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 |
|---|---|---|---|---|
740,323 | <p>This might be a little open-ended, but I was wondering: are there any natural connections between geometry and the prime numbers? Put differently, are there any specific topics in either field which might entertain relatively close connections?</p>
<p><strong>PS:</strong> feel free to interpret the term <em>natural... | Georgy | 139,717 | <p>Well, prime numbers are strongly related to the Riemann zeta function, $\zeta(s)$. This has a product representation which involves the roots of the function. The Riemann Hypothesis now states that all non-trivial roots in the complex plane lie on the "critical line":
$$Re(z)=\frac12$$
which can be thought of as a g... |
740,323 | <p>This might be a little open-ended, but I was wondering: are there any natural connections between geometry and the prime numbers? Put differently, are there any specific topics in either field which might entertain relatively close connections?</p>
<p><strong>PS:</strong> feel free to interpret the term <em>natural... | Patrick Das Gupta | 344,568 | <p>How about another imagery about prime numbers? A prime number $p = 1 \times p$ and hence, geometrically it is like a one-dimensional segment. On the other hand, a composite number $c= a \times b$, where $a$ and $b$ are its prime factors, is like a rectangle having an area $c$, with side lengths $a$ and $b$ . So,... |
740,323 | <p>This might be a little open-ended, but I was wondering: are there any natural connections between geometry and the prime numbers? Put differently, are there any specific topics in either field which might entertain relatively close connections?</p>
<p><strong>PS:</strong> feel free to interpret the term <em>natural... | Alexander Bogomolny | 42,291 | <p>A simple example: number p>2 is prime iff any equiangular p-gon with rational side lengths is regular, see, e.g., <a href="http://www.cut-the-knot.org/Outline/Geometry/EquiangularP-gon.shtml" rel="nofollow noreferrer">http://www.cut-the-knot.org/Outline/Geometry/EquiangularP-gon.shtml</a></p>
|
740,323 | <p>This might be a little open-ended, but I was wondering: are there any natural connections between geometry and the prime numbers? Put differently, are there any specific topics in either field which might entertain relatively close connections?</p>
<p><strong>PS:</strong> feel free to interpret the term <em>natural... | Елена Михалькова | 623,221 | <p>If I understand your question correctly, there are several geometrical models for primes (some already mentioned above), that connect algebraic deductions to geometric representations:
1. The Sieve of Eratosthenes, usually depicted on a grid of <span class="math-container">$10x \times 10y$</span>, but can be made cy... |
740,323 | <p>This might be a little open-ended, but I was wondering: are there any natural connections between geometry and the prime numbers? Put differently, are there any specific topics in either field which might entertain relatively close connections?</p>
<p><strong>PS:</strong> feel free to interpret the term <em>natural... | gnuvo | 942,086 | <p>Let there be an infinite triangular lattice. Sequentially enumerate the triangles in a constant direction and a 6 cell hexagon forms. However, if one enumerates the cells such that each time a prime number is encountered the direction of fill switches, the number line exits the 6 cell, but forms a new hexagon (24 ... |
595,865 | <p>A real number $r$ belonging to $[0,1]$ is said to be computable if there is a simple TM such that for each binary encoding of $n$ ($n$ is a natural number), returns the $n$th bit in the binary expansion of $r$. Give a proof that there are non-computable real numbers.</p>
| Xoff | 36,246 | <p>By a cardinal argument :</p>
<ol>
<li>You have $2^{\aleph_0}$ reals in $[0;1]$</li>
<li>You have $\aleph_0$ Turing machines.</li>
<li>There is no bijection between $\aleph_0$ and $2^{\aleph_0}$</li>
</ol>
|
915,218 | <p>I'd like some help figuring out how to calculate $n$ points of the form $(x,\sin(x))$ for $x\in[0,2\pi)$, such that the Cartesian distance between them (the distance between each pair of points if you draw a straight line between them) is the same.</p>
<p><strong>My background:</strong> I know math up to and throug... | ratchet freak | 17,442 | <p>It is impossible to have more than 3 points that are equidistant.</p>
<p>Having 3 points equidistant on a sine wave is impossible as it requires the points to form an equilateral triangle, and at least one of the sides would need a slope steeper than $60°$ and the steepest slope on a sine wave is $45°$.</p>
<p>The... |
4,126,124 | <p>I managed to come to the end of a proof regarding a determinant of a linear operator. However, I stuck at the end. I know it is kinda simple but I couldn't see it right away.</p>
<p>Here is my previous work:</p>
<p><span class="math-container">$F$</span> is a field. <span class="math-container">$V = M_{nxn}(F)$</spa... | Henno Brandsma | 4,280 | <p>There is a mapping from the set of metrics on <span class="math-container">$X$</span> to the set of topologies on <span class="math-container">$X$</span>: to a metric assign the topology <span class="math-container">$\mathcal{T}(d)$</span> generated by <span class="math-container">$d$</span>. A topology <span class=... |
819,805 | <p>The polynomial remainder theorem states that when a polynomial $P(x)$ of degree $> 0$ is divided by $x-r$ ($r$ being some constant) the remainder is equal to $P(r)$, that is:<br>
$$\begin{array}l If & \quad P(x) = (x-r)Q(x)+R \\ then & \quad P(r) = R \end{array}$$
The algebra and the graphic representatio... | Hakim | 85,969 | <p>Polynomial long division gives a solution of the equation</p>
<p>$$f(x)=q(x)g(x) + r(x)\,,$$</p>
<p>where the degree of $r(x)$, is less than that of $g(x)$.</p>
<p>If we take $g(x) = x-a$ as the divisor, giving the degree of $r(x)$ as $0$, i.e. $r(x) = R$:</p>
<p>$$f(x)=q(x)(x-a) + R\,.$$</p>
<p>Setting $x=a $,... |
291,284 | <p>Let $(\Omega, \cal{A}, \mathbb{P})$ be a probability space and $X$ a random variable on $\Omega$. Let, also, $f:\Omega\to\mathbb{R}$ be a Borel function. Then:<br>
$X$ and $f(X)$ are independent $\Longleftrightarrow$ there exists some $t\in\mathbb{R}$ such that $\mathbb{P}[f(X)=t]=1$, that is $f(X)$ is a degenerate ... | Did | 6,179 | <blockquote>
<p>If the random variable $Y$ is independent of itself, then there exists $y$ such that $Y=y$ almost surely.</p>
</blockquote>
<p>To prove this, consider the CDF $F_Y:x\mapsto\mathbb P(Y\leqslant x)$ and note that for every $x$ the event $[Y\leqslant x]$ is independent of itself hence $F_Y(x)$ is $0$ or... |
3,921,335 | <p>how can I formally proof that for a > 1: <span class="math-container">$$ a> \sqrt a > \sqrt[3]a > \sqrt[4]a ... $$</span>?
