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 |
|---|---|---|---|---|
66,187 | <p>I would like to remove facets from an <a href="http://mathworld.wolfram.com/Octahedron3-Compound.html" rel="nofollow noreferrer">octahedron 3-compound</a> - like in the picture below.</p>
<p><img src="https://i.stack.imgur.com/xvvj4.png" alt="octahedron 3-compound with holes"></p>
<p>I tried to combine these two g... | Silvia | 17 | <p>You can't get what you want with <code>EdgeForm[Thickness[.03]]</code>, that's not what <a href="http://reference.wolfram.com/language/ref/EdgeForm.html" rel="nofollow noreferrer"><code>EdgeForm</code></a> is meant for (that is, styles for the <em>1-dimensional</em> edges).</p>
<p>Here is a quick solution based on ... |
2,753,812 | <p>So far I have been looking for a possible divisor such that the sum of $7$ seventh powers will never leave a certain remainder but there seems not to be such a divisor below $400$ and the search is too computationally heavy to continue.</p>
| Will Jagy | 10,400 | <p>If you mean non-negative seventh powers, the answer is yes, infinitely many misses. Indeed, the volume of the region
$x_i \geq 0$ and $\sum_{i=1}^7 x_i^7 \leq 1$ is a constant $C$ smaller than $1.$ the volume of $$\sum_{i=1}^7 x_i^7 \leq N$$
is roughly $CN.$ This volume is a good approximation of the (maximum) cou... |
2,254,672 | <p>I'm trying to solve a differential equation for the modeling of an ultrasonic horn in wich his form is a catenary, and his differential equation for a wave involves a hyperbolic tangent. I have the solution for the differential equation, but I need to do the "steps by steps". I had transformed the trigonometric coef... | Michael Seifert | 248,639 | <p>Define $y(x)$ such that $v(x) = y(x)/\cosh(kx/r)$. (<strong>EDIT:</strong> see below for why you might want to do this.) Then we have
$$
v'(x) = \frac{1}{\cosh (kx/r)} y'(x) - \frac{k}{r} \frac{\sinh (kx/r)}{\cosh^2 (kx/r)} y(x) = \frac{1}{\cosh (kx/r)} \left[ y'(x) - \frac{k}{r} \tanh \left( \frac{kx}{r} \right)... |
43,650 | <p>Consider the following problem:</p>
<blockquote>
<p>Let $f:{\mathbb R}^3 \to{\mathbb R}$ be $$f(x,y,z)=x+4z$$ where $$x^2+y^2+z^2\leq 2.$$ Find the minimum of $f$. </p>
</blockquote>
<p>This is similar to the question <a href="https://math.stackexchange.com/q/41385/9464">here</a>. However, since this is not an a... | Luboš Motl | 10,599 | <p>The function $x+4z$ increases most quickly (measured by the increase per unit length) in the direction of the vector $(1,0,4)$ - the coefficients of the monomials. The inequality $x^2+y^2+z^2 \leq 2$ is a solid ball with radius $\sqrt{2}$. So the minimum value of the linear function is on the boundary of the ball in... |
43,650 | <p>Consider the following problem:</p>
<blockquote>
<p>Let $f:{\mathbb R}^3 \to{\mathbb R}$ be $$f(x,y,z)=x+4z$$ where $$x^2+y^2+z^2\leq 2.$$ Find the minimum of $f$. </p>
</blockquote>
<p>This is similar to the question <a href="https://math.stackexchange.com/q/41385/9464">here</a>. However, since this is not an a... | Per Alexandersson | 934 | <p>Here is an idea:
Consider the plane $x+4z=c$, for different values of c.
Changing c will shift the plane, and clearly, $c$ is the variable we want to minimize. Thus, we seek the minimal $c$ such that the plane intersects the sphere with radius 2. Therefore, the minimizing plane must be a tangent plane to the sphere,... |
89,176 | <p>The Hahn–Banach theorem states that: Given a sublinear functional $S: V \rightarrow \mathbb R$, if $T: U \rightarrow \mathbb R$ is a linear functional on a linear subspace $U \subseteq V$ that is dominated by $S$ on $U$, then there exists a linear extension of $T$ to $V$ that is dominated by $S$ on $V$.</p>
<p>Now,... | Matthias Ludewig | 16,702 | <p>In this answer I made more or less the same mistake over and over again. It turns out that sublinear functions are not as nice as I believed. I thought it would be best to delete it.</p>
|
3,567,228 | <p>I am really struggling and have spent hours proving this theorem so I would greatly appreciate some help. I think I have nearly proved the theorem except for showing that <span class="math-container">$f^{-1}(-\infty,0]$</span> is a regular sublevel set of <span class="math-container">$f$</span>. </p>
<p>My attempt ... | Li-Chung Wang | 743,599 | <p>The fact that <span class="math-container">$0$</span> is a regular value of <span class="math-container">$f$</span> follows from the formula given in Lee's Smooth Manifolds, p.119, l.5.
The case when <span class="math-container">$D$</span> is compact:
By Lee's Smooth Manifolds, p.46, Proposition 2.28, we may obtain ... |
1,775,166 | <p>If $$T:V\rightarrow V$$ is a bijective linear transformation then can $T^2$ also be a bijective linear transformation? Can $T^n$ also be bijective linear transformation?</p>
| Patrick Abraham | 337,503 | <p>To start an easy example:
$$I_n = I_n^2 = I_n^k$$</p>
<p>But the most easy way to think about this question is, if you look at the determinant and the determinant of matrix products.</p>
<p>$$det(A*B)=det(A)*det(B)$$</p>
<p>So if you have a bijective (non-singular,$\ det(T)\neq 0$) Transformation T follows:</p>
... |
1,775,166 | <p>If $$T:V\rightarrow V$$ is a bijective linear transformation then can $T^2$ also be a bijective linear transformation? Can $T^n$ also be bijective linear transformation?</p>
| Henry Swanson | 55,540 | <p>In fact, $T^n$ <em>must</em> be a bijective linear transformation.</p>
<p>A function has an inverse iff it's bijective. Since $T$ is bijective, it has an inverse, $T^{-1}$. So consider $(T^{-1})^n$ as a candidate for the inverse of $T^n$.
$$ (T^{-1})^n \circ T^n = (T^{-1})^{n-1} \circ T^{-1} \circ T \circ T^{n-1} =... |
314,281 | <p>Let <span class="math-container">$B=-B$</span> be a nowhere dense bounded closed convex set in the Hilbert space <span class="math-container">$\ell_2$</span> such that the linear hull of <span class="math-container">$B$</span> is dense in <span class="math-container">$\ell_2$</span>.</p>
<blockquote>
<p><strong>Q... | Nik Weaver | 23,141 | <p>No, take <span class="math-container">$B$</span> to be the set of vectors in the unit ball of <span class="math-container">$l^2$</span> whose first coordinate is zero. If <span class="math-container">$T: l^2 \to l^2$</span> is a bounded operator and <span class="math-container">$T(B)$</span> is compact, then letting... |
3,114,873 | <p>Does the function <span class="math-container">$\frac{1-2xy}{x^2 +y^2}$</span> have a max or min value for <span class="math-container">$(x,y)=/=0$</span>?</p>
<p>What I've tried so far is to take the the partial derivatives:</p>
<p><span class="math-container">$$\frac{\partial f}{\partial x} = \frac{2(-x+x^2*y -... | Daniel Schepler | 337,888 | <p><strong>Hint:</strong> Observe that for any <span class="math-container">$\sigma \in S_{n-1}$</span>, we have <span class="math-container">$\sigma(n) = n$</span>. Now, if <span class="math-container">$\sigma \in (1~n) \cdot S_{n-1}$</span>, then in that case what can we say about <span class="math-container">$\sigm... |
1,040,846 | <p>I want to prove, that $a_n$ is a null sequence if $$\lim_{n \to \infty}|\frac{a_{n+1}}{a_n}|= c < 1$$</p>
<p>That means that $\forall \epsilon > 0: \exists N \in \mathbb{N}: n \ge N: |\frac{a_{n+1}}{a_n} - c| < \epsilon$</p>
<p>How can I get rid of the $a_{n+1}$ and the $c$, to show $\forall \epsilon >... | Martín-Blas Pérez Pinilla | 98,199 | <p>Alternative solution: for $n$ large enough
$$0\le|a_{n+1}|\le|a_n|,$$
i.e., $(|a_n|)_{n\ge n_0}$ is bounded and monotonic. Let be $l=\lim_{n\to\infty}|a_n|\ge 0$. If $l\ne 0$, then
$$\lim_{n\to\infty}\frac{|a_{n+1}|}{|a_n|} = \frac{l}{l} = 1\ne c.$$
Finally,
$$
0 = \lim_{n\to\infty}|a_n|=|\lim_{n\to\infty}a_n|\impli... |
1,665,931 | <p>Let $a,c \in \mathbb{R}$ and $b \in \mathbb{C}$. We consider the equation $$a \bar{z}z+ b\bar{z} + \bar{b}z+c=0.$$ What curve it represents in the complex plane?</p>
<p>I think it a circle, but I am not able to conclude. Any help?</p>
| egreg | 62,967 | <p>If $a\ne0$, write $B=-b/a$ and $C=c/a$, so the equation is
$$
z\bar{z}-B\bar{z}-\bar{B}z+B\bar{B}=B\bar{B}-C
$$
that can be written
$$
|z-B|^2=B\bar{B}-C
$$
which is a circle, with center $B$ and radius $B\bar{B}-C$, if $B\bar{B}-C>0$, the point $B$ if $B\bar{B}-C=0$, the empty set if $B\bar{B}-C<0$.</p>
<p>I... |
348,276 | <p>I am not an expert, thus I apologize if my question is very naive. Let <span class="math-container">$\mathsf{M}$</span> be a model category (I do not assume any functoriality on the factorization),</p>
<blockquote>
