just <span class="math-container">$\sin(2\pi t)$</span>, I thought of playing with this function to better understand what fourier transform is doing. </p>
<p>As a star... | nmasanta | 623,924 | <p><strong>Alternative approach:</strong></p>
<p>Given equation can be written as<span class="math-container">$$\left(\frac{dy}{dx}\right)^2-4x\frac{dy}{dx}+4y=0$$</span>
<span class="math-container">$$\implies y=x\frac{dy}{dx}-\frac{1}{4}\left(\frac{dy}{dx}\right)^2\tag1$$</span>
which is <a href="https://en.wikiped... |
3,030,317 | <p>I have four points on a rectangular grid <span class="math-container">$(x_1,y_1)$</span>, <span class="math-container">$(x_1,y_2)$</span>, <span class="math-container">$(x_2,y_1)$</span> and <span class="math-container">$(x_2,y_2)$</span>. I also have the value of a third variable <span class="math-container">$z$</... | Adrian | 357,064 | <p>Actually regular 2-d cubic splines does a good job, without any need for Hermite or Coons, or invention of new functions. All partials here are zero, grid points have function values 2, 3, 4, 5.
<a href="https://i.stack.imgur.com/xolcV.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/xolcV.jpg" al... |
2,819,652 | <p>I know that my function $f$ must satisfy the following condition for $x \geq 0:$
$$\frac{d\left(xf(x)-x\right)}{dx} \geq 0.$$
What can I say about $f?$ I am curious about its possible sign or variations with respect to $x$.</p>
<p>I have investigated Grönwall's inequality to find an upper bound for $f$ but it seems... | Dr. Sonnhard Graubner | 175,066 | <p>Hint: Write</p>
<p>$$x^3-(\sqrt[3]{2})^3$$ and use that
$$a^3-b^3=(a-b)(a^2+ab+b^2)$$</p>
|
971,167 | <p>I'm having issues with this problem. I have solved for the eigenvalues but am having trouble finding the bases for both eigenvalues. The pictures below contain my work for solving for the eigenvalues and I solved for one of the basis but I think it is incorrect so I stopped. Can someone please help me. Thanks</p>
<... | mvggz | 167,171 | <p>I think I know where you made a mistake; when you calculated $A-(2+\sqrt{3}*i)I$
, you wrote that the term on the up-left was $(\sqrt{3}*i-1)$ whereas it was $(-\sqrt{3}*i-1) $, you could have tried anything from there it would have been confusing at best :).</p>
<p>I'll show you a practical way of dealing with thi... |
971,167 | <p>I'm having issues with this problem. I have solved for the eigenvalues but am having trouble finding the bases for both eigenvalues. The pictures below contain my work for solving for the eigenvalues and I solved for one of the basis but I think it is incorrect so I stopped. Can someone please help me. Thanks</p>
<... | egreg | 62,967 | <p>The eigenvalues $\lambda_1=2+i\sqrt{3}$ and $\lambda_2=2-i\sqrt{3}$ are correct.</p>
<p>Now you have to solve
$$
(A-\lambda_1I_2)x=0
$$
which is the system having matrix
$$
\begin{bmatrix}
1-\lambda_1 & 2 \\
-2 & 3-\lambda_1
\end{bmatrix}
=
\begin{bmatrix}
-1-i\sqrt{3} & 2 \\
-2 & 1-i\sqrt{3}
\end{b... |
116,220 | <p>Say two graphs are not isomorphic but are both strongly regular with the same set of parameters. Are there any parameters (other than the usual such as order, degrees, eigenvalues and multiplicities, etc.) that are determined, e.g., independence number, chromatic number, etc.?</p>
<p>Thanks for any help</p>
| Louis Deaett | 2,502 | <p>It seems the girth of a strongly regular graph would be determined by its parameters in the following way. If $\lambda > 0$, then the girth is 3. If $\lambda=0$ and $\mu > 1$, then the girth is 4. If $\lambda=0$ and $\mu=1$ then the girth is 5. That last case is a little unusual...</p>
|
4,502,175 | <p>I want to create a simple demo of moving an eye (black circle) inside a bigger circle with a black stroke when moving a mouse. I have cursor positions mouseX and mouseY on a canvas and I need to map the value of mouse position into a circle so the eye is moving inside the circle.</p>
<p>This should be trivial but I ... | Glorious Nathalie | 948,761 | <p>I think, from my understanding of your question, that you are trying to clip coordinates in the plane, to be inside or on the boundary of the big circle.</p>
<p>For that, suppose the given mouse position is <span class="math-container">$x_1 = mouseX , y_1 = mouseY $</span> and the equation of the circle in the plane... |
2,192,992 | <p>I'm looking for an easily understandable interpretation for a transpose of a square matrix A. An intuitive visual demonstration, how $A^{T}$ relates to A. I want to be able to instantly visualize in my mind what I'm doing to the space when transposing the vectors of a matrix. </p>
<p>From experience, understanding ... | levap | 32,262 | <p>One geometric description of $A^T$ can be obtained from the <a href="https://en.wikipedia.org/wiki/Singular_value_decomposition" rel="noreferrer">SVD decomposition</a> (this will be similar to your third point). Any square matrix $A \in M_n(\mathbb{R})$ can be written as a product $A = S \Lambda R^T$ where $\Lambda... |
2,192,992 | <p>I'm looking for an easily understandable interpretation for a transpose of a square matrix A. An intuitive visual demonstration, how $A^{T}$ relates to A. I want to be able to instantly visualize in my mind what I'm doing to the space when transposing the vectors of a matrix. </p>
<p>From experience, understanding ... | Bart Vanderbeke | 711,371 | <p>When trying to grasp the relation between <span class="math-container">$A$</span>, <span class="math-container">$A^T$</span> and <span class="math-container">$A^{-1}$</span>, I created the attached plot<br>
For <span class="math-container">$A^T$</span> this reads:<br></p>
<ul>
<li><span class="math-container">$\ma... |
81,016 | <p>I'm trying to learn about derived categories of algebraic stacks. To be honest, as of now, I don't need anything fancy nor deep. In my setup I have a scheme $X$ (well, a smooth and projective variety over $\mathbb{C}$) and a finite group $G$ acting on it. I would like to understand simple things like: what does a co... | algori | 2,349 | <p>A very readable reference for constructible derived categories on quotient stacks is Bernstein-Lunts, Equivariant sheaves and functors. In particular, Bernstein and Lunts construct the bounded derived category of sheaves on $[X/G]$ as the limit of truncated bounded derived categories of the "Borel constructions" $X\... |
178,828 | <p>Let $\left\{ x_n \right\}_{n\geq0}$ be a sequence of real numbers such that $$x_{n+1}=\lambda x_n+(1-\lambda)x_{n-1},\ n\geq 1,$$for some $0<\lambda<1$ </p>
<p>(a) Show that $x_n=x_0+(x_1-x_0)\sum_{k=0}^{n-1}(\lambda -1)^k$ </p>
<p>(b) Hence, or otherwise, show that $x_n$ converges and find the limit.</p>
<... | lab bhattacharjee | 33,337 | <p>$x_{n+1}=\lambda x_n+(1-\lambda)x_{n-1}$</p>
<p>If $x_n=a^n =>a=\lambda,1=>x_n=A+B\lambda^n $ where A,B are finite indeterminate constants.</p>
<p>Clearly, the convergence of $x_n$ depends whether $\lambda^n$ tends to a finite value or ∞ as $n→∞$. We know about infinity G.P. like $\sum_{0≤r<∞} b a^r$ wh... |
81,261 | <p>We have $Z_1,Z_2,Z_3$ are independent standard normal random variables.
