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 |
|---|---|---|---|---|
2,720,460 | <p>I have a question regarding a improper double integral which will diverge but I cannot seem to understand how to reach that conclusion. The integral is the following: $$\iint_{\mathbf{ℝ^2}}\frac{x}{1+x^2+y^2}dxdy$$ </p>
<p>I can see that $f(x,y) \geq 0 \space \forall x\geq0$ and $f(x,y)\leq 0 \space \forall x\leq 0... | J.G. | 56,861 | <p>Let's ask a related question: what's $\int_{-1}^1\frac{dx}{x}$? Naively, it's $[\ln x]^1_{-1}=0$. But you're adding a $+\infty$ and a $-\infty$, and we say the integral diverges. In the case at hand, the same thing happens. The $x\ge 0$ region can be written as $|\theta|\le \pi/2$, if you'll permit me to use the int... |
3,096,526 | <p>The book "basic category theory" states that in the category <strong>Mon</strong>, epimorphisms are not necessarily surjections, but doesn't explain why. Why is this the case?</p>
| Angina Seng | 436,618 | <p>Let <span class="math-container">$\Bbb Z$</span> be the additive group of integers, and <span class="math-container">$\Bbb N_0=\{0,1,2,\ldots\}$</span>.
Then the inclusion <span class="math-container">$\Bbb N_0\to\Bbb Z$</span> is an epimorphism in <strong>Mon</strong>.</p>
|
833,627 | <p>Find the general solution to the following recurrence: $$nC_n=anC_{n-1}+bC_{n-1}$$
where a and b are constants.</p>
| Claude Leibovici | 82,404 | <p>Continuing from Anurag's answer, we could go one step further and show that $$C_m=C_0\prod_{n=1}^{n=m}\frac{an+b}{n}=\frac{C_0}{m!}\prod_{n=1}^{n=m}(an+b)=\frac{C_0}{m!}\frac{a^m \Gamma \left(\frac{b}{a}+m+1\right)}{ \Gamma
\left(\frac{b}{a}+1\right)}$$ which would simplify a lot if $a$ is a divisor of $b$, whate... |
1,151,489 | <p>Is there a efficient test (formula or inequality) of whether a given point is in a bulb of period n? In other words, something other than running the iteration a lot of times to see if it converges to a period n cycle. I would like to apply such a method (if it exists) to writing some code.</p>
| PMay | 94,428 | <p>Thanks for the answer. So it appears that one way or the other one needs to do iterations, in this case Newton's method. I will have to study your blog to understand this approach. In the equations you give, c is the point one is testing, and Zo is one of the points in the period n cycle, correct? What about an ... |
4,229,521 | <blockquote>
<p>Let <span class="math-container">$A\in M_{n \times n}(\Bbb R)$</span> and suppose that for every <span class="math-container">$u, v \in \Bbb R^{n}$</span> <span class="math-container">$$(Av,Au) = (v,u)$$</span> where <span class="math-container">$(\cdot,\cdot)$</span> is the standard inner product on <s... | fwd | 897,162 | <p>I can complete your solution. Fix <span class="math-container">$u$</span>. From <span class="math-container">$v^TA^TAu = v^Tu$</span>, we have
<span class="math-container">$$
v^T(A^TA - I)u = 0,
$$</span>
or <span class="math-container">$(v, (A^TA - I)u) = 0$</span> for every vector <span class="math-container">$v$<... |
3,642,143 | <p>Let <span class="math-container">$\ T:\Bbb R^3\rightarrow \Bbb R^3$</span> be the linear tranformation defined by
<span class="math-container">$\ T(a,b,c)=(2a-b,a+b+c,-a+c)$</span>,</p>
<p>Find a basis for the Range (T).</p>
<p>I already solved the standard matrix <span class="math-container">$\ A=
$$ \left[
... | DonAntonio | 31,254 | <p><span class="math-container">$$T^{-1}\begin{pmatrix}x\\y\\z\end{pmatrix}=\begin{pmatrix}
1/4 & 1/4 & -1/4 \\
-1/2 & 1/2 & -1/2 \\
1/4 & 1/4 & 3/4 \\
\end{pmatrix}\begin{pmatrix}x\\y\\z\end{pmatrix}=\frac14\begin{pmatrix}x+y-z\\-2x+2y-2z\\x+y+3z\end{pmatrix}$$</span></p>
<p>Of... |
4,096,806 | <p>I'm given by symmetry matrix <span class="math-container">$A$</span> and <span class="math-container">$3$</span> eigenvalues of it: <span class="math-container">$1,2,3$</span>. And I know two of it's eigenvectors: <span class="math-container">$(3,2,1)$</span> and <span class="math-container">$(2,-1,-4)$</span>, whic... | Marc van Leeuwen | 18,880 | <p>In general you can find a last eigenvalue of a diagonalisable matrix<span class="math-container">$~A$</span> by subtracting the known eigenvalues from the trace of the matrix (which equals the sum of the eigenvalues taken with their multiplicities). Once you know the eigenvalue<span class="math-container">$~\lambda$... |
455,060 | <p>I always see questions on here that deal with this modular stuff, and I have no idea what any of it means, so I figured I would ask here.</p>
<p>So lets say we have
$$a \equiv b\pmod n$$
The example on wiki is $$38\equiv 14\mod 12$$
This is because 38-14 = 24, which has a factor of 12. Why is it 12 instead of 24,... | hasnohat | 46,166 | <p>I like to think of modular arithmetic as the arithmetic you obtain when you set a particular number equal to zero. Suppose $4=0$. Then $9 = 2(4)+1 = 2(0)+1=1$. So you say that 9 is congruent to 1 mod 4, or $9\equiv 1(\mod 4)$. If you're doing math on a clock, then you would use $12=0$.</p>
|
1,170,937 | <p>I was given a task to prove
$$(\vec{A}\times \nabla)\times \vec{B} = (\vec A \cdot \nabla)\vec B + \vec A \times \operatorname{rot} \vec B - \vec A \operatorname{div} B$$
using tensorial notation i.e. Kroneker delta and Levi-Civita symbol. Here are my wrong calculations:
$$\epsilon_{ijk}A_j(\nabla \times B)_k = \ep... | James S. Cook | 36,530 | <p>Ok, so the question is what is meant by $\vec{A} \times \nabla$. Notice:
$$ \vec{A} \times \nabla = \hat{x_1}(A_2 \partial_3-\partial_2 A_3)+\hat{x_2}(A_3 \partial_1-\partial_3 A_1)+\hat{x_3}(A_1 \partial_2-\partial_2 A_1)$$
when you feed this operator $\vec{B}$ under the cross product you have to think about produc... |
1,100,857 | <p>If I have a set $S$ with operation $*$ as a monoid. Would I say I have a monoid $S$ with the binary operation $*$ or would I say I have a monoid $(S,*)$ where the binary operation $*$ does blahblahblah?</p>
<hr>
<p>My book referred to $S$ as a monoid, which is strange considering $S$ is a set, and <strong>needs</s... | hjhjhj57 | 150,361 | <p>I believe it's a matter of choice (and formality), both ways are right, given that the operation in the set is obvious from the context. Just note that if you already defined what a monoid is, in the second way you'd only need to say that $(S,*)$ is a monoid. </p>
<p>What you'd need to do sometimes is explicitly de... |
206,391 | <p>I expect the following code to count the number of function calls in <code>NIntegrate</code>.</p>
<pre><code>i = 0;
f[x_] := (i += 1; Tanh[x] Sin[Exp[x]] Exp[-0.55 x^2 Exp[x^2]])
Print[NIntegrate[f[x], {x, -2, 1}]]
Print[i]
</code></pre>
<p>However, the output is</p>
<pre><code>0.0901049
2
</code></pre>
<p>whic... | mikado | 36,788 | <p>A very simple modification (adding <code>?NumericQ</code>) achieves the result you expected.</p>
<pre><code>i = 0;
f[x_?NumericQ] := (i += 1; Tanh[x] Sin[Exp[x]] Exp[-0.55 x^2 Exp[x^2]])
Print[NIntegrate[f[x], {x, -2, 1}]]
Print[i]
(* 0.0901049*)
(*122*)
</code></pre>
<p>The issue is that <code>NIntegrate</code>... |
672,165 | <p>1) If there are $5$ vectors found in $\mathbb{R}^7$ will these vectors Span $\mathbb{R}^7$? Please explain.</p>
<p>2) Give an example of a $3$ by $5$ matrix for which all systems, $Ax=b$ for any $b$ in $\mathbb{R}^3$ is found to be consistent.</p>
<p>3) Given the last question, is it possible to find a $5$ by $3$ ... | Wei Zhou | 106,010 | <p>Let $x_1, \cdots, x_r$ be the representives for these conjugacy classes of finite orders. Since $G$ is residually finite, there exists $N_i \lhd G$ of finite index in $G$ such that $x_i \not \in N_i$ for each $i$. Let $N=\cap N_i$, then $N$ is what you need.</p>
|
19,119 | <p>I need to estimate $\pi$ using the following integration:</p>
<p>$$\int_{0}^{1} \!\sqrt{1-x^2} \ dx$$</p>
<p>using monte carlo </p>
<p>Any help would be greatly appreciated, please note that I'm a student trying to learn this stuff so if you can please please be more indulging and try to explain in depth..</p>
| Shai Covo | 2,810 | <p>Let's also elaborate on Ross Millikan's answer, adapted to the case $f(x)=\sqrt{1-x^2}$, $0 \leq x \leq 1$. Suppose that $(X_1,Y_1),(X_2,Y_2),\ldots$ is a sequence of independent uniform vectors on $[0,1] \times [0,1]$, so that for each $i$, $X_i$ and $Y_i$ are independent uniform$[0,1]$ random variables. Define $Z_... |
