qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
4,118,149 | <p>Let's say I have the following matrix:
<span class="math-container">\begin{bmatrix}
\frac{1}{3} & \frac{2}{3} & 0 & \frac{2}{3} \\
\frac{2}{3} & -\frac{1}{3} & \frac{2}{3} & 0 \\
a & b & c & d \\
e & f & g & h
\end{bmatrix}</span></p>
<p>How do I find t... | 2'5 9'2 | 11,123 | <p>Putting aside the given initial condition, the constant functions <span class="math-container">$y=1$</span> and <span class="math-container">$y=-1$</span> are solutions.</p>
<p>You could not have another continuous solution that takes a value inside <span class="math-container">$(-1,1)$</span> and then also takes a ... |
3,412,418 | <blockquote>
<p>You have been chosen to play a game involving a 6-sided die. You get
to roll the die once, see the result, and then may choose to either
stop or roll again. Your payoff is the sum of your rolls, unless this
sum is (strictly) greater than 6. If you go over <span class="math-container">$6$</span>,... | Henry | 6,460 | <p>Suppose <span class="math-container">$f(x,t)$</span> is your expected return if so far you have rolled <span class="math-container">$x$</span> and you have a threshold <span class="math-container">$t$</span> where you stop when <span class="math-container">$x \ge t$</span>. Then </p>
<ul>
<li><p><span class="math-... |
2,651,537 | <p>Prove that $(5m+3)(3m+1)=n^2$ is not satisfied by any <strong>positive</strong> integers $m,n$.</p>
<p>I have been staring at this for some time (it's the difficult part of a competition problem, I won't spoil it by naming the problem). I tried looking at it modulo 3,4,5,7,8,16 for a contradiction, as well as looki... | Yong Hao Ng | 31,788 | <p><strong>1. Solving $(5m+3)(3m+1)=n^2$</strong><br>
Here is an elementary approach that only uses modular arithmetic.<br>
We do not assuming $m\geq 0$ initially, to highlight where it is used exactly in this approach. </p>
<p>It was shown above that since
$$3(5m+3)-5(3m+1)=4$$
We have $d=\gcd(5m+3,3m+1)$ divides $4... |
10,722 | <p>I notice that geometry students frequently have difficulty with representations of 3-dimensional objects in 2 dimensions. Today, we worked with physical manipulatives in order to help visualize where right triangles can occur in 3 dimensions in both pyramids and rectangular prisms (the focus is on fluency with the P... | Daniel R. Collins | 5,563 | <p>Here's a piece comparing virtual manipulatives to traditional teaching without manipulatives, in the context of community college remedial courses:</p>
<blockquote>
<p>Violeta Menil and Eric Fuchs, "Teaching Pre-Algebra and Algebra
Concepts to Community College Students through the Use of Virtual
Manipulative... |
637,199 | <p>If $K^T=K$, $K^3=K$, $K1=0$ and $K\left[\begin{matrix}1\\2 \\-3\end{matrix}\right]=\left[\begin{matrix}1\\2 \\-3\end{matrix}\right]$,</p>
<p>how can I find the trace of $K$ and the determinant of $K$?</p>
<p>I think for determinant of $K$, since $K^3-K=(K^2-I)K=0$, then $K^2=I$ since $K$ is nonzero. Then this impl... | Cameron Williams | 22,551 | <p>$K$ has a nontrivial kernel so it is not one-to-one. What does this tell you about its determinant? To determine the trace, you could use the property that the trace is the sum of the eigenvalues.</p>
|
1,440,522 | <p>For a function $f:[0,1]\to \mathbb{R}$, let $C$ be the set of points where $f$ is
continuous. Prove that $C$ is in the Borel $\sigma$-algebra.</p>
<p>I know that for $A=\{f(x): f(x)<a\}$ is open for each real number a, and since openness is preserved by continuity, the set $f^{-1}(A)\cap C$ should also be op... | Calvin Khor | 80,734 | <p>Hint. A point $x$ is a continuity point if $\lim_{y→ x} f(y) = f(x)$. Hence the set of continuity points is
$$ S = \left\{ x: \lim_{y→ x} f(y) = f(x) \right\} $$
We know how to express $\lim_{y→ x} f(y) = f(x)$ as $∀ ε>0,\dots$ we can then turn '$∀$'s into '$∩$'s and '$∃$'s into '$∪$'s, and conclude with the den... |
4,019,956 | <p><strong>Preface</strong></p>
<p>We will use the following facts</p>
<p>i) The sequence <span class="math-container">$ \left\lbrace a_n \right\rbrace $</span> is convergent to <span class="math-container">$a$</span> if for each <span class="math-container">$ \varepsilon >0$</span> there exists <span class="math-co... | sirous | 346,566 | <p>Another approach:</p>
<p><span class="math-container">$p=a^2+ab+b^2-a-2b=a^2+a(b-1)+b^2-2b$</span></p>
<p>We solve polynomial p for <span class="math-container">$a$</span>:</p>
<p><span class="math-container">$a=\frac{-(b-1)\pm \sqrt {\Delta}}2$</span></p>
<p><span class="math-container">$\Delta=(b-1)^2-4b^2+8b$</sp... |
1,998,244 | <p>Given the equation of a damped pendulum:</p>
<p>$$\frac{d^2\theta}{dt^2}+\frac{1}{2}\left(\frac{d\theta}{dt}\right)^2+\sin\theta=0$$</p>
<p>with the pendulum starting with $0$ velocity, apparently we can derive:</p>
<p>$$\frac{dt}{d\theta}=\frac{1}{\sqrt{\sqrt2\left[\cos\left(\frac{\pi}{4}+\theta\right)-e^{-(\the... | Deepak Suwalka | 371,592 | <p>We can also solve it using Venn diagram, make diagram-
<a href="https://i.stack.imgur.com/jbn0N.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/jbn0N.png" alt="enter image description here"></a></p>
<p>Now, according to diagram,</p>
<p>$B=17-6\;\implies11$</p>
<p>$S=18-6\;\implies12$</p>
<p>$B... |
50,362 | <p>I have a question about the basic idea of singular homology. My question is best expressed in context, so consider the 1-dimensional homology group of the real line $H_1(\mathbb{R})$. This group is zero because the real line is homotopy equivalent to a point. The chain group $C_1(\mathbb{R})$ contains all finite ... | MartianInvader | 8,309 | <p>Your construction works, but I think it's a little more complicated than it needs to be. In particular, you can just define a map from a simplex directly, without having to go through a disc or anything.</p>
<p>I'd do it by taking a triangle (again, let's call the vertices 0,1,2) and mapping [0,1] and [0,2] via th... |
2,005,649 | <p>I am struggling to find a parameterization for the following set : </p>
<p>$$F=\left\{(x,y,z)\in\mathbb R^3\middle| \left(\sqrt{x^2+y^2}-R\right)^2 + z^2 = r^2\right\}
\quad\text{with }R>r$$</p>
<p>I also have to calculate the area. </p>
<p>I know its a circle so we express it in terms of the angle but my prob... | Olivier Moschetta | 369,174 | <p>Here's a somewhat simpler approach. We need to use 1 uppercase, 1 digit and 1 special character. There are $26\cdot 10\cdot 16$ ways to choose which 3 symbols we want to use. Then we have to choose a spot for them in the password. There are 8 possible spots for the uppercase, 7 for the digit and 6 for the special ch... |
2,005,649 | <p>I am struggling to find a parameterization for the following set : </p>
<p>$$F=\left\{(x,y,z)\in\mathbb R^3\middle| \left(\sqrt{x^2+y^2}-R\right)^2 + z^2 = r^2\right\}
\quad\text{with }R>r$$</p>
<p>I also have to calculate the area. </p>
<p>I know its a circle so we express it in terms of the angle but my prob... | CakeMaster | 385,422 | <p>Starting from a similar point as you, but with slightly different logic from there, I got...</p>
<p>$$78^8-68^8-52^8-62^8+42^8+52^8+36^8-26^8 = 706905960284160 \approx 7.07E14$$</p>
<p>I started with the total number of possibilities.
