qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
1,487,992 | <blockquote>
<p>Determine: $(0.064)^{\frac{1}{3}}$ is transcendental or algebraic </p>
</blockquote>
<p>To show a number is transcendental/algebraic do I need to show there is a monic polynomial with integer coefficients such that the number is/is not its root ?</p>
<p>in this case:</p>
<p>$x=(0.064)^{\frac{1}{3}}... | Robert Lewis | 67,071 | <p>A number $y$ is algebraic if it satisfies a polynomial $p(\alpha)$ having rational numbers as coefficients; that is, if $p(y) = 0$ where $p(\alpha) \in \Bbb Q[\alpha]$. Note we may in fact assume $p(\alpha)$ is monic, since $p(y) = 0$ if and only if $\lambda^{-1} p(y) = 0$, where $\lambda \in \Bbb Q$ is the leading... |
1,487,992 | <blockquote>
<p>Determine: $(0.064)^{\frac{1}{3}}$ is transcendental or algebraic </p>
</blockquote>
<p>To show a number is transcendental/algebraic do I need to show there is a monic polynomial with integer coefficients such that the number is/is not its root ?</p>
<p>in this case:</p>
<p>$x=(0.064)^{\frac{1}{3}}... | Michael Hardy | 11,667 | <p>Either a <b>monic</b> polynomial with <b>rational</b> coefficients, or any polynomial with integer coefficients.</p>
<p>For example $5x^3 -29x^2+3x+11 = 0$ has integer coefficients, and is equivalent to $x^3 - \frac{29}5 x^2 + \frac 3 5 x + \frac{11}5 =0\vphantom{\dfrac11}$, which is <b>monic</b> and has <b>rationa... |
3,707,880 | <p>Let <span class="math-container">$W_{1}$</span> and <span class="math-container">$W_{2}$</span> be subspaces of a vector space <span class="math-container">$V$</span>.</p>
<p>(a) Prove that <span class="math-container">$W_{1}+W_{2}$</span> is a subspace of <span class="math-container">$V$</span> that contains both ... | auspicious99 | 178,721 | <p>For part (b), it might be better to start with:</p>
<p>Let us suppose that <span class="math-container">$W \subseteq V$</span>, and <span class="math-container">$W$</span> is a subspace of <span class="math-container">$V$</span>, where <span class="math-container">$W\supseteq W_{1}$</span> and <span class="math-con... |
4,047,049 | <p>Assuming <span class="math-container">$\log x$</span> is normally distributed. How do I get the distribution of <span class="math-container">$x$</span>? Also how can I get the standard deviation of <span class="math-container">$x$</span> assuming I know some asymmetric error on <span class="math-container">$x$</span... | JMP | 210,189 | <p>I think the authors are subtly trying to point out that the probability of throwing two dice both even is the same as throwing two dice that then sum to <span class="math-container">$2,4$</span> or <span class="math-container">$6$</span>, which is easy to see by any mapping of:</p>
<p><span class="math-container">$$... |
1,432,729 | <p>I know that $\pi \approx \sqrt{10}$, but that only gives one decimal place correct. I also found an algebraic number approximation that gives ten places but it's so cumbersome it's just much easier to just memorize those ten places.</p>
<p>What's a good approximation to $\pi$ as an irrational algebraic number (or a... | Will Jagy | 10,400 | <p>If you want to stay with degree two or three but no larger, find an implementation of <a href="https://en.wikipedia.org/wiki/Integer_relation_algorithm">PSLQ</a> and feed it the quadruple (at incredible decimal accuracy) $$ \left(\pi^3, \; \pi^2, \; \pi, \; 1 \right) $$
so as to ask for integer relations, that is in... |
1,432,729 | <p>I know that $\pi \approx \sqrt{10}$, but that only gives one decimal place correct. I also found an algebraic number approximation that gives ten places but it's so cumbersome it's just much easier to just memorize those ten places.</p>
<p>What's a good approximation to $\pi$ as an irrational algebraic number (or a... | Jack D'Aurizio | 44,121 | <p>Some nice approximations can be produced by exploiting the ideas of Archimedes. The difference between a unit circle and an inscribed regular <span class="math-container">$n$</span>-agon is made by <span class="math-container">$n$</span> circle segments. If we approximate them with <em>parabolic</em> segments and ca... |
555,921 | <p>Apparently, I'm not understanding this simple concept. What are the differences between the two? Can a person have multiple pure strategies that change throughout the game?</p>
| Řídící | 56,801 | <p>A pure strategy determines <em>all</em> your moves during the game (and should therefore specify your moves for all possible other players' moves).</p>
<p>A mixed strategy is a probability distribution over all possible pure strategies (some of which may get zero weight). After a player has determined a mixed strat... |
363,910 | <p>Given two positive rational number $a,b$. How to show that almost surely Brownian motion
attains a local maximum at some time in $(a,b)$?</p>
| Abel | 71,157 | <p>Let $U=\mathbb{S}^3$ and consider
$$U\cap\mathbb{R}^3 = \{(x,y,v,w)\in\mathbb{R}^4|x^2+y^2+v^2+w^2 = 1\}\cap\{(x,y,v,w)\in\mathbb{R}^4|w=0\}$$
$$ = \{(x,y,v,w)\in\mathbb{R}^4|w=0\mbox{ and }x^2+y^2+v^2=1\}\cong \mathbb{S}^2.$$</p>
<p>$\mathbb{S}^2$ does indeed contain some copies of $\mathbb{S}^1\times\mathbb{R}$ u... |
1,738,050 | <p>I know that the series converges. My questions is to what. I tried seeing if it was a telescoping series:
$\sum_{n=2}^\infty \frac{2}{n^3-n} = 2\sum_{n=2}^\infty (\frac{1}{n^2-1}-\frac{1}{n})$ but it doesn't seem to cancel any terms. Thoughts?</p>
| marty cohen | 13,079 | <p>That partial fraction
is incorrect.</p>
<p>$\frac1{n^3-n}
=\frac1{n(n^2-1)}
=\frac1{n(n-1)(n+1)}
=\frac{a}{n}+\frac{b}{n-1}+\frac{c}{n+1}
$.</p>
<p>Find $a, b, $ and $c$
and then see if things
cancel out in the sum.</p>
|
2,433,051 | <p>Following <a href="http://functions.wolfram.com/ElementaryFunctions/Sin/29/" rel="nofollow noreferrer">Wolfram Sine inequalities</a> I found that</p>
<p>$$|\sin(x)| \le |x| \quad \text{for} \quad x \in \mathbb{R}$$
How can I prove this relation?</p>
| The 7th sense | 352,390 | <p>For x>=0 you can use corollary of Lagrange mean value theorem. Take f(x)= sinx -x . Take derivative f'(x)= cosx -1 which is less than 0 and f(0)=0 which means f(x) is decreasing for all positive x. Sinx -x<0</p>
|
441,994 | <p>In thinking about my question <a href="https://mathoverflow.net/questions/441982">here</a> for the Linial arrangement, the following limit arose:
<span class="math-container">$$ \lim_{n\to\infty}\frac{(n-1)\sum_{k=0}^n {n\choose k}(k+1)^{n-2}}
{\sum_{k=0}^n {n\choose k}(k+1)^{n-1}}. $$</span>
Is this limit ... | Fedor Petrov | 4,312 | <p>The same value as in Nemo's answer may be obtained in a slightly more elementary way: denote <span class="math-container">$c_k={n\choose k} (k+1)^{n-2}$</span>. We want to find the limit of <span class="math-container">$\frac{\sum (n-1)c_k}{\sum (k+1)c_k}$</span>. Both sums are concentrated around <span class="math-... |
3,729,197 | <p>For this problem suppose that the <span class="math-container">$x_i$</span>'s must be non-negative integers, i.e., <span class="math-container">$x_i∈{0,1,2,⋯}$</span> for <span class="math-container">$i=1,2,3$</span>. How many distinct solutions does the following equation have such that at least one of the <span cl... | Alexey Burdin | 233,398 | <p>If we denote <span class="math-container">$x_1=x,\,x_2=y,\,x_3=100-x-y$</span> then looking at the inequalities plot we realize that all the values fits except for the shaded region
<span class="math-container">$$\begin{cases}
x\le 40\\ y\le 40\\ x+y\ge 60
\end{cases}$$</span>
<img src="https://i.imgur.com/Z1zTssx.p... |
215,846 | <p>Are proofs by induction limited to cases where there is an explicit dependance on a integer, like sums? I cannot grasp the idea of induction being a proof in less explicit cases. What if you have a function that suddenly changes behavior? If a function is positive up to some limit couldn't I prove by induction that ... | Jemmy | 36,250 | <p>This math.SE <a href="https://math.stackexchange.com/questions/4202/induction-on-real-numbers">question</a> has a lot of great answers as far as induction over the reals goes. And as Austin mentioned, there are many cases in graph theory where you can use induction on the vertices or edges of a graph to prove a resu... |
215,846 | <p>Are proofs by induction limited to cases where there is an explicit dependance on a integer, like sums? I cannot grasp the idea of induction being a proof in less explicit cases. What if you have a function that suddenly changes behavior? If a function is positive up to some limit couldn't I prove by induction that ... | Brian M. Scott | 12,042 | <p>All forms of mathematical induction $-$ ordinary induction, so-called strong induction, structural induction, transfinite induction, etc. $-$ can usefully be thought of in the following way, in terms of the non-existence of a minimal counterexample to the theorem being proved. This view makes it easier to avoid the ... |
1,336,175 | <p>I would like to study <strong>category of sets and multi-valued functions</strong>: A category whose objects are sets and morphisms are multi-valued functions. </p>
<p>By a multi-valued function $f:A\rightarrow B$, from set A to set B, I mean a function that assigns to each element of A, a subset of B. There might ... | Giorgio Mossa | 11,888 | <p>What you are describing is the category of $\mathbb {Rel}$ of sets and relations.</p>
<p>The objects of this category are sets and the morphisms are binary relations. This comes from the fact that there is a bijection between the sets of function of the form $X \to \mathcal P(Y)$ and (relations) subsets of the cart... |
2,770,523 | <p>I'm struggling with the following problem,</p>
<p>Let $g(z)=\sum^k_1 m_{\alpha}(z-z_{\alpha})^{-1}$. Show that if $g(z)=0$, then $z_1,\cdots,z_k$ cannot all lie on the same sie of a straight line through $z$.</p>
<p><strong>What I did:</strong></p>
<p>The book says that I should use the fact that if $z_1,\cdots,z... | Community | -1 | <p>Two odd numbers may be co-prime but their sum will always be an <em>even</em> number. Hence, the given statement does not hold true for co-prime numbers that are both odd.</p>
|
2,633,975 | <p>Let $u\in C^2(\Omega)\cap C^0(\overline{\Omega})$, $\Omega\subset\mathbb{R}^n$ open and bounded and $\lambda>0$ sufficiently small so that $2\lambda u<1$. Define $w$ by $w\leq\frac{1}{\lambda}$ and
$$
u(x)=w(x)-\frac{\lambda}{2}w(x)^2.