Find</p>
<p>a) $\mathbb P(Z_1<Z_2+Z_3)$</p>
<p>b) $\mathrm{Var}(Z_1Z_2^2)$</p>
<p>c) $\mathbb P(Z_1/Z_2>1)$</p>
<p>d) $\mathbb P(Z_1^2>Z_2^2+Z_3^2)$</p>
| André Nicolas | 6,312 | <p>Many questions, from easy to not so easy! Almost all the questions deal with basic items that must be covered by whatever book you are using. Summaries, and further links, are, as usual, available at the suitable <a href="http://en.wikipedia.org/wiki/Cauchy_distribution" rel="nofollow">Wikipedia entry.</a></p>
<p>a... |
155,860 | <p>I often stumble over the term "Lie superalgebra" (= "Lie algebra with a $\mathbb{Z}_2$ grading"). Obvious question: What about $\mathbb{Z}_3$ grading (and so on)? Is a Lie algebra with $\mathbb{Z}_n$ grading just the special case of a quantum Lie algebra $L(q)$ with $q$ being an $n$-th root of 1 (I only looked at th... | Dietrich Burde | 32,332 | <p>Such generalizations exist. It is well known that the classical
Clifford algebras can be used to construct Lie superalgebras. The main
tool of the construction is the notion of the $\mathbb{Z}_2$-graded commutator which includes both the ordinary commutator and anticommutator. This can be generalized. For the case o... |
3,898,904 | <p>I'm currently making a start on group theory and have hit a roadblock with a relatively basic theorem on finite cyclic groups. The specific relation killing me is:
<span class="math-container">$$\mathbb{Z}_n \times \mathbb{Z}_m \cong \mathbb{Z}_{mn} \Leftrightarrow \text{gcd}(m,n) = 1$$</span>
So, the most straightf... | Bart Michels | 43,288 | <p>You can identify <span class="math-container">$\mathbb Z_0 = \mathbb Z / 0 \mathbb Z$</span> with <span class="math-container">$\mathbb Z$</span>. And then, it is of course true that when <span class="math-container">$n$</span> is coprime to <span class="math-container">$0$</span>, then
<span class="math-container">... |
1,628,616 | <p>I need to prove that the set of functions of the form $\{g\colon [-1,1]\to \mathbb{R}\, |$ there exists a constant $c$ such that $g(x) = c\cdot x\}$ is complete. How shall I proceed? I am taking a Cauchy sequence and want to prove that it converges to a function in this set, but how can I guarantee the limit will ha... | MPW | 113,214 | <p><strong>Hint:</strong> You can project $B$ straight down into the $x$-$y$ plane to obtain a closed quarter-disk. Do you see that this is a homeomorphism, and that the closed quarter-disk is homeomorphic to the closed disk (your "balls" must be closed balls in this context)?</p>
|
1,913,394 | <p>I need to prove that rank($\mathrm{A}$) is not continuous everywhere but is lower semi-continuous everywhere, where $\mathrm{A}\in \mathbb{C}^{n\times m} $</p>
| Arin Chaudhuri | 404 | <p>We have to show the set $S = \{A \in \mathbb{C}^{m \times n} : \texttt{rank}(A) > r \} \tag{1}$ is open for all real $r.$</p>
<p>We need the following fact : given a matrix $A_0 = (a_{ij}^{0})$ of rank $r_0$ there is a $\delta > 0$ such that for any matrix $A=(a_{ij})$ with $\|A - A_0\|_{\texttt{max}} < \... |
2,495,789 | <p>\begin{pmatrix}4&-6&2&-6\\ \:-2&3&-1&3\end{pmatrix}</p>
<p>Am I supposed to move the $0 $ to the other side of the equation and find the inverse of the matrix above or is there another way?</p>
| operatorerror | 210,391 | <p>You are solving for the kernel of the matrix
\begin{pmatrix}4&-6&2&-6\\ \:-2&3&-1&3\end{pmatrix}
which is the same as solving the augmented system
$$
\begin{pmatrix}4&-6&2&-6&0\\ \:-2&3&-1&3&0\end{pmatrix}\rightarrow
\begin{pmatrix}0&0&0&0&0\\ ... |