$$78^8$$
Then, I subtracted all possibilities without a digit, without an upper... |
2,087,107 | <p>In the following integral</p>
<p>$$\int \frac {1}{\sec x+ \mathrm {cosec} x} dx $$</p>
<p><strong>My try</strong>: Multiplied and divided by $\cos x$ and Substituting $\sin x =t$. But by this got no result.</p>
| Community | -1 | <p><strong>A big partial fraction decomposition!!</strong>:</p>
<p>We use <a href="https://en.m.wikipedia.org/wiki/Tangent_half-angle_substitution" rel="nofollow noreferrer">Weierstrass substitution</a> and use $u =\tan \frac {x}{2} $ to get $$I =\int \frac {1}{\sec x+\mathrm {cosec}x } dx =\int \frac {\sin x\cos x}{\... |
1,548,130 | <p>Let's say that $F$ is a nice well-behaved function. How would I compute the following derivative?</p>
<p>$\frac{\partial}{\partial t} \left\{ \int_{0}^{t} \int_{x - t + \eta}^{x + t - \eta} F(\xi,\eta) d\xi d\eta \right\}$</p>
<p>I'm guessing I need the fundamental theorem of calculus, but the double integral is R... | Dustan Levenstein | 18,966 | <p>Hint: You have that $g$ is bijective. Use the following facts about bijective, injective and surjective functions:</p>
<ul>
<li>Every bijective function $h: C \to D$ has an inverse $h^{-1}: D \to C$ so that $h \circ h^{-1} = id_D$ and $h^{-1} \circ h = id_C$.</li>
<li>If you compose a bijective function with an inj... |
1,548,130 | <p>Let's say that $F$ is a nice well-behaved function. How would I compute the following derivative?</p>
<p>$\frac{\partial}{\partial t} \left\{ \int_{0}^{t} \int_{x - t + \eta}^{x + t - \eta} F(\xi,\eta) d\xi d\eta \right\}$</p>
<p>I'm guessing I need the fundamental theorem of calculus, but the double integral is R... | goblin GONE | 42,339 | <p>Use the following facts to do your heavy lifting:</p>
<blockquote>
<p><strong>Proposition A.</strong> Given functions $g : Z \leftarrow Y$ and $f : Y \leftarrow X$...</p>
<ol>
<li><p>If $g \circ f$ is surjective, then so too is $g$.</p></li>
<li><p>If $g \circ f$ is injective, then so too is $f$.</p></li... |
1,220,502 | <p>The problem is this. Given that $\int_0^a f(x) dx = \int_0^a f(a-x)dx$, evaluate $$\int_0^\pi \frac{x\sin x}{1+\cos^2x} dx$$</p>
<p>I write the integral as $$\int_0^\pi \frac{(\pi-x)\sin(\pi-x)}{1+\cos^2(\pi-x)} dx$$
but I don't see how that helps to do it.</p>
| Mercy King | 23,304 | <p>Using the property
$$
\int_0^af(x)\,dx=\int_0^af(a-x)\,dx,
$$
with
$$
f(x)=\frac{x\sin x}{1+\cos^2x},\quad a=\pi
$$
we have
\begin{eqnarray}
\int_0^\pi f(x)\,dx&=&\int_0^\pi=\int_0^\pi f(\pi-x)\,dx=\int_0^\pi\frac{(\pi-x)\sin(\pi-x)}{1+\cos^2(\pi-x)}\,dx\\
&=&\int_0^\pi\frac{(\pi-x)\sin x}{1+\cos^2x... |
1,955,591 | <p>I have to prove that ' (p ⊃ q) ∨ ( q ⊃ p) ' is a tautology.I have to start by giving assumptions like a1 ⇒ p ⊃ q and then proceed by eliminating my assumptions and at the end i should have something like ⇒(p ⊃ q) ∨ ( q ⊃ p) but could not figure out how to start.</p>
| Nitin Uniyal | 246,221 | <p>$(p\rightarrow q)\lor (q\rightarrow p)\equiv (\lnot p\lor q)\lor (\lnot q\lor p)\equiv \lnot p\lor (q\lor\lnot q)\lor p\equiv \lnot p\lor T\lor p\equiv \lnot p\lor p\equiv T$</p>
|
816,088 | <blockquote>
<p>The sum of two variable positive numbers is $200$.
Let $x$ be one of the numbers and let the product of these two numbers be $y$. Find the maximum value of $y$.</p>
</blockquote>
<p><em>NB</em>: I'm currently on the stationary points of the calculus section of a text book. I can work this out in my... | lab bhattacharjee | 33,337 | <p>For positive real $a,b$</p>
<p>$$\frac{a+b}2\ge \sqrt{ab}\implies ab\le\frac{(a+b)^2}4$$</p>
<p>which can also be shown as follows $$(a+b)^2-4ab=(a-b)^2\ge0$$</p>
<p>Alternatively if $\displaystyle a+b=k, ab=a(k-a)=\frac{k^2-(k-2a)^2}4\le\frac{k^2}4$</p>
|
67,929 | <p>Joel David Hamkins in an answer to my question <a href="https://mathoverflow.net/questions/67259/countable-dense-sub-groups-of-the-reals">Countable Dense Sub-Groups of the Reals</a> points out that "one can find an uncountable chain of countable dense additive subgroups of $\mathbb{R}$ whose subset relation has the ... | André Henriques | 5,690 | <p>A countable abelian group can be encoded by a <i>presentation</i>.
Such a presentation has countably many generators, and countably many relations (each one being a finite word in the generators).</p>
<p>The isomorphism type of the group is encoded by a countably infinite word in a countable alphabet. The cardinali... |
1,871,103 | <blockquote>
<p>Given that $a_{1}=0$, $a_{2}=1$ and
$$a_{n+2}=\frac{(n+2)a_{n+1}-a_{n}}{n+1}$$
prove that $\lim\limits_{n\to\infty} a_n=e$</p>
</blockquote>
<p>What I did:</p>
<p>It was hinted to prove that $a_{n+1}-a_{n}=\frac{1}{n!}$ which I did inductively.
But then using this information now I get:</p>
<... | Community | -1 | <p>Instead, you should have got:</p>
<p>$$a_{n+1} = a_n + \frac1{n!} = a_{n-1} + \frac1{(n-1)!} + \frac1{n!}= \cdots = a_1 + \frac1{1!} + \frac1{2!} + \cdots + \frac1{(n-1)!} + \frac1{n!}$$</p>
<p>i.e.</p>
<p>$$a_{n+1} = \sum_{k=1}^n \frac1{k!} = \sum_{k=0}^n \frac1{k!} - 1$$</p>
<p>Hence, the limit is actually $e... |
96,211 | <p>A modulus of continuity for a function $f$ is a continuous increasing function $\alpha$ such that $\alpha(0) = 0$ and $|f(x) - f(y)| < \alpha(|x-y|)$ for all $x$ and $y$. I am trying to prove that an equicontinuous family $\mathcal F$ of functions has a common modulus of continuity. This seems intuitively obvious... | Beni Bogosel | 7,327 | <p>Suppose $\delta_n$ is increasing and $\delta_n \to \delta$. Denote $\alpha(\delta)=\sup \{ |f(x)-f(y)| : d(x,y)<\delta, f \in \mathcal F\}$ (with strict inequality). Then pick $x,y$ with $d(x,y)<\delta$. This means that there exists $n$ such that $d(x,y)<\delta_n$ and therefore $|f(x)-f(y)| \leq \alpha(\del... |
100,955 | <p>I'm trying to find the most general harmonic polynomial of form $ax^3+bx^2y+cxy^2+dy^3$. I write this polynomial as $u(x,y)$. </p>
<p>I calculate
$$
\frac{\partial^2 u}{\partial x^2}=6ax+2by,\qquad \frac{\partial^2 u}{\partial y^2}=2cx+6dy
$$
and conclude $3a+c=0=b+3d$. Does this just mean the most general harmon... | Semiclassical | 137,524 | <p>An alternative approach is to 'guess' the form of the final analytic function. We want $f(z)=u(x,y)+i v(x,y)$ where $z=x+i y$ and $u(x,y)$ is a homogeneous cubic polynomial. Since $u(x,y)$ is cubic, we infer that $f(z)$ should also be some cubic polynomial; since it is homogeneous, we further conclude that $f(z)=\al... |
660,461 | <p>$A = \{a,b,c,d,e\}$</p>
<p>$B = \{a,b,c\}$</p>
<p>$C = \{0,1,2,3,4,5,6\}$</p>
<p>The first few iterations are as follows:</p>
<p>$1.$ $a,a,0$</p>
<p>$2.$ $b,b,1$</p>
<p>$3.$ $c,c,2$</p>
<p>$4.$ $d,a,4$</p>
<p>$5.$ $e,b,5$</p>
<p>$...$</p>
<p>I'm trying to figure out at which iterations we will have $x,y,z$... | Casteels | 92,730 | <p>Firstly, I think your program is incorrect as the $32$nd iteration is $(b,b,3)$ and the first iteration of $(x,y,z)$ with $x=y$ and $z=5$ is $(c,c,5)$ at iteration 48. </p>
<p>In any case, the iterations you desire will occur when $n$ is a solution to the system \begin{align*}
n &\equiv 1,2\text{ or }3\pmod{15}... |
458,779 | <p>I need to find the volume of an object restricted with the $x^{2}+z^{2} < 8$ and $0 < y < 2$ planes. It would be easy if the cylinder were "parallel" to the XY plane, because then:</p>
<p>$$0 < r < 2\sqrt{2}$$
$$0 < \phi < 2\pi$$
$$0 < z < 2$$</p>
<p>But well, how should I handle this he... | Mikasa | 8,581 | <p>You can also use the following limits indicating that we are using Cartesian coordinates. I use the symmetric of the solid volume as well.</p>
<p>$$4\int_{x=0}^{\sqrt{8}}\int_{z=0}^{\sqrt{8-x^2}}\int_{y=0}^2dydzdx$$</p>
<p><img src="https://i.stack.imgur.com/UQxeY.png" alt="enter image description here"></p>
|
525,326 | <p>If all elements of $S$ are irrational and bounded from below by $\sqrt 2$ then $\inf S$ must be irrational .</p>
<p>I would say this statement is true since $S=\{ \sqrt 2, \sqrt 3, \sqrt 5,\ldots\}$ the greatest lower bound is $\sqrt 2$ which is irrational and bounded from below the sequence. </p>