$$
I have to prove that $w\rightarrow u$ uniformly as $\lambda\rightarrow0$... | Hagen von Eitzen | 39,174 | <p>Find orthogonal matrices $U,V$ such that $V^{-1}x\sim e_1$, $Uv\sim e_1$. As
$ \|A-B\|_2=\|U(A-B)V\|_2$, we now want to find $B'=UBV$ with $B'e_1=\lambda e_1$ such that $\|A-B\|_2$ is minimized, where $A'=UAV$. As the first column of $B'$ is uniquely determined whereas all other entries of $B'$ are free to our likin... |
2,168,524 | <p>I know a solution to this question having to do with the fact that the $\gcd(15, 21) = 3$, so the answer is no.</p>
<p>But I can't figure out what is the reasoning behind this. Any help would be really appreciated! </p>
| user421604 | 421,604 | <p>It's really simple, imagine the two numbers as $15n$ and $21k$.</p>
<p>Suppose that it's possible, then: $15n-21k = 1$</p>
<p>As you said $\gcd(15, 21) = 3$, so you can factor out: $3(5n-7k) = 1$.</p>
<p>On the left side, you have a multiple of 3, and in the other you have a 1, which isn't, so you have a contradi... |
2,659,001 | <p>I would like to compute the subdifferential of the function</p>
<p>$$ f(x)=a^\text{T}x+\alpha\sqrt{x^\text{T}Bx} $$
where $\alpha>0$ and $B$ is symmetric positive definite. </p>
<p><strong>Attempted Solution</strong> (I am brand new to subdifferentiability)</p>
<p>Since subderivatives, like normal derivatives,... | lab bhattacharjee | 33,337 | <p>Hint:</p>
<p>$$(-7)^3=(u^{2/3}-5u^{1/3})^3=u^2-125u-15u(-7)$$</p>
|
2,659,001 | <p>I would like to compute the subdifferential of the function</p>
<p>$$ f(x)=a^\text{T}x+\alpha\sqrt{x^\text{T}Bx} $$
where $\alpha>0$ and $B$ is symmetric positive definite. </p>
<p><strong>Attempted Solution</strong> (I am brand new to subdifferentiability)</p>
<p>Since subderivatives, like normal derivatives,... | Daniel Schepler | 337,888 | <p>Here is a solution inspired by the mechanical process to calculate a Groebner basis of the ideal $\langle x^2 - 5x + 7, x^3 - u \rangle$ in $\mathbb{Q}[x, u]$.</p>
<p>Let $x := u^{1/3}$; then we have the system of equations
$$x^2 - 5x + 7 = 0, \quad (1) \\ x^3 - u = 0. \quad (2)$$
Now, let us try to eliminate $x$. ... |
1,013,692 | <p>I want to determine which group $(\mathbb{Z}/24\mathbb{Z})^{*}$ is isomorphic to.</p>
<p>$\mathbb{Z}/24\mathbb{Z}$ contains the 24 residue classes $z + 24\mathbb{Z}$ of the division mod 24. For brevity, I will identify them with $z$, so $\mathbb{Z}/24\mathbb{Z} = \{ 0, 1, ..., 23\}$. For $(\mathbb{Z}/24\mathbb{Z})^... | Community | -1 | <p>Every finite <em>abelian</em> group is a product of cyclic groups. If every element has order two, then it has to be a product of copies of $\mathbf{Z} / 2$.</p>
<p>I believe your conjecture is true for finite abelian groups, but false for general finite groups (and false for infinite abelian groups). But it's been... |
3,196,706 | <p>Let <span class="math-container">$K = \mathbb{Q}(\theta)$</span> be a numberfield and <span class="math-container">$[K:\mathbb{Q}]=n$</span>. When <span class="math-container">$\mathbb{Q}_p$</span> is the field of <span class="math-container">$p$</span>-adic numbers and <span class="math-container">$K_p=\mathbb{Q}_... | Lubin | 17,760 | <p>I’ll give the results, with no hint of a proof:</p>
<p>The general situation is that if <span class="math-container">$G(X)=\text{Irr}(\theta,\Bbb Q[X])$</span> splits as a product of <span class="math-container">$\Bbb Q_p$</span> irreducibles <span class="math-container">$G=g_ig_2\cdots g_m$</span>, then there are ... |
4,250,632 | <p>Calculating <span class="math-container">$81^{3/2}$</span>, I got <span class="math-container">$729$</span> (not saying it is correct, but I am trying :) ). Would <span class="math-container">$-81^{3/2}$</span> just be the opposite (<span class="math-container">$-729$</span>) and does it make a difference if <span ... | thebluepandabear | 968,164 | <p>I'm also new to Mathematics but I will try and explain in simple terms:</p>
<p>Using exponent rules we can say that <span class="math-container">$\left(-81\right)^{\frac{3}{2}}$</span> is equal to <span class="math-container">$\sqrt[2]{\left(-81\right)^{3}}$</span>. In this case <span class="math-container">$\left(-... |
3,096,741 | <p><span class="math-container">$X, Y$</span> are two independent <span class="math-container">$\mathcal{N}(0,1)$</span> random variables </p>
<p>this question was a follow up question of this <a href="https://math.stackexchange.com/questions/3096530/show-that-e-fracx22-in-l1-iff-exy-in-l1-iff-exy-in-l1">one</a></p>
... | Alecos Papadopoulos | 87,400 | <p>Invoking the convexity of the exponential function appears to be the simplest proof here. Since the exponential function is convex we have that</p>
<p><span class="math-container">$$E[\exp\{XY\} \mid X] \geq \exp\{E(XY \mid X)\} = \exp\{XE(Y)\} = \exp\{X\cdot 0\}=1.$$</span></p>
<p>We have used the zero mean propert... |
2,408,954 | <p>In this webpage, <a href="https://plus.maths.org/content/friends-and-strangers" rel="nofollow noreferrer">https://plus.maths.org/content/friends-and-strangers</a> under the section, FINDING $R(3,4)$,</p>
<p>The author assumes that 10 people/points are necessary and takes out one point, say A. This A is connected to... | Franz | 281,855 | <p>If I am understanding the problem based on the very quick skim I gave it, it's about connected graphs with coloring. It doesn't have to be at least 5 blue. For example, all 9 could be red. We know, however that there has to be at least 5 of one of the colours, be it blue or red. Whatever configuration we have, it is... |
3,674,924 | <p>The angular momentum components in Cartesians are
<span class="math-container">$$\hat L_x=\hat y\hat p_z-\hat z\hat p_y$$</span>
<span class="math-container">$$\hat L_y=\hat z\hat p_x-\hat x\hat p_z$$</span>
<span class="math-container">$$\hat L_z=\hat x\hat p_y-\hat y\hat p_x$$</span></p>
<p>Starting from
<span cl... | Ninad Munshi | 698,724 | <p>Swapping the order of the summation we get</p>
<p><span class="math-container">$$S \equiv \sum_{j=1}^\infty \sum_{i=j}^\infty \frac{1}{i^2}\frac{1}{j^2} = \sum_{j=1}^\infty \sum_{i=1}^\infty \frac{1}{i^2}\frac{1}{j^2} - \sum_{j=1}^\infty \sum_{i=1}^{j-1} \frac{1}{i^2}\frac{1}{j^2}$$</span></p>
<p>Since the <span c... |
211,865 | <p>The given matrix is </p>
<p>$$
\begin{pmatrix}
2 & 2 & 2 \\
2 & 2 & 2 \\
2 & 2 & 2 \\
\end{pmatrix}
$$</p>
<p>so, how could i find the eigenvalues and eigenvector without computation?