286,742 | <p>I'm searching for a proof of Witt's result that a biquadratic extension $K(\sqrt{a},\sqrt{b})/K$ extends to a Galois extension $L/K$ with quaternion group $Q_8$ iff the quadratic forms $<a,b,\frac{1}{ab}>, <1,1,1>$ are equivalent iff $(a,b)(a,a)(b,b) = 0 \in Br(K)$.</p>
<p>I know there is a proof in his... | Mike Shulman | 49 | <p>"<a href="https://ncatlab.org/nlab/show/topos" rel="noreferrer">Topos</a>" sometimes means "elementary topos" and sometimes "Grothendieck topos".</p>
|
286,742 | <p>I'm searching for a proof of Witt's result that a biquadratic extension $K(\sqrt{a},\sqrt{b})/K$ extends to a Galois extension $L/K$ with quaternion group $Q_8$ iff the quadratic forms $<a,b,\frac{1}{ab}>, <1,1,1>$ are equivalent iff $(a,b)(a,a)(b,b) = 0 \in Br(K)$.</p>
<p>I know there is a proof in his... | Mike Shulman | 49 | <p>What is a "set"? ZFC has one answer; <a href="https://ncatlab.org/nlab/show/ETCS" rel="noreferrer">ETCS</a> has another; <a href="https://ncatlab.org/nlab/show/Bishop+set" rel="noreferrer">Bishop</a> had another; <a href="https://ncatlab.org/nlab/show/h-set" rel="noreferrer">HoTT</a> has yet another.</p>
|
286,742 | <p>I'm searching for a proof of Witt's result that a biquadratic extension $K(\sqrt{a},\sqrt{b})/K$ extends to a Galois extension $L/K$ with quaternion group $Q_8$ iff the quadratic forms $<a,b,\frac{1}{ab}>, <1,1,1>$ are equivalent iff $(a,b)(a,a)(b,b) = 0 \in Br(K)$.</p>
<p>I know there is a proof in his... | Zach Teitler | 88,133 | <p><em>Tensor:</em> for some people, a tensor is an element of a tensor product of vector spaces. For others it’s a section of a tensor product of tangent and cotangent bundles on a manifold. Members of the first group would call that latter notion a tensor field. </p>
<p>More interestingly:</p>
<p><em>Tensor rank:</... |
286,742 | <p>I'm searching for a proof of Witt's result that a biquadratic extension $K(\sqrt{a},\sqrt{b})/K$ extends to a Galois extension $L/K$ with quaternion group $Q_8$ iff the quadratic forms $<a,b,\frac{1}{ab}>, <1,1,1>$ are equivalent iff $(a,b)(a,a)(b,b) = 0 \in Br(K)$.</p>
<p>I know there is a proof in his... | Anton Fetisov | 10,605 | <p>Antisymmetrization and symmetrization of tensors. Should we divide it by $(n!)$ ? This affects the relations between tensor and (anti-)symmetric algebra, the theory over $\mathbb Q$ and $\mathbb Z$ and is generally a mess. A special case: is a quadratic form over $\mathbb Z$ represented by a polynomial with integer ... |
286,742 | <p>I'm searching for a proof of Witt's result that a biquadratic extension $K(\sqrt{a},\sqrt{b})/K$ extends to a Galois extension $L/K$ with quaternion group $Q_8$ iff the quadratic forms $<a,b,\frac{1}{ab}>, <1,1,1>$ are equivalent iff $(a,b)(a,a)(b,b) = 0 \in Br(K)$.</p>
<p>I know there is a proof in his... | Philippe Gaucher | 24,563 | <ol>
<li>The <strong>category of models</strong> of a (finitary or infinitary) first-order theory</li>
<li>A <strong>model category</strong> which is an abstract setting for doing homotopy theory</li>
</ol>
<p>The first notion is expounded for example in the Adamek and Rosicky book "Locally Presentable and Accessible ... |
286,742 | <p>I'm searching for a proof of Witt's result that a biquadratic extension $K(\sqrt{a},\sqrt{b})/K$ extends to a Galois extension $L/K$ with quaternion group $Q_8$ iff the quadratic forms $<a,b,\frac{1}{ab}>, <1,1,1>$ are equivalent iff $(a,b)(a,a)(b,b) = 0 \in Br(K)$.</p>
<p>I know there is a proof in his... | Pablo Zadunaisky | 17,353 | <p>I believe the number of proper definitions of the term "quantum group" is some element of $\mathbb R \setminus \{1\}$. Even the simplest example, the quantized enveloping algebra of $\mathfrak{sl}_2(\mathbb C)$, has at least two definitions (more if you count the coproduct as part of the definition). </p>
<p>If you... |
286,742 | <p>I'm searching for a proof of Witt's result that a biquadratic extension $K(\sqrt{a},\sqrt{b})/K$ extends to a Galois extension $L/K$ with quaternion group $Q_8$ iff the quadratic forms $<a,b,\frac{1}{ab}>, <1,1,1>$ are equivalent iff $(a,b)(a,a)(b,b) = 0 \in Br(K)$.</p>
<p>I know there is a proof in his... | Fedor Petrov | 4,312 | <p>Are Hermite polynomials $H_n(x)$ orthogonal w.r.t. the weight $e^{-x^2}$ or $e^{-x^2/2}$?</p>
|
286,742 | <p>I'm searching for a proof of Witt's result that a biquadratic extension $K(\sqrt{a},\sqrt{b})/K$ extends to a Galois extension $L/K$ with quaternion group $Q_8$ iff the quadratic forms $<a,b,\frac{1}{ab}>, <1,1,1>$ are equivalent iff $(a,b)(a,a)(b,b) = 0 \in Br(K)$.</p>
<p>I know there is a proof in his... | Etienne Vouga | 36,939 | <p>Differential geometry seems to have many cases of conflicting conventions:</p>
<p>1) Is the Laplacian a non-negative operator? Or a non-positive one?</p>
<p>2) Is the mean curvature the... mean... of the principal curvatures? Or their sum?</p>
<p>3) Is the directional derivative linear in the magnitude of the dir... |
286,742 | <p>I'm searching for a proof of Witt's result that a biquadratic extension $K(\sqrt{a},\sqrt{b})/K$ extends to a Galois extension $L/K$ with quaternion group $Q_8$ iff the quadratic forms $<a,b,\frac{1}{ab}>, <1,1,1>$ are equivalent iff $(a,b)(a,a)(b,b) = 0 \in Br(K)$.</p>
<p>I know there is a proof in his... | Chain Markov | 110,691 | <p>One of the most classical examples is the word «algebra», which denotes not only a branch of mathematics, but also the following mathematical objects:</p>
<p>In linear algebra an algebra is a vector space, equipped with a bilinear operator (called product). </p>
<p>In set theory an algebra is a collection of sets ... |
286,742 | <p>I'm searching for a proof of Witt's result that a biquadratic extension $K(\sqrt{a},\sqrt{b})/K$ extends to a Galois extension $L/K$ with quaternion group $Q_8$ iff the quadratic forms $<a,b,\frac{1}{ab}>, <1,1,1>$ are equivalent iff $(a,b)(a,a)(b,b) = 0 \in Br(K)$.</p>
<p>I know there is a proof in his... | Port | 88,538 | <p>In set theory a forcing notion can be a pre order with a largest element or lowest element, depending on the style of the author. </p>
|
64,414 | <p>Show that every prime $p>3$ is either of the form $6n+1$ or of the form $6n+5$, where $n=0,1,2, \dots$</p>
| Brian M. Scott | 12,042 | <p>This is elementary algebra. For what value(s) of $n$ is $6n$ prime? $6n+2$? $6n+3$? $6n+4$? Are there any other possibilities besides these and the two that you already mentioned?</p>
|
795,138 | <p>Let $X_1$ and $X_2$ are independent $N(0, \sigma^2)$ which means (mean = 0, variance = $\sigma^2$) random variables. What is the distribution of $X_1^2 + X_2^2$?</p>
<p>My approach is that
$X_1\sim N(0, \sigma^2)$ and $X_2\sim N(0, \sigma^2)$. </p>
<p>Transforming $X_1$ and $X_2$ into standard normal,
$X_1/\sigma... | gt6989b | 16,192 | <p>Not really - recall that the mgf is
$$
m_X(t) = \mathbb{E}\left[e^{tX}\right]
$$
and if you rescale $X$ by a constant $\sigma$, what happens to the result?</p>
|
795,138 | <p>Let $X_1$ and $X_2$ are independent $N(0, \sigma^2)$ which means (mean = 0, variance = $\sigma^2$) random variables. What is the distribution of $X_1^2 + X_2^2$?</p>
<p>My approach is that
$X_1\sim N(0, \sigma^2)$ and $X_2\sim N(0, \sigma^2)$. </p>
<p>Transforming $X_1$ and $X_2$ into standard normal,
$X_1/\sigma... | Bombyx mori | 32,240 | <p>You have
$$
X_{i}^{2}=\sigma^{2}(Z^{2})=\sigma^{2}\Gamma(\frac{1}{2},2)=\Gamma(\frac{1}{2},2\sigma^{2})
$$</p>
<p>Therefore we have
$$
X_{1}^{2}+X_{2}^{2}=\Gamma(1,2\sigma^{2})
$$</p>
<p>where we used property of $\Gamma$-distribution. </p>
|
794,303 | <p><em>Disclaimer: The story given below is purely fictional and does not, in any way, relate to a prison break.</em> :P </p>
<blockquote>
<p>Four roads lead away from a jail. A prisoner is trying to escape from
the jail and selects one road at random. If road A is selected, the
probability of escaping is (1/8)... | nadia-liza | 113,971 | <p>$$\sum_{n=0}^{\infty} \frac{5^{2n-1}}{10^{2n -1}}=\sum_{n=0}^{\infty} (\frac{5}{10})^{2n-1}=2\sum_{n=0}^{\infty} (\frac{1}{2})^{2n}= 2 \sum_{n=0}^{\infty} (\frac14)^n=2 \frac{1}{1-\frac14}$$</p>
|
729,387 | <p>I hope this is the right section for this kind of questions.</p>
<p>I am trying to simulate, with MATLAB, a diffusion model starting from a Random Walk.