1,622,124 | <p>Assume $b>0$. Show that the Lebesgue intergral $\int_{1}^{\infty} x^{-b} e^{\sin {x}} \sin {(2x)}\, dx$ exists iff $b>1$.</p>
<p>We know if $b>1$ the integrand can be bounded and it's just an integral $\int_{1}^{\infty} x^{-b}\,dx$. I don't know how to show the other direction?</p>
| 5xum | 112,884 | <p>The absorption law says that $p\lor (p\land q)$ is equal to $p$ <strong>no matter what $q$ is</strong>. In particular, you can replace $q$ with $(r\lor(s\land t))$, and the expression will still be equal to $p$.</p>
<p>In your case, you can also replace $q$ with $\neg q$ and the expression is still equal to $p$.</p... |
1,622,124 | <p>Assume $b>0$. Show that the Lebesgue intergral $\int_{1}^{\infty} x^{-b} e^{\sin {x}} \sin {(2x)}\, dx$ exists iff $b>1$.</p>
<p>We know if $b>1$ the integrand can be bounded and it's just an integral $\int_{1}^{\infty} x^{-b}\,dx$. I don't know how to show the other direction?</p>
| DanielWainfleet | 254,665 | <p>You can use $\land$ and $\lor$ as $+$ and $\times$ with each of them being distributive over the other : $a\land (b\lor c) \equiv a\times (b+c)=(a\times b)+(a\times c),$ and $a\lor (b\land c)\equiv a+(b\times c)=(a+b)\times (a+c).$ Also $a+a=a\times a=a=a\times 1$ where $1$ is "True". Also $a+1=1.$ We also use $0$... |
4,236,875 | <p>Today, I was solving a problem which is described below.</p>
<blockquote>
<p>Show that the equation of any circle passing through the points of intersection of the ellipse <span class="math-container">$(x + 2)^2 + 2y^2 = 18$</span> and the ellipse <span class="math-container">$9(x - 1)^2 + 16y^2= 25$</span> can be w... | Lalit Tolani | 913,165 | <p>Note that center will lie on <span class="math-container">$x-$</span>axis, therefore let the required equation of circle be of type</p>
<p><span class="math-container">$(x-a)^2+y^2=r^2\implies x^2-2ax+a^2+y^2=r^2$</span></p>
<p>Now it passes through <span class="math-container">$(2,1) $</span> and <span class="math-... |
4,236,875 | <p>Today, I was solving a problem which is described below.</p>
<blockquote>
<p>Show that the equation of any circle passing through the points of intersection of the ellipse <span class="math-container">$(x + 2)^2 + 2y^2 = 18$</span> and the ellipse <span class="math-container">$9(x - 1)^2 + 16y^2= 25$</span> can be w... | Tryst with Freedom | 688,539 | <h2>Method without solving the equations:</h2>
<blockquote>
<p><span class="math-container">$(x + 2)^2 + 2y^2 = 18$</span> , <span class="math-container">$9(x - 1)^2 + 16y^2= 25$</span></p>
</blockquote>
<p>There is a fast way to solve this question... but it requires a little cleverness:</p>
<p><span class="math-conta... |
1,506,719 | <p>Let us have measurable spaces $(S_1, \Sigma_1)$ and $(S_2, \Sigma_2)$.
Idea of measurable function $f$ with respect to $\Sigma_1,\Sigma_2$ is the following. $f:$ $S_1 \to S_2$ has to be such that:
$$\forall E_2: f^{-1}(E_2) \in \Sigma_1, $$
where
$ E_2 \subset \Sigma_2, E_1 \subset \Sigma_1 $.</p>
<hr>
<p>Anoth... | celtschk | 34,930 | <p>Let me pick up where A. Alp Uzman left off: Why are we using the preimage function and not the image function?</p>
<p>OK, remember we have sets $S$ with an additional structure $\Sigma$ which is given by a set of subsets. Let's ignore the specific case and only remember that $\Sigma$ gives us special subsets.</p>
... |
764,973 | <p>here is the question: </p>
<p>$$
{\rm y}''\left(t\right) + 2\,{\rm y}'\left(t\right) + 5\,{\rm y}\left(t\right)
=
0;
\qquad\qquad
{\rm y}\left(0\right) = 2\,,\quad {\rm y}'\left(0\right) = -1.