<p><strong>Q1.</strong> Is there a reference where it is proven that (all) homotopy colimits exist?</p... | Andrea Gagna | 42,137 | <p>This is mostly a comment to Simone's thorough answer. It addresses a different, arguably more natural, way to deal with categories of functor <span class="math-container">$\mathbf{M}^I$</span>, with <span class="math-container">$\mathbf{M}$</span> a model category; but I have nothing to add with respect to Ivan's qu... |
348,276 | <p>I am not an expert, thus I apologize if my question is very naive. Let <span class="math-container">$\mathsf{M}$</span> be a model category (I do not assume any functoriality on the factorization),</p>
<blockquote>
<p><strong>Q1.</strong> Is there a reference where it is proven that (all) homotopy colimits exist?</p... | Ivan Di Liberti | 104,432 | <p>I recently found a very convincing and elegant proof of <strong>Q1</strong> in <em>Balzin</em>'s <strong><a href="https://arxiv.org/abs/1803.00681" rel="nofollow noreferrer">Reedy model structures in families</a></strong>. In the paper, it appears as <strong>Cor. 3.41.</strong></p>
<p>The <em>idea of the proof</em... |
2,762,951 | <p>Let $u$ be a $W^{1,p}$-Sobolev function defined on the half disk $D^2_+ := \{(x,y)\in \mathbb{R}^2| x^2+y^2 < 1, y > 0\}$.</p>
<p>If I define a function $\tilde u$ by
$$ \tilde u\colon (x,y) \mapsto \begin{cases} u(x,y) & y > 0 \\ u(x,-y) & y < 0\end{cases}, $$
is this $\tilde u$ a Sobolev $W^{1... | Klaus Niederkrüger | 333,322 | <p>Thank you Revzora for your answer. I'll just add in a few more details to your solution.</p>
<p>Indeed by a <a href="https://en.wikipedia.org/wiki/Sobolev_space#Absolutely_continuous_on_lines_(ACL)_characterization_of_Sobolev_functions" rel="nofollow noreferrer">theorem due to Nykodim</a> a function $f$ is Sobole... |
2,906,282 | <p>I want calculate the reflection from a straight <strong>R</strong> on the surface from the straight <strong>BC</strong>: </p>
<p><a href="https://i.stack.imgur.com/bOINy.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/bOINy.png" alt="enter image description here"></a></p>
<p>The triangle with th... | PM. | 416,252 | <p>Each elementary matrix $E_i$ is invertible so $\det(E_i)\ne0$.</p>
<p>As $$\det(E_iE_j)=\det(E_i)\det(E_j)\ne0$$ then $E_iE_j$ is invertible.</p>
|
3,224,765 | <p>The following question was asked on a high school test, where the students were given a few minutes per question, at most:</p>
<blockquote>
<p>Given that,
<span class="math-container">$$P(x)=x^{104}+x^{93}+x^{82}+x^{71}+1$$</span>
and,
<span class="math-container">$$Q(x)=x^4+x^3+x^2+x+1$$</span>
what is t... | Bill Dubuque | 242 | <p>The key idea employed here is the <strong>method of simpler multiples</strong> - a very widely used technique. Note that <span class="math-container">$\,Q\,$</span> has a "simpler" multiple <span class="math-container">$\,QR = x^5\!-\!1,\,$</span> so we can first reduce <span class="math-container">$P$</s... |
3,224,765 | <p>The following question was asked on a high school test, where the students were given a few minutes per question, at most:</p>
<blockquote>
<p>Given that,
<span class="math-container">$$P(x)=x^{104}+x^{93}+x^{82}+x^{71}+1$$</span>
and,
<span class="math-container">$$Q(x)=x^4+x^3+x^2+x+1$$</span>
what is t... | nonuser | 463,553 | <p>Let <span class="math-container">$a$</span> be zero of <span class="math-container">$x^4+x^3+x^2+x+1=0$</span>. Obviously <span class="math-container">$a\ne 1$</span>. Then <span class="math-container">$$a^4+a^3+a^2+a+1=0$$</span>
so multiply this with <span class="math-container">$a-1$</span> we get <span class="ma... |
101,566 | <p>How do I generalize the equation to be able to plug in any result for $\phi(n)=12$ and find any possible integer that works?</p>
| Pete L. Clark | 299 | <p>Note that $\lim_{n \rightarrow \infty} \varphi(n) = \infty$, so for any $k \in \mathbb{Z}^+$, the equation $\varphi(n) = k$ has at most finitely many solutions. How to find all of them?</p>
<p>Solutions to $\varphi(n) = k$ for very small values of $k$ can easily be worked out by hand and probably should be as a go... |
10,674 | <p>Let $p$ be a rational prime and $K$ a number field.
Dedekind's discriminant theorem tells us that
$p$ ramifies in $K$ $\iff$ $p$ divides the discriminant of $K$.
Hence if $p$ does not divide discriminant of $K$,
$(p)$ either splits, i.e., </p>
<p>(i) $(p)=P_1 \cdots P_g$ for $P_i \neq P_j$ and $g \geq 2$
or </p>... | tkr | 389 | <p>Adding a few more words to Pete's answer...In the Galois case, at least, the inertia degrees $f$ and ramifications indices $e$ of each prime $P_i$ lying over $p$ are the same. If $p$ is unramified then $e = 1$ and so the only question is what is $f$ and what is $g$? The common degree $f$ is the order of any (and eac... |
2,692,957 | <p>I am interested in the isometries of the hyperbolic plane in the Beltrami–Klein disk model. (<a href="https://en.wikipedia.org/wiki/Beltrami%E2%80%93Klein_model" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Beltrami%E2%80%93Klein_model</a>) The Wikipedia article does not say anything about the structure o... | Alan C | 5,894 | <p>In addition to Lee Mosher's very nice answer, here are some texts that some of my friends have recommended:</p>
<ul>
<li>John Ratcliffe: <em>Foundations of Hyperbolic Manifolds</em> </li>
<li>Cannon, Floyd,
Kenyon, Parry: <em>Hyperbolic Geometry</em></li>
</ul>
|
29,530 | <p>On the questions tab can we have a filter feature (I mean like frequented votes etc) of watched tags. I am aware that the tags that I have watched are demarked in a different colour. But I would like to discuss the pros and cons of getting a dedicated watched tags filter because I feel most people are well versed in... | Andrew T. | 174,736 | <p><a href="https://meta.stackexchange.com/questions/330326/custom-filters-release-announcement">Custom Question Lists</a> is now available on all SE sites.</p>
<p>One of the features is a filter to list only watched tags:</p>
<blockquote>
<p><img src="https://i.stack.imgur.com/rKGxt.png" alt="Custom Question Lists... |
960,026 | <p>I'm having some trouble finding out how to calculate the probability of the following problem:</p>
<p><img src="https://i.stack.imgur.com/1VdCX.png" alt="enter image description here"></p>
<p>I'm confused as to how I should be calculating the probability of B. I'm not sure how to factor in the possibility of A win... | Graham Kemp | 135,106 | <p>Let $K$ be the event that party A wins in Kentucky. Let $T$ be the event that party A wins in Tennessee. The events of B winning in each state is the complementary events $K^c, T^c$</p>
<p>We are given: $\mathsf P(K)=0.50,\, \mathsf P(T)=0.50,\, \mathsf P(T\mid K)=0.72$</p>
<p>We wish to find the probability tha... |
718,590 | <p>Is it true that in general complete metric space $(M,d)$, a closed ball of radius $r$ centered at $p\in M$ is always compact? That is, the ball is the set of all points $\left\{x:d(x,p)\leq r\right\}$.</p>
| Vincent Boelens | 94,696 | <p>For a Banach space $X$ the closed unit ball is compact iff dim $X<\infty$.</p>
|
2,940,072 | <p><span class="math-container">$$\left(\frac{f}{g}\right)'(x_{0})=\frac{f'(x_0)g(x_0)-f(x_0)g'(x_0)}{g^2(x_0)}$$</span></p>
<p>So, <span class="math-container">$\frac{1}{g}.f=\frac{f}{g}$</span>, then <span class="math-container">$$\frac{f}{g}'(x_0)=\frac{f(x)\frac{1}{g(x)}-f(x_0)\frac{1}{g(x_0)}}{x-x_0}=f(x)\frac{\f... | Neeyanth Kopparapu | 591,204 | <p>You are close. Note that instead of cancelling the term <span class="math-container">$\frac{\frac{f(x)}{g(x_0)}}{x-x_0}$</span> you actually subtract it twice. Second, note that the <span class="math-container">$$\lim_{x \rightarrow x_0} \frac{\frac{1}{g(x)} - \frac{1}{g(x_0)}}{x-x_0} = \left(\frac{1}{g(x_0)}\right)... |
2,298,359 | <p>Which of the following has an element that is less than any another element in that set?</p>
<p>I. The set of positive rational numbers</p>
<p>II. The set of positive rational numbers $r$ such that $r^2$ is greater than or equal to $2$</p>
<p>III. The set of positive rational numbers such that $r^2 > 4$. </p>
... | Toby Mak | 285,313 | <p>HINT:</p>
<p>An important point for these types of questions are understanding the properties of rational numbers. For example, by contradiction, question I is not true. Let us suppose there is a smallest positive rational number $\frac{p}{q}$. We can multiply by $\frac{1}{2}$ to get $\frac{p}{2q}$. However, this i... |
1,174,433 | <p>I'm trying to prove $\mathbb Z_p^*$ is a group if and only if $p$ is prime. I know that if $p$ is prime $\mathbb Z_p^*$ is a group, but how can I do the converse? In another words, if the equation $ax\equiv 1 (\mod m)$ has solution for every integer $a$ which is not divisible by $m$, then $m$ is prime.</p>