<p>Is this corre... | Community | -1 | <p>Consider $$S = \left\{2 + \frac{\sqrt{2}}{n} : n = 1, 2, 3, \dots\right\}$$</p>
<p>Then every element of $S$ is larger than $\sqrt{2}$, $S$ contains no rational entries, and $\inf S = 2$ is rational.</p>
|
201,173 | <p>I have problem solving this equation:
$$
\left(\frac{1+iz}{1-iz}\right)^4 = \frac12 + i {\sqrt{3}\over 2}
$$
I know how to solve equations that are like:
$$
w^4 = \frac12 + i {\sqrt{3}\over 2}
$$
And I have solved it to:
$$
w = \cos(-\frac{\pi}{12} + \frac{\pi k}{2})) + i\sin(-\frac{\pi}{12} + \frac{\pi k}{2})... | Michael Hardy | 11,667 | <p>$$
w=\frac{1+iz}{1-iz}
$$
First, multiply both sides by $1-iz$:
$$
w(1-iz) = 1+iz
$$
Expand the left side:
$$
w-wiz = 1+iz
$$
Put all terms involving $z$ on one side and those not involving $z$ on the other side:
$$
w-1=iz+wiz
$$
Factor
$$
w-1 = iz(1+w)
$$
Divide both sides by $i(1+w)$:
$$
z= \frac{w-1}{i(w+1)}
$$
M... |
2,919,266 | <p>Let $(x_n)$ be a sequence in $(-\infty, \infty]$. </p>
<p>Could we define the sequence $(x_n)$ so that limsup$(x_n) = -\infty$? </p>
<p>My intuitive thought is no, but I’m not 100% sure. </p>
| Nosrati | 108,128 | <p>Usiing telescopic sum
$$\sum_{n\geq1}^k\dfrac{n}{(n+1)(n+2)}2^n=\sum_{n\geq1}^k\left(\dfrac{2^{n+1}}{n+2}-\dfrac{2^{n}}{n+1}\right)=\dfrac{2^{k+1}}{k+2}-1$$</p>
|
3,337,440 | <p>Suppose that, in a memoryless way, an object A can suddenly transform into object B or object C. Once it transforms, it can no longer transform again (so if it becomes B, it cannot become C, and visa versa) </p>
<p>Suppose that the pdf of an object A becoming object B is </p>
<p><span class="math-container">$$\lam... | Henry | 6,460 | <p>The way you have set up the question does not work, and you have demonstrated that it does not work.</p>
<p>So let's create a system that does work involving memorylessness and your two rates of <span class="math-container">$\lambda$</span> and <span class="math-container">$\mu$</span>:</p>
<ul>
<li><p>Suppose you... |
2,450,245 | <p>A set $Q$ contains $0$, $1$ and the average of all elements of every finite non-empty subset of $Q$. Prove that $Q$ contains all rational numbers in $[0,1]$.</p>
<p>This is the exact wording, as it was given to me.
Obviously, the elements that correspond to the average, are rational, since they can be expressed as... | Michael Burr | 86,421 | <p>Let's explore the first few possibilities to see what's going on.</p>
<ul>
<li><p>You already know that $Q$ contains $0$ and $1$.</p></li>
<li><p>The average of $0$ and $1$ is $\frac{1}{2}$, so this is in $Q$ too.</p></li>
<li><p>The average of $0$ and $\frac{1}{2}$ is $\frac{1}{4}$ and the average of $\frac{1}{2}$... |
2,189,123 | <p>There are 36 gangsters, and several gangs these gangsters belong to. No two gangs have identical roster, and no gangster is an enemy of anyone in their gang. However, each gangster has at least one enemy in every gang they are <strong>not</strong> in. What is the greatest possible number of gangs? </p>
| Aravind | 4,959 | <p>I show that $3^{12}$ is the optimal value as found by bof and Makholm.
Let A,B be enemies. Let $N_A$ be the number of gangs containing A for which there is a gangster C such that A is the only enemy of C in that gang. Similary, let $N_B$ be the number for B. If $N_A \leq N_B$, then replace the enemies of A with the ... |
2,459,123 | <p>My attempt:</p>
<p>Step 1 $n=4 \quad LHS = 4! = 24 \quad RHS=4^2 = 16$</p>
<p>Therefore $P(1)$ is true.</p>
<p>Step 2 Assuming $P(n)$ is true for $n=k, \quad k!>k^2, k>3$</p>
<p>Step 3 $n=k+1$</p>
<p>$LHS = (k+1)! = (k+1)k! > (k+1)k^2$ (follows from Step 2)</p>
<p>I am getting stuck at this s... | Ian | 83,396 | <p>The Baire category theorem says that if $A=\bigcap_{n=1}^\infty A_n$ and $A_n$ are open and dense then $A$ is dense again. So it suffices to choose a sequence of open dense sets whose measure tends to zero, then take their intersection. You have basically found such a sequence: you have $U(\epsilon)$, so take $A_n=U... |
1,929,698 | <p>Let $f(x)=\chi_{[a,b]}(x)$ be the characteristic function of the interval $[a,b]\subset [-\pi,\pi]$. </p>
<p>Show that if $a\neq -\pi$, or $b\neq \pi$ and $a\neq b$, then the Fourier series does not converge absolutely for any $x$. [Hint: It suffices to prove that for many values of $n$ one has $|\sin n\theta_0|\ge... | Disintegrating By Parts | 112,478 | <p>Suppose $a \ne -\pi$ or $b \ne \pi$ and $a\ne b$. Then the function you are talking about must be discontinuous.</p>
<p>Suppose the series did converge absolutely for some $x$. That would mean
$$
\sum_n \left|\frac{e^{-ina}-e^{-inb}}{2\pi in}\right| < \infty.
$$
But that would force the <em>uniform</em>... |
2,268,947 | <p><a href="https://i.stack.imgur.com/6hCd2.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/6hCd2.png" alt="enter image description here"></a></p>
<p>($\dot I = \{0,1\}$)</p>
<p>The homotopy I've constructed is:
$$G(t_1, t_2) =
\begin{cases}
\alpha(t_1), & \text{if $(t_1,t_2) \in I \times \{0... | De Yang | 368,308 | <p>You are correct that the gluing lemma doesn't work: i.e. $G$ is not continuous unless $\gamma$ and $\delta$ are constant paths. The question you ask in the bottom is 'can I do something differently to make this map continuous'. I think the answer is no - the information given $F$ is not used in constructing the h... |
2,012,223 | <p>I've got a problem with finding main argument of these complex number. How can i evaluate this two examples?</p>
<p>$$\sin \theta - i\cos \theta$$</p>
<p>$$\frac{(1-i\tan \theta)}{1+\tan \theta}$$</p>
| user90369 | 332,823 | <p>It's not so clear what you mean, perhaps it's meant (where $z$ is a complex variable, here $z:=r(\cos\phi +i\sin\phi)$ with $r\in\mathbb{R}$): </p>
<p>$\displaystyle \sin\phi-i\cos\phi=-i\frac{z}{|z|}$ </p>
<p>$\displaystyle \frac{1-i\tan\phi}{1+\tan\phi}=\frac{2\overline{z}}{(1-i)z+(1+i)\overline{z}}$</p>
|
2,249,109 | <p><strong>Question:</strong> Three digit numbers in which the middle one is a perfect square are formed using the digits $1$ to $9$.Then their sum is?</p>
<p>$A. 134055$<br>
$B.270540$<br>
$C.170055$<br>
D. None Of The Above</p>
<p>Okay, It's pretty obvious that the number is like $XYZ$ where $X,Z\in[1,9] $ and $Y\i... | lhf | 589 | <p>Induction is really the easiest route:</p>
<p>Just sum</p>
<p>$\quad E_{n}= F_{n-1}A + F_{n}B$</p>
<p>$\quad E_{n+1}= F_nA + F_{n+1}B$</p>
<p>to get </p>
<p>$\quad E_{n+2}= F_{n+1}A + F_{n+2}B$</p>
|
3,796,216 | <p>Use a group-theoretic proof to show that <span class="math-container">$\mathbb{Q}^*$</span> under multiplication is
not isomorphic to <span class="math-container">$\mathbb{R}^*$</span> under multiplication.</p>
<p><strong>I have tried this:</strong></p>
<p>Suppose <span class="math-container">$$ \phi: \mathbb{Q}^*\t... | tomasz | 30,222 | <p>The squaring map is not onto <span class="math-container">$\mathbf R^\times$</span>, so it does not quite work. However, it is pretty close: the image is a subgroup of index <span class="math-container">$2$</span>. In <span class="math-container">$\mathbf Q^\times$</span>, the index is infinite. Thus, the two groups... |
4,142,540 | <p>Let <span class="math-container">$V$</span> a vector subspace of dimension <span class="math-container">$n$</span> on <span class="math-container">$\mathbb R$</span> and <span class="math-container">$f,g \in V^* \backslash \{0\}$</span> two linearly independent linear forms. I want to show that <span class="math-con... | azif00 | 680,927 | <p>Another way:</p>
<p>Consider the linear maps <span class="math-container">$h : V \to \mathbb R^2$</span> and <span class="math-container">$\tilde h : \mathbb R^2 \to V^*$</span> given by <span class="math-container">$h(v) = (f(v),g(v))$</span> for all <span class="math-container">$v \in V$</span> and <span class="ma... |
243,903 | <p>I have posted this question on mathstack echange but did not get any answer. It mam trying my luck here. </p>
<p>The only simple finite groups admitting an irreducible character of degree 3 are $\mathfrak{A}_5$