I am using a Random Walk with information increment X normally distributed ($\mu, \sigma$ ).
I also have a boundary $\alpha $, and $\alpha > \mu$. The startin... | horchler | 80,812 | <p>Your simulation code looks okay. Some suggestions. You're never going to get reliable estimates of your parameters by taking such large time steps when your boundary is so close. Also, only 500 runs is is pretty small, you might increase it to 3,000 or more. You might find this paper helpful in terms of learning abo... |
513,626 | <p>I know that for any $n$, $n^2 - 2$ is never a multiple of $3$. I feel like this is a rather simple proof, but I cannot figure out how to manipulate the definition of a multiple of $3$: $n$ is a multiple of $3$ if it can be written $n = 3k$ for some integer $k$.</p>
| OR. | 26,489 | <p>$$(n^2-2)^2=\left[(n-1)n^2(n+1)+3-3n^2\right]+1$$</p>
|
513,626 | <p>I know that for any $n$, $n^2 - 2$ is never a multiple of $3$. I feel like this is a rather simple proof, but I cannot figure out how to manipulate the definition of a multiple of $3$: $n$ is a multiple of $3$ if it can be written $n = 3k$ for some integer $k$.</p>
| Community | -1 | <p>Suppose $n$ is multiple of $3$ then $n=3k$ and $n^2-2=9k^2-2$ if this is divisible by $3$, then as $9k^2$ is divisible by $3$ this should imply $-2$ is divisible by $3$.(???)</p>
<p>Suppose $n$ is not a multiple of $3$ and $n=3k+1$ and $n^2-2=9k^2+6k-1$ if this is divisible by $3$, for similar reasons this would i... |
328,923 | <p>If $(A,\tau{_1})$ is a compact Hausdorff space and $\tau{_2}$ is a strictly finer topology on $X$, can $(A, \tau_{2})$ be compact?</p>
| Brian M. Scott | 12,042 | <p>Consider the identity map $e$ from $\langle A,\tau_2\rangle$ to $\langle A,\tau_1\rangle$. Since $\tau_2\supseteq\tau_1$, $e$ is continuous. Since $\tau_2\supsetneqq\tau_1$, there is a set $K\subseteq A$ such that $K$ is closed in $\tau_2$ but not in $\tau_1$. If $\langle A,\tau_2$ were compact, $K$ would be compact... |
250,940 | <p>i found this interesting question on the web but i am not quite sure if my solution is accurate. Honestly i would appreciate few opinions.</p>
<p>Given Question:</p>
<blockquote>
<p>At a subway station, eastbound trains and northbound trains arrive
independently, both according to a Poisson process. On average... | Cameron Buie | 28,900 | <p>Since $e^x>0$ for all real $x$, then $y=e^x$ does not intersect $y=kx$ at all if $k=0$.</p>
<p>If $k<0$, then observe that $e^x-kx$ is strictly increasing, with $$\lim_{x\to\infty}(e^x-kx)=\infty$$ and $$\lim_{x\to-\infty}(e^x-kx)=-\infty.$$ From the strict monotonicity, there is at most one solution to $e^x=... |
301,778 | <p>I am not a mathematician (I am an engineer who is working on improving his mathematics), so I apologize in advance if my question is trivial.</p>
<p>Consider a graph of $N$ nodes, with some defined criterion as to whether two nodes are connected or not. What are some measures of the graph's connectedness? I can thi... | Community | -1 | <p>The "score" you speak of in (4) is typically called the <em>degree</em> or <em>valence</em> of a node. The average degree is simply $2E/N$, where $E$ is the number of edges in the graph, because adding up the degrees of all the nodes amounts to counting all the edges twice. This is certainly a nice measure: at the s... |
482,475 | <p>What is the following limit?
$$
\lim_{x\rightarrow0}\frac{1-\sin\left(\frac{\pi}{2}-x\right)}{x}
$$</p>
| Cameron Buie | 28,900 | <p><strong>Hint</strong>: $$\sin\left(\frac\pi2-x\right)=\cos x.$$</p>
|
482,475 | <p>What is the following limit?
$$
\lim_{x\rightarrow0}\frac{1-\sin\left(\frac{\pi}{2}-x\right)}{x}
$$</p>
| njguliyev | 90,209 | <p>Hint: $1-\sin\left(\frac{\pi}{2} - x\right) = 1-\cos x=2\sin^2 \frac{x}{2}$.</p>
|
270,703 | <p><strong>Question</strong>: Is it known if the mapping class groups (of surfaces of finite type) are similar to Gromov-hyperbolic groups in the following senses: </p>
<p>1) Does every finite generating set give us a finite presentation?</p>
<p>2) Are there finitely many cone types with respect to any (some) finite ... | user111 | 89,429 | <p>Concerning the other coefficients $c_k$, Branan [1] showed that $(k+1)c_{k+1}=(n-k)\bar c_{n-k}$ is a necessary condition for univalence in $|z|<1$
if $c_1=1$ and $c_n=1/n$.</p>
<p>For univalent polynomials <strong>with real coefficients and such that $c_{n}=1/n$</strong>, Suffridge [2] has proved that the coeff... |
308,750 | <p>In the Art Gallery Problem, we have given
a polygon $P$ on $n$ vertices and a number $k$ and we
want to know if there exists $k$ guards
such that every point inside the polygon
is seen by at least one of the guards.
We say a point $p$ sees a point $q$
if the entire segment $pq$ is contained inside $P$.
Let us ... | Till | 104,681 | <p>I have found an algorithm in the literature [1] running in
$O(kn\log n \log k)$ time.</p>
<p>I briefly repeat the argument.</p>
<p>Compute the visibility region of each guard in $O(n)$ time.
