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 |
|---|---|---|---|---|
1,849,003 | <p>The following question was asked in an theoretical computer science entrance exam in India:</p>
<blockquote>
<p>A spider is at the bottom of a cliff, and is n inches from the top.
Every step it takes brings it one inch closer to the top with
probability $1/3$, and one inch away from the top with probability
... | joriki | 6,622 | <p>The recurrence for the expectation $a_n$ is</p>
<p>$$
a_k=1+\frac13a_{k-1}+\frac23a_{k+1}\;,
$$</p>
<p>with solution $a_k=c_1+c_22^{-k}-3k$. The boundary conditions are $a_0=0$ and $a_n=1+a_{n-1}$, yielding $c_1+c_2=0$ and $c_22^{-n}-3n=1+c_22^{-(n-1)}-3(n-1)$. Solving the second equation for $c_2$ yields $c_2=-4\... |
2,036,591 | <p>Male basketball players at the high school, college, and professional ranks use a regulations basketball that weighs an average of 22.0 ounces with a std dev of 1.0 ounce. Assume that the weights of basketballs are approx. normally distributed. (Round to nearest tenth of a percent)</p>
<p>a. What % of regulation ba... | Techie91 | 393,973 | <p>It looks like you need to find the z-score and use the Z table. where x_bar = 22 ounces, sigma = 1 ounce. you need to use the equation Z = (x-x_bar)/sigma</p>
<p>a) you need to find P(Z). x = 19.5. solve for Z, then find P(Z) using the Z table, where P(Z) is the area under the curve. look up how to use the Z table.... |
2,036,591 | <p>Male basketball players at the high school, college, and professional ranks use a regulations basketball that weighs an average of 22.0 ounces with a std dev of 1.0 ounce. Assume that the weights of basketballs are approx. normally distributed. (Round to nearest tenth of a percent)</p>
<p>a. What % of regulation ba... | Brandon | 378,998 | <p>I will demonstrate the usage of normCdf and invNorm in a graphing calculator to solve problems like these, though they can be solved like the answer given by Techie91.</p>
<p>The notation for normalcdf is <em>NormalCDF(lower bound, upper bound, mean, standard deviation)</em></p>
<p>Part A:</p>
<p>$$NormalCDF (-10... |
210,071 | <p>I have some lines of code that generate numerical solutions to equations. Then I want to combine two of these in a piecewise function. The way I did it is the following -lf1[r] and lf4[r] are the aforementioned numerical solutions</p>
<pre><code>test[r_] :=
Piecewise[{{lf1[r], 0.688199 <= r <= 10}, {lf4[r],... | Bob Hanlon | 9,362 | <pre><code>Clear["Global`*"]
test[r_] :=
Piecewise[{{Exp[r], r < -1}, {1 - r^2, -1 < r < 1}, {Sin[Pi r],
r > 1}}];
plotRng = {-3, 3};
</code></pre>
<p>Extract plot intervals from <a href="https://reference.wolfram.com/language/ref/Piecewise.html" rel="noreferrer"><code>Piecewise</code></a> and t... |
2,247,445 | <p>Find isomorphism between <span class="math-container">$\mathbb F_2[x]/(x^3+x+1)$</span> and <span class="math-container">$\mathbb F_2[x]/(x^3+x^2+1)$</span>.</p>
<hr />
<p>It is easy to construct an injection <span class="math-container">$f$</span> satisfying <span class="math-container">$f(a+b)=f(a)+f(b)$</span> an... | mrnovice | 416,020 | <p>We have $$\tan\alpha =\frac{1}{3} =\frac{\text{opposite side}}{\text{adjacent side}}$$</p>
<p>Then construct a right angled triangle, with hypotenuse equal to $\sqrt{1^2+3^2} =\sqrt{10}$</p>
<p>Then $$\cos\alpha = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{3}{\sqrt{10}}$$</p>
<p>and $$\sin\alpha =\frac{\te... |
3,817,760 | <p>In geometry class, it is usually first shown that the medians of a triangle intersect at a single point. Then is is explained that this point is called the centroid and that it is the balance point and center of mass of the triangle. Why is that the case?</p>
<p>This is the best explanation I could think of. I ho... | Random | 316,502 | <p>The centroid is the center of mass of the configuration where we have three point masses of mass 1 at each of the three vertices of the triangle.</p>
<p>Notice that we can replace two of those point masses by a mass of 2 at their midpoint (that is, at their center of mass). Therefore, the total center of mass is cle... |
1,395,093 | <blockquote>
<p>Prove that if $f\in L^1([0,1],\lambda)$ is not constant almost
everywhere then there exists an interval so that
$\int_I\!f\,\mathrm{d}\lambda\neq 0$. Here $\lambda$ is the Lebesgue
measure.</p>
</blockquote>
<p>Since this is obviously true for continuous functions, I've been trying to use the f... | peterwhy | 89,922 | <p>Setting $f(x) = 0$,
$$\begin{align*}
1+\frac ax + \frac a{x^2} &= 0\\
x^2+ax+a &= 0\\
x &= \frac{-a\pm\sqrt{a^2-4a}}{2}
\end{align*}$$
So the $x$-intercepts, if any, depend on $a$.</p>
<hr>
<p>Differentiating $f(x)$ w.r.t. $x$,
$$f'(x) = -\frac a{x^2}-\frac {2a}{x^3} = -\frac{ax(x+2)}{x^4}$$
Setting $f... |
3,963,517 | <blockquote>
<p>Write to power series of <span class="math-container">$$f(x)=\frac{\sin(3x)}{2x}, \quad x\neq 0.$$</span></p>
</blockquote>
<p>I try to get a series for <span class="math-container">$\sin(3x)$</span> with <span class="math-container">$x=0$</span> and multiplying the series with <span class="math-contain... | vlad333rrty | 862,985 | <p>Just write taylor series for sin(3x) and the divide by 2x
<span class="math-container">$$\frac{1}{2} \sum_{n=0}^{\infty} (-1)^n \frac{3^{2n+1}x^{2n}}{(2n+1)!}$$</span></p>
|
484,095 | <p>When taking the log of a matrix we have various choices, but fixing a particular choice, we should have</p>
<p>$$P^{-1}\log{(A)} P = \log(P^{-1}AP),$$</p>
<p>right? (Here $P \in GL$.)</p>
<p>It is supported by the notion that we can exponentiate both sides and it comes out true. Is there some snag here that I'm m... | André Nicolas | 6,312 | <p>It is not correct to divide. If it is useful, you might rewrite as
$$\sum_{i=0}^kn_i(b^i-1)\equiv 0\pmod{b-1}.$$</p>
<p>It is useful, for example, if you want to <strong>prove</strong> that your relationship holds for all choices of the $n_i$. That is because one can show that $b^i-1\equiv 0\pmod{b-1}$.</p>
<p>Th... |
17,285 | <p>I have problems in understanding few concepts of elementary set theory. I've choosen a couple of problems (from my problems set) which would help me understand this concepts. To be clear: it's not a homework, I'm just trying to understand elementary set's theory concepts by reading solutions.