$$</p>
<ul>
<li>$\mathcal{L} (y''(t)) = s^2y(s) -s y(0) -y'(0)$</li>
<li>$\mathcal{L} (+2y'(t)) = 2(sy(s) -y(0))$</li>
<li... | Matt L. | 70,664 | <p>With the correct initial values you get</p>
<p>$$y(s)=\frac{2s+3}{s^2+2s+5}=2\frac{s+1}{(s+1)^2+4}+\frac{1}{2}\frac{2}{(s+1)^2+4}$$</p>
<p>which corresponds to</p>
<p>$$y(t)=2e^{-t}\cos(2t)+\frac{1}{2}e^{-t}\sin(2t)$$</p>
|
55,853 | <p>Given a function $f:X\to X$, let $\text{Fix}(f)=\{x\in X\mid x=f(x)\}$.</p>
<p>In a recent <a href="https://math.stackexchange.com/questions/55846/is-the-set-of-fixed-points-in-a-non-hausdorff-space-always-closed/55847#55847">comment</a>, I wondered whether
$X$ is Hausdorff $\iff$ $\text{Fix}(f)\subseteq X$ is clos... | Auguste Hoang Duc | 4,204 | <p>This is a remark related to the question.</p>
<p><strong>Theorem :</strong> let $X$ be a topological space. Suppose that for every space $Y$ and pair of maps $f,g : Y \rightarrow X$, the subset $\{ y \in Y : f(y)=g(y) \}$ is closed. Then $X$ is Hausdorf.</p>
<p>It is easy to prove by considering the $2$ projection... |
2,018,631 | <p>Let C(x, y) : "x and y have chatted over the Internet"</p>
<p>where the domain for the variables x and y consists of all students in your class.</p>
<p>a ) There are two students in your class who have not
chatted with each other over the Internet.</p>
<p><strong>My answer:</strong> $\exists x \exists y[(x \not =... | JKEG | 217,837 | <p>I was going to comment on the ambiguity from the lack of braces but now that you've edited, I believe it is correct. I would remove the equivalence and just leave it as an implication as it is redundant and makes it a bit harder to read. </p>
<p>$\exists x \exists y[(x \not =y) \land \lnot C(x, y) \land \forall a ... |
1,001,913 | <p>If we want to graph a horizontal line, we will do the following:</p>
<pre><code>y = 0x + 3
</code></pre>
<p>No matter the domain for x, the range for y will always be 3. Therefore, we have a horizontal line.</p>
<pre><code>y = 0(0) + 3 = (0,3)
y = 0(1) + 3 = (1,3)
y = 0(2) + 3 = (2,3)
</code></pre>
<p>Now the fo... | Greg Martin | 16,078 | <p>The key is to recognize the given expression as the convolution of two simpler series. Using the known generating function
$$
\sum_{j=0}^\infty \binom{j+q}j x^j = (1-x)^{-q-1},
$$
we see that
\begin{align*}
\sum_{j=0}^\infty x^j \sum_{t=0}^j \binom{t+k}t \binom{j-t-(k-m)}{j-t} &= \sum_{j=0}^\infty \sum_{t=0}^j \... |
748,083 | <p>I'm trying to find the Jordan Normal Form of the matrix $A = \begin{bmatrix} 1 & 0 & 1 & -1 \\0 & 1 & 1 & 0\\ 0 & 0 & 1 & 1\\ 0 & 0 & 0 & 1\end{bmatrix}$ </p>
<p>(That is, a matrix J and a matrix S such that $A = S J S^{-1})$.</p>
<p>Finding the eigenvalues and eigen... | Ewan Delanoy | 15,381 | <p>Here is a partial answer : I show below that inequality (1) holds
when $x\geq 0$ or $x \leq c_1=-\frac{1}{1-\big(\frac{1}{1000}\big)^{\frac{1}{13}}}$
(note that $c_1 \approx -2.42 \ldots$).</p>
<p>Let $g(x)=(1+x)^a-1-ax-x^a$ for $x\geq 0$. Then $g'(x)=a(1+x)^{a-1}-a-ax^{a-1}$,
$g''(x)=a(a-1)\big((1+x)^{a-2}-x^{a-2}... |
1,645,211 | <p>The problem statement is,</p>
<p>Find a linear transformation $T: \mathbb R^3 \to \mathbb R^3$ such that the set of all vectors satisfying $4x_1-3x_2+x_3=0$ is the (i) null space of $T$ (ii) range of $T$.</p>
<p>For (i),</p>
<p>I found out the basis of the null space for the system,</p>
<p>$$\begin{pmatrix}4&... | Ramiro | 190,563 | <p>If $s$ is just a collection of subsets of $X$, without any additional condition on $s$, then the result is FALSE, even if $\mu$ and $\nu$ are finite $\sigma$-additive measures. </p>
<p>The issue is that $\{A\in M(s) : \mu(A)=\nu(A) \}$ may not be closed under intersection. </p>
<p>Example: </p>
<p>Let $\Omega =\... |
3,702,520 | <p>I've used cosine & sine rule, and after playing around with them, I'll always end up with an equation involving <span class="math-container">$\sin\theta,\cos\theta$</span> and other weird (and wonderful expressions). I've approached the question in many different ways: by showing <span class="math-container">$A... | Jan Eerland | 226,665 | <p>Well, notice that:</p>
<p><span class="math-container">$$-2-\sqrt{3}i=\left|-2-\sqrt{3}i\right|\exp\left(\arg\left(-2-\sqrt{3}i\right)i\right)=$$</span>
<span class="math-container">$$\sqrt{\left(-2\right)^2+\left(-\sqrt{3}\right)^2}\exp\left(\left(\pi+\arctan\left(\frac{\sqrt{3}}{2}\right)\right)i\right)=$$</span>... |
2,829,121 | <p>I get how to derive the ellipse equation, but I'm struggling to understand what it means intuitively. </p>
<p>You see, a circle equation can be understood very intuitively. The circle equation models how the radius of the circle can be represented using the Pythagorean theorem. But I don't understand what the ellip... | Nij | 226,212 | <p>The circle equation is simply one specific ellipse equation, where the scaling of each direction ($x$ and $y$) is identical.</p>
<p>Division by some factor of just one variable matches scaling by that factor in that direction. Hence, $\frac{x^2}{3^2}+y^2=r^2$ will make the ellipse twice as large in the $x$ directio... |
2,829,121 | <p>I get how to derive the ellipse equation, but I'm struggling to understand what it means intuitively. </p>
<p>You see, a circle equation can be understood very intuitively. The circle equation models how the radius of the circle can be represented using the Pythagorean theorem. But I don't understand what the ellip... | user674871 | 674,871 | <p>I think it is foci and the semi axes confusion. Intuitively you understand now equation with a and b mentioned above. That is ellipse. Now, because of symmetry, there MUST be a distance from center i. e. foci distance, where the "constant rope length" condition is satisfied. </p>
<p>In reverse you can think ... |
1,609,443 | <p>So my textbook states that it is $(-\infty,0)$ that is concave down, but why not $(-\infty,0]$? Could someone help?</p>
| User12345 | 133,683 | <p>Most undergrad textbooks use the following definition:</p>
<p>A function $f$ is concave down on $(a,b)$ when $f''(x)<0$ for all $a<x<b$.</p>
<p>In your example $f''(x)=0$ at $0$, so the function $f(x)=x^3$ is not concave down at $x=0$ according to this definition.</p>
|
2,840,425 | <p>I am trying to solve the assignment below with the quotient rule. Every time I get stuck on a similar assignment which has a square root inside the numerator of a fraction. What do I do with the square root in the numerator and how do I find the derivative?</p>
<p>$$\frac{\sqrt x+3}{x}$$</p>
| Bernard | 202,857 | <p><strong>Another hint</strong>:</p>
<p>$$(\operatorname{argtanh})'(x)=\dfrac1{1-x^2},$$
so you can obtain $\;\displaystyle\int\frac{\mathrm dx}{a^2-x^2}$ with the substitution $\; x=at$.