<p>I'm ... | Asinomás | 33,907 | <p>The equation in $x$ : $ax\equiv 1 \bmod n$ has a solution if and only if $(a,n)=1$. If $n$ is not prime take $a$ a proper divisor of $n$. Then $a$ is not divisible by $n$ , while $ax\equiv 1$ has no solution.</p>
<p>Although note that $\mathbb Z_p^*$ is the group in which the elements are the congruence classes rel... |
1,984,190 | <p>$X \in \mathbb{R}$ is the number of coin tosses it takes to get the first head.</p>
<p><strong>Question</strong>: find the probability of getting the first head at an even number of tosses.</p>
<p><strong>Official solution</strong>:
$P(X_{even}) = \frac{1}{2^{2}} + \frac{1}{2^{4}} + ... = \frac{1}{3}$ (sum of a ge... | martini | 15,379 | <p>Because $X_{\rm odd} \ne X_{\rm even}$. The most probable result is to get a head with the first toss (we have $P(X = 1) = \frac 12$), and $1$ is odd.</p>
|
1,984,190 | <p>$X \in \mathbb{R}$ is the number of coin tosses it takes to get the first head.</p>
<p><strong>Question</strong>: find the probability of getting the first head at an even number of tosses.</p>
<p><strong>Official solution</strong>:
$P(X_{even}) = \frac{1}{2^{2}} + \frac{1}{2^{4}} + ... = \frac{1}{3}$ (sum of a ge... | Mees de Vries | 75,429 | <p>Another way of looking it is as follows: suppose that you throw pairs of coins, one for odd, one for even, the odd one first, until at least one of them comes up heads. Then, when your experiment has ended, the outcome of the last two coins is either HH, HT, or TH. In two out of those three, the odd one gets the fir... |
2,379,881 | <p>I want to know if this integral converges or not : $\int _{1}^{\infty }\!{\frac {\sin \left( \cos \left( x \right) +\sin\left( x\sqrt {3} \right) \right) }{x}}{dx}.$ I tried to integrate by parts or to use dirichlet's test, but it seems impossible to prove that $\int _{1}^{x}\!\sin \left( \cos \left( t \right) +\si... | cdamle | 352,854 | <p>Here is my idea to solve this problem: </p>
<p>Note that $\cos x$ has positive values on $I_n = (\frac{2n-1}{2} \pi, \frac{2n+1}{2} \pi)$ for every $n \in \mathbb{Z}$ and that $\sin (\sqrt{3} x)$ has positive values on $J_k = (\frac{2k}{\sqrt{3}} \pi, \frac{2k+1}{\sqrt{3}} \pi)$ for every $k \in \mathbb{Z}$. </p>
... |
353,630 | <p>Consider a uniform random tournament with <span class="math-container">$n$</span> vertices. (Between any two vertices <span class="math-container">$x,y$</span>, with probability <span class="math-container">$0.5$</span> draw an edge from <span class="math-container">$x$</span> to <span class="math-container">$y$</sp... | Seva | 9,924 | <p>The outdegree of every fixed vertex is distributed binomialy with the mean <span class="math-container">$n/2$</span> and the variance <span class="math-container">$n/4$</span>. Hence, the probability that the outdegree deviates from <span class="math-container">$n/2$</span> by <span class="math-container">$cn/3$</sp... |
1,499,822 | <p>So we were shown this problem from our first functional analysis lecture and I was wondering if someone could help or give a hint:</p>
<blockquote>
<p>Show that the unit sphere $\mathbb{S}^{n-1} := \{ x \in \mathbb{R}^n: \| x \| = 1 \}$ is a complete metric space equipped with $d(x,y):= \arccos \langle x,y \rangl... | Christian Blatter | 1,303 | <p>For $x$, $y\in S^{n-1}$ write $\arccos\langle x,y\rangle=:\theta_{xy}$. Then $$d(x,y):=\theta_{xy}\in[0,\pi]$$ is nothing else but the usual spherical distance on $S^{n-1}$. Using elementary geometry one immediately verifies that
$$|x-y|=2\sin{\theta_{xy}\over 2}\qquad(x, \>y\in S^{n-1})\ .\tag{1}$$
In particula... |
89,987 | <p>In the sequence: </p>
<p>$$\lim _{n\rightarrow \infty }{\frac {n+1}{2\,n+3}}\neq 3$$</p>
<p>I know how to prove that the limit is actually $1/2$, but is there another way to prove that 1 is NOT the limit? </p>
<p>I tried to prove by negation showing that if 1 is the limit I can't find an $N$ that for every epsilo... | GEdgar | 442 | <p>For example, show the limit is $\le 1/2$ like this:</p>
<p>$$
\frac{n+1}{2n+3} < \frac{n+1}{2n+2} = \frac{1}{2}
$$</p>
|
4,374,739 | <p>Spivak's chapter 6, question 1 asks: for which of the following functions <span class="math-container">$f$</span> is there a continuous function <span class="math-container">$F$</span> with domain <span class="math-container">$\mathbb{R}$</span> such that <span class="math-container">$F(x) = f(x)$</span> for all <sp... | Cameron Buie | 28,900 | <p>The general idea is to pick some positive <span class="math-container">$\varepsilon$</span> and some irrational <span class="math-container">$z,$</span> then show that there is an open interval containing <span class="math-container">$z$</span> such that for every rational <span class="math-container">$x$</span> (in... |
1,398,464 | <p>Folks - Please help. What's the gradient for the cost function below?</p>
<p>$ D(Y||AX)=\frac{1}{2} ||Y-AX||^2_F $</p>
<p>Additional info - </p>
<p>-need to get the derivative of that with respect to A.
-Multiplicative NMF Algorithms Based on the Squared Euclidean Distance</p>
| john316 | 262,158 | <p>Let $M=(AX-Y)$, then the function and its differential can be expressed in terms of the Frobenius (:) product as
$$\eqalign{
f &= \frac{1}{2}\,M:M \cr\cr
df &= M:dM \cr
&= (AX-Y):dA\,X \cr
&= (AX-Y)X^T:dA \cr
}$$
Since $df=(\frac{\partial f}{\partial A}):dA\,\,$ the gradient must be
$$... |
4,610,313 | <p>I am working on AoPS Vol. 2 exercises in Chapter 1 and attempting to solve the below problem:</p>
<blockquote>
<p>Given that <span class="math-container">$\log_{4n} 40\sqrt{3} = \log_{3n} 45$</span>, find <span class="math-container">$n^3$</span> (MA<span class="math-container">$\Theta$</span> 1991).</p>
</blockquot... | lone student | 460,967 | <p>In this answer, we provide <em>another</em> method that uses the <em>definition</em> and <em>basic properties</em> of logarithm.</p>
<p>The original equation states :</p>
<p><span class="math-container">$$
\begin{align}&\log_{4n}{40}\sqrt{3}=\log_{3n}{45}\\
\implies &{40}\sqrt 3=\left(4n\right)^{\log_{3n}45}... |
2,618,813 | <p>What is the image of the set {$ { {z ∈ C : z = x + iy, x ≥ 0, y ≥ 0} } $} under
the mapping $ z \to z^2$</p>
<p>my answer : $f(z) = z^2 =(x+iy)^2 = x^2-y^2 +2ixy$,</p>
<p>here I get $u=x^2-y^2$ and $v=2xy$. After that I am confused that how can I find the image of the set.</p>
<p>Please help me, as any Hints can... | Sahiba Arora | 266,110 | <p><strong>Hint:</strong> The map $z \mapsto z^2$ is just the map $$re^{ i\theta}\mapsto r^2e^{ i2\theta}.$$</p>
|
1,481,627 | <p>I have been given homework in which the $2\times 2$ matrices of determinant $1$ are equipped with the subspace topology of $\mathbb{R}^4$. However, $\mathbb{R}^4$ is a space of 4-tuples, while 2x2 matrices are not n-tuples. How do I get the open sets of this topology?</p>
<p>How is this set of matrices a subset of ... | Community | -1 | <p>Any matrix
$$\begin{pmatrix}
a & b\\
c & d
\end{pmatrix}\in\mbox{Mat}^{2\times 2}(\mathbb{Z})$$
can be interpreted as the touple $(a,b,c,d)\in\mathbb{R}^4$, so $\mbox{Mat}^{2\times 2}(\mathbb{Z})$ can be thought of a subspace of $\mathbb{R}^4$ and be given the subspace topology.</p>
|
2,798,338 | <p>Let $G$ be a group of order 20, Prove that $G$ has a normal subgroup of order 5.</p>
<p>Obviously by Sylow theorem there is a subgroup of order 5, and since all Sylow p-subgroups are conjugate the only problem is to show that there is only one sungroup of order 5.</p>
<p>any help would be appreciated</p>
| alper akyuz | 483,006 | <p>Or we can say just use the fact that there is only 1 group of order 5 which is cyclic group of order 5 and it is normal, we also know that all p subgroups are conjugate, but cyclic group of order 5 is normal so it only conjugates with itself.</p>
|
3,271,507 | <p>Given a ring <span class="math-container">$R$</span> and a multiplicatively closed system <span class="math-container">$S\subset R$</span>, we define the ring <span class="math-container">$S^{-1}R=\{\frac rs | r\in R, s\in S\}$</span>. However if I have <span class="math-container">$R=\Bbb Z$</span> and <span class=... | Alekos Robotis | 252,284 | <p>Be careful: <span class="math-container">$3\mathbb{Z}=\{m\in \mathbb{Z}: m=3k\:\text{for}\:k\in \mathbb{Z}\}.$</span> <span class="math-container">$S=\mathbb{Z}\setminus 3\mathbb{Z}$</span> is indeed a multiplicative system, but actually consists of all integers not divisible by <span class="math-container">$3$</spa... |
1,915,273 | <p>$$\int_0^r\sqrt{x-x^2}.dx$$</p>
<p>I only have basic calculus and would like to know how would one go about integrating an expression of this form.