<hr/>
<strong>Problem 1... | Ross Millikan | 1,827 | <p>For problem 3, $\mathbb{Q^N}$ means the set of functions from $\mathbb{N}$ into $\mathbb{Q}$. So you need to decide how many of them there are. Certainly there are at least as many as functions from $\mathbb{N}$ into 2, which is $2^\mathbb{N}$, as you could just restrict the functions values to 0 and 1. Have you ... |
41,175 | <p>Suppose that I have two linear functions</p>
<pre><code>f[x_] := f0 + f1 x
g[x_] := g0 + g1 x
</code></pre>
<p>and a (possibly rather complicated) set of conditional expressions, obtained through Reduce. For example, we might have something like this:</p>
<pre><code>conditions = (f0 == f1 && g0 == 0) || ... | TransferOrbit | 3,236 | <p>First convert your conditions to a list of <a href="http://reference.wolfram.com/mathematica/ref/Rule.html" rel="nofollow noreferrer"><code>Rule</code></a>s</p>
<pre><code>myrules = Apply[List, conditions /. {Equal -> Rule}, {0, 1}]
</code></pre>
<p>which gives</p>
<p><img src="https://i.stack.imgur.com/HO9YV.... |
3,586,001 | <p>Let <span class="math-container">$k$</span> be a positive integer. Is it true that there are infinitely many primes of the form <span class="math-container">$p=6r-1$</span> such that <span class="math-container">$r$</span> and <span class="math-container">$k$</span> are coprime? Feel free to assume any well known (e... | Mohammed M. Zerrak | 471,639 | <p>Assume the contrary, then there is only finitely many primes of the form <span class="math-container">$6r-1$</span> , where gcd<span class="math-container">$(k,r )=1$</span>, then there is infinitely many primes of the form <span class="math-container">$6qr-1$</span> where the integer <span class="math-container">$q... |
109,061 | <p>Most people learn in linear algebra that its possible to calculate the eigenvalues of a matrix by finding the roots of its characteristic polynomial. However, this method is actually very slow, and while its easy to remember and its possible for a person to use this method by hand, there are many better techniques a... | J. M. ain't a mathematician | 498 | <p>(This was supposed to be a comment, but it got too long.)</p>
<p>Robert raises a good question in the comments. Of course, general analytical expressions for roots of polynomials of high degree aren't really used much in practice, precisely because the expressions themselves are a bit unwieldy, and the special func... |
855,463 | <p>I have the following question regarding support vector machines: So we are given a set of
training points $\{x_i\}$ and a set of binary labels $\{y_i\}$.</p>
<p>Now usually the hyperplane classifying the points is defined as:</p>
<p>$ w \cdot x + b = 0 $</p>
<p><strong>First question</strong>: Here $x$ does not d... | nullgeppetto | 100,951 | <p>I suggest to you to thoroughly study a really insightful <a href="http://research.microsoft.com/pubs/67119/svmtutorial.pdf" rel="nofollow">tutorial</a> about Support Vector Machines for Pattern Recognition due to C. Burges. It isn't elementary, but it gives significant information about SV learning. It really helped... |
855,463 | <p>I have the following question regarding support vector machines: So we are given a set of
training points $\{x_i\}$ and a set of binary labels $\{y_i\}$.</p>
<p>Now usually the hyperplane classifying the points is defined as:</p>
<p>$ w \cdot x + b = 0 $</p>
<p><strong>First question</strong>: Here $x$ does not d... | Whadupapp | 162,069 | <p>To answer both questions at once: The set $\{x \in \mathbb R^n \mid w'x+b=0\}$ defines the separating hyperplane. This hyperplane separates the space $\mathbb R^n$ into two halfspaces $H_+ = \{x \in \mathbb R^n \mid w'x+b>0\}$ and $H_- = \{x \in \mathbb R^n \mid w'x+b<0\}$.</p>
<p>Thus, for an unknown example... |
297,251 | <p>In his well-known <a href="http://matwbn.icm.edu.pl/ksiazki/fm/fm101/fm101110.pdf" rel="nofollow noreferrer">paper</a> Bellamy constructs an indecomposable continua with exactly two composants. The setup is as follows:</p>
<p>We have an inverse-system $\{X(\alpha); f^\alpha_\beta: \beta,\alpha < \omega_1\}$ of m... | Balazs | 6,107 | <p>The author of <a href="http://people.bath.ac.uk/masgks/Theses/kasprzyk.pdf" rel="nofollow noreferrer">this</a> and <a href="https://arxiv.org/pdf/math/0311284.pdf" rel="nofollow noreferrer">this</a> might be able to send you lists of vertices, from which you can generate the pictures yourself, if you ask nicely. </p... |
3,625,739 | <p>My question is from Humphreys <em>Introduction to Lie Algebras and Representation Theory</em>. There is a lemma in section 6.3 which says, if <span class="math-container">$\phi: L \to \mathfrak{gl}(V)$</span> is a representation of a semisimple Lie algebra <span class="math-container">$L$</span> then <span class="ma... | wasatar | 641,315 | <p>I like the answer above but I just realized this is very clear: <span class="math-container">$\mathfrak{sl}(V)$</span> are exactly the trace zero matrices and when dim<span class="math-container">$V=1$</span> we just have <span class="math-container">$\mathfrak{sl}(V)=0$</span>.</p>
|
2,416,910 | <p>Let $A:=\{-k,\ldots,-2,-1,0,1,2,\ldots,k\}$, $k<\infty$, $k\in \mathbb{N}$.</p>
<p>Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be such that $f(-x)=-f(x)$ for all $x\in A$ and $f(x+y)=f(x)+f(y)$ for all $x,y\in A$ such that $x+y\in A$.</p>
<p>Does it follow that there is $\alpha\neq 0$ such that $f(x)=\alpha x$ for ... | Davide Giraudo | 9,849 | <p>Let <span class="math-container">$X$</span> be a random variable taking the value <span class="math-container">$a$</span> with probability <span class="math-container">$p$</span> and <span class="math-container">$-a$</span> with probability <span class="math-container">$1-p$</span>, with <span class="math-container"... |
50,479 | <p>Can the number of solutions $xy(x-y-1)=n$ for $x,y,n \in Z$ be unbounded as n varies?</p>
<p>x,y are integral points on an Elliptic Curve and are easy to find using enumeration of divisors of n (assuming n can be factored).</p>
<p>If yes, will large number of solutions give moderate rank EC?</p>
<p>If one drops $... | Aaron Meyerowitz | 8,008 | <p>$n=938995200$ also has $22$ solutions. It might be nicer to put $a=y$ and $b=x-y-1$ so $x=a+b+1$ and one has $ab(a+b+1)=n$. For $n=391287046550400$ there are $26$ solutions with $a,b>0$ which grows to $78$ if one allows negative values and shrinks to $13$ if $[a,b]$ and $[b,a]$ are considered the same (and must b... |
4,290,960 | <p>In machine learning (and not only), it is very common to see concatenation of different feature vectors into a single one of higher dimension which is then processed by some function. For example, feature vectors computed for an image at different scales are concatenated to form a multi-scale feature vector which is... | Ewan Delanoy | 15,381 | <p>If we solve the quadratic in <span class="math-container">$c$</span>, we obtain</p>
<p><span class="math-container">$$
c=ab \pm \sqrt{(a^2-1)(b^2-1)} \tag{1}
$$</span></p>
<p>We see that <span class="math-container">$(a^2-1)(b^2-1)$</span> is a perfect square.
Let <span class="math-container">$d \gt 0$</span> be the... |
51,903 | <p>Does anyone know of a tool which</p>
<ol>
<li>Can display formulas neatly, preferably like this website without hassle. (Unlike wikipedia with <code>:<math></code>) </li>
<li>Has a wiki like structure: i.e categories of pages, individual articles with hyperlinked sections, subsections etc. </li>
<li>Preferrab... | kuch nahi | 8,365 | <p>After spending a lot of time searching I have finally set it up. The answer is <a href="http://drupal.org/drupal-7.0" rel="nofollow noreferrer">Drupal 7</a>. It redresses all complaints except one (it requires an offline installation). I am currently maintaining it locally (with remote desktop) but I will probably g... |
180,323 | <p>apologies if this is a naive question. Consider two Galois extensions, K and L, of the rational numbers. For each extension, consider the set of rational primes that split completely in the extensions, say Split(K) and Split(L).</p>
<p>If Split(K) = Split(L), then is it necessarily true that K and L are isomorphic ... | Ari Shnidman | 949 | <p>For your first question, it is true that $L$ and $K$ are isomorphic. This follows from the Chebotarev density theorem, for example. </p>
<p>For the second question, it's not really known how to do this in general. But class field theory lets you describe the sets $S$ which arise from abelian extensions (over any... |
9,880 | <p>The ban on edits that change only a few characters can be quite annoying in some cases. I don't have a problem with trying to keep trivial spelling corrections and such out of the peer review queue, but sometimes a small edit is semantically significant. For example, it makes perfect sense to edit an otherwise good ... | Community | -1 | <p>In most cases, if there is one error to fix there are more of them in the post and you should fix them all. Each small edit bumps the post to the frontpage, too many unnecessary edits are harmful due to that.</p>
<p>The minimum of 6 character edits is only active for suggested edits, which additionally require revi... |
481,673 | <p>Find all functions $g:\mathbb{R}\to\mathbb{R}$ with $g(x+y)+g(x)g(y)=g(xy)+g(x)+g(y)$ for all $x,y$.</p>
<p>I think the solutions are $0, 2, x$. If $g(x)$ is not identically $2$, then $g(0)=0$. I'm trying to show if $g$ is not constant, then $g(1)=1$. I have $g(x+1)=(2-g(1))g(x)+g(1)$. So if $g(1)=1$, we can show i... | Arash | 92,185 | <p>It is much more complicated than it seems. If you put $x=y=0$, you get either $g(0)=0$ or $g(0)=2$. If $g(0)\neq 0$ then put $y=0$ in the equation and you get for all other $x$, $g(x)=2$.</p>
<p>First solution: $g(x)=2$</p>
<p>Now if $g(0)=0$ there are other possibilities. Choose $x=y=2$ and then either $g(2)=0$ o... |
481,673 | <p>Find all functions $g:\mathbb{R}\to\mathbb{R}$ with $g(x+y)+g(x)g(y)=g(xy)+g(x)+g(y)$ for all $x,y$.</p>
<p>I think the solutions are $0, 2, x$. If $g(x)$ is not identically $2$, then $g(0)=0$. I'm trying to show if $g$ is not constant, then $g(1)=1$. I have $g(x+1)=(2-g(1))g(x)+g(1)$. So if $g(1)=1$, we can show i... | Caleb Stanford | 68,107 | <p>Here's a much less murky solution with strong influence from Calvin Lin's excellent answer.
$$
g(x + y) + g(x)g(y) = g(xy) + g(x) + g(y) \tag{0}
$$
First of all, plugging in $y = 1$ gives
$$ g(x + 1) + g(x)g(1) = 2g(x) + g(1)$$</p>
<p>For readability let $g(1) = k$ and we have
$$
g(x + 1) = (2-k)g(x) + k \tag{1}
$$... |
2,535,656 | <p><a href="https://i.stack.imgur.com/sjkal.png" rel="nofollow noreferrer">This is the example to numbers to the 3rd.</a></p>
<p><a href="https://i.stack.imgur.com/kzdHS.png" rel="nofollow noreferrer">Here is the same thing just with numbers to the 4th</a></p>
<p>Well I was messing around with numbers I made this dis... | Qiaochu Yuan | 232 | <p>The relevant keyword here is <a href="https://en.wikipedia.org/wiki/Finite_difference" rel="nofollow noreferrer">finite differences</a>. In general, if <span class="math-container">$f(n)$</span> is a sequence, we can construct from it a new sequence, the <strong>forward difference</strong></p>
<p><span class="math-c... |
2,535,656 | <p><a href="https://i.stack.imgur.com/sjkal.png" rel="nofollow noreferrer">This is the example to numbers to the 3rd.</a></p>
<p><a href="https://i.stack.imgur.com/kzdHS.png" rel="nofollow noreferrer">Here is the same thing just with numbers to the 4th</a></p>
<p>Well I was messing around with numbers I made this dis... | Michael Hardy | 11,667 | <p>The binomial theorem says</p>
<p>$$
(a+b)^x = a^x + xa^{x-1}b + \frac{x(x-1)}2 a^{x-2} b^2 + \frac{x(x-1)(x-2)} 6 a^{x-3} b^3 + \cdots.
$$
The differences in the column to the right of the one that lists values of $a^x$ are
$$
(a+1)^x - a^x = \left( a^x + x a^{x-1} + \frac{x(x-1)} 2a^{x-2} + \cdots \right) - a^x.
