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 |
|---|---|---|---|---|
112,320 | <p>What is the number of strings of length $235$ which can be made from the letters A, B, and C, such that the number of A's is always odd, the number of B's is greater than $10$ and less than $45$ and the Number of C's is always even?</p>
<p>What I can think of is </p>
<p>$$\left(\binom{235}{235} - \left\lfloor235 -... | Gerry Myerson | 8,269 | <p>It's the coefficient of $x^{235}$ in $$(x+x^3+x^5+\cdots)(x^{11}+x^{12}+\cdots+x^{44})(1+x^2+x^4+\cdots)$$ Use the formula for sum of a geometric progression, then use the binomial theorem, it should fall right out. </p>
|
4,821 | <p>A quick bit of motivation: recently a question I answered quite a while ago ( <a href="https://math.stackexchange.com/questions/22437/combining-two-3d-rotations/178957">Combining Two 3D Rotations</a> ) picked up another (IMHO rather poor) answer. While it was downvoted by someone else and I strongly concur with the... | Qiaochu Yuan | 232 | <p>Bad answers should be downvoted if you feel you have the expertise to conclude that they are bad with some confidence. This is useful information you are communicating to other users, who may not have such expertise, and it is worth communicating. I agree that there is some conflict of interest here, but it's not en... |
761,726 | <p>We know that the Banach space $X$ is infinite-dimensional,</p>
<p>theconclusion we want to show is: then $X'$ is also infinite-dimensional. </p>
<p>$X'$: the space of linear bdd functions</p>
| Prahlad Vaidyanathan | 89,789 | <p>Again, Hahn-Banach to the rescue : Choose an infinite basis $\mathcal{B}$ of $X$. Start with $v_1\in \mathcal{B}$, and choose $f_1 \in X'$ such that $\|f_1\| = 1$ and $f_1(v_1) = 1$.</p>
<p>Now choose $v_2 \in \mathcal{B}$ and use Hahn-Banach to produce $f_2 \in X'$ such that $f_2(v_1) = 0$ and $f_2(v_2) = 1$.</p>
... |
4,042,741 | <p>I'm really struggling to understand the literal arithmetic being applied to find a complete residue system of modulo <span class="math-container">$n$</span>. Below is the definition my textbook provides along with an example.</p>
<blockquote>
<p>Let <span class="math-container">$k$</span> and <span class="math-conta... | David | 893,427 | <p>You can draw a picture for modulo 6. Get a piece of paper and place it with the longer dimension horizontal. With a dark pen and a ruler, draw seven horizontal lines completely across the page. In about the middle of the page write the integers from 0 to 5 ONE PER REGION, starting at the bottom.</p>
<p>Enter all th... |
252,272 | <p>I'm working with trace of matrices. Trace is defined for square matrix and there are some useful rule, i.e. <span class="math-container">$\text{tr}(AB) = \text{tr}(BA)$</span>, with <span class="math-container">$A$</span> and <span class="math-container">$B$</span> square, and more in general trace is invariant unde... | martini | 15,379 | <p>Yes, it holds true. Let $A$ be a $n\times m$ and $B$ be a $m \times n$ matrix over the commutative ring $R$, we have
\begin{align*}
\mathrm{tr}(AB) &= \sum_{i=1}^n (AB)_{ii}\\
&=\sum_{i=1}^n \sum_{j=1}^m A_{ij}B_{ji}\\
&= \sum_{j=1}^m \sum_{i=1}^n B_{ji}A_{ij}\\
&= \sum_{j=1}^m (BA)_{jj... |
4,393,925 | <p>Is the integral of <span class="math-container">$\tan(x)\,\mathrm{d}x$</span> equal to the negative <span class="math-container">$\ln$</span> of absolute value of <span class="math-container">$\cos(x)$</span>, the same as integral of <span class="math-container">$\tan(x)\,\mathrm{d}x$</span> equal to the <span class... | Átila Correia | 953,679 | <p>As @Randall has mentioned, the following property of logarithms is useful in the present context:</p>
<p><span class="math-container">\begin{align*}
\alpha\ln|x| = \ln|x|^{\alpha}
\end{align*}</span></p>
<p>At the given case, the proposed integral is given by
<span class="math-container">\begin{align*}
\int\tan(x)\m... |
2,351,883 | <p>I am learning about tensor products.
In trying to understand the definitions, I seem to be getting
some contradiction.</p>
<p>Consider the differential form</p>
<p>$$
d x^{1} \wedge d x^{2} = d x^{1} \otimes d x^{2} - d x^{2} \otimes d x^{1}.
$$</p>
<p>If I use the symmetry property of the tensor product</p>
<p>... | Community | -1 | <p>The symmetry property of the tensor product is not that $u \otimes v = v \otimes u$: it is that the linear map $S$ satisfying $S(u \otimes v) = v \otimes u$ is invertible (and the inverse has the same form).</p>
<p>That is, on <em>vector spaces</em> (or vector bundles or other such things), $S$ is an isomorphism $U... |
141,115 | <p>I know only some basics about mathematica. However I need to write down the following sum: </p>
<p>$\sum_{\{m_k\}_N}\prod_{k=1}^N\frac{1}{m_k}[T_k(Z(\tau))]^{m_k}$. </p>
<p>Where $\{m_k\}_N$ denotes partitions of $N$ i.e. $\sum_{k=1}^Nkm_k=N$. The argument in brackets [..] is the Hecke Operator, for now not that i... | Szabolcs | 12 | <p>This happens because texturing is done triangle by triangle. Polygons with more sides are broken down into triangles, and each triangle is textured individually. I believe your example is effectively equivalent to</p>
<pre><code>pic = ExampleData[{"TestImage", "Mandrill"}];
Graphics[{Texture[pic], EdgeForm[Black... |
3,108,847 | <p>I am trying to prove that
If <span class="math-container">$z\in \mathbb{C}-\mathbb{R}$</span> such that <span class="math-container">$\frac{z^2+z+1}{z^2-z+1}\in \mathbb{R}$</span>. Show that <span class="math-container">$|z|=1$</span>.</p>
<p>1 method , through which I approached this problem is to assume <span cla... | Bello Bello | 643,633 | <p>Let <span class="math-container">$w=\frac{z^2+z+1}{z^2-z+1}=1+\frac{2z}{z^2-z+1}$</span></p>
<p>Since <span class="math-container">$\frac{z^2+z+1}{z^2-z+1}\in \mathbb{R}$</span> , so <span class="math-container">$\operatorname{Im}(w)=0$</span>
<span class="math-container">$$\iff w-\bar{w}=0$$</span></p>
<p>Now, le... |
696,869 | <p>Question:
Show that if a square matrix $A$ satisfies the equation $A^2 + 2A + I = 0$, then $A$ must be invertible.</p>
<p>My work: Based on the section I read, I will treat I to be an identity matrix, which is a $1 \times 1$ matrix with a $1$ or as an square matrix with main diagonal is all ones and the rest is zer... | Mitchell | 529,914 | <p>Since $A^2 + 2A + I = 0$, $(A + I)^2 = 0$.
Hence $A = -I$, for any size square matrix.
