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 |
|---|---|---|---|---|
1,072,561 | <p>can you help me with this quest?</p>
<p>About composition $f$ and vector space $\mathbf{V}=\mathbb{Z^4_2}$ we know the following:</p>
<p>$f \circ f = id_V$,$~~f $
$
\left(\begin{array}{ccc}
1\\
0\\
1\\
0\\
\end{array}\right)=
\left(\begin{array}{ccc}
1\\
1\\
0\\
0\\
\end{array}\right)
$,
$~~f $
$
\left(\begin{ar... | Ilmari Karonen | 9,602 | <p>Let $f(\mathbf v_E) = \mathbf v_B$. Can you show that the map $f$ satisfies the <a href="//en.wikipedia.org/wiki/Linear_map#Definition_and_first_consequences" rel="nofollow">definition of a linear map</a>, i.e. that $f(\mathbf u + \mathbf v) = f(\mathbf u) + f(\mathbf v)$ and that $f(\alpha\,\mathbf u) = \alpha\, f... |
446,959 | <p>I am searching two simple/efficient/generic algorithms to generate a uniform distribution of random points:</p>
<ul>
<li>in the volume of a n-dimensional hypersphere</li>
<li>on the surface of a n-dimensional hypersphere</li>
</ul>
<p>knowing the dimension $n$, the center of the hypersphere $\vec{x}$ and its radiu... | sheß | 86,591 | <p>For the volume I'd say the simlest algorithm would be a standard accept-reject algorithm.</p>
<pre><code>(1) draw a uniform random point within an n-dimensional hypercube
(2) repeat (1) if the distance exceeds $r$, otherwise done
</code></pre>
<p>if im not mistaken though, accept-rates decrease as $n$ increases, m... |
1,680,210 | <p>Three cards are drawn at random from a full deck. what is the probability of getting a three , a seven, and an ace?</p>
<p>Can someone help me with this question please, thanks!</p>
| Ojas | 154,392 | <p>Substituting <span class="math-container">$x = 1 \implies a \mid 92$</span></p>
<p>Substituting <span class="math-container">$x = -1 \implies (a+2)\mid 88$</span></p>
<p>Divisors of <span class="math-container">$92 = 1,2,4,23,46,92$</span></p>
<p>Divisors of <span class="math-container">$88 = 1,2,4,8,11,22,44,88... |
1,680,210 | <p>Three cards are drawn at random from a full deck. what is the probability of getting a three , a seven, and an ace?</p>
<p>Can someone help me with this question please, thanks!</p>
| Jean Marie | 305,862 | <p>A remark and an alternative proof.</p>
<p><strong>Remark</strong>: this problem as already been posed in the 1963 Putnam competition</p>
<p><a href="https://mks.mff.cuni.cz/kalva/putnam/psoln/psol637.html" rel="nofollow noreferrer">https://mks.mff.cuni.cz/kalva/putnam/psoln/psol637.html</a></p>
<p><strong>An alte... |
4,070,996 | <p>I'm a bit lost in this integral: <span class="math-container">$$\int \frac{1}{1+\sin^4(x)} \, \mathrm dx$$</span>
I have tried solving with Wolfram, but I was getting a cosecant solution which doesn't seem as the correct method.</p>
<p>Do you have any ideas? :)</p>
<p>EDIT:
Do you please have step-by-step solution, ... | Claude Leibovici | 82,404 | <p>For <span class="math-container">$n=4$</span>, you can make it shorter since
<span class="math-container">$$A=\frac 1{1+\sin^4(x)}=\frac 1{(\sin^2(x)+i)(\sin^2(x)-i)}$$</span></p>
<p>Using partial fractions and double angle formula
<span class="math-container">$$A=\frac{i}{\cos (2 x)-(1-2 i)}-\frac{i}{\cos (2 x)-(1+... |
918,049 | <p>Let $f: X \to Y$ be a map of sets. We are given that $X$ is a topological space. We are to show that there is a topology on $Y$ making $f$ continuous, and moreover, determine if this topology is unique. </p>
<p>Should I read it as "$X$ is just a set on which there is given a topology, i.e sets that are open in $X$ ... | almagest | 172,006 | <p>The first thing is to establish some notation so that you can apply induction.</p>
<p>Put $a_1=\sqrt{\alpha},a_n=\sqrt{\alpha+a_n}$. Check that that is the series you want. </p>
<p>Now you go about proving it by induction in the usual way. Start by showing that $a_1<(1+\sqrt{4\alpha+1})/2$. Then show that if $a... |
137,813 | <p>Let $k$ be an algebraically closed field of characteristic $0$, let $C_{/k}$ be a nice (smooth, projective, geometrically integral curve), let $K = k(C)$, and let $\overline{K}$ be an algebraic closure of $K$. Let $E_{/K}$ be an elliptic curve with $j(E) \notin k$. Let $P \in E(\overline{K}) \setminus E(K)$ be a p... | Remke Kloosterman | 8,621 | <p>I will sketch a counterexample for the modified question. The idea behind the construction is similar to the counterexample for the original question. Only the geometric details of this construction are trickier and does not integrate so well in my previous answer. For this reason I will post this a new answer.</p>
... |
2,338,508 | <p>Given this matrix that stretches to infinity to the right and up:
$$
\begin{matrix}
...&...&...\\
\frac{1}{4}& \frac{1}{8}& \frac{1}{16}&... \\
\frac{1}{2} & \frac{1}{4}& \frac{1}{8}&... \\
1 & \frac{1}{2}& \frac{1}{4}&... \\
... | Bob Jones | 370,252 | <p>Notice that the second matrix you come up with is not absolutely convergent: you have an infinite number of $2$'s whose sum is $\infty$, and same with an infinite number of $-1$'s. So you can't just rearrange terms like that, and if you do, you get a wrong answer. What you need to do is not write $1$ as $2-1$, but j... |
3,520,538 | <blockquote>
<p>Let <span class="math-container">$X$</span> be a Hausdorff topological space.
Prove that if <span class="math-container">$\{ C_i | i \in I \}$</span> is an infinite collection of compact subsets of <span class="math-container">$X$</span> such that <span class="math-container">$\cap_{i \in I} C_i = ... | Kavi Rama Murthy | 142,385 | <p>How about <span class="math-container">$u_n=\frac 1 n$</span> for <span class="math-container">$n$</span> even and <span class="math-container">$\frac 1 {n^{2}}$</span> ( or <span class="math-container">$\frac 1 {2^{n}})$</span> for <span class="math-container">$n$</span> odd?</p>
|
1,927,599 | <p>In first-order logic, there is normally a formal distinction between <em>constants</em> and <em>variables</em>. Namely:</p>
<ul>
<li><p>A <em>constant symbol</em> is a $0$-ary function symbol in a language $\mathcal{L}$.</p></li>
<li><p>A <em>variable</em> is one of countably many special symbols used for first-or... | Eric Wofsey | 86,856 | <p>The obvious reason to distinguish constants from variables is that sometimes we really do want constants as a part of our structure, and not constants that can be uniquely determined from the other structure like the identity of a group. For instance, a perfectly natural sort of mathematical structure to study is a... |
4,333,566 | <p>Say that <span class="math-container">$\{a_n\}^\infty_\mathrm{n=1}$</span> is a sequence of real numbers so that for all odd <span class="math-container">$n$</span>, <span class="math-container">$a_n \in (0, 1)$</span> and for all even <span class="math-container">$n$</span>, <span class="math-container">$a_n \not\i... | Henry | 6,460 | <p>You do not need any property for <span class="math-container">$a$</span>, since its square is a square.</p>
<p>If <span class="math-container">$b=v^2+w^2+z^2$</span> (copying Simon Fox's description) then</p>
<p><span class="math-container">$$a^2b =a^2v^2+a^2w^2+a^2z^2 = (av)^2+(aw)^2+(az)^2$$</span></p>
|
3,818,445 | <p>I'm trying to compute <span class="math-container">$\operatorname{Ext}_{\mathbb{Z}}^1(\mathbb{Z}[1/p],\mathbb{Z})\cong \mathbb{Z}_p/\mathbb{Z}$</span>.</p>
<p>Now I have the projective resolution
<span class="math-container">$$0\rightarrow \bigoplus_{i>1}\mathbb{Z}\xrightarrow{\alpha} \mathbb{Z}\oplus \bigoplus_{... | Angina Seng | 436,618 | <p>Let's do this by injective resolutions instead. I'll write <span class="math-container">$A$</span> for <span class="math-container">$\Bbb Z[1/p]$</span>.