$... |
3,663,267 | <p>in a linear algebra class i was given a theorem: If <span class="math-container">$\left\{\boldsymbol{u}_{1}, \boldsymbol{u}_{2}, \ldots, \boldsymbol{u}_{n}\right\}$</span> is a linearly independent subset of <span class="math-container">$\mathbb{R}^{n}$</span> then
<span class="math-container">$
\mathbb{R}^{n}=\oper... | Naba Kumar Bhattacharya | 556,399 | <p>Define, <span class="math-container">$e_{ij}$</span> to be the <span class="math-container">$n \times n$</span> matrix having <span class="math-container">$ij$</span>-th entry 1 and all others 0. Try to find out the products of several forms of those matrices.
You can refer to the problem 6 of chapter 7.2 (pg 238-2... |
416,514 | <p>I really think I have no talents in topology. This is a part of a problem from <em>Topology</em> by Munkres:</p>
<blockquote>
<p>Show that if $A$ is compact, $d(x,A)= d(x,a)$ for some $a \in A$. </p>
</blockquote>
<p>I always have the feeling that it is easy to understand the problem emotionally but hard to expr... | not all wrong | 37,268 | <p><em>Hint</em>: Compact $\iff$ sequentially compact for metric spaces. Can you construct a sequence which must tend to the $a$ you want to find?</p>
|
3,476,181 | <p>I'm trying to solve this group theory problem, and I'm really not sure how to approach this. The question is:</p>
<p>Show that <span class="math-container">$\mathbb{F}_7[\sqrt{-1}] := \{a+b\sqrt{-1}$</span> | <span class="math-container">$a,b \in \mathbb{F}_7\}$</span> is a ring.</p>
<p>I have been stuck on this p... | acupoftea | 726,513 | <p>This answer is translated (with small modifications) from <a href="https://www.mimuw.edu.pl/~rytter/TEACHING/JAO/higman.pdf" rel="nofollow noreferrer">here</a>.</p>
<p><span class="math-container">$\Sigma$</span> is a finite alphabet.<br>
<span class="math-container">$\Sigma^\ast$</span> is the set of finite string... |
176,167 | <p>$\newcommand{\scp}[2]{\langle #1,#2\rangle}\newcommand{\id}{\mathrm{Id}}$
Let $f$ and $g$ be two proper, convex and lower semi-continuous functions (on a Hilbert space $X$ or $X=\mathbb{R}^n$) and let $g$ be continuously differentiable. Consequently, the subdifferential $\partial f$ and the gradient $\nabla g$ are m... | Connor Mooney | 16,659 | <p>This inequality is not true. Here is a counterexample:</p>
<p>Let ${\bf x},{\bf y} \in \mathbb{R}^2$ with ${\bf y} = 0$ and ${\bf x} = (1,\epsilon)$ for $\epsilon$ small positive. Let $g(x,y) = Cy^2$ with $C(\epsilon)$ very large, chosen say so that
$${\bf x} - \nabla g({\bf x}) = (1, -M)$$
for some $M$ large.</p>
... |
2,418,181 | <p><strong>Question:</strong>
Let $P_1,\ldots,P_n$ be propositional variables. When is the statement $P_1 \oplus \cdots \oplus P_n$ true?</p>
<p>I'm currently learning the basics of discrete math. I am stuck on this last question of my assignment... not really sure how to go about solving it.</p>
<p>I do know that a ... | Anurag A | 68,092 | <p>Idea: You may think of $\oplus$ as acting like mod $2$ addition by taking $T$ as $1$ and $F$ as $0$. Then
\begin{align*}
T \oplus F &=1+0=1 \pmod{2}\\
T \oplus T &=1+1=0 \pmod{2}\\
F \oplus F &=0+0=0 \pmod{2}
\end{align*}</p>
<p>Let $k$ be the number of statements that are true among $P_1, P_2, \ldots ... |
3,684,158 | <p>You are given two circles:</p>
<p>Circle G: <span class="math-container">$(x-3)^2 + y^2 = 9$</span></p>
<p>Circle H: <span class="math-container">$(x+3)^2 + y^2 = 9$</span></p>
<p>Two lines that are tangents to the circles at point <span class="math-container">$A$</span> and <span class="math-container">$B$</span... | K. Miyamoto | 991,812 | <p>This is not a complete answer to your original question, but there is a nice geometrical explanation on why any point <span class="math-container">$P$</span> on the ellipse <span class="math-container">$16x^2+25y^2=625$</span> satisfies <span class="math-container">$|\overline{AP}|+|\overline{BP}|=10$</span>. This c... |
1,147,773 | <p>Do you have any explicit example of an infinite dimensional vector space, with an explicit basis ?</p>
<p>Not an Hilbert basis but a family of linearly independent vectors which spans the space -any $x$ in the space is a <strong>finite</strong> linear sum of elements of the basis.</p>
<p>In general the existence ... | Surb | 154,545 | <p>What do you think about
$$X=\text{span}\{e_1,e_2,e_3,...\}$$</p>
<p>where $$e_i=(0,...,0,\underset{\underset{i^{th}\ place}{\uparrow}}{1},0,...).$$</p>
|
3,225,151 | <p><span class="math-container">$ZFC+V=L$</span> implies that <span class="math-container">$P(\mathbb{N})$</span> is a subset of <span class="math-container">$L_{\omega_1}$</span>. But I’m wondering what layer of the constructible Universe contains a smaller set.</p>
<p>My question is, what is the smallest ordinal <s... | Noah Schweber | 28,111 | <p>It is consistent that there is no such <span class="math-container">$\alpha$</span>.</p>
<p>More precisely, it is consistent with ZFC that there is a formula <span class="math-container">$\varphi$</span> in the language of second-order arithmetic such that <span class="math-container">$\{x:\varphi(x)\}$</span> is n... |
1,638,490 | <p>Yes, I know the title is bizarre. I was urinating and forgot to lift the seat up. That made me wonder: assuming I maintain my current position, is it possible for the toilet seat (assume it is a closed, but otherwise freely deformable curve) to be moved/deformed such that stream does not pass through the hole anymor... | dxiv | 291,201 | <p>Here is a topologically equivalent reformulation.</p>
<blockquote>
<p>Suppose you are ice fishing on a frozen lake. Is it possible for the hole in the ice to deform in such a way that the rod and line no longer pass through it, without breaking either?</p>
</blockquote>
<p>The answer is yes, but (in both cases) ... |
289,405 | <p>Consider an elementary class $\mathcal{K}$. It is quite common in model theory that a structure $K$ in $\mathcal K$ comes with a closure operator $$\text{cl}: \mathcal{P}(K) \to \mathcal{P}(K), $$ which establishes a <a href="https://en.wikipedia.org/wiki/Pregeometry_(model_theory)" rel="nofollow noreferrer">pregeo... | Ivan Di Liberti | 104,432 | <p>The following is valid when $A\subseteq M\prec N$, $\text{cl}(A)$ in $M$ equals $\text{cl}(A)$ in $N$, i.e. closures don't grow in elementary extensions.</p>
<hr>
<p>The answer to my questions looks to me to be <strong>yes</strong>.</p>
<p>Let $K = \bigcup K_i$ We know that $K_i$ has a basis $A_i$.</p>
<p>Step 1... |
2,967,626 | <p>I have a graph that is 5280 units wide, and 5281 is the length of the arc.</p>
<p><img src="https://i.stack.imgur.com/4rFK4.png"></p>
<p>Knowing the width of this arc, and how long the arc length is, how would I calculate exactly how high the highest point of the arc is from the line at the bottom?</p>
| Nicholas Parris | 566,184 | <p>The final solution comes out to be very nice, so I suspect this is a homework problem. As such, I will just give you an outline.
