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 |
|---|---|---|---|---|
9,302 | <p>Say I have a symmetric matrix. I have the concept of 2-norm as defined on wikipedia. Now I want to prove (disprove?) that the norm of a symmetric matrix is maximum absolute value of its eigenvalue. I would really appreciate if this can be done only using simple concepts of linear algebra.</p>
<p>I am quite new to m... | Jonas Meyer | 1,424 | <p>Using the <a href="http://en.wikipedia.org/wiki/Spectral_theorem#Hermitian_matrices" rel="nofollow">spectral theorem</a> to obtain an orthogonal basis of eigenvectors for the matrix is probably the best approach, similar to what Ross said, given your criterion of "simple concepts of linear algebra". Here is another... |
4,560,624 | <p>Is there any elementary way to prove that <span class="math-container">$(\sin x + \cos x)(6 - \sin x)<9$</span>?</p>
<p>I've noticed that <span class="math-container">$(\sin x + \cos x)$</span> has to be positive so <span class="math-container">$x \in\left(-\dfrac{\pi}{4}, \dfrac{3\pi}{4}\right)$</span> and then ... | Zhanxiong | 192,408 | <p>Using the product-to-sum identity <span class="math-container">$\sin\theta\sin\phi = \frac{1}{2}(\cos(\theta - \phi) - \cos(\theta + \phi))$</span> to simplify:
<span class="math-container">\begin{align*}
& (\sin x + \cos x)(6 - \sin x) \\
= & \sqrt{2}\sin(x + \pi/4)(6 - \sin x) \\
= & 6\sqrt{2}\sin(x ... |
4,560,624 | <p>Is there any elementary way to prove that <span class="math-container">$(\sin x + \cos x)(6 - \sin x)<9$</span>?</p>
<p>I've noticed that <span class="math-container">$(\sin x + \cos x)$</span> has to be positive so <span class="math-container">$x \in\left(-\dfrac{\pi}{4}, \dfrac{3\pi}{4}\right)$</span> and then ... | River Li | 584,414 | <p><strong>Proof 1</strong>: Note that <span class="math-container">$\sin x + \cos x = \sqrt 2 \, \sin (x + \pi/4) \le \sqrt 2$</span>
and <span class="math-container">$6 - \sin x > 0$</span>.</p>
<p>If <span class="math-container">$\sin x \ge 0$</span>, we have
<span class="math-container">$(\sin x + \cos x)(6 - \s... |
917,365 | <p>I hope to get the exact value of the following double series:</p>
<p>$$ \sum_{k \geq 0} \sum_{n \geq 0} \binom{2k}{ k} \frac{(-1)^n}{n! (2k+2n+1) 2^{4k+2n+1}}. $$</p>
<p>I am not sure it is possible or not.
I need your comments.</p>
| hkju | 28,848 | <p>This problem came from the question asked by one of my friends. What is the exact
value of the following definite integral </p>
<p>$$ \int_0^{1/2} \frac{exp(-x^2)}{\sqrt{1-x^2}} dx ? $$ </p>
<p>So, I got the corresponding Taylor series, multiply both of them.
I need closed form for the value, that is, which doesn... |
2,351,629 | <p>First of all I don't understand why we need Banach's theorem, as a result I can't make it intuitive for me to understand how it works but I tried to solve an example.</p>
<p><a href="https://en.wikipedia.org/wiki/Banach_fixed-point_theorem#Statement" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Banach_fi... | Rodrigo de Azevedo | 339,790 | <p>Let</p>
<p><span class="math-container">$$\mathrm A := 1_n 1_n^\top - \mathrm I_n + \mathrm x \mathrm y^\top = - \mathrm I_n + \begin{bmatrix} | & |\\ \mathrm x & 1_n\\ | & |\end{bmatrix} \begin{bmatrix} | & |\\ \mathrm y & 1_n\\ | & |\end{bmatrix}^\top = - \left( \mathrm I_n - \begin{bmatr... |
2,706,504 | <p>In Lang's Algebra, I had hard times with this question:</p>
<blockquote>
<p>Let $K$ be field with characteristic $p$ (a prime). Let $L|K$ be a finite extension of $K$, and suppose $gcd([L:K],p)=1$. Show $L$ is separable over $K$.</p>
</blockquote>
<p>In order to have $L$ separable over $K$, one needs to have $K(... | faradawn | 1,152,902 | <p>For every <span class="math-container">$\alpha \in L$</span>, the minimum polynomial <span class="math-container">$f_\alpha(x)$</span> has a degree dividing the degree of the extension. As <span class="math-container">$[L : K]$</span> is relatively prime to <span class="math-container">$p$</span>, we get that the de... |
1,249,730 | <p>let $ x \in G$ such that $(a^{-1})*(x^2)*(a) = x^3$ for some self inverse $a.$ Prove that $x$ has order $5.$</p>
<p>I don't know how to start this proof. Seems really difficult. </p>
| Alvaro Fuentes | 110,677 | <p>$a^{-1}x^2a=x^3 \implies a^{-1}x^4a=x^6 \implies a^{-1}x^6a=x^9$ but $x^6=x^3x^3=(a^{-1}x^2a)(a^{-1}x^2a)=a^{-1}x^4a \implies a^{-1}(a^{-1}x^4a)a=x^9 \implies a^{-1}a^{-1}x^4aa=x^9$ now using $a^2=a^{-2}=e$ we have $x^4=x^9$ so $x^5=e$ and because $5$ is prime $x=e$ or $x$ has order $5$.</p>
|
2,001,441 | <p>I am having the hardest time solving the following trigonometric equation. Can anyone help please? Thank you.</p>
<p>Solve for x. [Hint: Let $\\u = \tan^{-1}(x)$ and $\\v = tan^{-1}(2x)$. Solve the equation $\\u+v = \frac{π}{4}$ by taking the tangent of each side.]</p>
<p>$\tan^{-1}(x) +\tan^{-1}(2x)= \frac{π}{... | Claude Leibovici | 82,404 | <p>Among the useful trigonometric identities, try to memorize $$\tan^{-1}(a)\pm\tan^{-1}(b)=\tan^{-1}\frac{a\pm b}{1\mp ab}$$ So, using $a=x$ and $b=2x$ $$\tan^{-1}(x)+\tan^{-1}(2x)=\tan^{-1}\left(\frac{3x}{1-x^2}\right)$$ Then, the equation reduces to $$\frac{3x}{1-x^2}=1$$</p>
|
335,611 | <p>find this limit:
$$\displaystyle\lim_{n\to+\infty}\left[\sum_{k=1}^{n}\left(\dfrac{1}{\sqrt{k}}- \int_{0}^{\large {1/\sqrt k}}\dfrac{t^2}{1+t^2}dt\right)-2\sqrt{n}\right]$$</p>
| Community | -1 | <p>We have
$$\int_{0}^{\large {1/\sqrt k}}\dfrac{t^2}{1+t^2}dt=\frac{1}{\sqrt k}-\arctan\frac{1}{\sqrt k}$$
Now if we denote by
$$u_n=\sum_{k=1}^n\arctan\frac{1}{\sqrt k}-2\sqrt{n}$$
we have
$$u_n-u_{n-1}=\arctan\frac{1}{\sqrt n}-2\sqrt{n}+2\sqrt{n-1}\sim\frac{-7}{12n\sqrt{n}},$$
and since the series $\sum\frac{1}{n\sq... |
3,992,972 | <p>I want to draw the set <span class="math-container">$M_{4}$</span>={(x, y) ; |xy|< 1/4}</p>
<p>My attempt is that I evaluate the expression of |xy|<1/4 depending on the values of x an y, i.e depending on the quadrant.</p>
<p>For the second quadrant I get the following:</p>
<p>x < 0 <span class="math-contain... | DonAntonio | 31,254 | <p>An idea: draw first the two hyperbolas <span class="math-container">$\;y=\pm\cfrac1{4x}\;$</span> , and then "bound" the points with both coordinates different from zero belonging to <span class="math-container">$\;M_4\;$</span> by them. Of course, the above involves assuming <span class="math-container">$... |
4,448,525 | <p>To do a proof by induction, it is necessary to prove</p>
<p>(1) the proposition <span class="math-container">$P(n)$</span> is true for <span class="math-container">$n=1$</span></p>
<p>(2) if the proposition <span class="math-container">$P(n)$</span> is true for <span class="math-container">$n=k$</span>, then <span c... | Shambhala | 673,139 | <p>(1) The base case is not necessarily <span class="math-container">$n=1$</span>. One might want to prove a statement is true for all numbers <span class="math-container">$n$</span> greater than or equal to a certain number <span class="math-container">$m$</span>.</p>
<p>To understand the importance of the base case y... |
4,448,525 | <p>To do a proof by induction, it is necessary to prove</p>
<p>(1) the proposition <span class="math-container">$P(n)$</span> is true for <span class="math-container">$n=1$</span></p>
<p>(2) if the proposition <span class="math-container">$P(n)$</span> is true for <span class="math-container">$n=k$</span>, then <span c... | Curious Goblin | 1,056,948 | <p>I'm not sure I can answer ##3 and 4 authoritatively. But I've never heard of proof by induction being used for n other than integers. And I've never heard of a circumstance where proof by contradiction is not a valid method of argument.</p>
<p>Regarding #1, you need to show that that <span class="math-container">$... |
4,299,417 | <p>Let <span class="math-container">$f: \mathbb{R} \rightarrow [0, \infty)$</span> be a function that:
<span class="math-container">$$f^2(x+y)+f^2(x-y)=2f^2(x)+2f^2(y) \forall x,y \in \mathbb{R}$$</span>
Prove that <span class="math-container">$f(x+y) \leq f(x)+f(y) \forall x,y \in \mathbb{R}$</span></p>
<p>This proble... | DreamAR | 983,565 | <p>For an inner product space, we can define <span class="math-container">$\|x\|=\sqrt{(x,x)}$</span> to be a norm, which fulfill <em>triangular inequallity</em>: <span class="math-container">$\|x+y\|\le \|x\|+\|y\|$</span> by Cauchy-Schwartz inequallity. We have <em>parallelogram identity</em>:
<span class="math-conta... |
3,755,638 | <p>Given a point <span class="math-container">$A$</span>, a circle <span class="math-container">$O$</span> and conic section <span class="math-container">$e$</span>, if <span class="math-container">$BC$</span> is a moving chord of the circle <span class="math-container">$O$</span> tangent to <span class="math-container... | Blue | 409 | <p>Here's something of a brute-force vector proof.</p>
<p>In the figure, <span class="math-container">$P$</span> is a point on our given conic, <span class="math-container">$\bigcirc K$</span> of radius <span class="math-container">$r$</span> is our given circle, and <span class="math-container">$A$</span> is our given... |
3,543,060 | <p>The ODE is defined on <span class="math-container">$[0,b]$</span> with Neumann boundary conditons.