I have tried substituting say $u=x-x^2$ but I'm still left with an $x$.</p>
<p>The method is not in my book and I can't find a similar example anywhere.</p>
<p>A hint... | egreg | 62,967 | <p>First of all, the function is defined for $0\le x\le 1$, so we have to assume the same for $r$.</p>
<p>Consider the curve $y=\sqrt{x-x^2}$ that we can square leading to $x^2+y^2-x=0$, which is the equation of a circle with center at $(1/2,0)$ and radius $1/2$. Thus a substitution that can work is
$$
x-\frac{1}{2}=\... |
3,636,563 | <p>How to find the values of <span class="math-container">$x,y$</span> and <span class="math-container">$z$</span> if
<span class="math-container">$3x²-3(1126)x=96y²+24(124)y=8z²-4(734)z $</span>?</p>
<p>I dont have any idea!! I think we can have many values of <span class="math-container">$x,y$</span> and <span clas... | manifold | 363,892 | <p>let partial sum be
<span class="math-container">$$S_n = \sum_{k=2}^n \int_{\frac{1}{k}}^{\frac{1}{k-1}}f$$</span> then we have to show that <span class="math-container">$\lim_{n \rightarrow \infty} S_n = \int_0^1 f$</span> and key point we will use is that <span class="math-container">$ \exists M>0 \,\,\, |f| <... |
406,966 | <p>Define a graph with vertex set <span class="math-container">$\mathbb{R}^2$</span> and connect two vertices if they are unit distance apart. The famous <a href="http://en.wikipedia.org/wiki/Hadwiger%E2%80%93Nelson_problem" rel="nofollow noreferrer">Hadwiger-Nelson problem</a> is to determine the <a href="http://en.wi... | MJD | 25,554 | <p>Let $S$ be a set of integers, each greater than 1, and $m(S)$ be the largest positive integer that is <em>not</em> expressible as a sum of elements of $S$, if it exists. (Elements of $S$ may be used more than once, of course, so that $7 = 2+2+3$ is considered to be a sum of 2's and 3's.) <a href="https://en.wikiped... |
118,185 | <blockquote>
<p>Given a polynomial f with only real coefficients, the zeros of f are conjugates of zeros of f.</p>
</blockquote>
<p>(Sorry my English left me, when trying to formulate this theorem.) </p>
<p>So I would like to know how to refer to this theorem in a scientific text.</p>
| Stefan Hansen | 25,632 | <p>There is an equivalent definition of uniform integrability which goes: Let $(\Omega,\mathcal{F},P)$ be a probability space. A set $\mathcal{H}$ of random variables is uniformly integrable if and only if
$$
\sup_{X\in \mathcal{H}}\int_\Omega |X| dP < \infty
$$
and
$$
\forall \epsilon >0\, \exists \delta >... |
118,185 | <blockquote>
<p>Given a polynomial f with only real coefficients, the zeros of f are conjugates of zeros of f.</p>
</blockquote>
<p>(Sorry my English left me, when trying to formulate this theorem.) </p>
<p>So I would like to know how to refer to this theorem in a scientific text.</p>
| Did | 6,179 | <p>Three conditions are involved:</p>
<blockquote>
<p>(C) For every <span class="math-container">$\varepsilon\gt0$</span>, there exists a finite <span class="math-container">$c$</span> such that, for every <span class="math-container">$X$</span> in <span class="math-container">$\mathcal H$</span>, <span class="math-con... |
1,861,292 | <p>I'm reading Osbourne's Basic Homological Algebra, and on page 18 he has this situation where we've got a ring $R$ and a right-ideal $I$, and some left $R$-module $B$. He says $I\otimes B$ is not a subgroup of $R\otimes B$, but he doesn't give an example, just states "the free abelian groups and equivalence relations... | Rob Arthan | 23,171 | <p>One example is the case $R = \Bbb{Z}_4$, $B = I = \{0, 2\}$ that Osbourne is talking about at that point in the book. In this example, you find that $I \otimes B \simeq R \otimes B \simeq \Bbb{Z}_2$, but the natural mapping from $I\otimes B$ to $R \times B$ is not an injection. In fact, $2 \otimes 2$ generates $I \o... |
325,890 | <p>To evaluate this integral I have got to use Parseval's Theorem and the fourier transform of</p>
<p>$$s(x)=\begin{cases}
0 & x\leq -a \\
1 & -a<x<a \\
0 & x \geq a.\end{cases}$$</p>
<p>This works out to be</p>
<p>$$\tilde{s}(k)=\frac{e^{ika}}{ik}-\frac{e^{-ika}}{ik}$$</p>
<p>I have then changed... | TCL | 3,249 | <p>Take $a=1$ in your function $s(x)$. The Fourier transform of $s$ is $$\frac{1}{\sqrt{2\pi}}\int_{-1}^1 e^{-iwx}\,dx=\sqrt{\frac{2}{\pi}}\frac{\sin w}{w} $$ Then Parseval (also called Plancherel) relation says $$\int_{-\infty}^{\infty}\frac{2}{\pi}\frac{\sin^2w}{w^2}\,dw = \int_{-1}^1 1^2\,dx=2.$$</p>
|
177,643 | <p>Let $I=\{0, 1, \ldots \}$ be the multiplicative semigroup of non-negative integers. It is possible to find a ring $R$ such that the multiplicative semigroup of $R$ is isomorphic (as a semigroup) to $I$?</p>
| robjohn | 13,854 | <p>With all these simple solutions available, how about a complex one?</p>
<ol>
<li>The argument of the product of two complex numbers is the sum of their arguments.</li>
<li>The imaginary part of a complex number divided by its real part is the tangent of its argument.</li>
</ol>
<p>Consider the product
$$
\begin{al... |
3,603,066 | <p>My question: How does one distinguish between two embeddings of the same graph on the plane? </p>
<p>For instance, are two such embeddings considered the same if the degree sequence of their faces are the same? Or does even the slightest change, such as drawing an edge more crooked in one embedding than the other, ... | User49582934 | 106,207 | <p>This can be solved by taking log on both sides.</p>
<p>Growth formula is given by
<span class="math-container">$$ P = P_0 e^{rt} $$</span></p>
<p>Here, <span class="math-container">$P_0$</span> and <span class="math-container">$P$</span> are initial and final length of the tree, respectively, <span class="math-co... |
2,586,466 | <p>Write the value of $n$ if the sum of n terms of the series $1+3+5+7...n =n^2$.</p>
<p>I'm not getting the right value if I proceed with the general formula for finding sum of n terms of a arithmetic series. The general summation formula for arithmetic series is $\frac{n(2a+(n-1)d)}{2}$, where $a$ is the first term,... | Community | -1 | <p>Note that the sum of an arithmetic progression with $n$ terms having a starting term $a$ and common difference $d$ is: $$S_n = \frac{n}2 \left[2a + (n-1)d\right]$$ Now, if $a=1, d=2$, what do you get?</p>
<p>$$S_n = \frac{n}{2}\left[2+(n-1)2\right] = \frac{n}{2}\left[2n \right] = n^2$$</p>
|
2,759,112 | <p>Are there cases where two events have high probability (i.e. each of them has high probability) but, at the same time, the probability that they both obtain is low? If yes, could you please provide a simple, daily-life example?</p>
| trancelocation | 467,003 | <p>There is always a restriction to the probability of $A \cap B$:</p>
<p>$$1 \geq P(A \cup B) = P(A) + P(B) - P(A \cap B) \Rightarrow$$
$$P(A \cap B) \geq P(A) + P(B) -1$$
For example:</p>
<ul>
<li>$P(A) = 0.9$ and $P(B) = 0.9 \Rightarrow P(A \cap B) \geq 0.9+0.9-1 = 0.8$</li>
</ul>
<p>So, the higher the probabilit... |
2,738,679 | <p>I am having trouble to prove this exercise for a Real Analysis course. I tried two approaches. However, I am insecure about both of them.</p>
<p>This is the question:</p>
<p>If $X$ is a partition of ℕ* such that:</p>
<p>i) $1 ∈ X$;</p>
<p>ii) For all $n ∈ ℕ$*, if $n ∈ X$, then $2n ∈ X$;</p>
<p>iii) For all $n ∈... | Rai | 548,890 | <p>Your second approach is correct.<br>