(total = $O(kn)$)
Split the guarding set $G$ into two guarding
set $G_1 \cup G_2$ of roughly
equal size. Recursively comput... |
4,092,291 | <p>Let <span class="math-container">$R$</span> be an integral domain and <span class="math-container">$R((x))$</span> be the ring of formal Laurent series over <span class="math-container">$R$</span>. (The answer to <a href="https://math.stackexchange.com/questions/605211/ring-of-formal-laurent-series-units-and-definin... | Hagen Knaf | 2,479 | <p>If I am not mistaken the following sequence of arguments eventually shows that if <span class="math-container">$R$</span> is a principal ideal domain, then <span class="math-container">$R((x))$</span> is a principal ideal domain too. I thought that from this fact through an appropriate localization argument one coul... |
509,349 | <p>The problem I am having is figuring out the way show the following sequence is monotone:</p>
<p>let $x_1 = \frac{3}{2}$ and $x_{n+1} = {x_n}^2-2x_n+2$, show that the sequence $x_n$ is monotone and bounded and find the limit.</p>
<p>I have found the first three terms, and found that the sequence is decreasing, I ha... | Abel | 71,157 | <p>If your second equation is supposed to be $x_{n+1} = x_n^2-2x_n+2$, it might be useful to note that this is equivalent to $x_{n+1}-1 = (x_n-1)^2$. Thus, you might want to study the sequence $y_n = x_n-1$.</p>
|
568,923 | <p>what is the name of the curve made up of the points $(x,x^2...x^n)$ in $\mathbb {R}^n$ for all $x\in \mathbb R$??</p>
<p>For example: in $\mathbb R^2$ it would just be a parabola.</p>
| copper.hat | 27,978 | <p>A function $f:S \to T$ is completely specified by the values $f(a),f(b),f(c),f(d)$.</p>
<p>Each of these values can be any of the 7 elements of $T$.</p>
<p>Hence there are $7 \cdot 7 \cdot 7 \cdot 7 = 7^4 = 2401$ possible functions.</p>
|
349,653 | <p>I'm trying to evaluate the following limit:</p>
<p>$\displaystyle\lim_{x \to \infty} \displaystyle\frac{\ln(2x)}{\ln(x)}$</p>
<p>Using L'Hospital's rule, I end up with:</p>
<p>$\displaystyle\lim_{x \to \infty} \displaystyle\frac{\frac{1}{x}}{\frac{2}{x}}$</p>
<p>= $\displaystyle\lim_{x \to \infty} x/2x$</p>
<p>... | Brian M. Scott | 12,042 | <p>HINT: No differentiation is needed:</p>
<p>$$\frac{\ln 2x}{\ln x}=\frac{\ln 2+\ln x}{\ln x}=1+\frac{\ln 2}{\ln x}$$</p>
<p>You <strong>can</strong> use l’Hospital’s rule here, but you have to differentiate $\ln 2x$ correctly.</p>
|
349,653 | <p>I'm trying to evaluate the following limit:</p>
<p>$\displaystyle\lim_{x \to \infty} \displaystyle\frac{\ln(2x)}{\ln(x)}$</p>
<p>Using L'Hospital's rule, I end up with:</p>
<p>$\displaystyle\lim_{x \to \infty} \displaystyle\frac{\frac{1}{x}}{\frac{2}{x}}$</p>
<p>= $\displaystyle\lim_{x \to \infty} x/2x$</p>
<p>... | Calvin Lin | 54,563 | <p>No. You forgot to apply the chain rule when differentiating $\ln 2x$.</p>
<p>Also, I'm not sure where $n$ came from.</p>
|
75,781 | <p>This is my first question on the forum. I'm wondering if the following proof is valid.</p>
<hr>
<p><strong>Proof:</strong>
Let $\{A_\lambda\}_{\lambda \in L}$ be an arbitrary collection of disjoint non-empty open subsets of $\mathbb{R}$. Since every non-empty open subset of $\mathbb{R}$ can be written uniquely as ... | Weltschmerz | 1,313 | <p>To each of those open disjoint subsets you can associate one and only one rational number (just pick a rational number in the set). Thus you obtain an injection from your family of subsets <strong>into</strong> the set of rational numbers, which is countable. The conclusion follows that your family must indeed be co... |
1,555,741 | <p>I started integrating this but do not know how to finish it, here is what I got up to:
$$\int_{-\pi}^\pi \lvert x\rvert \cos(nx) dx$$
for $n=1, 2, 3, ...$<br>
the first thing I did was break it up into two integrals:<br>
$$\int_{-\pi}^0 -x \cos(nx) dx + \int_0^\pi x \cos(nx) dx$$<br>
I began to solve the first inte... | Graham Kemp | 135,106 | <p>You need to solve: $$f_{X+Y}(z) = \int_{x^2+(z-x)^2\leq 1} f_{X,Y}(x, z-x)\operatorname d x$$</p>
<p>To find the integral bounds, solve $z^2-2xz+2x^2\leq 1$ for min/max $x$ where $z\in[-\sqrt 2;\sqrt 2]$</p>
|
1,415,556 | <p>I'm solving some exercises from my class notes on Commutative Algebra,On the following exercise I got stuck:</p>
<p>Describe $Spec( \mathbb C[x,y]/x(x-a))$ where $a$ is some complex number.</p>
<p>As far as I know the maximal ideals of $\mathbb C[x,y]$ are of the form $(x-a,y-b)$ where $a,b \in \mathbb C$.But I do... | Sabino Di Trani | 82,009 | <p>Let's observe that if $a \neq 0$ the ideals $(x)$ and $(x-a)$ are coprime, by CRT you obtain:</p>
<p>$$\frac{\mathbb{C}[x,y]}{x(x-a)} \simeq \frac{\mathbb{C}[x,y]}{(x)} \oplus \frac{\mathbb{C}[x,y]}{(x-a)} \simeq \mathbb{C}[y]\oplus \mathbb{C}[y]$$</p>
<p>Now is easy to look at the prime ideals of $\mathbb{C}[x]\... |
1,944,929 | <p>I have done some digging and I cannot find any posts addressing limits with exponentials and <em>without</em> L'Hôpital's rule.</p>
<p>I have one of these questions for my assignment, but for ethical reasons I have made up a similar function: </p>
<blockquote>
<p>Find the following limit without L'Hôpital's rule... | hmakholm left over Monica | 14,366 | <p>$$ \lim_{x\to 0}\frac{2^x-7^x}{2x} = \frac12\lim_{x\to 0}\frac{2^x-7^x-0}{x-0}$$
and the second limit is <em>by definition</em> the derivative of $x\mapsto 2^x-7^x$ at $x=0$. Differentiate this function symbolically and you're done.</p>
|
2,000,342 | <p>If some function like $f$ depends on just one variable like $x$, we denote its derivative with respect to the variable by:
$$\frac{\mathrm{d} }{\mathrm{d} x}f(x)$$
Now if the function happens to depend on $n$ variables we denote its derivative with respect to the $i$th variable by:
$$\frac{\partial }{\partial x_i}f(... | Mauro ALLEGRANZA | 108,274 | <p>Mainly historical; see <a href="http://jeff560.tripod.com/calculus.html" rel="nofollow noreferrer">Earliest Uses of Symbols of Calculus : Partial Derivative</a> :</p>
<blockquote>
<p>The "curly <span class="math-container">$\mathrm{d}$</span>" was used in 1770 by <a href="https://en.wikipedia.org/wiki/Marq... |
53,641 | <p>This should be true in a more general setting, but for simplicity consider billiards that are connected, compact subsets of the plane with boundary $C^2$ except at finitely many points. A ball (or a ray of light) rolls inside, going in straight lines, and upon collision with the wall, the orbit is reflected.</p>
<p... | Bill Thurston | 9,062 | <p>Do you mean to ask whether the trajectories in almost all cases (in {shapes X trajectories}
are dense in the set of {positions, directions} on the table, or just in the set positions?
The first question seems more natural to me; the answer is <em>no</em>:
If there are
two convex portions of the boundary curve point... |
2,416,485 | <p>Prove that: $(A-B)\cap B=\phi$.</p>
<p>My Attempt:</p>
<p>Let $x$ be an element of $(A-B)\cap B$. Then,
$$x\in (A-B)\cap B \iff x\in (A-B) \textrm {and} x\in B$$
$$\iff (x\in A \textrm {and} x\notin B) \textrm {and} x\in B$$.</p>
| Tsemo Aristide | 280,301 | <p>Hint: consider <span class="math-container">$g:\mathbb{C}\rightarrow \mathbb{C}^*$</span> defined by <span class="math-container">$g(z)=\exp(2\pi i z)$</span>, its kernel is <span class="math-container">$\mathbb{Z}$</span> and it is surjective.</p>
|
125,324 | <p>When I enter <code>f + 0</code> where <code>f</code> is an undefined symbol, I get <code>f</code>. So far so good. When I enter <code>f + 0.</code>, I get <code>f + 0.</code>. How can I get Mathematica to simplify 0. just like it does for 0? I tried using <code>Simplify</code>, I got the same result. </p>
| Szabolcs | 12 | <p>Another reason why <code>0.</code> is not removed is that it is an <a href="https://reference.wolfram.com/language/tutorial/ExactAndApproximateResults.html">inexact</a> zero. In other words, it represents something that is zero to a certain precision, but unknown beyond that.</p>
<p>The usual way to convert very s... |
4,621,711 | <p>Find the first number such that the average of the sum of the squares from <span class="math-container">$1$</span> to <span class="math-container">$n$</span> (where <span class="math-container">$n > 1$</span>) equals <span class="math-container">$k^2$</span>.</p>