<span class="math-container">$$y''(x)=\frac{c_1y(x)}{c_2+y(x)}$$</span>
<span class="math-container">$$y'(0)=0; y'(b)=0$$</span></p>
<p>How to solve the above ODE? Any help is appreciated!</p>
| Jan | 529,121 | <p>Reduce the order of the equation by substituting <span class="math-container">$w = w(y)$</span>. Then it yields </p>
<p><span class="math-container">$$ww' = \frac{c_1y}{c_2 + y},$$</span></p>
<p>which is a separable ODE. This equation can be integrated in order to get</p>
<p><span class="math-container">$$\frac{w... |
2,461,773 | <p>I would like some clarification about the Cantor Set:</p>
<ul>
<li>What are the elements in the Cantor Set?</li>
<li>How do I write the Cantor Set in mathematical terms (i.e in a summation)? I have seen online a formula but I do not understand how they got it so would be grateful if you could explain why it is this... | Austin Weaver | 480,825 | <p>The Cantor Set $\mathcal C$ can be defined by the following recursive sequence of sets:</p>
<p>$$\mathcal C_0 = [0, 1]$$
$$\mathcal C_{n+1}= \frac{\mathcal C_n}3\cup\left(\frac23+\frac{C_n}3\right)$$
$$\mathcal C := \bigcap^\infty_{n=0}\mathcal C_n$$</p>
<p>This definition should reflect the intuitive notion of wh... |
29,601 | <p>I know the order of the group is the number of elements in the set. For example the group of $U_{10}$ (units of congruence class of 20) has order 4. </p>
<p>Major Edit, kinda changed the question.
Lets say my element $a$ has a finite order $n$. Then what is the order of $a, a^2, a^3...a^{11}$?</p>
| MichalisN | 8,432 | <p><strong>Answer to the edited question:</strong>
That depends on n.
Generally the order of $a^i$ is $\frac{n}{\text{gcd}(n,i)}$. If you want to prove this, you have to check two things: </p>
<p>Firstly that $(a^i)^{\frac{n}{\text{gcd}(n,i)}}=a^{\frac{in}{\text{gcd}(n,i)}}=1$. This holds because we have a multiple ... |
804,414 | <p>A Fair Dice is Thrown Repeatedly. Let $X$ be number of Throws required to get a '$6$' and $Y$ be number of throws required to get a '$5$'. Find $$E(X|Y=5)$$</p>
| James Dow Allen | 152,519 | <p>Since Y=5, we know the first four shakes are not 5, and the fifth is not 6.
The expectation is thus
(1/5) * 1 + (4/25) * 2 + (16/125) * 3 + (64/625) * 4 + (256/625) * (E(X) + 5)
with E(X) = 6.
This is 3637/625 = 5.8192</p>
|
223,087 | <p>Given a list of numbers in decimal form, what is the most efficient way to determine if there are any consecutive 1s in the binary forms of those numbers? My solution so far:</p>
<pre><code>dim = 3;
declist = Range[0, 2^dim - 1];
consecutiveOnes[binary_] := AnyTrue[Total /@ Split[binary], # > 1 &];
consecuti... | eyorble | 52,935 | <p>For the sake of getting answers for higher values of <code>dim</code>, I present to you some bit manipulation hacking for <code>dim=20</code>:</p>
<pre><code>dim = 20;
</code></pre>
<p>Find the binary numbers which encompass the range of interest in <code>dim</code> that are alternating 1s and 0s, one of which end... |
275,151 | <p><a href="https://i.stack.imgur.com/XxGxa.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/XxGxa.png" alt="enter image description here" /></a>I'm a beginner at Mathematica, and I'm trying to figure out how to fill between two lines horizontally. Consider the toy example</p>
<pre><code>Plot[{Conditi... | Bob Hanlon | 9,362 | <pre><code>$Version
(* "13.1.0 for Mac OS X x86 (64-bit) (June 16, 2022)" *)
Clear["Global`*"]
</code></pre>
<p>Split each of the lines into two segments and cycle the <code>PlotStyle</code></p>
<pre><code>Plot[{
ConditionalExpression[2*x - 2, 2 < x < 4],
ConditionalExpression[2*x + 2, 0... |
4,036,117 | <p>In Joe Silverman's "The Arithmetic of Ecliptic Curves" he talks a lot about the integral domain <span class="math-container">$\overline{K}(C)$</span>. On page <span class="math-container">$27$</span> he suddenly decides to chose an element <span class="math-container">$f$</span> from some set <span class="... | user10354138 | 592,552 | <p><span class="math-container">$\bar{K}(C)$</span> is the function field of <span class="math-container">$C$</span> over the algebraically closure of <span class="math-container">$K$</span> (as recalled in the beginning of chapter).</p>
<p><span class="math-container">$\bar{K}(C)^*$</span> is the multiplicative group ... |
4,036,117 | <p>In Joe Silverman's "The Arithmetic of Ecliptic Curves" he talks a lot about the integral domain <span class="math-container">$\overline{K}(C)$</span>. On page <span class="math-container">$27$</span> he suddenly decides to chose an element <span class="math-container">$f$</span> from some set <span class="... | Álvaro Lozano-Robledo | 14,699 | <p>There is already a correct answer, but I will just add here that the notation is defined in Chapter I, in a Definition right after Example 1.3.3. where Silverman defines the affine coordinate ring and the function field of a variety.</p>
|
96,198 | <p>I'm trying to prove that the variance of a RV whose values are discrete 1's or 0's is greater than the variance of a RV who's values are 0's or continuous on the domain (0,1], where any "1" in the Bernoulli RV corresponds to a value on (0,1] in the other RV. Intuitively, I think this is the case, but I'm trying to d... | Robert Israel | 8,508 | <p>I presume you're assuming $X$, $Y$, $Z$ independent.
Note that $(X+Y+Z)/\sqrt{3}$, $(Y - Z)/\sqrt{2}$ and $(2X-Y-Z)/\sqrt{6}$ are uncorrelated (and therefore independent) standard normal random variables.