Thank you</p>
| DonAntonio | 31,254 | <p>Try to solve the following matrix equation, taking into account that your matrix is singular:</p>
<p>$$\begin{pmatrix}2&2&2\\2&2&2\\2&2&2\end{pmatrix}\begin{pmatrix}x\\y\\z\end{pmatrix}=\begin{pmatrix}2(x+y+z)\\2(x+y+z)\\2(x+y+z)\end{pmatrix}=2\begin{pmatrix}x+y+z\\x+y+z\\x+y+z\end{pmatrix}=... |
69,471 | <p>Suppose I am looking at $GL(4,K)$ acting on a cubic form in say four variables $x,y,z,w$ over $K$ via the usual induced action on a polynomial. Does anyone know what is/where I can find how to compute the ring of invariants? The case of personal interest is when $K$ is a finite field but the the answer over $\mathbb... | Abdelmalek Abdesselam | 7,410 | <p>According to Dolgachev in
<a href="http://arxiv.org/abs/math/0408283" rel="nofollow">http://arxiv.org/abs/math/0408283</a>
the ring of invariants is generated by 6 invariants of degrees 6, 16, 24, 32, 40 and 100
the last one being a polynomial in the other ones.
In addition to the references to Salmon and Clebsch in... |
69,471 | <p>Suppose I am looking at $GL(4,K)$ acting on a cubic form in say four variables $x,y,z,w$ over $K$ via the usual induced action on a polynomial. Does anyone know what is/where I can find how to compute the ring of invariants? The case of personal interest is when $K$ is a finite field but the the answer over $\mathbb... | Chirag Lakhani | 2,565 | <p>There is also a nice book by Sturmfels called <a href="http://books.google.com/books/about/Algorithms_in_invariant_theory.html?id=3lF0cyOHwgQC" rel="nofollow">Algorithms in Invariant Theory</a> that discusses the problem of finding invariants. It may be worth a look.</p>
|
62,967 | <p><code>CoefficientRules</code> acts like the following.</p>
<pre><code>In[1]:= CoefficientRules[2 x^3 + 3 x^2 y + 4 x y^2 - 5 x + 1]
Out[1]= {{3, 0} -> 2, {2, 1} -> 3, {1, 2} -> 4, {1, 0} -> -5, {0, 0} -> 1}
</code></pre>
<p>My question is how one can "extend" this function so that it may allow the n... | ubpdqn | 1,997 | <pre><code> fun[ex_] := Module[{v, p, m, mn, cp},
v = Variables[ex];
p = Cases[ex, Power[#, a_] :> a, Infinity] & /@ v;
m = (Min /@ p);
mn = (1 - Sign[#]) #/2 & /@ m;
{If[Min[m] > 0, CoefficientRules[ex],
cp = Times @@ MapThread[Power[#1, -#2] &, {v, mn}] ex;
(#1 + mn -> #2) &... |
123,494 | <p>In <em>Mathematica</em>, almost everything is notebook: your "Untitled-1.nb" is a notebook, Help documentation are a series of notebooks, even those windows helping you to draw things or format your notebooks are, themselves, notebooks.</p>
<p>But I occasionally find some exceptions and I want to know what are they... | b3m2a1 | 38,205 | <p>One other note, you can find every single notebook with:</p>
<pre><code>FrontEndExecute@
FrontEnd`ObjectChildren[$FrontEnd]
</code></pre>
|
43,231 | <p>I'm trying to see why my textbook's solution is correct and mine isn't.</p>
<p>"Find an expression in terms of $x$ and $y$ for $\displaystyle \frac{dy}{dx}$, given that $x^2+6x-8y+5y^2=13$</p>
<p>First, the textbook's solution, which I understand and agree with fully:
<img src="https://i.stack.imgur.com/uyDlo.png... | FUZxxl | 5,282 | <p>I don't know, what's wrong with your solution. Notice, that</p>
<p>$${-x-3\over5y-4}={3+x\over4-5y}$$</p>
|
31,571 | <p>In his paper "Smooth models for elliptic threefolds" (In: The Birational Geometry of Degenerations, Progress in Mathematics, v. 29, Birkhauser, (1983), 85-133), Rick Miranda mentions in the example of section 8 (page 101-102) that it is an unfortunate fact of life that there are no small resolutions for the singular... | Francesco Polizzi | 7,460 | <p>A <span class="math-container">$3$</span>-dimensional hypersurface singularity of type <span class="math-container">$$y^2=u^2+v^2+w^k$$</span> admits a small resolution if and only if <span class="math-container">$k$</span> is even.
If <span class="math-container">$k$</span> is odd the corresponding singularity is f... |
31,571 | <p>In his paper "Smooth models for elliptic threefolds" (In: The Birational Geometry of Degenerations, Progress in Mathematics, v. 29, Birkhauser, (1983), 85-133), Rick Miranda mentions in the example of section 8 (page 101-102) that it is an unfortunate fact of life that there are no small resolutions for the singular... | Sándor Kovács | 10,076 | <p><a href="https://mathoverflow.net/questions/31696/best-strategy-for-small-resolutions/45193#45193">This</a> answer to a more general question might also be relevant for similar questions.</p>
|
4,618,932 | <p>The <a href="https://en.wikipedia.org/wiki/Sequence" rel="nofollow noreferrer">wikipedia article</a> for a sequence defines it as such:</p>
<blockquote>
<p>In mathematics, a sequence is an <strong>enumerated</strong> collection of objects in which repetitions are allowed and order matters. [...] The number of elemen... | Joe | 623,665 | <p>In analysis, the term "sequence" usually means a function from <span class="math-container">$\mathbb N$</span> to <span class="math-container">$\mathbb R$</span>. Of course, people are free to use the term "sequence" in another way, e.g. to refer to a function from <span class="math-container">$\... |
2,654,507 | <blockquote>
<p>Find the residue of $\dfrac{z^2}{(z-1)(z-2)(z-3)}$ at $\infty$.</p>
</blockquote>
<p>We know that $\text{Res} (f)_\infty +\text{Res} (f)_{\text{ at other poles}}=0$</p>
<p>Now $f$ has poles at $1,2,3$ of order $1$.</p>
<p>Sum of residues of $f$ at $1,2,3=\dfrac{1}{2}+(-4)+\dfrac{9}{2}=1\implies \t... | farruhota | 425,072 | <p>It is:
$$16((\cos^2t+\sin^2t)^3-3\sin^2t\cos^2t(\cos^2t+\sin^2t)+3\sin^2t\cos^2t)=16(1^3-0)=16.$$</p>
|
369,585 | <p>Let <span class="math-container">$X$</span> be a smooth, projective ireducible scheme over an algebraically closed field <span class="math-container">$k$</span>. I'm trying to understand when there exists an abelian variety <span class="math-container">$A$</span> such that <span class="math-container">$X$</span> is ... | Ari Shnidman | 949 | <p>Any curve of genus greater than two, whose Jacobian <span class="math-container">$J$</span> is simple, will do. If it were a divisor on an abelian surface <span class="math-container">$S$</span>, then there would be a surjection <span class="math-container">$J\to S$</span> with positive dimensional kernel, contradi... |
369,585 | <p>Let <span class="math-container">$X$</span> be a smooth, projective ireducible scheme over an algebraically closed field <span class="math-container">$k$</span>. I'm trying to understand when there exists an abelian variety <span class="math-container">$A$</span> such that <span class="math-container">$X$</span> is ... | Samir Canning | 124,840 | <p>Here's another answer using the Albanese that's of a slightly different flavor. Let <span class="math-container">$X$</span> be <span class="math-container">$n$</span>-dimensional and suppose that <span class="math-container">$h^0(X,\Omega^1_X)<n$</span>. Then any map <span class="math-container">$X\rightarrow A$<... |
2,043,418 | <p>Suppose $1 \leq p < \infty$. Let $f: \mathbb{R} \to \mathbb{R}$ be continuous and has compact support and let $t \in \mathbb{R}$. Define $f_t(x) = f(x-t).$</p>