Then
<span class="math-container">$$0\to\Bbb Z\to\Bbb Q\to\Bbb Q/\Bbb Z\to0$$</span>
gives an injective resolution of <span class="math-container">$\Bbb Z$</span>.... |
3,414,072 | <blockquote>
<p>Let <span class="math-container">$U \subset \mathbb{R}^N$</span> be an open set, let <span class="math-container">$f : U \times [a, b] \to \mathbb{R}$</span> be a continuous function. Consider the function <span class="math-container">$$g(x):= \int_a^b f(x,y) \,dy$$</span>
with <span class="math-con... | Marios Gretsas | 359,315 | <p>Let <span class="math-container">$x \in U$</span> such that <span class="math-container">$x_n \to x$</span></p>
<p>Then <span class="math-container">$x_n$</span> is bounded,thus exists a closed ball <span class="math-container">$B:=\overline{B(0,M)}$</span> such that <span class="math-container">$x_n \in B,\forall... |
3,969,679 | <p><strong>The letters in the word GUMTREE and KOALA are rearranged to form a 12-letter word where KOALA appears precisely in order but not necessarily together. How many ways can this happen?</strong></p>
<p>So I attmepted it via this method:</p>
<p>Firstly arrange like this since KOALA must be in order but not necess... | heropup | 118,193 | <p>You are assuming that the letters for the two words are interleaved, when they do not need to be. For example, "KOALAGUMTREE" is a permissible arrangement, as is "GUMTREEKOALA" or "GUMKOALATREE." The problem only stipulates that the letters in "KOALA" appear in sequence.</p>... |
758,158 | <p>I am trying to use <span class="math-container">$f(x)=x^3$</span> as a counterexample to the following statement. </p>
<p>If <span class="math-container">$f(x)$</span> is strictly increasing over <span class="math-container">$[a,b]$</span> then for any <span class="math-container">$x\in (a,b), f'(x)>0$</span>. <... | evil999man | 102,285 | <p>$$\frac{d x^3}{dx}=3x^2\geqslant0, \forall x \in[1,2]$$</p>
<p>Function is strictly increasing in $[1,2]$ and...</p>
|
758,158 | <p>I am trying to use <span class="math-container">$f(x)=x^3$</span> as a counterexample to the following statement. </p>
<p>If <span class="math-container">$f(x)$</span> is strictly increasing over <span class="math-container">$[a,b]$</span> then for any <span class="math-container">$x\in (a,b), f'(x)>0$</span>. <... | André Nicolas | 6,312 | <p>Consider $a^3-b^3$, where $a\gt b$. We want to show this is <strong>positive</strong>.
We have
$$a^3-b^3=(a-b)(a^2+ab+b^2).$$
Note that $a^2+ab+b^2$ is positive unless $a=b=0$. There are many ways to do this. For example,
$$a^2+ab+b^2=\frac{1}{2}\left(a^2+b^2+(a+b)^2\right).$$
More conventionally, complete the squa... |
4,694 | <p>While teaching the concept of vector spaces, my professor mentioned that addition and multiplication aren't necessarily what we <em>normally</em> call addition and multiplication, but any other function that complies with the eight axioms needed by the definition of a vector space (for instance, associativity, commu... | whuber | 1,489 | <p>Many of the operations commonly encountered in vector spaces, such as modular arithmetic, functional convolution, or even p-adic arithmetic, ultimately are related to the addition and multiplication with which you are familiar. One interesting exception is Conway's Field of <em>nimbers</em> (which is effectively a ... |
4,694 | <p>While teaching the concept of vector spaces, my professor mentioned that addition and multiplication aren't necessarily what we <em>normally</em> call addition and multiplication, but any other function that complies with the eight axioms needed by the definition of a vector space (for instance, associativity, commu... | Qiaochu Yuan | 232 | <p>My favorite example is the set of subsets of a set under the operation of <a href="http://en.wikipedia.org/wiki/Symmetric_difference">symmetric difference</a> (otherwise known as bitwise XOR). This forms a vector space over the <a href="http://en.wikipedia.org/wiki/Finite_field">finite field</a> $\mathbb{F}_2$. Th... |
148,853 | <p>Here is my code;</p>
<pre><code>f =
Exp[-2*x^2]*(1 - Exp[-1/y])*Sqrt[2/π]*
Sum[
Exp[-n/y]*Exp[4*n*χ*x*Sin[θ]]*Exp[-n^2*χ^2*x*(Sin[θ])^2],
{n, 0, 10}];`
g = D[f, y];
f1 = g^2/f;
f2 = Integrate[f1, {x, -∞, ∞}]
</code></pre>
<p>When I evaluate the expression for <code>f2</code>, Mathematica jus... | Bill | 18,890 | <p>Let's see if we can demonstrate a little progress while it is working. We can use <code>Simplify</code> to sometimes make integrals a little easier. Then because you have a fraction we can use <code>Apart</code> to see if we can break this into a sum of simpler terms. Then we use <code>Expand</code> to try to really... |
148,853 | <p>Here is my code;</p>
<pre><code>f =
Exp[-2*x^2]*(1 - Exp[-1/y])*Sqrt[2/π]*
Sum[
Exp[-n/y]*Exp[4*n*χ*x*Sin[θ]]*Exp[-n^2*χ^2*x*(Sin[θ])^2],
{n, 0, 10}];`
g = D[f, y];
f1 = g^2/f;
f2 = Integrate[f1, {x, -∞, ∞}]
</code></pre>
<p>When I evaluate the expression for <code>f2</code>, Mathematica jus... | m_goldberg | 3,066 | <p>I don't think you have a chance of finding a symbolic result for your integral. I think where you went astray was in not exploring the behavior of the integrand before you tried to evaluate the integral. If you had done that, I think you would have gone for a numerical analysis of <code>f2</code>'s behavior, which i... |
725,241 | <p>Where s=circumcenter, H= orthocenter, and A'= midpoint of one side of triangle.
<img src="https://i.stack.imgur.com/Pg6rv.png" alt="enter image description here"></p>
<p>How can can I determine the location of the three vertices of the triangle? </p>
| robjohn | 13,854 | <p>Compare the coefficients of $x^{j-1}$ from
$$
\begin{align}
(1+x)^{k-1}(1+x)^n
&=\sum_{i=1}^k\binom{k-1}{i-1}x^{i-1}\sum_{j=0}^n\binom{n}{j}x^j\\
&=\sum_{i=1}^k\sum_{j=i}^{n+i}\binom{k-1}{i-1}\binom{n}{j-i}x^{j-1}\\
&=\sum_{j=1}^{n+k}\sum_{i=1}^j\binom{k-1}{i-1}\binom{n}{j-i}x^{j-1}
\end{align}
$$
and
$$... |
1,912,660 | <p>I am trying to really understand why the gradient of a function gives the direction of steepest ascent intuitively.</p>
<p>Assuming that the function is differentiable at the point in question,<br>
a) I had a look at a few resources online and also looked at this
<a href="https://math.stackexchange.com/questions/2... | jnyan | 365,230 | <p>This answer is not as detailed as other answers. Start from single dimension. In y=f(x), gradient gives derivative. Now derivative gives direction if steepest ascent.
In two dimensions, the partial derivatives are taken to get the direction in both directions individually. So the movement should be in such a directi... |
1,912,660 | <p>I am trying to really understand why the gradient of a function gives the direction of steepest ascent intuitively.</p>
<p>Assuming that the function is differentiable at the point in question,<br>
a) I had a look at a few resources online and also looked at this
<a href="https://math.stackexchange.com/questions/2... | amd | 265,466 | <p>Let’s try coming at it from a different direction, so to speak. </p>
<p>Consider the plane in $\mathbb R^3$ given by $ax+by=z$. The vector $\mathbf n=\langle a,b,-1\rangle$ is normal to this plane. A bit of thought should convince you that the projection of $\mathbf n$ onto the $xy$ plane, $\langle a,b\rangle$, po... |
3,144,398 | <p>Q) Suppose the rank of the matrix</p>
<p><span class="math-container">$\begin{pmatrix} 1 &1 &2 &2 \\ 1&1 &1 &3 \\ a&b &b &1 \end{pmatrix}$</span></p>
<p>is <span class="math-container">$2$</span> for some real numbers <span class="math-container">$a$</span> and <span class="math... | Michael Albanese | 39,599 | <p>Let <span class="math-container">$M$</span> be a manifold with boundary. Here's a trick you can use if <span class="math-container">$\partial M$</span> is connected.</p>
<p>The double of <span class="math-container">$M$</span> is defined as <span class="math-container">$D(M) = M\cup_{\partial M} M$</span>; it is a ... |
3,144,398 | <p>Q) Suppose the rank of the matrix</p>
<p><span class="math-container">$\begin{pmatrix} 1 &1 &2 &2 \\ 1&1 &1 &3 \\ a&b &b &1 \end{pmatrix}$</span></p>
<p>is <span class="math-container">$2$</span> for some real numbers <span class="math-container">$a$</span> and <span class="math... | Moishe Kohan | 84,907 | <p>Let me answer the second part of your question, namely, about the uniqueness of the boundary. More precisely, the setup for Siebenmann's compactification is that you have an open manifold <span class="math-container">$M$</span> (every component is noncompact). Then Siebemnann gives you some conditions under which th... |
1,506,805 | <p>$\lim _{x\to 0}\left(\frac{\cos\left(x+\frac{\pi }{2}\right)}{x}\right)\:$</p>
| Olivier Oloa | 118,798 | <p><strong>Hint.</strong> You may observe that
$$
\cos\left(x+\frac{\pi }{2}\right)=-\sin x
$$ then your limit is equal to
$$
-\lim _{x\to 0}\left(\frac{\sin x}{x}\right)=?