Can someone help? ;)</p>
| Mostafa Ayaz | 518,023 | <p>To show <span class="math-container">$\sqrt[n]a>\sqrt[n+1]a$</span>, raise both sides to <span class="math-container">$n(n+1)$</span> to obtain
<span class="math-container">$$
a^{n+1}>a^n
$$</span>
and prove the latter inequality from <span class="math-container">$a>1$</span>.</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... | Igor Rivin | 109,865 | <p>Any complex number $q$ satisfies $q i = i q,$ so it is not true that $q=i$ However, just writing $q = a + b i + c j + d k,$ and writing down $q i$ and $i q$ ought to do the trick.</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... | OgvRubin | 468,471 | <p>You should write <span class="math-container">$\int_{u(x)}^{v(x)}f(t)dt$</span>. Then use the fundamental theorem of calculus by first rewriting</p>
<p><span class="math-container">$$\int_{u(x)}^{v(x)}f(t)dt = \int_{c}^{v(x)}f(t)dt+\int_{u(x)}^{c}f(t)dt=\int_{c}^{v(x)}f(t)dt-\int_{c}^{u(x)}f(t)dt$$</span>
for any co... |
3,868,523 | <blockquote>
<p>Given the bases <em>a</em> = {(0,2),(2,1)} and <em>b</em> = {(1,0),(1,1)} compute the change of coordinate matrix from basis <em>a</em> to <em>b</em>.<br />
Then, given the coordinates of <em>z</em> with respect to the basis <em>a</em> as (2,2), use the previous question to compute the coordinates of <e... | Community | -1 | <p>What you can do is use the changes of coordinates between each basis and the standard basis. So, you want: <span class="math-container">$A=\begin{pmatrix}0&2\\2&1\end{pmatrix}$</span> and <span class="math-container">$B=\begin{pmatrix}1&1\\0&1\end{pmatrix}$</span>.</p>
<p>Then what is normally calle... |
3,868,523 | <blockquote>
<p>Given the bases <em>a</em> = {(0,2),(2,1)} and <em>b</em> = {(1,0),(1,1)} compute the change of coordinate matrix from basis <em>a</em> to <em>b</em>.<br />
Then, given the coordinates of <em>z</em> with respect to the basis <em>a</em> as (2,2), use the previous question to compute the coordinates of <e... | tossimmar | 835,571 | <p>Let <span class="math-container">$\mathcal{B} = \{b_1, b_2\}$</span> and <span class="math-container">$\mathcal{B}^{\prime} = \{b_1^{\prime}, b_2^{\prime}\}$</span> be two bases.</p>
<p>If <span class="math-container">$v$</span> is a vector expressed with respect to <span class="math-container">$\mathcal{B}$</span>,... |
2,671,384 | <p>I want to solve following difference equation:</p>
<p>$a_i = \frac13a_{i+1} + \frac23a_{i-1}$, where $a_0=0$ and $a_{i+2} = 1$</p>
<p><strong>My approach:</strong>
Substituting $i=1$ in the equation,<br>
$3a_1 = a_2 + 2a_0$ <br>
$a_2 = 3a_1$<br>
Similarly substituting $i = 2, 3 ...$<br>
$a_3 = 7a_1$<br>
$a_4 = 15a... | Ian | 83,396 | <p>If you proceed this way, $a_1$ is an unknown, to be solved for by substituting in the boundary condition at the other endpoint and solving for $a_1$. (This is actually a fairly common trick, for instance it is also used to derive the invariant distribution of a birth-death process by writing all the $\pi_i$'s in ter... |
4,645,440 | <blockquote>
<p>Question: Let <span class="math-container">$X$</span> be a random variable then is the statement <span class="math-container">$\mathbb{P}(B|\{X=i\})$</span> = 0.6(random number), equivalent of saying: <span class="math-container">$X = i \implies \mathbb{P}(B) = 0.6$</span>?</p>
</blockquote>
<blockquote... | drhab | 75,923 | <p>If you would have said something like:</p>
<p>"Based on the info that random variable <span class="math-container">$X$</span> has taken value <span class="math-container">$i$</span> we conclude that the probability of occurrence of event <span class="math-container">$B$</span> is now <span class="math-container... |
4,645,440 | <blockquote>
<p>Question: Let <span class="math-container">$X$</span> be a random variable then is the statement <span class="math-container">$\mathbb{P}(B|\{X=i\})$</span> = 0.6(random number), equivalent of saying: <span class="math-container">$X = i \implies \mathbb{P}(B) = 0.6$</span>?</p>
</blockquote>
<blockquote... | ryang | 21,813 | <p><span class="math-container">$$P(B|A)=k\tag1$$</span> <span class="math-container">$$A \text{ happens}\implies P(B)=k\tag2$$</span></p>
<p>It is false that for every pair of events <span class="math-container">$A$</span> and <span class="math-container">$B$</span> and every <span class="math-container">$k\in[0,1],$<... |
3,379,899 | <p>We have two coins. The first one is a fair coin and has <span class="math-container">$50\%$</span> chance of landing on head and <span class="math-container">$50\%$</span> of landing on tails. for the other one it is <span class="math-container">$60\%$</span> for head and <span class="math-container">$40\%$</span> f... | David G. Stork | 210,401 | <p>Call the fair coin <span class="math-container">$A$</span> and the biased coin <span class="math-container">$B$</span>.</p>
<p><span class="math-container">$$P(THT|A) = (.5)^3$$</span></p>
<p><span class="math-container">$$P(THT|B) = (.4)(.6)(.4)$$</span></p>
<p><span class="math-container">$$P(A| THT) = \frac{P(... |
884,342 | <p>$N$ is a normal subgroup of $G$ if $aNa^{-1}$ is a subset of $N$ for all elements $a $ contained in $G$. Assume, $aNa^{-1} = \{ana^{-1}|n \in N\}$.</p>
<p>Prove that in that case $aNa^{-1}= N.$</p>
<p>If $x$ is in $N$ and $N$ is a normal subgroup of $G$, for any element $g$ in $G$, $gxg^{-1}$ is in $G$. Suppose $... | Adriano | 76,987 | <p>I'm a little confused as to how you got to your last step. Here's my version.</p>
<hr>
<p>Suppose that $aNa^{-1} \subseteq N$ for all $a \in G$. Then to show that equality holds, it suffices to show that $bNb^{-1} \supseteq N$ for all $b \in G$. To this end, choose any $b \in G$ and choose any $x \in N$. We want t... |
1,075,505 | <p>I have the following function:</p>
<p>$$f(x+iy) = x^2+iy^2$$</p>
<p>My textbook says the function is only differentiable along the line $x = y$, can anyone please explain to me why this is so? What rules do we use to check where a function is differentiable?</p>
<p>I know the Cauchy-Riemann equations, and that $... | Community | -1 | <p>Being complex differentiable at a point is <a href="http://en.wikipedia.org/wiki/Cauchy%E2%80%93Riemann_equations#Complex_differentiability" rel="nofollow">equivalent</a> to the combination of</p>
<ol>
<li>Being real differentiable at that point, and</li>
<li>Satisfying the Cauchy-Riemann equations</li>
</ol>
<p>T... |
2,234,737 | <p>$f:\mathbb{R}^m\to\mathbb{R}$ differentiable such that $f(x/2) = f(x)/2$, show $f$ is linear.</p>
<p>I tried to do:</p>
<p>$f(a+b) = f(2(a+b)/2) = f(2(a+b))/2$</p>
<p>but that wouldn't turn into $f(a)+f(b)$</p>
<p>In the same way: $f(a)/2 + f(b)/2 = f((a+b)/2)$ which won't help.</p>
<p>Also, I need to use the d... | Misha Lavrov | 383,078 | <p>We actually only need differentiability at $0$ for this to hold.</p>
<p>First, note that $f(0) = 0$, since the functional equation tells us that $f(0) = f(0)/2$. The existence of a derivative at $0$ tells us that
$$\lim_{v \to 0} \frac{|f(v) - \operatorname{grad}(f) \cdot v|}{\|v\|} = 0.$$
In particular, if we choo... |
1,789,742 | <p>I'm reading the book generatingfunctionology by Herbert Wilf and I came across a partial fraction expansion on page 20 that I cannot understand. The derivation is as follows:</p>
<p>$$
\frac{1}{(1-x)(1-2x)...(1-kx)} = \sum_{j=1}^{k} \frac{\alpha_j}{1-jx}
$$</p>
<p>The book says to fix $r, 1 \leq r \leq k$, and mul... | Brian M. Scott | 12,042 | <p>My edition of the book doesn’t have</p>
<p>$$\alpha_r = \frac{1}{(1-x)\ldots(1-(r-1)x)(1-(r+1)x)\ldots(1-kx)}\;,$$</p>
<p>which in any case makes no sense, since $\alpha_r$ is a constant and the righthand side is not. However, after letting $x=\frac1r$ it does show</p>
<p>$$\begin{align*}
\alpha_r&=\frac1{(1-... |
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... | Michael E2 | 4,999 | <p>Here's a workaround I've used for <code>Sinc</code>. It defines <code>mySinc[k, x]</code> to be the k-th derivative of <code>Sinc[x]</code>, for <code>k</code> a nonnegative integer. (<em>Update notice:</em> Thanks to J.M. for pointing out a simple formula for the derivatives at zero.)</p>
<pre><code>ClearAll[mySin... |
3,426,550 | <p>Are there parts in functional analysis that would be easier to learn without real analysis or is this just not possible?</p>
<p>When I say functional analysis I mean</p>
<ol>
<li>Lebesgue measure and integral</li>
<li>spaces of Lebesgue integrable functions</li>
<li>Banach spaces</li>
<li>duality</li>
<li>bounded ... | clm | 591,572 | <p>In my experience, what differentiates "Analysis" from other math regarding functions and the real or complex numbers etc, is the level of depth and rigor. You could probably learn something in each of the topics you listed, but it would be hard to understand the proofs or justification behind a lot of results. Conse... |
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.... | Henry | 6,460 | <p>Your calculation has a minor error in that case 4 should end with <code>*(10-3)^4</code> rather than <code>*(10-2)^4</code>. </p>