<span class="math-container">$$0 < \frac{2 b^2 r^2}{z} - \left(2 r ^2 - 2 b r \sqrt{1 - \frac{b^2}{z^2}}\right) z$$</span>
Simplifying terms, dividing both sides by <span class="math-co... |
55,071 | <p><strong>Bug introduced in 10.0.0 and persisting through 10.3.0 or later</strong></p>
<hr>
<p>I've upgraded my home installation of <em>Mathematica</em> from version 9 to 10 today on a Windows 8.1 machine, and I'm getting a weird font issue - the fonts are not anti-aliased, and look unbalanced and weird. Just look:... | Tetsuo Ichii | 18,579 | <p>This problem is probably due to the MathematicaMono font which is introduced in v10.</p>
<p>Defining the problem:<br>
Some fonts ("[","_","]","=", etc.) are rendered badly with strange thinning in v10 in some notebook magnifications.
This is obvious when you compare the renderings from v10 with those from v9. </p>
... |
1,866,801 | <p>Let $A$ be an infinite set, $B\subseteq A$ and $a\in B$. Let $X\subseteq \mathcal{P}(A)$ be an infinite family of subsets of $A$ such that $a\in \bigcap X$.</p>
<p>Suppose $\bigcap X\subseteq B$. Is it possible that, for every non-empty finite subfamily $Y\subset X$, $\bigcap Y \not\subseteq B$ ? </p>
<p>Thanks fo... | Piquito | 219,998 | <p>$$\sin x + \sin y = 1\Rightarrow 2\sin\frac{x+y}{2}\cos \frac{x-y}{2}=1\\\cos x+\cos y=0\Rightarrow 2\cos\frac{x+y}{2}\cos \frac{x-y}{2}=0$$
It follows $$ \cos \frac{x+y}{2}=0\text{ and } \cos \frac{x-y}{2}=0\iff $$ The first ($\cos \frac{x+y}{2}=0$) gives, from the first given equation,</p>
<p>$$\begin{cases}x+y=... |
1,866,801 | <p>Let $A$ be an infinite set, $B\subseteq A$ and $a\in B$. Let $X\subseteq \mathcal{P}(A)$ be an infinite family of subsets of $A$ such that $a\in \bigcap X$.</p>
<p>Suppose $\bigcap X\subseteq B$. Is it possible that, for every non-empty finite subfamily $Y\subset X$, $\bigcap Y \not\subseteq B$ ? </p>
<p>Thanks fo... | fleablood | 280,126 | <p>There are so many different ways to solve it the real question is which way.</p>
<p>What reaches out to grab me is:</p>
<p>$\cos x + \cos y = 0$</p>
<p>$\cos x = - \cos y$ which means either $y = \pi - x$ (within a period of $2\pi$) or $y = x + \pi$ (within a period of $2\pi$).</p>
<p>If $y = x + \pi$ then $\sin... |
31,948 | <p>A Finsler manifold is defined as a differentiable manifold with a metric defined on so that any well-defined curves of finite arc length is given by a generalized arc length integral of an asymmetric norm over each tangent space defined at a point. This generalizes the Riemannian manifold structure since the norm is... | Will Jagy | 3,324 | <p>The answer is no. Any smooth manifold admits a Riemannian metric using paracompactness and partitions of unity: in short, a convex sum of positive definite symmetric matrices is positive definite symmetric. So any manifold has such a structure. But there are topological obstructions to the existence of global pseudo... |
651,707 | <p>Let $A=(a_{ij})\in \mathbb{M}_n(\mathbb{R})$ be defined by</p>
<p>$$
a_{ij} =
\begin{cases}
i, & \text{if } i+j=n+1 \\
0, & \text{ otherwise}
\end{cases}
$$
Compute $\det (A)$</p>
<hr>
<blockquote>
<p>After calculation I get that it may be $(-1)^{n-1}n!$. Am I right?</p>
</blockquote>
| user76568 | 74,917 | <p>These are matrices with only off-diagonal elements, with values being the row numbers. So the number of negative factors is the same for if it was a diagonal matrix: <strong>even</strong> for an <strong>even</strong> $n$, and <strong>odd</strong> for an <strong>odd</strong> $n$.
So looks like the answer is:</p>
<p... |
2,617,467 | <p>I have been trying to solve this limit for more than an hour and I'm stuck.</p>
<blockquote>
<p>$$ \lim_{n\to\infty} \frac{3^{n}+\sqrt{n}}{n!+2^{(n+1)}}$$</p>
</blockquote>
<p>What I've so far is:</p>
<p>$$ \lim_{n\to\infty} \frac{3^{n}+\sqrt{n}}{n!+2^{(n+1)}}\\ \lim_{n\to\infty} \frac{3^{n}}{n!+2^{(n+1)}}+\lim... | Arnaud Mortier | 480,423 | <p>My advice: don't systematically replace part of an expression by its limit as soon as you know it, while keeping the $n$ in the rest of the expression. </p>
<p>For instance when you have $\sqrt{n} \dfrac{1}{n!} $, you rush into replacing $\dfrac{1}{n!} $ by $0$ and then you're stuck. Analyse the situation: $\sqrt{n... |
2,617,467 | <p>I have been trying to solve this limit for more than an hour and I'm stuck.</p>
<blockquote>
<p>$$ \lim_{n\to\infty} \frac{3^{n}+\sqrt{n}}{n!+2^{(n+1)}}$$</p>
</blockquote>
<p>What I've so far is:</p>
<p>$$ \lim_{n\to\infty} \frac{3^{n}+\sqrt{n}}{n!+2^{(n+1)}}\\ \lim_{n\to\infty} \frac{3^{n}}{n!+2^{(n+1)}}+\lim... | Guy Fsone | 385,707 | <p>
<p>we have
$$ \frac{3^{n}+\sqrt{n}}{n!+2^{(n+1)}}\le \frac{3^{n}+\sqrt{n}}{n!} =\frac{3^{n}}{n!} + \frac{1}{\sqrt{n}(n-1)!}\to0$$</p>
<p>Indeed, the convergence of the series $$e^3 =\sum_{n=0}^{\infty}\frac{3^{n}}{n!}\implies \frac{3^{n}}{n!} \to0$$</p>
|
3,792,242 | <p>A large part of the set theory is devoted to infinities of various kinds, and this has been built on Cantor's groundbreaking work on uncountable sets. However, even Cantor's proof is based on the assumption that certain sets exist, namely that the power set of a countably infinite set exists. Sure, after assuming th... | Andreas Blass | 48,510 | <p>This should perhaps be a comment, but it's too long. The question emphasizes a dichotomy among the subsets of <span class="math-container">$\mathbb N$</span>: computable sets versus all the others. But there's actually a much richer spectrum here: computable sets, computably enumerable sets (like the set of code num... |
220,907 | <p>If I have these three lists:</p>
<pre><code>list1={{0.01,87.,0.},{0.03,87.,0.18353},{0.1,87.,0.494987},{0.3,87.,0.899803},{1.,87.,1.08076},{3.,87.,1.10593},{10.,87.,1.04781},{10.,87.,1.02449},{10.,87.,0.964193},{30.,87.,1.0602},{30.,87.,1.04075},{30.,87.,1.05987},{100.,87.,1.14661},{100.,87.,1.00639},{100.,87.,1.09... | SuperCiocia | 35,368 | <p>From the description that you give, this what you want:</p>
<pre><code>listtotal = Sort[Join[list1, list2, list3], #1[[1]] < #2[[1]] &]
</code></pre>
<p>Even </p>
<pre><code>listtotal =
Sort[Join[list1, list2, list3], #1[[1]] < #2[[1]] &] // Flatten
</code></pre>
<p>if you want the list flattened... |
4,180,344 | <p>I started learning from Algebra by Serge Lang. In page 5, he presented an equation<br><i></p>
<blockquote>
<p>Let <span class="math-container">$G$</span> be a commutative monoid, and <span class="math-container">$x_1,\ldots,x_n$</span> elements of <span class="math-container">$G$</span>. Let <span class="math-contai... | Rob Arthan | 23,171 | <p>Suggestion: consider the special case when <span class="math-container">$n = 3$</span> and write <span class="math-container">$a, b, c$</span> for <span class="math-container">$x_1, x_2, x_3$</span> respectively. What Lang is saying in that case is that we have all of the following equalities:
<span class="math-con... |
1,742,418 | <p>So far I have this:</p>
<p>What is the chance that at least one is a 2?</p>
<p>There is $\frac{5}{6}$ that you will not get any 2 on the four dices. From this we get $\frac{5^4}{6^4}$</p>
<p>The probability of getting at least one 2 is $1 - \frac{5^4}{6^4} $</p>
<p>Now I have no idea how to proceed from here.</p... | ervx | 325,617 | <p>Your answer to the first question is correct. For the second question, we use the formula for conditional probability. </p>
<p>Let $A$ be the event that the first die is a $1$.</p>
<p>Let $B$ be the event that at least one is a $2$.</p>
<p>Then,</p>
<p>$$
P(A|B)=\frac{P(A\cap B)}{P(B)}.