$$ which is easier to obtain.</p>
|
2,849,303 | <p>Given two distinct factorizations of a positive integer with the same number of factors (not necessarily prime or all distinct), must the sums of the respective sets of factors also be distinct? This question arises frequently in puzzles of the KenKen or Killer Sudoku type. I have found no obvious counter examples ... | dxiv | 291,201 | <p>The sums can match, for example $\,144 = 8 \cdot 6 \cdot 3 = 4 \cdot 4 \cdot 9\,$ with $\,8+6+3=4+4+9\,$.</p>
<p><hr>
[ <em>EDIT</em> ] Also, $144 = 2\cdot8\cdot9 = 3 \cdot 4 \cdot 12$ with $\,2+8+9=3+4+12\,$, so multiple such factorizations may exist for the same number.</p>
<p>Morevover, there exist such ... |
3,557,398 | <p>Here are two second-order differential equations.</p>
<p><span class="math-container">$$ y''+9y=\sin(2t) \tag 1 $$</span></p>
<p><span class="math-container">$$ y'' +4y =\sin(2t) \tag 2 $$</span></p>
<p>I am told to use undetermine coefficients method to solve.</p>
<p>For 1), I use <span class="math-container">... | Surajit | 528,430 | <p>The relation is symmetric in <span class="math-container">$x$</span> and <span class="math-container">$y$</span>. Now divide it in three cases, and deduce the inequality. </p>
<p><strong>Case I:</strong> <span class="math-container">$x+y<z$</span>.</p>
<p><strong>Case II:</strong> <span class="math-container">$... |
1,614,899 | <p>Construct Context-free Grammar for integers. Integer can begin with + or - and after that we have non-empty string of digits. Integer must not contain unnecessary leading zeros and zero should not be preceded by + or -.
For example: 0; 123; -15; +9999 are correct, but +0; 01; +-3; +09; + are incorrect.</p>
<p>I hav... | Elle Najt | 54,092 | <p>Hint: (0,1), or any bounded open set.</p>
|
1,832,515 | <p>I know that you almost always set domains for a summation function $\left( \sum \right)$, but can you also set an interval for that domain? Say the domain was 1 to 10, could I set an increment of 0.5 instead of the standard 1? I know that there are ways around this such as setting another variable for the previous n... | lab bhattacharjee | 33,337 | <p>As $8-3=x^2-2x+8-(x^2-2x+3)$
$=(\sqrt{x^2-2x+8}+\sqrt{x^2-2x+3})(\sqrt{x^2-2x+8}-\sqrt{x^2-2x+3})$</p>
<p>$$\sqrt{x^2-2x+8}+\sqrt{x^2-2x+3}=125\iff(\sqrt{x^2-2x+8}-\sqrt{x^2-2x+3})=\dfrac{8-3}{125}$$</p>
<p>Adding we get $2\sqrt{x^2-2x+8}=125+\dfrac1{25}=?$</p>
<p>Now square both sides. </p>
|
2,653,713 | <p>Let $\varphi:\mathbb{R}\rightarrow \mathbb{R}$ be a convex function. If $y<\varphi(x)$ why does there exist a line through $(x,y)$ which lies strictly below the graph of $\varphi$? I ask because this is a step in the proof of Jensen's inequality for conditional expectation.</p>
| BoolHool | 532,226 | <p>Okay so, for your first question, all the points in the form of the line $y=x$ is in the form $(a,a)$ (So this is our best alternative, keep in mind however that the $x$'s in $y=x$ is not the same as the $x$'s in $y=e^x$, as discussed in the comments). As you mentioned, you are using the Euclidean Metric. So we need... |
2,283,367 | <p>I know product of nth roots of unity is 1 or -1 depending whether n is odd or even.
But in this way I am getting 1. Where am I wrong?</p>
<p>$ \text{Let }\alpha = \cos \frac{2 \pi}{n} + \iota \sin \frac{2 \pi}{n} \text{ be a root of }x^n=1 \\
\text{Then product of nth roots will be } 1\cdot \alpha \cdot \alpha^2 ..... | florence | 343,842 | <p>Consider the following:
$$(-1)^1 = (-1)^{2\cdot \frac{1}{2}} = ((-1)^2)^\frac{1}{2}=1^{\frac{1}{2}} = 1$$
You did something like that. $n(n-1)/2$ is always an integer when $n$ is a natural number, but when you take out the $n$ you are left with $(n-1)/2$ which is not always an integer, which messes things up when $... |
1,255,215 | <p>How to prove that:</p>
<p>$$\cos{n\theta}=\cos^n{\theta}-
\binom {n} {2}\cos^{n-2} \theta \cdot \sin^2 \theta+
\binom {n} {4}\cos^{n-4} \theta \cdot \sin^{4} \theta -\cdots$$</p>
<p>$$\sin n\theta =
\binom {n} {1}\cos^{n-1} \theta \cdot \sin \theta -
\binom {n} {3}\cos^{n-3} \theta \cdot \sin^3\theta +\cdots$$</p>... | Pauly B | 166,413 | <p>$$\begin{align}
\frac{dy}{dx}&=\frac{\sin t-\cos t}{\cos t+\sin t}\\
&=\frac{\sqrt2(\sin t \cos\frac\pi4-\cos t\sin\frac\pi4)}{\sqrt2(\cos t \cos\frac\pi4+\sin t\sin\frac\pi4)}\\
&=\frac{\sin(t-\frac\pi4)}{\cos(t-\frac\pi4)}\\
&=\tan(t-\frac\pi4)
\end{align}$$</p>
|
101,256 | <p>Can you please help me finding an exact description of the set:</p>
<p>$$ E_{R}=\{\cos{z} | z \in \mathbb{C}, |z|>R\} $$</p>
<p>For any $0<R \in \mathbb{R}$.</p>
<p>My feeling is the $E_R = \mathbb{C}$, for any $R$, but I don't know how to show it, if it's true.</p>
| Christian Blatter | 1,303 | <p>The function $\cos$ is periodic with period $2\pi$; therefore any vertical strip of width $2\pi$ in the $z$-plane, $z=x+iy$, will give rise to the full set of values of $\cos$. By excluding the $z$'s with $|z|\leq R$ there is an infinity of such strips left, so there are no values excluded. It follows that we may as... |
118,680 | <p>The question is motivated by this one <a href="https://mathoverflow.net/questions/118626/real-symmetric-matrix-has-real-eigenvalues-elementary-proof">real symmetric matrix has real eigenvalues - elementary proof</a>: </p>
<p>Are there other fields $F$ than $\mathbb{R}$ (maybe some valued fields or real closed field... | Will Sawin | 18,060 | <p>This is a countable family of first-order statements, so it holds for every real-closed field, since it holds over $\mathbb R$. </p>
<p>From a square matrix, we immediately derive that such a field must satisfy the property that the sum of two perfect squares is a perfect square. Indeed, the matrix:</p>
<p>$ \left... |
1,483,343 | <p>For equation
$$
x_1+x_2+x_3 = 15
$$
Find number of positive integer solutions on conditions:
$$
x_1<6, x_2 > 6
$$
Let: $y_1 = x_1, y_2 = x_2 - 6, y_3 = x_3$
than, to solve the problem, equation $y_1+y_2 +y_3 = 9$ where $y_1 < 6,0<y_2, 0<y_3 $ has to be solved. <strong>Is this correct?</strong></p>
<p... | Klaramun | 189,687 | <p>If such a function $f\colon \mathbb{N} \rightarrow \mathbb{N}$ exists, consider the image set of it
\begin{equation*}
Im(f) = \{ f(n) \}_{n \in \mathbb{N}}
\end{equation*}
Which is a subset of $\mathbb{N}$, so there exists an element there $f(n_0)$ which is the smallest, i.e. $f(n_0) \leq f(n)$ for all $n \in \mathb... |
125,393 | <p>As we know, Linux’s Bash shell is coming to Windows 10. Now, I want to know, how to using Run[] command or other commands to run some commands like gfortran installed in Linux of Win10 ? And, is it possible to call mathematica using Bash shell in Windows 10 ? </p>
| Emilio Pisanty | 1,000 | <p>As we <a href="http://www.howtogeek.com/249966/how-to-install-and-use-the-linux-bash-shell-on-windows-10/" rel="noreferrer">also</a> <a href="http://www.howtogeek.com/265900/everything-you-can-do-with-windows-10s-new-bash-shell/" rel="noreferrer">know</a> <a href="https://msdn.microsoft.com/en-us/commandline/wsl/abo... |
3,572,721 | <p>A finite sum of cosine functions weighted with different amplitude and phase, but with a fixed frequency,<span class="math-container">$$f(x) = \sum_{n=1}^{N}A_{n}cos(x+\phi_{n})$$</span>
the question is if I were to fit <span class="math-container">$f(x)$</span> with <span class="math-container">$cos(x)$</span>, wha... | G Cab | 317,234 | <p>You just obtain a cosine
<span class="math-container">$$
\eqalign{
& \sum\limits_{n = 1}^N {A_{\,n} \cos \left( {x + \phi _{\,n} } \right)} = {\mathop{\rm Re}\nolimits} \left( {\sum\limits_{n = 1}^N {A_{\,n} e^{ix + i\phi _{\,n} } } } \right) = \cr
& = {\mathop{\rm Re}\nolimits} \left( {\left( {\sum... |
240,620 | <p>I would like to train some recreational probability (Puzzles). </p>
<p>Does any of you know a good collection? Preferably with hints or answers.</p>
<p>I've been studying quite a bit of probability theory, but I don't suspect that will do me any real good.</p>
<p>Thanks in advance,</p>
| broccoli | 50,577 | <p>There is a book, <a href="http://books.google.com/books/about/Fifty_Challenging_Problems_in_Probabilit.html?id=QiuqPejnweEC" rel="nofollow">Fifty Challenging Problems in Probability With Solutions</a> by Frederick Mosteller which is quite a good read.