<p>Prove: $\lim_{t \to 0} ||f-f_t||_p = 0$.</p>
<p>Edit: I already got the answer. I'll post it as a formal answer soon!</p>
| Lutz Lehmann | 115,115 | <p>Your proof is nearly right.</p>
<p>You only need to insert at the appropriate place that $f$ is uniformly continuous so that you can select $δ$ independent of $x_0$.</p>
<p>You can also connect $N$ more explicitly to the support of $f$, so that not only the difference, but the function values themselves are zero.<... |
3,484,791 | <p>.I am trying to prove the following statement (Linear Algebra Done Right, Section 2.A, #17):</p>
<p>Suppose <span class="math-container">$p_0,p_1,\ldots,p_m$</span> are polynomials in <span class="math-container">$P_m(F)$</span> such that <span class="math-container">$p_j(2)=0$</span> for each <span class="math-con... | Doug M | 317,162 | <p>The set <span class="math-container">$\{p_0,\cdot ,p_m\}$</span> has <span class="math-container">$m+1$</span> members. Each can be written as <span class="math-container">$(x-2)p$</span> with <span class="math-container">$p$</span> an element in <span class="math-container">$P_{m-1}.$</span> As <span class="math-... |
1,488,831 | <p>Written in Abstract Algebra by T. W. Judson :</p>
<p><strong>Theorem</strong></p>
<blockquote>
<p>Let R be a ring with identity and suppose that I is an ideal in R such that 1 is in I. Since for any r∈R, r1=r∈I by the definition of an ideal, I=R. </p>
</blockquote>
<p>and considering the definition of the ideal... | drhab | 75,923 | <p>Because for any $r\in R$ we have $r=r1\in I\subseteq R$.</p>
|
3,349,312 | <p>I have some confusion in this question
<a href="https://math.stackexchange.com/questions/624611/a-problem-on-comparison-of-dimension-between-two-subspace-of-polynoamial-vector">A problem on comparison of dimension between two subspace of polynomial vector space.</a></p>
<blockquote>
<p>Let <span class="math-contain... | egreg | 62,967 | <p>Note first that, for a polynomial <span class="math-container">$P\in V_e$</span>, <span class="math-container">$P(1)=P(-1)$</span>, so that the only polynomials in <span class="math-container">$V_0\cap V_e$</span> are those in <span class="math-container">$V_e$</span> for which <span class="math-container">$P(1)=0$<... |
3,547,816 | <p>I have such an equation where I need to find x</p>
<p><span class="math-container">$$y =\frac{1}{x+1}$$</span></p>
<p>And I know that answer is
<span class="math-container">$$x = \frac{1}{y}-1$$</span></p>
<p>But I did not understand how to get it?</p>
| azif00 | 680,927 | <p>Multiplying both sides of
<span class="math-container">$$y = \frac{1}{x+1}$$</span>
by <span class="math-container">$x+1$</span> we get
<span class="math-container">$$(x+1)y = 1$$</span>
and multiplying the previous one by <span class="math-container">$\frac{1}{y}$</span> we obtain
<span class="math-container">$$x+... |
3,547,816 | <p>I have such an equation where I need to find x</p>
<p><span class="math-container">$$y =\frac{1}{x+1}$$</span></p>
<p>And I know that answer is
<span class="math-container">$$x = \frac{1}{y}-1$$</span></p>
<p>But I did not understand how to get it?</p>
| fleablood | 280,126 | <p>Just manipulate:</p>
<p><span class="math-container">$y = \frac {1}{x+1}$</span> and as a fraction <span class="math-container">$\frac ab \ne 0$</span> if <span class="math-container">$a \ne 0$</span> we know <span class="math-container">$y\ne 0$</span> so </p>
<p>Method 1) "flip both sides"</p>
<p><span class="... |
198,204 | <p>Is complex valued function like $y(t) = t^2 + i\cdot t^2$ a periodic function?</p>
| celtschk | 34,930 | <p>You can simply factorize your function as $y(t)=(1+\mathrm i)t^2$. So the question boils down to whether $t^2$ is a periodic function. That would mean that there's an $a\ne 0$ so that for all $t$, $(t+a)^2 = t^2$. Now applying the binomial formula and subtracting $t^2$ on both sides gives the equation $2ta+a^2=0$ or... |
2,393,625 | <p>Any subgroup of order $2$ will be cyclic subgroup and so will be generated by single element of order $2$ in $S_4$, so to count number of subgroups of order $2$ we need to count number of elements of order $2$ in $S_4$, I tried counting them but I got answer $8$ but is $9$ actually.</p>
| JJMalone | 471,924 | <p>Your starting point sounds good; we only need to consider elements of order 2. If we consider a sequence of four numbers, these elements can be thought of as the "swapping" permutations. </p>
<p>There are $ {4 \choose 2} = 6$ ways to choose a pair of numbers which swap and leave the other values alone. Then for ea... |
98,402 | <blockquote>
<p><strong>Theorem :</strong></p>
<p>If an odd number <span class="math-container">$n$</span> , <span class="math-container">$n > 1$</span> can be uniquely expressed as : <span class="math-container">$n=x^2-y^2$</span> ; <span class="math-container">$x,y \in \mathbb{Z}^{*}$</span> then</p>
<p><span clas... | Bill Dubuque | 242 | <p>While Fermat's difference of squares technique is generally not an efficient way to prove primality, as Gauss noted, one can speed up the basic brute force search by employing quadratic exclusions. Below is an example from W. Bosma and M. van der Hulst, <a href="http://www.math.ru.nl/~bosma/pubs/prit.pdf" rel="nofol... |
64,392 | <p>thank you for reading my question.</p>
<p>I have a problem trying to export mixed data from Mathematica. I have different matrices and vectors, which should be combined to a output file.</p>
<p>Here a minimal-working-example:</p>
<pre><code>a = {{1}, {2}, {3}};
b = {{4}, {5}, {6}};
c = {7, 8, 9};
d = Transpose[{a... | rhermans | 10,397 | <p>If the lists are all the same length</p>
<pre><code>Transpose[Flatten /@ {a, b, c}]
</code></pre>
<p>But beware that for ragged lists (i.e., those with sub-lists that are not all the same length) Transpose will not work and you have to use <code>Flatten</code> again.</p>
<pre><code>a = {{0}, {1}, {{2}}, {3}};
b =... |
4,306,339 | <p>(I have already prove that <span class="math-container">$\mathbb{Q}(\sqrt{3},i\sqrt{5})=\mathbb{Q}(\sqrt{3}+i\sqrt{5})$</span> in case is useful); now I am asked to prove that if <span class="math-container">$v\in \mathbb{Q}(\sqrt{3},i\sqrt{5})$</span> has the property that every image of <span class="math-container... | José Carlos Santos | 446,262 | <p>Consider the map<span class="math-container">$$\begin{array}{rccc}f\colon&\Bbb R^2&\longrightarrow&\Bbb R\\&(x,y)&\mapsto&xy.\end{array}$$</span>Then <span class="math-container">$A=f^{-1}\bigl(\{1\}\bigr)$</span> and so, since <span class="math-container">$f$</span> is continuous and <span c... |
3,631,710 | <p>I'm wondering what the best method of sketching a curve is, if you know the coordinates of a point on the line and its intrinsic equation in the form:
<span class="math-container">$$
s=f(\psi)
$$</span>
where s is arc length from the origin to a point on the curve and where <span class="math-container">$$tan(\psi )... | Amit Bendkhale | 440,224 | <p>It is the National Council of Educational Research and Training, that decides the syllabus, and publishes content of secondary and senior-secondary students.