You want to prove that $X = \mathbb{N}$<br>
For being $X$ a subset of $\mathbb{N}$ , then $X \subset\mathbb{N}$<br>
Now we have to proof that $\mathbb{N}\subset X$<br>
So let be $n \in\mathbb{N}$, then it exist some $k \in\mathbb{N}$ with $n=2^{k+1}-p$, where p$\leq k$<br>
For i)... |
6,155 | <p>I have a category $C$, which is equipped with a symmetric monoidal structure (tensor product $\otimes$, unit object $1$). My category also has finite coproducts (I'll write them using $\oplus$, and write $0$ for the initial object), and $\otimes$ distributes over $\oplus$.</p>
<p>By an <em>exponential monad</em>, ... | Peter Teichner | 4,625 | <p>There is a beautiful explanation of the Chern character that my student Fei Han proved in his thesis: The Chern character is given by the map that "crosses with the circle". The hard part is to explain why domain and range of this map are K-theory respectively de Rham cohomology. This uses isomorphisms</p>
<p>$K^0(... |
23,639 | <p>Prove that for every positive integer $x$ of exactly four digits, if the sum of digits is divisible by $3$, then $x$ itself is divisible by 3 (i.e., consider $x = 6132$, the sum of digits of $x$ is $6+1+3+2 = 12$, which is divisible by 3, so $x$ is also divisible by $3$.)</p>
<p>How could I approach this proof? I'm... | Qiang Li | 6,016 | <p>Actually, this is true for an integral number with any digits. The proof is quite easy. Let's denote the integral number by $\overline{a_n a_{n-1} \ldots a_1}$. If the sum of its digits $\sum_{i=1}^n{a_i}$ is divisible by 3, then $\sum_{i=1}^n{(1+\overline{9...9}_{i-1})*a_i}$ is too. Here $\overline{9...9}_{i-1}$ de... |
179,322 | <p>I'm interested in classes C of $R^1$-valued random variables which possess the following properties:</p>
<p>1) the sum of two independent random variables from class C belongs to class C;</p>
<p>2) for any $\lambda \in R^1$, $\xi \in C$ we have $\lambda\xi \in C$; </p>
<p>3) any random variable from class $C$ has... | Vivek Shende | 4,707 | <p>I think at some level this is a convention having to do with the fact that our exponential is $x \mapsto e^x$ rather than $x \mapsto e^{2\pi i x}$. But having made this choice, here's one way that factor appears:</p>
<p>Over $\mathbb{C}$, you have the exponential sequence </p>
<p>$$0 \to 2\pi i \mathbb{Z} \to \ma... |
50,923 | <p>I have two questions:</p>
<ol>
<li><p>How can I calculate compound interest when I want to reinvest a percentage of the sum of my principal and interest periodically?</p>
<p>To make it clearer,</p>
<p>Usually I use this formula for calculating compound interest: <span class="math-container">$M = P( 1 + i )^n$</span>... | Ross Millikan | 1,827 | <p>For 2, you just regard each day as a separate term. So the interest rate is the rate per day and you have 365 (or 366 or 360!?) terms. For 365, if the interest rate were 10%/year, it would be 10/365$\approx $0.0274%/day, and the final value would be $1.000274^{365}$ times the original value, or about 1.105 times t... |
50,923 | <p>I have two questions:</p>
<ol>
<li><p>How can I calculate compound interest when I want to reinvest a percentage of the sum of my principal and interest periodically?</p>
<p>To make it clearer,</p>
<p>Usually I use this formula for calculating compound interest: <span class="math-container">$M = P( 1 + i )^n$</span>... | AndyS | 13,294 | <p><strong>1.</strong> For question 1 you just have to redefine the amount you want to invest after the end of every period. If $x$ is the daily interest rate (if, for example, the daily interest rate is 5% then $x=0.05$), then after 30 days (1 month) your initial capital, $c$, has become
$$
c_1=(1+x)^{30} c.
$$</p>
... |
187,486 | <p>Let $SL(2,{\mathbb R})$ act on ${\mathbb R}^2$ by matrix multiplication.</p>
<p>What is known about group cohomology $H^*(SL(2,{\mathbb R}),{\mathbb R}^2)$?</p>
<p>And about $$H^*(\Gamma,{\mathbb R}^2)$$ for a cocompact lattice $$\Gamma\subset SL(2,{\mathbb R})?$$</p>
| Oscar Randal-Williams | 318 | <p>It is zero. This is an application of the "centre kills" trick, which I will state in homology.</p>
<p><strong>Trick.</strong> Let $M$ be a $G$-module for which there is an element $z$ in the centre of $G$ which acts as $-1$ on $M$. Then $2H_*(G;M)=0$.</p>
<p>In your situation the homology is a real vector space, ... |
200,298 | <p>How to find the sum of the absolute values for the roots of this equation:</p>
<p>$$x^4-4x^3-4x^2+16x-8=0$$</p>
| Clive Newstead | 19,542 | <p>Notice that
$$\begin{align}x^4-4x^3-4x^2+16x-8 &= (x-1)^4 - 10(x-1)^2 + 1 \\
&= ((x-1)^2-5)^2-24 \end{align}$$
so you can actually calculate the roots explicitly and sum their absolute values.</p>
|
1,687,206 | <p>Do I multiply $4$ to the power of $2$ or do I multiply $8$ to the power of $2$? What is the answer? I'm thinking it's $76$.</p>
| Soham | 242,402 | <p>$$V=3b+2b^2$$</p>
<p>So, when $b=4$, we get $$V=3\times4+2\times4^2=12+(2\times16)=12+32=44$$</p>
|
1,702,885 | <blockquote>
<blockquote>
<p>Question: Prove $$ \frac{\sin^3(x)-\cos^3(x)}{\sin(x)+\cos(x)} = \frac{\csc^2(x) -\cot(x) -2\cos^2(x)}{1-\cot^2(x)} $$</p>
</blockquote>
</blockquote>
<p>RHS: $$ \frac{\csc^2(x) -\cot(x) -2\cos^2(x)}{1-\cot^2(x)} $$</p>
<p>$$ ⇔ \frac{\frac{1}{\sin^2(x)} +\frac{\cos(x)}{\sin(x)}-2\... | lab bhattacharjee | 33,337 | <p>HINT:</p>
<p>Writing $\cos x\sin x=u$</p>
<p>$$1-u-2u^2=(1-2u)(1+u)$$</p>
<p>$1-2u=(\sin x-\cos x)^2$</p>
|
925,005 | <p>Show that $$\|a+b+c\|^2\leq 3\|a\|^2+3\|b\|^2+3\|c\|^2$$
where $a,b,c$ are in some Hilbert space $(H,\langle\cdot,\cdot \rangle)$?</p>
<p>I see that we have $$\|a+b\|^2\leq2\|a\|^2 +2 \|b\|^2$$ due to the parallelogram law. What about the first inequality?</p>
| Kim Jong Un | 136,641 | <p>You can expand the LHS and use $\left<x,y\right>+\left<y,x\right>\leq ||x||^2+||y||^2$ 3 times. This latter inequality follows by expanding the non-negative entity $\left<x-y,x-y\right>$.</p>
|
3,490,198 | <p>For all <span class="math-container">$a,b \in \mathbb{Z}$</span> and for all <span class="math-container">$m,n \in \mathbb{N}\setminus \left\lbrace0\right\rbrace$</span>,</p>
<p>is <span class="math-container">$a^{48m+1}+b^{48n+1} \equiv 0 \pmod{39} \iff a+b \equiv 0 \pmod{39}$</span>?</p>
<p>I think the answer is... | nonuser | 463,553 | <p>Hint: Use Euler theorem <span class="math-container">$$a^{\varphi(n)}\equiv 1\pmod n$$</span>
if <span class="math-container">$\gcd(a,n )= 1$</span>. Notice that <span class="math-container">$$\varphi (39) = 2\cdot 12 =24$$</span></p>
<hr>
<p><strong>Edit: 12. 29. 2019</strong></p>
<ul>
<li>a) If <span class="mat... |
1,281,142 | <p>I am trying to find all the closed points of $\mathbb{A}_{\mathbb{R}}^2$.</p>
<p>After a quick google research, I found that $\mathbb{A}_{\mathbb{R}}^2 = \operatorname{Spec}(\mathbb{R}[x,y])$ and then all we need to find is the maximal ideals of $\mathbb{R}[x,y]$.</p>
<p>However, set-theoretically this equalty doe... | Daniel McLaury | 3,296 | <p>Re: </p>
<blockquote>
<p>However, set-theoretically this equalty does not make sense, as we are talking about points and prime ideals.</p>
</blockquote>
<p>The points of a scheme are*, by definition, prime ideals of a ring. If this is new to you, you're probably not ready to be looking at this question until yo... |