You can look up this <a href="http://bayesianthink.blogspot.com/"... |
139,818 | <p>I'm currently working with large datasets of "discrete data". Each of these is a <em>multiset</em> of real numbers of the form $kS + B$, where $k$ is an integer, and $S$ (the <em>"scale</em>") and $B$ (the <em>"bias"</em>) are real values.</p>
<p>These datasets are such that it is often economical to represent the... | gwr | 764 | <h2>Making discrete Data an abstract data type</h2>
<p>I would try to do something like the following:</p>
<pre><code>(* write a constructor function for the data type 'discrete data' *)
discreteData[
scale_?NumericQ ,
bias_?NumericQ ,
tally : { { _?NumericQ , _?NumericQ } .. }
] := discreteDa... |
2,702,347 | <blockquote>
<p>Prove that $$\left|\dfrac{1}{z^4-4z^2+3}\right|\leq \dfrac{1}{3},\quad\text{ if}\quad |z|=2$$</p>
</blockquote>
<p>I know that I can complete squares on the denominator ($(z^2-2)^2-1$). I know the triangular inequality and some other few things... but I can't work so well with absolute value on $\mat... | Martin R | 42,969 | <p>For $|z|=2$ the triangle inequality gives
$$
|z^2 -3| \ge |z^2| - 3 = |z|^2 - 3 = 1 \, ,\\
|z^2 -1| \ge |z^2| - 1 = |z|^2 - 1 = 3 \, ,
$$
and therefore
$$
|z^4 - 4z^2 +3| = |(z^2-3)(z^2-1)| = |z^2-3 | \cdot | z^2-1 | \ge 1 \cdot 3 = 3 \, .
$$</p>
<p><em>Alternatively:</em> You already found that
$$
z^4 - 4z... |
949,320 | <p>how can I solve this:</p>
<blockquote>
<p>How many different ways can 8 children be divided into two groups of 3
and one group of 2?</p>
</blockquote>
<p>My method was:</p>
<p>8C3*5C3*2C2+8C3*5C2*3C3+8C2*6C3*3C3=560+560+560=1680</p>
<p>But the answer is:</p>
<p>8C3*5C3*2C2=560</p>
<p>I feel close to gettin... | Varun Iyer | 118,690 | <p>The logic behind here is the set of actions that take place.</p>
<p>First, we have $8$ people choosing $3$ kids for the first group. That is $8C3$</p>
<p>Next, we have $5$ kids left, who need to be divided into $3$ kids to form the second group.</p>
<p>Finally, we only have $2$ kids left and one group $2$, so tha... |
4,132,907 | <p>I'm trying to figure out the identity above though I'm having difficulties towards figuring it out and would kindly appreciate your support!</p>
<p><span class="math-container">$n\binom{n-1}{r-1} = r\binom{n}{r}$</span></p>
<p>What I have tried:
Given that</p>
<p><span class="math-container">$$\binom{n}{r}=\binom{n-... | Vishu | 751,311 | <p><span class="math-container">$$\frac nr \binom{n-1}{r-1} = \frac nr \frac{(n-1)!}{(r-1)! (n-r)!} = \frac{n!}{r!(n-r)!} = \binom nr$$</span></p>
|
41,940 | <p>For example, the square can be described with the equation $|x| + |y| = 1$. So is there a general equation that can describe a regular polygon (in the 2D Cartesian plane?), given the number of sides required?</p>
<p>Using the Wolfram Alpha site, this input gave an almost-square:
<code>PolarPlot(0.75 + ArcSin(Sin(2x... | jimjim | 3,936 | <p>Depends what you accept as "a general equation" and what is "a plot". For example, under the $l_1$ metric, the equation of what is perceived as a circle ( points having the same distance to a center) ends up looking like a square. Now would that count as genuine plot? Or, are we only allowed the Euclidean metric for... |
2,440,478 | <p>So, I understand that for something to be <em>rigorous</em> it is inclined to meet strict and occasionally extreme criteria. Within the scope of mathematics, this therefore implies that anything that is truly rigorous may not defy any of the rules of mathematics, and as it should with everything in mathematics, but ... | Theo Bendit | 248,286 | <p>There's nothing fundamentally different about a geometric proof compared to other proofs. It refers to a style of proof, where the steps are designed to be pictured, operating on some kind of prototypical example. When people don't follow along with a picture, the proofs tend to be more dense and less intelligble. S... |
3,753,205 | <p>In my general topology textbook there is the following exercise:</p>
<blockquote>
<ul>
<li>(i) - Let <span class="math-container">$\mathcal B$</span> be a basis for a topology <span class="math-container">$\tau$</span> on a non-empty set <span class="math-container">$X$</span>. If <span class="math-container">$\math... | Henno Brandsma | 4,280 | <p>One base for <span class="math-container">$\Bbb R$</span> is the countable family <span class="math-container">$\mathcal{B}=\{(a,b): a < b, a,b \in \Bbb Q\}$</span>.</p>
<p>Note that for any irrational <span class="math-container">$w$</span>, <span class="math-container">$\mathcal{B}(w):=\mathcal{B} \cup \{(0,w)\... |
703,758 | <p>I need help with a set problem</p>
<p>Given:</p>
<p>$$A=\{(\sqrt{n}+2) \in \Bbb Z \ /\ \ 16\le n^2 \le 1296 \}$$</p>
<p>$$B=\{({3m-2}) \in A \ /\ \ 4 \le 4m+3 \le 17 \}$$</p>
<p>Calculate the value of :
$$n(A)\times n(B)$$</p>
<p>So far I've got into $$ A = \{-8;-7;-6;-5;-4;4;5;6;7;8\} $$
$$B= \{4;5;6;7;8\}$$<... | Community | -1 | <p>Not quite: The symbol $\sqrt{n}$ usually refers to the nonnegative square root of $n$, so $A$ should only contain positive numbers. Now</p>
<p>$$\sqrt{1296} = \sqrt{6^4} = 36$$</p>
<p>so the smallest element of $A$ corresponds to $\sqrt{4} + 2$, while the largest corresponds to $\sqrt{36} + 2$; that is, it's equiv... |
703,758 | <p>I need help with a set problem</p>
<p>Given:</p>
<p>$$A=\{(\sqrt{n}+2) \in \Bbb Z \ /\ \ 16\le n^2 \le 1296 \}$$</p>
<p>$$B=\{({3m-2}) \in A \ /\ \ 4 \le 4m+3 \le 17 \}$$</p>
<p>Calculate the value of :
$$n(A)\times n(B)$$</p>
<p>So far I've got into $$ A = \{-8;-7;-6;-5;-4;4;5;6;7;8\} $$
$$B= \{4;5;6;7;8\}$$<... | David | 119,775 | <p><strong>Hint</strong>. The symbol $\sqrt{\ }$ means the positive square root. It doesn't mean plus- or- minus unless you actually write $\pm$. </p>
|
578,248 | <p>Let $X$ be a random variable taking values in some metric space $M$. Let $\{\phi_n\}$ be a sequence of measurable functions from $M$ to another metric space $\tilde M$. Suppose that $\phi_n(X)$ converges in law to a random variable $Y$. Must it be the case that the pairs $(X, \phi_n(X))$ converge in law to a pair $(... | grand_chat | 215,011 | <p>A more elementary counterexample: Let $X$ have uniform distribution over $[0,1]$ and define $\phi_n(x):=x$ when $n$ is odd, and $\phi_n(x):=1-x$ when $n$ is even. It's clear that every $\phi_n(X)$ has the same distribution (namely, that of $X$), hence we have convergence in law. However, the pair $(X,\phi_n(X))$ doe... |
2,323,235 | <p>When I was trying to evaluate a definite integral (given below the question for those who're curious), I came across a paper of Ramanujan pertaining to the evaluation of the very same integral. Amidst the paper's proofs, Ramanujan mentions the following:</p>
<hr>
<p><em>It's easy to see that</em>:
$$\prod^{n=\alph... | Angina Seng | 436,618 | <p>Use the infinite product formula for $\Gamma$:
$$\frac1{\Gamma(z)}=ze^{\gamma z}\prod_{n=1}^\infty\left(1+\frac{z}{n}\right)e^{-z/n}$$
which yields
$$\frac1{\Gamma(z+1)}=\frac1{z\Gamma(z)}=e^{\gamma z}\prod_{n=1}^\infty\left(1+\frac{z}{n}\right)e^{-z/n}$$
etc.</p>
|
2,323,235 | <p>When I was trying to evaluate a definite integral (given below the question for those who're curious), I came across a paper of Ramanujan pertaining to the evaluation of the very same integral. Amidst the paper's proofs, Ramanujan mentions the following:</p>
<hr>
<p><em>It's easy to see that</em>:
$$\prod^{n=\alph... | robjohn | 13,854 | <p>I assume that the upper limit of the product is supposed to be $\infty$.</p>
<p><strong>Hint:</strong></p>
<p>By <a href="https://math.stackexchange.com/a/103028/13854">Gautschi's Inequality</a>, $\lim\limits_{m\to\infty}\frac{\Gamma(m+\alpha+1)}{m^\alpha\,\Gamma(m+1)}=1$. Therefore,
$$
\begin{align}
\lim_{m\to\in... |
343,993 | <p>Let $I$ be an ideal of $\mathbb{C}[x,y]$ such that its zero set in $\mathbb{C}^2$ has cardinality $n$. Is it true that $\mathbb{C}[x,y]/I$ is an $n$-dimensional $\mathbb{C}$-vector space (and why)?</p>
| Martin Brandenburg | 1,650 | <p>Here is a positive result (see Fulton, <em>Algebraic curves</em>, Corollary I.7.4 on page 23). Let $k$ be some algebraically closed field. If $I \subseteq k[x_1,\dotsc,x_n]$ is an ideal, then $V(I) \subseteq \mathbb{A}^n(k)$ is finite if and only if $k[x_1,\dotsc,x_n]/I$ is a finite-dimensional $k$-vector space. In ... |