Since $X = (X+Y+Z)/3 + (2X-Y-Z)/3$, $\text{Var}(X|X+Y+Z=1) = (1/9) \text{Var}(2X-Y-Z) = 2/3$.</p>
|
25,556 | <p>Why is the zero set in <span class="math-container">$\mathbb{C}\times\mathbb{C}$</span> of a polynomial <span class="math-container">$f(x,y)$</span> in two complex variables always non-discrete (no zero of <span class="math-container">$f$</span> is isolated)?</p>
| Mariano Suárez-Álvarez | 274 | <p>Write $f(x,y)$ as a polynomial in $y$ with coefficients in $\mathbb C[x]$, so that $f(x,y)=\sum_{i=0}^nf_i(x)y^i$. If $x\in\mathbb C$ is such that $f_n(x)\neq0$, then there exist $n$ values of $y$ such that $f(x,y)=0$. It follows that if $X\subseteq \mathbb C\times\mathbb C$ is the zero locus of $f$, then the first ... |
1,134,323 | <p>Say you pick a number $x$, like $\frac 43$. Its inverse is of course $\frac 34$. $x$ is a distance of $\frac 13$ away from 1, and its inverse is a distance of $\frac 14$ away from 1. Is there any number $x$ that is a distance $d$ away from 1, whose inverse $\frac 1x$ is also a distance $d$ away from 1? I came up... | malo_x | 213,118 | <p>Yes. Your equation only has the solution $x=1$ but that doesn't mean that the only solutions to the problem are solutions to said equation. You can easily check that -1 also satisfies your problem. Maybe consider the following equation, since you're measuring distance:
\begin{equation*}
|x-1| = \left|1-\frac{1}{x}\r... |
4,243,005 | <p>I was having trouble with this integral</p>
<blockquote>
<p>Prove
<span class="math-container">$$\int_{-\infty}^{\infty} e^{2x}x^2 e^{-e^{x}}dx=\gamma^2 -2\gamma+\zeta(2)$$</span></p>
</blockquote>
<p>Where <span class="math-container">$\gamma$</span> is the euler mascheroni constant. Let <span class="math-container... | Diger | 427,553 | <p><span class="math-container">$$\Gamma''(2)=\Gamma''(s+2)\Big|_{s=0}=\left((s+1)\Gamma(s+1)\right)''\Big|_{s=0}=\left(\Gamma(s+1)+(s+1)\Gamma'(s+1)\right)'\Big|_{s=0} \\
=2\Gamma'(s+1)+(s+1)\Gamma''(s+1)\Big|_{s=0} = -2\gamma + \Gamma''(1) \, .$$</span>
Now <span class="math-container">$$\Psi(s+1)+\gamma = \sum_{n=1}... |
3,434,656 | <p>We are given a <span class="math-container">$3 \times 3$</span> real matrix <span class="math-container">$A$</span>, and we know it has three eigenvalues. One eigenvalue is <span class="math-container">$\lambda_1=-1$</span> with corresponding eigenvector <span class="math-container">$v_1=\left[\begin{matrix}
0 \... | Bernard | 202,857 | <p><strong>Hint</strong>:</p>
<p>The characteristic polynomial <span class="math-container">$\chi_A$</span> has real coefficients, hence its nonreal roots are pairwise conjugate, so <span class="math-container">$1-i$</span> is the third eigenvalue. Furthermore, if <span class="math-container">$v_2$</span> is an eigenv... |
95,341 | <p>I know that the complement of the zero set of a polynomial $P: \mathbb{C}^n \rightarrow \mathbb{C}$ is connected in $\mathbb{C}^n$ (by the way, can anybody suggest a reference?).</p>
<p>Is it possible to extend the proof also to polynomials
$P: SL(N,\mathbb{C})^n \rightarrow \mathbb{C}$ ?</p>
<p>Thanks!</p>
| ACL | 10,696 | <p>An answer to the more general question could be in three steps.</p>
<p>1) If $X$ is a complex algebraic variety, connected for the Zariski topology, then it is connected for the usual topology, an important and nontrivial fact that can be found in Mumford's Red Book, or in Shafarevich's one. </p>
<p>2) The complem... |
415,150 | <blockquote>
<p>Let $x$ and $y$ be positive numbers. Let $a_0=y$, and let $$a_n=\frac{(x/a_{n-1})+a_{n-1}}{2}$$Prove that the sequence $\{a_n\}$ has limit $\sqrt{x}$. Generalize to $n$th roots.</p>
</blockquote>
<p>I already solved the first part <a href="https://math.stackexchange.com/questions/415125/limit-of-recu... | Evariste | 29,168 | <p>The integral you've done is absolutely right, although you might always want to include the upper and lower limits before you evaluate the integral(i.e. never leave it in the indefinite form like this: $$-\frac{4}{25}\frac{x^{2+1}}{2+1}+x$$
).</p>
<p>You were probably wondering if you did everything correctly. I th... |
4,482,285 | <p>I'm currently making a game and have run into a problem I'm not quite sure how to solve, I'll try to lay it out as a maths question. None of the values are fixed, so I'm looking for an equation that solves the below question:</p>
<p>A plane lies on the position vector <code>p0</code> <code><x0, y0, z0></code> ... | Glorious Nathalie | 948,761 | <p>The center of the sphere lies on the line</p>
<p><span class="math-container">$ C(t) = (s_x, 0, s_z) + t (0, 1, 0) = (s_x, t, s_z) $</span></p>
<p>We want the distance of a point on this line to be equal to the known given value of <span class="math-container">$R$</span> (the radius of the sphere). The formula for ... |
1,838,596 | <p>I need to compute the surface area of the torus $$T^2=\{(x,y,z)\subseteq\mathbb R^3 \left(\sqrt {x^2+y^2}- R\right)^2+z^2=r^2\}$$
where $0<r<R$.</p>
<p>I know I need to compute the metric tensor and the Gramian determinant etc, but in order to so, I need a regular global parametrization of $T^2$, I guess? How... | Mark Viola | 218,419 | <p>We can describe the surface of the torus using parameters $(\phi,\alpha)$ by the position vector $\vec r(\phi,\alpha)$</p>
<p>$$\begin{align}
\vec r(\phi,\alpha)&=\hat \rho(\phi) R+(\hat \rho(\phi) r\cos(\alpha)+\hat z r\sin(\alpha))\\\\
&=\hat \rho(\phi)\,(R+r\cos(\alpha))+\hat z r\sin(\alpha)
\end{align}$... |
1,838,596 | <p>I need to compute the surface area of the torus $$T^2=\{(x,y,z)\subseteq\mathbb R^3 \left(\sqrt {x^2+y^2}- R\right)^2+z^2=r^2\}$$
where $0<r<R$.</p>
<p>I know I need to compute the metric tensor and the Gramian determinant etc, but in order to so, I need a regular global parametrization of $T^2$, I guess? How... | David Jaramillo | 349,591 | <p>Note you have a sum of squares equal a constant. Hence you should use trigonometric functions.
$$
\left(\sqrt{x^2+y^2}-R\right)^2+z^2=r^2
$$
So we set $z=r\cos(\phi),\sqrt{x^2+y^2}-R=r\sin(\phi)$. Squaring the second relation we get
$$
x^2+y^2=(R+r\sin(\phi))^2
$$
So we may introduce another angle an set $x=(R+r\sin... |
2,545,813 | <p>The question is:</p>
<p>If $R$ has an identity and $A$ is an $R$-module, then there are submodules $B$ and $C$ of $A$ such thatt $B$ is unitary $RC = 0$ and $A = B \oplus C.$
And if $A_{1}$is another $R$-module, with $A_{1} = B_{1} \oplus C_{1}$, $B_{1}$ unitary ,$RC_{1} = 0.$</p>
<p>Why if $f:A\rightarrow A_{1}$ ... | 1ENİGMA1 | 255,913 | <p>Hint:Assume that $f(B) \not\subset B_{1}$. We have some $0\neq b\in B$ such that that $f(b)\in C_{1}$ (i.e., $f(b)\notin B_1$).<br/> So, $f(b)=f(1_Rb)=1_Rf(b)\in C_1$.But, we know that $RC_1=0$, so $1_Rf(b)=f(b)=0$.<br/> This is a contradiction of $b\neq 0$. ($f(b)=0=f(0)$).<br/>
You can use above idea for $f(C) \su... |
3,946,998 | <p>I was doing a question. Suddenly I got stuck at this last part of the problem. It was to prove
<span class="math-container">$ 2^r +2 = a^2 +b^2$</span> where <span class="math-container">$r \neq 2$</span>, r is a prime and <span class="math-container">$ a \neq b$</span>. Also <span class="math-container">$r^2 -1$</s... | NN2 | 195,378 | <p>The question is wrong. If <span class="math-container">$r=11$</span>, <span class="math-container">$2^{11}-7 = 2041$</span> is not a square number.</p>
|
4,645,950 | <p>The question is: Peter and Mary take turns rolling a fair die. If Peter rolls 1 or 2 he wins and the game stops. If Mary rolls 3,4,5, or 6, she wins and the game stops. They keep rolling until one of them wins. Suppose Peter rolls first.</p>
<p>a) What is the probability that Peter wins and rolls at most 4 times?</p... | user115350 | 334,306 | <p>For a roll, the probability that Peter fail is 4/6 and Mary fail 2/6.</p>
<p>If Peter wins on his kth roll, Peter should fail on his first k-1 rolls and Mary fail on her first k-1 rolls. The probability is <span class="math-container">$( \frac{2}{6} )^{k-1}(\frac{4}{6})^{k-1}$</span>.</p>
<p>And, on k-th roll, the p... |