$$</p>
<p>You already co... |
4,344,538 | <p><strong>Lemma.</strong> Let <span class="math-container">$X$</span> be a topological vector space over <span class="math-container">$\mathbb{K}$</span> and <span class="math-container">$v\in X$</span>. Then the following mapping is continuous <span class="math-container">$$\begin{align*}
\varphi _v:\mathbb{K}&\r... | Paul Frost | 349,785 | <p>Let <span class="math-container">$\xi \in \mathbb K$</span> and <span class="math-container">$U$</span> be an open neighborhood of <span class="math-container">$(\xi,v)$</span> in <span class="math-container">$ \mathbb K \times X$</span>. By definition of the product topology there exist open neigbhorhoods <span cl... |
1,110,411 | <p>How can I prove that a continuous map $f : \mathbb{R}P^2 \to S^1$ is homotopic to the constant map? I know that in the projective space every point is a line but I do not get why the above has to be true.</p>
| WWK | 115,619 | <p>Hint:</p>
<p>$\pi(\mathbb{R}P^2)=\mathbb{Z_2}$, $\pi(S^1)=\mathbb{Z}$. So the degree of $f_*:\pi(\mathbb{R}P^2) \rightarrow \pi(S^1)$ can only be zero. So by the lifting property, this map can be lift to $\tilde{f}:\mathbb{R}P^2 \rightarrow \mathbb{R}$. </p>
|
1,110,411 | <p>How can I prove that a continuous map $f : \mathbb{R}P^2 \to S^1$ is homotopic to the constant map? I know that in the projective space every point is a line but I do not get why the above has to be true.</p>
| Anthony Conway | 116,246 | <p>As $\pi_1(\mathbb{R}P^2)$ is finite, the subgroup $f_*(\pi_1(\mathbb{R}P^2))$ of $\pi_1(S^1)=\mathbb{Z}$ must be trivial: indeed there are no non-trivial finite subgroups of $\mathbb{Z}$. Denote by $p: \mathbb{R} \longrightarrow S^1$ the universal cover of the circle. Using the lifting criterion for covering spaces ... |
1,536,487 | <p>$A$ is a real symmetric matrix and $E_1$ is a matrix whose $11$ entry is $1$ and rest are $0$. Further $A$ and $E_1$ don't commute. What can be said about eigenvalues of $A+E_1$</p>
| Ron Gordon | 53,268 | <p>Use the binomial theorem:</p>
<p>$$(\bar{z_1} t + \bar{z_2} (1-t))^n = \sum_{k=0}^n \binom{n}{k} \bar{z_1}^k \bar{z_2}^{n-k} t^k (1-t)^{n-k} $$</p>
<p>Thus, the integral is</p>
<p>$$(z_2-z_1)\sum_{k=0}^n \binom{n}{k} \bar{z_1}^k \bar{z_2}^{n-k} \int_0^1 dt \, t^k (1-t)^{n-k} = (z_2-z_1)\frac1{n+1} \bar{z_2}^n \su... |
1,536,487 | <p>$A$ is a real symmetric matrix and $E_1$ is a matrix whose $11$ entry is $1$ and rest are $0$. Further $A$ and $E_1$ don't commute. What can be said about eigenvalues of $A+E_1$</p>
| DanielWainfleet | 254,665 | <p>When a function $f:C\to C$ has a continuous complex derivative $f'$, and you restrict its domain to $R$, it is easily shown that for $t\in R$ we have $$Re (f'(t))=\frac {d Re (f(t))}{dt} \land Im f'(t)=\frac {d Im (f(t))}{dt}$$ and that these are continuous in $t$. Therefore $$\int_0^1f'(t)dt=\int_0^1Re (f'(t))dt+i\... |
1,350,062 | <p>In almost all of the physics textbooks I have ever read, the author will write the oscillating function as</p>
<blockquote>
<p>$$x(t)=\cos\left(\omega t+\phi\right)$$</p>
</blockquote>
<p>My question is that, is there any practical or historical reason why we should prefer $\cos$ to $\sin$ here? One possible e... | tomi | 215,986 | <p>Another reason is to use the initial position as a parameter.</p>
<p>Compare $z=z_0\lambda^t$ and $h=h_0\cos \omega t$</p>
<p>You can't do the same with $\sin$ - at best you can say perhaps $d=d_{max}\sin \omega t$</p>
<p>By the way, this argument fails once you introduce a phase shift!</p>
|
2,983,519 | <p>I have to calculate the limit of this formula as <span class="math-container">$n\to \infty$</span>.</p>
<p><span class="math-container">$$a_n = \frac{1}{\sqrt{n}}\bigl(\frac{1}{\sqrt{n+1}}+\cdots+\frac{1}{\sqrt{2n}}\bigl)$$</span></p>
<p>I tried the Squeeze Theorem, but I get something like this:</p>
<p><span cla... | Jack D'Aurizio | 44,121 | <p>Rearrange it as
<span class="math-container">$$ \frac{1}{n}\left(\sqrt{\frac{n}{n+1}}+\sqrt{\frac{n}{n+2}}+\ldots+\sqrt{\frac{n}{n+n}}\right) = \frac{1}{n}\sum_{k=1}^{n}\frac{1}{\sqrt{1+\frac{k}{n}}}$$</span>
which is a Riemann sum for
<span class="math-container">$$ \int_{0}^{1}\frac{dx}{\sqrt{1+x}}=2\sqrt{2}-2.$$<... |
198,612 | <p>Let $(P,\leq)$ be a partially ordered set. A <em>down-set</em> is a set $d\subseteq P$ such that $x\in d$ and $x'\in P, x'\leq x$ imply $x'\in d$. If the down-set is totally ordered, we say it is a totally ordered down-set (tods).</p>
<p>Let $d_1, d_2$ be tods. We say that they are <em>incompatible</em> if neither ... | Bjørn Kjos-Hanssen | 4,600 | <p>Here's a simplified version of Dominic van der Zypen's counterexample: order finite binary strings by extension, with the empty string at the bottom. Consider the club $ D$ consisting of the tods generated by strings of the form $0^n 1$.</p>
|
3,379,756 | <p>Why does the close form of the summation
<span class="math-container">$$\sum_{i=0}^{n-1} 1$$</span> equals <span class="math-container">$n$</span> instead of <span class="math-container">$n-1$</span>?</p>
| heropup | 118,193 | <p>Suppose <span class="math-container">$n = 2$</span>. Then <span class="math-container">$n - 1 = 1$</span>, and the sum becomes <span class="math-container">$$\sum_{i=0}^1 1.$$</span> Clearly, there are two terms in this sum, corresponding to <span class="math-container">$i = 0$</span> and <span class="math-contain... |
2,322,481 | <p>Look at this limit. I think, this equality is true.But I'm not sure.</p>
<p>$$\lim_{k\to\infty}\frac{\sum_{n=1}^{k} 2^{2\times3^{n}}}{2^{2\times3^{k}}}=1$$
For example, $k=3$, the ratio is $1.000000000014$</p>
<blockquote>
<p>Is this limit <strong>mathematically correct</strong>?</p>
</blockquote>
| marty cohen | 13,079 | <p>Is
$\lim_{k\to\infty}\frac{\sum_{n=1}^{k} 2^{2\times3^{n}}}{2^{2\times3^{k}}}=1
$?</p>
<p>Let</p>
<p>$\begin{array}\\
s(k)
&={2^{-2\times3^{k}}}\sum_{n=1}^{k} 2^{2\times3^{n}}\\
&=\sum_{n=1}^{k} 2^{2\times3^{n}-2\times3^{k}}\\
&=1+\sum_{n=1}^{k-1} 2^{2\times3^{n}-2\times3^{k}}\\
&\ge 1\\
\text{and}... |
78,968 | <p>Let $K_0$ be a bounded convex set in $\mathbf{R}^n$ within which lie two sets $K_1$ and $K_2$. Assume that,</p>
<ol>
<li>$K_1\cup K_2=K_0$ and $K_1\cap K_2=\emptyset$.</li>
<li>The boundary between $K_1$ and $K_2$ is unknown. (To avoid the trivial case, we assume that the boundary is not a hyperplane.)</li>
<li>Eit... | Niels J. Diepeveen | 3,457 | <p>Given the constraints, I think the following approach will be close to optimal.</p>
<p>Start by randomly querying points until you have 1 point in one set and
2 in the other.