2,870,039 | <p>Just for the sake of simplicity, assume that the centre of rotation is $(0, 0)$. I know that I can find the destination point $(x', y')$ given angle of rotation $a$ and source point $(x, y)$ using
$$\begin{cases}
x' = x \cos a - y \sin a \\
y' = x \sin a + y \cos a \\
\end{cases}$$</p>
<p>However, I want to ask ... | md2perpe | 168,433 | <p>First we replace $\sin a$ with $\sqrt{1-\cos^2a}$:
$$x' = x \cos a - y \sqrt{1-\cos^2 a}$$</p>
<p>Then we eliminate the square root by isolating it and squaring:
$$y \sqrt{1-\cos^2 a} = x \cos a - x'$$
$$y^2 (1-\cos^2 a) = (x \cos a - x')^2 = x^2 \cos^2 a - 2 x x' \cos a + (x')^2$$</p>
<p>This can now be rewritten... |
2,110,111 | <p>For example, concerning de Rham cohomology as explained on the lower part of page 5 <a href="http://www.math.uni-hamburg.de/home/wockel/teaching/data/cohomology_of_lie_algebras_mavraj.pdf">here</a>, one considers the vector space of all forms on a manifold $M$, $A_{DR}(M)$ in the notation of the paper (?).</p>
<p>T... | Michael Albanese | 39,599 | <p>A <em>cochain complex</em> $(A^{\bullet}, d^{\bullet})$ is a sequence of abelian groups $\dots, A^{-2}, A^{-1}, A^0, A^1, A^2, \dots$ and group homomorphisms $d^k : A^k \to A^{k+1}$ such that $d^k\circ d^{k-1} = 0$. That is, we have a complex </p>
<p>$$\dots \xrightarrow{d^{-3}} A^{-2} \xrightarrow{d^{-2}} A^{-1} \... |
2,110,111 | <p>For example, concerning de Rham cohomology as explained on the lower part of page 5 <a href="http://www.math.uni-hamburg.de/home/wockel/teaching/data/cohomology_of_lie_algebras_mavraj.pdf">here</a>, one considers the vector space of all forms on a manifold $M$, $A_{DR}(M)$ in the notation of the paper (?).</p>
<p>T... | Andrew D. Hwang | 86,418 | <p>$\newcommand{\dd}{\partial}\newcommand{\Brak}[1]{\left\langle #1\right\rangle}$To give a more elementary answer intended to complement the existing answers, here are some multivariable calculus-type observations. Throughout, $M$ denotes a compact, smooth manifold of dimension $n$ (possibly disconnected), a... |
1,950,381 | <p>If $H$ is a subgroup of $S_n$ and if H is not contained in $A_n$ prove that precisely one half of the elements of H are even permutations. I know that multiplying two odd permutations gives a even, and that one even and one odd return an odd permutation but i have no idea where to go from there. </p>
| DonAntonio | 31,254 | <p>Hint:</p>
<p>Let $\;\sigma\in H\;$ be <em>an odd</em> permutation, and now define </p>
<p>$$\;f:A_N\cap H\to H\setminus(A_n\cap H)\;,\;\;f(x):=\sigma x\;$$ </p>
<p>Prove $\;f\;$ is a bijection.</p>
|
2,480,493 | <p>I have a problem with what looks like a very easy equation to solve $\left|\frac{1 + z}{1- i\overline z}\right| = 1$ . ($z$ is a complex number, $\overline z$ is a conjugate of $z$) I got stuck at the point when after replacing $\overline z = a-bi $ and $z = a +bi$ and getting rid of absolute value I end up with $... | ivanculet | 493,187 | <p>Take this
$$|1+z|=|1-i\bar{z}|$$
then square this babies
$$(1+z)(1+\bar{z})=(1-i\bar{z})(1+iz)$$
$$1+z+\bar{z}+z\bar{z}=1-i\bar{z}+iz+z\bar{z} $$
so
$$ z+\bar{z}=i(z-\bar{z})$$
$$2Re(z)=i(i2Im(z)$$
therefore $$Re(z)=-Im(z)$$</p>
|
485,340 | <p>My question is to to find the period of
$$\sin(8\pi\{x\}),$$
where $\{\cdot\}$-is the fractional part of function.</p>
<p>I know that the period of $\{\cdot\}$ is 1 and the period of $\sin(8\pi x)$ is $1/4$. But how to find the overall period of the given function? </p>
| An_Elephant | 1,020,656 | <p>Let <span class="math-container">$g(f(x))$</span> be the composition of <span class="math-container">$g$</span> and <span class="math-container">$f$</span>.</p>
<p>Further, suppose that <span class="math-container">$g(x)$</span> is an odd function, so that <span class="math-container">$g(f(x)) = -g(f(-x))$</span> .<... |
359,528 | <p>Let $P$ be the set of permutations all of whose cycles are of even length. Prove that the exponential generating function for $P$ is $\dfrac{1}{\sqrt{1-x^2}}$.</p>
| Marko Riedel | 44,883 | <p>This is definitely straightforward using symbolic combinatorics as formalized by Flajolet and Sedgewick. Let $\mathcal{Q}$ be the combinatorial class of permutations consisting only of even cycles.</p>
<p>We have the combinatorial specification
$$ \mathcal{Q} =
\mathfrak{P}(\mathfrak{C}_2(\mathcal{Z}) + \mathfrak... |
621 | <p>The OP of this question is asking if <a href="https://mathematica.stackexchange.com/questions/9639">https://mathematica.stackexchange.com/questions/9639</a> could be reopened. I did not want to act unilaterally, so I want to ask the rest of you whether you think the question should be reopened. Plead your case for o... | rm -rf | 5 | <p>I can speak for myself and why <em>I</em> voted to close. First, let me note that I voted to close as "Not a real question", as did 3 other voters. It just so happens that a mod's choice of close reason trumps that of others and hence it looks like we all voted to close as "Too localized". I do not think it fits the... |
1,869,981 | <p>I have a question in a book which says in how many ways can 6 different things be divided between 2 boys and (my understanding of) the explanation goes something along the lines of:</p>
<p>Items: 1 1 1 1 1 1</p>
<p>The first item can be distributed to either boy 1 or boy 2, i.e. 2 choices the second item could be ... | Kiran | 82,744 | <p>The second method does not take care of the fact that items are distinct.</p>
<p>Assume items are $A,B,C,D,E,F$</p>
<p>In the second method, you counted $(1,5)$ as just one.</p>
<p>In fact, it can be</p>
<p>$A~~(B,C,D,E,F)\\
B~~(A,C,D,E,F)\\
C~~(A,B,D,E,F)\\
D~~(A,B,C,E,F)\\
E~~(A,B,C,D,F)\\
F~~(A,B,C,D,E)$</p>
... |
1,057,172 | <p>How many 3digit numbers can be written with $2,4,4,6,6$ ? </p>
<p>I tried $\frac{5.4.3}{2!.2!} = 15$ but it's wrong. </p>
<p>when I solved the question "how many 3digit numbers can be written with $1,1,2$"
the solution $\frac{3.2.1}{2!}$ was correct but why this way doesn't work for above question?</p>
| Mark Joshi | 106,024 | <p>if you are allowed to wait forever: 1. </p>
<p>There are lots of ways to do this. Suppose we ask the question hit 1 or -n first and stop when this happens. </p>
<p>Then since hitting time is a finite stopping time, the value of the random walk stopped at the hitting is still a martingale and so has expectation 0. ... |
2,419,785 | <blockquote>
<p>Prove the statement is true using mathematical induction: $$2n-1 \leq n!$$</p>
</blockquote>
<p>My attempt: this is true for $n=1$. </p>
<p>Suppose it is true for $n$, i.e., $2n-1 \leq n!$</p>
<p>Now, $2n-1 \leq n!\implies 2n-1+2 \leq n!+2$ </p>
<p>From here, how do I proceed? </p>
| Robert Israel | 8,508 | <p>By the laws of exponents
$$x^{10n} x^5 \left(x^{-5}\right)^n = x^{10 n + 5 - 5 n} = x^{5 + 5 n}$$
so you want
$$ x^{5+5n} = x^{-10}$$
Assuming $x$ is not $0$ or $\pm 1$, $x^t$ is a one-to-one function of $t$, so this will
imply $$5 + 5 n = -10$$</p>
|
2,546,434 | <p>I have trouble calculating this limit algebraically (without L'Hospital's rule):</p>
<p>$$\lim_{x \to 1} \frac{x^{\pi} - x^e}{x-1}$$</p>
<p>Substituting 1 gives indeterminate form. If this was e.g. $(x^5 - x^2)$ in the numerator, one could easily factor out the $(x-1)$, because $(x^5 - x^2) = x^2 (x - 1) (x^2 + x ... | carmichael561 | 314,708 | <p>Hint: write
$$ \lim_{x\to 1}\frac{x^{\pi}-x^e}{x-1}=\lim_{x\to 1}\frac{x^{\pi}-1}{x-1}-\lim_{x\to 1}\frac{x^e-1}{x-1}$$
and then note that each term is a derivative.</p>
|
2,546,434 | <p>I have trouble calculating this limit algebraically (without L'Hospital's rule):</p>
<p>$$\lim_{x \to 1} \frac{x^{\pi} - x^e}{x-1}$$</p>
<p>Substituting 1 gives indeterminate form. If this was e.g. $(x^5 - x^2)$ in the numerator, one could easily factor out the $(x-1)$, because $(x^5 - x^2) = x^2 (x - 1) (x^2 + x ... | orion | 137,195 | <p>Try</p>
<p>$$\lim_{x\to 1}x^{e}\frac{x^{\pi-e}-1}{x-1}$$</p>
<p>Now it depends on what you are "allowed" to use. If you assume binomial expansion works for real exponends, then that's it (this is the Taylor expansion, or the well-known first order approximation $(1+\epsilon)^n\approx 1+n\epsilon$).</p>
|
427,663 | <p>If $|z|=|w|=1$, and $1+zw \neq 0$, then $ {{z+w} \over {1+zw}} \in \Bbb R $</p>
<p>i found one link that had a similar problem.