and $PSL(2,7)$. That seems to be a result coming from Blichfeldt's work on $GL(3,\mathbb{C})$, which I ca... | Geoff Robinson | 14,450 | <p>It depends on how much group theory you want to use. If $G$ is such a simple group and $\chi$ is a faithful complex irreducible character of degree $3$, then a Theorem of Feit and Thompson proves that $|G|$ is not divisible by any prime $p > 7$. It is easy to check (since $Z(G)$ contains no element of order $3$),... |
3,453,175 | <p>If <span class="math-container">$y=\dfrac {1}{x^x}$</span> then show that <span class="math-container">$y'' (1)=0$</span></p>
<p>My Attempt:</p>
<p><span class="math-container">$$y=\dfrac {1}{x^x}$$</span>
Taking <span class="math-container">$\ln$</span> on both sides,
<span class="math-container">$$\ln (y)= \ln \... | PierreCarre | 639,238 | <p>According to your own calculations, <span class="math-container">$y'(x) = - y(x)(1+ \ln x)$</span>, and, in particular, since <span class="math-container">$y(1)=1$</span>, you have that <span class="math-container">$y'(1)=-1$</span>. If you derive again,</p>
<p><span class="math-container">$$
y''(x) = (-y(x)(1+ \ln... |
66,009 | <p>Hi I have a very simple question but I haven't been able to find a set answer. How would I draw a bunch of polygons on one graph. The following does not work:</p>
<pre><code>Graphics[{Polygon[{{989, 1080}, {568, 1080}, {834, 711}}],
Polygon[{{1184, 1080}, {989, 1080}, {834, 711}, {958, 541}}],
Polygon[{{1379,... | MikeLimaOscar | 5,414 | <p>As @Szabolcs points out <code>Dispatch</code> does not interact well with <code>SameQ</code>, etc in Mathematica 10.</p>
<pre><code>Dispatch[1 -> 2] === Dispatch[1 -> 2]
</code></pre>
<blockquote>
<p>False</p>
</blockquote>
<pre><code>Dispatch[1 -> 2] == Dispatch[1 -> 2]
</code></pre>
<blockquote>
... |
491,926 | <p>Let I have $(S,\Sigma,\mu)$ be a probability space, then $X,Y \in \Sigma$. Define $\rho (X,Y)$ by $\rho (X,Y)$ = correlation between random variable $I_X$ and $I_Y$, where $I_X$ and $I_Y$ are the indicator function of $X$ and $Y$. Express $\rho (X,Y)$ in term of $\mu (X)$, $\mu (Y)$, $\mu(XY)$. Conclude that $\rho(X... | triple_sec | 87,778 | <p>To finish @JonathanY 's answer: $$(1)\quad \mu(XY^c)=\mu(X)-\mu(XY).$$
$$(2)\quad \mu(X^c Y)=\mu(Y)-\mu(XY).$$
$$(3)\quad\mu(X^c Y^c)=\mu(Y^c)-\mu(XY^c)=1-\mu(Y)-\mu(XY^c)=1-\mu(Y)-\mu(X)+\mu(XY),$$
using (1). Now, putting these together,
\begin{align*}
&\,\mu(XY)\mu(X^c Y^c)-\mu(XY^c)\mu(X^cY)\\
=&\,\mu(XY)... |
2,643,705 | <p>If $A,B,C,D$ are all matrices and $A=BCD$ (with dimensions such that all matrix multiplications are defined), how does one solve for $C$? </p>
<p>In the particular context I'm working in, $B$ and $D$ are both orthogonal, and $C$ is diagonal. I'm not sure if that's necessary to solve for $C$.</p>
| Robert Howard | 509,508 | <p>You would have to multiply by $B^{-1}$ on the left on each side of the equation which would cancel $B$ on the right, and then by $D^{-1}$ on the right on each side, which would cancel $D$, like this:
$$\begin{align}A&=BCD\\B^{-1}A&=B^{-1}BCD\\B^{-1}A&=CD\\B^{-1}AD^{-1}&=CDD^{-1}\\B^{-1}AD^{-1}&=C... |
3,353,826 | <p>All the vertices of quadrilateral <span class="math-container">$ABCD$</span> are at the circumference of a circle and its diagonals intersect at point <span class="math-container">$O$</span>. If <span class="math-container">$∠CAB = 40°$</span> and <span class="math-container">$∠DBC = 70°$</span>, <span class="math-c... | leftaroundabout | 11,107 | <p>Although I'm not sure how common this is in pure maths settings, I would say the best notation is simply <span class="math-container">$\operatorname{round}(x)$</span>. This is easily understood, albeit not completely unambiguous – but <em>definitely</em> better than <span class="math-container">$[x]$</span> which co... |
3,353,826 | <p>All the vertices of quadrilateral <span class="math-container">$ABCD$</span> are at the circumference of a circle and its diagonals intersect at point <span class="math-container">$O$</span>. If <span class="math-container">$∠CAB = 40°$</span> and <span class="math-container">$∠DBC = 70°$</span>, <span class="math-c... | John Thompson | 787,805 | <p>The short and sweet is that there is no short and sweet. Rounding has many different contexts and interpretations, which means that you will have to define what rounding means to you in your particular context before your use it. Now, there is standard notation for two specific types of rounding. <span class="math... |
2,756,139 | <p>Let $I=[0,1]$ and $$X = \prod_{i \in I}^{} \mathbb{R}$$
That is, an element of $X$ is a function $f:I→\mathbb{R}$.</p>
<p>Prove that a sequence $\{f_n\}_n ⊆ X$ of real functions converges to some $f ∈ X$ in the product topology on $X$, if and only if it converges pointwise, i.e. for every $x ∈ I$, $f_n(x) → f(x)$ i... | Logic_Problem_42 | 338,002 | <p>Every single element of $X$ is simply a function: $I\to \mathbb{R}$. They differ as normal functions differ. </p>
|
24,810 | <p>The title says it all. Is there a way to take a poll on Maths Stack Exchange? Is a poll an acceptable question?</p>
| Gerry Myerson | 8,269 | <p>Vote this answer up, if you think a poll is not an acceptable question. </p>
|
3,973,611 | <p>Let <span class="math-container">$$F(x)=\int_{-\infty}^x f(t)dt,$$</span>
where <span class="math-container">$x\in\mathcal{R}$</span>, <span class="math-container">$f\geq 0$</span> is complicated (it cannot be integrated analytically).</p>
<p>Can I used the Simpson's rule to approximate this integral, knowing that <... | Especially Lime | 341,019 | <p>First, your evaluation of <span class="math-container">$|D|$</span> is incorrect. This is the number of ways to choose a capital letter for the first character, and then choose the remaining seven characters with no capitals. But in fact the single capital letter could be in any of <span class="math-container">$8$</... |
4,090,416 | <blockquote>
<p>Suppose <span class="math-container">$f(x)$</span> be bounded and differentiable over <span class="math-container">$\mathbb R$</span>, and
<span class="math-container">$|f'(x)|<1$</span> for any <span class="math-container">$x$</span>. Prove there exists <span class="math-container">$M<1$</span> s... | Martin R | 42,969 | <p>The function <span class="math-container">$g: \Bbb R \to \Bbb R$</span>, defined as
<span class="math-container">$$
g(x) = \begin{cases}
\frac{f(x)-f(0)}{x-0} & \text{ if } x \ne 0 \\
f'(0) & \text{ if } x = 0
\end{cases}
$$</span>
is continuous, with <span class="math-container">$\lim_{x \to - \infty} ... |
1,027,330 | <p>How does one figure out whether this series:
$$\sum_{n=3}^{\infty}(-1)^{n-1}\frac{1}{\ln\ln n}$$
converges or diverges? And, what is the general approach behind solving for convergence/divergence in a series that seems to "oscillate" (thanks to the -1 in this case)? </p>
<p>I have so far tried to split the functio... | mathamphetamines | 126,882 | <p>$$
a_n = \frac{1}{lnln(n)}
$$</p>
<p>$$
for\ n>3, a_n>0
$$
Next you must show that this function is decreasing everywhere, and also that as n approaches infinity, a_n converges to zero.</p>
|
1,224,180 | <p>Q: evaluate $\lim_{x \to \infty}$ $ (x-1)\over \sqrt {2x^2-1}$</p>
<p>What I did:</p>
<p>when $\lim_ {x \to \infty}$ you must put the argument in the form of $1/x$ so in that way you know that is equal to $0$</p>
<p>but in this ex. the farest that I went was</p>
<p>$\lim_{x \to \infty}$ $x \over x \sqrt{2}$ $1-(... | mich95 | 229,072 | <p>$\frac{x-1}{\sqrt{2x^{2}-1}}=\frac{x-1}{x \sqrt{2-\frac{1}{x^{2}}}}=\frac{1-\frac{1}{x}}{\sqrt{2-\frac{1}{x^{2}}}}$. Hence the limit is $\frac{1}{\sqrt{2}}$</p>
|
3,551,030 | <p>Let the real sequence <span class="math-container">$x_n s.t. \Vert x_{n+2} - x_{n+1}\Vert = M \Vert x_{n+1} - x_{n} \Vert$</span> </p>
<p>If the <span class="math-container">$0< \vert M \vert <1$</span>, then <span class="math-container">$x_n$</span> surely convergent since it is a contractive. </p>
<p>But t... | Kavi Rama Murthy | 142,385 | <p>Suppose <span class="math-container">$x_1 \neq x_2$</span>. </p>