196,155 | <p>I have recently read some passage about nested radicals, I'm deeply impressed by them. Simple nested radicals $\sqrt{2+\sqrt{2}}$,$\sqrt{3-2\sqrt{2}}$ which the later can be denested into $1-\sqrt{2}$. This may be able to see through easily, but how can we denest such a complicated one $\sqrt{61-24\sqrt{5}}(=4-3\sqr... | Mr Pie | 477,343 | <p>There is also another radical identity, which generalises <span class="math-container">$\sqrt{a+\sqrt b}$</span> even further: <span class="math-container">\begin{align*}\sqrt{a\pm \sqrt{b+\sqrt{c}}}&= \frac{1}{\sqrt{2}}\left\{\sqrt{a^2+\sqrt{\frac{a^2-b+\sqrt{(a^2-b)^2-c}}{2}}-\sqrt{\frac{a^2-b-\sqrt{(a^2-b)^2-... |
513,239 | <p>I've recently heard a riddle, which looks quite simple, but I can't solve it.</p>
<blockquote>
<p>A girl thinks of a number which is 1, 2, or 3, and a boy then gets to ask just one question about the number. The girl can only answer "<em>Yes</em>", "<em>No</em>", or "<em>I don't know</em>," and after the girl ans... | Ben Millwood | 29,966 | <p>"I am thinking of a number which is either 0 or 1. Is the sum of our numbers greater than 2?"</p>
|
513,239 | <p>I've recently heard a riddle, which looks quite simple, but I can't solve it.</p>
<blockquote>
<p>A girl thinks of a number which is 1, 2, or 3, and a boy then gets to ask just one question about the number. The girl can only answer "<em>Yes</em>", "<em>No</em>", or "<em>I don't know</em>," and after the girl ans... | Oleg567 | 47,993 | <p><strong>Question 1</strong>:</p>
<p>I'm thinking of a huge integer number with last digit $7$.</p>
<p>Is this ratio $~~\dfrac{\mbox{my number}}{\mbox{your number}}~~$ integer?</p>
<ul>
<li>$1$ (yes)</li>
<li>$2$ (no)</li>
<li>$3$ (I don't know)</li>
</ul>
<hr>
<p><strong>Question 2</strong>:</p>
<p>Consider se... |
513,239 | <p>I've recently heard a riddle, which looks quite simple, but I can't solve it.</p>
<blockquote>
<p>A girl thinks of a number which is 1, 2, or 3, and a boy then gets to ask just one question about the number. The girl can only answer "<em>Yes</em>", "<em>No</em>", or "<em>I don't know</em>," and after the girl ans... | Neal | 20,569 | <p>This was suggested by a friend:</p>
<blockquote>
<p>If $k$ is your number, does $\mathbb{S}^{3k-2}$ have an exotic smooth structure?</p>
</blockquote>
|
513,239 | <p>I've recently heard a riddle, which looks quite simple, but I can't solve it.</p>
<blockquote>
<p>A girl thinks of a number which is 1, 2, or 3, and a boy then gets to ask just one question about the number. The girl can only answer "<em>Yes</em>", "<em>No</em>", or "<em>I don't know</em>," and after the girl ans... | Blue | 409 | <p>I get the sense that there's <a href="http://www.youtube.com/watch?v=kTcRRaXV-fg">a classic comedy routine</a> in here somewhere.</p>
<blockquote>
<p>"You know, they give baseball players these days very peculiar names. [...] Well, now, let's see ... On our team we have: <strong>NO</strong>'s on first, <strong>YE... |
513,239 | <p>I've recently heard a riddle, which looks quite simple, but I can't solve it.</p>
<blockquote>
<p>A girl thinks of a number which is 1, 2, or 3, and a boy then gets to ask just one question about the number. The girl can only answer "<em>Yes</em>", "<em>No</em>", or "<em>I don't know</em>," and after the girl ans... | Bennett Gardiner | 78,722 | <p>Along the lines of "the open problem" method - </p>
<p>Define $$ f(n) = \pi^{n-1}\mathrm{e}^{\pi(n-1)}.$$</p>
<p>Where $n$ is the number chosen by the girl. The question is, is $f(n)$ irrational? </p>
<p>If $n=1$, $f(1) = 1$, so the answer is <strong>"No"</strong>. </p>
<p>If $n=2$, $f(2) = \pi\mathrm{e}^{\pi}$,... |
4,036,704 | <blockquote>
<p><span class="math-container">$$(A ∧ (A → B)) ⇒ B$$</span></p>
</blockquote>
<p>I know that this is a tautology, but apart from setting up a truth-table I dont know how I would go about a formal proof in discrete mathematics. thanks</p>
<p><a href="https://i.stack.imgur.com/j6cHN.png" rel="nofollow noref... | Gio | 872,828 | <p><span class="math-container">$(A \land (A\rightarrow B)) \rightarrow B $</span> is true when <span class="math-container">$A \land (A\rightarrow B)$</span> is false or <span class="math-container">$B$</span> is true. Therefore if it was false then <span class="math-container">$A \land (A\rightarrow B)$</span> would ... |
910,414 | <p>My question is about visualizing projective space, in particular the real projective plane $\mathbb{P}^2(\mathbb{R})$. I know there are different ways to define this space, but in each we can say that "two parallel lines intersect." If you type projective space into Wikipedia, or look it up in a textbook, a lot of t... | John Hughes | 114,036 | <p>Suppose that your eye is at the origin, and the "canvas" on which you draw is in front of your eye, and is the $z = 1$ plane. Then the point $(2, 3, 5)$ in space will project, in your "drawing" or "seeing" of the world, to the point $(2/5, 3/5, 1)$ in the $z = 1$ plane. In general, any point $(x, y, z)$ will project... |
2,173,586 | <p>Let $R$ be an unique factorization domain. Being an integral domain, it has a field of quotients $F$. We can consider $R[x]$ to be a subring of $F[x]$. </p>
<blockquote>
<p>Given any polynomial $f(x)\in F[x]$, then $f(x)=(f_0(x)/a)$, where $f_0(x)\in R[x]$ and where $a\in R$.</p>
</blockquote>
<p>I don't underst... | Patrick Stevens | 259,262 | <p>The countable intersection and union are also limits. $$\bigcup_{n \in \mathbb{N}} A_n = \lim_{n \to \infty} \bigcup_{i=1}^n A_n$$</p>
<p>Set operations can be defined without reference to the size of the domain of "summation": $$\bigcup A := \{ x: x \in X \in A \}$$
$$\bigcap A := \{x : (\forall X \in A)(x \in x) ... |
2,173,586 | <p>Let $R$ be an unique factorization domain. Being an integral domain, it has a field of quotients $F$. We can consider $R[x]$ to be a subring of $F[x]$. </p>
<blockquote>
<p>Given any polynomial $f(x)\in F[x]$, then $f(x)=(f_0(x)/a)$, where $f_0(x)\in R[x]$ and where $a\in R$.</p>
</blockquote>
<p>I don't underst... | Asaf Karagila | 622 | <p>Sets are not real numbers. They are part of a much larger universe. The real numbers are bound by $\Bbb R$, they all have to be members of $\Bbb R$, and the summation has to obey the rules of summations which extend addition (under the usual definition of a sum, that is).</p>
<p>On the other hand, an infinite famil... |
308,014 | <p>First of all, on this Matlab exercise sheet that I am currently working through what does the term 'the lower $2 \times 2$ block' mean in the question below?</p>
<p>$A =
\left[\begin{array}\
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9 \\
\end{array}\right]$,</p>
<p>Define a $2 \times 2$ matrix tha... | Dennis Jaheruddin | 45,737 | <p>If the size of <code>A</code> can vary, you can always get the lower 2x2 block as such:</p>
<pre><code>A(end-1:end,end-1:end)
</code></pre>
<p>Note that this always works as long as the width and height of <code>A</code> are both at least 2.</p>
|
15,013 | <p>I am doing a plot where I have multiple shaded regions, and I want the line that separates the two regions to be dashed with dashes being alternating colors (so the demarcation stands out from both regions).</p>
<p>For example, say I am plotting the two regions shown here</p>
<pre><code>Plot[{1, Abs[BesselJ[1, x]]... | Michael E2 | 4,999 | <p>Here's a way to use the <code>"ArcLength"</code> setting to get equal-length segments. One has to scale the coordinates so the arc length in the coordinate geometry is proportional to the arc length in the screen geometry. This can be done with <a href="http://reference.wolfram.com/language/ref/ScalingFunctions.htm... |
867,938 | <p>I was reading the book Linear Algebra Done Right by Axler. In the chapter on inner product space (Ch.6), he defines the norm of x on $R^n$ space as:</p>
<p>$||x|| = \sqrt{x_1^2 + ... + x_n^2}$</p>
<p>and says:</p>
<p>"The norm is not linear on $R^n$. To inject linearity into the discussion, we introduce the dot ... | mdp | 25,159 | <p>Try checking linearity of the norm with $-3$ as the scalar instead, or by checking the additive property - you should see that it fails.</p>