Also, we know that
$$\operatorname{argtanh}x=\frac 12\ln\biggl(\frac{1+x}{1-x}\biggr).$$</p>
|
1,027,249 | <p>Prove that $2$ is the only even prime:</p>
<p>I have tried, this is what I have done. Is this is suffitient? </p>
<p>Proof: To prove that $2$ is the only even prime
number we need to prove following:
(1) $2$ is an even prime number
(2) If there existed another even prime
number say $m$, then $m=2$</p>
<p>Proof of... | John Marty | 71,806 | <p>Yes, it is sufficient. Although you really shouldn't say proof is "obvious" trivial might be better. The issue here is proportionality. One could easily say that the whole problem was trivial. </p>
<p>In general the best way to gauge what is trivial and what is not is to consider whether your assumption trivializes... |
67,039 | <p>I just noticed that <code>Exit</code> and <code>Quit</code> can work without brackets i.e. a single</p>
<pre><code>Exit
</code></pre>
<p>or</p>
<pre><code>Quit
</code></pre>
<p>will quit the kernel. Quite surprising!</p>
<p>This isn't mentioned in the document of them. Is there a list for functions with this fe... | xzczd | 1,871 | <p>Just noticed <code>Return</code> can work without brackets when it's inside a dialog:</p>
<p><a href="https://i.stack.imgur.com/NLwxU.gif" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/NLwxU.gif" alt="enter image description here"></a></p>
|
50,441 | <p>here is the silly question of mine,</p>
<p>if <code>mathematica</code> takes <code>3</code> seconds to evaluate the function <code>f[3]</code>
then I plug the this value <code>f[3]</code> into a new function <code>g[f[3],f[3]]</code>, Let's ignore the evaluation time for the new function <code>g</code>, how long do... | Szabolcs | 12 | <p>Use <a href="http://reference.wolfram.com/mathematica/ref/With.html" rel="nofollow noreferrer"><code>With</code></a> to define temporary constants.</p>
<pre><code>With[{a=f[3]}, g[a,a]]
</code></pre>
<p><a href="https://mathematica.stackexchange.com/q/2639/12">Memoization</a> is also worth mentioning but keep in m... |
76,493 | <p>i'm interested in geometric interpretations of many linear algebra notions (check also related <a href="https://mathoverflow.net/questions/75241/geometric-interpretation-of-matrix-minors">geometric interpretation of matrix minors</a>). it came to me recently that geometric description of adjugate matrix (for example... | Paul Bryan | 78,645 | <p>Copied from <a href="https://math.stackexchange.com/a/2042191/395665">math.stackexchange</a>: Another way to approach this avoids the use of duality via inner-products. Functorality should be clear. It may be that one requires finite rank free modules here so that the bilnear pairing below is <a href="https://en.wik... |
5,848 | <p>This is the question which i am attempting to solve and it seems to be difficult to get rid of the exponents.</p>
<blockquote>
<p>Show that a the two cubic curves $Y^3 = X^2 + X^3$ and $X^3 = Y^2 + Y^3$ intersect each other at nine points.</p>
</blockquote>
<p>Any help would be appreciated. By the way i would al... | T.. | 467 | <p>In addition to the usual generalities on Bezout's theorem and resultants, notice that for this system, $X^2 + Y^2 = 0$, so that there are 6 finite solutions (3 on each line $X = \pm iY$, organized as 1 transverse intersection point per line, plus one double point at (0,0) on each line, leading to a quadruple interse... |
5,848 | <p>This is the question which i am attempting to solve and it seems to be difficult to get rid of the exponents.</p>
<blockquote>
<p>Show that a the two cubic curves $Y^3 = X^2 + X^3$ and $X^3 = Y^2 + Y^3$ intersect each other at nine points.</p>
</blockquote>
<p>Any help would be appreciated. By the way i would al... | Laurent Lessard | 1,356 | <p>Suppose $(x,y)$ satisfies both equations. Substituting one equation into the other, we obtain $x^2+y^2=0$. The only real solution to this equation are $x=y=0$, as pointed out by Moron.</p>
<p>Now suppose $x\neq 0$, and so $x$ is a complex number. Then $y = \pm i x$. Eliminating $y$ from the other equation yields tw... |
2,425,669 | <p><em>Having fun with the calculator, I realized that :</em></p>
<p><code>(a^c)</code> and <code>(a^b)</code></p>
<p>with <code>c > b</code> and <code>c > 4</code> and <code>b = 2</code></p>
<pre><code>a^c / a^b = a^(c-2)
</code></pre>
<p><strong><em>So, for example:</em></strong></p>
<p><code>3^5 / 3^2 = 2... | Sean Roberson | 171,839 | <p>Congrats, you discovered the rule of division with powers:</p>
<p>$$ \frac{b^m}{b^n} = b^{m-n} $$</p>
|
495,487 | <p>Show that the difference of the number of cycles of even and odd permutations is $(-1)^n (n-2)!$, <strong>using a bijective mapping</strong> (combinatorial proof).</p>
<p>Suppose to convert a permutation from odd to even we always flip 1 and 2. Then the difference can be easily calculated.</p>
<p>Thus, if I can fi... | Brian M. Scott | 12,042 | <p>This isn’t quite a bijective proof in the usual sense, since one of the quantities involved can be negative. However, it uses a bijection to reduce the problem to a very simple counting problem, so I think that it qualifies. I’ve left the final step to you.</p>
<p>Let $$E_n=\{\langle\pi,\sigma\rangle:\pi\text{ is a... |
3,140,384 | <p>Assume you're stranded on an island, and you've got some really important maths to do. Obviously you don't have a calculator with you, so you need to find another way.</p>
<p>I know this sounds absurd, but it quite annoys me that, apart from maybe addition to division, computing any other value always ends up at "P... | NoChance | 15,180 | <p>The topic of this question is very interesting. It may deserve a book on its own. I am sure that there are books that discussed bits of this topic.</p>
<p>Computers rely on algorithms to perform function calculations. Mathematicians in the past must have relied on elementary devices, tools, tables and approximation... |
180,073 | <p>The function <code>FullSimplify</code> can easily reformat this expression</p>
<pre><code>FullSimplify[Cos[omegan t]^2 + Sin[omegan t]^2]
(* output: 1 *)
</code></pre>
<p>However, it cannot simplify the same expression if contained in a larger expression:</p>
<pre><code>sol = DSolve[m*(x''[t] + omegan^2 * x[t]) =... | Bill | 18,890 | <p>You can always do your own expression substitution</p>
<pre><code>sol = DSolve[m*(x''[t] + omegan^2*x[t]) == B*Exp[I*omegad*t], x, t] /.