Since the $det(-I) = 1$, the inverse exists. </p>
<p>Since $I* I = I$ and $-I * -I = I$</p>
<p>$A = -I * -I = A$</p>
<p>Thus, $A*A = I$, making $A$ its own inverse.</p>
<p>Another solution to this problem is as follows:</p>
... |
84,076 | <p>I think computation of the Euler characteristic of a real variety is not a problem in theory.</p>
<p>There are some nice papers like <em><a href="http://blms.oxfordjournals.org/content/22/6/547.abstract" rel="nofollow">J.W. Bruce, Euler characteristics of real varieties</a></em>.</p>
<p>But suppose we have, say, a... | F. C. | 10,881 | <p>One possible way to compute Euler characteristic is to use its properties:</p>
<ul>
<li><p>$\chi$ is additive on disjoint unions</p></li>
<li><p>$\chi$ is multiplicative on fibrations</p></li>
<li><p>$\chi$ of the point is $1$</p></li>
</ul>
<p>So one has to either decompose the variety as a disjoint union, or pro... |
791,372 | <p>Hi I am trying to solve this double integral
$$
I:=\int_0^\infty \int_0^\infty \frac{\log x \log y}{\sqrt {xy}}\cos(x+y)\,dx\,dy=(\gamma+2\log 2)\pi^2.
$$
Thank you.</p>
<p>The constant in the result is given by $\gamma\approx .577$, and is known as the Euler-Mascheroni constant. I was thinking to write
$$
I=\Re \... | Pranav Arora | 117,767 | <p>The integral is
$$I=\Re\left(\left(\int_0^{\infty} \frac{\ln x}{\sqrt{x}}e^{ix}\,dx\right)^2\right)$$
Evaluating the definite integral first:
$$J=\int_0^{\infty} \frac{\ln x}{\sqrt{x}}e^{ix}\,dx$$
Use the substitution $\sqrt{x}=e^{i\pi/4}t$ to obtain:
$$J=2e^{i\pi/4}\int_0^{\infty} e^{-t^2}\ln(t^2e^{i\pi/2})\,dt=2e^... |
280,500 | <p>I would like to pose a question about the range of validity of the expansion of Binomial Theorems. </p>
<p>I know that for non-positive integer, rational $n$
$$
\left(1+x\right)^{n}=1+nx+\frac{n\left(n-1\right)}{2!}x^{2}+\dots,
$$
where the range of validity is $\left|x\right|<1$.</p>
<p>My question is that if ... | Beer | 58,613 | <p>Sorry to resurrect this post again. But I was following the suggestion above to find the range of validity of $f(x)=\log(1+\sin(x))$, and I obtained that $|\sin(x)|<1 \implies |x|<\frac{\pi}{2} \text{ or } |x-2\pi|<\frac{\pi}{2}$ etc. </p>
<p>When I tried to plot $f(x)$ vs its series expansion with Mathema... |
1,823,187 | <blockquote>
<p>There are $n \gt 0$ different cells and $n+2$ different balls. Each cell cannot be empty. How many ways can we put those balls into those cells?</p>
</blockquote>
<p>My solution:</p>
<p>Let's start with putting one different ball to each cell.
for the first cell there are $n+2$ options to choose a b... | Joshhh | 285,105 | <p>A simpler example might be $f(x) = \frac{\sin{x}}{\sqrt{x}}$. Since $\frac{1}{\sqrt{x}}$ is continuously differentiable and monotonically decreasing to 0, and since $\sin{x}$ has a bounded and integrable anti-derivative, from Dirichlet's test $\int\limits_1^{\infty}\frac{\sin{x}}{\sqrt{x}}dx$ converges. Integration ... |
2,426,535 | <p>In the book <em>Simmons, George F.</em>, Introduction to topology and modern analysis, page no- 98, question no- 2, the problem is : <strong><em>Let $X$ be a topological space and a $Y$ be metric space and $f:A\subset X\rightarrow Y$ be a continuous map. Then $f$ can be extended in at most one way to a continuous ... | bc78 | 250,074 | <p>To follow your ides: Assume for a contradiction that <span class="math-container">$f(x_0)\neq g(x_0)$</span> and choose <span class="math-container">$\epsilon=\frac{d(f(x_0),g(x_0))}{4}>0$</span>, where <span class="math-container">$d$</span> is the metric of the metric space <span class="math-container">$Y$</spa... |
537,021 | <p>Say I divide a number by 6, will a number modulus by 6 always be between 0-5? If so, will a number modulus any number (N) , the result should be between $0$ and $ N - 1$?</p>
| Prahlad Vaidyanathan | 89,789 | <p>Yes, this is the <a href="http://en.wikipedia.org/wiki/Euclidean_division" rel="nofollow">Euclidean Algorithm</a> : For any $a, n \in \mathbb{Z}, n\neq 0$, there exist $q,r \in \mathbb{Z}$ such that
$$
a = qn + r, 0 \leq r < |n|
$$
and by definition, $a\equiv r\pmod{n}$</p>
|
1,742,982 | <p>I was trying to solve the equation using factorial as shown below but now I'm stuck at this level and need help.</p>
<p>$$C(n,3) = 2*C(n,2)$$</p>
<p>$$\frac{n!}{3!(n-3)!} = 2\frac{n!}{2!(n-2)!}$$</p>
<p>$$3! (n - 3)! = (n - 2)!$$</p>
| sayan | 312,099 | <p>First we have to find the zeros of the eqn $y=-(x^2)(x+5)(x-3)$. So from the
given eqn we get the zeros which are 0 of multiplicity 2,-5 of 1 and 3 of 1.</p>
<p>1.$y>0$ when $-5<x<3$</p>
<p>2.$y<0$ when $-\infty<x<-5\;\;and\;\;3<x<\infty$</p>
<p>3.As $x$ tends to $-\infty$, $y$ tends to $... |
1,978,035 | <p>In the Wikipedia page about quintics, there was a list of quintics that could be solved with trigonometric roots.</p>
<p>For example:$$x^5+x^4-4x^3-3x^2+3x+1\tag1$$ has roots of the form $2\cos \frac {2k\pi}{11}$
$$x^5+x^4-16x^3+5x^2+21x-9=0\tag2$$ has roots of the form $\sum_{k=0}^{7}e^{\frac {2\pi i 3^k}{41}}$</p... | Will Jagy | 10,400 | <p>ADDED: I remembered where I had seen this before, with lower degree; let $\alpha \neq 1$ be a seventh root of unity, $$ \alpha^7 = 1, $$ and take
$$ \gamma = \alpha + \alpha^6, $$
$$ \gamma^2 = \alpha^2 + 2 + \alpha^5, $$
$$ \gamma^3 = \alpha^3 + 3 \alpha + 3 \alpha^6 + \alpha^4. $$
Therefore
$$ \gamma^3 + \gamma^... |
1,978,035 | <p>In the Wikipedia page about quintics, there was a list of quintics that could be solved with trigonometric roots.</p>
<p>For example:$$x^5+x^4-4x^3-3x^2+3x+1\tag1$$ has roots of the form $2\cos \frac {2k\pi}{11}$
$$x^5+x^4-16x^3+5x^2+21x-9=0\tag2$$ has roots of the form $\sum_{k=0}^{7}e^{\frac {2\pi i 3^k}{41}}$</p... | Roman Chokler | 38,328 | <p>I am not sure if there is any simple formula for the root structure of polynomials having a solvable polynomial. The only thing is that the subnormal series tells you the order in which you adjoin radicals. It does not tell you which number in each field formed you take a radical of to make the next field, but if yo... |