<p>The dot product is linear in the sense that the map $x\mapsto x\cdot y$ for some fixed $y$ (independent of $x$) is linear; the map $x\mapsto x\cdot x$ is <em>not</em> linea... |
1,287,692 | <p>To show whether or not the 3 planes
$$x+y-2z=5\tag 1$$
$$x-y+3z=6 \tag2$$
$$x+5y-12z=12 \tag 3$$ all have a common line of intersection.</p>
<p>Can I do $(3)-(2)$ to get the line $6y-15z=6$ and $(1)-(2)$ to get the line $2y-5z=-1$ which is $6y-15z=-3$ , and say that as these aren't the same line, they don't have ... | Vlad | 229,317 | <p><strong>HINT</strong>: Find normal vectors of the planes and check if three of them are linearly independent. If there is a common line for all the planes, then their normal vectors will lie within the same plane, therefore three of them will <strong>not</strong> be linearly independent.</p>
<hr>
<p>In your specif... |
1,287,692 | <p>To show whether or not the 3 planes
$$x+y-2z=5\tag 1$$
$$x-y+3z=6 \tag2$$
$$x+5y-12z=12 \tag 3$$ all have a common line of intersection.</p>
<p>Can I do $(3)-(2)$ to get the line $6y-15z=6$ and $(1)-(2)$ to get the line $2y-5z=-1$ which is $6y-15z=-3$ , and say that as these aren't the same line, they don't have ... | zoli | 203,663 | <p><strong>An elementary solution and notes to the OP</strong></p>
<p>If $3$ planes have a unique common point then they don't have a common straight line. In order to see if there is a common line we have to see if we can solve the following system of equations:</p>
<p>$$
\begin{matrix}
x+y&-2z&=&5\\
x-y... |
4,584,609 | <p>Which is bigger</p>
<p><span class="math-container">$$ \int_0^{\frac{\pi}{2}}\frac{\sin x}{1+x^2}dx$$</span> or <span class="math-container">$$ \int_0^{\frac{\pi}{2}}\frac{\cos x}{1+x^2}dx~?$$</span></p>
<p>I let <span class="math-container">$x=\frac{\pi}{2}-t$</span> in the second integral, and I obtain this
<span ... | Claude Leibovici | 82,404 | <p>Suppose that you use the two <span class="math-container">$1,400^+$</span> years old approximations
<span class="math-container">$$\sin(x) \simeq \frac{16 (\pi -x) x}{5 \pi ^2-4 (\pi -x) x}\qquad (0\leq x\leq\pi)$$</span>
<span class="math-container">$$\cos(x) \simeq\frac{\pi ^2-4x^2}{\pi ^2+x^2}\qquad (-\frac \pi 2... |
4,584,609 | <p>Which is bigger</p>
<p><span class="math-container">$$ \int_0^{\frac{\pi}{2}}\frac{\sin x}{1+x^2}dx$$</span> or <span class="math-container">$$ \int_0^{\frac{\pi}{2}}\frac{\cos x}{1+x^2}dx~?$$</span></p>
<p>I let <span class="math-container">$x=\frac{\pi}{2}-t$</span> in the second integral, and I obtain this
<span ... | GEdgar | 442 | <p>Hardy-Littlewood-Polya ...<br />
Consider two nonnegative functions <span class="math-container">$f(x),g(x)$</span> on <span class="math-container">$[a,b]$</span>. Among all rearrangements
<span class="math-container">$f_1$</span> of <span class="math-container">$f$</span> and all rearrangements <span class="math-c... |
455,642 | <p>I'm coming from the programming world , and I need to create unique number for each element in a matrix. Say I have a $4\times4$ matrix $A$. I want to find a simple formula that will give each of the $16$ elements a unique number id. Can you suggest me where to start ? </p>
| pathfinder | 23,431 | <p>How about $$f(r,c) = Kr+c,$$ where $K$ is the number of columns in the matrix and $r$ and $c$ correspond to the row and column you want an ID for?</p>
<p>For your 4 by 4 matrix, we would have $$f(r,c) = 4r+c,$$ where we have $r,c\in\{0,1,2,3\}$. </p>
|
455,642 | <p>I'm coming from the programming world , and I need to create unique number for each element in a matrix. Say I have a $4\times4$ matrix $A$. I want to find a simple formula that will give each of the $16$ elements a unique number id. Can you suggest me where to start ? </p>
| Emily | 31,475 | <p>If you're coming from a programming background, you should know about column-major and row-major orderings.</p>
<p>In row-major order, the matrix $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ is laid out in memory as <code>[1, 2, 3, 4]</code>. In column-major order, it is <code>[1 3 2 4]</code>.</p>
<p>It... |
264,130 | <p>I have been trying to find an analytical expression for the following:</p>
<p>$\frac{\partial {X^{+}}}{\partial {X}}$</p>
<p>In my case, $X$ has a constant rank.
I've found the formula for differentiating a pseudoinverse in <a href="http://i.stanford.edu/pub/cstr/reports/cs/tr/72/261/CS-TR-72-261.pdf" rel="nofollo... | rych | 45,979 | <blockquote>
<p>but I can't see how to input the original matrix.</p>
</blockquote>
<p>I think when people write $\tfrac {\partial f} { \partial X}$ or $\tfrac {df}{dX}$ they really mean to find the Fréchet derivative, denoted by $df$, or $Df$. Your formula becomes (appears also in <a href="https://math.stackexchang... |
1,832,887 | <p>Consider the conjunction introduction and implication elimination rules of natural deduction:</p>
<p>$$\frac{\Gamma\vdash\alpha \quad \Gamma\vdash\beta}{ \Gamma\vdash \alpha \land \beta} (\land I)
\qquad
\frac{ \Gamma \vdash \alpha \to \beta \quad \Gamma \vdash \alpha} {\Gamma,\vdash\beta} (\to E) \qquad \text{(... | lab bhattacharjee | 33,337 | <p>HINT:</p>
<p>Let $$\dfrac{p(x)}{(x-1)(x-2)(x-3)(x-4)(x-5)(x-6)}=\sum_{i=1}^6\dfrac{A_i}{x-i}$$</p>
<p>Multiply both sides by $(x-1)(x-2)(x-3)(x-4)(x-5)(x-6)$ and put $x=1,2,3,4,5,6$ one by one in the resultant identity.</p>
|
1,832,887 | <p>Consider the conjunction introduction and implication elimination rules of natural deduction:</p>
<p>$$\frac{\Gamma\vdash\alpha \quad \Gamma\vdash\beta}{ \Gamma\vdash \alpha \land \beta} (\land I)
\qquad
\frac{ \Gamma \vdash \alpha \to \beta \quad \Gamma \vdash \alpha} {\Gamma,\vdash\beta} (\to E) \qquad \text{(... | Ghartal | 83,884 | <p>Let's do it in the most elementary way. Let $$Q(x)=P(x+1)-P(x)-x+2 \tag{1}$$Observe that $Q$ is of degree $4$ and $Q(3)=Q(4)=Q(5)=0$. Therefore we can write$$Q(x)=a(x-3)(x-4)(x-5)(x-b) \tag{2}$$You have also from $(1)$ that $Q(1)=Q(2)=1$, which after substitution in $(2)$ you get $a=-1/8$ and $b=2/3$. So $$Q(6)=-\fr... |
881,013 | <p>I am still an undergraduate student, and so perhaps I just haven't seen enough of the mathematical world. </p>
<p><strong>Question:</strong> What are some examples of mathematical logic solving open problem outside of mathematical logic?</p>
<p>Note that the <a href="//en.wikipedia.org/wiki/Ax%E2%80%93Grothendieck... | bof | 111,012 | <p>The <a href="https://en.wikipedia.org/wiki/James_Earl_Baumgartner" rel="nofollow noreferrer">Baumgartner</a>–<a href="https://en.wikipedia.org/wiki/Andr%C3%A1s_Hajnal" rel="nofollow noreferrer">Hajnal</a> theorem says that, for any countable ordinal $\alpha,$ and for any coloring of the edges of the complete graph o... |
672,265 | <p>Find the normal form of the matrix $A$: $$\begin{bmatrix}2 & 0 & 0 & 0\\1 & 2 & 0 & 0\\0 & 0 & 1 & 1\\0 & 0 & 0 & 1\end{bmatrix}$$</p>
<p>It looks like A's jordan decomposition should be: </p>
<p>$$\begin{bmatrix}2 & 1 & 0 & 0\\0 & 2 & 0 & 0\\... | EuYu | 9,246 | <p>A quick argument would be something along these lines.</p>
<p>You have a block matrix
$$C=\begin{pmatrix}A & 0 \\ 0 & B\end{pmatrix}$$
So the Jordan form of $C$ is just the direct sum of the Jordan forms of $A$ and $B$. Since $B$ is already in Jordan form, we only have to worry about $A$.</p>
<p>Now $A$ is... |
4,242,877 | <p>I was reading the book of Real Analysis by H.L.Royden and they have given the definition of Riemann sums and Riemann integral as follows:<a href="https://i.stack.imgur.com/cFfRc.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/cFfRc.jpg" alt="enter image description here" /></a></p>
<p>After this d... | Danny | 184,313 | <p>If <span class="math-container">$f$</span> is the Dirichlet function, then we get the same values for the upper and lower integrals if we use the first definitions of <span class="math-container">$m_i$</span> and <span class="math-container">$M_i$</span>. If <span class="math-container">$x_{i-1} < x_i$</span>, th... |
339,289 | <blockquote>
<p>Prove that if $f$ is defined for $x\ge 0$ by $f(x)=\sqrt x$, then $f$