Cos[omegan t]^2 + Sin[omegan t]^2 -> 1
</code></pre>
<p>which gives you your desired result</p>
<pre><code>(*{{x -> Function[{t}, C[1] Cos[omegan t] + C[2] Sin[omegan t... |
337,806 | <p>I'm stuck on this problem for few days and can't find the solution.Hope some one here can help me solve this. I'm so grateful for any any help:</p>
<blockquote>
<blockquote>
<p>Let $V$ be a finite-dimensional vector space and $f$ a non-degenerate symmetric bilinear form on $V$. With every basis $\beta = \{\al... | Elmar Zander | 10,076 | <p>Denote the bilinear $f$ represented in the basis $\beta$ by $U$ and in the basis $\beta'$ by $U'$. Then given any vector $x=\sum c_i \alpha_i=\sum c'_i\alpha'_i$ and $y=\sum d_i \alpha_i=\sum d'_i\alpha'_i$ we have
$$f(x,y)=c^TUd=c'^TU'd'$$
Further we have
$$f(x,y)=\sum_{ij} c_id'_j f(\alpha_i,\alpha'_j)=\sum_{ij} ... |
102,234 | <p>Our assignment was to enter in a series and check to see if it converges by graphing. </p>
<p>However, when I attempt to graph it, I get a bunch of errors that I'm not sure what they mean. Best way to show it is a picture <a href="https://i.stack.imgur.com/ZWMds.png" rel="nofollow noreferrer"><img src="https://i.st... | David G. Stork | 9,735 | <pre><code>ListPlot[Accumulate[Table[1/CubeRoot[n], {n, 1, 5000}]]]
</code></pre>
<p>strongly suggests this does not converge.</p>
|
628,254 | <p>I am searching for some good book which section is devoted to modular arithmetic. I am self learner so I strongly prefer that book has exercises best with answers or solutions. I have CS background and has taken course on discrete mathematics but besides some basic facts on modulo operation it lacked some introducti... | amWhy | 9,003 | <p>You might want to check out the <a href="https://www.khanacademy.org/math/applied-math/cryptography">Khan Academy</a> on modular arithmetic. (Scroll down to "modular arithmetic"). There's a substantial series of tutorials there; you'll also be able to find practice problems at the Academy. </p>
|
1,405,637 | <p>Given that $ U_n=[x(1-x)]^n$ and $n\geq2$,$V_n=\int_{0}^{1}e^xU_ndx$,prove that $V_n+2n(2n-1)V_{n-1}-n(n-1)V_{n-2}=0$<br></p>
<p>I tried to solve it by integration by parts,taking $U_n$ as first function and $e^x$ as second function.<br></p>
<p>$V_n=-\int_{0}^{1}n[x(1-x)]^{n-1}(1-2x)e^x dx=-\int_{0}^{1}nU_{n-1}(1-... | YINING DING | 825,053 | <p>I will continue @Nate Eldredge's answer and finish this proof.</p>
<p><span class="math-container">\begin{equation}
\sum\limits_{n\geq1}\mathbb{P}(X_n \neq 2) = \sum\limits_{n\geq1}\mathbb{P}(X_n =-n^2 or \, -n^3) \leq \sum\limits_{n\geq1}\frac{1}{n^2} < \infty
\end{equation}</span>
Then Borel-Cantelli lemma, we ... |
176,192 | <p>I am trying to solve a system of differential equations as follows:</p>
<pre><code>L1 = 5.0;
L2 = 0.43;
d = 0.0056;
g = 9.8;
h1 = 5.0;
h2 = 0.43;
A1 = 0.00001;
A2 = 0.001;
p0 = 100000;
v0 = 0.001;
NDSolve[{P[t] -
1*1000*10*
L2 == (1000/(2*0.0056)*y[t]^2*
L2*(0.316*(1000*y[t]*0.0056/0.00089))^(-0.25)) +
500*01.... | Michael E2 | 4,999 | <p>Process <code>cyclesperday</code> thus:</p>
<pre><code>cyclesperday = Developer`ToPackedArray[cyclesperday, Real];
</code></pre>
<p>Your data gets loaded as a mix of <code>Real</code> and <code>Integer</code> values. To take best advantage of the CPU, the data should all be of the same type and in a <a href="http... |
1,025,139 | <p>How to prove the following? Should I use induction or something else?</p>
<blockquote>
<p>Let $n$ and $r$ be positive integers with $n \ge r$. Prove that
$$\binom{r}{r} + \binom{r+1}{r} + \cdots + \binom{n}{r} = \binom{n+1}{r+1}.$$</p>
</blockquote>
<p><em>Attempted start:</em></p>
<p>Basis step: ${\binom{1}{... | David | 119,775 | <p>Here is a combinatorial proof. Consider the problem of choosing $r+1$ numbers from $1,2,\ldots,n+1$, where repetition is not allowed and order is not important.</p>
<ul>
<li>First do it the obvious way. The number of ways is the RHS.</li>