<p>If you correct that and add the numbers up then you would get <span class="math-container">$19550250$</span>. Dividing by <span class="math-container">$10^8$</span> would then give ... |
3,470,703 | <p><span class="math-container">$\int \left(\frac{1}{2}t+\frac{\sqrt{2}}{4} \right)e^{t^2-\sqrt{2}t}\ dt$</span></p>
| Fareed Abi Farraj | 584,389 | <p><span class="math-container">$\int u'e^u dt =e^u$</span></p>
<p>Try taking <span class="math-container">$u=t^2-\sqrt{2}t$</span> and show your work if you didn't get to your answer if you want us to help you.</p>
|
128,784 | <p>Consider the two lists</p>
<pre><code>list1={1,2,a[1],8,b[4],9};
list2={8,b[4],9,1,2,a[1]};
</code></pre>
<p>it is evident by inspection that <code>list2</code> is just a cyclic rotation of <code>list1</code>. Considering an equivalence class of lists under cyclic rotations, I would like to have a function <code>c... | Kagaratsch | 5,517 | <p>My own attempt at a solution is this</p>
<pre><code>cycRot[x_List] := Block[{p},
p = Position[x, Sort[x][[1]], 1][[1, 1]];
{x[[p ;;]], x[[1 ;; p - 1]]} // Flatten
]
</code></pre>
<p>However, I am not sure if this is going to be slow for larger lists, since the cyclic property is not being utilized to improve p... |
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... | Ivan Izmestiev | 98,590 | <p>Wikipedia is correct. This is discussed in Alexandrov's book "Convex polyhedra" in Section 3.6.5.</p>
|
2,204,217 | <p>I am attempting to find a formula for $$(\mu * \mu)(n)$$ where * represents the Dirichlet Convolution operator. I know this can be expressed as $$\sum_{d|n} \mu(d)\mu(\frac{n}{d})$$ but I'd like the formula to not include any sums over divisors. I know it will be necessary to include information about the factorizat... | Especially Lime | 341,019 | <p>As you say, this is multiplicative. This means that once you know how to calculate it for numbers of the form $p^a$ you can calculate it for arbitrary $n$ by writing $n$ as a product of powers of different primes $p^aq^b\cdots$, and then multiplying the corresponding values $(\mu*\mu)(n)=((\mu*\mu)(p^a))((\mu*\mu)(q... |
2,204,217 | <p>I am attempting to find a formula for $$(\mu * \mu)(n)$$ where * represents the Dirichlet Convolution operator. I know this can be expressed as $$\sum_{d|n} \mu(d)\mu(\frac{n}{d})$$ but I'd like the formula to not include any sums over divisors. I know it will be necessary to include information about the factorizat... | mds | 445,155 | <p><strong>Note on the Previous Solution</strong>: Let $\operatorname{rad}(n)$ denote the <em>radix</em>, or squarefree part, of $n$, i.e., so that if $n = p_1^{\alpha_1} \cdots p_k^{\alpha_k}$ is the factorization of $n$ into powers of distinct primes then $\operatorname{rad}(n) = p_1 p_2 \cdots p_k$. Then we see that... |
2,279,392 | <p>I could use some help again.
Let $f,g$ be functions from $\mathbb{N}\to \mathbb{N}$.
Also known is that $f(n) = g(2n)$ for every $n\in\mathbb{N}$.
Assuming that $f$ is a surjective function, how do you prove that $g$ is not a one-to-one function?</p>
<p>Cheers</p>
| Siong Thye Goh | 306,553 | <p>Since $f$ is a surjection, $\exists k \in \mathbb{Z}$ such that $f(k)=g(1)$.</p>
<p>$$g(2k)=g(1)$$.</p>
<p>Hence $g$ is not injective.</p>
|
1,115,389 | <p>Given: </p>
<p>$$\sum_{n = 0}^{\infty} a_nx^n = f(x)$$</p>
<p>where:</p>
<p>$$a_{n+2} = a_{n+1} - \frac{1}{4}a_n$$</p>
<p>is the recurrence relationship for $a_2$ and above ($a_0$ and $a_1$ are also given).</p>
<p>Is there a nice closed form to this pretty recurrence relationship?</p>
| drhab | 75,923 | <p>First of all: your approach is okay and your answer is <em>almost</em> okay. There is a factor $72$ in your denominator that must be left out and that's all. As you will understand it concerns $9$ factors (and not $10$). An alternative way:</p>
<p>For selecting $9$ spots in such a way that the rooks placed on them ... |
4,412,700 | <p>I've been working on a problem but have been stuck for several hours finishing it.</p>
<p>The problem is to show that
<span class="math-container">$$
\frac{x}{(e+x) \ln(e+x)} \leq \ln(\ln(e+x)) \leq \frac{x}{e}
$$</span>
for all <span class="math-container">$x > 0$</span>.</p>
<p>I proceeded by first integratin... | Calvin Lin | 54,563 | <p>Let <span class="math-container">$ f(x) = \ln ( \ln (e+x) ) $</span>.</p>
<ol>
<li>Show that <span class="math-container">$ f'(x) = \frac{1}{(e+x)( \log e+x)} $</span>.</li>
<li>Show that <span class="math-container">$f(0) = 0 $</span>.</li>
<li>Show that for <span class="math-container">$X> x$</span>, <span clas... |
243,510 | <p>So this question is seriously flooring me.</p>
<p>Let $G$ be drawn in the plane so that it satisfies: </p>
<ol>
<li>The boundary of the infinite region is a cycle $C$</li>
<li>Every other region has boundary cycle of length $3$</li>
<li>Every vertex of $G$ not in $C$ has even degree</li>
</ol>
<p>Show that $\chi(... | William Macrae | 49,817 | <p>This came up on CS Theory a while back: <a href="https://cstheory.stackexchange.com/questions/4027/coloring-planar-graphs">https://cstheory.stackexchange.com/questions/4027/coloring-planar-graphs</a></p>
<p>Unfortunately some of the links in that post are dead / not free.</p>
<p><a href="http://www.google.com/url?... |
801,574 | <p>Let $A$ be a complex $n\times m$ matrix ($n<m$), which is row full rank. For any $m\times m$ matrix $M$, such that $AM$ is still row full rank, can we find an invertible matrix $X$, such that $XA=AM$?</p>
| fuglede | 10,816 | <p><strong>Hint:</strong> $A = \begin{pmatrix} 1 & 0 \end{pmatrix}$, and $M = \begin{pmatrix} 1 & 1 \\ 0 & 1\end{pmatrix}$.</p>
|
2,149,006 | <p>While learning the power rule, one thing popped up in my mind which is confusing me. We know what the power rule states :</p>
<p>$$\frac{\mathrm{d}}{\mathrm{d}x}(x^n) = nx^{n-1}$$ where $n$ is a real number.</p>
<blockquote>
<p>But instead of $n$, if we have a trig function like $\sin(x)$, <strong>will the powe... | R.W | 253,359 | <p>No. It will not maintain the rule as you pointed out. The reason for that is that the derivative act on a function like an operator much more generalized as the power rule: It is a limit of some quantity. Because of that if you change the power to some function you will have to understand how this new functions chan... |
654,263 | <p>My intention is neither to learn basic probability concepts, nor to learn applications of the theory. My background is at the graduate level of having completed all engineering courses in probability/statistics -- mostly oriented toward the applications without much emphasis on mathematical rigor.</p>
<p>Now I am ve... | arsmath | 4,880 | <p>I personally liked Davidson's <i>Stochastic Limit Theory</i>. It rigorously introduces measure-theoretic probability, and then proves the key convergence results. Towards the end it rigorously develops continuous-time stochastic processes.</p>
<p>You mention inference, so if you are particularly interested in the... |
2,372,035 | <p>$ABCD$ is a square with side-length $1$ and equilateral triangles $\Delta AYB$ and $\Delta CXD$ are inside the square. What is the length of $XY$?</p>
| lioness99a | 401,264 | <p>First draw a diagram of the problem:</p>
<p><a href="https://i.stack.imgur.com/LEyyy.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/LEyyy.png" alt="Diagram"></a></p>
<p>Remember that angles in an equilateral triangle are all $60^\circ$ and each point is directly above the midpoint of the opposi... |
1,849,809 | <p>Let $n$ be a positive integer. Define $$\textbf{A}_n(x):= \left[\frac{1}{x+i+j-1}\right]_{i,j\in\{1,2,\ldots,n\}}$$ as a matrix over the field $\mathbb{Q}(x)$ of rational functions over $\mathbb{Q}$ in variable $x$. </p>
<p>(a) Prove that the <em>Hilbert matrix</em> $\textbf{A}_n(0)$ is an invertible matrix over ... | Batominovski | 72,152 | <p>For Part (b), according to i707107's answer, the $(i,j)$-entry of $\textbf{H}_n:=\big(\textbf{A}_n(0)\big)^{-1}$ is given by $$h_{i,j}:=(-1)^{i+j}\,n\,\binom{n+i-1}{i-1}\,\binom{n-1}{j-1}\,\binom{n+j-1}{i+j-1}\,\binom{i+j-2}{i-1}\,.$$
Hence, $n$ is a divisor of the greatest common divisor $g_n$ over $\mathbb{Z}$ of ... |
3,382,561 | <p>I just encountered the random variable <span class="math-container">$Y = |X|$</span>, where <span class="math-container">$X \sim \text{N}(\mu, \sigma^2)$</span>. Now, based on what we know about the absolute value function, this random variable is still continuous; however, the absolute value function means that the... | Community | -1 | <p><span class="math-container">$$P(\lvert X\rvert\le\alpha)=\begin{cases}P(-\alpha\le X\le\alpha)&\text{if }\alpha>0\\ 0&\text{if }\alpha<0\end{cases}$$</span>
Therefore the cdf is <span class="math-container">$$F_{\lvert X\rvert}(\alpha)=\begin{cases}0&\text{if }\alpha<0\\ F_X(\alpha)-\sup_{\bet... |
1,170,708 | <p>What functions satisfy $f(x)+f(x+1)=x$?</p>
<p>I tried but I do not know if my answer is correct.