is continuous at every point of its domain.</p>
</blockquote>
<p>Definition of a continuous function is:</p>
<p>Let $A\subseteq\mathbb{R}$ and let $f:A\to\mathbb{R}$. Denote $c\in A$.</p>
<p>Then $f(x)$ is continuous at $c$ iff... | user62089 | 62,089 | <p>Let us go from the definition you gave:</p>
<p><span class="math-container">$f(x)$</span> is continuous at <span class="math-container">$c$</span> iff for every <span class="math-container">$\varepsilon>0$</span>, <span class="math-container">$\exists$</span> <span class="math-container">$\delta>0$</span> suc... |
3,118,282 | <p>I'm trying to do the following problem in my book, but I don't understand how the book got their answer.</p>
<p>The problem:
Determine whether the following relations are equivalence relations:<span class="math-container">$\newcommand{\relR}{\mathrel{R}}$</span></p>
<p>The relation <span class="math-container">$\... | Nico | 101,332 | <p>Let <span class="math-container">$xRy$</span> if and only if <span class="math-container">$|x-y|\le 1$</span>.</p>
<p><strong>Reflexivity</strong>: Notice <span class="math-container">$|x-x|=0\le 1$</span>. Thus <span class="math-container">$xRx$</span>.</p>
<p><strong>Symmetry</strong>: Let <span class="math-cont... |
907,055 | <p>I've begun a course in "Real Analysis" recently and I have this trivial exercise. Could someone check if my proof is correct?</p>
<p>Proposition: There exists Injective function $ f: A \rightarrow B \iff $ there exists function $ g: B \rightarrow A $ is surjective</p>
<p>Proof: Firstly, we prove injective function... | Steve Kass | 60,500 | <p>No, this proof is not correct. First of all, you misstate what you set out to prove in each half. “Firstly, we prove...” says you will prove that something implies $g$ is surjective. But at this point, $g$ is not something that has been defined. There is a big difference between “$g$ is surjective” and “There exists... |
2,329,475 | <p>Google returns that response, not sure why is a complex number.</p>
| Evargalo | 443,536 | <p>If you remember the famous 'Euler' formula:</p>
<p>$e^{\pi i}+1=0$</p>
<p>And assuming $\ln$ verify $\ln(\exp(z)) = z$, you get the result $\ln(-1)=\pi i$.</p>
|
1,428,319 | <p>If we have a non-zero matrix $A\in\mathrm{Mat}_n(\mathbb{R})$ that is non-invertible. How do we prove that $A$ is both a left and right zero divisor?</p>
| ajotatxe | 132,456 | <p>If $a$ is not a multiple of $2$ or $3$, then $a=6m\pm 1$ and then
$$a^2+23=36m^2\pm12m+1+23=12(3m^2\pm m+2)$$
so it suffices to show that $3m^2\pm m$ is even. And indeed it is, because $3m^2\pm m=m(3m\pm 1)$, and one of these two factors is even.</p>
|
1,428,319 | <p>If we have a non-zero matrix $A\in\mathrm{Mat}_n(\mathbb{R})$ that is non-invertible. How do we prove that $A$ is both a left and right zero divisor?</p>
| Gabriel Romon | 66,096 | <p>$a^2+23=(a-1)(a+1)+24$</p>
<p>Since $a$ is odd; $a-1$ and $a+1$ are even. </p>
<p>Moreover, since $4$ does not divide $a$, 4 divides either $a-1$ or $a+1$.</p>
<p>Since $3$ does not divide $a$, $3$ divides either $a-1$ or $a+1$.</p>
<p>Hence $2\times 4\times 3$ divides $(a-1)(a+1)$.</p>
|
2,465,803 | <p>Given a random permutation <span class="math-container">$\sigma \in S_n$</span> from <span class="math-container">$[n] \to [n]$</span> in a uniform probability space, what is the probability that <span class="math-container">$\sigma $</span> has exactly <span class="math-container">$k$</span> fixed points for a give... | Jaroslaw Matlak | 389,592 | <ol>
<li>Number of dearrangements of $k$-elements set is
$$!k = k!\sum\limits_{m=0}^{k}\frac{(-1)^m}{m!}$$</li>
<li>In n-elements set we can select $(n-k)$ elements to dearrange them (all remaining $k$ points are fixed) in $\binom{n}{k}$ ways</li>
</ol>
<p>Thus
$$A_k^n = \binom{n}{k}(n-k)!\sum\limits_{m=0}^{n-k}\frac{... |
3,132,404 | <p>For the purpose of this question, <span class="math-container">$A'$</span> is the derived set of set <span class="math-container">$A$</span>, <span class="math-container">$A^\circ$</span> is the interior of set <span class="math-container">$A$</span>, and <span class="math-container">$A^{c}$</span> is the complement... | little o | 543,867 | <p><span class="math-container">$A' \subseteq \overline A.$</span> So <span class="math-container">$(\overline A)^c \subseteq (A')^{c}.$</span> Now we know that <span class="math-container">$(\overline A)^c = (A^c)^{\circ}.$</span> So we have <span class="math-container">$(A^c)^{\circ} \subseteq (A')^{c}.$</span></p>
|
2,915,786 | <p>I started writing a proof using the method of proof by contradiction and encountered a situation which was true. More specifically, the hypothesis that I set out to prove was:</p>
<p>If the first 10 positive integer is placed around a circle, in any order, there exists 3 integer in consecutive locations around the ... | Siong Thye Goh | 306,553 | <p>Your goal is to show that $p$ is false. </p>
<p>If $p \implies q$ and $q$ is true. </p>
<p>We can't conclude if $p$ is true or false. Hence, we get an inconclusive situation. </p>
|
2,296,968 | <p>So far I have this: Let P(n) be the statement "$n2^n \lt n!$". $k_{0}=6$. $(6)2^6 = 384 <720=6!$. $P(k_{0})$ is true. Let $n \geq 6$ and assume P(n) to be true. By the induction hypothesis, $(n+1)2^{n+1}=(n+1)(2)2^n ...$ Somehow this gets to be $n!(n+1)(2)2^n$. I am clearly missing a few steps in my proof. </p>
| Community | -1 | <p>Assume this holds:</p>
<p>$$
n2^n < n!\qquad(1)
$$</p>
<p>Multiplying (1) by $n+1$:</p>
<p>$$n(n+1)2^n < (n+1)!\qquad(2)$$</p>
<p>But $n > 2$ so from (2):</p>
<p>$$(n+1)2*2^n < n(n+1)2^n < (n+1)!$$</p>
|
3,845,756 | <p>I am studying sequences in <span class="math-container">$\mathbb{R}$</span>. I would like to know if a sequence satisfies that
"For all <span class="math-container">$n \in \mathbb{N}, |a_{n+1} - a_{n}| \leq \frac{1}{3^{n}}$</span>" then sequence <span class="math-container">$\left\{a_{n}\right\}_{n \in \... | J. W. Tanner | 615,567 | <p>It's not true.</p>
<p>A counterexample is <span class="math-container">$a_n=\sqrt n$</span>.</p>
|
1,415,214 | <p>I'm supposed to calculate the derivative of $\frac{d}{dx}\int_{x^{2}}^{x^{8}}\sqrt{8t}dt$ the answer I got is $8x^7\cdot \sqrt{8x^8}$ but when I put this into the grading computer it is marked wrong. I've tried solving in several different ways and always got the same answer so I used my graphing calculator to test... | randomgirl | 209,647 | <p>$g(x)=\int_{x^2}^{x^8} f(t) dt=F(t)|_{x^2}^{x^8}=F(x^8)-F(x^2)$ and so differentiating we have $g'(x)=(x^8)'f(x^8)-(x^2)'f(x^2)$</p>
|
1,415,214 | <p>I'm supposed to calculate the derivative of $\frac{d}{dx}\int_{x^{2}}^{x^{8}}\sqrt{8t}dt$ the answer I got is $8x^7\cdot \sqrt{8x^8}$ but when I put this into the grading computer it is marked wrong. I've tried solving in several different ways and always got the same answer so I used my graphing calculator to test... | Bernard | 202,857 | <p>The derivative is
$$8x^7\sqrt{8x^8}-2x\sqrt{8x^2}=4\sqrt 2\bigl(4x^{11}-x\lvert x\rvert\bigr).$$
You have to derive also w.r.t. the lower bound of integration.</p>
|
252,511 | <p>I am interested in the details of Elie Cartan's thesis, and, more specifically the explicit construction of the exceptional Lie groups as groups of symmetries of some specific homogeneous polynomials (according to what I have read in many places). I am interested in the details. For instance, what does one such a po... | Alexandre Eremenko | 25,510 | <p>All details about Cartan's thesis can be found in the thesis itself:</p>
<p><a href="https://archive.org/details/surlastructured00bourgoog" rel="nofollow">https://archive.org/details/surlastructured00bourgoog</a></p>
<p>There is also a German translation for those who do not read French:</p>
<p>Ueber die einfache... |
3,623,724 | <p>The game works as follows: assume that there are two players, 1 and 2. 1 decides to lie or tell the truth. If 1 tells the truth, 2 needs to decide to take her medications or not. Later, in case 2 did not take her medications or 1 lied to her, 2 needs to decide to go to the hospital or not.