<p><span class="math-container">$\|x_{n+2}-x_{n+1}\|=M^{n-1} \|x_2-x_1\|$</span> for all <span class="math-container">$n$</span>. RHS tends to <span class="math-container">$\infty$</span>. If the sequence is convergent then LHS tends to <span class="... |
2,563,303 | <blockquote>
<p><strong><em>Question:</em></strong> If $z_0$ and $z_1$ are real irrational numbers I write $$q=z_0+z_1\sqrt{-1}$$ Surely $q$ is just a complex number. Under what condition will the <a href="https://en.wikipedia.org/wiki/Absolute_value#Complex_numbers" rel="nofollow noreferrer">number</a> $|q|$ be an ... | Fred | 380,717 | <p>If $(A_n)$ is a positive sequence with $\lim_{n \to \infty}\frac{A_{n+1}}{A_n}= 0$, then by the ratio test: $\sum_{n=1}^{\infty}A_n$ converges, hence $A_n \to 0$.</p>
|
2,933,383 | <p>Cauchy's theorem of limits states that if <span class="math-container">$\ \lim_{n \to \infty} a_n=L ,$</span> then <span class="math-container">$$ \lim_{n \to \infty} \frac{a_1+a_2+\cdots+a_n}{n}=L $$</span>
If I apply this in the series <span class="math-container">$$S = \lim_{n\to\infty} \dfrac{1}{n}[e^{\frac{1}... | asdf | 436,163 | <p>The main issue is that your series UP TO <span class="math-container">$n$</span> are <span class="math-container">$a_i=e^{\frac{i}{n}}$</span> but this defines the sequence only UP TO some <span class="math-container">$n$</span> which is fixed and thus taking this <span class="math-container">$n$</span> fixed tells... |
2,664,370 | <p>I don't know how to start, i've noticed that it can be written as $$\lim_{x\to 0} \frac{2^x+5^x-4^x-3^x}{5^x+4^x-3^x-2^x}=\lim_{x\to 0} \frac{(5^x-3^x)+(2^x-4^x)}{(5^x-3^x)-(2^x-4^x)}$$</p>
| egreg | 62,967 | <p>This is a very special case of l’Hôpital’s theorem, possibly what gave him (or Bernoulli) the idea for the general case.</p>
<p>If you have two functions $f$ and $g$ which are differentiable at $0$ (or any other point), then
$$
\lim_{x\to0}\frac{f(x)-f(0)}{g(x)-g(0)}=\frac{f'(0)}{g'(0)}
$$
provided $g'(0)\ne0$ (if ... |
1,004,303 | <p>Let $S=\{(x,0)\} \cup\{(x,1/x):x>0\}$. Prove that $S$ is not a connected space (the topology on $S$ is the subspace topology)</p>
<p>My thoughts: Now in the first set $x$ is any real number, and I can't see that this set in open in $S$. I can't find a suitable intersection anyhow.</p>
| Crostul | 160,300 | <p>Call $X = \{ (x, 0) : x \in \mathbb{R} \} \subset S$. Clearly $X$ is closed in $S$ since $X$ is closed in $\mathbb{R}^2$.</p>
<p>But $X$ is also open in $S$, since $X= S \cap T$, where $T= \{ (x, y) \in \mathbb{R}^2 : y< \frac{1}{2x} , x>0\} \cup \{ (x,y) \in \mathbb{R}^2 : y < 1-x\}$ is open in $\mathbb{R... |
2,141,082 | <p>Let me first put down a couple definitions, two of which have terminology I make up for this post. If you already know about sheaf theory, you can safely skip Definitions 1-3 and 7-8, and the Construction. Definitions 4-6 introduce notation and terminology that is probably nonstandard, so I recommend reading those i... | MickG | 135,592 | <p>Let me make a little sum-up and (hopefully) complete the proofs of the WBFs.</p>
<p>There is a big trap I fell right into: <strong>$\mathcal{F}$ (presheaf) is a sub-presheaf of $\mathcal{F}^+$ (sheafification)</strong> is <strong>false</strong>. Indeed, as shown in <strong>Fact U2</strong>, that holds iff $\mathcal... |
1,685,423 | <p>$$
x_n=\begin{cases}\frac{1}{n^2} & \text{if n is even} \\ \frac{1}{n} & \text{if n is odd}\end{cases}$$.</p>
<p>How can I show that $$ \sum x_n$$ is convergent?</p>
| Ahmed S. Attaalla | 229,023 | <p>An even $n$ contributes a positive amount to the sum. However if $n$ is odd then $n=2k+1$ will contribute:</p>
<p>$$\sum_{k=0}^{\infty} \frac{1}{2k+1}$$</p>
<p>What can you say about this sum?</p>
|
964,999 | <p>If A and B are two closed sets of $R$ is A.B closed?
By A.B I mean the set $\sum_{i=1}{^ n} a_ib_i$ where $a_i \in A,b_i\in B,n\in N$
How to view A.B geometrically?
I am new to this subject.Sorry if the question sounds something wrong</p>
| John | 105,625 | <p>Without further clarification from the OP, I am interpreting the question as "$A,B$ are subsets of $\mathbb{R}$, and $A \cdot B =\{ab \,| \, a\in A,b \in B\}$".</p>
<p>If $A,B$ are closed sets in $\mathbb{R}$ with standard topology, then $A\cdot B$ may not be closed in $\mathbb{R}$. An example is $A=\{0\} \cup \{\f... |
2,950,813 | <blockquote>
<p>Take <span class="math-container">$G$</span> to be a group of order <span class="math-container">$600$</span>. Prove that for any element <span class="math-container">$a$</span> <span class="math-container">$\in$</span> G there exist an element <span class="math-container">$b$</span> <span class="math... | Nicky Hekster | 9,605 | <p><strong>Proposition</strong> Let <span class="math-container">$G$</span> be a finite group and <span class="math-container">$n$</span> a positive integer. Then the map <span class="math-container">$f: G \mapsto G$</span> defined by <span class="math-container">$f(g)=g^n$</span> is a bijection if and only if gcd<span... |
302,179 | <p>The question I am working on is:</p>
<blockquote>
<p>"Use a direct proof to show that every odd integer is the difference of two squares."</p>
</blockquote>
<p>Proof:</p>
<p>Let n be an odd integer: $n = 2k + 1$, where $k \in Z$</p>
<p>Let the difference of two different squares be, $a^2-b^2$, where $a,b \in Z... | Herng Yi | 34,473 | <p>Another approach, a graphical proof:
$$\underbrace{\begin{array}{ccccc}
1\odot & 3\otimes & 5\odot & \cdots & (2k - 1)\otimes\\\hline
\bigodot & \bigotimes & \bigodot & \cdots & \bigotimes\\
\bigotimes & \bigotimes & \bigodot & \cdots & \bigotimes\\
\bigodot & \big... |
302,179 | <p>The question I am working on is:</p>
<blockquote>
<p>"Use a direct proof to show that every odd integer is the difference of two squares."</p>
</blockquote>
<p>Proof:</p>
<p>Let n be an odd integer: $n = 2k + 1$, where $k \in Z$</p>
<p>Let the difference of two different squares be, $a^2-b^2$, where $a,b \in Z... | Ben Millwood | 29,966 | <p>What's going on here, I think, is you're confused about what you're being asked to prove.</p>
<p>The statement is "every odd integer is the difference of two squares", or, more precisely,
"<strong>for all</strong> odd integers $n$, <strong>there exist</strong> $a$ and $b$ such that $a^2 - b^2 = n$". Think for a b... |
874,404 | <p>I'm reading through Basic Algebra I (which I enjoy so far. Thoughts on this for self-studying?) and am having a difficult time proving surjection. I believe I understand the concept, but when it comes to the proof I'm not sure how I'm supposed to prove it. Is this an acceptable proof, or is it at least part of the f... | Magician | 155,759 | <p>It is not correct to say that $$\forall y\in Y \texttt{ and }\forall z\in Z, \exists x \in X \mid y = f(x), z = g(y) = g(f(x)) = (f \circ g)(x)$$
You cannot guarantee that $z=g(y),\forall y,z$. For example, let $X=Y=Z$ and $f,g$ be the identity map. It is clearly incorrect that $\forall y,z(\exists x(y=x,z=y=x))$.</... |
874,404 | <p>I'm reading through Basic Algebra I (which I enjoy so far. Thoughts on this for self-studying?) and am having a difficult time proving surjection. I believe I understand the concept, but when it comes to the proof I'm not sure how I'm supposed to prove it. Is this an acceptable proof, or is it at least part of the f... | Andrew D'Addesio | 61,123 | <p>You write:</p>
<p>$$
\text{Therefore, }
\forall (y,z) \in Y \times Z, \exists x \in X \mid y = f(x), z = g(y) = g(f(x)) =: (g \circ f)(x).
$$</p>
<p>What this is saying is that if $f : X \to Y$ and $g : Y \to Z $ are surjective, then there always exists an $x \in X$ such that the calculation $(f(x), g(f(x)))$ will... |
3,853,509 | <blockquote>
<p>prove <span class="math-container">$$\sum_{cyc}\frac{a^2}{a+2b^2}\ge 1$$</span> holds for all positives <span class="math-container">$a,b,c$</span> when <span class="math-container">$\sqrt{a}+\sqrt{b}+\sqrt{c}=3$</span> or <span class="math-container">$ab+bc+ca=3$</span></p>
</blockquote>
<hr />
<p><str... | saulspatz | 235,128 | <p>Instead of calculating derivatives, try to restate it in terms of a series you know.