You can check their site for E-Books <a href="http://ncert.nic.in/textbook/textbook.htm?lemh1=0-6" rel="nofollow noreferrer">http://ncert.nic.in/textbook/textb... |
3,631,710 | <p>I'm wondering what the best method of sketching a curve is, if you know the coordinates of a point on the line and its intrinsic equation in the form:
<span class="math-container">$$
s=f(\psi)
$$</span>
where s is arc length from the origin to a point on the curve and where <span class="math-container">$$tan(\psi )... | Confused Simpleton | 722,188 | <p>These books covers the essence of what you are looking for. Almost everybody in 11th and 12th study from these books. </p>
<p><a href="https://jeemain.guru/best-books-for-mathematics-part-1/" rel="nofollow noreferrer">https://jeemain.guru/best-books-for-mathematics-part-1/</a></p>
<p>PS- The books written by NCERT... |
3,631,710 | <p>I'm wondering what the best method of sketching a curve is, if you know the coordinates of a point on the line and its intrinsic equation in the form:
<span class="math-container">$$
s=f(\psi)
$$</span>
where s is arc length from the origin to a point on the curve and where <span class="math-container">$$tan(\psi )... | Community | -1 | <p>I think that there is a misunderstanding of how math is taught in our country, it's not as intensive as you think. JEE is an exam which is taken by million students but only <span class="math-container">$11,000$</span> get selected. So if you require books of <strong>JEE Advanced</strong> standard then better go for... |
343,894 | <p>I've been helping my siblings with their GCSE and A Level maths and I've come across a question where they have just taken the positive square root. It's a pure maths question and there's no (obvious) reason to ignore the negative square root.</p>
<p>I always thought that the square root always gave two values, a p... | mau | 89 | <p>It depends. Sometimes there is an implicit constraint for which you only want the positive answer.</p>
|
4,373,267 | <p>In view of the paper "Forcing As A Computational Process" by J. Hamkins, R. Miller and K. Williams, I have revised my original question, part (a) about the computability of the Forcing Truth Definition, <a href="https://math.stackexchange.com/questions/3465785/looking-into-the-future-within-forcing">Lookin... | Community | -1 | <p>Let the graph have <span class="math-container">$m$</span> vertices of degree <span class="math-container">$6$</span> and <span class="math-container">$n$</span> vertices of degree <span class="math-container">$5$</span>.</p>
<p>Then <span class="math-container">$6m+5n$</span> is twice the number of edges and so <sp... |
124,291 | <p>I've been working on this all day long. Here's what I've done until now.The denominator is easy. It's $n^{2n}$. I compute the numerator as follows. </p>
<p>All $n$ bins have at least one ball = $n$ bins must have one of the $2n$ balls each + the remaining $n$ balls are placed in any of the bins in any fashion.</p>
... | André Nicolas | 6,312 | <p>The approach that you started on is a good one. The "total" count was right, but the count of the "favourable" cases was not. Let us generalize slightly. We throw $m$ (numbered) balls, one at a time, at a line of $x$ (numbered) buckets, and ask for the probability that none of the buckets ends up empty. </p>
<p>Th... |
206,421 | <p>If $4 \tan(\alpha - \beta) = 3 \tan \alpha $, then prove that
$$\tan \beta = \frac{\sin(2 \alpha)}{7 + \cos(2 \alpha)}$$</p>
<p>This is not homework and I've tried everything so I would just like a straight answer thank you in advance. </p>
| lab bhattacharjee | 33,337 | <p>$$\frac{\tan(\alpha-\beta)}{\tan \alpha}=\frac 3 4$$</p>
<p>Or, $$\frac{\sin(\alpha-\beta)\cos\alpha}{\cos(\alpha-\beta)\sin \alpha}=\frac 3 4$$</p>
<p>Applying <a href="http://en.wikipedia.org/wiki/Componendo_and_dividendo" rel="nofollow">Componendo and dividendo</a>,</p>
<p>$$\frac{\cos(\alpha-\beta)\sin \alpha... |
3,163,067 | <p><strong>Definition 1</strong> (Formal Language). A <em>language</em> <span class="math-container">$L$</span> over an <em>alphabet</em> <span class="math-container">$\Sigma$</span> (any nonempty finite set) is a subset of the set of all finite sequences of elements of <span class="math-container">$\Sigma$</span>, i.e... | noctusraid | 185,359 | <p>Here's a certainly much cleaner way (tbh, I didn't check your solution properly):</p>
<p>We want to show that <span class="math-container">$g_n$</span> converges uniformly to <span class="math-container">$g$</span>, which is the same as asking that <span class="math-container">$$\lim_{n \to + \infty} \sup_{y \in [0... |
1,790,032 | <p>Say you have a number $x^{\sqrt 2}$.</p>
<p>Is there any way to represent this number so that there's no root (or irrational) as the exponent (so that it's easier to understand for me)? I just can't wrap my head around this.</p>
<p>I was thinking something like $$x^{2^{1/2}}$$ and extending on that idea, but I don... | Roman83 | 309,360 | <p>$$\sqrt2= 1.4142135623730951...$$
Let $(a_n):1;1.4;1.41;1.412; 1.4121,...$
Then $$x_1=x^{1}$$
$$x_2=x^{a_2}=x^{1.4}=x^{\frac{14}{10}}=x^{\frac{7}{5}}=\sqrt[5]{x^7}$$
$$...$$
$$x_n=x^{a_n}$$
$$...$$
Then $$x^{\sqrt2}=\lim_{n\rightarrow \infty}x_n$$</p>
|
16,848 | <p>Each orientable 3-manifold can be obtained by doing surgery along a framed link in the 3-sphere. Kirby's theorem says that two framed links give homeomorphic manifolds if and only if they are obtained from one another by a sequence of isotopies and Kirby moves.</p>
<p>The original proof by R. Kirby (Inv Math 45, 35... | Bruce Westbury | 3,992 | <p>The first reference attempts to solve this problem but only gives a partial answer.
The second reference shows that every 3-manifold can be obtained by surgery on a link (but does not discuss Kirby calculus).</p>
<ul>
<li><p><a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1075370" rel="nofollow noreferrer"... |
94,213 | <p>You want to access a particular smartphone which has a 4-digit numeric pin, entered by tapping the screen. One day you see the owner wipe the screen, unlock the device, and then get distracted and walk off. You rush over, grab the phone but alas it has auto-locked again. The screen is clean except for smudge marks o... | Will Sherwood | 124,497 | <p>If you know that the pin is made up of only $x=2$ smudge marks, and it is a $n=4$ digit pin, the maximum number of combinations you will need to try is $x^n=2^4=16$, and the average number of pins you will need to try is $x^n/2=8$. This can be generalized for all $x,n\in\mathbb{N}$.</p>
|
3,546,801 | <p>I have often heard (both online and in person) people say that "<span class="math-container">$\mathbb{R}^2$</span> can't be totally ordered." I would like to understand this statement. </p>
<p>Of course, on the face of it, this is false: Pick your favorite bijection <span class="math-container">$f:\mathbb{R}^2 \to ... | Dark Malthorp | 532,432 | <p>I have not seen a standard definition of a nice total ordering, but I like the one you provided.</p>
<p>Neither <span class="math-container">$\mathbb{R}^2$</span> nor <span class="math-container">$S^1$</span> admit a nice ordering by your definition. The proof is straightforward:</p>
<p><strong>Claim 1:</strong> I... |
1,863 | <p>Problem / background: consider the following code snippet.</p>
<pre><code>pnt[fig_, n_] := {fig[[1, n]], fig[[2, n]]}
hor := {.025, .1, .25, .4, .475, .525, .6, .75, .9, .975}
ver := {.05, .20, .80, .95}
fig4 := Transpose[{
{hor[[1]], ver[[2]], 1}, {hor[[2]], ver[[1]], 1}, {hor[[4]],
ver[[1]], 1}, {hor[[5]]... | István Zachar | 89 | <p>Also, you can use <code>FaceForm</code> if you want to specify more than one directive (i.e. not just color, but opacity, etc.) for the primitive:</p>
<pre><code>Graphics[{FaceForm[{Pink, Opacity[.2]}], EdgeForm[Black],
FilledCurve[BezierCurve[{{-1, 0}, {-1, 2}, {1, 2}, {1, 0}}]]},
ImageSize -> 100]
</code>... |
869,892 | <p>The polynomials $p(x) = 5x^3 - 27x^2 + 45x - 21$ and $q(x) = x^4 - 5x^3 + 8x^2 - 5x + 3$ both interpolate the points $(1,2) , (2,1) , (3,6), (4,47)$. Even though these polynomials are of different degree, I do not understand how this is possible when the Lagrange interpolation theorem states there is only one polyno... | JP McCarthy | 19,352 | <p>There is a unique polynomial of degree $n$ or less through $n+1$ points but as many as you want of higher degree.</p>
|
2,026,143 | <p>What is need to be proven, in a proof by contradiction that a set is closed?</p>
<p>If we have to show that $K$ is closed that mean that we need to show that $K^{C}$ is open.