You then have the endpoints of two segments crossing the boundary.</p>
<p>By using bisection on each of those segments, you can find pairs o... |
78,968 | <p>Let $K_0$ be a bounded convex set in $\mathbf{R}^n$ within which lie two sets $K_1$ and $K_2$. Assume that,</p>
<ol>
<li>$K_1\cup K_2=K_0$ and $K_1\cap K_2=\emptyset$.</li>
<li>The boundary between $K_1$ and $K_2$ is unknown. (To avoid the trivial case, we assume that the boundary is not a hyperplane.)</li>
<li>Eit... | whuber | 1,489 | <p>Let $K_0$ be the unit disk in $\mathbb{R}^2$. $K_1$ consists of the point at $(1,0)$ and some other point $\zeta$ uniformly and randomly chosen on the boundary of $K_0$ (the unit circle). It is not convex. $K_2$ is the complement of $K_1$ in $K_0$. It <em>is</em> convex. Therefore this is a valid instance of th... |
675,857 | <p>I am trying to figure out easy understandable approach to given small number of $n$, list all possible is with proof, I read this paper but it is really beyond my level to fathom,</p>
<p>attempt for $\phi(n)=110$, </p>
<p>$$\phi(n)=110=(2^x)\cdot(3^b)\cdot(11^c)\cdot(23^d)\quad\text{ since }\quad p-1 \mid \phi(n)=... | David | 119,775 | <p>If
$$n=\prod_{p\,{\rm prime}}p^{\alpha(p)}\ ,$$
then you need
$$\prod_{p\,{\rm prime}}p^{\alpha(p)-1}(p-1)=110\ .$$
Since $11\mid110$ you must have $p=11$ as one of the factors on the LHS. (It can't be $p-1=11$ as $12$ is not prime.). Then the exponent must be $\alpha(11)-1=1$ since $11^2\not\mid110$, and the $p-1... |
2,335,627 | <p>Is the following statement true? If so, how can I prove it?</p>
<blockquote>
<p>If the power series $\sum_{n=0}^{\infty} a_n x^n$ converges for all $x
\in (x_0, x_1)$ then it converges absolutely for all $x \in
(-\max\{|x_0|, |x_1|\}, \max\{|x_0|, |x_1|\})$.</p>
</blockquote>
| hamam_Abdallah | 369,188 | <p>All power series are based on this</p>
<p>$$0 <a <b\implies 0 < \frac {a}{b}<1$$</p>
<p>and</p>
<p>$|q|<1\implies \sum q^n $ converges.</p>
<p>The rest is artistic.</p>
|
509,236 | <p>I had previously solved the problem of proving that $n^3-n-4$ must be a multiple of $5$, given that $n-3$ is a multiple of $5$. I did so by algebraically manipulating $n^3-n-4$ into:</p>
<p>$$
2(n-3)+(n-1)(n-1)((n-3)+5)
$$</p>
<p>Given that the first term is a given multiple of $5$, and the second term is a produ... | Brian M. Scott | 12,042 | <p>Write $n$ in the form $5k+r$, where $r\in\{0,1,2,3,4\}$. Show that $n^3-r^3$ is a multiple of $5$, as of course is $n-r$, so that</p>
<p>$$(n^3-n-4)-(r^3-r-4)=(n^3-r^3)-(n-r)$$</p>
<p>is a multiple of $5$. Thus, if $n^3-n-4$ is a multiple of $5$, so is $r^3-r-4$. Now show that $3$ is the only value of $r$ for whic... |
509,236 | <p>I had previously solved the problem of proving that $n^3-n-4$ must be a multiple of $5$, given that $n-3$ is a multiple of $5$. I did so by algebraically manipulating $n^3-n-4$ into:</p>
<p>$$
2(n-3)+(n-1)(n-1)((n-3)+5)
$$</p>
<p>Given that the first term is a given multiple of $5$, and the second term is a produ... | Cameron Buie | 28,900 | <p>We can write $n=5k+r,$ where $r\in\{0,1,2,3,4\}$ by division with remainder. Then $$\begin{align}n^3-n-4 &= (5k+r)^3-(5k+r)-4\\ &= (125k^3+75k^2r+15kr^2+r^3)-5k-r-4\\ &= 5(25k^3+15k^2r+3kr^2-5k)+r^3-r-4.\end{align}$$ All that remains, then, is to show that $r^3-r-4$ is not a multiple of $5$ when $r=0,1,2... |
2,006,676 | <p>I'm trying to show that for a non-empty set $X$, the following statements are true for logical statements $P(x)$ and $Q(x)$:</p>
<ul>
<li>$∃x∈X$, ($P(x)$ or $Q(x)$) $\iff$ $(∃x∈X, P(x))$ or $(∃x∈X, Q(x))$</li>
<li>$∃x∈X$, ($P(x)$ and $Q(x)$) $\implies$ $(∃x∈X, P(x))$ and $(∃x∈X, Q(x))$</li>
</ul>
<p>Is it possib... | rschwieb | 29,335 | <p>$T$ is the complement of the prime ideal $(x)$, so you are just localizing at a prime ideal, and therefore $T^{-1}R$ is a local ring. The units are those things outside the unique maximal ideal. Describe those elements.</p>
|
3,167,062 | <p>Recently friend of mine showed me a way to write <span class="math-container">$f(x)=0$</span>, however I could not find anything about that in the Internet. Is it correct?</p>
<p><span class="math-container">$f(x)=x^2-1\stackrel{set}{=}0 \\
x = 1 \lor x = -1$</span></p>
| Lazar Ljubenović | 24,269 | <p>Apparently, this is known as <strong>perp product</strong>; however, the only online reference I could find after a (rather quick) search is <a href="http://geomalgorithms.com/vector_products.html" rel="nofollow noreferrer">this <em>geomalgorithms</em> page</a>. It's also <a href="http://mathworld.wolfram.com/PerpDo... |
2,159,136 | <p>I arrived to this question while solving a question paper. The question is as follows:</p>
<blockquote>
<p>If $f_k(x)=\frac{1}{k}\left(\sin^kx + \cos^kx\right)$, where $x$ belongs to $\mathbb{R}$ and $k>1$, then $f_4(x)-f_6(x)=?$</p>
</blockquote>
<p>I started as </p>
<p>$$\begin{align}
f_4(x)-f_6(x)&=\f... | kotomord | 382,886 | <p>Hint.
Let $x=\frac{\pi}{4}$</p>
<p>In your answer $f4(x)-f(6x) = 0$</p>
<p>By brute-forse, $f4(x)-f6(x) = \frac{0.5}{4} - \frac{0.25}{6} = \frac{1}{8}-\frac{1}{24} =\frac{1}{12}$, so, you make a typo :(</p>
|
2,159,136 | <p>I arrived to this question while solving a question paper. The question is as follows:</p>
<blockquote>
<p>If $f_k(x)=\frac{1}{k}\left(\sin^kx + \cos^kx\right)$, where $x$ belongs to $\mathbb{R}$ and $k>1$, then $f_4(x)-f_6(x)=?$</p>
</blockquote>
<p>I started as </p>
<p>$$\begin{align}
f_4(x)-f_6(x)&=\f... | Fred | 380,717 | <p>Hint: let $f(x):=f_4(x)-f_6(x)$. Then show that $f'(x)=0$ for all $x$. Hence $f$ is constant. Furthermore: $f(0)=\frac{1}{12}$</p>
|
2,159,136 | <p>I arrived to this question while solving a question paper. The question is as follows:</p>
<blockquote>
<p>If $f_k(x)=\frac{1}{k}\left(\sin^kx + \cos^kx\right)$, where $x$ belongs to $\mathbb{R}$ and $k>1$, then $f_4(x)-f_6(x)=?$</p>
</blockquote>
<p>I started as </p>
<p>$$\begin{align}
f_4(x)-f_6(x)&=\f... | lab bhattacharjee | 33,337 | <p>HINT:</p>
<p>$$\sin^6x+\cos^6x=(\sin^2x+\cos^2x)^3-3\sin^2x\cos^2x(\sin^2x+\cos^2x)=1-3\sin^2x\cos^2x$$</p>
<p>$$\sin^4x+\cos^4x=(\sin^2x+\cos^2x)^2-2\sin^2x\cos^2x=1-2\sin^2x\cos^2x$$</p>
|
1,629,592 | <p>I tried to prove this question by first considering the possible last digit of $p$ when $p=n^2+5$, but that reasoning got me nowhere. Then I tried to prove it by contrapositive, and however I just couldn't really find where to start.<br>
Hence I'm here asking for some hints (only hints, no solution please).</p>
<p>... | quid | 85,306 | <p>Consider the last digits of squares, so $n^2$, these are $0,1,4,5,6,9$. </p>
<p>From there you get the last digits of $n^2 +5$. </p>
<p>Now, drop those from the list that cannot be the last digit of a prime. </p>
|
1,629,592 | <p>I tried to prove this question by first considering the possible last digit of $p$ when $p=n^2+5$, but that reasoning got me nowhere. Then I tried to prove it by contrapositive, and however I just couldn't really find where to start.<br>
Hence I'm here asking for some hints (only hints, no solution please).</p>
<p>... | John | 7,163 | <p>Hint: The possible values of $n^2$ (mod $10$) are $0, 1, 4, 5, 6, 9$. Which ones can you eliminate since $n^2 + 5$ is not prime?</p>
|
58,912 | <p>In the board game <a href="http://en.wikipedia.org/wiki/Hex_%28board_game%29" rel="nofollow">Hex</a>, players take turns coloring hexagons either red or blue. One player tries to connect the top and bottom edges of the board, colored red; the other tries to connect the left and right edges, colored blue. It is known... | Tanner Swett | 13,524 | <p>An answer to this came to me today as I was thinking about how I might describe connectedness to a layperson. Path-connectedness is easy enough: a set is path-connected if, given two points in the set, you can get from one to the other by following a path within the set. If I'm not mistaken, connectedness in the pla... |
638,922 | <p>If $q \in \mathbb{H}$ satisfies $qi = iq$, prove that $q \in \mathbb{C}$</p>
<p>This seems kinda of intuitive since quaternions extend the complex numbers. I am thinking that $q=i$ because i know that $ij = k , ji = -k$, which is expand to all combinations of $i,j,k,$ which I think means that I have to use $ijk = i... | David Holden | 79,543 | <p>let $q=a+bi+cj+dk = (a,b,c,d)$ so</p>
<p>$$
qi = (-b,a,d,-c) \\
iq = (-b,a,-d,c)
$$</p>
|
638,922 | <p>If $q \in \mathbb{H}$ satisfies $qi = iq$, prove that $q \in \mathbb{C}$</p>
<p>This seems kinda of intuitive since quaternions extend the complex numbers. I am thinking that $q=i$ because i know that $ij = k , ji = -k$, which is expand to all combinations of $i,j,k,$ which I think means that I have to use $ijk = i... | DonAntonio | 31,254 | <p>If you put $\;q=a+bi+cj+dk\;$ , then</p>
<p>$$\begin{align*}qi=ai-b-ck+dj\\
iq=ai-b+ck-dj\end{align*}$$</p>
<p>Well, what do you deduce about the coefficients $\;a,b,c,d\in\Bbb R\;$ above?</p>
|
3,995,913 | <p>We are looking at the following expression:</p>
<p><span class="math-container">$$\frac{d}{dx}\int_{u(x)}^{v(x)}f(x) dx$$</span></p>
<p>The solution is straightforward for this: <span class="math-container">$\frac{d}{dx}\int_{u(x)}^{v(x)}f(t) dt$</span>. Do we evaluate the given expression in like manner? Do we trea... | EDX | 763,728 | <p>Be aware of the name of variables !</p>
<p>The <span class="math-container">$x$</span> in <span class="math-container">$v(x)$</span> is not the same as <span class="math-container">$dx$</span> lets call it <span class="math-container">$dt$</span></p>
<p>Consider your integral as a primitive</p>
<p><span class="math-... |
1,470,378 | <p>I am not sure i formatted it right but I was studying for calculus and came across a problem I couldn't compute. </p>
<p>$$\int \frac{x^2}{x^2-2}dx$$</p>
<p>I have not learned partial fractions yet so if this is a case where that is used, the techniques might not work for me. </p>
<p>What I have tried. to do i... | Empty | 174,970 | <p><strong>Hint 1 :</strong></p>
<p>Put , $x=\sqrt 2\sec \theta$. Then , $\,dx=\sqrt 2 \sec\theta \tan \theta$.</p>
<p><strong>Hint 2:</strong></p>
<p>$\frac{x^2}{x^2-2}=1+\frac{2}{x^2-2}$</p>
|
4,371,541 | <p>The series representation <span class="math-container">$\sum1/n^s$</span> of the Riemann zeta function <span class="math-container">$\zeta(s)$</span> is known to converge for <span class="math-container">$\sigma>1$</span>, where <span class="math-container">$s=\sigma+it$</span>.</p>
<p>In order to show an infinit... | reuns | 276,986 | <p>As you said the series converges uniformly on <span class="math-container">$\Re(s) \ge 1+\epsilon$</span> (so that it is analytic there, by say Morera's theorem) and only locally uniformly on <span class="math-container">$\Re(s) >1$</span> (so that it is analytic there as well).