<span class="math-container">$$\frac1{2+3x^2}=\frac12\frac1{1+\frac{3x^2}2}$$</span>
Now if you set <span class="math-container">$y=\sqrt\frac32x$</span> you'll see an expression with a familiar series that has radius of convergence... |
1,055,832 | <p>How can one prove distributivity of a <a href="http://ncatlab.org/nlab/show/Heyting+algebra" rel="nofollow"> Heyting Algebra</a> via the <a href="http://ncatlab.org/nlab/show/Yoneda+lemma" rel="nofollow"> Yoneda lemma</a>?</p>
<p>I'm able to prove it using the Heyting algebra property $(x \wedge a) \leq b$ if and o... | Sina Hazratpour | 278,320 | <p>Use full and faithfulness of Yoneda embedding to prove every cartesian closed category C with finite coproducts must be distributive, i.e. $ a × (b + c)$ and
$(a × b) + (a × c)$ are isomorphic objects in C. </p>
|
4,550,991 | <p>This is question is taken from an early round of a Norwegian national math competition where you have on average 5 minutes to solve each question.</p>
<p>I tried to solve the question by writing every number with four digits and with introductory zeros where it was needed. For example 0001 and 0101 would be the numb... | Jaap Scherphuis | 362,967 | <p>Here is what I would think is the quickest way to do it.</p>
<p>As you already wrote, use leading zeroes so that all numbers have 4 digits. You can even include <span class="math-container">$0000$</span> to make things easier as that will not affect the result.</p>
<p>Let's count how many times a 1 appears in the fi... |
3,060,250 | <p>Let <span class="math-container">$f : \mathbb{R} \to \mathbb{R}$</span> be differentiable. Assume that <span class="math-container">$1 \le f(x) \le 2$</span> for <span class="math-container">$x \in \mathbb{R}$</span> and <span class="math-container">$f(0) = 0$</span>. Prove that <span class="math-container">$x \le f... | S. Senko | 361,834 | <p>Presuming that you meant to write <span class="math-container">$1 \le f'(x) \le 2$</span> rather than <span class="math-container">$1 \le f(x) \le 2$</span>, you can prove it as follows. First, fix an arbitrary non-zero <span class="math-container">$x \in \Bbb{R}$</span> (the result is obvious if <span class="math-c... |
58,926 | <p>Is it well known what happens if one blows-up $\mathbb{P}^2$ at points in non-general position (ie. 3 points on a line, 6 on a conic etc)? Are these objects isomorphic to something nice? </p>
| known google | 392 | <p>I'll just add to Francesco's answer by saying that general position of the points on the plane is equivalent to ampleness of the anticanonical sheaf $\omega_X^{\otimes -1}$.</p>
<p>The key observation is that on a del Pezzo surface, an irreducible negative curve ($C^2 < 0$) must be an exceptional curve (i.e. $C^... |
3,335,060 | <blockquote>
<p>The numbers of possible continuous <span class="math-container">$f(x)$</span> defiend on <span class="math-container">$[0,1]$</span> for which <span class="math-container">$I_1=\int_0^1 f(x)dx = 1,~I_2=\int_0^1 xf(x)dx = a,~I_3=\int_0^1 x^2f(x)dx = a^2 $</span> is/are</p>
<p><span class="math-container"... | Green05 | 650,645 | <p>How about you simply use the Taylor series for <span class="math-container">$e^x$</span>?
<span class="math-container">$$e^x = \sum_{r=0}^\infty \frac {x^r}{r!} \\
e^3 \gt 1+3/1+9/2+9/2+27/8+ 81/40+81/80+243/560+(243/560)*3/8 + ((243*3)/(560*8))*1/3 = 1+3+9+3.375+2.025+1.0125+0.4339..+0.1627...+0.0542... = 20.0633 ... |
3,981,458 | <p>A star graph <span class="math-container">$S_{k}$</span> is the complete bipartite graph <span class="math-container">$K_{1,k}$</span>. One bipartition contains 1 vertex and the other bipartition contains <span class="math-container">$k$</span> vertices. <a href="https://en.wikipedia.org/wiki/Star_(graph_theory)" re... | qualcuno | 362,866 | <p>By direct computation,</p>
<p><span class="math-container">$$
\widetilde{d}(S_k) = \frac{k + \overbrace{1 + \cdots + 1}^{k}}{k+1} = 2k/(k+1).
$$</span></p>
<p>Let's enumerate the vertices of <span class="math-container">$S_k$</span> as <span class="math-container">$v_0, \ldots, v_k$</span> with <span class="math-co... |
2,620,032 | <p>Find the derivative of $y=(\tan (x))^{\log (x)}$</p>
<p>I thought of using the power rule that:
$$\dfrac {d}{dx} u^n = n.u^{n-1}.\dfrac {du}{dx}$$
Realizing that the exponent $log(x)$ is not constant, I could not use that. </p>
| user326210 | 326,210 | <p>You can rewrite $y = (\tan{(x)})^{\log{(x)}}$ as $$y=\exp\left(\log{(\tan{(x)})}\log(x)\right)$$ using the rule $a^b = \exp(\log(a))^b = \exp{(b\cdot\log{(a)})}$. </p>
<p>In this form, you can find the derivative of $y$ using the chain rule and product rule.</p>
|
127,225 | <p>I got stuck solving the following problem:</p>
<pre><code>Table[Table[
Table[
g1Size = x; g2Size = y;
vals =
FindInstance[(a1 - a2) - (b1 - b2) == z && a1 + b1 == g1Size &&
a2 + b2 == g2Size && a1 + a2 == g1Size && b1 + b2 == g2Size &&
a1 > 0 &am... | mikado | 36,788 | <p>The (edited) OP gives the following implementation of the time-consuming part of the calculation. (I will ignore the problem of finding the maximum as trivial).</p>
<pre><code>refimplementation = Coefficient[Times @@ (1 + #1 x) // Expand, x^#2] &;
</code></pre>
<p>I offer the following slightly different impl... |
830,111 | <p>We have the following set of lines:
$$L_1: \frac{x-2}{1}=\frac{y-3}{-2}=\frac{z-1}{-3}$$
$$L_2:\frac{x-3}{1}=\frac{y+4}{3}=\frac{z-2}{-7}$$</p>
<p>This leads to the following parametric equations: $$L_1:x=t+2,\space y=-2t+3,\space z=-3t+1$$
$$L_2: x=s+3,\space y=3s-4,\space z=-7s+2$$
The $x$ line looked pretty simp... | user2357112 | 91,416 | <p>Your error is in going from</p>
<p>$$-2(s+1)+3=3s-4$$</p>
<p>to</p>
<p>$$s=5,\space t=6$$</p>
<p>I don't know how you got that, but it should be $s=1$, $t=2$. Maybe you dropped a term or misplaced a sign while solving the equation.</p>
|
1,831,191 | <p>I am confused about the following Theorem:</p>
<p>Let <span class="math-container">$f: I \to \mathbb{R}^n$</span>, <span class="math-container">$a \in I$</span>. Then the function <span class="math-container">$f$</span> is differentiable at <span class="math-container">$a$</span> if and only if there exists a functi... | anomaly | 156,999 | <p>The theorem is simply stating that the function
\begin{align*}
\varphi(x) &=
\begin{cases}
\frac{f(x) - f(a)}{x - a} & \text{if}\;x\not = a; \\
f'(a) & \text{if}\;x = a
\end{cases}
\end{align*}
is continuous. And it clearly is; the only point to check is $x = a$, and the condition $\lim_{x\to a} \varph... |
227,311 | <p>It is known that the roots of Chebyshev polynomials of the second kind, denote it by $U_n(x)$, are in the interval $(-1,1)$. I have noticed that, by looking at the low values of $n$, the roots of $(1-x)U_n(x)+U_{n-1}(x)$ are in the interval $(-2,2)$. However, I don't have a clear idea how to start proving this, co... | John Jiang | 4,923 | <p>My previous answer shows that if all roots are real, then they must be contained with $[-1,\sqrt{2})$. To show all roots are real, use <a href="https://books.google.com/books?id=FzFEEVO3PXYC&pg=PA199&lpg=PA199#v=onepage&q&f=false" rel="nofollow">lemma 6.3.9 from the book Analytic Theory of Polynomial... |