Let there be $x\in K^{C}$ we need to show that $B(x,\delta)\subset K^{C}$</p>
<p>What is the contrary assumption?</p>
<p>For all $\delta>... | Alex Provost | 59,556 | <p>In any topological space $X$, the set $K \subset X$ is <em>not</em> closed if there exists a limit point of $K$ that isn't in $K$. (This is the negation of "$K$ is closed if it contains all its limit points.") For a metric space (in terms of balls), this means that there exists $x \in X\setminus K$ such that every o... |
1,830,799 | <p>1There are $\frac{21!}{2!3!} = 120$ total positions (disregarding order within same colour). I imagine labelling the people Y (yellow) and NY (not yellow), so I imagine I have $4$ copies of the letter Y and $5$ of NY. So I draw out $9$ slots and want to arrange so that at least $2$ Y are together.
I get $30$ arrang... | Vera | 169,789 | <p>There are $\binom{5+4}4$ sums $a+b+c+d+e=5$ where the $a,b,c,d,e$ are nonnegative integers. Here $a$ stands for the number of non-yellows on the left of the utmost left yellow, $b$ for the number of non-yellows between the utmost left yellow and the yellow closest to the utmost left yellow, et cetera. </p>
<p>There... |
355,262 | <p>Is there a closed-form expression for the sum $\sum_{k=0}^n\binom{n}kk^p$ given positive integers $n,\,p$? Earlier I thought of this series but failed to figure out a closed-form expression in $n,\,p$ (other than the trivial case $p=0$).</p>
<p>$$p=0\colon\,\sum_{k=0}^n\binom{n}kk^0=2^n$$</p>
<p>I know that $\sum_... | Brian M. Scott | 12,042 | <p>Use the identity $k\dbinom{n}k=n\dbinom{n-1}{k-1}$: for $p=1$ you get</p>
<p>$$\sum_k\binom{n}kk=n\sum_k\binom{n-1}{k-1}=n\sum_k\binom{n-1}k=n2^{n-1}\;.$$</p>
<p>For $p=2$:</p>
<p>$$\begin{align*}
\sum_k\binom{n}kk^2&=n\sum_k\binom{n-1}{k-1}k\\
&=n\sum_k\binom{n-1}k(k+1)\\
&=n\sum_k\binom{n-1}kk+n\sum... |
58,914 | <p>Question:</p>
<p>If a square matrix $A$ satisfies $A^2=I$ and $\det A>0$, show that $A+I$ is non-singular.</p>
<p>I have tried to suppose a non-zero vector $x$ s.t. $Ax=x$ but fail to make a contradiction.</p>
<p>And I tried to find the inverse matrix of $A+I$ directly, suppose $(A+I)^{-1}=\alpha I +\beta A$, ... | Community | -1 | <p>$-I$ with even size is a counterexample.</p>
|
214,007 | <p>I have an experimental data set: </p>
<pre><code>data1 = {{71.6`, 0.41`}, {27.2`, 4.96`}, {59.3`, 0.18`}, {46.`,2.72`},
{42.2`, 1.06`}, {89.1`, 3.75`}, {88.6`, 1.9`}, {62.3`,1.8`},
{35.5`,1.84`}}
</code></pre>
<p>In order to eliminate unrealistic data points step by step automatically, I fit and ... | Mark R | 65,931 | <p>As others have said, the original data is too far off a linear fit to have any points survive your constraint. I think this function will do what you want:</p>
<pre><code>TrimDataWithLinearFit[{data_, relativeAbsDifferenceLimit_,
absoluteDifferenceLimit_}] :=
Block[{linearFit = Fit[data, {1, x}, x], newData}... |
1,040,505 | <p>Apparently,
$$(1-\cot 37^\circ)(1-\cot 8^\circ)=2.00000000000000000\cdots$$<br>
Since it is a $2.0000000000\cdots$ instead of $2$, it isn't exactly $2$.<br>
Why is that?</p>
| Michael Hardy | 11,667 | <p>$$
1 = \cot45^\circ = \cot(37^\circ+8^\circ) = \frac{\cot37^\circ\cot8^\circ-1}{\cot37^\circ+\cot8^\circ}.
$$
Therefore
$$
\cot37^\circ+\cot8^\circ = \cot37^\circ\cot8^\circ-1
$$
so
$$
2 = 1-\cot37^\circ-\cot8^\circ +\cot37^\circ\cot8^\circ=(1-\cot37^\circ)(1-\cot8^\circ).
$$</p>
|
161,780 | <p>I've written the code that follows:</p>
<pre><code>tails = Function[l, If[ l == {}, {}, Prepend[tails[Drop[l, 1]], l]]]
Te = Composition[AllTrue[PrimeQ], Map[FromDigits], tails, IntegerDigits]
Timing[Select[Range[2, 10^6], Te]]
</code></pre>
<p>This takes around 21seconds on my computer, but the equivalent Haske... | Carl Woll | 45,431 | <p><em>Updated to use PrimePi per @MichaelE2's helpful comment!</em></p>
<p>Here's another approach. First, let's get a list of all the primes, and then define a prime vector where 1 indicates the index is a prime:</p>
<pre><code>primeList = Prime[Range[PrimePi[10^7]]]; // AbsoluteTiming
primeVector = ConstantArray[... |
776,615 | <p>Consider a simple function $f(x,y)=\frac{x}{y}, x,y \in (0,1]$, the Hessian is not positive semi definite and hence it is a non convex function. However, when we plot the function using Matlab/Maxima, it "appears" convex. For the sake of clarity we want to find points which violate the definition of convexity, in ot... | Kirillvh | 33,078 | <p>I think if you only look at one quadrant of $f(x)=\frac{x}{y}$ then it appears to be convex but if you look at all the quadrants then you should see that its non-convex.</p>
|
776,615 | <p>Consider a simple function $f(x,y)=\frac{x}{y}, x,y \in (0,1]$, the Hessian is not positive semi definite and hence it is a non convex function. However, when we plot the function using Matlab/Maxima, it "appears" convex. For the sake of clarity we want to find points which violate the definition of convexity, in ot... | AndreaCassioli | 130,183 | <p>If you write down the Hessian matrix you'll see that for $x,y\in (0,1]$ the Hessian is indeed positive semi-definite. This is call <em>conditional</em> positive semi-definiteness. You can see the Hessian as a matrix parametrized by $x$ and $y$: it is positive semi-definite depending on their values. Try to compute t... |
1,292,836 | <p>I am trying to evaluate $$\oint _C \frac{-ydx+xdy}{x^2+y^2}$$</p>
<p>clockwise around the square with vertices (−1,−1), (−1,1), (1,1), and (1,−1).</p>
<p>So from the question,
$$\vec{F}=<\frac{-y}{x^2+y^2},\frac{x}{x^2+y^2}>$$</p>
<p>I first conducted the gradient test $\frac{\partial F_2}{\partial x}=\fra... | Mark Viola | 218,419 | <p>The reason that Stokes' Theorem does not apply directly here is that $F$ is not continuous (and thus, not differentiable) throughout the region bounded by $C$. The singularity at the origin is the source the issue. </p>
<p><strong>METHOD 1</strong>: Brute Force</p>
<p>The integration is comprised of the sum of ... |
1,060,011 | <p>Given <strong>a, b, c</strong> 3 real numbers. Prove that if a < b < c, then |b|$\leqslant$max (|a|,|c|).</p>
<p>I orginally proved this by discussing four cases as </p>
<p>1) a,b,c<0 </p>
<p>2) a<0, b,c>0</p>
<p>3) a,b<0, c>0</p>
<p>4) a,b,c>0</p>
<p>Here I wonder if there is an easier way to d... | angryavian | 43,949 | <p>Some casework is unavoidable; what you have is fine.</p>
<p>If $b=0$, the inequality is clear. If $b\ne 0$, then at least one of $a$ and $c$ has the same sign as $b$ and is farther away from zero. [Explicitly, if $b>0$, then $c>b>0 \implies |b| <|c|$; if $b<0$, then $a<b<0 \implies |b| <|a|$... |
4,346,363 | <p>I'm interested in the equation <span class="math-container">$$AX=B$$</span> where <span class="math-container">$A, X,$</span> and <span class="math-container">$B$</span> are rectangular matrices with suitable dimensions, for example <span class="math-container">$m\times n$</span>, <span class="math-container">$n\tim... | Ninad Munshi | 698,724 | <p>If <span class="math-container">$m < n$</span> then <span class="math-container">$A^TA$</span> will always be uninvertible. The reason is since <span class="math-container">$A$</span> was a map from <span class="math-container">$\Bbb{R}^n$</span> to <span class="math-container">$\Bbb{R}^m$</span> (in this case, a... |
4,186,977 | <p>The challenge is to prove <span class="math-container">$$\left(1-\dfrac{a}{b}\right)\left(1+\dfrac{c}{d}\right)=4.$$</span></p>
<p><a href="https://i.stack.imgur.com/mTFF9.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/mTFF9.jpg" alt="enter image description here" /></a></p>
<p>Apart from the Pyt... | QLimbo | 929,588 | <p>So what I want to say is that I found what could be said to be the "best" way to solve this problem. First, consider the following drawing:</p>