137,571 | <p>As the title, if I have a list:</p>
<pre><code>{"", "", "", "2$70", ""}
</code></pre>
<p>I will expect:</p>
<pre><code>{"", "", "", "2$70", "2$70"}
</code></pre>
<p>If I have</p>
<pre><code>{"", "", "", "3$71", "", "2$72", ""}
</code></pre>
<p>then:</p>
<pre><code>{"", "", "", "3$71", "3$71", "2$72", "2$72"}
... | Chris Degnen | 363 | <p>Borrowing kglr's pattern</p>
<pre><code>x = {"", "", "1$71", "3$71", "", "", "2$72", "", "", "", ""};
Prepend[
Map[Last, Partition[x, 2, 1] /. {p : Except[""], ""} :> {p, p}],
First[x]]
</code></pre>
<blockquote>
<p>{"", "", "1\$71", "3\$71", "3\$71", "", "2\$72", "2\$72", "", "", ""}</p>
</blockquote>
<p... |
792,390 | <p>Is there a name or short way of writing of $\frac{n!}{m!}$? I've searched and the closest I could find was binomial coefficient. Is there any other way?</p>
| Mikotar | 149,637 | <p>One way that I've seen is $(n)_m$. This is a common way to write this notion in combinatorics.</p>
|
792,390 | <p>Is there a name or short way of writing of $\frac{n!}{m!}$? I've searched and the closest I could find was binomial coefficient. Is there any other way?</p>
| Jack M | 30,481 | <p>That tends to be called a <em>falling</em> or <em>rising</em> factorial, and there are <a href="http://en.wikipedia.org/wiki/Pochhammer_symbol">multiple notations</a>, though none of them are standard enough that you could use one in a paper without defining it first.</p>
|
51,096 | <p>Is it possible to have a countable infinite number of countable infinite sets such that no two sets share an element and their union is the positive integers?</p>
| long trail | 13,270 | <p>Let p_n be the nth prime and let A_n be the positive integers whose smallest prime divisor is p_n (throw 1 in with A_1). This is basically applying the sieve of Eratosthenes to the entire set of positive integers. </p>
|
51,096 | <p>Is it possible to have a countable infinite number of countable infinite sets such that no two sets share an element and their union is the positive integers?</p>
| André Nicolas | 6,312 | <p>Maybe we can start slowly, by doing a decomposition of $\mathbb{N}$ into $2$ disjoint sets, say the <strong>odds</strong> and the <strong>evens</strong>.</p>
<p>Let's now go for a decomposition into $3$ disjoint sets. Leave the odds alone, and decompose the evens into those divisible by $2$ but no higher power of $... |
585,808 | <p>As part of showing that
$$
\sum_{n=1}^\infty \left|\sin\left(\frac{1}{n^2}\right)\right|
$$
converges, I ended up with trying to show that
$$
\left|\sin\left(\frac{1}{n^2}\right)\right|<\frac{1}{n^2}, \quad n=1, 2, 3,\dots
$$
since I know that the sum of the right hand side converges. But I can't show this. I've ... | Community | -1 | <p>Consider $f(x)=\sin x -x\Rightarrow f'(x)=\cos x-1 < 0 \text { for x > 0 } \Rightarrow f(x)\text{ is strictly decreasing function}$</p>
<p>But then, $f(0)=0$ which would imply $f(x)< 0 $ for $x > 0$ i.e., $\sin x < x$ for $x >0$ </p>
<p>Thus, $\sin \big(\frac{1}{n^2}\big)< \frac{1}{n^2}$</p>
... |
1,219,462 | <p>Proposition: Any polynomial of degree $n$ with leading coefficient $(-1)^n$ is the characteristic polynomial of some linear operator.</p>
<p>I do not want to construct an 'explicit matrix' corresponding to the polynomial $(-1)^n(\lambda_n x^n+\cdots+ \lambda_0)$. However, I want to use induction to prove the existe... | K_user | 221,871 | <p>You may get an idea from <a href="http://en.wikipedia.org/wiki/Companion_matrix" rel="nofollow">Companion matrix</a>. </p>
|
2,253,645 | <p>$(1,2)$ intersection $(2,3)=\{2\}$</p>
<p>$(1,2)$ intersection $[2,3]=\{2\}$</p>
<p>$\{1,2\}$ intersection $[1,2]=[1,2]$</p>
<p>$\{1,2\}$ union $[1,2]=[1,2]$</p>
<p>$\{1,2\}$ intersection $(1,3)$ intersection $[1,3)=(1,3)$</p>
<p>$\{1,2\}$ union $(1,3)$ union $[1,3)=(1,3)$</p>
<p>Is my answer correct? I find... | niksirat | 133,629 | <p>Thanks for your answer @rych. Consider spline $s$ of degree $k=2m+1$ for nodes $x_0<\cdots<x_n$ in <strong>whole line</strong> $\mathbb{R}$ instead of $[x_0,x_n]$.</p>
<p>Definition of spline:
\begin{array}{l}
1.\qquad s_i(x)=s(x)|_{[x_i,x_{i+1}]}\in\mathbb{P}_{2m+1},\qquad i=0,1,\ldots,n-1,\\
2.\qquad s\in\m... |
543,938 | <p>Can anyone share a link to proof of this?
$${{p-1}\choose{j}}\equiv(-1)^j(\text{mod}\ p)$$ for prime $p$.</p>
| lab bhattacharjee | 33,337 | <p>$$\binom {p-1}j=\prod_{1\le r\le j}\frac{p-r}r$$</p>
<p>Now, $\displaystyle p-r\equiv -r\pmod p\implies \frac{p-r}r\equiv-1\pmod p$</p>
|
543,938 | <p>Can anyone share a link to proof of this?
$${{p-1}\choose{j}}\equiv(-1)^j(\text{mod}\ p)$$ for prime $p$.</p>
| Marc van Leeuwen | 18,880 | <p>It is well known that $\binom pi\equiv0\pmod p$ for $0<i<p$. Now Pascal's recurrence gives $\binom{p-1}i\equiv-\binom{p-1}{i-1}\pmod p$ for those$~i$, and so $\binom{p-1}i\equiv(-1)^i\pmod p$ for $0\leq i<p$ follows by an immediate induction on$~i$, with $\binom{p-1}0=1$ as base case.</p>
<p>More generally... |
4,202,490 | <p>Trying to construct an example for a Business Calculus class (meaning trig functions are not necessary for the curriculum). However, I want to touch on the limit problem involved with the <span class="math-container">$\sin(1/x)$</span> function.</p>
<p>I am sure there is a simple function, or there isn't... But woul... | Toby Bartels | 63,003 | <p>I endorse Ethan's answer: define the function by drawing a graph. <em>You</em> know that it's <span class="math-container">$ x \mapsto \sin ( 1 / x ) $</span>, but <em>they</em> don't need to know that.</p>
<p>If you don't like that, hardmath has given an answer; but you can make it look more like <span class="math... |
2,744,068 | <p>Theorem:
The only idempotent matrix whose eigenvalues are all zero is the null matrix.</p>
<p>Then how to prove this?</p>
| quasi | 400,434 | <p>Suppose $A$ is idempotent, and $A\ne 0$.
<p>
Let $x$ be such that $Ax\ne 0$, and let $y=Ax$.