<p>Here is what I have done so far:</p>
<p>The sen... | Desco | 1,141,062 | <p>Even tough the other answer provided is far more elegant, I would like to show another possible approach:</p>
<p>First of all, we need <span class="math-container">$\frac{(n+1)(2n+1)}{6}$</span> to be an integer, and checking the possibilities modulo <span class="math-container">$2$</span> and <span class="math-cont... |
1,727,144 | <blockquote>
<p>Suppose you roll a fair dice $12$ times in a row. What is the probability of the event "exactly $k$ of the rolls are a $5$ or a $6$" ?</p>
</blockquote>
<p>I'm just asking for some verification of my counting. Let $X$ be the random variable that counts the number of $5$ and $6$ rolled.</p>
<p>$$\beg... | barak manos | 131,263 | <p>The probability of "5" or "6" in a single roll is $\frac26$.</p>
<p>The result of each roll is independent of all the other rolls.</p>
<p>Hence the probability of exactly $k$ out of $12$ rolls giving "5" or "6" is:</p>
<p>$$\binom{12}{k}\cdot\left(\frac26\right)^{k}\cdot\left(1-\frac26\right)^{12-k}$$</p>
|
109,003 | <p>Let $m,n$ be integers. I want to find the possible values of $m,n$ such that $4(m+n)\over (2m+n)^2+3n^2$ and $4n\over (2m+n)^2+3n^2$ are both integers too. Would someone please help? Of course letting $(2m+n)^2+3n^2=4$ gives some good values, but is this all the $m,n$ I can get? </p>
<p>Added: I can see that the pr... | Arturo Magidin | 742 | <p>Since $(2m+n)^2 + 3n^2 = 4m^2 + 4mn + n^2 + 3n^2 = 4m^2 + 4mn + 4n^2$, then
$$\frac{4(m+n)}{(2m+n)^2+3n^2} = \frac{4(m+n)}{4m^2+4mn+4n^2} = \frac{m+n}{m^2+mn+n^2}.$$
Similarly,
$$\frac{4n}{(2m+n)^2+3n^2} = \frac{4n}{4m^2+4mn+4n^2} = \frac{n}{m^2+mn+n^2}.$$
For both to be integers, you need $m^2+mn+n^2$ to divide bot... |
923,373 | <p>Consider the matrix $$A=\begin{pmatrix}-1 & 3& 3& 3\\ 3& 1& -1& 5\\ 3& -1& 7& -1\\ 3&5& -1&1\end{pmatrix}.$$</p>
<p>How do I show that $1$ is not an eigenvalue for $A$, by showing that there are no eigenvectors for $\lambda = 1$?</p>
<p>This what I've done so ... | Jack D'Aurizio | 44,121 | <p>Your series is just the Fourier sine series of the (sawtooth wave) periodic continuation of the function $f(\theta)=-\frac{\theta}{2}$ defined over $[-\pi,\pi]$, hence:
$$\sum_{n=1}^{+\infty}\frac{(-1)^n \sin(n\theta)}{n}=\left\{\begin{array}{rcl}-\frac{1}{2}(\theta\operatorname{mod} 2\pi)&\text{if}&\theta\n... |
1,448,128 | <p>Prove $\lim\limits_{n\to\infty}\sqrt{2+\sqrt{2+...+\sqrt{2}}}=2$</p>
<p>How to evaluate this limit?</p>
| Bernard | 202,857 | <p>It is the limit of the sequence defined by $u_0=0$ and $\;u_{n+1}=\sqrt{2+u_n}$. As the function $f(x)=\sqrt{2+x}$, is continuous the limit $\ell$ is a nonnegative fixed point of the function, i.e. is satisfies the equation:
$$\ell=\sqrt{\ell+2}\iff \ell^2-\ell-2=0\enspace\text{and}\enspace \ell\ge 0 $$
Now this eq... |
1,588,351 | <blockquote>
<p>Let $\mu$ be a finite signed measure on the Borel sets of $\mathbb{R}$, and suppose that $\mu \ll m$ where $m$ is Lebesgue measure. Prove that the function $t\mapsto \mu\{t+x:x\in A\}$ is continuous in $t$ for every Borel set $A$.</p>
</blockquote>
<p><strong>An Attempt:</strong>
Using the sequence ... | Brian M. Scott | 12,042 | <p>Yes, the greatest lower bound of $A$ is $0$. First, it’s clear that $0<\frac1n$ for each $n\in\Bbb Z^+$, so $0$ is <em>a</em> lower bound for $A$. Secondly, if $x>0$, then there is an $n\in\Bbb Z^+$ such that $\frac1n<x$, so $x$ is not a lower bound for $A$. Thus, $0$ must be the largest lower bound for $A$... |
1,588,351 | <blockquote>
<p>Let $\mu$ be a finite signed measure on the Borel sets of $\mathbb{R}$, and suppose that $\mu \ll m$ where $m$ is Lebesgue measure. Prove that the function $t\mapsto \mu\{t+x:x\in A\}$ is continuous in $t$ for every Borel set $A$.</p>
</blockquote>
<p><strong>An Attempt:</strong>
Using the sequence ... | Pedro | 70,305 | <p><em>I am studying the following set
$$ A=\left\{\frac{1}{n},\;n\in \mathbb N\right\}$$
where $\mathbb N$ begins from $1$ not $0$. I know that the upper bound of this set is $1$ but what is the lower bound?</em></p>
<p><strong><em>A</em> upper bound is $1$ and <em>a</em> lower bound is $0$, but there others.</strong... |
1,459,424 | <p>$$y'+y\cos x=3\cos x$$
When I find the integration factor it is $e^{\sin x}$, but as far as I know that has no solution when I try to complete this by integration by parts.</p>
| MrYouMath | 262,304 | <p>I would rather solve this ODE using the idea that the general solution of a linear ODE is given by the sum of the homogenous solution and the particular solution.</p>
<ol>
<li>Homogenous Solution:
$$y_h'+\cos(x)y_h=0$$</li>
</ol>
<p>Solving this yields $y_h=c_1e^{-\sin(x)}$.</p>
<ol start="2">
<li>Particular Sol... |
1,459,424 | <p>$$y'+y\cos x=3\cos x$$
When I find the integration factor it is $e^{\sin x}$, but as far as I know that has no solution when I try to complete this by integration by parts.</p>
| mathreadler | 213,607 | <p>I would solve this with Fourier Transforms. </p>
<ol>
<li>Differentiation is multiplication with a "ramp". </li>
<li>Multiplication in ordinary domain is convolution in Fourier domain.</li>
<li>$\cos(x)$ is a sum of dirac impulses in the Fourier domain.</li>
</ol>
<p>1,2 and 3 together will give you a linear equat... |
3,273,431 | <blockquote>
<p>Let <span class="math-container">$T:V \rightarrow W$</span> be a linear transformation. If <span class="math-container">$V$</span> is infinite-dimensional, prove that at least one of the range or null space of <span class="math-container">$T$</span> is infinite-dimensional.</p>
</blockquote>
<p>The q... | Theo Bendit | 248,286 | <p>Here's a slicker way to approach this than the hint suggests, though it's much the same logic underneath.</p>
<p>Suppose <span class="math-container">$U$</span> is a subspace of <span class="math-container">$V$</span>, and consider <span class="math-container">$S = T|_U : U \to W$</span>. Note that <span class="mat... |
1,891,398 | <blockquote>
<p>Let $a$ belong to a group and $|a|=m$. If $n$ is relatively prime to $m$ show that $a$ can be written as the $n^{th}$ power of some element in the group.</p>
</blockquote>
<p>We need to show that if $a\in G$ and $a^m=e\implies \exists \ b\in G $ such that $b^n=a$ i.e.$b^{nm}=e=({b^m})^n$ i.e. $\exis... | Noah Schweber | 28,111 | <p>HINT: Let $H$ be the subgroup of $G$ generated by $a$ (so $H=\{a^1, a^2, . . . , a^m=e\}$). Consider the map $f: H\rightarrow H$ given by $f(x)=x^n$. Can you show that $f$ is injective? Do you see why the injectivity of $f$ solves your problem?</p>
<p>SUBHINT: Suppose $f$ <em>weren't</em> injective. Then $a^{xn}=a^... |
4,005,776 | <p>I'd like to find all 3rd roots of this number z = i - 1. Now I've found formulas on how to do it; First we transform the complex number into this form</p>
<p><span class="math-container">$$ \sqrt[n]{r} * e^{i\frac {\phi + 2k\pi}{n}} $$</span> Where n should be 3 (because of 3rd root) and k should be k = n - 1 (Inclu... | G Tony Jacobs | 92,129 | <p>I find these much easier to think about visually. The number you start with is <span class="math-container">$i-1$</span>, or in the more standard notation <span class="math-container">$-1+i$</span>. That number is located in the second quadrant of the complex plane. Its modulus (i.e., its absolute value, or distance... |
96,840 | <p>I want to show the convergence of the following improper integral $\int_0^\infty e^{-x^2}dx$.
I try to use comparison test for integrals
$x≥0$, $-x ≥0$, $-x^2≥0$ then $e^{-x^2}≤1$. So am ending with the fact that $\int_0^\infty e^{-x^2}dx$ converges if $\int_0^\infty dx$ converges but I don’t appreciate this. Thank... | Calvin McPhail-Snyder | 10,104 | <p>I'm assuming you're asking about the convergence of $\int_0^\infty e^{-x^2}dx$. The easiest way that I can think of to prove this is to note that $e^{-x^2}$ is continuous and bounded, and hence integrable, on the interval $[0,1]$, and that on the remaining unbounded interval $[1,\infty)$ it is a function everywhere... |
96,840 | <p>I want to show the convergence of the following improper integral $\int_0^\infty e^{-x^2}dx$.