<a href="https://math.stackexchange.com/questions/343982/prove-if-z-1-and-w-1-then-1-zw-neq-0-and-z-w-over-1">Prove if $|z| < 1$ and $ |w| < 1$, then $|1-zw^*| \neq 0$ and $| {{z-w}... | vadim123 | 73,324 | <p>We have $$\frac{z-w}{1-zw}=\frac{(z-w)(1-\overline{zw})}{(1-zw)(1-\overline{zw})}$$
Note that $(1-zw)(1-\overline{zw})=1-(zw+\overline{zw})+|zw|^2$ is real, so we need only consider the numerator (as Berci hinted).</p>
<p>We calculate $(z-w)(1-\overline{zw})=z-w-|z|^2\overline{w}+|w|^2\overline{z}=(z+\overline{z})-... |
3,715,674 | <p>Let <span class="math-container">$X$</span> be a scheme and <span class="math-container">$U$</span> an open subset of <span class="math-container">$X$</span>. Let <span class="math-container">$s \in \Gamma(U, O_X)$</span> and <span class="math-container">$x \in U$</span>.<br>
I am getting confused with what the diff... | Jean-Armand Moroni | 1,064,750 | <p>An <em>exact</em> differential has the property that its integral over a path is path-independent: it does not depend upon the taken path, but only upon the origin and end of the path.</p>
<p>In mechanics, the work of a force can be <em>path-dependent</em> (e.g. friction), or <em>path-independent</em> (e.g. gravity)... |
2,315,236 | <blockquote>
<p>Let $I := [0, \frac{\pi}{2}]$ and let $f : I \to \Bbb{R}$ be defined by $f(x) = \sup \{x^2, \cos(x)\}$ for $x \in I$. Show there exists an absolute minimum point $x_0 \in I$ for $f$ on $I$. Show that $x_0$ is a solution to the equation $x^2 = \cos(x)$. </p>
</blockquote>
<p>I am having a little troub... | Hagen von Eitzen | 39,174 | <p>The argument works for the general case of $f(x)=\sup\{g(x),h(x)\}$ where $g,h\colon [a,b]\to \Bbb R$ are continuos, $g$ strictly increasing, $h$ strictly decreasing, $h(a)>g(a)$, and $g(b)>h(b)$:</p>
<p>A minimum point $x_0\in [a,b]$ exists because $[a,b]$ is compact and $f$ is continuous.</p>
<p>If $f(x_0)... |
1,623,796 | <p>In the structure $\langle \mathbb{Q}, < \rangle$ which of the following axioms hold? How about when we use the weak versions of the axioms (all $\leftrightarrow$ replaced with $\rightarrow$ )? </p>
<p>Extensionality, Empty Set, Power Set, Infinity</p>
<p>Find an instance of Separation that is true in $\langle \... | egreg | 62,967 | <p>You have to interpret $\in$ with $<$. In order to check whether extensionality holds you have to consider the sentence</p>
<blockquote>
<p>for all $x$ and $y$, $x=y$ if and only if, for every $z$, $z<x$ if and only if $z<y$</p>
</blockquote>
<p>or, in symbolic form,
$$
\forall x\forall y
(x=y \leftright... |
3,577,098 | <p>In metric space there is a well-known result that <span class="math-container">$X$</span> is a separable metric space <span class="math-container">$\implies$</span>Every uncountable set in <span class="math-container">$X$</span> has a limit point.My question is does the result hold if <span class="math-container">$(... | Aiden Chow | 743,034 | <p>The number of ways that Carlos getting a <span class="math-container">$6$</span> while still being higher than Eric is <span class="math-container">$5$</span>(Eric can get <span class="math-container">$1$</span> to <span class="math-container">$5$</span>). The total number of ways to pick out of the urn when Carlos'... |
1,713,841 | <p>I think $\pi$ is algebraic of degree 3 over $\mathbb{Q}(\pi^3)$. To prove it, I need to show that $\pi \notin \mathbb{Q}(\pi^3)$ (which implies that $x^3-\pi^3$ is the minimum polynomial of $\pi$ over $\mathbb{Q}(\pi^3)$). I have no experience solving this kind of problem for transcendental elements like $\pi$.</p>
| may | 1,061,720 | <p><span class="math-container">$|(\mathbb{Q}(\pi^3))(\pi):\mathbb{Q}(\pi^3)|=irr(\mathbb{Q}(\pi),\mathbb{Q}(\pi^3)$</span></p>
<p><span class="math-container">$irr(\pi,\mathbb{Q}(\pi^3)=x^3-\pi^3$</span></p>
<p><span class="math-container">$\pi$</span> is the root of <span class="math-container">$x^3-\pi^3$</span></p>... |
2,755 | <p>As Akhil had great success with his <a href="https://mathoverflow.net/questions/1291/a-learning-roadmap-for-algebraic-geometry">question</a>, I'm going to ask one in a similar vein. So representation theory has kind of an intimidating feel to it for an outsider. Say someone is familiar with algebraic geometry enou... | Ben Webster | 66 | <p>I second the suggestion of Fulton and Harris. It's a funny book, and definitely you want to keep going after you finish it, but it's a good introduction to the basic ideas.</p>
<p>You specifically might be happier reading a book on algebraic groups.</p>
<p>While I third the suggestion of Ginzburg and Chriss, I wo... |
2,755 | <p>As Akhil had great success with his <a href="https://mathoverflow.net/questions/1291/a-learning-roadmap-for-algebraic-geometry">question</a>, I'm going to ask one in a similar vein. So representation theory has kind of an intimidating feel to it for an outsider. Say someone is familiar with algebraic geometry enou... | Chirag Lakhani | 2,565 | <p>I ran across an excellent book by Lakshmibai and Brown called <a href="http://www.ams.org/bookstore?fn=20&arg1=hinseries&ikey=HIN-40">Flag Varieties: an Interplay of Geometry, Combinatorics, and Representation Theory</a>. It seems like an excellent book for an algebraic geometer who is interested in represe... |
2,925,320 | <p>I was wondering how did Spectral graph theory, this multidisciplinary area between Linear Algebra and Graph theory start?</p>
<p>How did it emerge? </p>
<p>Was there a certain problem (maybe graph isomorphism problem?) that evoked its formation? </p>
| Mike | 544,150 | <p>Well for one thing, certain combinatorial properties of a graph that are of interest to graph theorists and theoretical computer scientists, can be gleaned from analysing the adjacency matrix of the graph. For example, (put informally) a <span class="math-container">$k$</span>-regular graph <span class="math-contain... |
2,167,598 | <p>I've seen the answer for gcd$(a,b)$ but never for the lcm$(a,b)$?</p>
| edupppz | 217,861 | <p>Well this actually follows from the answer for the gcd. Here's a big hint, suppose x' and y' are such that gcd(a,b) = ax'+by'. Now, we know the gcd divides both a and b and that lcm is divisible by both a and b. So try working from there toward a solution.</p>
|
1,072,634 | <p>I need to solve the differential equation $$y\frac{dy}{dx} = (x+7)(y^2+6)$$</p>
<p>I know that the first step is to isolate both term each side and then integrate...</p>
<p>But I can't figure out how to isolate term on this one, I would probably be able to solve the rest of the equation, just need to know how to s... | Asaf Karagila | 622 | <p>The definition of a function $f$ from $A$ to $B$, regardless to injectivity or surjectivity, is that the domain of $f$ is $A$, in its entirety.</p>
<p>This means that if $f\colon A\to B$, then for every $a\in A$, there is a unique $b\in B$ such that the pair $(a,b)\in f$.</p>
<p>So the converse holds just for it t... |
3,258,256 | <p>I know that if we have a matrix A defined for a space V with dimension n that has n distinct eigenvalues, we can write it as <span class="math-container">$X^{-1}DX$</span> where D is a diagonal matrix with the eigenvalues of A on its diagonal and X is a matrix of eigenvectors. </p>
<p>It seems that what we are doin... | Osvaldo Paniccia | 352,453 | <p>Because in that basis, the algebra of matrices becomes really simple and immediate and you can get lot of information just by looking at the matrix itself.