$f(x)=y$</p>
<p>$y+f(x+1)=x$</p>
<p>$f(x+1)=x-y$</p>
<p>$f(x)=x-1-y$</p>
<p>$2y=x-1$</p>
<p>$f(x)=(x-1)/2$</p>
| N. S. | 9,176 | <p>Replacing $x$ by $x+1$ you get</p>
<p>$$f(x+1)+f(x+2)=x+1=f(x)+f(x+1)+1$$</p>
<p>This shows that
$$f(x+2)=f(x)+1$$</p>
<p>Let $g(x)=f(x)-\frac{x}{2}$. Then $g(x)$ is periodic with period $2$, and </p>
<p>$$g(x)+g(x+1)=-\frac{1}{2}$$</p>
<p><strong>This gives your answer:</strong></p>
<p>Define $g(x)$ to be an... |
1,170,708 | <p>What functions satisfy $f(x)+f(x+1)=x$?</p>
<p>I tried but I do not know if my answer is correct.
$f(x)=y$</p>
<p>$y+f(x+1)=x$</p>
<p>$f(x+1)=x-y$</p>
<p>$f(x)=x-1-y$</p>
<p>$2y=x-1$</p>
<p>$f(x)=(x-1)/2$</p>
| Akiva Weinberger | 166,353 | <p>The simplest answer is:
$$f(x)=\frac x2-\frac14$$
If you have any function $g$ of period 2, such that $g(x)=-g(x+1)$ for all $g$ ($\sin(\pi x)$ is a good example), then $f(x)=g(x)+\frac x2-\frac14$ also works. For example, the function $\sin(\pi x)+\frac x2-\frac14$ also satisfies it.</p>
|
1,734,819 | <p>I think I'm on the right track with constructing this proof. Please let me know.</p>
<p>Claim: Prove that there exists a unique real number $x$ between $0$ and $1$ such that
$x^{3}+x^{2} -1=0$</p>
<p>Using the intermediate value theorem we get
$$r^{3}+r^{2}-1=c^{3}+c^{2}-1$$
......
$$r^{3}+r^{2}-c^{3}-c^{2}=0$$</... | User8128 | 307,205 | <p>You just need the intermediate value theorem and monotonicity. Put $p(x) = x^3 + x^2 - 1$. Since $p(0) = -1$ and $p(1)= 1$, the intermediate value theorem says that there is at least one value $r \in (0,1)$ such that $p(r) = 0$. </p>
<p>Consider $p'(x) = 3x^2 + 2x > 0$ for $x \in (0,1]$. Hence $p$ is increasing.... |
1,966,128 | <p>I know that by De Morgan's law that it is false. But how to disprove it?</p>
| Siong Thye Goh | 306,553 | <p>Hint:</p>
<p>Let the universal set be $\left\{ 0,1 \right\}$</p>
<p>Let $A=\left\{ 0\right\}$ and $B=\left\{ 1\right\}$</p>
<p>Verify that it works.</p>
|
1,966,128 | <p>I know that by De Morgan's law that it is false. But how to disprove it?</p>
| Alexis Olson | 11,246 | <p>All you need is a single counterexample.</p>
<p>Let $A = \{1,2,3\}$ and $B = \{3,4,5\}$ and the universal set $U = \{1,2,3,4,5,6\}$.</p>
<p>Then $A^c = \{4,5,6\}$ and $B^c = \{1,2,6\}$ and</p>
<p>$$A^c \cup B^c = \{1,2,4,5,6\} = (A \cap B)^c$$</p>
<p>but</p>
<p>$$(A \cup B)^c = \{6\} = A^c \cap B^c$$</p>
<p>an... |
3,788,283 | <p>I am searching for the solution to</p>
<p><span class="math-container">$$\int_{0}^{\infty}\int_{0}^{\infty}\frac{1}{2\pi\sigma_{X}\sigma_{Y}\sqrt{1-\rho^{2}}}\exp\left(-\frac{1}{2(1-\rho^{2})}\left[\frac{(\bar{x}-\mu_{X})^{2}}{\sigma_{X}^{2}}+\frac{(\bar{y}-\mu_{Y})^{2}}{\sigma_{Y}^{2}}-\frac{2\rho(\bar{x}-\mu_{X})(... | Benedict W. J. Irwin | 330,257 | <p>I've thought about this for a few days now, I didn't originally intend to answer my own question but it seems best to write this as an answer rather than add to the question. I think there is nice interpretation in the following:
<span class="math-container">$$
f(x) = \lim_{h \to 0} \frac{e^{h f(x)}-1}{h}
$$</span>
... |
3,788,283 | <p>I am searching for the solution to</p>
<p><span class="math-container">$$\int_{0}^{\infty}\int_{0}^{\infty}\frac{1}{2\pi\sigma_{X}\sigma_{Y}\sqrt{1-\rho^{2}}}\exp\left(-\frac{1}{2(1-\rho^{2})}\left[\frac{(\bar{x}-\mu_{X})^{2}}{\sigma_{X}^{2}}+\frac{(\bar{y}-\mu_{Y})^{2}}{\sigma_{Y}^{2}}-\frac{2\rho(\bar{x}-\mu_{X})(... | Tom Copeland | 27,786 | <p>Seems like you have happened upon some relations similar to ones I've written about over several years. Try for starters the <a href="https://math.stackexchange.com/questions/125343/lie-group-heuristics-for-a-raising-operator-for-1n-fracdnd-betan-fra">MSE-Q&A</a> "Lie group heuristics for a raising operator... |
551,252 | <p>I'm sure I've made a trivial error but I cannot spot it.</p>
<p>Fix R>0
Consider the cube $C_R$ as the cube from (0,0,0) to (R,R,R) (save me from listing the 8 vertices) </p>
<p>Consider $S_R$ as the surface of $C_R$</p>
<p>Consider the vector field $v:\mathbb{R}^3\rightarrow\mathbb{R}^3$
given by $v(x,y,z) = (3x... | AlexR | 86,940 | <p><strong>Hint</strong><br>
You can do
$$\iint_{S_R} v\cdot n\ dA = \int_0^R \int_0^R v(0,x',y') \cdot -e_1 + v(R, x', y') \cdot e_1 + v(x', 0, y')\cdot-e_s + \ldots dx' dy'$$
The double integral will yield a $R^2$-like term and the integrand should give another (By Gauss, the results of (1) and (2) should be equal).<... |
4,566,926 | <p><span class="math-container">$a_n$</span> converges to <span class="math-container">$a \in \mathbb{R}$</span>, find <span class="math-container">$\lim\limits_{n \rightarrow \infty}\frac{1}{\sqrt{n}}(\frac{a_1}{\sqrt{1}} + \frac{a_2}{\sqrt{2}} +\frac{a_3}{\sqrt{3}} + \cdots + \frac{a_{n-1}}{\sqrt{n-1}} + \frac{a_n}{... | Oliver Díaz | 121,671 | <p>Given <span class="math-container">$\varepsilon>0$</span>, there is <span class="math-container">$N$</span> such that <span class="math-container">$\sup_{n\geq N}|a_n-a|<\varepsilon$</span>
<span class="math-container">$$\frac{1}{\sqrt{n}}\sum^n_{k=1}\frac{1}{\sqrt{k}}=\frac{1}{\sqrt{n}}\sum^n_{k=1}\frac{a_k-a... |
90,130 | <p>So now that my term's over, I've been brushing up on my quantum field theory, and I came across the following line in my textbook without any justification:</p>
<p>$$\frac{1}{4\pi^2}\int_m^{\infty}\sqrt{E^2-m^2}e^{-iEt}dE \sim e^{-imt}\text{ as }t\to\infty$$</p>
<p>Well, I can see intuitively that <em>if</em> most... | Sasha | 11,069 | <p>Clearly the integral as stated diverges, so one needs to regularize it. To that end, consider $t$ complex with small negative imaginary part, to ensure that it converges at $E\to\infty$.</p>