243,849 | <p>I was working on an examples from my textbook concerning transforming formulae into disjunctive-normal form (DNF) until I found an expression that I cannot solve. I hope somebody can help me transform the following statement into DNF:</p>
<p>$$ (\lnot q \lor r) \land ( q \lor \lnot r)$$</p>
| Dominik | 50,527 | <p>By the distibution law one obtains $(\lnot q \lor r) \land ( q \lor \lnot r) \leftrightarrow ((\lnot q \lor r) \land q) \lor ((\lnot q \lor r) \land \lnot r) \leftrightarrow (\lnot q \land q) \lor (r \land q)\lor (\lnot q \land \lnot r) \lor (r \land \lnot r)\leftrightarrow (r \land q)\lor (\lnot q \land \lnot r)$... |
1,368,180 | <p>Here is a tricky compass and straightedge construction problem. </p>
<blockquote>
<p>Given triangle $\triangle ABC$ and point $D$ on segment $\overline{AB}$, construct point $P$ on line $\overleftrightarrow{CD}$ such that $\angle APB = \angle BAC$. </p>
</blockquote>
<p>This configuration appears frequently in O... | Tae Hyung Kim | 94,401 | <p>I misread the question the first time I answered this, so let me try again. </p>
<p>The condition that you gave $\angle APB = \angle BAC$ is the degenerate case of the "two angles that inscribe the same arc are equal" theorem. More specifically, if you were to draw the circumcircle of $\triangle APB$, you would see... |
1,368,180 | <p>Here is a tricky compass and straightedge construction problem. </p>
<blockquote>
<p>Given triangle $\triangle ABC$ and point $D$ on segment $\overline{AB}$, construct point $P$ on line $\overleftrightarrow{CD}$ such that $\angle APB = \angle BAC$. </p>
</blockquote>
<p>This configuration appears frequently in O... | PRITHU PURKAIT | 635,014 | <p>Construct line <span class="math-container">$AK$</span> with <span class="math-container">$K$</span> on <span class="math-container">$BC$</span> (produced, if necessary) and <span class="math-container">$\angle BAK = \angle ACB$</span>. The circumcircle of <span class="math-container">$\triangle ABK$</span> intersec... |
698,460 | <p>I've literally tried everything including proofs by cases. If I were to try a an odd integer, then I would get $8k^3 + 4k^2 + 8k^2 + 4k + 1$, which obviously couldn't convert to a $\mod 7$, and I'm honestly not sure what I can do! Does anyone here have any tips?</p>
<p>Note, this is a soft question.</p>
| Sangchul Lee | 9,340 | <p>Euler's identity will help you expand $\cos^{2N}\theta$ in terms of trigonometric polynomial. Then use the identity back to simplify them. For example, if $N = 1$, then</p>
<p>$$ \cos^{2}\theta = \left( \frac{e^{i\theta} + e^{-i\theta}}{2} \right)^{2} = \frac{e^{2i\theta} + 2 + e^{-2i\theta}}{4}= \frac{1}{2} + \fra... |
432,262 | <p>Let <span class="math-container">$x_0=1$</span> and
<span class="math-container">$$x_{k+1} = (1-a_k)\left(\frac{3}{2}-\frac{1}{2}\frac{1}{x_k}\right)$$</span>
where <span class="math-container">$a_n$</span> is a known sequence satisfying that <span class="math-container">$a_k\in(0,1)$</span> for all <span class="mat... | Alapan Das | 156,029 | <p>Taking <span class="math-container">$b_k=(1-a_k), y_k=\sqrt{2}x_k, c=\frac{3}{\sqrt{2}}$</span> we have <span class="math-container">$y_{k+1}=b_k(c-\frac{1}{y_k})$</span>. Now as <span class="math-container">$b_k \rightarrow 1$</span> assuming it's variation to be sufficiently small for large <span class="math-conta... |
718,664 | <p>I'm trying to find a small primitive root modulo $p^k$, where $p$ is prime. My strategy is to test small numbers $g=2,3,\ldots$ until I find a primitive root modulo $p$. That is, until $ord_pg=\phi(p)=p-1$. There are results that suggest such a search won't take long.</p>
<p>To go from modulo $p$ to modulo $p^k$... | mercio | 17,445 | <p>With computer search I found that $g=5$ for $p = 40487$ is an example of (2) .
<a href="http://www.wolframalpha.com/input/?i=5%5E40486+mod+1639197169" rel="nofollow">http://www.wolframalpha.com/input/?i=5%5E40486+mod+1639197169</a></p>
<p>I would guess that this should happen with probability $1/p$.
$\sum_{2 < p... |
2,760,605 | <p>$||x-1|-2|=|x-3|$.
Find the value of $x$.<br>
In my attempt I got the critical values of the expression as $1$ and $3$.
But I’m not sure is we can just not consider the $2$ in the LHS.</p>
<p>My steps are Case 1: when $x>3$, $x-1-2=x-3$, $0=0$</p>
<p>Case 2: $1<x<3$, $X-1-2=-x+3 \implies 2x=6 \implies x... | Cesareo | 397,348 | <p>Hint.</p>
<p>Use the Stirling asymptotic approximation formula</p>
<p>$$
n! = \sqrt{2\pi n}\left(\frac{n}{e}\right)^n(1+O(1/n))
$$</p>
<p>because</p>
<p>$$
\frac{n}{(n!)^{1/n}}= \frac{n}{(2\pi n)^{1/(2n)}(n/e)}(1+O(1/n)) = \frac{e}{(2\pi n)^{1/(2n)}}(1+O(1/n))
$$</p>
<p>hence </p>
<p>$$
\lim_{n\rightarrow \inf... |
85,984 | <p>I was doing good at school in plane geometry and trigonometry - especially in geometric proofs like proving the equality of two line segments or two angles - more than I was doing in analytic geometry.</p>
<p>I am considering doing research in mathematics to be my career (and my life) someday. and I am wondering ab... | Paul | 16,158 | <p>I am not sure this is something you are looking for. But I find it interesting: it is easy to understand and is kind of related to plane geometry: </p>
<p>A graph is planar if it can be drawn on the plane without any crossing. It is well known that every planar graph can be drawn on the plane such that every edge i... |
31,544 | <p>For example, how to make Mathematica display only the lines beginning with <code>In[some number]</code>?</p>
<pre><code>In[8]:= m1 = {{1, 1, 1}, {8, 4, 2}, {64, 16, 4}}
</code></pre>
<blockquote>
<pre><code>Out[8]= {{1, 1, 1}, {8, 4, 2}, {64, 16, 4}}
</code></pre>
</blockquote>
<pre><code>In[9]:= m1.{b1, b2, b3} ... | Szabolcs | 12 | <p>I see three ways:</p>
<ol>
<li><p>Copy and paste everything into a new notebook (to ensure we don't destroy the output for good). Then select Cell -> Delete All Output from the menu.</p>
</li>
<li><p><kbd>alt</kbd>-click (<kbd>⌥</kbd>-click on a Mac) on an input cell's bracket. All input cells will get selected... |
1,350,030 | <blockquote>
<p>Let's consider a field $F$ and $\alpha,\beta\in\overline{F}-F$ (where $\overline{F}$ is an algebraic closure). If $F(\alpha)=F(\beta)$, is it true that $\alpha$, $\beta$ have the same minimal polynomial over $F$? </p>
</blockquote>
<p>For reference, I explain my way. (From here forward, all field iso... | Zev Chonoles | 264 | <p>Let $a$ and $b$ be any two distinct elements of $F$. Then we have $F(a)=F(b)=F$ but the minimal polynomial of $a$ over $F$ is $x-a$, whereas the minimal polynomial for $b$ over $F$ is $x-b$, which are different.</p>
<p>This is just a trivial <em>example</em> of the general behavior. Let $F$ be <strong>any</strong>... |
80,242 | <p>Let $K_0$ be a bounded convex set in $\mathbb{R}^n$ within which lie two sets $K_1$ and $K_2$. $K_0,K_1,K_2$ have nonempty interior. 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, I assume that the bounda... | Joel David Hamkins | 1,946 | <p>You haven't said precisely what you mean by an algorithm here, and this is actually a nontrivial issue since your question is outside the usual finitary context of computability theory, so it isn't clear what you might mean, and there are choices to be made about it. </p>
<p>One way to make the concept precise woul... |
855,686 | <p>In a triangle, the sum of two sides is $x$ and the product of the same two sides is $y$.