Also, the representative matrix of <span class="math-container">$L$</span> is exactly <span class="math-container">$X^{-1}$</span>.</p>
<p>At the end you will h... |
4,013,630 | <blockquote>
<p>The population of bacteria triples each day on a petri dish. If it takes 20 days for the population of bacteria to fill the entire dish, how many days will it take bacteria to fill half of the petri dish?</p>
</blockquote>
<ul>
<li><p>My doubt 1: if we don't have a initial population can be solve this p... | José Carlos Santos | 446,262 | <p>The natural way is induction. You have <span class="math-container">$a_2=-3$</span>. Now take <span class="math-container">$k\in\Bbb N$</span> and assume that<span class="math-container">$$a_k=\begin{cases}-3&\text{ if $k$ is even}\\7&\text{ if $k$ is odd.}\end{cases}$$</span>Then,<span class="math-container... |
4,013,630 | <blockquote>
<p>The population of bacteria triples each day on a petri dish. If it takes 20 days for the population of bacteria to fill the entire dish, how many days will it take bacteria to fill half of the petri dish?</p>
</blockquote>
<ul>
<li><p>My doubt 1: if we don't have a initial population can be solve this p... | cosmo5 | 818,799 | <p>Use <a href="https://en.wikipedia.org/wiki/Iverson_bracket" rel="nofollow noreferrer">Iverson bracket</a>.</p>
<p><span class="math-container">$$a_n=-3[n \equiv 0\pmod 2]+7[n \equiv 1\pmod 2]$$</span></p>
|
325,236 | <blockquote>
<p>Is there a bijection between $\mathbb N$ and $\mathbb N^2$?</p>
</blockquote>
<p>If I can show $\mathbb N^2$ is equipotent to $\mathbb N$, I can show that $\mathbb Q$ is countable. Please help. Thanks,</p>
| Josephine Moeller | 950 | <p>Yes. Imagine starting at $(1,1)$ and then zig-zagging diagonally across the quadrant. I'll leave you to formulate it.</p>
<p>Hint: for every natural number $>1$ there's a set of elements of $\mathbb{N}^2$ that add up to that number. For $2$, there's $(1,1)$. For $3$, there's $(2,1)$ and $(1,2)$...</p>
|
4,623,427 | <p><span class="math-container">$f$</span> is a twice differentiable function from <span class="math-container">$R^n $</span> to <span class="math-container">$R$</span>. I want to show that
<span class="math-container">$$ \| \nabla^2 f(x) \| \le L \implies \| \nabla f(x) - \nabla f(y) \| \le L \| x-y \| $$</span> f... | Tomer | 578,689 | <p>Based on the hint given in the comment below:</p>
<p>From the mean value theorem we know that
<span class="math-container">$$g(y) - g(x) = (\nabla g (z))^T (x-y) $$</span> for some <span class="math-container">$z$</span> on the line going thru <span class="math-container">$x$</span> and <span class="math-container">... |
3,799,421 | <p>While I was finding the potential at a point in the plane of a uniformly charged ring, I got the following integral as the solution.
<span class="math-container">$$\int_{0}^{2\pi} \frac{d\theta}{\sqrt{R^2+x^2-2Rx\cos\theta}},$$</span>
where <span class="math-container">$R$</span> and <span class="math-container">$x$... | SarGe | 782,505 | <p>Once even I tried to derive this result and got stuck on the same step. After searching many articles and abstracts on internet, I concluded that there is no simple analytical solution of this integral rather a lot of heavy computation is required.</p>
<p>The potential equation was like <span class="math-container">... |
1,137,914 | <p>Can Someone please help me prove that the number 1729 is a Pseudoprime?</p>
<p>So a pseudoprime is a composite $n$ such that $n |(2^n − 2)$. And every prime
number also has this property.</p>
| Derek Holt | 2,820 | <p>Let $G$ be the semidirect product ${\mathbb Q}^2 \rtimes {\rm SL}(2,{\mathbb Q})$ with the natural action. Then $G$ has a chief series $1 < {\mathbb Q}^2 < {\mathbb Q}^2 \rtimes \{\pm I_2\} < G$ and, since the action on ${\mathbb Q}^2$ is irreducible and ${\rm PSL}(2,{\mathbb Q})$ is simple, these are the o... |
4,133,172 | <p>There are special arguments to trigonometric functions which produce closed-form results, e.g. <span class="math-container">$\sin(\frac{\pi}{4}) = \frac{1}{\sqrt 2}$</span>. I also recently learned that <span class="math-container">$\cos(\frac{\pi}{5})=\frac{\phi}{2}$</span> as shown <a href="https://www.youtube.com... | TomKern | 908,546 | <p>Wikipedia <a href="https://en.wikipedia.org/wiki/Trigonometric_constants_expressed_in_real_radicals#Scope_of_this_article" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Trigonometric_constants_expressed_in_real_radicals#Scope_of_this_article</a> claims that:</p>
<blockquote>
<p>expressions in real radicals... |
4,133,172 | <p>There are special arguments to trigonometric functions which produce closed-form results, e.g. <span class="math-container">$\sin(\frac{\pi}{4}) = \frac{1}{\sqrt 2}$</span>. I also recently learned that <span class="math-container">$\cos(\frac{\pi}{5})=\frac{\phi}{2}$</span> as shown <a href="https://www.youtube.com... | Tyma Gaidash | 905,886 | <p>Technically, every value of sin can be expressed in closed form by means of using the exponential definition of sine:
<span class="math-container">$$\mathrm{sinθ=\frac{e^{iθ}-e^{-iθ}}{2i}}$$</span></p>
<p>Simply substitute the value of θ wanted and you will get a form in terms of the natural base exponential functio... |
137,915 | <blockquote>
<p>Given two continuously differentiable functions $f,\
g:\mathbb{R}\rightarrow \mathbb{R}$ such that $f(x_0) = g(x_0)$ and
$f'(x_0) < g'(x_0)$, there exists $\epsilon > 0$ such that $f(x) <
g(x)$ for $x \in (x_0, x_0 + \epsilon)$.</p>
</blockquote>
<p>The above result is not too difficult... | leslie townes | 18,076 | <p>You are right, it would be enough to assume that $f$ and $g$ are differentiable at $x_0$.</p>
<p>By considering a translate of the difference $g - f$ it suffices to prove that if $h$ is differentiable at $0$ and $h(0) = 0$ and $h'(0) > 0$, then there is $\delta > 0$ with the property that $h(x) > 0$ for al... |
2,235,304 | <blockquote>
<p>Let <span class="math-container">$f(z) = \frac{z^2}{(z^2+1)^2(z^2+2z+2)}$</span>.</p>
<p>Find singularities of this function. For each singularity determine if it is removable, a pole (if a pole determine it's order,) or essential.</p>
</blockquote>
<p>I have worked out that the singularities of this fu... | Davide F. | 208,720 | <p>Let start with the poles: as your function is rational the pole are the zero of the denominator, so being $z^2+1=0 \iff z=\pm 1$ and $z^2+2z+2=0\iff z=-1\pm i$ we have $4$ singularities: $\{i,-i,-1+i,-1-i\}$.</p>
<p>Now note that $z^2+2z+2\neq 0$ if $z=\pm i$, and the same is true for $z^2$.</p>
<p>Let be $g(z)=\f... |
2,877,032 | <p>$\int y^{2}dx+x^{2}dy$ and $C\left( r\right) =\left(a \cos t,b\sin t\right)$ ,$0<t< \pi$</p>
<p>Could someone give me a hint to evaluate this integral?</p>
<p>my efforts:
$C^{'}\left( r\right)=\left(-a \sin t,b\cos t\right)$ then $\left\| C^{'}\left( r\right) \right\|$=$\sqrt {a^{2}\sin ^{2}t+b^{2}\cos ^{2}t... | Fred | 380,717 | <p>Let $x(t)=a \cos t$ and $y(t)= b \sin t$.</p>
<p>Then $\int_C y^{2}dx+x^{2}dy=\int_0^{\pi}(y(t)^2 x'(t)+x(t)^2 y'(t)) dt.$</p>
|
1,826,435 | <p>I was given a problem:
(Edited) </p>
<blockquote>
<p>Show $$ S=\bigl\{S_p:p\in\mathbb{P}\bigr\}\cup\bigl\{\{1\}\bigr\} $$
where $\mathbb{P}$ is the set of prime numbers, and $S = \{n \in
\mathbb{N}: n \text{ is a multiple of } p\}$ is a subbasis on
$\mathbb{N}$</p>
</blockquote>