<p>The integral directly matches the following <a href="http://functions.wolfram.com/Bessel-TypeFunctions/BesselK/07/01/01/00... |
90,130 | <p>So now that my term's over, I've been brushing up on my quantum field theory, and I came across the following line in my textbook without any justification:</p>
<p>$$\frac{1}{4\pi^2}\int_m^{\infty}\sqrt{E^2-m^2}e^{-iEt}dE \sim e^{-imt}\text{ as }t\to\infty$$</p>
<p>Well, I can see intuitively that <em>if</em> most... | Felix Marin | 85,343 | <p>$\newcommand{\bbx}[1]{\,\bbox[8px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\n... |
1,679,044 | <blockquote>
<p>Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle. Then find the area.</p>
<p>$$y = 2x^2, y = 8x^2, 3x + y = 5, x ≥ 0$$ </p>
</blockquote>
<p>Here is my drawing: <a href="http://www.webassign.net/waplots/d/a... | DylanSp | 308,461 | <p>You'll use one integral from $x = 0$ to $x = \frac{5}{8}$, where $y = 8x^2$ intersects $3x + y = 5$, and another integral from $x = \frac{5}{8}$ to $x = 1$. Put together, this gives the expression
$$
\left[\int_{0}^{5/8} 8x^2 - 2x^2\,dx\right] + \left[\int_{5/8}^{1} (-3x + 5) - 2x^2\,dx \right]
$$</p>
|
27,951 | <p>Something I notice is when there's an advanced/specialized question, it often receives very few upvotes. Even if it is seemingly well written. I try to upvote advanced questions <strong>that I might not even understand</strong>, if they appear well written. </p>
<p>Is this good behaviour? Should we encourage upvoti... | Stella Biderman | 123,230 | <p>I think that the general guidelines for "asking a good question" on this site are nearly content agnostic. I might write a horribly thought out and poorly reasoned question about category theory which belies a fundamental misunderstanding of what a morphism is, but that isn't grounds to call it a bad question. The o... |
1,753,263 | <p>I've been struggling with understanding how surface integrals come to be. Can someone please explain why they have such a format with a normal vector to the surface?
thanks!</p>
| Chris Rackauckas | 244,186 | <p>Part of it is because of the fact that these methods were first used to understand physics, and there are clear physical interpretations. The vector coming out of the surface is the "flow", and so a lot of physical laws like Guass' law and Fick's ask about balancing flows (i.e. for Gauss' law, if a ball contains ele... |
1,753,263 | <p>I've been struggling with understanding how surface integrals come to be. Can someone please explain why they have such a format with a normal vector to the surface?
thanks!</p>
| spinoza | 310,540 | <p>Let $F$ be a vector field, $n$ a normal vector to the surface, with $u=\frac{n}{|n|}$ the unit normal vector. </p>
<p>The integral of the flux of a vector field through a surface requires an orientation (direction) that defines an "inside" and an "outside" relative to the surface that is the boundary between the "i... |
3,605,368 | <p>Imagine a <span class="math-container">$9 \times 9$</span> square array of pigeonholes, with one pigeon in each pigeonhole. Suppose that all at once, all the pigeons move up, down, left, or right by one hole. (The pigeons on the edges are not allowed to move out of the array.) Show that some pigeonhole winds up with... | Christian Blatter | 1,303 | <p>A hint: Color the <span class="math-container">$9^2$</span> squares checkerboard like!</p>
|
38,915 | <p>It has to be the silliest question, but it’s not clear to me how to calculate eigenvectors quickly. I am just talking about a very simple 2-by-2 matrix.</p>
<p>When I have already calculated the eigenvalues from a characteristic polynomial, I can start to solve the equations with $A\mathbf{v}_1 = e_1\mathbf{v}_1$ a... | Jiangwei Xue | 9,430 | <p>Please goes through this list one by one. </p>
<ol>
<li>Your method will work as long as either $A$ or $B$ is not zero. </li>
<li>When doesn't it work? Try calculate the eigenvector of $\left(\begin{smallmatrix} 1 & 0 \\ 1 & 1 \end{smallmatrix} \right )$.</li>
<li>Does the method of taking $\left(\begin{sm... |
70,500 | <p>I am trying to find $\cosh^{-1}1$ I end up with something that looks like $e^y+e^{-y}=2x$. I followed the formula correctly so I believe that is correct up to this point. I then plug in $1$ for $x$ and I get $e^y+e^{-y}=2$ which, according to my mathematical knowledge, is still correct. From here I have absolutely n... | Michael Hardy | 11,667 | <p>$$
e^y+e^{-y}=2
$$
Letting $u = e^y$, this becomes
$$
u + \frac 1u = 2
$$
Multiplying both sides by $u$:
$$
u^2 + 1 = 2u
$$
That's just a quadratic equation.</p>
|
1,976,733 | <p>What does the following statement mean?</p>
<blockquote>
<p>Suppose that $X$ is a random variable for the experiment, taking values in a set $T$.
The function $B↦P(X∈B)$ defines a probability measure on $T$.</p>
</blockquote>
<p>Let, </p>
<p>$Experiment$ = a dice rolls.</p>
<p>$\Omega = \{1,2,3,4,5,6\}$</p>
... | Anthony P | 377,521 | <p>Consider the Real interval $[0,\epsilon]$. How many points are inside that interval? Divide the interval so that the new interval is $[0,\epsilon/2] $. How many points are inside this interval? Still infinitely many. We say that $\mathbb{Q}$ is dense in $\mathbb{R}$ in the language of Real Analysis. This means you c... |
1,976,733 | <p>What does the following statement mean?</p>
<blockquote>
<p>Suppose that $X$ is a random variable for the experiment, taking values in a set $T$.