If 1 lied to her and she w... | Hagen von Eitzen | 39,174 | <p>The strategy of 2 consists of a few "bits": </p>
<ul>
<li>When told the truth, will she take her medicine? Or not take her medicine, but go to the hospital? Or neither? Symbolize these by <span class="math-container">$M, H_T, 0_T$</span>.</li>
<li>When lied to, will she go to the hospital or not? Symbolize these by... |
10,992 | <p>Alright, I'm trying to figure out how to calculate a critical value using t-distribution in Microsoft Excel... ex. a one-tailed area of 0.05 with 39 degrees of freedom: t=1.685</p>
<p>I know the answer, but how do I get this? I've tried TDIST() TINV() and TTEST() but they all give me different answers. This web ca... | Mike Spivey | 2,370 | <p>The right function is <code>TINV()</code>, but the problem is that <code>TINV()</code> assumes that you're inputting the two-tailed probability, not the one-tailed probability. In other words, <code>TINV(p,d)</code> outputs the value of $x$ such that $P(|X| \geq x) = p$, where $X$ has a $t$ distribution with $d$ de... |
1,709,915 | <p>Given an algebraic category, Birkhoff's Variety Theorem gives a categorical characterization of the full subcategories whose object-class forms a variety (i.e. can be defined by equations in the sense of Model Theory).</p>
<p>The theorem is often stated as being of fundamental importance to Universal Algebra. As fa... | Alex Kruckman | 7,062 | <p>This may not be the best answer, but it is an interesting sort of application of Birkhoff's theorem.</p>
<p>Let's say you have some class $V$ of algebras which is defined "semantically" (e.g. as all the algebras arising from some construction), and you want to give an equivalent "syntactic" description (i.e. an axi... |
1,709,915 | <p>Given an algebraic category, Birkhoff's Variety Theorem gives a categorical characterization of the full subcategories whose object-class forms a variety (i.e. can be defined by equations in the sense of Model Theory).</p>
<p>The theorem is often stated as being of fundamental importance to Universal Algebra. As fa... | J.-E. Pin | 89,374 | <p>Birkhoff's Variety Theorem is indeed of fundamental importance to Universal Algebra, notably for the stream of research it has generated, inside and outside of Universal Algebra. Here are a few examples.</p>
<p>The book [4] is entirely devoted to varieties of groups. See also the paper <a href="http://projecteuclid... |
982,259 | <p>Let $n\in\mathbb{N}$.</p>
<p>So far I have: If the sum of the digits of $n$ is $k$, then $n = 9m + k$, where $m$ element of an integer (not sure why). Now consider $5n-n$.</p>
<p>Help?</p>
| mookid | 131,738 | <p><strong>Hint:</strong> start to prove that if $S(n)$ is the sum of the digits of $n$, then
$
n - S(n)
$ is a multiple of 9.</p>
|
2,894,315 | <p>Let $a;b;c>0$ such that $a+b+c=6$. Find the maximum value of $A=a^2bc+a^2+2b^2+2c^2$</p>
<hr>
<p>WLOG $b\ge c$. I see maximum value of $A=36$ at $(a;b;c)=(2;1;3)$</p>
<p>So i need to prove $A\le 36$. Or I will prove </p>
<p>$(a+b+c)^4\ge 36a^2bc+(a^2+2b^2+2c^2)(a+b+c)^2$</p>
<p>Or $(2a-b-c)(b^3+c^3+a^2b+a^2... | Cesareo | 397,348 | <p>With the help of the Lagrange multipliers and $f(a,b,c) = a^2 b c + a^2 + 2 b^2 + 2 c^2$</p>
<p>$$
L(a,b,c,\lambda) =f(a,b,c)+\lambda(a+b+c-6)
$$</p>
<p>the stationary points are the solutions for</p>
<p>$$
\nabla L = \left\{
\begin{array}{c}
2 b c a+2 a+\lambda =0 \\
c a^2+4 b+\lambda =0 \\
b a^2+4 c+\lambda... |
1,700,457 | <p>Assume $H \leq G$ and $K \leq G$ and assume $aH=bK$ for some $a,b \in G$. Prove that $H=K$</p>
<p>would someone give me a hint. I know that definition of being a subgroup. I know that the element look like this $ah_1=bk_1$ for all $h_1 \in H$ and for all $k_1 \in K$. I do not have an idea from where should I get st... | Mark Fischler | 150,362 | <p>Assume $H \subseteq G$ and $K \subseteq G$ and assume $aH=bK$ for some $a,b \in G$. </p>
<p>Then $\forall h \in H \exists k \in k :ah = bk
$ and of course that $k$ may depend on $h$, so let us label one of the (possibly multiple) values of $k$ that satisfies $ah = bk$ as $k(h)$.</p>
<p>Consider the $e$ (the group... |
99,018 | <p>If $g$ is Lie algebra over field char(k)=0,
then the following facts are well-known:</p>
<p>1) S(g) and U(g) are isomorphic as $g$-modules. (Symmetrization map S(g)->U(g) gives isomorphism).</p>
<p>2) S(g)^g and ZU(g)=U(g)^g are isomorphic as commutative algebras. (The <a href="http://en.wikipedia.org/wiki/Duflo_i... | Alexander Premet | 24,386 | <p>Question 2 has a negative answer. Indeed, let $L =sl(2,k)$ where $k$ is an algebraically closed field of characteristic $p>3$ and let {$e,h,f$} be the standard basis of $L$. Then it is well-known (and was first proved by Rudakov-Shafarevich in the late 60s) that the centre $Z(L)$ of $U(L)$ is generated by $E=e^p$... |
3,214,255 | <p>How to simplify <span class="math-container">$$\frac{\sqrt{6+4\sqrt{2}}}{4+2\sqrt{2}}?$$</span></p>
<p>Rationalise the denominator </p>
<p><span class="math-container">$$\frac{\sqrt{6+4\sqrt{2}}}{4}(2-\sqrt{2})$$</span></p>
<p>This is still not simplify.</p>
| Dr. Sonnhard Graubner | 175,066 | <p>We have <span class="math-container">$$6+4\sqrt{2}=4+2+4\sqrt{2}$$</span> so <span class="math-container">$$2\sqrt{6+4\sqrt{2}}=4+2\sqrt{2}$$</span> and we get</p>
<p><span class="math-container">$$\frac{\sqrt{6+4\sqrt{2}}}{4+2\sqrt{2}}=\frac{1}{2}$$</span></p>
|
2,386,864 | <p>Six persons P, Q, R, S, T and U play in a tournament called "High Rollers". Every game involved two players. Each of the participants played with every other participant exactly once. In the game both the players rolled an unbiased die each. The player who gets the larger number on the top surface of the die wins th... | Piquito | 219,998 | <p>COMMENT.-This is another way, maybe interesting for some people, of stating the same problem.</p>
<p>Given an odd natural number, $2n + 1$, there are $n$ different ways to express it as the sum of two natural $$2n+1=(2n-k)+(k+1);\space k=1,2,....,n$$ Then the problem can be stated as follows equivalently: $$\text{... |
803,488 | <p>Imagine Rock Paper Scissors, but where winning with a different hand gives a different reward.</p>
<ul>
<li><p>If you win with Rock, you get \$9. Your opponent loses the \$9.</p>
</li>
<li><p>If you win with Paper, you get \$3. Your opponent loses the \$3.</p>
</li>
<li><p>If you win with Scissors, you get \$5. Your... | mjqxxxx | 5,546 | <p>Let $(x_1,x_2,x_3)$ be the first player's strategy (i.e., his probabilities of playing rock, paper, and scissors respectively), and let $(y_1,y_2,y_3)$ be the second player's strategy. The expected payoff to the first player is
$$
P(x,y)=9(x_1y_3-x_3y_1)+3(x_2y_1-x_1y_2)+5(x_3y_2-x_2y_3).