<li>Then do it by initially choosing the largest of the $r+1$ numbers.</li>... |
951,414 | <p>Consider the absolute value equation:</p>
<p>|x| + |x-2| +|x-4|= 6</p>
<p>How to find the solution(s)?</p>
<p>My attempt:</p>
<p>For |x|, we got x, for x>=0 and -x, for x <0</p>
<p>For |x-2|, we got x-2, for x >= 0 and -(x-2), for x<0</p>
<p>For |x-4|, we got x-4, for x>=0 and -(x-4), for x<0</p>
<p>... | mathlove | 78,967 | <p>I think you misunderstand the absolute value. </p>
<p>Note that $$|x|=\begin{cases}x &\text{if $x\ge 0$}\\-x &\text{if $x\lt 0$}\end{cases}$$
So, for example,
$$|x-2|=\begin{cases}x-2 & \text{if $\color{red}{x-2}\ge 0$}\\-(x-2)& \text{if $\color{red}{x-2}\lt 0$}\end{cases}$$</p>
<p>So, for your qu... |
1,483,191 | <p>It's now known that to reach the solved position of the rubik cube from any other position you need at most $20$ moves (a $180^{\circ}$ of a face is counted as one move , not two ) . </p>
<p>See for example here : <a href="http://www.cube20.org/" rel="nofollow">http://www.cube20.org/</a> </p>
<blockquote>
<p>Now... | Community | -1 | <p>By symmetry, no two positions can be more than 20 steps apart. There's nothing special about the "solved" configuration: the same argument that shows you can get from any starting $A$ to the solved state in no more than 20 steps shows that you can get from any starting $A$ to any fixed $B$ in no more than 20 steps.<... |
211,052 | <p>I want to modify a list of table row by adding an extra column. For that I <a href="https://reference.wolfram.com/language/ref/Map.html" rel="nofollow noreferrer"><code>Map</code></a> the data with a pure function that evluate the new column value from exisiting one and reconstruct a new list from that and the init... | Mark R | 65,931 | <p>Another alternative:
<code>Join[#[[1 ;; 3]], {#[[1]] > 2 b, #[[4]]}] & /@ data</code></p>
|
2,846,928 | <blockquote>
<p>If I have the a complex number $z \in \mathbb{C}$ with absolute value $|z| = 1$, how do I show that $-i \frac {z-1}{z+1}$ is real? </p>
</blockquote>
| dxiv | 291,201 | <p>A number $w$ is real iff $\operatorname{Im}(w) = \frac{1}{2i}(w - \bar w) = 0\,$. In this case:</p>
<p>$$\require{cancel}
\begin{align}
-i \frac {z-1}{z+1} - \overline{\left(-i \frac {z-1}{z+1}\right)} &= -i \frac {z-1}{z+1} - \left(+i \frac{\bar z - 1}{\bar z + 1}\right) \\
&= -i \cdot\frac{(z-1)(\bar z +... |
1,800,611 | <p>I've setup a recurrence relation as part of a numerical analysis problem, and found that $$x_{n+1} = \frac{x_n+1}{2}$$</p>
<p>The notes then say that for $x_0=0$, it is easy to show that $$x_k = 1 - 2^{-k}$$</p>
<p>Plugging in a few numbers shows this is clearly correct, and equating $x_{n+1}=x_n$ shows it converg... | sqtrat | 210,479 | <p><strong>Hint:</strong> It's not too difficult to show (by induction if you want) that if $x_{n+1}=ax_n +b$ for all $n \geq 0$ and $a\not=1$, then $$x_n = a^n\left(x_0-\frac{b}{1-a}\right)+\frac{b}{1-a}.$$</p>
<p>Then, in your case $a=\frac{1}{2}=b$. Therefore $x_n= (\frac{1}{2})^n(x_0-1)+1.$</p>
|
1,800,611 | <p>I've setup a recurrence relation as part of a numerical analysis problem, and found that $$x_{n+1} = \frac{x_n+1}{2}$$</p>
<p>The notes then say that for $x_0=0$, it is easy to show that $$x_k = 1 - 2^{-k}$$</p>
<p>Plugging in a few numbers shows this is clearly correct, and equating $x_{n+1}=x_n$ shows it converg... | Will Orrick | 3,736 | <p>Here's one possible process for discovering the general formula: try applying the recurrence a few times:
$$
\begin{aligned}
x_1&=\frac{x_0+1}{2}=\frac{x_0}{2}+\frac{1}{2}\\
x_2&=\frac{x_1+1}{2}=\frac{x_1}{2}+\frac{1}{2}=\frac{x_0}{4}+\frac{1}{4}+\frac{1}{2}=\frac{x_0}{4}+\frac{3}{4}\\
x_3&=\frac{x_2+1}{... |
245,853 | <p>The graphs of my provided code look approximately like the one in the picture below. How is it possible to plot such graphs? Please help how to break the axis scale as shown. Any suggestion appreciated. Thanks a lot.