<p>
Then $Ay= A(Ax) = A^2x = Ax = y$, so $y$ is an eigenvector with eigenvalue $1$.</p>
|
2,744,068 | <p>Theorem:
The only idempotent matrix whose eigenvalues are all zero is the null matrix.</p>
<p>Then how to prove this?</p>
| copper.hat | 27,978 | <p>If all eigenvalues are zero then $T^n = 0$. Since $T=T^2 = \cdots = T^n$ then we see that $T=0$.</p>
|
1,334,680 | <p>How to apply principle of inclusion-exclusion to this problem?</p>
<blockquote>
<p>Eight people enter an elevator at the first floor. The elevator
discharges passengers on each successive floor until it empties on the
fifth floor. How many different ways can this happen</p>
</blockquote>
<p>The people are <... | DRF | 176,997 | <p>What you are butting your head against here IMO is the fact that you need a meta-language at the beginning. Essentially at some point you have to agree with other people what your axioms and methods of derivation are and these concepts cannot be intrinsic to your model. </p>
<p>Usually I think we take axioms in pro... |
1,334,680 | <p>How to apply principle of inclusion-exclusion to this problem?</p>
<blockquote>
<p>Eight people enter an elevator at the first floor. The elevator
discharges passengers on each successive floor until it empties on the
fifth floor. How many different ways can this happen</p>
</blockquote>
<p>The people are <... | Stefan Perko | 166,694 | <p>As already pointed out, this is indeed circular. The only thing you can do is pretend it is not. </p>
<p>A simple reason is for example: How are you going to explain what a proof is? Well, you might just a give a philosophical description, but I turns out that you can study proofs mathematically (as sequences of fo... |
1,334,680 | <p>How to apply principle of inclusion-exclusion to this problem?</p>
<blockquote>
<p>Eight people enter an elevator at the first floor. The elevator
discharges passengers on each successive floor until it empties on the
fifth floor. How many different ways can this happen</p>
</blockquote>
<p>The people are <... | Pepijn Schmitz | 249,930 | <p>It's turtles all the way down.<a href="https://en.wikipedia.org/wiki/Turtles_all_the_way_down" rel="nofollow">*</a></p>
<p>In other words: there is nothing at the bottom of mathematics but philosophy. It's a set of rules we came up with because it seemed useful, but it has no absolute grounding in reality or the un... |
1,195,092 | <p>The book I am using for my Introduction of Topology course is Principles of Topology by Fred H. Croom.</p>
<p>I was given the following problem: </p>
<blockquote>
<p>Prove that a set $E$ is countable if and only if there is a surjection from $\mathbb{N}$ to $E$. </p>
</blockquote>
<p>I have a rough idea on how ... | bobbym | 77,276 | <p>The absorbing chain matrix is</p>
<p>$$\left(
\begin{array}{ccccc}
\text{} & o & 5 & 55 & 6 \\
o & \frac{2}{3} & \frac{1}{6} & 0 & \frac{1}{6} \\
5 & \frac{2}{3} & 0 & \frac{1}{6} & \frac{1}{6} \\
55 & 0 & 0 & 1 & 0 \\
6 & 0 & 0 & 0 &a... |
220,736 | <p>I have reduced a problem I'm working on to something resembling a graph theory problem, and my limited intuition tells me that it's not so esoteric that only I could have ever considered it. <strong>I'm looking to see if someone knows of any related work.</strong> Here's the problem:</p>
<hr>
<p>Given a roadway ... | Joseph O'Rourke | 6,094 | <p>Just to emphasize Thomas Richard's remark about smoothness, unless I've miscalculated, a $\frac{1}{4} L$-square leads to area
$$2 \epsilon L - \epsilon^2 (4-\pi) < 2 \epsilon L \;.$$
<hr />
<a href="https://i.stack.imgur.com/9jfoc.jpg" rel="nofollow noreferrer"><img src="https... |
13,705 | <p>Let $m$ be a positive integer. Define $N_m:=\{x\in \mathbb{Z}: x>m\}$. I was wondering when does $N_m$ have a "basis" of two elements. I shall clarify what I mean by a basis of two elements: We shall say the positive integers $a,b$ generate $N_m$ and denote $N_m=<a,b>$ if every element $x\in N_m$ can be wri... | Aryabhata | 1,102 | <p>(Now that you have relaxed the last condition of paragraph 1)</p>
<p>I suppose you mean non negative $\alpha$, $\beta$? (You seem to be allowing them to be zero).</p>
<p>In which case, this looks like the <a href="http://mathworld.wolfram.com/FrobeniusNumber.html" rel="nofollow">Frobenius Problem</a> which says th... |
69,208 | <p>Consider $f:\{1,\dots,n\} \to \{1,\dots,m\}$ with $m > n$.
Let $\operatorname{Im}(f) = \{f(x)|x \in \{1,\dots,n\}\}$.</p>
<p>a.)
What is the probability that a random function will be a bijection when viewed as
$$f':\{1,\dots,n\} \to \operatorname{Im}(f)?$$</p>
<p>b.)
How many different function f are the... | zyx | 14,120 | <p>$x^4=0$ is correct if interpreted to mean "the identity holds up to terms of degree 4 or higher". This is because terms of degree $\geq 4$ were dropped from the power series for $\sin$ and $\cos$ at the start of the calculation. If terms of degree $\geq n$ are dropped from the series then this order $n$ approxima... |
2,992,454 | <p>Prove :</p>
<blockquote>
<p><span class="math-container">$f : (a,b) \to \mathbb{R} $</span> is convex, then <span class="math-container">$f$</span> is bounded on every closed subinterval of <span class="math-container">$(a,b)$</span></p>
</blockquote>
<p>where <span class="math-container">$f$</span> is convex if... | RRL | 148,510 | <p>Here is an analytic proof that a convex function <span class="math-container">$f:[\alpha,\beta] \to \mathbb{R}$</span> is bounded on the closed interval.</p>
<p>Take <span class="math-container">$M = \max (f(\alpha),f(\beta))$</span>. Note that any <span class="math-container">$x \in [\alpha,\beta]$</span> is of th... |
1,480,511 | <p>I have included a screenshot of the problem I am working on for context. T is a transformation. What is meant by Tf? Is it equivalent to T(f) or does it mean T times f?</p>
<p><a href="https://i.stack.imgur.com/LtRS1.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/LtRS1.png" alt="enter image desc... | Gondim | 237,793 | <p>You have shown that $3$ and $5$ divide $a^2$, so $3$ and $5$ also divide $a$ (remember that if a prime divides a product, then it divides at least one of the factors). Now $a=15k$, so you get $b^2 = 15k^2$. Do the same thing to show that $15|b$ and get a contradiction.</p>
|
1,480,511 | <p>I have included a screenshot of the problem I am working on for context. T is a transformation. What is meant by Tf? Is it equivalent to T(f) or does it mean T times f?</p>
<p><a href="https://i.stack.imgur.com/LtRS1.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/LtRS1.png" alt="enter image desc... | Ennar | 122,131 | <p>I'll give a proof for more general statement: Let $p$ be prime and $a$ a positive integer coprime with $p$. Then $\sqrt{ap}$ is irrational.</p>
<p>Assume that $ap = \frac{x^2}{y^2}$ where $x,y\in\mathbb Z$ are coprime. Then we have $apy^2 = x^2\implies p|x^2\implies p|x$, because $p$ is prime. Thus, $x = px'$ and $... |
93,458 | <blockquote>
<p>Let <span class="math-container">$n$</span> be a nonnegative integer. Show that <span class="math-container">$\lfloor (2+\sqrt{3})^n \rfloor $</span> is odd and that <span class="math-container">$2^{n+1}$</span> divides <span class="math-container">$\lfloor (1+\sqrt{3})^{2n} \rfloor+1 $</span>.</p>
</... | lhf | 589 | <p><em>Hint for the first part:</em> Consider $u_n = (2+\sqrt{3})^{n} + (2-\sqrt{3})^{n}$. Prove that $u_n$ is always an even integer and that $u_n = \lceil (2+\sqrt{3})^n \rceil$. Use that $(2-\sqrt{3})^{n}\to 0$.</p>
<p>(This has now been incorporated into the edited question.)</p>
<p><em>Hint for the second part:<... |
853,308 | <p>I want to calculate the expected number of steps (transitions) needed for absorbtion. So the transition probability matrix $P$ has exactly one (lets say it is the first one) column with a $1$ and the rest of that column $0$ as entries.</p>
<p>$P = \begin{bmatrix} 1 & * & \cdots & * \\ 0 & \vdots &am... | Mauro ALLEGRANZA | 108,274 | <p>You have not necessarily to think in term of "definition" as a sort of "replacement" of the intuitive notion of function with its set-theoretic counterpart. </p>
<p>We can say instead that set theory provides a "model" for the mathematical concept of function. </p>
<p>Functions was already known in mathematics wel... |
2,390,538 | <p>The problem says:</p>
<p>If every closed ball in a metric space $X$ is compact, show that $X$ is
separable.</p>
<p>I'm trying to use an equivalence in metric spaces that tells us: let X be the matric space, the following are equivalent</p>
<p>X is 2nd countable</p>
<p>X is Lindeloff</p>
<p>X is separable</p>
<... | Matematleta | 138,929 | <p>For a different approach, choose $x\in X$ so that $X = \cup_{n \in \mathbb{N}} \overline B_x(n).$ Now consider, for fixed $n\in \mathbb N,$ the fact that the compact ball $\overline B_x(n)$ is totally bounded. This means that, for each $m\in \mathbb N$, there is a $\textit{finite}$ set of points $x_{m,n}\in \overlin... |