If $x^2 - c^2=y$ where c is the third side, then what is the ratio of the inradius to the circumradius of the triangle?</p>
<p>I guess I have found half of it: if the two sides of the triangle are $a$ and $b$, then $x=a+b$ and... | Peter Woolfitt | 145,826 | <p>We have equations for the inradius $r$ and the circumradius $R$, namely
$r=\dfrac{\triangle}{s}$, and $R=\dfrac{abc}{4rs}$ where $\triangle$ denotes the area of the triangle and $s$ denotes the semiperimeter. Hence we have
$$\frac{r}{R}=\frac{4r^2s}{abc}=\frac{4\triangle^2}{sabc}$$
We have $c=\sqrt{x^2-y}$, $s=\dfra... |
1,920,624 | <p>Prove that if 1 is a linear combination of two integers $a, b$, then $a$ and $b$
are relatively prime. Use only basic divisibility properties. (Do not use GCDLC).</p>
<p>I can only figure out how to prove this using the GCDLC.</p>
| Fly by Night | 38,495 | <p>Assume that $ma+nb=1$, and that $a$ and $b$ are not coprime, i.e. there is an integer $d>1$ which divides both $a$ and $b$.</p>
<p>If $d$ divides $a$ and $b$ then there exist integers $s$ and $t$ for which $a=ds$ and $b=dt$. In that case</p>
<p>$$ma+nb=1 \iff m(ds)+n(dt)=1 \iff d(ms+nt)=1$$</p>
<p>It follows t... |
1,920,624 | <p>Prove that if 1 is a linear combination of two integers $a, b$, then $a$ and $b$
are relatively prime. Use only basic divisibility properties. (Do not use GCDLC).</p>
<p>I can only figure out how to prove this using the GCDLC.</p>
| Vidyanshu Mishra | 363,566 | <p>the combination possible is a-b(for any two consecutive integers with a>b).
a-b=1 so a=b+1, which means that a and b are consecutive integers.then one must be even and other must be odd, since odd and even numbers are coprime thus so are a and b.</p>
|
2,785,464 | <p>It's clear that every orthonormal basis is a frame, but is <strong>every</strong> basis a frame? In particular, is a non-orthogonal basis always a frame? If yes, why?</p>
<p>I'm asking about general Hilbert spaces, in particular infinite dimensional spaces.</p>
| pcp | 430,515 | <p>Let $(e_n)_{n\in \mathbb N}$ be an ONB for a Hilbert space. $(\frac{1}{n}e_n)_{n\in \mathbb N}$ is spanning and representation is unique, i.e., it is a basis. But it is not a frame, since
$$
\sum_{n\in \mathbb N} \left|\langle e_k, \frac{1}{n}e_n \rangle\right|^2 \leq \frac{1}{k^2}.
$$</p>
|
356,594 | <p>The roots of the quadratic equation $ax^ 2-bx+c=0,$ $a>0$, both lie within the interval $[2,\frac{12}{5}]$. Prove that:
(a) $a \leq b \leq c <a+b$.
(b) $\frac{a}{a+c}+\frac{b}{b+a}>\frac{c}{c+b}$</p>
<p>So we can use the quadractic formula and obtain
$2 \leq \frac{b \pm \sqrt{b^2-4ac}}{2a} \leq \frac{12}{5... | 23rd | 46,120 | <p>As shown in other answers, let $\alpha,\beta$ be the two real roots of $ax^2-bx+c=0$, i.e. $$ax^2-bx+c=a(x-\alpha)(x-\beta)\iff \alpha+\beta=\frac{b}{a},\ \alpha\beta=\frac{c}{a}.$$</p>
<p>Since $\alpha,\beta\in[2,2.4]$ and $a>0$,
$$\frac{b}{a}=\alpha+\beta\ge 4\Rightarrow a\le \frac{b}{4}<b;$$
$$\frac{b}{c}... |
2,916,600 | <p>So I understand the requirements for an orthonormal basis and everything around it. However, there's one thing I am missing:</p>
<p>Suppose you have two vectors which are orthonormal $u_1$ and $u_2$. According to the answerbook the multiplication of vector $u_1$ and $u_2$ results in another orthonormal vector $u_3$... | user | 505,767 | <p>Here we are using the property of <a href="https://en.wikipedia.org/wiki/Cross_product" rel="nofollow noreferrer"><strong>cross product</strong></a> which is defined only for $v\in \mathbb{R^3}$.</p>
<p><a href="https://i.stack.imgur.com/ACugs.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/ACugs... |
2,916,600 | <p>So I understand the requirements for an orthonormal basis and everything around it. However, there's one thing I am missing:</p>
<p>Suppose you have two vectors which are orthonormal $u_1$ and $u_2$. According to the answerbook the multiplication of vector $u_1$ and $u_2$ results in another orthonormal vector $u_3$... | José Carlos Santos | 446,262 | <p>The cross-product $u_1\times u_2$ of any two vectors $u_1$ and $u_2$ is always orthogonal to both of them. Furthermore,$$\lVert u_1\times u_2\rVert=\lVert u_1\rVert.\lVert u_2\rVert.\sin\theta,$$where $\theta$ is the angle between them. Therefore, if $u_1$ and $u_2$ are orthogonal and both of them have norm $1$, $u_... |
4,188,147 | <p>I'm trying to show the momentum operator <span class="math-container">$P : \mathcal D(P)\ \dot{=}\ C^\infty_0(\Bbb R)\to L^2(\Bbb R)$</span> with <span class="math-container">$P=-i\hbar\partial_x$</span> is essentially self-adjoint, which is equivalent to saying that it is symmetric (densely defined, which is true b... | Michał Miśkiewicz | 350,803 | <p>Jose27's approach is probably the simplest, but I suppose someone may find another approach useful too. The main point I'd like to stress is: there's a number of ways in which one manipulate weak derivatives in the same way as strong derivatives.</p>
<p><span class="math-container">$\newcommand{\R}{\mathbb{R}}$</spa... |
4,224,303 | <p><strong>Question</strong></p>
<p>Let <span class="math-container">$X_0, X_1, X_2, \dots$</span> be independent and identically distributed continuous random variables with density <span class="math-container">$f(x)$</span>. Let <span class="math-container">$N$</span> be the first index <span class="math-container">$... | tommik | 791,458 | <blockquote>
<p>N should follow a geometric distribution... what should the parameter for my geometric distribution be?</p>
</blockquote>
<p>That's correct.</p>
<p>To get the parameter observe that</p>
<p><span class="math-container">$$P(Y>X)=\int_{-\infty}^{\infty}f(x)\left[ \int_{x}^{\infty}f(y)dy\right]dx=$$</spa... |
706,250 | <p>For all $n\ge3\in \mathbb N$, if $n \equiv 3 \pmod{4}$ then ${3^n} \equiv 2 \pmod{5}$.</p>
<p>I tried to set $n = 3+4k$ but it doesn't help.</p>
<p>Any hints first please?</p>
| William Chang | 133,204 | <p>Simply: $3^n=3^{3+4k}$ for some $k \in \mathbb{Z}$, so $3^n=27*3^{4k} \equiv 2*(3^4)^k \equiv 2*1^k \equiv2 \pmod{5}.$</p>
<p>We use Fermat's Little Theorem, $a^{p-1} \equiv 1 \pmod{p}$ for any prime number $p$.</p>
|
262,509 | <p>I have a list of data <code>{{x1,c11}, {x1,c12}, {x2,c21},{x2,c22}, {x3,c31},{x3,c32}, {x4,c4}, {x5,c51},{x5,c52}, ...}</code>, in which <code>x</code> is real value and <code>c</code> is complex value. For some <code>x</code>, there is only a <code>c</code> value, for example, <code>{x4,c4}</code> pair, while for t... | yarchik | 9,469 | <p>Let <code>d</code> be your data. Then</p>
<pre><code>d1 = Cases[d, {x_, y_} /; Im[y] > -5]
d2 = Cases[d, {x_, y_} /; Im[y] < -5]
ListLinePlot[{Re[d1], Re[d2]}]
</code></pre>
<p>It happens that the <code>Im[y] > -5</code> criterion works just fine for the data in the OP. In more complicated situations, a mor... |
262,509 | <p>I have a list of data <code>{{x1,c11}, {x1,c12}, {x2,c21},{x2,c22}, {x3,c31},{x3,c32}, {x4,c4}, {x5,c51},{x5,c52}, ...}</code>, in which <code>x</code> is real value and <code>c</code> is complex value. For some <code>x</code>, there is only a <code>c</code> value, for example, <code>{x4,c4}</code> pair, while for t... | Edmund | 19,542 | <p>You may use <a href="https://reference.wolfram.com/language/ref/FindClusters.html" rel="nofollow noreferrer"><code>FindClusters</code></a> to separate the <code>Complex</code> curves. Then display as needed.</p>
<p>Using file in OP</p>
<pre><code>test = ToExpression /@ Import[FileNameJoin[{$HomeDirectory, "Down... |
147,917 | <blockquote>
<p>Let $A,B\in M_{n\times n} (\mathbb{C})$. Is it possible that $ABA-BAB=I$?</p>
</blockquote>
<p>I came across this interesting problem as I was studying for an exam. I guess in the case when $A$ and $B$ commute we have $A(A-B)B=I$ and I am not sure if it can happen.</p>
<p>Any ideas?</p>
| Robert Israel | 8,508 | <p>Indeed, you could have $A = I$ and then you need $B - B^2 - I = 0$, which is true e.g. for $B = \pmatrix{0 & 1\cr -1 & 1\cr}$.</p>
<p>EDIT: For a non-commuting example, try $A = \pmatrix{2 & 0\cr 0 & 1\cr}$, $B = \pmatrix{-1/2 & 1\cr -7/2 & 3\cr}$</p>
|
2,529,990 | <p>Find the area of the shape surrounded by $y = \sin(x)$, $y = -\cos(x)$, $x = 0$, $x = π/4$.