Analyticity is also immediate from... |
1,383,613 | <p>Give recursive definition of sequence $a_n = 2^n, n = 2, 3, 4... where $ $a_1 = 2$</p>
<p>I'm just not sure how to approach these problems. </p>
<p>Then it asks to give a def for:</p>
<p>$a_n = n^2-3n, n = 0, 1, 2...$</p>
<p>Thanks for all the help!</p>
| David | 119,775 | <p>Note that there are lots of answers to these questions. IMO those given by Terra Hyde are probably what your instructor is expecting, but you could also say:</p>
<p>for the first one,
$$a_{n+1}=a_n+2^n$$
and for the second
$$a_{n+1}=a_n+a_{n-1}-n^2+7n-6$$
among many, many other possibilities.</p>
|
4,102,035 | <p>I'm trying to solve this question and reach a proof, I have already proven that <span class="math-container">$P(B|A)>P(B)$</span>, and <span class="math-container">$P(B^c|A)<P(B^c)$</span>, using the information from the question, but I am struggling to find out how to prove that: <span class="math-container">... | Rahul Madhavan | 439,353 | <p>You have said that you already showed <span class="math-container">$P(B|A)>P(B)$</span>. I will use this.
<span class="math-container">\begin{align*}
P(B|A)&>P(B)\\
\frac{P(B\cap A)}{P(A)}&>P(B)\\
P(B\cap A)&>P(B)P(A)\tag 1
\end{align*}</span>
Now consider <span class="math-container">$P(A^C)... |
4,102,035 | <p>I'm trying to solve this question and reach a proof, I have already proven that <span class="math-container">$P(B|A)>P(B)$</span>, and <span class="math-container">$P(B^c|A)<P(B^c)$</span>, using the information from the question, but I am struggling to find out how to prove that: <span class="math-container">... | João Víctor Melo | 852,373 | <p>A solution could be the following:</p>
<p><span class="math-container">$$P(B^c | A^c) = P(A^c \cap B^c)/P(A^c) = [1-(P(A)+P(B)-P(A \cap B))]/P(A^c) $$</span></p>
<p>Now we observe that the above value is bigger than:</p>
<p><span class="math-container">$$[1-P(A) - P(B) + P(A)\times P(B)]/P(A^c)$$</span></p>
<p>since... |
4,117,377 | <p>In calculating the integral</p>
<p><span class="math-container">$$\int_{-\infty}^{\infty} \frac{\sin x}{x}dx$$</span></p>
<p>by contour integration, we use</p>
<p><span class="math-container">$$\int_{-\infty}^{\infty} \frac{\sin x}{x}dx = \int_{-\infty}^{\infty} \frac{\operatorname{Im}(e^{ix})}{x}dx =\operatorname{I... | saulspatz | 235,128 | <p>The value of the improper integral is <span class="math-container">$$\lim\limits_{\substack{z\to&\,\,\,\infty \\ y\to&-\infty}}\int_y^z\frac{\sin x}{x}\,\mathrm{d}x$$</span> Clearly, if this limit exists, then so does the Cauchy principal value, and it has the same value as the integral.</p>
<p>I would appr... |
4,117,377 | <p>In calculating the integral</p>
<p><span class="math-container">$$\int_{-\infty}^{\infty} \frac{\sin x}{x}dx$$</span></p>
<p>by contour integration, we use</p>
<p><span class="math-container">$$\int_{-\infty}^{\infty} \frac{\sin x}{x}dx = \int_{-\infty}^{\infty} \frac{\operatorname{Im}(e^{ix})}{x}dx =\operatorname{I... | Community | -1 | <p>I don't think there is a way in which we can rigorously use the expression <span class="math-container">$$\operatorname{Im}\left(\int_{-\infty}^{\infty} \frac{e^{ix}}{x}dx \right)$$</span></p>
<p>If you test the following in wolfram alpha for example</p>
<ul>
<li><code>integral from 0.0000001 to inf of e^(ix)/x dx</... |
1,548,841 | <p>As you can see, In the image a rectangle gets translated to another position in the coordinates System.</p>
<p>The origin Coordinates are <code>A1(8,2) B1(9,3)</code> from the length <code>7</code> and the height <code>3</code> you can also guess the vertices of the rectangle.</p>
<p><b>Now the Rectangle gets move... | mathlove | 78,967 | <p>This answer uses complex numbers.</p>
<p>We want to find $a,b,\theta\in\mathbb R$ where $0\lt\theta\lt \pi$ such that
$$(16+9i)-(a+bi)=\sqrt 2(8+2i)(\cos\theta+i\sin\theta)$$
$$(16+11i)-(a+bi)=\sqrt 2(9+3i)(\cos\theta+i\sin\theta)$$
Then, solving
$$16-a=\sqrt 2(8\cos\theta-2\sin\theta)$$
$$9-b=\sqrt 2(8\sin\theta+2... |
567,668 | <p>In my analysis class we are discussing integrable functions and I need help thinking through something. I don't want any answers, just what should I be looking at? (hints)</p>
<p><img src="https://i.stack.imgur.com/qKte6.png" alt="enter image description here"></p>
| user729424 | 729,424 | <p>If <span class="math-container">$G_1$</span>, <span class="math-container">$G_2$</span> have order <span class="math-container">$p$</span>, for some prime <span class="math-container">$p$</span>, then <span class="math-container">$G_1\times G_2$</span> will have <span class="math-container">$p+3\ge5$</span> subgroup... |
1,789,742 | <p>I'm reading the book generatingfunctionology by Herbert Wilf and I came across a partial fraction expansion on page 20 that I cannot understand. The derivation is as follows:</p>
<p>$$
\frac{1}{(1-x)(1-2x)...(1-kx)} = \sum_{j=1}^{k} \frac{\alpha_j}{1-jx}
$$</p>
<p>The book says to fix $r, 1 \leq r \leq k$, and mul... | John Wayland Bales | 246,513 | <p>It appears that the $\alpha_j$ coefficients are being computed by the standard Heaviside method (sometimes called the "cover up" method).</p>
<p><a href="https://en.wikipedia.org/wiki/Heaviside_cover-up_method" rel="nofollow">Heaviside's Method for partial fraction expansion</a></p>
|
1,789,742 | <p>I'm reading the book generatingfunctionology by Herbert Wilf and I came across a partial fraction expansion on page 20 that I cannot understand. The derivation is as follows:</p>
<p>$$
\frac{1}{(1-x)(1-2x)...(1-kx)} = \sum_{j=1}^{k} \frac{\alpha_j}{1-jx}
$$</p>
<p>The book says to fix $r, 1 \leq r \leq k$, and mul... | Ismail Bello | 267,865 | <p>To find the coefficients of the expansion, you can follow the usual "cover" up rule which is essentially the results you have shown albeit buried in indices and what not: $$
\frac{1}{(1-x)(1-2x)...(1-kx)} = \sum_{j=1}^{k} \frac{\alpha_j}{1-jx}
$$
say we would want to find an arbitrary $\alpha_j$ for some value of $... |
1,686,904 | <blockquote>
<p>How can we prove that <span class="math-container">$$\binom{n}{1}\binom{n}{2}^2\binom{n}{3}^3\cdots \binom{n}{n}^n \leq \left(\frac{2^n}{n+1}\right)^{\binom{n+1}{2}}$$</span></p>
</blockquote>
<p><span class="math-container">$\bf{My\; Try::}$</span> Using <span class="math-container">$\bf{A.M\geq G.M\;,... | Macavity | 58,320 | <p>You have the right approach, just need a small detour. </p>
<p><strong>Hint</strong> Start with:
$$\sum_k k \binom{n}{k} = n 2^{n-1} \tag{why?}$$</p>
<p>Now apply AM-GM and watch your exponentiation.</p>
|
244,489 | <p>Given:</p>
<p>${AA}\times{BC}=BDDB$</p>
<p>Find $BDDB$:</p>
<ol>
<li>$1221$</li>
<li>$3663$</li>
<li>$4884$</li>
<li>$2112$</li>
</ol>
<p>The way I solved it:</p>
<p>First step - expansion & dividing by constant ($11$):
$AA\times{BC}$=$11A\times{BC}$</p>
<ol>
<li>$1221$ => $1221\div11$ => $111$</li>
<li>$3... | aditsu quit because SE is EVIL | 51,012 | <p>For the "square in triangle" subproblem, I think the solution always looks like <img src="https://i.stack.imgur.com/MAqih.png" alt="test"> </p>
<p>where $0 \leq a \leq b$. If $a > b$, then just flip the figure around the $x=y$ line.</p>
<p>With the notations in the figure (if you don't like them, you can file a... |
128,408 | <p>The following:</p>
<pre><code>mat = {{1., -3, -2}, {-1, 3, -2}, {0., 0., 0.}}
MatrixFunction[Sinc, mat]
</code></pre>
<p>returns:</p>
<pre><code>MatrixFunction::drvnnum: Unable to compute the matrix function because
the subsequent derivative of the function Sinc[#1] & at a numeric value is
not numeric.