The function $B↦P(X∈B)$ defines a probability measure on $T$.</p>
</blockquote>
<p>Let, </p>
<p>$Experiment$ = a dice rolls.</p>
<p>$\Omega = \{1,2,3,4,5,6\}$</p>
... | Mauro ALLEGRANZA | 108,274 | <p>See the full discusssion in <a href="https://books.google.it/books?id=jF1j0y2-hM0C&pg=PA65" rel="nofollow noreferrer"><strong>CH:VII RATIONAL, REAL, AND COMPLEX NUMBERS</strong></a>; he is defining <em>fractions</em> :</p>
<blockquote>
<p>We shall define the <em>fraction</em> <span class="math-container">$\dfrac... |
389,425 | <blockquote>
<p>What kind of math topics exist?</p>
</blockquote>
<p>The question says everything I want to know, but for more details: I enjoy studying mathematics but the problem is that I can't find any information with a summary of all math topics, collected together. I also googled this and took a look at other... | amWhy | 9,003 | <p>You might want to start at the <a href="http://www.math-atlas.org/">Mathematical Atlas</a>. It gives a visual map of domains of mathematics, and how they are inter-related: you can click on any "bubble" to learn more about each area. There is also a menu to the right-hand side: for example, see <a href="http://www.m... |
3,645,263 | <p>I have recently started studying Set Theory in a self-thaught way, for that purpose I have been following Kunen's book: Set Theory: An Introduction to Independence Proofs. I'm in Chapter I section 7 and it has been defined the ordinals addition but I don't quite understand that definition. I have seen that in other ... | Ross Millikan | 1,827 | <p>As Kunen explains it you are putting a copy of <span class="math-container">$\beta$</span> after <span class="math-container">$\alpha$</span> and looking at the resulting order type. Whether <span class="math-container">$\alpha \lt \beta$</span> is not important. If they are both finite it is just regular addition... |
3,645,263 | <p>I have recently started studying Set Theory in a self-thaught way, for that purpose I have been following Kunen's book: Set Theory: An Introduction to Independence Proofs. I'm in Chapter I section 7 and it has been defined the ordinals addition but I don't quite understand that definition. I have seen that in other ... | Brian M. Scott | 12,042 | <p>The definition that Ken is using amounts to placing a copy of the ordinal <span class="math-container">$\beta$</span> after the ordinal <span class="math-container">$\alpha$</span>. Since the sets <span class="math-container">$\alpha$</span> and <span class="math-container">$\beta$</span> are not actually disjoint (... |
1,090,658 | <p>I'm doing some previous exams sets whilst preparing for an exam in Algebra.</p>
<p>I'm stuck with doing the below question in a trial-and-error manner:</p>
<p>Find all $ x \in \mathbb{Z}$ where $ 0 \le x \lt 11$ that satisfy $2x^2 \equiv 7 \pmod{11}$</p>
<p>Since 11 is prime (and therefore not composite), the Ch... | kingW3 | 130,953 | <p>$$2x^2\equiv 7\pmod{11}\\2x^2\equiv -4\pmod{11}\\x^2\equiv -2\pmod{11}\\x^2\equiv 9\pmod{11}\\x^2\equiv\pm3\pmod{11}$$
As HSN mentioned since $11$ is prime than there are at most 2 solutions,</p>
|
46,446 | <p>I'm an highschool graduate who is currently waiting for college.</p>
<p>Meanwhile, I'm trying to do a little project by myself. (Computer stuff) And yesterday, I found that I needed to deal with something called "De Casteljau algorithm"</p>
<p>I know calculus (single-variable, but I'm thinking about learning multi... | Edison | 11,857 | <p>What exactly do you need the De Casteljau algorithm for? You might be able to use other methods and get good results. Anyway, it believe you have the required mathematical tools to work and understand De Casteljau's algorithm. </p>
<p>Here is a similar question
<a href="https://stackoverflow.com/questions/6271817/c... |
197,393 | <p>Playing around on wolframalpha shows $\tan^{-1}(1)+\tan^{-1}(2)+\tan^{-1}(3)=\pi$. I know $\tan^{-1}(1)=\pi/4$, but how could you compute that $\tan^{-1}(2)+\tan^{-1}(3)=\frac{3}{4}\pi$ to get this result?</p>
| Robert Israel | 8,508 | <p>Note that
$$ \tan \left(\arctan(1+z) + \arctan\left(2 + z + z^2 \right) + \arctan \left( 3+3\,z+4\,{z}^{2}+2\,{z}^{3}+{z}^{4} \right) \right)=z
$$
so that $$\arctan(1+z) + \arctan\left(2 + z + z^2 \right) + \arctan \left( 3+3\,z+4\,{z}^{2}+2\,{z}^{3}+{z}^{4} \right) = \arctan(z) + n \pi $$
for the appropriate inte... |
1,653,806 | <p>This is a but of a more mathematically juvenile question but I'm trying to get all my intuition in order. When taking a limit we can cancel things that might be zero because in taking a limit, we allow ourselves to avoid trouble spots. If the limit exists, it is very much a number/function which could be zero. So, i... | Markus Heinrich | 127,899 | <p>Since you said you have a mathematical background and know about the unification in terms of the exterior derivative, I thought I should add some information about the connection to the exterior derivative here.</p>
<p>As it is often the case, the generalisation of concepts can be performed in various ways, because... |
503,827 | <p>I need help to prove the following:</p>
<p>Let $a$, $b$, and $c$ be any integers. If $a \mid b$, then $a \mid bc$</p>
<p>My brain is in overload and just not working.</p>
| user96646 | 96,646 | <p>If $a\mid b$ then $b=an$ for $n\in$ $\mathbb{Z}$. Then, $bc = (an)c$ and so $a\mid bc$. </p>
|
3,754,225 | <p>Let S be the circle with centre <span class="math-container">$(0,0)$</span>, radius <span class="math-container">$r$</span> units. The chord <span class="math-container">$C$</span> of the circle S subtends an angle of 2π/3 at its center. If R represents the region consisting of all points inside S which are closer t... | Vishu | 751,311 | <p>The circle of interest is <span class="math-container">$x^2+y^2=r^2$</span>. Since there are infinitely many chords which subtend an angle <span class="math-container">$\frac{2\pi}{3}$</span> at the center, we might as well assume the chord to be horizontal (slope <span class="math-container">$0$</span>). By some si... |
1,435,189 | <p>A map $T:X \rightarrow Y$ is $A, B$- measurable if the pre-image of every set in $B$ is a set in $A$, where $A,B$ are $\sigma$-algebras in $X,Y$. </p>
<p>But I was thinking that isn't any map $T: X \rightarrow Y$ measurable then? Because since pre-images behave so nicely, the pre-image of all sets from $B$ make up... | parsiad | 64,601 | <p><strong>Hint</strong>: The more sets you have in the $\sigma$-algebra of $X$ and the less sets you have in the $\sigma$-algebra of $Y$, the more "likely" your function is to be measurable.</p>
<p>I'm sure you can think of some trivial nonmeasurable examples now.</p>
|
1,160,076 | <p>Consider $f:\left[a,b\right]\rightarrow \mathbb{R}$ continuous at $\left[a,b\right]$ and differentiable at $\left(a,b\right)$.<br>
$\forall x\:\ne \frac{a+b}{2}$, $f'\left(x\right)\:>\:0$, but at $x\:=\frac{a+b}{2}$, $f'\left(x\right)\:=\:0$.<br>
Show that $f$ is <strong>strictly increasing</strong> at $\left[a,b... | Timbuc | 118,527 | <p>Suppose there are $\;x,y\in [a,b]\;,\;\;x,y\;$ with $\;f(x)=f(y)\;$ . Since the function is <strong>at least</strong> monotonic increasing "weakly", as $\;f'(x)\ge 0\;$ , the above means </p>
<p>$$f(w)=0\;\;\;\forall\,w\in (x,y)\implies f'(w)=0\;\;\;\forall\,w\in (x,y)\implies\;\text{contradiction}$$</p>