$$
To constrain the probab... |
758,135 | <p>A visiting speaker in Economics recently happened to mention that John Maynard Keynes' <a href="http://www.gutenberg.org/ebooks/32625">A Treatise on Probability</a> revolutionized probability theory. I have not heard any such claim before and it struck me as strange. The <a href="http://en.wikipedia.org/wiki/A_Treat... | Michael Emmett Brady | 160,180 | <p>J M Keynes did not revolutionize probability theory.J M Keynes revolutionized decision theory by showing that the mathematical laws of the calculus of probabilities were a special case requiring that the weight of the evidence,w,where w is defined on the unit interval[ 0,1],must equal ,approximate ,tend,or approach... |
2,828,636 | <p>I wanted to show that both sets are equal. My Textbook says following:</p>
<p>$\in$ means "is element of", $\land$ is the and operator, $\lnot$ is the not operator, $\notin$ means "is not element of", $\lor$ is the or operator</p>
<p>$$\def\-{\setminus}\begin{split} x \in A\-A\-B &\iff\\
(x \in A) \land (\lnot... | user | 505,767 | <p>In $x-y$ plane the equation of the line is given by</p>
<p>$$ax+by=d$$</p>
<p>beeing the normal $n=(a,b)$ such that $|n|=1$, indeed by dot product between $OP$ and $n$ for any point $P=(x,y)$ on the line we have</p>
<p>$$(x,y)\cdot (a,b)=ax+by=|OP|\cos \theta=d$$</p>
|
3,132,039 | <p>I am trying to create a set of 10-tuples, where each tuple consists of elements taken from a set {defective, nondefective). Each sample must have precisely 1 defective part and 9 nondefective parts. My attempt</p>
<p>Let the sample be a 10-tuple <span class="math-container">$$S:=(X,\Omega,F)$$</span> <span class="m... | Exit path | 161,569 | <p>A sequence of sheaves
<span class="math-container">$$0 \to \mathcal{F} \to \mathcal{G} \to \mathcal{H} \to 0$$</span>
is exact if and only if the induced sequence on stalks is exact for every <span class="math-container">$x \in X$</span>. It's easy to show this fact is equivalent to the statement that taking stalks... |
3,477,152 | <p>I am trying to prove that</p>
<p><span class="math-container">$$\sum_{n=1}^\infty \frac{n}{\sqrt{n+1}}$$</span></p>
<p>diverges without checking the limit, bounds or doing any other lengthy steps, as it should be seen as divergent "immediately", but I have no clue about how I would quickly prove this.</p>
<p>So f... | Francesco_Trig | 731,440 | <p>You can notice that this series has the same behavior of <span class="math-container">$$\sum_{n=1}^\infty \frac{n}{\sqrt n} =\sum_{n=1}^\infty \sqrt n$$</span> that diverges.</p>
|
3,477,152 | <p>I am trying to prove that</p>
<p><span class="math-container">$$\sum_{n=1}^\infty \frac{n}{\sqrt{n+1}}$$</span></p>
<p>diverges without checking the limit, bounds or doing any other lengthy steps, as it should be seen as divergent "immediately", but I have no clue about how I would quickly prove this.</p>
<p>So f... | Deepesh Thakur | 639,689 | <p><span class="math-container">$$n+1 \le 2n\ \ \forall \ n \ge 1$$</span>
<span class="math-container">$$\Rightarrow \frac{n}{\sqrt{2n}} \le \frac{n}{\sqrt{n+1}}$$</span>
<span class="math-container">$$\Rightarrow \sum_{n=1}^{\infty}\frac{n}{\sqrt{2n}} \le \sum_{n=1}^{\infty}\frac{n}{\sqrt{n+1}}$$</span>
<span class="... |
3,321,324 | <p>In doing some old exam questions I came across this. The results are all excruciatingly obvious intuitively, but I have no idea how to formalize a proof for any of them.</p>
<p>Let <span class="math-container">$(M,d)$</span> be a metric space. For <span class="math-container">$A\subseteq M, A \ne \emptyset$</span> ... | Lázaro Albuquerque | 85,896 | <p>a) Take an arbitrary <span class="math-container">$z \in U_r(y)$</span> and play with the triangle inequality.</p>
<p>b) For bounded sets <span class="math-container">$A$</span> and <span class="math-container">$B$</span>, <span class="math-container">$\inf A \cup B = \min (\inf A, \inf B)$</span>. Apply induction.... |
3,321,324 | <p>In doing some old exam questions I came across this. The results are all excruciatingly obvious intuitively, but I have no idea how to formalize a proof for any of them.</p>
<p>Let <span class="math-container">$(M,d)$</span> be a metric space. For <span class="math-container">$A\subseteq M, A \ne \emptyset$</span> ... | Eric Towers | 123,905 | <p>Hints: a) uses the triangle inequality, b) uses that the infimum of a finite set is the minimum of that set, and c) proceed by contradiction using an open cover.</p>
<p>For a), you want to use the idea that <span class="math-container">$d(x,a) + d(a,y) \leq d(x,y)$</span> is true for every <span class="math-contain... |
1,829,086 | <p>So far, I've tried out to reformulate: $$\int{\frac{1}{\cos(x)}}dx$$
to: $$\int{\frac{\sin(x)}{\cos(x)\sin(x)}}dx$$</p>
<p>which is basically: $$\int{\frac{\tan(x)}{\sin(x)}}dx$$
But I'm not sure if this is the right way to go, or if I try something else.</p>
<p>Any tips or methods would be very helpful.</p>
| Sophie Agnesi | 346,183 | <p><strong>Hint:</strong></p>
<p>If $u=\tan\left(x\over2\right)$, then $\cos x={1-u^2\over1+u^2}$ and $dx={2\ du\over1+u^2}$. Hence</p>
<p>\begin{equation}
\int {1\over\cos x}\ dx=\int{2\over1-u^2}\ du=\int\left[\frac{1}{1+u}+\frac{1}{1-u}\right]\ du
\end{equation}</p>
|
1,829,086 | <p>So far, I've tried out to reformulate: $$\int{\frac{1}{\cos(x)}}dx$$
to: $$\int{\frac{\sin(x)}{\cos(x)\sin(x)}}dx$$</p>
<p>which is basically: $$\int{\frac{\tan(x)}{\sin(x)}}dx$$
But I'm not sure if this is the right way to go, or if I try something else.</p>
<p>Any tips or methods would be very helpful.</p>
| Strategy Thinker | 112,976 | <p>These are the things I would try.
I would find $\cos(x)$ in terms of $x/2$ because I would have to do that eventually anyway, here $\cos(x) = 2\cos^2(x/2) - 1$.</p>
<p>Then I would replace the $dx$ with an expression in $du$ as that also must be done eventually.
\begin{align*}
du = & 0.5 \sec^2(x/2)\, dx\\
dx =... |
1,829,086 | <p>So far, I've tried out to reformulate: $$\int{\frac{1}{\cos(x)}}dx$$
to: $$\int{\frac{\sin(x)}{\cos(x)\sin(x)}}dx$$</p>
<p>which is basically: $$\int{\frac{\tan(x)}{\sin(x)}}dx$$
But I'm not sure if this is the right way to go, or if I try something else.</p>
<p>Any tips or methods would be very helpful.</p>
| JamesJ | 474,394 | <p>\begin{align}
I&=\int\frac{dx}{\cos x}=\int\frac{1-\sin x}{\cos x}\frac{dx}{1-\sin x}=\int\frac{1-\sin x}{\cos x}d\left(\frac{\cos x}{1-\sin x}\right)=\ln\left|\frac{\cos x}{1-\sin x}\right|+C
\end{align}</p>
|
969,601 | <p>For the function whose graph is a paraboloid given by</p>
<p>$z = x^2 + y^2/4$</p>
<p>I know that the level curve represents an ellipse. I also know that the parametrization of this curve in the form $x = x(t)$, $y = y(t)$ is </p>
<p>$x(t) = \cos(t)$
and<br>
$y(t) = 2\sin(t)$</p>
<p>In class, my professor said t... | Community | -1 | <p>My attempt at calculating the probability that there is a mine in $D$:</p>
<p>If there is one mine in $\lbrace A, B, C, D \rbrace$ and one mine in $\lbrace D, E \rbrace$, then there are two cases:</p>
<ol>
<li>There are two mines, placed in either E and A, or E and B, or E and C</li>
<li>There is one mine, placed ... |
729,373 | <p>I am to evaluate $\displaystyle\int_0^{\infty} \dfrac{\sin x}{x(x^2+1)}dx$ via contour integration.</p>
<p>Now I used an indented semicircular contour, and the parts lying on the real line and the big arc were no problem, but the small arc is being resistant, and I'm not sure what to do. Usually, on the small arc f... | Ellya | 135,305 | <p>My guess is you should have two integrals, one for the residue at $i$ and the other one at $-i$</p>
<p>For the $i $ residue take the angle $3\pi/4$ for the other take $-3\pi/4$</p>
|
146,973 | <p>I want to find the expected value of $\text{max}\{X,Y\}$ where $X$ ist $\text{exp}(\lambda)$-distributed and $Y$ ist $\text{exp}(\eta)$-distributed. X and Y are independent.
I figured out how to do this for the minimum of $n$ variables, but i struggle with doing it for 2 with the maximum.</p>
<p>(The context in whi... | Julius | 24,547 | <p>Let <span class="math-container">$V=\max\{X,Y\}$</span>. Then
<span class="math-container">$$\mathbb{P}(V\leq t)=\mathbb{P}(X\leq t,Y\leq t)=\mathbb{P}(X\leq t)\mathbb{P}(Y\leq t).$$</span>
Now find <span class="math-container">$f_V(t)$</span> and then <span class="math-container">$\int_{0}^{+\infty}tf_V(t)dt$</span... |
3,938,933 | <p>I am interested in the general case, but let us start with a smaller example. Suppose that cars arrive to a street with intensity <span class="math-container">$\lambda$</span> per minute. We would like to know the probability that at least two cars have arrived on the street during any five minute interval in the ne... | robjohn | 13,854 | <p><strong>Generating Function Approach</strong></p>
<p>Let's compute the probability that no <span class="math-container">$5$</span> minute span has more than <span class="math-container">$1$</span> car entry.</p>
<hr />
<p><strong>Poisson with <span class="math-container">$\bf{1}$</span> Minute Resolution</strong></p... |
3,938,933 | <p>I am interested in the general case, but let us start with a smaller example. Suppose that cars arrive to a street with intensity <span class="math-container">$\lambda$</span> per minute. We would like to know the probability that at least two cars have arrived on the street during any five minute interval in the ne... | robjohn | 13,854 | <p><strong>Conditional Probability Approach</strong></p>
<p>Poisson says that in an hour, the probability that <span class="math-container">$n$</span> cars have entered is
<span class="math-container">$$
e^{-60\lambda}\frac{(60\lambda)^n}{n!}\tag1
$$</span>
Given that <span class="math-container">$n$</span> cars have e... |
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