Do I subtract $S_2$ from the $S_1$? How do I find the shape's area?
<a href="https://i.stack.imgur.com/67G7B.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/67G7B.png" alt="Picture of the graph: "></a></p>
| John Wayland Bales | 246,513 | <p>If you added $1$ to both the upper and the lower function, that would move both graphs one unit higher so that the region between them would lie entirely above the $x$-axis. Yet the area of the region would remain the same. So the position of the $x$-axis makes no difference. You merely integrate the difference of t... |
4,518,339 | <p>Before I start, please excuse my jargon and any preconceived notions. I am trying to fix how I think about this. It feels as if I am missing something.</p>
<p>So I understand that matrix multiplication linearly maps a vector or matrix to that basis. It describes a linear transformation, as if the coordinate plane sc... | Aiden Chow | 743,034 | <p>Let's say that you choose the starting number to be some number <span class="math-container">$k$</span>, and the difference to be <span class="math-container">$d$</span>. Then, the sum of the <span class="math-container">$n$</span> numbers will be:
<span class="math-container">$$k+(k+T_1d)+(k+T_2d)+\cdots+(k+T_{n-1}... |
3,291,975 | <p>Transpose this formula to make <span class="math-container">$y$</span> the subject.</p>
<p><span class="math-container">$$x=\sqrt{x^2y^2+1-y}$$</span></p>
<p>My try:</p>
<p><span class="math-container">$$x^2=x^2y^2+1-y$$</span></p>
<p><span class="math-container">$$x^2-x^2y^2=1-y$$</span></p>
<p><span class="ma... | Dr. Sonnhard Graubner | 175,066 | <p>You can write <span class="math-container">$$0=y^2-\frac{1}{x^2}y+\frac{1-x^2}{x^2}$$</span> using the quadratic formula we obtain
<span class="math-container">$$y_{1,2}=\frac{1}{2x^2}\pm\sqrt{\left(\frac{1}{2x^2}\right)^2-\frac{1-x^2}{x^2}}$$</span></p>
|
3,246,533 | <p>ABC is a triangle such that AB = AC. Let D be the foot of the perpendicular from C to AB and E the foot of the perpendicular from B to AC. Then</p>
<p>(a) <span class="math-container">$BC^3 < BD^3 + BE^3$</span></p>
<p>(b) <span class="math-container">$BC^3 = BD^3 + BE^3$</span></p>
<p>(c) <span class="math-co... | amitava | 566,278 | <p>Note that, triangles BDC and CEB are congruent as @SS_C4 mentioned. Then <span class="math-container">$CD = BE$</span>, Now</p>
<p><span class="math-container">$BC^2 = BD^2+CD^2$</span>, </p>
<p>So <span class="math-container">$BC^3 = BD^{2}BC+CD^{2}BC = BD^{2}(BD+d_1)+CD^{2}(CD+d_2)\\
= BD^3+CD^3+BD^{2}d_1 + CD^{... |
23,306 | <p>I am trying to perform a simple arithmetic in complex algebra.</p>
<pre><code> a := Subscript[a, 1] + I*Subscript[a, 2];
b := Subscript[b, 1] + I*Subscript[b, 2];
r[z_] := a/(b - z) + a\[Conjugate]/(b\[Conjugate] - z);
</code></pre>
<p>when I try to evaluate r[0], mathematica gives me the follo... | amr | 950 | <p>I did this and it seems to work:</p>
<pre><code>SetAttributes[Subscript, HoldAll];
a := Subscript[a, 1] + I*Subscript[a, 2];
b := Subscript[b, 1] + I*Subscript[b, 2];
r[z_] := a/(b - z) + a\[Conjugate]/(b\[Conjugate] - z);
r[0] // ComplexExpand // Simplify
</code></pre>
<p>I'm not really sure if this will work i... |
23,306 | <p>I am trying to perform a simple arithmetic in complex algebra.</p>
<pre><code> a := Subscript[a, 1] + I*Subscript[a, 2];
b := Subscript[b, 1] + I*Subscript[b, 2];
r[z_] := a/(b - z) + a\[Conjugate]/(b\[Conjugate] - z);
</code></pre>
<p>when I try to evaluate r[0], mathematica gives me the follo... | m_goldberg | 3,066 | <p>Attributes are powerful, but so is <code>With</code>, which is what I would use here.</p>
<pre><code>With[{a = Subscript[a, 1] + I*Subscript[a, 2],
b = Subscript[b, 1] + I*Subscript[b, 2]},
r[z_] := a/(b - z) + a\[Conjugate]/(b\[Conjugate] - z)]
r[0] // ComplexExpand // Simplify
</code></pre>
<blockquot... |
3,446,691 | <p>[Please excuse ignorance, I'm not a mathematician, though I do data analysis.]</p>
<p>The question is strictly for <span class="math-container">$2\times2$</span> real non-symmetric matrix <span class="math-container">$\bf A$</span>. Let us assume, having real eigenvalues.</p>
<p>I've learned how to compute the eig... | Devansh Bisla | 686,238 | <p>Looks like Schur decomposition to me. We can do it by hand at least for 2x2 matrices here is an example:</p>
<p>For any arbitrary matrix, A, Schur decomposition is defined as <span class="math-container">$A = UTU^{*}$</span> where T is upper triangular, U is a unitary matrix and <span class="math-container">$*$</s... |
2,055,546 | <p>So I stumbled across this math problem:</p>
<p>$1015$ USD is to be shared between $3$ people: Person A, Person B and Person C.
Person A gets double of what Person B gets and Person B gets $100$ USD more than Person C</p>
<p>Basically:</p>
<p>$A = 2B$</p>
<p>$B = C + 100$</p>
<p>$A + B + C = 1015$</p>
<p>Can so... | Community | -1 | <p>Basically, $A=2B$ and $B=C+100$. We have $$A+B+C =1015$$ $$\Rightarrow 2B+B+(B-100) =1015$$ because $B+C=100$. Hence, $$4B-100 =1015 \Rightarrow B=\frac{1115}{4}$$ Then we can find $A=\frac{1115}{2}$ and $C=\frac{715}{4}$. Hope it helps.</p>
|
647 | <p>For example, I gave an exam earlier today with a problem that ended in the sentence</p>
<blockquote>
<p>Use the chain rule to find $(f\circ g)'(3)$.</p>
</blockquote>
<p>During the exam, one of the students asked me what the circle between the $f$ and $g$ means, and I answered that it represents the composition ... | Dave L Renfro | 745 | <p>For questions like "what does the circle between $f$ and $g$ mean" in a calculus course, I would tell the student and then write on the board (for everyone else) something like:</p>
<blockquote>
<p>In problem 5, $(f \circ g)(x)$ stands for $f(g(x))$ (composition of functions).</p>
</blockquote>
<p>I wouldn't do... |
1,063,382 | <p>Is there any proper way to know whether a function has the same domain and range $[a,b]$ where $a,b<\infty$ i.e. $f:[a,b] \to [a,b]$ ?</p>
<p>For example:
$$
f(x) = e^{−x} ,\qquad [\ln(1.1), \ln(3)]
$$</p>
| Ahaan S. Rungta | 85,039 | <p>So, first, you realize that the equation of the circle must be of the form $$ (x-h)^2 + (y-k)^2 = r^2. $$Then, since $(1,1)$ is on the circle, we get $$ (1-h)^2 + (1-k)^2 = r^2. $$Then, we note that $(2,3)$ is the <em>only</em> intersection of $$ (x+2)^2 + (y-3)^2 = 16 $$ and $$ (x-h)^2 + (y-k)^2 = r^2. $$So let us ... |
82,070 | <p>I reading on Sums and I am reading about the difference between using a generalized Sigma notation and the delimited form. Ok, I understand that the generalized form is more expressive. </p>
<p>But I found the following example confusing to me:<br>
It says: </p>
<blockquote>
<p>In the following we can change ... | Phira | 9,325 | <p>A complex index in delimited form is never used, the main reason being that this is highly problematic for non-linear variable changes if you think of the delimited form as running through values 1 to n and the non-delimited form as a system of conditions that happen to be inequalities in your case.</p>
<p>If the v... |
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