1,278,329 | <p>Solve the recurrence $a_n = 4a_{n−1} − 2 a_{n−2}$</p>
<p>Not sure how to solve this recurrence as I don't know which numbers to input to recursively solve?</p>
| Demosthene | 163,662 | <p>You can make an "educated guess" and propose the following ansatz: $a_n=p^n$. Your recurrence relation now has the following characteristic equation:
$$p^2-4p+2=0\Longleftrightarrow (p-2-\sqrt{2})(p-2+\sqrt{2})=0$$
Therefore there are two roots, at $p=2\pm\sqrt{2}$, and we get:
$$a_n=\alpha\left(2+\sqrt{2}\right)^n+... |
205,049 | <p><a href="http://1d4chan.org/wiki/Most_Excellent_Adventure" rel="noreferrer">Most Excellent Adventure</a> is a home brew roleplaying game system based on the Bill & Ted Films, <em>plays gnarly air guitar riff</em>. </p>
<p>In this game system, when you draw from your dice pool you need to connect the results as ... | Hagen von Eitzen | 39,174 | <p>It is remarkable, though maybe not trivial, that for almost every connected set of phone keys, there exists a Eulerian path through the digits (that is you can walk from digit to adjacent digit and reach all digits exactly once). By simply repeating multiple digits, you can thus dial a number (almost) whenever the s... |
205,049 | <p><a href="http://1d4chan.org/wiki/Most_Excellent_Adventure" rel="noreferrer">Most Excellent Adventure</a> is a home brew roleplaying game system based on the Bill & Ted Films, <em>plays gnarly air guitar riff</em>. </p>
<p>In this game system, when you draw from your dice pool you need to connect the results as ... | Alan Gee | 43,196 | <p>I'm not sure that this qualifies as a complete answer to the question, but it will certainly provide some useful ideas and information.</p>
<p>I wrote a program to solve this using the following <i>brute force</i> technique.</p>
<ol>
<li>Create a Possibe_Next_Key integer set for each of the digits 0 to 9 E.g. For ... |
1,262,174 | <p>I am currently teaching Physics in an Italian junior high school. Today, while talking about the <a href="http://en.wikipedia.org/wiki/Dipole#/media/File:Dipole_Contour.svg" rel="noreferrer">electric dipole</a> generated by two equal charges in the plane, I was wondering about the following problem:</p>
<blockquote>... | David H | 55,051 | <p>Consider this post an appendix to @achillehui's brilliant solution, which arrives at the following elliptic integral for the overall area:</p>
<blockquote>
<p>$$\mathcal{A}=\sqrt{8}+\int_{1}^{\phi}\sqrt{\frac{1+x-x^{2}}{x\left(1+x\right)}}\,\mathrm{d}x=:\sqrt{8}+\mathcal{B},\tag{1}$$</p>
</blockquote>
<p>where a... |
4,408,507 | <p>We study the definition of Lebesgue measurable set to be the following:</p>
<p>Let <span class="math-container">$A\subset \mathbb R$</span> be called Lebesgue measurable if <span class="math-container">$\exists$</span> a Borel set <span class="math-container">$B\subset A$</span> such that <span class="math-container... | L. F. | 620,160 | <p>As <a href="https://math.stackexchange.com/users/1021258">@Esgeriath</a>'s <a href="https://math.stackexchange.com/a/4523017">answer</a> mentioned, the correct inequality should be
<span class="math-container">$$
\sin \frac{A}{2} \; \sin \frac{B}{2} \; \sin \frac{C}{2} \le \frac{1}{8}.
$$</span>
That answer used mul... |
2,278,798 | <p>The converse statement, "A metric space on which every continuous, real valued function is bounded is compact" is dealt with on this site, as it is in Greene and Gamelin's monograph, "Introduction to Topology", where a hint to its proof is offered. I see no discussion of the direct statement in my title. Is it tru... | Henno Brandsma | 4,280 | <p>If $f$ is continuous on $X$ to $\mathbb{R}$. Then define $U_n = f^{-1}[(-n,n)]$ for $ n \in \mathbb{N}^+$, the the $U_n$ are open by continuity and $U_k \subseteq U_l$ wheneven $ k \le l$, i.e. the family is increasing.
Also, $\mathcal{U} = \{U_n: n \in \mathbb{N}^+ \}$ is an open cover of $X$, as every $y$ is in so... |
2,150,886 | <p>I want to find a first order ode, an initial value problem, that has the solution
$$y=(1-y_0)t+y_0$$
where $y_0$ is the initial value.The ode has to be of first order, that is:
$$y'=f(y).$$
I need this to test a special solver I am building.
The main objective is to find an ode that has the property that the end-... | Christian Blatter | 1,303 | <p>This is maybe not what you intended. </p>
<p>Note that you want a differential equation of the form $y'=f(y)$, and at the same time the proposed solution has constant derivative with respect to $t$. This implies that $f$ is necessarily of the form $f(y)=c$ for some constant $c$. Given $y_0$ an IVP satisfying your r... |
2,901,734 | <p>As title says find the minimum value of $(1+\frac{1}{x})(1+\frac{1}{y})$when given that $x+y=8$ using AGM inequality including Arithmetic Mean, Geometric Mean, and Harmonic Mean.</p>
| farruhota | 425,072 | <p>Using all three means:
$$(1+\frac{1}{x})(1+\frac{1}{y})=1+\color{red}{\frac1x+\frac1y}+\color{blue}{\frac1{xy}}\ge 1+\color{red}{\frac12}+\color{blue}{\frac1{16}}=\frac{25}{16},$$
where:
$$\text{AM-HM:} \ \ \frac{x+y}{2}\ge \frac{2}{\frac1x+\frac1y} \iff \frac1x+\frac1y\ge \frac4{x+y}=\frac48=\frac12;\\
\text{AM-GM:... |
182,346 | <p>Let's call a polygon $P$ <em>shrinkable</em> if any down-scaled (dilated) version of $P$ can be translated into $P$. For example, the following triangle is shrinkable (the original polygon is green, the dilated polygon is blue):</p>
<p><img src="https://i.stack.imgur.com/M0LOu.png" alt="enter image description here... | Beni Bogosel | 13,093 | <p>Here is my variant, a bit more geometrical.</p>
<p>Denote by $P$ the original polygon, and $P_\lambda$ the contracted polygon with a factor $\lambda \in (0,1)$ which lies inside $P$. Note that $P_\lambda$ is obtained from $P$ after a dilation and a translation, and therefore there exists a point $O_\lambda$ such th... |
1,928,439 | <p>Is there a space whose dual is $F^m$? ($F$ is the field w.r.t. which the original set is a vector space)</p>
<p>I'm trying to do the following exercise:</p>
<p><a href="https://i.stack.imgur.com/j9WMP.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/j9WMP.png" alt="enter image description here"><... | Alex Ortiz | 305,215 | <p>Given a vector space $\mathbf F^m$, this space has its double dual space ${({\mathbf F^m})^{\ast\ast}} = \mathcal L(({\mathbf F^m})^{\ast},\mathbf F),$ the set of linear functionals on $({\mathbf F^m})^{\ast}$. This double dual is <em>naturally isomorphic</em> to $\mathbf F^m$. Let $v \in \mathbf F^m$, $\varphi \in ... |
883,620 | <p>$a,b,c \geq 0$ and $a^2+b^2+c^2+abc=4$ prove that $ab+bc+ac-abc \leq 2$
can any one help me with this problem,I believe Dirichlet's theorem is the key for this
sorry for making mistake over and over again,but i'm certain that the inequality is true now.</p>
| Joonas Ilmavirta | 166,535 | <p>I assume you want $ab+bc+ca-abc\leq2$. If this is not the case, let me know. This is not the best inequality you could get, but it's true.</p>
<p>Since $(a-b)^2\geq0$, we have $a^2+b^2\geq2ab$, and similarly $b^2+c^2\geq2bc$ and $c^2+a^2\geq2ca$.