I try to use comparison test for integrals
$x≥0$, $-x ≥0$, $-x^2≥0$ then $e^{-x^2}≤1$. So am ending with the fact that $\int_0^\infty e^{-x^2}dx$ converges if $\int_0^\infty dx$ converges but I don’t appreciate this. Thank... | kwadr4tic | 22,153 | <p>It does, and you can also compute its value:</p>
<p>$\int_{[0, \infty) \times[0, \infty)} e^{-(x^2+y^2)} dx dy = \int_{[0, \infty)}\big(\int_{[0, \infty)} e^{-(x^2+y^2)} dy \big)dx = \int_{[0, \infty)} e ^ {-x^2}\big(\int_{[0, \infty)} e^{-y^2} dy \big)dx =$
$= \int_{[0, \infty)} e ^ {-x^2}dx \ \ \int_{[0, \infty)}... |
3,367,980 | <p>This question has bugging me since it came in previous test.</p>
<p>You are given the set of integers from <span class="math-container">$1$</span> to <span class="math-container">$N$</span>, i.e. <span class="math-container">$1, 2, 3, \ldots, N$</span>.</p>
<p>Find the total number of proper <em>ordered</em> subse... | robjohn | 13,854 | <p>We will only consider subsets that contain no consecutive items.</p>
<p>Let <span class="math-container">$a_{n,k}$</span> be the number of <span class="math-container">$k$</span>-subsets of <span class="math-container">$n$</span> items containing item <span class="math-container">$n$</span> (so does not contain ite... |
2,581,714 | <p>I have been trying to solve this <a href="https://math.stackexchange.com/q/2581210/144766">recent linear algebra problem</a>:</p>
<blockquote>
<p>Let $A, B$ be $3 \times 3$ matrices such that $(A-B)^2 = 0$. Prove that $\det (AB - BA) = 0$.</p>
</blockquote>
<p>This was my approach:$\DeclareMathOperator{\Tr}{Tr}$... | epi163sqrt | 132,007 | <blockquote>
<p>The notation $\sqrt[-100]{100}$ is correct, albeit not commonly used. In fact the whole equality chain
\begin{align*}
\color{blue}{\sqrt[-100]{100}}=100^{\frac{1}{-100}}=100^{\frac{-1}{100}}=\sqrt[100]{100^{-1}}=\sqrt[100]{\frac1{100}}\tag{1}
\end{align*}
is correct.</p>
</blockquote>
<p>Sometime... |
4,369,535 | <blockquote>
<p>Let <span class="math-container">$$f(x) = \sin^{-1}\left(\frac{2x}{1+x^2}\right) ~~ -\infty<x<\infty.$$</span></p>
<p>Show that,</p>
<p>(a) <span class="math-container">$f(x) = 2\tan^{-1}(x)$</span> for <span class="math-container">$-1\leq x \leq 1$</span> and</p>
<p>(b) <span class="math-contai... | Michael Hoppe | 93,935 | <p>Show that <span class="math-container">$f'(x)=\frac{2}{1+x^2}$</span>, hence <span class="math-container">$f(x)$</span> and <span class="math-container">$2\arctan(x)$</span> differ only by a constant. Then substitute <span class="math-container">$x=0$</span> to show that this constant is <span class="math-container"... |
1,613,426 | <blockquote>
<p>Prove that $$|\frac{a-b}{1-\bar ab}|=1$$ if $|a|=1$ or $|b|=1$</p>
</blockquote>
<p>I assumed $|a|=1$. Then tried to show that our statement holds.</p>
<p>I wrote $a=a_1+ia_2$ and $b=b_1+ib_2$ and $\bar a=a_1-ia_2$</p>
<p>Also $$|a|=|\bar a|=a_1^2+a_2^2=1$$</p>
<p>However, after multiplying it all... | Unit | 196,668 | <p>Some hints: $a\bar{a} = 1$ and $a-b = a(1-b/a)$. And for $|b| = 1$, note that $1 = \bar{1}$.</p>
|
4,452,636 | <p>Given two points, A and B;
Given two circles, having 2 points in common, I1 and I2:</p>
<ul>
<li>one circle at center C1, with radius r1, with the point A on to it</li>
<li>and another circle at center C2, with radius r2, with the point B on it.</li>
</ul>
<p><a href="https://i.stack.imgur.com/0rWdF.jpg" rel="nofoll... | robjohn | 13,854 | <p>This is a <a href="https://en.wikipedia.org/wiki/Telescoping_series" rel="nofollow noreferrer">telescoping series</a>. The cancellation of terms you noticed gives
<span class="math-container">$$
\sum_{k=0}^n\left(\tan^{-1}(k+1)-\tan^{-1}(k)\right)=\tan^{-1}(n+1)-\tan^{-1}(0)
$$</span>
Then note that the limit at <sp... |
1,726,026 | <p>I need help to factorize $x^4-x^2+16$. I have tried to take $x^4$ as $(x^2)^2$ and factorize it in the typical way of factorizing a quadratic expression but that did not help. Can someone help me to factor this and also introduce me to the procedure that i need to follow to factorize expressions with degree higher t... | Soham | 242,402 | <p>$$x^4-x^2+16$$</p>
<p>$$=[(x^2)^2+2\cdot x^2\cdot 4+4^2]-9x^2$$</p>
<p>$$=(x^2+4)^2-(3x)^2$$</p>
<p>$$=(x^2+4-3x)(x^2+4+3x)$$</p>
<p>by using $a^2-b^2=(a+b)(a-b)$.</p>
|
1,726,026 | <p>I need help to factorize $x^4-x^2+16$. I have tried to take $x^4$ as $(x^2)^2$ and factorize it in the typical way of factorizing a quadratic expression but that did not help. Can someone help me to factor this and also introduce me to the procedure that i need to follow to factorize expressions with degree higher t... | Jimmy Kudo | 327,285 | <p><em>Can be done by making perfect squares</em>
$$
Let\ ax^2 +bx + c=0
\\Try\ to\ make\ the\ equation\ look\ in\ the\ form
\\ax^2 +2\sqrt {ca}x +c-(2\sqrt{ca}-b)x=0
\\You\ will\ see\ that\ ax^2 +2\sqrt {ca}x +c\ makes\ a \ perfect\ square
\\\therefore \quad ax^2 +2\sqrt {ca}x +c=(\sqrt ax + \sqrt c )^2
\\Thus\ the\ ... |
59,079 | <p>Let $(X,d)$ be a complete metric linear space whose balls are convex. Let $Y\subseteq X$ be a bounded, closed and convex subset that verifies the following property: for all $y_0\in Y$, the distance function $y\rightarrow d(y_0,y)$ attains its sup in $Y$.
Does $Y$ have extreme points?</p>
<p>I was trying to adapt t... | Sergei Ivanov | 4,354 | <p>There is a counter-example.</p>
<p>Note that in any normed space, its unit ball satisfies your supremum-attaining property. Indeed, for any $x_0\in X$ the supremum of $d(x_0,\cdot)$ on the ball is attained at the point $-x_0/\|x_0\|$ if $x_0\ne 0$, and at any point of the sphere if $x_0=0$.</p>
<p>It remains to co... |
1,189,397 | <p>I want to make a question about this exercise.</p>
<blockquote>
<p>Let $G$ be a group, and let $a$, $b$, be elements of $G$. We define the commutator of $a$ and $b$ as follows: </p>
<p>\begin{equation}
[a,b]:=aba^{-1}b^{-1}.
\end{equation}</p>
<p>Let $C:= \langle \{[a,b] \mid a,b\in G\} \rangle$ be th... | Matthew Towers | 5,316 | <p>There's something missing from your argument. $C$ is normal if for all $c \in C$ and all $g \in G$ we have $g^{-1}cg \in C$. But $C$ is the <em>subgroup generated by</em> all commutators: it is (in general) not just the set of commutators. Elements of the subgroup generated by a set $S$ can be written as products... |
11,743 | <p>As part of many hobbies (origami, sculpting, construction toys) I often find myself building polyhedra from regular polygons. I am intimately familiar with all of the Archimedean and Platonic solids, and can construct most of the other isohedra, deltahedra, and Johnson solids from memory. The smaller prisms, antipr... | David Eppstein | 440 | <p>There is a collection of models in <a href="http://www.eg-models.de/" rel="nofollow">Electronic Geometry Models</a>. But they are less models a hobbyist would be likely to want to build and more objects with properties of interest to mathematicians; many of them have more than three dimensions, for instance.</p>
|
3,052,512 | <blockquote>
<p><strong>Problem:</strong> Consider the optimization problems
<span class="math-container">$$\min_\beta \|y-X\beta\|^2+\alpha\|\beta\|^2 \tag 1$$</span>
and
<span class="math-container">$$\min_\beta \|\beta\|^2 \text{ subject to } \|y-X\beta\|^2 \le c \tag 2$$</span>