<a href="https://i.stack.imgur.com/8hNeO.jpg" rel="nofollow noreferrer"><img src="https://i.stack.im... | kglr | 125 | <p>Use the functions <code>Te</code> and <code>Ti</code> from Bob Hanlon's answer to get two plots for each function with desired plot ranges:</p>
<pre><code>{xrangea, xrangeb} = {{0, 10^-14}, {10^-13, 10^-12}};
{yrange1, yrange2} = {{0, 8 10^6}, {0, 350}};
{plot1a, plot1b, plot2a, plot2b} = Quiet@Plot[
Evaluat... |
3,164,343 | <p>Machine generates one random integer in range <span class="math-container">${[0;40)}$</span> on every spin.<br>
You should choose 5 numbers in that range. </p>
<p>Then the machine will spit out <strong>5</strong> numbers (<em>numbers are independent of each other</em>). </p>
<p><strong>what is the probability t... | Ethan Bolker | 72,858 | <p>Here is the argument if neither you nor the machine can repeat.</p>
<p>Since all the numbers are equally likely let's assume you chose <span class="math-container">$\{1,2,3,4,5\}$</span>. </p>
<p>There are <span class="math-container">$T = \binom{40}{5}$</span> total ways the machine can choose its five numbers. L... |
3,164,343 | <p>Machine generates one random integer in range <span class="math-container">${[0;40)}$</span> on every spin.<br>
You should choose 5 numbers in that range. </p>
<p>Then the machine will spit out <strong>5</strong> numbers (<em>numbers are independent of each other</em>). </p>
<p><strong>what is the probability t... | lonza leggiera | 632,373 | <p>There are a few different errors in your calculation. </p>
<ul>
<li>If the numbers you choose are <span class="math-container">$\ n_1, n_2, \dots, n_5\ $</span> (I'm presuming these have to be all different), and the numbers generated by the machine are <span class="math-container">$\ m_1, m_2, \dots, m_5\ $</span... |
1,652,381 | <p>Suppose $f$ derivable on $\mathbb R$ and that $\lim_{x\to a}f'(x)=\ell$. Show that $$\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}=\ell.$$</p>
<p>The proof of my course goes like this:</p>
<p>By mean value theorem, there is $y_h\in ]x,x+h[$ s.t. $$f(x+h)-f(x)=f'(y_h)h$$ and thus $$\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}=\lim_{h\t... | Bobson Dugnutt | 259,085 | <p>One possibility is that the $1$st, $3$rd, $5$th, and so on, numbers are just increasing by one, so
$$7,\_,8,\_,9,\_,10,\_,...$$</p>
<p>Now notice that $16=2 \cdot 8$, and $27=3 \cdot 9$. Therefore it would be reasonable to assume the next number to be $40=4 \cdot 10$.</p>
|
3,378,655 | <p>I have an assignment about epsilon-delta proofs and I'm having trouble with this one. I have worked it through using some of the methods I've picked up for similar proofs but it's just something about this particular expression that doesn't sit right with me. Any feedback would be very helpful. This is how far I've ... | user | 505,767 | <p>Assuming wlog <span class="math-container">$\frac32\le x\le \frac52$</span> we have</p>
<p><span class="math-container">$$\left|\frac{2 - x}{x - 1}\right| < \varepsilon \iff \left|2 - x\right| < \varepsilon \left|x - 1\right|\le \frac 32 \varepsilon $$</span></p>
<p>then it suffices to take <span class="mat... |
2,065,027 | <p>$$\newcommand{\gcd}{\text{gcd}}$$</p>
<blockquote>
<p>Prove: if $d=\gcd(m,n)$ so $\gcd\left(\frac{m}{d},\frac{n}{d}\right)=1$</p>
</blockquote>
<p>Intuitively it is obvious, but I am having a hard time to formalize the proof, what I have came to this:</p>
<p>$d=\gcd(m,n)$ so $d|m$ and $d|n$ therefore $m=dx$ and... | gowrath | 255,605 | <p>Your proof seems fine although you could show the contradiction more evidently.</p>
<p>We want to prove that if $d = \gcd(m,n)$ then $\gcd(m/d, n/d) = 1$.</p>
<p>Assume for the sake of contradiction that this is not the case i.e. $d = \gcd(m,n)$ but $\gcd(m/d, n/d) = k$ where $k > 1$. Then we have:</p>
<p>$$\f... |
1,736,502 | <p>I start with a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t),P)$. I assume that the filtration is right-continuous. On this probability space I define a supermartingale $M$. Now suppose that I can find another process $N$ such that $N$ is a cadlag modification of $M$. Hence it is also a supermartin... | TheBridge | 4,437 | <p>I would rather go for this decomposition $M_{t_n}-M_t=M_{t_n}-N_{t_n}+N_{t_n}-N_t+N_t-M_t$. Then as $E[M_{t_n}-N_{t_n}]$ and $E[M_{t}-N_{t}]$ are null by the modification hypothesis, we only have to prove the result for the process $N$, i.e. :</p>
<p>$lim_{n\to\infty} E[N_{t_n}-N_t]=0$ </p>
<p>But $N_t$ is a super... |
168,495 | <p>I am being asked to calculate the area of a triangle with a side-length of 15.5 inches. </p>
<p>The formula for calculating a regular polygon's area is 1/2Pa</p>
<p>Where P is the perimeter of the polygon, and a is the apothem. I am completely lost.</p>
| Robert Israel | 8,508 | <p>A simpler approach:
from $a x_1^2 + b x_1 + c = y_1$ and $a x_2^2 + b x_2 + c = y_2$ you get
$a (x_1^2 - x_2^2) + b (x_1 - x_2) = y_1 - y_2$, so (noting that $x_1^2 - x_2^2 = (x_1 - x_2)(x_1 + x_2)$), $b = \dfrac{y_1 - y_2}{x_1 - x_2} - a (x_1 + x_2)$.<br>
Similarly $b = \dfrac{y_1 - y_3}{x_1 - x_3} - a (x_1 + x_3)... |
3,187,628 | <p>On a keyboard 1-9 and A-D , how many possible way to have a 5 digit code with 3 different numbers and two identical letters (order isn't a problem)?</p>
<p>My try : 9C3 x 4C1 = 84 x 4 = 336</p>
| lab bhattacharjee | 33,337 | <p><span class="math-container">$$\dfrac{(-1)^n\pi^{2n+1}}{3^{2n}}=-3i\left(\dfrac{i\pi}3\right)^{2n+1}$$</span></p>
<p>So, the required sum <span class="math-container">$$S=-3i\dfrac{e^{i\pi/3}-e^{-i\pi/3}}2$$</span></p>
<p>using
<a href="https://proofwiki.org/wiki/Power_Series_Expansion_for_Exponential_Function" r... |
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