<p><a href="https://i.stack.imgur.com/RhLkw.png" rel="noreferrer"><img src="https://i.stack.imgur.com/RhLkw.png" alt="enter image description here" /></a></p>
<p>A... |
3,934,819 | <p>I am wondering if we can find a linear transformation matrix <span class="math-container">$A$</span> of size <span class="math-container">$3\times 3$</span> over the field of two elements <span class="math-container">$\mathbb{Z}_2$</span> i.e. a matrix <span class="math-container">$A$</span> of zeros and ones s.t.</... | mathcounterexamples.net | 187,663 | <p>The images of the vector <span class="math-container">$e_1, e_2,e_3$</span> of the <a href="https://en.wikipedia.org/wiki/Canonical_basis?wprov=sfti1" rel="nofollow noreferrer">canonical basis</a> of <span class="math-container">$\mathbb Z_2^3$</span> defines entirely <span class="math-container">$A$</span>.</p>
<p>... |
3,934,819 | <p>I am wondering if we can find a linear transformation matrix <span class="math-container">$A$</span> of size <span class="math-container">$3\times 3$</span> over the field of two elements <span class="math-container">$\mathbb{Z}_2$</span> i.e. a matrix <span class="math-container">$A$</span> of zeros and ones s.t.</... | J. W. Tanner | 615,567 | <p>It's not possible because <span class="math-container">$A\begin{bmatrix}0\\1\\1\end{bmatrix}\ne A\begin{bmatrix}0\\0\\1\end{bmatrix}+A\begin{bmatrix}0\\1\\0\end{bmatrix}$</span>.</p>
|
81,435 | <p>Normally we consider simple arithmetic to be related to the world of objects. So the sum $3+2=5$ means $3$ three apples and $2$ apples gives $5$ apples. But is there an alternative interpretation which does not have anything to do with discrete objects?</p>
| Michael Hardy | 11,667 | <p>"Discrete" objects? Or do you mean <em>physical</em> objects? If the latter, I could say that 43, 21, and 50 are three numbers, and 20 and 100 are two numbers distinct from those, and three plus two is five, so 43, 21, 50, 20, and 100 are five numbers. This time they're not physical objects like apples, but they'... |
57,914 | <p>I have the following code to show a red area defined by inequalities:</p>
<pre><code>ClearAll["Global`*"];
p = Reduce[y <= 3/10 x + 18 && y > x^2/8, {x, y}]
r = RegionPlot[p, {x, -15, 18}, {y, -5, 25},
GridLines -> {Table[i, {i, -15, 18}], Table[j, {j, -5, 25}]},
PlotStyle -> Directive[{... | kglr | 125 | <pre><code>eqn = y <= 3/10 x + 18 && y > x^2/8;
sol = Reduce[eqn, {x, y}, Integers];
Length @ sol
(* 286 *)
points = {x, y} /. {ToRules[sol]}; (* thanks: BobHanlon *)
RegionPlot[eqn, {x, -15, 18}, {y, -5, 25},
GridLines -> {Range[-15, 18], Range[-5, 25]},
PlotStyle -> Directi... |
1,276,206 | <p>I am writing a computer program that involves generating 4 random numbers, a, b, c, and d, the sum of which should equal 100. </p>
<p>Here is the method I first came up with to achieve that goal, in pseudocode:</p>
<pre><code>Generate a random number out of 100. (Let's say it generates 16).
Assign this value as th... | David Holden | 79,543 | <p>there may be a need for slightly more precise specification of what kind of sample you want. but to begin with you may feel less uneasy if you sample by picking <i>three</i> numbers at random in $[0,100] \cap \mathbb{Z}$, let us call them $a,b,c$ supposing you have ordered them so that $0 \le a \le b \le c \le 100$... |
1,276,206 | <p>I am writing a computer program that involves generating 4 random numbers, a, b, c, and d, the sum of which should equal 100. </p>
<p>Here is the method I first came up with to achieve that goal, in pseudocode:</p>
<pre><code>Generate a random number out of 100. (Let's say it generates 16).
Assign this value as th... | Thomas Andrews | 7,933 | <p>No, this is not a good approach - half the time, the first element will be $50$ or more, which is way too often. Essentially, the odds that the first element is $100$ should not be the same as the odds that the first elements is $10$. There is only one way for $a=100$, but there are loads of ways for $a=10$.</p>
<p... |
1,736,405 | <p>I was told to consider the degrees but I'm not sure how the degrees of the polynomial so can help me here. </p>
| Hagen von Eitzen | 39,174 | <p>With $\alpha=\sqrt p$, $\beta=\sqrt[3]p$, $\gamma =\sqrt[6]p$ we have $\alpha=\gamma^3$, $\beta=\gamma^2$ and $\gamma = \frac 1p\alpha\beta^2$.</p>
|
3,098,350 | <p>Recently I was able to find a result to a common definite integral:
<span class="math-container">\begin{equation}
J(n_1, k_1, m_1) = \int_0^{\infty} \frac{x^{k_1}}{\left(x^{n_1} + a_1 \right)^{m_1}}\:dx = \frac{a^{\frac{k_1 + 1}{n_1} - m_1}}{n_1}\Gamma\left(m_1 - \frac{k_1 + 1}{n_1}\right)\Gamma\left(\frac{k_1 + 1}... | omegadot | 128,913 | <p>This is <em>not</em> a solution. It is an approach that shows where some of the potential roadblocks lay. Of course, this is not to say they are not insurmountable, but <span class="math-container">$\ldots$</span></p>
<p>The approach to be use will make use of the so-called <a href="https://en.wikipedia.org/wiki/Sc... |
257 | <p>In my graduate courses, I often have my students write term papers on original mathematical topics. I explain the process in <a href="https://mathoverflow.net/a/133643/1946">this answer over at MathOverflow</a>. It works fairly well for me. </p>
<p>But I'd be very interested in hearing about other experiences or ap... | dtldarek | 42 | <p>One professor (who I greatly respect) did as follows:</p>
<ul>
<li>He made the papers a part of the exam, the second part being the talk/presentation.</li>
<li>The papers had to be in some particular conference format (there were style files at the conference page).</li>
<li>There were two dates in which students c... |
2,304,714 | <p>In our introductory course on groups, we defined cosets and quotient groups in the following way.</p>
<blockquote>
<p>Let $N\trianglelefteq G$ be a normal subgroup of a group $(G,\,\cdot\,)$. Then the quotient group $G/N$ contains all the cosets of $N$ with respect to the elements of $G$, i.e. $G/N=\{Ng:g\in G\}$... | Trevor Gunn | 437,127 | <p><em>This addresses the question in the comments about taking cosets of (subgroups of) $R^\times$.</em></p>
<p>A ring is a group with a multiplication. A quotient ring is a quotient group with a multiplication. Since the group structure on a ring is additive, it only makes sense to look at additive cosets.</p>
<p>T... |
26,313 | <p>In page 21 of <em>A Problem seminar</em>, D. J. Newman presents a novel way (at least for me) to determine the expectation of a discrete random variable. He refers to this expression as the <strong>failure probability formula</strong>. His formula goes like this</p>
<p>$f_{0}+f_{1}+f_{2}+\ldots$</p>
<p>where $f_{n... | Leandro Vendramin | 17,845 | <p>I believe that it is important to mention the notion of Weyl groupoid. </p>
<p>The abstract Weyl groupoid was defined by Heckenberger and Yamane. (In their paper you will find some basic material about Weyl groupoids, generalized root systems and some ideas about the role played by the Weyl groupoid in Lie superalg... |
26,313 | <p>In page 21 of <em>A Problem seminar</em>, D. J. Newman presents a novel way (at least for me) to determine the expectation of a discrete random variable. He refers to this expression as the <strong>failure probability formula</strong>. His formula goes like this</p>
<p>$f_{0}+f_{1}+f_{2}+\ldots$</p>
<p>where $f_{n... | Alexander Chervov | 10,446 | <p>As it is clear from the other answers there are several viewpoints on Weyl group in super case.</p>
<p>Let me mention papers by Sergeev and Veselov, who also stands on the point of the Weyl groupoid in the super case.