<p>The easiest way to do th... | egreg | 62,967 | <p>The formulation is quite ambiguous; I believe your set $S$ consists of the sets $\{1\}$ and of the sets $S_p=\{n\in\mathbb{N}: p\mid n\}$ as $p$ varies through the prime numbers. In other terms,
$$
S=\bigl\{S_p:p\in\mathbb{P}\bigr\}\cup\bigl\{\{1\}\bigr\}
$$
where $\mathbb{P}$ is the set of prime numbers.</p>
<p>Yo... |
951,524 | <p>How do I prove, that $(\mathbb R×\mathbb R;+,*) $ is a ring, but not a field, where the $+$ and $*$ operations are: $(a,b)+(c,d):=(a+c,b+d)$ and $(a,b)*(c,d):=(ac,bd)$?</p>
<p>For the solution: so I would have to first show, that $(\mathbb R×\mathbb R;+,*) $ is a ring, I have to prove that using the definition o... | lab bhattacharjee | 33,337 | <p>From
<a href="https://math.stackexchange.com/questions/346368/sum-of-tangent-functions-where-arguments-are-in-specific-arithmetic-series">Sum of tangent functions where arguments are in specific arithmetic series</a>,</p>
<p>$$\tan180x=\frac{\binom{180}1t-\binom{180}3t^3+\cdots+\binom{180}{177}t^{177}-\binom{180}{... |
1,552,501 | <p><a href="https://i.stack.imgur.com/zFz40.jpg" rel="nofollow noreferrer">My Question</a></p>
<p>First I let u = y' and employed the Chain Rule to obtain du/dx = du/dy * u</p>
<p>But I am not sure where to go from there. Any tips, suggestions, or solutions to the problem would be much appreciated!</p>
<p>I suspect ... | Claude Leibovici | 82,404 | <p>You already received the explanation : close to $x=0$, both $\sin(x)$ and $\tan(x)$ are $\simeq x$.</p>
<p>Let me go a bit further, using more terms in Taylor expansion $$\tan(x)=x+\frac{x^3}{3}+\frac{2 x^5}{15}+O\left(x^6\right)$$ $$\sin(x)=x-\frac{x^3}{6}+\frac{x^5}{120}+O\left(x^6\right)$$ $$\sin^2(x)=x^2-\frac{... |
1,178,969 | <p>$$\sum_{k=1}^{\infty}\ln\frac{k+5}{k+4}$$</p>
<p>$$
\frac{\ln\frac{k+6}{k+5}}{\ln\frac{k+5}{k+4}}
=\cdots
$$</p>
<p>(ln(k+6/k+5)/ln(k+5/k+4))</p>
<p>ln((k+6/k+5)*(k+4/k+5))</p>
<p>simplify</p>
<p>lim as k approaches infinity of ln(kˆ2 +10k +24/ kˆ2 +10k+25)= ln(1) = 0.</p>
<p>why doesn't the series converge?</... | Bernard | 202,857 | <p>With equivalents (we can use equivalents because we have a series with positive termes): $$\ln \Bigl(\dfrac{k+5}{k+4}\Bigr)=\ln\Bigl(1+\dfrac 1{k+4}\Bigr)\sim_{+\infty} \dfrac 1{k+4}\sim_{+\infty} \dfrac 1k .$$
The series has a general term equivalent to the general term of the harmonic series, which diverges, hen... |
2,476,267 | <p>Here's a problem presented in my proofs class that I cannot for the life of me figure out. Apparently the answer is P2 according to the answer book but I have no idea how Help is much appreciated, thanks in advance!</p>
<p>Anna wrote down the following 3 statements, denoted P1, P2, P3, on a blank piece of paper:</p... | Brian Tung | 224,454 | <p><strong>Hint.</strong> If exactly one of the above statements is TRUE, and there are exactly three statements in all, then exactly how many are FALSE?</p>
|
1,160,726 | <p>I am trying to figure out why the equation,</p>
<p>$2x^2\:-5y^2\:=\:1$</p>
<p>has no solutions. I have tried taking the equation mod $2$,$4$, and $5$, but when I do this I am not sure what I am supposed to look for. I would really appreciate an explanation on how to solve this.</p>
<p>Thanks!</p>
| Lucian | 93,448 | <p>Modulo $5$, <em>x</em> can be only $0,\pm1,\pm2.~$ Squaring and then doubling, we have $0,2,3$, all of which are $\neq1\bmod5.~$ $($We were required to find $2x^2\equiv1\bmod5)$.</p>
|
2,212,946 | <blockquote>
<p><strong>Question:</strong>
Show that if $f(x)$ is a continuous function, then:
$$\int_{0}^{1} \int_{0}^{y} f(x) \, dx \, dy = \int_{0}^{1}(1-x)\ f(x) \ dx$$</p>
</blockquote>
<p>How can this be done? I was taught to visualise double integrals as either a summation of horizontal or vertical strips... | Rab | 417,087 | <p>The area you're looking for is the one highlighted in the image below.<a href="https://i.stack.imgur.com/wzMYV.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/wzMYV.png" alt="enter image description here"></a></p>
<p>Your integral integral is one that runs from x=0 to x=y, the external one runs f... |
18,174 | <p>I am an undergraduate secondary math education major. In <span class="math-container">$2$</span> weeks I have to give a <a href="https://www.mathmammoth.com/lessons/number_talks.php" rel="nofollow noreferrer">Number Talk</a> in my math ed class on the problem "<span class="math-container">$3.9$</span> times <span cl... | TheSimpliFire | 8,598 | <p>Consider the following solution. <span class="math-container">\begin{align*}7.5\times3.9&=(0.1\times75)\times(0.1\times39)\\\\&=0.01\times75\times39\\\\&=75\%\,\,\text{of}\,\,39\quad(\text{as}\,\,\%\,\,\text{means}\,\,1/100=0.01)\tag{1}\\\\&=\frac34\times39\quad\small{\left[=\frac{4-1}4\times39=\left... |
112,575 | <p>If we set $\exp(x)=\sum x^k/k!$, then $\exp(x+y)=\exp(x)\cdot \exp(y)$. In terms of coefficients it means that $(x+y)^n=\sum \frac{n!}{k!(n-k)!} x^ky^{n-k}$, i.e. just binomial expansion.</p>
<p>Now consider logarithm. Set $L(x):=\sum_{k>0} x^k/k$, then $L(x)=-\log(1-x)$ in a sense, and hence $$L(u+v-uv)=L(u)+L(... | Ira Gessel | 10,744 | <p>As David noted, since the summands aren't in general integers, it's difficult to give a combinatorial interpretation to the formula. However, if we multiply by $a$ or $b$ we get integers and we can give a combinatorial interpretation to the identity that we obtain (though doing this destroys the symmetry between $a$... |
1,735,653 | <p>So I need to use the fact that: $$\cos(4x) + i\sin(4x) = \left(\cos(x) + i\sin(x)\right)^4$$ to derive identities for $\cos(4x)$ and $\sin(4x)$ in terms of $\cos(x)$ and $\sin(x)$. I'm not sure how to go about this, could I please get some help.</p>
| nbubis | 28,743 | <p><strong>Hint</strong>:</p>
<p>Try replacing $x$ by $-x$, and then adding the expressions. Using the fact that $\sin(-x) = -\sin(x)$ and that $\cos(-x) = \cos(x)$:</p>
<p>$$\cos(4x) + i \sin(4x) = \left(\cos(x) + i\sin(x)\right)^4$$
$$\cos(4x) - i \sin(4x) = \left(\cos(x) - i\sin(x)\right)^4$$
So:
$$2\cos(4x) = \le... |
1,843,369 | <p>Consider the congruence </p>
<p>$$2x+7y \equiv 5\pmod{12}$$</p>
<p>Here $(2,7,12)=1$. Since $(2,12)=2$, we must have
$$7y \equiv 5\pmod{2}$$</p>
<p>Which clearly gives $y \equiv 1\pmod{2}$, or $y \equiv 1,3,5,7,9,11\pmod{12}$</p>
<p>Why does the previous statement follow?. This is not a problem. It's something ... | Pjotr5 | 157,405 | <p>The statement $2x + 7y \equiv 5 \pmod{12}$ means that $$2x + 7y = 5 + 12k$$ for some $k \in \mathbb{Z}$. We can consider this equation modulo $2$ to get $7y \equiv 5 \pmod{2}$, since both $2x$ and $12k$ are divisible by $2$. Because $7 \equiv 5 \equiv 1 \pmod{2}$ we get $y \equiv 1 \pmod{2}$, which means that we can... |
58,009 | <p>Let $f: X \to Y$ be a morphism of varieties such that its fibres are isomorphic to $\mathbb{A}^n$. Since the definition of a vector bundle stipulates that $f$ be locally the projection $U \times \mathbb{A}^n \to U$, it is likely that there exist morphisms that are not locally of that form, but I can't come up with a... | Sasha | 4,428 | <p>I think such map is locally trivial if and only if it is smooth. The "only if" part is clear. So, assume $f$ is smooth. Take any point $y_0 \in Y$ and $n+1$ affine-linearly independent points in the fiber over $y_0$. Then choose local sections through this points (this is where you need smoothness), denote them by $... |
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