167,904 | <p>In one the the answers to this thread " <a href="https://mathoverflow.net/questions/119850/">Can one embedd the projectivezed tangent space of CP^2 in a projective space? </a> " it was mentioned that " $\mathbb{P}(T\mathbb{P}^2)$ isomorphic to the variety of complete flags in the vector space $\mathbb{C^3}$ ". </p>
... | Steven Sam | 321 | <p>I will take $P^2$ to mean the space of lines in $C^3$. The tangent space at a particular point in $P^2$ (say represented by a line $L$) is a linear map from $L$ to $C^3/L$. Let $\mathcal{L}$ be the line bundle whose total space is $\{(L,x) \in P^2 \times C^3 \mid x \in L\}$ (this is $O(-1)$ but it doesn't matter). T... |
720,823 | <p>I was solving some linear algebra problems and I have a quick question about one problem. I'm given the matrix $A = \{a_1,a_2\}$ where $a_1=[1,1]$ and $a_2=[-1,1]$. I need to solve for the eigenvalues of the matrix $A$ over the complex numbers $\mathbb{C}$. I solved and got the eigenvalue $1-i$ and $1+i$. Now this i... | Studentmath | 125,085 | <p>Regarding your first question: as you have said, it's exactly the same. I have seen [xI-A] used mostly for finding the Eigenvalues, and the other way around for finding the Ker of a given eigenvalue (for the matching vectors, etc.). It's more comftorble to work with positive x's and negetive numbers than the other w... |
1,429,853 | <p>In a number sequence, I've figured the $n^{th}$ element can be written as $10^{2-n}$.</p>
<p>I'm now trying to come up with a formula that describes the sum of this sequence for a given $n$. I've been looking at the geometric sequence, but I'm not sure how connect it.</p>
| Community | -1 | <p>If it is $10^{2-n}$ then, the initial term is $10$ and common ratio is $\frac 1{10}$, Now, can you get it?</p>
|
1,775,787 | <p>How many different numbers can be obtained by rearranging the digits of 1,273,421,695?</p>
<p>Would it be C(10,2)*C(10,2)*P(8,6) = 40 million, 824 thousand</p>
<p>Or would it be (10*10*8*8*6*5*4*3*2*1)/(2!*2!) = 1 million 152 thousand</p>
<p>Thanks in advance for the help.</p>
| Justin | 337,806 | <p>Using permutations, I'm pretty sure that it would be 10! which is equal to (10*9*8*7*6*5*4*3*2*1). </p>
<p>Hope that helps,
Justin</p>
|
1,775,787 | <p>How many different numbers can be obtained by rearranging the digits of 1,273,421,695?</p>
<p>Would it be C(10,2)*C(10,2)*P(8,6) = 40 million, 824 thousand</p>
<p>Or would it be (10*10*8*8*6*5*4*3*2*1)/(2!*2!) = 1 million 152 thousand</p>
<p>Thanks in advance for the help.</p>
| Ross Millikan | 1,827 | <p>You must place the two <span class="math-container">$1$</span>s, two <span class="math-container">$2$</span>s and <span class="math-container">$6$</span> other digits into <span class="math-container">$10$</span> positions. You can place the <span class="math-container">$1$</span>s in <span class="math-container">$1... |
4,600,992 | <p>I have two sequences of random variables <span class="math-container">$\{ X_n\}$</span> and <span class="math-container">$\{Y_n \}$</span>. I know that
<span class="math-container">$X_n \to^d D, Y_n \to^d D$</span>. Can I conclude that <span class="math-container">$X_n - Y_n \to^p 0$</span>?</p>
<p>If I cannot, what... | donaastor | 251,847 | <p>Here is a different way to prove that <span class="math-container">$f(x)$</span> approaches infinity. This one doesn't have any series which could confuse you. Your intuition "failed" (tricked you) because you forgot to multiply your brackets with that <span class="math-container">$x$</span> outside. If yo... |
78,243 | <p>A positive integer $n$ is said to be <em>happy</em> if the sequence
$$n, s(n), s(s(n)), s(s(s(n))), \ldots$$
eventually reaches 1, where $s(n)$ denotes the sum of the squared digits of $n$.</p>
<p>For example, 7 is happy because the orbit of 7 under this mapping reaches 1.
$$7 \to 49 \to 97 \to 130 \to 10 \to 1$$
B... | Gerry Myerson | 3,684 | <p>Guy, Unsolved Problems In Number Theory, 3rd edition, problem E34, writes, "It seems that about 1/7 of all numbers are happy, but what bounds on the density can be proved?" He doesn't give an answer, so I suppose nothing was known as of the publication of the book. Helen Grundman has written several papers on happy ... |
118,545 | <p>I gather that the question whether the Bruck-Chowla-Ryser condition was sufficient used to top the list, but now that that's settled - what is considered the most interesting open question?</p>
| Peter Dukes | 45,255 | <p>Personally, I am most interested in design theory with an "asymptotic flavor", and I think there are (edit: <em>were</em>, pre-Keevash) some very interesting open questions in this direction.</p>
<p>To cut to the chase, I think asymptotic design theory today is effectively searching for a constructive proof of Gust... |
332,993 | <p>How do I approach the problem?</p>
<blockquote>
<p>Q: Let $ \displaystyle z_{n+1} = \frac{1}{2} \left( z_n + \frac{1}{z_n} \right)$ where $ n = 0, 1, 2, \ldots $ and $\frac{-\pi}{2} < \arg (z_0) < \frac{\pi}{2} $. Prove that $\lim_{n\to \infty} z_n = 1$.</p>
</blockquote>
| achille hui | 59,379 | <p>One common trick to deal with recurrence equation involving reciprocal is to rewrite
$z_n$ as ratio of two sequence $p_n, q_n$ to be determined. Notice,</p>
<p>$$\frac{p_{n+1}}{q_{n+1}} = z_{n+1} = \frac12 (z_n + \frac{1}{z_n}) = \frac12(\frac{p_n}{q_n} + \frac{q_n}{p_n}) = \frac{p_n^2 + q_n^2}{2p_nq_n}$$</p>
<p>W... |
2,699,753 | <p>$a,b$ and $c$ are all natural numbers, and function $f(x)$ always returns a natural number. If$$
\sum_{n=b}^{a} f(n) = c,$$
in terms of $b,c$ and $f$, how would you solve for $a$? Do I require more information to solve for $a$?</p>
<p>EDIT: If $x$ increases $f(x)$ increases</p>
| Cye Waldman | 424,641 | <p>In problems of this type I prefer to convert the sequence to a generalized Fibonacci form. I have solved this exact problem elsewhere and the solution to the general form $f(n) = Af(n-1)+B$ is given <a href="https://math.stackexchange.com/questions/2357418/solving-fibonacci-recurrence-relation/2358956#2358956">here<... |
817,680 | <p><strong>Question:</strong></p>
<blockquote>
<p>Assume that $a_{n}>0,n\in N^{+}$, and that $$\sum_{n=1}^{\infty}a_{n}$$ is convergent. Show that
$$\sum_{n=1}^{\infty}\dfrac{a_{n}}{(n+1)a_{n+1}}$$ is divergent?</p>
</blockquote>
<p>My idea: since
$\sum_{n=1}^{\infty}a_{n}$ converges, then there exists $M>... | vadim123 | 73,324 | <p>This is essentially the same as Erick Wong's lovely <a href="https://math.stackexchange.com/a/281028/73324">solution</a> to a similar problem.</p>
<p>Set $\displaystyle b_n=\frac{a_n}{(n+1)a_{n+1}}$. Since $a_n\to 0$ and $a_n>0$ there is some decreasing subsequence $a_{n_1}>a_{n_2}>a_{n_3}>\cdots$. By... |
666,503 | <p>How to isolate $x$ in this equation: $px+(\frac{b}{a})px=m$</p>
<blockquote>
<p>Blockquote</p>
</blockquote>
<p>And get $\frac{a}{a+b}*\frac{m}{p}$</p>
| Community | -1 | <p>Notice that $p$ and $x$ are two common factors of the two terms of the sum so we can factor:
$$px+\frac b a px=p\left(1+\frac b a\right)x=p\frac{a+b}ax$$
Can you take it from here?</p>
|
1,690,854 | <p>Solve the equation
$$-x^3 + x + 2 =\sqrt{3x^2 + 4x + 5.}$$
I tried. The equation equavalent to
$$\sqrt{3x^2 + 4x + 5} - 2 + x^3 - x=0.$$
$$\dfrac{3x^2+4x+1}{\sqrt{3x^2 + 4x + 5} + 2}+x^3 - x=0.$$
$$\dfrac{(x+1)(3x+1)}{\sqrt{3x^2 + 4x + 5} + 2}+ (x+1) x (x-1)=0.$$
$$(x+1)\left [\dfrac{3x+1}{\sqrt{3x^2 + 4x + 5} + 2}+... | chenbai | 59,487 | <p>hint:
edit 2:
when $x>1$, there is no solution, when $-\dfrac{1}{3} \le x \le 0$, there is no solution also.</p>
<p>when $0\le x\le 1, x(1-x) < f(x)=\dfrac{x}{3}+\dfrac{1}{2+\sqrt{5}}$ </p>
<p>and
$\dfrac{3x+1}{\sqrt{3x^2 + 4x + 5} + 2}\ge \dfrac{x}{3}+\dfrac{1}{2+\sqrt{5}}$ </p>
<p>so only possible is $... |
315,235 | <p>I am learning about vector-valued differential forms, including forms taking values in a Lie algebra. <a href="http://en.wikipedia.org/wiki/Vector-valued_differential_form#Lie_algebra-valued_forms">On Wikipedia</a> there is some explanation about these Lie algebra-valued forms, including the definition of the operat... | Adam Marsh | 525,270 | <p>Although this is an old post, it comes up prominently in searches and so another viewpoint might be helpful. I’ll view the finite-dimensional real Lie algebra $\mathfrak{g}$ as real matrices under the Lie commutator (such a faithful rep always exists). The reference the treatment is taken from is available online <a... |
520,285 | <p>I've been going through this representation theory <a href="http://math.berkeley.edu/~serganov/math252/notes1.pdf" rel="nofollow">lecture notes</a>, and I've found the ''hungry knights'' problem very interesting.