where <span class="math-con... | Cesareo | 397,348 | <p>Here from (1)</p>
<p><span class="math-container">$$
f(\beta) = y'\cdot y-2\beta'\cdot X'y+\beta'\cdot X'\cdot X\cdot\beta+\alpha \beta'\cdot\beta
$$</span></p>
<p>so the minimum condition gives</p>
<p><span class="math-container">$$
-X'\cdot y+X'\cdot X\beta+\alpha\beta = 0
$$</span></p>
<p>and then</p>
<p><sp... |
2,338,369 | <p>Assume $∃xy∈N$ st. $x^2-y^2 =1$</p>
<p>$(x-y)(x+y) = 1$ so;</p>
<p>$(x-y)∈N$ and $(x+y)∈N$</p>
<p>How do I proceed from the information I know</p>
| Vidyanshu Mishra | 363,566 | <p>$(y+1)^2-y^2=2y+1 $ since $x,y$ are natural the smallest value of $y$ is $1$ for which the sequence $2y+1$ have a value of $3$ and for other values it keeps on increasing.</p>
|
2,617,884 | <p>Let $\,f,g:\,[a,b]\rightarrow \mathbb R$ be differentiable, postive functions with $f(a)=g(a)$ and $\frac{f'(x)}{f(x)}\le \frac{g'(x)}{g(x)}\,\forall\,x\in[a,b]$ $$Prove,\,that:
\frac{f'(x)}{f(x)}\le \frac{g'(x)}{g(x)} \Rightarrow f(x)\le g(x) \,\forall\,x\in[a,b]$$ </p>
<p>I suppose it has something to do with Rol... | nonuser | 463,553 | <p>We have $$(\ln f(x))' \leq (\ln g(x))'$$ so $$ \Big(\ln g(x)/f(x)\Big)'\geq 0 = c'$$</p>
<p>so $$g(x) /f(x) \geq e^c$$ and thus $$g(x) \geq e^cf(x)$$ Since $f(a)=g(a)$ we get $c=0$ so $$g(x) \geq f(x)$$</p>
|
143,092 | <p>If $N$ is an $n\times n$ nilpotent matrix such that $N^k=0$ for some integer $k$. Is it true that $(DN)^k=0$ for any diagonal matrix $D$?</p>
| N. S. | 9,176 | <p>Let $a$ be your digit. Let $d = \gcd(a,n)$. </p>
<p>Then </p>
<p>$$n | aaa.a \Leftrightarrow n| a \cdot 111..1 \Leftrightarrow \frac{n}{d} | \frac{a}{d} \cdot 111....11$$</p>
<p>Since $\frac{n}{d}$ and $\frac{a}{d}$ are relatively prime, we get</p>
<p>$$n| aaa...aa \Leftrightarrow \frac{n}{d}|1111...1$$ </p>
<p... |
1,719,772 | <p>I need to prove that $(-x)(-y) = xy$ using only the field axioms. I tried starting with $ since$ $(-(-x)(-y)) + (-x)(-y) = 0$ by A6, or the additive inverse. And then adding $xy$ to both sides by A2 or transitivity. But I'm sure that to say $(-(-x)(-y))=-(xy)$ is outside the field axioms. So I'm stuck and w... | almagest | 172,006 | <p>It depends how picky you are, and how minimal your axioms are. The below is middle of the road.</p>
<ol>
<li><p>We have $(-x)y+xy=(-x+x)y$ (distributive law) $=0y$ (defn of additive inverse)</p></li>
<li><p>We want $0y=0$, but this takes a surprising amount of effort. We have $1y=y$ (defn of mult identity) so $0y+1... |
798,721 | <blockquote>
<p>Let $a,b,c,d$ positive real numbers with $d= \max(a,b,c,d)$. Proof
that </p>
<p>$$a(d-c)+b(d-a)+c(d-b)\leq d^2$$</p>
</blockquote>
<ul>
<li>I believe that the GM-AM inequality with $n=4$ variables might be helpful. </li>
</ul>
<p>$$\sqrt[n]{x_1 x_2 \dots x_n} \le \frac{x_1+ \dots + x_n}{n}$$... | Hagen von Eitzen | 39,174 | <p>Consider the polynomial $$f(x)=x^3-(a+b+c)x^2+(ab+bc+ac)x-abc$$
having $a,b,c$ as roots. We have
$$f(d)=d^3-(a+b+c)d^2+(ab+bc+ac)d-abc=d\cdot(RHS-LHS)-abc $$
and $f(d)=(d-a)(d-b)(d-c)\ge0$.</p>
<p>Or in short
$$ a(d-c)+b(d-a)+c(d-b)=(a+b+c)d-(ac+ab+bc)\\=\frac{d^3-(d-a)(d-b)(d-c)-abc}{d}\le d^2$$ </p>
|
4,124,652 | <p>A few quick questions from an older text on complex analysis:</p>
<p><span class="math-container">$|e^{it}|^2=e^{it}\cdot\overline{e^{it}}=e^{it}\cdot e^{-it}$</span></p>
<p>I'm not sure about the notation <span class="math-container">$|e^{it}|^2$</span>, but I'm assuming it indicates the magnitude of the vector squ... | José Carlos Santos | 446,262 | <p>If <span class="math-container">$z\in\Bbb C$</span>, and if <span class="math-container">$a,b\in\Bbb R$</span> are such that <span class="math-container">$z=a+bi$</span>, then<span class="math-container">\begin{align}z\cdot\overline z&=(a+bi)(a-bi)\\&=a^2+b^2\\&=\sqrt{a^2+b^2}^2\\&=|z|^2.\end{align}<... |
4,124,652 | <p>A few quick questions from an older text on complex analysis:</p>
<p><span class="math-container">$|e^{it}|^2=e^{it}\cdot\overline{e^{it}}=e^{it}\cdot e^{-it}$</span></p>
<p>I'm not sure about the notation <span class="math-container">$|e^{it}|^2$</span>, but I'm assuming it indicates the magnitude of the vector squ... | CyclotomicField | 464,974 | <p>Given <span class="math-container">$z=x+iy$</span> we have that <span class="math-container">$z \overline{z} = (x+yi)(x-yi)=x^2 + xyi - xyi -(yi)^2 = x^2 + y^2$</span> which is the magnitude squared.</p>
|
1,493,817 | <p>My date of birth as DDMMYYYY is a prime number. I have a bit of a thing for prime numbers so I thought that was pretty cool. I wondered whether that's really special.</p>
<p>I already figured that less than $50\%$ of the people have that because even birth years are never prime. So let's only consider people born i... | Eric Brooks | 247,802 | <p>Here is some very simple R code if anyone wants to run the various possible permutations of this query:</p>
<pre><code>library(lubridate)
library(gmp)
dates = seq(as.Date("1900-01-01"),as.Date("1999-12-31"),1)
prime1 = isprime(as.numeric(strftime(dates,"%d%m%Y")))
table(prime1)
</code></pre>
|
2,312,592 | <p>How to solve next integral:
$$\int\frac{x^3+4x^2-5}{x^2}dx$$
I am using power rule for top part wich produce this:
$$\int\frac{\frac{x^2}{2}+4x-5x}{x^2}dx$$
Does my calculation right? How can I continue from here? Please describe all steps and rules used for solving this integral.</p>
| Dashi | 437,620 | <p>What you already have is incorrect.
Try this:</p>
<p>$$
\int \frac{x^3+4x^2-5}{x^2}dx = \int \frac{x^3}{x^2} + \frac{4x^2}{x^2} -\frac{5}{x^2} dx
$$</p>
<p>$$
=\int (x) dx + \int 4 dx + \int \frac{5}{x^2} dx
$$</p>
<p>$$
=\frac{x^2}{2} + 4x + \frac{5}{x} + C
$$</p>
<p>Overview of Integration rules:</p>
<p>$... |
89,115 | <p>I have some high dimensional high rank tensors, let's say $$F_{ijkl}$$ and I need to find $$F^{abcd}=g^{ai}g^{bj}g^{ck}g^{dl}F_{ijkl}.$$
Here $g^{ij}$ is the contravariant metric.</p>
<p>Simple summation in Mathematica takes way to much time</p>
<pre><code>Do[Fup[[a,b,c,d]]=Sum[F[[j,j,k,l]]g[[a,i]]g[[b,j]]g[[c,k]... | QuantumDot | 2,048 | <p>My keyboard is broken. So here is fast answer (on Mathematica 9); more later...</p>
<p>Here is your input:</p>
<pre><code>dim = 3; g = RandomReal[{0, 1}, {dim, dim}];
F = RandomReal[{0, 1}, {dim, dim, dim, dim}];
</code></pre>
<p>Now multiply four <code>g</code>'s and the <code>F</code>. Use <code>TensorProduct... |
89,115 | <p>I have some high dimensional high rank tensors, let's say $$F_{ijkl}$$ and I need to find $$F^{abcd}=g^{ai}g^{bj}g^{ck}g^{dl}F_{ijkl}.$$
Here $g^{ij}$ is the contravariant metric.</p>
<p>Simple summation in Mathematica takes way to much time</p>
<pre><code>Do[Fup[[a,b,c,d]]=Sum[F[[j,j,k,l]]g[[a,i]]g[[b,j]]g[[c,k]... | Federico | 6,478 | <h1>Edit</h1>
<p>My solution is faster than the accepted one, even with the <code>Activate</code>/<code>Inactive</code> trick suggested in the comments.</p>
<h1>Original answer</h1>
<p>You can define your tensor contraction routine using the builtins <code>Dot</code> and <code>Transpose</code>. Here is an example:</p>
... |
2,329,355 | <p>In an Math Exam there are 80 more men than women. The result showed that the women's average is 20% higher than men's, and that the total average is 75%. what is the women's average?