|
1,682,961 | <p>I was making a few exercises on set proofs but I met an exercise on which I don't know how to start:</p>
<blockquote>
<p>If $A \cap C = B \cap C $ and $ A-C=B-C $ then $A = B$</p>
</blockquote>
<p>Where should I start? Should I start from $ A \subseteq B $ or should I start from this $ ((A\cap C = B\cap C) \land... | user 1 | 133,030 | <p>$$A=(A \cap C)\cup(A-C)=(B \cap C)\cup(B-C)= B.$$
1st and last equality proved <a href="https://math.stackexchange.com/questions/1360411/how-to-prove-a-a-setminus-b-cup-a-cap-b">here</a>, and the middle one is your assumptions.</p>
|
785,844 | <p>I've got the following limit to solve:</p>
<p>$$\lim_{s\to 1} \frac{\sqrt{s}-s^2}{1-\sqrt{s}}$$</p>
<p>I was taught to multiply by the conjugate to get rid of roots, but that doesn't help, or at least I don't know what to do once I do it. I can't find a way to make the denominator not be zero when replacing $s$ fo... | Mark Bennet | 2,906 | <p>Try putting $t=\sqrt s$ to get $$\frac {t-t^4}{1-t}=\frac {t(1-t^3)}{1-t}=t(1+t+t^2)$$You can notice this without the substitution, of course, but sometimes a substitution like this helps to clarify what is going on.</p>
|
2,835,290 | <p>I am given two variables $Y_1$ and $Y_2$ obeying an exponential distribution with mean $\beta= 1$</p>
<p>We are asked what the distribution of their average is and the solution must be found using moment generating functions.</p>
<p>The solution to this exercise says:</p>
<p><a href="https://i.stack.imgur.com/eOi... | Jack D'Aurizio | 44,121 | <p>Assuming $a>1$, the original integral equals
$$ \frac{1}{2}\int_{0}^{1}\log\left(\frac{a-x}{a+x}\right)\frac{dx}{x\sqrt{x(1-x)}}\,dx=-\int_{0}^{1}\frac{\text{arctanh}(x/a)}{x\sqrt{x(1-x)}}\,dx \tag{1}$$
or
$$ -\sum_{n\geq 0}\int_{0}^{1}\frac{x^{2n-1/2}}{(2n+1)a^{2n+1}\sqrt{1-x}}\,dx=-\sum_{n\geq 0}\frac{\pi\binom... |
3,367,903 | <p>Let <span class="math-container">$(\Omega,\leq)$</span> be a dense linear order, with
<span class="math-container">$\Omega \subset \mathbb{Q}$</span>, and where "<span class="math-container">$\leq$</span>" is induced by the usual order relation in <span class="math-container">$\mathbb{Q}$</span>. </p>
<p>Is there ... | Mark Kamsma | 661,457 | <p>Yes, after possibly deleting its endpoints, it is isomorphic to <span class="math-container">$\mathbb{Q}$</span>. That is, <span class="math-container">$\Omega$</span> will be isomorphic to one of the following:</p>
<ul>
<li><span class="math-container">$\mathbb{Q}$</span>,</li>
<li><span class="math-container">$\m... |
3,367,903 | <p>Let <span class="math-container">$(\Omega,\leq)$</span> be a dense linear order, with
<span class="math-container">$\Omega \subset \mathbb{Q}$</span>, and where "<span class="math-container">$\leq$</span>" is induced by the usual order relation in <span class="math-container">$\mathbb{Q}$</span>. </p>
<p>Is there ... | Alex Kruckman | 7,062 | <p>Following up on Mark Kamsma's answer, there are <span class="math-container">$2^{\aleph_0}$</span>-many suborders of <span class="math-container">$\mathbb{Q}$</span> which are isomorphic to <span class="math-container">$\mathbb{Q}$</span>, even up to automorphisms of <span class="math-container">$\mathbb{Q}$</span>.... |
3,291,303 | <p>this very same problem appeared in a different thread but the questions was slightly different. In my case, I'm looking precisely for the answer.</p>
<p>This is how I solved it, I only need confirmation of whether this actually is correct:</p>
<p>So I assumed that 1111... (100 ones) is going to be exactly divided ... | Ross Millikan | 1,827 | <p>Yes, it is correct. If you envision setting up the long division, the first subtraction strips off the first seven <span class="math-container">$1$</span>s, the next strips the next seven, and so on.</p>
|
1,282,420 | <p>In his book of Differential Topology, Hirsch starts a little detour in his theory in order to present a way to see things in a general perspective. The precise fragment I'm referring to is the following:</p>
<p><img src="https://i.stack.imgur.com/rfzFr.png" alt="enter image description here"></p>
<p>I have never e... | Thomas Andrews | 7,933 | <p>The first cases of inverse limits I saw were of the form:</p>
<p>$$X_1\mathop{\leftarrow}_{f_1} X_2\mathop{\leftarrow}_{f_2} X_3\mathop{\leftarrow}_{f_3}\cdots$$</p>
<p>Then the inverse limit of this sequence of sets is the set of tuples:</p>
<p>$$\left\{(x_1,x_2,\cdots,x_n,\cdots)\in X_1\times X_2\cdots\mid \for... |
1,374,660 | <p>The IQ of actuarial science majors is assumed to be normally distributed with mean 112 and standard deviation of 14. In a class of 19 students, find the probability that the mean IQ of all 19 students is greater than 120. </p>
<p>I know this question is based of the CLT, but I don't know what to change when everyon... | gogurt | 29,568 | <p>If $X_1, \ldots, X_{19}$ are the 19 students you sample from the class, then you can assume they're iid with distribution $N(112, 14)$ as stated in the problem. Then write $\bar{X} = (X_1 + \ldots + X_{19})/19$ for the sample mean you're dealing with.</p>
<p>Just from rules for means and variances of sums of indepe... |
3,657,106 | <blockquote>
<p>Sam was adding the integers from <span class="math-container">$1$</span> to <span class="math-container">$20$</span>. In his rush, he skipped one of the numbers and forgot to add it. His final sum was a multiple of <span class="math-container">$20$</span>. What number did he forget to add?</p>
</block... | fleablood | 280,126 | <blockquote>
<p>but it seems that I'm missing something?</p>
</blockquote>
<p>Oh, I can't resist. </p>
<p>You aren't missing anything. That's the problem.</p>
<p>(....<em>laughs in the corner to himself for hours</em>....)</p>
<p>You were supposed to skip a number but you included them all.</p>
<p>The numbers a... |
3,163,355 | <p>I found that the Wieferich prime <span class="math-container">$1093$</span> divides <span class="math-container">$3^{1036}-1$</span>.
Does <span class="math-container">$1093$</span> divides infinitely many <span class="math-container">$3^{k}-1$</span>, with k a positive integer? And what features must <span class="m... | Mostafa Ayaz | 518,023 | <p>Note that if <span class="math-container">$$r_k\triangleq 3^k-1\mod 1093$$</span>then <span class="math-container">$$r_{k+1}{=3^{k+1}-1\mod 1093\\=3^{k+1}-3+2\mod 1093\\=\Big(3\cdot(3^{k}-1)\mod 1093\Big)+2\mod 1093\\=3r_k+2\mod 1093}$$</span>Based on this and with <span class="math-container">$r_1=2\mod1093=2 $</sp... |
581,958 | <p>I am working on a project about Mathematics and Origami. I am working on a section about how origami can be used to solve cubic equations. This is the source I am looking at:<br>
<a href="http://origami.ousaan.com/library/conste.html">http://origami.ousaan.com/library/conste.html</a><br>
I understand the proof they ... | amorvincomni | 95,697 | <p>For the trisection of an angle note that $\cos3\alpha=4\cos^3\alpha-3\cos\alpha$ with $3\alpha$ constructible (this is equivalent to say that $\cos 3 \alpha$ is constructible). It is an equation of degree 3. In order to obtain the desired $\alpha$ angle, take a squared paper and do the crease $L_1$ corresponding t... |
2,309,613 | <p>Find a general solution to $x^2-2y^2=1$</p>
<p>I found that (3,2) is a solution.
Now what should I do?