Do you know any interesting problems similar to that one?</p>
<p>To define ''similar'': problems which ... | hengxin | 51,434 | <p>Here is a paper: <a href="http://www.jstor.org/stable/2689726" rel="nofollow">A Theorem about Primes Proved on a Chessboard</a>. It is </p>
<blockquote>
<p>"An elementary treatment of a class of solutions to the $n$-queens problem leads to a proof of Fermat's theorem on primes which are sums of two squares."</p>
... |
2,843,822 | <p>How do I rewrite $(1\,2)(1\,3)(1\,4)(1\,5)$ as a single cycle? I have tried questions in the form: $(1\,4\,3\,5\,2)(4\,5\,3\,2\,1)$.</p>
| TheSimpliFire | 471,884 | <p>For a product of transpositions, just reverse the order! $$(1\,2)(1\,3)(1\,4)(1\,5)=(1\,5\,4\,3\,2)$$</p>
|
191,673 | <p>If I input:</p>
<pre><code>data = RandomVariate[ProbabilityDistribution[x/8, {x, 0, 4}], 10];
{EstimatedDistribution[data, ProbabilityDistribution[x/8, {x, 0, θ}],
ParameterEstimator -> "MaximumLikelihood"], data}
</code></pre>
<p>Mathematica returns:</p>
<pre><code>{ProbabilityDistribution[\[FormalX]/
8, ... | m0nhawk | 2,342 | <p>Your <code>data</code> has only 10 samples of random values. This can result in discrepancy between the the PDF and estimation.</p>
<p>You will have almost 4 for 1000 samples:</p>
<pre><code>data =
RandomVariate[ProbabilityDistribution[x/8, {x, 0, 4}], 1000];
{EstimatedDistribution[data, ProbabilityDistribution[... |
1,497,232 | <p>Prove or disprove. If $f(A) \subseteq f(B)$ then $A \subseteq B$</p>
<p>Let y be arbitrary. </p>
<p>$f(A)$ means $\exists a \in A (f(a)=y)$</p>
<p>$f(B)$ means $\exists b \in B (f(b)=y)$ </p>
<p>but $\forall a \in A \exists ! y \in f(a)(f(a)=y)$ </p>
<p>and $\forall b \in B \exists ! y \in f(b)(f(b)=y)$</p>
<p... | Weaam | 1,746 | <p>Suppose $f$ is <em>injective</em> and that $f(A) \subseteq f(B)$. Let $x \in A$. Then $f(x) \in f(A) \subseteq f(B)$ and $f(x) = f(y)$ for some $y \in B$. But since $f$ is injective, $x = y$ and $x \in B$.</p>
<p>On the other hand, if $f$ is <em>not injective</em>, then there is distinct $x \neq y$ with $f(x) = f(y... |
1,497,232 | <p>Prove or disprove. If $f(A) \subseteq f(B)$ then $A \subseteq B$</p>
<p>Let y be arbitrary. </p>
<p>$f(A)$ means $\exists a \in A (f(a)=y)$</p>
<p>$f(B)$ means $\exists b \in B (f(b)=y)$ </p>
<p>but $\forall a \in A \exists ! y \in f(a)(f(a)=y)$ </p>
<p>and $\forall b \in B \exists ! y \in f(b)(f(b)=y)$</p>
<p... | drhab | 75,923 | <p>If codomain $Y$ of function $f:X\to Y$ is a singleton then $f(A)=Y$ for <strong>any</strong> non-empty set $A\subseteq X$. </p>
<p>So $f(A)\subseteq f(B)$ is true for any pair of nonempty subsets of $X$. </p>
<p>If $X$ is has more than one element then nonempty sets $A, B\subseteq X$ exist with $A\nsubseteq B$.</... |
2,870,956 | <p>Assume that f is a non-negative real function, and let $a>0$ be a real number.</p>
<p>Define $I_a(f)$ to be</p>
<p>$I_a(f)$=$\frac{1}{a}\int_{0}^{a} f(x) dx$ </p>
<p>We now assume that $\lim_{x\rightarrow \infty} f(x)=A$ exists.</p>
<p>Now I want to proof whether $\lim_{a\rightarrow \infty} I_a(f)=A$ is true ... | José Carlos Santos | 446,262 | <p>Your approach is fine, including the fact that you noticed that there's a problem with $0$. But that problem is easy to solve: define $f(x)=A+\frac1{x+k}$, for some $k>0$. Then the problem will go away.</p>
|
4,611,065 | <p>This question is motivated by curiosity and I haven't much background to exhibit .</p>
<p>Going through a couple of books dealing with real analysis, I've noticed that 2 definitions can be given of the exponential function known in algebra as <span class="math-container">$f(x)= e^x$</span>.</p>
<p>One definition sa... | peek-a-boo | 568,204 | <p><strong>Title question:</strong></p>
<p>Yes.</p>
<hr />
<p><strong>1. Do these definitions exhaust the ways the exponential function can be defined?</strong></p>
<p>No. There are lots of other ways. For example:</p>
<ol>
<li>for any <span class="math-container">$a>0$</span> you can define <span class="math-contai... |
1,576,836 | <p>I was solving a question related to functions and i come across a limit which i cannot understand.The question is <br>
If $a$ and $b$ are positive real numbers such that $a-b=2,$ then find the smallest value of the constant $L$ for which $\sqrt{x^2+ax}-\sqrt{x^2+bx}<L$ for all $x>0$<br></p>
<hr>
<p>First i f... | Adhvaitha | 228,265 | <p>The short answer to your question is that $x=\sqrt{x^2}$ only for $x > 0$. For $x<0$, we have $x=-\sqrt{x^2}$.</p>
<p>In the first method, the fact that
$$\dfrac{2x}{\sqrt{x^2+ax}+\sqrt{x^2+bx}} = \dfrac2{\sqrt{1+a/x}+\sqrt{1+b/x}}$$
is true only for $x > 0$. For $x<0$, we have $x = -\sqrt{x^2}$, and he... |
1,576,836 | <p>I was solving a question related to functions and i come across a limit which i cannot understand.The question is <br>
If $a$ and $b$ are positive real numbers such that $a-b=2,$ then find the smallest value of the constant $L$ for which $\sqrt{x^2+ax}-\sqrt{x^2+bx}<L$ for all $x>0$<br></p>
<hr>
<p>First i f... | Harish Chandra Rajpoot | 210,295 | <p>Notice, here is an easier method to solve </p>
<p>let $x=-\large \frac{1}{t}$, $$\lim_{x\to -\infty}(\sqrt{x^2+ax}-\sqrt{x^2+bx})$$
$$=\lim_{t\to 0^+}\left(\sqrt{\left(\frac{-1}{t}\right)^2+a\left(\frac{-1}{t}\right)}-\sqrt{\left(\frac{-1}{t}\right)^2+b\left(\frac{-1}{t}\right)}\right)$$
$$=\lim_{t\to 0^+}\frac{1}{... |
713,732 | <p>I want to know how one would go about solving an <em>unfactorable cubic</em>. I know how to factor cubics to solve them, but I do not know what to do if I cannot factor it. For example, if I have to solve for $x$ in the cubic equation:
$$2x^3+6x^2-x+4=0$$
how would I do it?</p>
<p>Edit: I have heard people telling ... | Git Gud | 55,235 | <p>I don't get the <code>Then I sum out the two matrices</code> part. You're supposed to compute $\begin{bmatrix} 1 & 1 & 0 & 1\end{bmatrix} \times \text{Parity matrix}^T$ to find the syndrome.</p>
<p>Then you get the syndrome $\begin{bmatrix} 1 & 1\end{bmatrix}^T$ which yields the correction $0101$.</... |
1,043 | <p>Hi all,</p>
<p>The short-time fourier transform decomposes a signal window into a sin/cosine series.</p>
<p>How would one approximate a signal in the same way, but using a set of arbitrary basis functions instead of sin/cos? These arbitrary basis functions are likely in my case to be very small discrete chunks of ... | Vasile Moșoi | 1,093 | <p>If we talk about the time periodic signals, may be used every set of functions with scalar product (Hilbert space), and optimise to minimize mean squared error relative to the original signal.