So far I did ...
$$H=W+80$$ $$\overline{W}=1,2\overline{H}\Rightarrow \overline{H}=\frac{\overline{W}}{1,2}$$<br>
but $$\overline{H}=... | Tucker | 256,305 | <p>$W$, average score from the women.</p>
<p>$M$, average score of the men.</p>
<p>$x$, total number of female students.</p>
<p>$x+80$, total number of male students</p>
<p>$$
\frac{M(x+80)+W(x)}{x+(x+80)}=.75
$$</p>
<p>$$
W=.2+M
$$</p>
<p>$$
(W-.2)(x+80)+xW=.75(2x+80)
$$</p>
<p>$$
xW+80W-.2x-16+xW=1.5x+60
$$</p... |
2,414,620 | <p>Lets have two known vectors $x_1'$ and $x_2'$ in XY ($\mathbb{R}^2$) representing the orthogonal projection of two unknown vectors $x_1$ and $x_2$ in $\mathbb{R}^3$, with known lengths $|x_1|=l_1$, $|x_1|=l_2$, and $x_1\cdot x_2=0$. </p>
<p>Let be the orthogonal projection made through $\hat z$, thus projecting ort... | Joseph O'Rourke | 237 | <p>This is <em>not</em> an answer to your question, but rather just pointing
out (tangentially), as reflected in
<a href="https://mathoverflow.net/a/210588/6094">this post</a>,
that projecting a $90^\circ$ rectangle angle
can result in a much smaller planar angle:
<hr />
<img src="h... |
308,909 | <p>Can we show that if $\operatorname{abs}(x) \lt 1$, then $$\ln(1+x^2) \leq x^2\;,$$
using Taylor's Theorem?</p>
<p>I am thinking of expanding it about $x=0$ but I got something like
$$f(x) = -x^2 + \frac{x^4}{2} - \dots$$</p>
<p>Is my approach correct? Could you give me some hints/guides here?</p>
<p>Thanks.</p>
| hkBattousai | 21,550 | <p>$$
\begin{array}{rcl}
\ln(1 + x^2) & \le & x^2 \\
e^{\ln(1 + x^2)} & \le & e^{x^2} \\
1 + x^2 & \le & e^{x^2} \\
1 + x^2 & \le & 1 + x^2 + \frac{x^4}{2!} + \frac{x^6}{3!} + \dots \\ {}
0 & \le & \frac{x^4}{2!} + \frac{x^6}{3!} + \frac{x^8}{4!} + \dots \\ {}
\end{array}
$$<... |
857,177 | <p>This is for my benefit and curiosity and not homework.<br>
How do you calculate the number of numbers between $1$ and $100$? How do you calculate the number of even and odd numbers between $1$ and $100$?</p>
<p>How can you use the grouping method (e.g. sum from $1$ to $100$ = $(1+101)x50$ or $(2+99)50))$ to find th... | amWhy | 9,003 | <blockquote>
<p>What is the sum of numbers from 5 to n? </p>
</blockquote>
<p>$$\sum_{k = 1}^n k - \sum_{k=1}^4 k = \frac{n(n+1)}2 - \frac{4(4+1)}{2} = \frac{n(n+1)}{2} -10$$ </p>
|
857,177 | <p>This is for my benefit and curiosity and not homework.<br>
How do you calculate the number of numbers between $1$ and $100$? How do you calculate the number of even and odd numbers between $1$ and $100$?</p>
<p>How can you use the grouping method (e.g. sum from $1$ to $100$ = $(1+101)x50$ or $(2+99)50))$ to find th... | André Nicolas | 6,312 | <p>Alice and Bob started work on the same day. Alice's wage the first day was $5$ dollars, the next day (she is a good worker) it was $6$ dollars, the next day it was $7$ dollars, and so on. On the last day she worked, she earned $n$ dollars. </p>
<p>Note that this means her total income $A$ was given by
$$A=5+6+7+\c... |
857,177 | <p>This is for my benefit and curiosity and not homework.<br>
How do you calculate the number of numbers between $1$ and $100$? How do you calculate the number of even and odd numbers between $1$ and $100$?</p>
<p>How can you use the grouping method (e.g. sum from $1$ to $100$ = $(1+101)x50$ or $(2+99)50))$ to find th... | joeytwiddle | 55,415 | <blockquote>
<p>Explain the intuition behind the formula for sums from 1 to n:
(n(n+1)/2).</p>
<p>What is the sum of numbers from 5 to n?</p>
</blockquote>
<p>Between 5 and n (inclusive) there are C = (n - 5 + 1) numbers.</p>
<p>The average of those numbers is: M = (5 + n) / 2</p>
<p>We can safely multiple ... |
3,940,252 | <p>, where A and B are square Matrices, exp(A) is the matrix exponential</p>
<p>I can see where the <span class="math-container">$\lVert A-B \rVert$</span> comes from, but can't figure out how to get the <span class="math-container">$ e^{max\{\lVert A \rVert, \lVert B \rVert\}} $</span>. The definition for the matrix e... | Henry Lee | 541,220 | <p>if we try letting <span class="math-container">$x=iy\Rightarrow dx=idy$</span> and <span class="math-container">$(0,\infty)\rightarrow(0,\infty)$</span>. We then know that:
<span class="math-container">$$\tan(iy)=i\tanh(y)$$</span>
<span class="math-container">$$\frac{1}{(iy)^2+b^2}=\frac{1}{b^2-y^2}$$</span>
so our... |
3,940,252 | <p>, where A and B are square Matrices, exp(A) is the matrix exponential</p>
<p>I can see where the <span class="math-container">$\lVert A-B \rVert$</span> comes from, but can't figure out how to get the <span class="math-container">$ e^{max\{\lVert A \rVert, \lVert B \rVert\}} $</span>. The definition for the matrix e... | Diger | 427,553 | <p>The integrand has poles at <span class="math-container">$x=\pm ib$</span> and singularities on the real axis at <span class="math-container">$x=(2n+1)\frac{\pi}{2}$</span> with cuts where the tangent becomes negative. Nevertheless, the integral can be viewed as a complex contour in the upper half-plane with the segm... |
2,209,770 | <p>Find with proofs the infimum, supremum, maximum and minimum of the following sets or prove non-existence.</p>
<p>$$E = \{x \in \mathbb{R} - \{0\}: x < \frac{1}{x}\}$$ </p>
<p>I do not know even how to think, Could anyone help me please? </p>
| lebelinoz | 430,256 | <p>Solve that condition: $x < \frac{1}{x}$ equates to $\frac{x^2 - 1}{x} < 0$. This can only hold under two conditions:</p>
<ul>
<li>$x^2 - 1 < 0$ and $x > 0$, which is equivalent to $0 < x < 1$;</li>
<li>$x^2 - 1 > 0$ and $x < 0$, which is equivalent to $x < -1$.</li>
</ul>
<p>So $E = (-... |
415,928 | <p>Determine whether $X^4-16X^2+4$ is irreducible in $\mathbb{Q}[X]$.</p>
<p>To solve this problem, I reasoned that since $X^4-16X^2+4$ has no rational roots hence irreducible.</p>
<p>But there is a hint to this question that uses different approach:<br>
Try supposing it is reducible, then it must factor into a produ... | Geoff Robinson | 13,147 | <p>You have already answered some of this yourself. You say that the polynomial has no rational root. If (and only if) it had a rational root, it could be factored as a product of a degree 1 polynomial and a degree 3 polynomial, both with rational coefficients. So this possibility has been discounted. But there is stil... |
1,630,686 | <p>I'm trying to prove the non-existance of three positive integers $x,y,z$ with $x\geq z$ such that\begin{align}
(x-z)^2+y^2 &\text{ is a perfect square,}\\
x^2+y^2 &\text{ is a perfect square,}\\
(x+z)^2+y^2 &\text{ is a perfect square.}
\end{align}
I failed tying to find such triple numerically. I tried ... | Piquito | 219,998 | <p>It is impossible that non-existance because there are infinitely many counterexamples. In fact, we must have by the Pythagorean triples
$$x-z=t^2-s^2;\space y=2ts\qquad (*)$$ $$x=t_1^2-s_1^2; \space y=2t_1s_1$$ $$x+z=t_2^2-s_2^2;\space \space y=2t_2s_2\qquad (**)$$ so we have from $(*)$ and $(**)$ $$x=\frac{t^2-s^2... |
981,211 | <p>Let $G$ be a topological group, $H$ be its normal subgroup of $G$, and $G/H$ be the quotient space induced by the natural map.(We know that $G/H$ is again a topological group) </p>
<p>If $\pi_1(G)$ is trivial, how to prove $ \pi_1(G/H)=\pi_0(H)/\pi_0(G) $?
The author of this article use this fact directly in the pa... | Dan Rust | 29,059 | <p>Matthew Leingang gave the answer in a comment but I'll flesh it out a little. The quotient map $q\colon G\to G/H$ is a fibration with fiber topologically isomorphic to $H$, and so we get a long exact sequence in homotopy groups
$$\cdots \to \pi_{k+1}(G) \to \pi_{k+1}(G/H) \stackrel{\delta_k}{\to} \pi_k(H) \stackrel{... |
42,879 | <p>Let $G$ be a finite group and $\chi$ a character of $G$. The values of $\chi$ generate an abelian Galois extension $K$ of $\mathbb{Q}$, and so one can consider the conjugate $\sigma(\chi)$ of $\chi$ by any element $\sigma$ of the Galois group. What's the shortest way to prove that $\sigma(\chi)$ is also a character ... | Pete L. Clark | 299 | <p>Well, wait a minute. I guess we can agree that every complex representation <span class="math-container">$\rho$</span> of <span class="math-container">$G$</span> is defined over <span class="math-container">$\mathbb{C}$</span>! Let <span class="math-container">$\sigma$</span> be an automorphism of the character fi... |
832,125 | <p>How do i prove there are only 8 elements of order 3 in $S_4$ and they are all in $A_4$?</p>
<p>Should i prove this by considering all 24 cases?</p>
| Alex G. | 130,309 | <p>No, there's a much easier way. First note that of the cycle types present in $S_4$, only one is of order three, namely the $3$-cycle. Now you only need to count the number of $3$-cycles in $S_4$. To do this, we note that a $3$-cycle is determined by picking $3$ distinct numbers in $\{1, 2, 3, 4\}$ and ordering them.... |
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