I can not catch what the question really want.</p>
<p>It is about pell's equation. Would you give me a form of general solution? </p>
| Leviathan | 448,426 | <p>The function is holomorphic and can therefore be decomposed in a power series;
\begin{align*}
f(z) = \sum_{n=0}^{\infty} a_n z^{n}
\end{align*}
Furthermore, it is known that $f(i\mathbb{R}) \in i \mathbb{R}$ and $f(\mathbb{R}) \in \mathbb{R}$ . That is why
\begin{align}
f(r) &= \sum_{n= 0}^{\infty} a_n r^{n... |
89,669 | <p>If we fix a locally profinite group $G$ , we note $R(G)$ the category of smooth representations of $G$, $\mathcal{E}$ the set of equivalence classes of $R(G)$, and finaly $Irr(G)$ the set of irreducible equivalence calasses. I recall that the theorem of Cantor-Bernstein says : If $E$ and $F$ two sets. If there is an... | Rami | 4,690 | <p>Your group has compact center (in fact trivial center). So supercuspidal representation indeed form a semi-simple category, as pm said (see <a href="http://www.math.uchicago.edu/~mitya/langlands/Bernstein/Bernstein93new.dvi" rel="nofollow">http://www.math.uchicago.edu/~mitya/langlands/Bernstein/Bernstein93new.dvi</a... |
1,602,392 | <p>I think I have a proof using Pythagoras for $\sqrt{a_1^2} + \sqrt{a_2^2} > \sqrt{a_1^2 + a_2^2}$.</p>
<p><em>I'm interested in whether there's a way to use that proof with Pythagoras to prove the general $a_n$ case (for this, hints are appreciated rather than complete proofs), and <strong>also</strong> in other ... | Kibble | 277,229 | <p>Here is a geometrical proof. Consider a square with side length $L = |a_1|+|a_2| + \ldots + |a_n|$. </p>
<p>$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$<a href="https://i.stack.imgur.com/DzcrF.png" rel="noreferrer"><img src="https://i.stack.imgur.com/DzcrF.png" alt="enter image description here"></a></p>
<p... |
3,714,418 | <p>I need to calculate the angle between two 3D vectors. There are plenty of examples available of how to do that but the result is always in the range <span class="math-container">$0-\pi$</span>. I need a result in the range <span class="math-container">$\pi-2\pi$</span>.</p>
<p>Let's say that <span class="math-conta... | Saaqib Mahmood | 59,734 | <p>Let <span class="math-container">$x$</span> be an arbitrary element. In the table below, let <span class="math-container">$0$</span> denote that <span class="math-container">$x$</span> is not in the set while <span class="math-container">$1$</span> denote that <span class="math-container">$x$</span> is in the set.<... |
29,323 | <p>You're hanging out with a bunch of other mathematicians - you go out to dinner, you're on the train, you're at a department tea, et cetera. Someone says something like "A group of 100 people at a party are each receive hats with different prime numbers and ..." For the next few minutes everyone has fun solving the p... | Anton Geraschenko | 1 | <p>You and infinitely many other people are wearing hats. Each hat is either red or blue. Every person can see every other person's hat color, but cannot see his/her own hat color; aside from that, you cannot share any information (but you are allowed to agree on a strategy before any of the hats appear on your heads).... |
29,323 | <p>You're hanging out with a bunch of other mathematicians - you go out to dinner, you're on the train, you're at a department tea, et cetera. Someone says something like "A group of 100 people at a party are each receive hats with different prime numbers and ..." For the next few minutes everyone has fun solving the p... | rgrig | 840 | <p>When they came to diner some shook hands. Ask them to prove that that two of them shook hands the same number of times.</p>
|
29,323 | <p>You're hanging out with a bunch of other mathematicians - you go out to dinner, you're on the train, you're at a department tea, et cetera. Someone says something like "A group of 100 people at a party are each receive hats with different prime numbers and ..." For the next few minutes everyone has fun solving the p... | Kiochi | 4,241 | <p>Here's an easy but fun one (probably suitable for a class or for mathematicians who have had some drinks). You have ten bags of coins, one of which contains fake coins. We may of course assume that each bag contains infinitely many coins. The real coins weigh 1 gram each, while the fake coins weigh .9 grams each. Yo... |
29,323 | <p>You're hanging out with a bunch of other mathematicians - you go out to dinner, you're on the train, you're at a department tea, et cetera. Someone says something like "A group of 100 people at a party are each receive hats with different prime numbers and ..." For the next few minutes everyone has fun solving the p... | BlueRaja | 2,883 | <p>Oldie but a goodie (Monty-hall problem): </p>
<p>You are on a game show with three doors, behind one of which is a car and behind the other two are goats. You pick door #1. Monty, who knows what’s behind all three doors, reveals that behind door #2 is a goat. Before showing you what you won, Monty asks if you want ... |
29,323 | <p>You're hanging out with a bunch of other mathematicians - you go out to dinner, you're on the train, you're at a department tea, et cetera. Someone says something like "A group of 100 people at a party are each receive hats with different prime numbers and ..." For the next few minutes everyone has fun solving the p... | JBL | 4,658 | <p>Take a convex polyhedron and any point inside of it. To every face, drop a normal line from the point. Note that it is both possible to land inside the face or outside. Construct a polyhedron where every such normal line drops outside of the corresponding face, or prove such a polyhedron cannot exist.</p>
|
29,323 | <p>You're hanging out with a bunch of other mathematicians - you go out to dinner, you're on the train, you're at a department tea, et cetera. Someone says something like "A group of 100 people at a party are each receive hats with different prime numbers and ..." For the next few minutes everyone has fun solving the p... | Daniel Krenn | 8,153 | <p>There is a plane with 100 seats and we have 100 passengers entering the plane one after the other. The first one cannot find his ticket, so chooses a random (uniformly) seat. All the other passengers do the following when entering the plane (they have their tickets). If the seat written on the ticket is free, one si... |
29,323 | <p>You're hanging out with a bunch of other mathematicians - you go out to dinner, you're on the train, you're at a department tea, et cetera. Someone says something like "A group of 100 people at a party are each receive hats with different prime numbers and ..." For the next few minutes everyone has fun solving the p... | Tony Huynh | 2,233 | <p>Here's one that I like that I just heard a few days ago. Alice and Bob play the following game. Alice is randomly dealt 5 cards from an ordinary deck of cards. She is allowed to show Bob 4 of the 5 cards (in order). Bob must then guess what the 5th card is. </p>
<p>Prove that Alice and Bob have a strategy where... |
2,623,621 | <p>I'm struggling to find the derivative of this function :</p>
<p>$$y=\bigg\lfloor{\arccos\left(\frac{1}{\tan\left(\sqrt{\arcsin x}\right)}\right)}\bigg\rfloor$$</p>
<p>I've been told it should be 0 but how can I find that ?!</p>
| Community | -1 | <p>For the "trivial" inclusion: note that for every $i\in I$ you have $S_i\subseteq\bigcup_{i\in I}S_i$ so $\overline{S_i}\subseteq\overline{\bigcup_{i\in I}S_i}$. Taking the union on the left side, we have $\bigcup_{i\in I}\overline{S_i}\subseteq\overline{\bigcup_{i\in I}S_i}$.</p>
<p>For the converse, it is enough t... |
3,648,462 | <p>I need to find a solution to this equation, being <span class="math-container">$z\in\mathbb{C}$</span>,
<span class="math-container">$$2e^3\cos (6z)-e^6=1.$$</span>
I don't know what method can I follow to solve it.</p>
| J. W. Tanner | 615,567 | <p><strong>Hint:</strong></p>
<p>The sum of coefficients of a polynomial <span class="math-container">$f(x)$</span> is <span class="math-container">$f(1)$</span>.</p>
|
2,004,895 | <p>In my textbook there is a question like below:</p>
<p>If $$f:x \mapsto 2x-3,$$ then $$f^{-1}(7) = $$</p>
<p>As a multiple choice question, it allows for the answers: </p>
<p>A. $11$<br>
B. $5$<br>
C. $\frac{1}{11}$<br>
D. $9$</p>
<p>If what I think is correct and I read the equation as:</p>
<p>$$f(x)=2x-3$$
th... | Chill2Macht | 327,486 | <p>This is to just to elaborate on why someone would use the notation $$f: x \mapsto 2x-3$$ When we treat a function $f$ as an object, i.e. do more with it than just evaluate it at points, then we need to be able to distinguish between the <em>object</em>, the function $f$, and its rule of assignment determining its v... |
2,932,799 | <p><strong>Notations.</strong></p>
<p>Let <span class="math-container">$\xi=(\xi_1,\xi_2,\xi_3,\xi_4)\in\mathbb( R\setminus\mathbb Q)^4$</span> such that <span class="math-container">$\xi_1\xi_4-\xi_2\xi_3\ne 0$</span> and the <span class="math-container">$\xi_i$</span> are linearly independent over <span class="math-... | Exodd | 161,426 | <p>Take for example
<span class="math-container">$$
A =
\begin{pmatrix}
1&1&1\\
1&1&1\\
1&3&2\\
\end{pmatrix}
\quad
E =
\begin{pmatrix}
0&1&0\\
0&0&0\\
0&0&0\\
\end{pmatrix}
$$</span>
and <span class="math-container">$B_\delta = A+\delta E$</span>. </p>
<p>You get (aft... |
150,295 | <p>I will appreciate any enlightenment on the following which must be an exercise in a certain textbook. (I don't recognize where it comes from.)
I understand that the going down property does not hold since $R$ is not integrally closed (in fact, it is not a UFD), but I have no idea how to show that $q$ is such a count... | wxu | 4,396 | <p>We show that there is not a prime $Q\subset P$ such that $Q\cap R=q$.</p>
<p>Let us compute $p,q$. The prime $q=(X)\cap R=\{Xh\mid h(1,1)=0,h(X,Y)\in k[X,Y]\}$, the prime $p=P\cap R=\{g(X,Y)\mid g(0,0)=g(1,1)=0, g\in k[X,Y]\}$.
Now it is clear $q\subset p$ and not equal $p$, since $X-Y\in p\setminus q$. </p>
<p>We... |
1,535,376 | <p>I have a hard time understanding that when and under what conditions we can use Gauss elimination with complete pivoting, and when with partial pivoting, and when with no pivoting? (I mean what is the exact feature of a matrix that will tell us which one to choose?)</p>
| Community | -1 | <blockquote>
<p>I have a hard time understanding that when and under what conditions
we can use Gauss elimination with complete pivoting, and when with
partial pivoting, and when with no pivoting? (I mean what is the exact
feature of a matrix that will tell us which one to choose?)</p>
</blockquote>
<p>My prof... |
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