In signal theory there are some functions frequently used in this scope such us the Legendre, Tchebyshev ore Hermite polyno... |
256,373 | <p>I've not been able to find a package which will deal with Geometric Algebra. Perhaps somebody can help?</p>
| Bob Hanlon | 9,362 | <pre><code>Clear["Global`*"]
$Version
"12.3.1 for Mac OS X x86 (64-bit) (June 19, 2021)"
sol = {Reduce[{x^3 + y^3 + z^3 == n, x >= 0, y >= 0, z >= 0,
0 < n < 100}, {n, x, y, z}, Integers] // ToRules}
(* {{n -> 1, x -> 0, y -> 0, z -> 1}, {n -> 1, x -> 0, y ... |
4,282,006 | <p><strong>Evaluate the limit</strong></p>
<p><span class="math-container">$\lim_{x\rightarrow \infty}(\sqrt[3]{x^3+x^2}-x)$</span></p>
<p>I know that the limit is <span class="math-container">$1/3$</span> by looking at the graph of this function, but I struggle to show it algebraically.</p>
<p>Is there anyone who can ... | TheSilverDoe | 594,484 | <p>You have <span class="math-container">$$\sqrt[3]{x^3+x^2} - x = x \left(\sqrt[3]{1 + \frac{1}{x}} - 1 \right) = x \left( 1 + \frac{1}{3x} + o\left(\frac{1}{x} \right) - 1\right) = \frac{1}{3} + o(1)$$</span></p>
<p>and you are done.</p>
|
3,118 | <p>Can anyone help me out here? Can't seem to find the right rules of divisibility to show this:</p>
<p>If $a \mid m$ and $(a + 1) \mid m$, then $a(a + 1) \mid m$.</p>
| Sayem Rahman | 977,607 | <p>If <span class="math-container">$a|m$</span> and <span class="math-container">$a+1|m$</span> then <span class="math-container">$lcm(a,a+1)|m$</span>. And we know that <span class="math-container">$lcm(a,a+1)\cdot gcd(a,a+1) = a(a+1)$</span> . But <span class="math-container">$a$</span> and <span class="math-containe... |
3,310,027 | <p>Is the expression <span class="math-container">$\forall a,b,c \in M : \varphi(a,b,c)$</span> equivalent to <span class="math-container">$\forall a \forall b \forall c : (a \in M \land b \in M \land c \in M) \rightarrow \varphi(a,b,c)$</span> ?</p>
| md2perpe | 168,433 | <p>The formula <span class="math-container">$\forall a \in M : \varphi(a)$</span> is syntactic sugar for <span class="math-container">$\forall a \ (a \in M \to \varphi(a))$</span> and <span class="math-container">$\forall a, b, c \in M : \varphi(a,b,c)$</span> is syntactic sugar for <span class="math-container">$\foral... |
2,189,818 | <p>I am having trouble finding the natural parameterization of these curves:</p>
<blockquote>
<p>$$\alpha(t)=\left(\sin^2\left(\frac{t}{\sqrt{2}}\right),\frac{1}{2}\sin \left(t\sqrt{2}\right), \left(\frac{t}{\sqrt{2}}\right)\right)$$</p>
</blockquote>
<p>The thing is when finding $$\|\alpha'(t)\|=\sqrt{\frac{3}{2}\... | Joffan | 206,402 | <p>I'll illustrate one of the techniques at <a href="https://math.stackexchange.com/questions/5877/efficiently-finding-two-squares-which-sum-to-a-prime">the question lulu linked</a> although due to limitations on the tools I have available I'll use smaller numbers:</p>
<p>Say we want to find the squares that add to th... |
748,489 | <p>I have a measure ($x$) which the domain is $[0, +\infty)$ and measure some sort of variability. I want to create a new measure ($y$) that represents regularity and is inverse related to $x$.</p>
<p>It is easy, just make $y = -x$. However I want this new measure to be positive, to make it more interpretable. Linearl... | Angel Moreno | 327,493 | <p>Try more generally: $$ y = f_a(x) = (1+x^a)^{-1/a} , a \in R^+ $$
For example: $$ f_3(x) = \frac {1} {\sqrt[3]{1+x^3}} $$
$$ f_1(x) = \frac {1} {1+x} $$
$$ f_{5/3}(x) = \frac {1} {\sqrt[5]{(1+x^{5/3})^3}} $$</p>
<p>The previous functions have a polynomial decrease. If you want an exponential decrease from 1 to 0 yo... |
748,489 | <p>I have a measure ($x$) which the domain is $[0, +\infty)$ and measure some sort of variability. I want to create a new measure ($y$) that represents regularity and is inverse related to $x$.</p>
<p>It is easy, just make $y = -x$. However I want this new measure to be positive, to make it more interpretable. Linearl... | J.G. | 56,861 | <p>As Angel Moreno has noted, we can use $1-\tanh x$. Of course, that also means we can use $1-\tanh^2 x=\operatorname{sech}^2 x$, which in turn means we could instead use $\operatorname{sech} x$.</p>
<p>To be even more general, let $F$ denote your favourite cumulative distribution function of support $[0,\,\infty)$; ... |
3,362,916 | <p>I'm trying to graph <span class="math-container">$|x+y|+|x-y|=4$</span>. I rewrote the expression as follows to get a function that resembles the direction of unit vectors at <span class="math-container">$\pi/4$</span> to the horizontal axis (take it to be <span class="math-container">$x$</span>)<span class="math-co... | Ninad Munshi | 698,724 | <p>I would take cases. When <span class="math-container">$x > y$</span></p>
<p><span class="math-container">$$|x+y| + |x-y| = 4 \implies x = 2$$</span></p>
<p>Try to figure out the rest of the cases on your own then graph everything together.</p>
|
3,362,916 | <p>I'm trying to graph <span class="math-container">$|x+y|+|x-y|=4$</span>. I rewrote the expression as follows to get a function that resembles the direction of unit vectors at <span class="math-container">$\pi/4$</span> to the horizontal axis (take it to be <span class="math-container">$x$</span>)<span class="math-co... | Parcly Taxel | 357,390 | <p>The lines <span class="math-container">$x+y=0$</span> and <span class="math-container">$x-y=0$</span> define a coordinate system rotated <span class="math-container">$45^\circ$</span> from the canonical axes and scaled by a factor of <span class="math-container">$\sqrt2$</span>. Accordingly, <span class="math-contai... |
925,746 | <p>I don't really understand Tautologies or how to prove them, so if someone could help, that would be great! </p>
| MPW | 113,214 | <p>If Q is true, the first implication is true, so the disjunction is true.</p>