As far as I understand such viewpoint agrees with their works on
Calogero-Moser integrable syste... |
3,214,136 | <p>I found a "fun algebra problem" that asks you to find the area of a triangle whose sides are <span class="math-container">$\sqrt{5}$</span>, <span class="math-container">$\sqrt{10}$</span>, <span class="math-container">$\sqrt{13}$</span>. After some algebra hell trying to work with Heron's formula, I plugged the qu... | J. W. Tanner | 615,567 | <p><strong>Hint:</strong></p>
<p>This <a href="https://en.wikipedia.org/wiki/Heron%27s_formula#History" rel="nofollow noreferrer">formula equivalent to Heron's </a></p>
<p><span class="math-container">$$\frac12 \sqrt{a^2c^2-\left(\dfrac{a^2+c^2-b^2}{2}\right)^2}$$</span></p>
<p>is useful in your situation.</p>
|
597,845 | <p>I am almost embarrassed writing this. But can someone tell me why this may not be true (so, please give me a counter example) for a power series where $x \in [-1,1]$</p>
<p>$|\sum_{n \geq 0} a_n x^n| \leq \sum_{n \geq 0} a_n $</p>
<p>Where $\sum_{n\geq 0} a_n $ is known to be convergent. </p>
<p>What if $a_n \geq... | alpacahaircut | 113,555 | <p>How about $a_n=\frac{(-1)^n}{n}$ then at $x=-1$ the sum on the left diverges and the sum on the right is $log(2)$. </p>
|
51,552 | <p>I am not sure if what I want to do is possible in <em>Mathematica</em>. I've provided something simple below to convey what I want to do.
Basically what I want to happen is that when the <code>Locator</code> crosses over the blue line, the <code>Disk[]</code> will turn red and stay red until the <code>Locator</code... | Kuba | 5,478 | <p>I don't know how to handle parallel <code>Events</code> but if there is single <code>Locator</code> you can try this:</p>
<pre><code>DynamicModule[{col = Blue, acc = 0, p = {1, 1}},
EventHandler[
Show[
Graphics[{Dynamic@Disk[p, Scaled@.03]}],
Plot[x^3, {x, 0, 1}] /. l_Line :> EventHandler[ l, {"MouseEnt... |
51,552 | <p>I am not sure if what I want to do is possible in <em>Mathematica</em>. I've provided something simple below to convey what I want to do.
Basically what I want to happen is that when the <code>Locator</code> crosses over the blue line, the <code>Disk[]</code> will turn red and stay red until the <code>Locator</code... | C. E. | 731 | <p>One way to detect crossings is to form a straight line between the previous position of the locator and the new position, and then check if there is an intersection between that straight line and $x^3$. I wrote this function to count the number of line intersection of a straight line with endpoints <code>p1</code> a... |
1,400,436 | <p>This is a question we asked on a second semester calculus test.</p>
<p>For what values of $p$ does this series converge?
$$\sum_{n=1}^{\infty}\frac{\sin(1/n)}{n^p}$$</p>
<p>I believe that it actually can be shown that $p> 0$ is a valid answer. </p>
<p>However. I am interested in finding a proof that is simple ... | Paolo Leonetti | 45,736 | <p>$$
\frac{\sin 1/n}{n^p} \sim \frac{1/n}{n^{p}} \,\,\, \text{if }n\to \infty
$$
so I guess every $p>0$ :)</p>
|
927,480 | <p>Consider a univariate function $f(x)$. I know the graphical intuition behind why $f'(x)=0$ at the extrema of $f$. But how do you prove it mathematically? </p>
<p>I start with the assumption of $x^*$ being a minimum (the maximum case can be proved likewise), then
$f(x^*+h) \geq f(x^*)$ where $h\in \mathcal{N}(x)$, $... | Harald Hanche-Olsen | 23,290 | <p><strong>Hint:</strong> For which numbers $n$ is $\lg n$ an integer?</p>
|
24,990 | <p>Usually I write equations in questions/answers, but writing the equation in LaTeX takes up the most time. It takes me nearly three or four times as long to write them in LaTeX. Is there any way to make my equation writing faster? I already use a visual LaTeX editor to speed things up, but I'm wondering if there are ... | Martin Sleziak | 8,297 | <p>It seemed to me as a good idea to make a CW answer where Math.SE users could add the problems they stumbled upon while using this search engine. (EDIT: And we can also use <a href="http://chat.stackexchange.com/rooms/info/46148/in-the-search-of-a-question-using-approach0-or-by-other-means?tab=general">this chatroom<... |
269,474 | <p>If $A$ is a $m \times n$ matrix and $B$ a $n \times k$ matrix, prove that</p>
<p>$$\text{rank}(AB)\ge\text{rank}(A)+\text{rank}(B)-n.$$</p>
<p>Also show when equality occurs.</p>
| salfaris | 453,441 | <p>Because I can't comment (yet), I will answer @user185640's question in a comment to @Babak Miraftab's answer which is "Why <span class="math-container">$\{B(\alpha_{r+1}),\ldots,B(\alpha_n)\}$</span> is linear independent" in this post.</p>
<p>Claim: <span class="math-container">$\{B(\alpha_{r+1}),\ldots,... |
2,214,287 | <p>My exam review states that I need to utilize the difference formula for sine to solve the equation on the interval $0 \leq \theta < 2\pi $</p>
<p>$$\sqrt3\sin \theta- \cos\theta = 1$$</p>
<p>I know that: $\sin \frac\pi3 = \frac{\sqrt3}{2}$
and $\cos\frac\pi3 = \frac12 $, so I divide each term by 2 and rewrite t... | lab bhattacharjee | 33,337 | <p>$$\sin\left(\theta-\dfrac\pi6\right)=\sin\dfrac\pi6$$</p>
<p>Now $\sin x=\sin A,x=n\pi+(-1)^nA$ where $n$ is any integer</p>
|
2,214,287 | <p>My exam review states that I need to utilize the difference formula for sine to solve the equation on the interval $0 \leq \theta < 2\pi $</p>
<p>$$\sqrt3\sin \theta- \cos\theta = 1$$</p>
<p>I know that: $\sin \frac\pi3 = \frac{\sqrt3}{2}$
and $\cos\frac\pi3 = \frac12 $, so I divide each term by 2 and rewrite t... | Jaideep Khare | 421,580 | <p>$$\sqrt3\sin \theta- \cos \theta = 1$$</p>
<p>$$\frac{\sqrt3}{2} \sin \theta- \frac{1}{2}\cos \theta = \frac{1}{2}$$</p>
<p>Now, $\sin \dfrac{\pi}{6}=\dfrac{ 1}{2}$ and $\cos \dfrac{\pi}{6}=\dfrac{ \sqrt{3}}{2}$</p>
<p>$$\cos \dfrac{\pi}{6} \sin \theta- \sin \dfrac{\pi}{6}\cos \theta = \frac{1}{2}$$</p>
<p>Let $... |
1,123,694 | <p>Prove that $az^n+b\overline{z}^n=0$ when $|a|\ne|b|$ and $n\in\mathbb{N_1}$does not have any complex solutions except for $0$. What happens if $n\in\mathbb{C}$?</p>
<p>The first one seems very obvious, but is there any way to show it very formally? </p>
| Timbuc | 118,527 | <p>$$az^n+b\overline z^n=0\implies az^{2n}+b|z|^{2n}=0$$</p>
<p>Suppose $\;z=re^{it}\;,\;\;r\in\Bbb R^+\;,\;\;t\in[0,2\pi]\;$ , then the above is</p>
<p>$$0=ar^{2n}e^{2nit}+br^{2n}\stackrel{\text{de Moivre}}= r^{2n}\left[\left(a\cos2nt+b\right)+ai\sin 2nt\right]\implies$$</p>
<p>$$\begin{cases}a\cos2nt+b=0\\{}\\a\si... |
1,228,290 | <p>We have the system $y''=-7y'-12y-u'-2u$</p>
<p>If we choose $x_1=y,x_2=y'$ we can write the system as</p>
<p>$x'=Ax + Bu \\ y= Cx$</p>
<p>Finding A is easy, but how do I find expressions for $B$ and $C$ when we have derivatives of the input in the expression?</p>
| Lutz Lehmann | 115,115 | <p>Set $x_1=y$, $x_2=y'+u$ then
\begin{align}
x_1'&=y'=x_2-u
\\
x_2'&=y''+u'=−7y′−12y−2u\\&=-7(x_2-u)-12x_1-2u\\&=-12x_1-7x_2+5u
\end{align}</p>
|
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