Summing up the three inequalities gives
$$
2(a^2+b^2+c^2)\geq2(ab+bc+... |
3,891,336 | <p>I have a problem with this question:</p>
<p>we have a language with alphabet {a, b, c}, all strings in this language have even length and does not contain any substring "ab" and "ba" for example these strings acceptable: "accb", "aa", "bb", "bcac", and thes... | J.-E. Pin | 89,374 | <p>Your language is regular. All you need to know is that regular languages are closed under Boolean operations (union, intersection, complement, set difference, ...)</p>
<p>Let <span class="math-container">$A = \{a,b,c\}$</span> be the alphabet. The set <span class="math-container">$E$</span> of words of even length i... |
1,386,367 | <p>I'm interested in the definite integral</p>
<p>\begin{align}
I\equiv\int_{-\infty}^{\infty} \frac{1}{x^2-b^2}=\int_{-\infty}^{\infty} \frac{1}{(x+b) (x-b)}.\tag{1}
\end{align}</p>
<p>Obviously, it has two poles ($x=b, x=-b$) on the real axes and is thus singular. I tried to apply the contour integration methods me... | Emilio Novati | 187,568 | <p>Hint:
$$
(I+P)(P-2I)=P-2I+P-2P=-2I
$$</p>
|
3,944,628 | <p>I'm reading a book and, in its section on the definition of a stopping time(continuous), the author declares at the start that for the whole section every filtration will be complete and right-continuous.</p>
<p>So, in the definition of a Stopping Time, how important are these conditions? Why would they matter?</p>
| B. Goddard | 362,009 | <p>You can do it with Lagrange multipliers. Maximize <span class="math-container">$f=\sin x/2 + \sin y/2+\sin z/2$</span> under the constraint <span class="math-container">$g=x+y+z = \pi$</span>.</p>
<p>Then</p>
<p><span class="math-container">$$\nabla f = \langle \cos(x/2)/2, \cos(y/2)/2, \cos(z/2)/2
\rangle =\lamb... |
3,944,628 | <p>I'm reading a book and, in its section on the definition of a stopping time(continuous), the author declares at the start that for the whole section every filtration will be complete and right-continuous.</p>
<p>So, in the definition of a Stopping Time, how important are these conditions? Why would they matter?</p>
| Z Ahmed | 671,540 | <p>In a triangle ABC, <span class="math-container">$A+B+C=\pi$</span>
<span class="math-container">$$f(x)=\sin(x/2) \implies f''(x)=-\frac{1}{4}\sin(x/2)<0, x\in[0,2\pi].$$</span> So by Jemsen's inequality <span class="math-container">$$\frac{f(A/2)+f(B/2)+f(C/2)}{3} \le f(\frac{A+B+C}{6}).$$</span>
<span class="mat... |
4,090,408 | <p>Show that <span class="math-container">$A$</span> is a whole number: <span class="math-container">$$A=\sqrt{\left|40\sqrt2-57\right|}-\sqrt{\left|40\sqrt2+57\right|}.$$</span>
I don't know if this is necessary, but we can compare <span class="math-container">$40\sqrt{2}$</span> and <span class="math-container">$57$<... | José Carlos Santos | 446,262 | <p>That number is <span class="math-container">$-10$</span>. In fact, if you try to express <span class="math-container">$\sqrt{57-40\sqrt2}$</span> as <span class="math-container">$a+b\sqrt2$</span>, you solve the system<span class="math-container">$$\left\{\begin{array}{l}a^2+2b^2=57\\2ab=-40.\end{array}\right.$$</sp... |
3,416,600 | <p>Show that <span class="math-container">$|{\sqrt{a^2+b^2}-\sqrt{a^2+c^2}}|\le|b-c|$</span> where <span class="math-container">$a,b,c\in\mathbb{R}$</span></p>
<p>I'd like to get an hint on how to get started. What I thought to do so far is dividing to cases to get rid of the absolute value. <span class="math-containe... | nonuser | 463,553 | <p>Use the formula: <span class="math-container">$${\sqrt{x}-\sqrt{y}}= {x-y\over \sqrt{x}+\sqrt{y}}$$</span></p>
<p>in your case you get:</p>
<p><span class="math-container">$${\sqrt{a^2+b^2}-\sqrt{a^2+c^2}}= {a^2+b^2-(a^2+c^2)\over \sqrt{a^2+b^2}+\sqrt{a^2+c^2}}$$</span></p>
<p>so you have to prove (if you assume ... |
152,405 | <p>This question complement a previous MO question: <a href="https://mathoverflow.net/questions/95837/examples-of-theorems-with-proofs-that-have-dramatically-improved-over-time">Examples of theorems with proofs that have dramatically improved over time</a>.</p>
<p>I am looking for a list of</p>
<h3>Major theorems in ma... | Vít Tuček | 6,818 | <p>Every $L^2$ function on $\mathbb{R}$ is almost everywhere the point-wise limit of its Fourier series. These days known as <a href="https://en.wikipedia.org/wiki/Carleson%27s_theorem">Carleson's theorem</a>.</p>
|
152,405 | <p>This question complement a previous MO question: <a href="https://mathoverflow.net/questions/95837/examples-of-theorems-with-proofs-that-have-dramatically-improved-over-time">Examples of theorems with proofs that have dramatically improved over time</a>.</p>
<p>I am looking for a list of</p>
<h3>Major theorems in ma... | Gil Kalai | 1,532 | <p><strong><a href="http://en.wikipedia.org/wiki/Classification_of_finite_simple_groups">The classification of finite simple groups</a></strong></p>
<p>This theorem describes completely all finite simple groups: A finite simple group is either cyclic groups of prime order, alternating groups, groups of Lie type (inclu... |
27,490 | <h2>Motivation</h2>
<p>The common functors from topological spaces to other categories have geometric interpretations. For example, the fundamental group is how loops behave in the space, and higher homotopy groups are how higher dimensional spheres behave (up to homotopy in both cases, of course). Even better, for nic... | David Roberts | 4,177 | <p>For an even more geometric application, the fundamental groupoid tells you all about parallel transport in bundles, as long as the transport is independent of the actual path and only relies on the homotopy class of the path. This is the case for flat connections (in particular on vector bundles). For appropriate sp... |
2,115,199 | <p>Let $A \in \text{End}(V)$ be an endomorphism, and $\mathbb Q[A]$ a subalgebra in $\text{End}(V)$ generated by $A$.</p>
<p>Is $\mathbb Q[A]$ always at most dim$V$-dimensional? How to prove it</p>
| martini | 15,379 | <p>Note that $\def\QA{\mathbf Q[A]}\QA$ is generated as a vector space by the powers
$$ \{ A^k: k \in \mathbf N \} $$
of $A$. Recall that by the Cayley-Hamilton theorem, we have
$$ \chi_A(A) = 0$$
where $\chi_A$ is the characteristic polynomial of $A$. Writing
$$ \chi_A(X) = \det(A-X) = (-1)^n X^n + \sum_{i=0}^{n-1}... |
2,069,507 | <p><a href="https://i.stack.imgur.com/B4b88.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/B4b88.png" alt="The image of parallelogram for help"></a></p>
<p>Let's say we have a parallelogram $\text{ABCD}$.</p>
<p>$\triangle \text{ADC}$ and $\triangle \text{BCD}$ are on the same base and between two... | N. S. | 9,176 | <p>This is a completion to the answer by TheLoneWolf.</p>
<p><strong>Lemma</strong> Let $ABC$ and $DEF$ be two triangles such that $AB=DE$ and $AC=DF$.
Then the two triangles have the same area if and only if $\angle A=\angle D$ or $\angle A+\angle D=180^\circ$.</p>
<p><strong>Proof:</strong></p>
<p>$$\mbox{Area}(AB... |
1,043,266 | <p>Carefully see this problem(I have solved them on my own, I'm only talking about the magical coincidence):</p>
<blockquote>
<p>A bag contains 6 notes of 100 Rs.,2 notes of 500 Rs., 3 notes of 1000 Rs..Mr. A draws two notes from the bag then Mr. B draws 2 notes from the bag.<br>
(i)Find the probability that A has... | Graham Kemp | 135,106 | <blockquote>
<p>Why doesn't it makes any difference? Think Intutively, if A has taken some money there must be less notes so there must be difference in probability, why doesn't order matter here?</p>
</blockquote>
<p>Think counterintuitively. A has taken two notes but we <em>don't know what they are</em> wh... |
3,481,348 | <p>Consider <span class="math-container">$H = \mathbb{Z}_{30}$</span> and <span class="math-container">$G = \mathbb{Z}_{15}$</span> as additive abelian groups. Then how do I show that <span class="math-container">${\rm Aut}(H) \cong {\rm Aut}(G)$</span>?</p>
<p>By the Chinese remainder theorem, I know that <span class... | Shaun | 104,041 | <p><strong>Hint:</strong> <span class="math-container">$${\rm Aut}(\Bbb Z_n)\cong U(n),$$</span></p>
<p>where <span class="math-container">$U(n)$</span> is the group of units modulo <span class="math-container">$n$</span>.</p>
|
3,481,348 | <p>Consider <span class="math-container">$H = \mathbb{Z}_{30}$</span> and <span class="math-container">$G = \mathbb{Z}_{15}$</span> as additive abelian groups. Then how do I show that <span class="math-container">${\rm Aut}(H) \cong {\rm Aut}(G)$</span>?</p>
<p>By the Chinese remainder theorem, I know that <span class... | Community | -1 | <p>For your first question, we have <span class="math-container">$\Bbb Z_{30}^×\cong(\Bbb Z_2\times\Bbb Z_{15})^×\cong\Bbb Z_2^×\times\Bbb Z_{15}^×\cong\Bbb Z_{15}^×$</span>.</p>
<p>And <span class="math-container">$\operatorname {Aut}(\Bbb Z_n)\cong\Bbb Z_n^×$</span>.</p>
|
2,437,026 | <p>Suppose that:</p>
<p>$$
X \sim Bern(p)
$$</p>
<p>Then, intuitively $X^2 = X \sim Bern(p)$. However, when I try to think of it logically, it doesn't make any sense. </p>
<p>As an example, $X$ is $1$ with probability $p$ and $0$ with probability $1-p$. Then, $X^2 = X\cdot X$ is $1$ only when both $X$'s are $1$, whi... | dezdichado | 152,744 | <p>Your logic is flawed. When $X$ is Bernoulli, the events $\{\omega: X^2(w) = 1\}$ and $\{w:X(w) = 1\}$ are precisely the same events. </p>
|
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