</co... | J. M.'s persistent exhaustion | 50 | <p>Two things come into play:</p>
<ol>
<li><p><code>Sinc'[0]</code> yields <code>Indeterminate</code>, when it should be $0$.</p></li>
<li><p>This value needs to be evaluated in the computation of <code>MatrixFunction[Sinc, mat]</code>.</p></li>
</ol>
<p>In details: for the computation of <code>MatrixFunction[]</code... |
128,408 | <p>The following:</p>
<pre><code>mat = {{1., -3, -2}, {-1, 3, -2}, {0., 0., 0.}}
MatrixFunction[Sinc, mat]
</code></pre>
<p>returns:</p>
<pre><code>MatrixFunction::drvnnum: Unable to compute the matrix function because
the subsequent derivative of the function Sinc[#1] & at a numeric value is
not numeric.
</co... | Dr. Wolfgang Hintze | 16,361 | <p><strong>EDIT #2</strong></p>
<p>The regularization method is more generally applicable than the Cauchy method. </p>
<p>Here's an example for which (according to J.M.) the Cauchy method fails but the regularization method works.</p>
<p>The matrix is that of the OP (integer form)</p>
<pre><code>matx = {{1, -3, -2}... |
3,304,542 | <p>Let <span class="math-container">$X,Y$</span> be Banach Spaces, show that <span class="math-container">$x_{n} \xrightarrow{w} x$</span> and <span class="math-container">$T \in BL(X,Y)\Rightarrow Tx_{n} \xrightarrow{w} Tx$</span></p>
<p>Question: Does sequential continuity (which <span class="math-container">$T$</sp... | S. Maths | 515,871 | <p>As in the previous answer you can prove sequential continuity. A stronger result is the following: <strong>bounded linear operators are continuous in the weak topology.</strong></p>
<p>Remark that the weak topology is not metrizable (no sequential characterization of continuity). </p>
|
3,522,681 | <blockquote>
<p>There are 8 people in a room. There are 4 males(M) and 4 females(F). What is the probability that there are no M-F pairs that have the same birthday ? It is OK for males to share a birthday and for females to share a birthday. Assume there are <span class="math-container">$10$</span> total birthdays.... | ironX | 534,898 | <p>Let <span class="math-container">$A$</span> be the event no M-F pair share the same birthday. Let <span class="math-container">$B_1$</span> be the event all females share ONE birthday. </p>
<p>Let <span class="math-container">$N(A \cap B_1)$</span> be the number of possible configurations realizing the event <span ... |
2,812,472 | <p>I am trying to show that</p>
<p>$$
\frac{d^n}{dx^n} (x^2-1)^n = 2^n \cdot n!,
$$ for $x = 1$. I tried to prove it by induction but I failed because I lack axioms and rules for this type of derivatives. </p>
<p>Can someone give me a hint?</p>
| Logic_Problem_42 | 338,002 | <p>Since $x^2-1=(x-1)(x+1)$ one have to differentiate $(x-1)^n(x+1)^n$. It's clear that the result would have the form $\sum a_{km}(x-1)^k(x+1)^m$ with $k,m\le n$. If we insert $x=1$, then all summands with $k>0$ became zero. So it remains to consider the summands of the form $\sum_{k=0} a_{km}(x+1)^m$. There is act... |
1,060,213 | <p>I have searched the site quickly and have not come across this exact problem. I have noticed that a Pythagorean triple <code>(a,b,c)</code> where <code>c</code> is the hypotenuse and <code>a</code> is prime, is always of the form <code>(a,b,b+1)</code>: The hypotenuse is one more than the non-prime side. Why is this... | Ross Millikan | 1,827 | <p>You don't need the formulas for generating Pythagorean triples. If you have $a^2=c^2-b^2=(c+b)(c-b)$ with $a$ prime, the only way to factor $a^2$ with distinct factors is $a^2 \cdot 1$, so $c-b=1, c+b=a^2$ </p>
<p>Added: this shows how to find all the triangles with a side of $a$, prime or not. You factor $a^2$... |
2,197,915 | <p>Suppose $X$ is a scheme. I have been studying (finite rank) locally free $\mathcal{O}_{X}$-modules, and more generally, quasi-coherent sheaves on $X$ mainly from Ravi Vakil's excellent <a href="http://math.stanford.edu/~vakil/216blog/FOAGfeb0717public.pdf" rel="noreferrer">notes</a> as well as Hartshorne. Let $\math... | Alex Mathers | 227,652 | <p>I don't think this is true. There exist (finitely generated) $A$-modules $M$ which are locally free but not free themselves, which means that $\widetilde M$ is a locally free $\mathcal O_{Spec(A)}$-module on $Spec(A)$, but not a free $\mathcal O_{Spec(A)}$-module. That is, there is a cover of $Spec(A)$ by affine ope... |
2,197,915 | <p>Suppose $X$ is a scheme. I have been studying (finite rank) locally free $\mathcal{O}_{X}$-modules, and more generally, quasi-coherent sheaves on $X$ mainly from Ravi Vakil's excellent <a href="http://math.stanford.edu/~vakil/216blog/FOAGfeb0717public.pdf" rel="noreferrer">notes</a> as well as Hartshorne. Let $\math... | Georges Elencwajg | 3,217 | <p>It is just not true that given a locally free sheaf $\mathcal G$ of rank $r$ and an arbitrary affine open subset $U\subset X$ the restriction $\mathcal G\vert U$ is isomorphic to the free $\mathcal O_U$-Module $\mathcal O_U^{\oplus r}$.<br>
Here is a counterexample: </p>
<p>Consider a smooth projective curve $\ove... |
332,001 | <p><a href="https://en.wikipedia.org/wiki/Cauchy%27s_theorem_(geometry)#Generalizations_and_related_results" rel="nofollow noreferrer">Wikipedia states</a> that A. D. Alexandrov generalized <em>Cauchy's rigidity theorem for polyhedra</em> to higher dimensions. </p>
<p>The relevant statement in the article is not linke... | Joseph O'Rourke | 6,094 | <p>This may help:</p>
<blockquote>
<p>Bauer, C.
"Infinitesimal Rigidity of Convex Polytopes."
<em>Discrete Comput Geom</em> (1999) 22: 177. <a href="https://doi.org/10.1007/PL00009453" rel="noreferrer">https://doi.org/10.1007/PL00009453</a></p>
</blockquote>
<p>"Aleksandrov [1] proved that a simple convex <spa... |
3,677,303 | <p>Is there an example of topological space being both sequentially compact and Lindelof but noncompact?</p>
| J.-E. Pin | 89,374 | <p>There is no such space. Indeed,
every <a href="https://en.wikipedia.org/wiki/Sequentially_compact_space" rel="nofollow noreferrer">sequentially compact space</a> is <a href="https://en.wikipedia.org/wiki/Countably_compact_space" rel="nofollow noreferrer">countably compact</a>. And a countably compact space is compac... |
3,677,303 | <p>Is there an example of topological space being both sequentially compact and Lindelof but noncompact?</p>
| MAS | 417,353 | <p>We know every sequentially compact space is Countably compact. And we also know countably compact <span class="math-container">$+$</span> Lindeloff space means Compact space. So we don't have an example of your question. </p>
<p>See, page <span class="math-container">$2$</span> graph, <a href="https://cpb-us-w2.wpm... |
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