<p>If Q is false, the second implication is true, so the disjunction is true.</p>
<p>In either case, the disjunction is true, <em>quod erat demonstrandum.</em></p>
|
3,058,019 | <blockquote>
<p>Two numbers <span class="math-container">$297_B$</span> and <span class="math-container">$792_B$</span>, belong to base <span class="math-container">$B$</span> number system. If the first number is a factor of the second number, then what is the value of <span class="math-container">$B$</span>?</p>
</... | vadim123 | 73,324 | <p>The long division is the source of the error; you can't have <span class="math-container">$7/2$</span> as the quotient. The quotient needs to be an integer, that's what "factor" means.</p>
<p>If the quotient is <span class="math-container">$2$</span>, then the base is <span class="math-container">$4$</span>. This ... |
1,604,573 | <p>Consider the random graph $G(n,\frac{1}{n})$. I'm trying to estimate the size of the maximum matching in $G$. </p>
<p>If we look at one vertex, the expected value of its degree is $\frac{n-1}{n}$ so it seems like with high prob it should be 1.</p>
<p>So if I can show that with high probability half of the vertices... | user2316602 | 187,745 | <p>I would like to expand on the answer of D Poole, specifically to show the concentration of <span class="math-container">$Y$</span>.</p>
<p>It is actually really easy if we use the method of bounded differences:</p>
<blockquote>
<p><strong>Theorem</strong> Suppose that <span class="math-container">$X_1$</span>, <... |
2,869,442 | <blockquote>
<p>Check whether the series
$$\sum_{n=1}^{\infty}\int_0^{\frac{1}{n}}\frac{\sqrt{x}}{1+x^2}\ dx$$
is convergent.</p>
</blockquote>
<p>I tried to sandwich the function by $\dfrac{1}{1+x^2}$ and $\dfrac{x}{1+x^2}$ , but this did not help at all.
Any other way of approaching?</p>
| A. Pongrácz | 577,800 | <p>I think you tried to estimate the wrong part of the fraction. If you think about it, $\sqrt{x}$ is much different from 1 and from $x$ in a close neighborhood of 0. However, $1+x^2$ is very close to 1 there.
So simply estimate by $\int\limits_{0}^{1/n} \sqrt{x}= \frac{2}{3}n^{-3/2}$, whose sum is convergent. </p>
|
2,869,442 | <blockquote>
<p>Check whether the series
$$\sum_{n=1}^{\infty}\int_0^{\frac{1}{n}}\frac{\sqrt{x}}{1+x^2}\ dx$$
is convergent.</p>
</blockquote>
<p>I tried to sandwich the function by $\dfrac{1}{1+x^2}$ and $\dfrac{x}{1+x^2}$ , but this did not help at all.
Any other way of approaching?</p>
| Andrei | 331,661 | <p>$$\sum_{n=1}^\infty\int_0^{\frac{1}{n}}\frac{\sqrt x}{1+x^2}dx<\sum_{n=1}^\infty\int_0^{\frac{1}{n}}\frac{\sqrt x}{1}dx=\sum_{n=1}^\infty\frac{2}{3n^{3/2}}=\frac{2}{3}\zeta\left(\frac{3}{2}\right)$$</p>
|
1,750,104 | <p>I've had this question in my exam, which most of my batch mates couldn't solve it.The question by the way is the Laplace Transform inverse of </p>
<p>$$\frac{\ln s}{(s+1)^2}$$</p>
<p>A Hint was also given, which includes the Laplace Transform of ln t.</p>
| xpaul | 66,420 | <p>Let $f(t)=\ln t$, then $F(s)=L\{f\}=-\frac{\gamma+\ln s}{s}$. So $\ln s=-sL\{f\}-\gamma$. Let $G(s)=\frac{s}{(s+1)^2}$ and then $g(t)=L^{-1}\{G\}=(t-1)e^{-t}$. Thus
$$ \frac{\ln s}{(s+1)^2}=-L\{f\}\frac{s}{(s+1)^2}-\frac{\gamma}{(s+1)^2}=-F(s)G(s)-\frac{\gamma}{(s+1)^2}. $$
Using
$$ F(s)G(s)=L\{\int_0^tf(\tau)g(t-\t... |
3,978,378 | <p>The question asks Find an efficient proof for all the cases at once by first demonstrating</p>
<p><span class="math-container">$$
(a+b)^2 \leq (|a|+|b|)^2
$$</span></p>
<p>My attempt at the proof:</p>
<p>for <span class="math-container">$a,b\in\mathbb{R}$</span></p>
<p><span class="math-container">$$
\begin{align*}
... | Martin Argerami | 22,857 | <p>All you need to say is that if <span class="math-container">$0<r^2<s^2$</span>, then <span class="math-container">$r<s$</span>. This follows from the fact that <span class="math-container">$r\geq s>0$</span> implies <span class="math-container">$r^2\geq s^2$</span>.</p>
<p>Also, you are missing a square ... |
519,764 | <p>Question: show that the following three points in 3D space A = <-2,4,0>, B = <1,2,-1> C = <-1,1,2> form the vertices of an equilateral triangle.</p>
<p>How do i approach this problem?</p>
| DonAntonio | 31,254 | <p>Another, fancy approach: </p>
<p>Calculate the directed vectors</p>
<p>$$\underline u:=\vec{AB}=B-A=(3,-2,-1)\;,\;\;\underline v:=\vec{AC}=C-A=(1,-3,2)$$</p>
<p>Now calculate the angle $\;\theta:=\angle BAC\;$ using the usual inner product</p>
<p>$$\cos\theta:=\frac{\underline u\cdot\underline v}{||\underline u|... |
37,804 | <p>I'm trying to gain some intuition for the usefullness of the spectral theory for bounded self adjoint operators. I work in PDE and any interesting applications/examples I've ever encountered are concerning <em>compact operators</em> and <em>unbounded operators</em>. Here I have the examples of $-\Delta$, the laplaci... | Helge | 3,983 | <p>Simple answer: bounded operators are simpler than unbounded ones, so it's better to study them.</p>
<p>Discretized models lead to bounded operators. For example consider the discrete Laplacian $\Delta$ on $\mathbb{Z}^d$ given by
$$
\Delta u(n) = \sum_{| m - n|_1 = 1} u(m),
$$
where $u:\mathbb{Z}^d\to \mathbb{C}$. ... |
3,765,225 | <p>I have a matrix:
<span class="math-container">$$\left(\begin{array}{lll}
a & 0 & 0 \\
0 & b & 0 \\
0 & 0 & c
\end{array}\right)$$</span>
Which I want to change to:
<span class="math-container">$$\left(\begin{array}{lll}
a & 0 & 0 \\
0 & c & 0 \\
0 & 0 & b
\end{array}\r... | Siddharth Bhat | 261,373 | <p>The solution proposed seems wrong. Recall that when building a turing machine that uses non-determinism, if <em>any</em> path accepts,</p>
<p>So we cannot <code>ACCEPT</code> if <em>none</em> is equivalent: we can only take <code>ACCEPT</code> decisions from <em>one</em> of the branches of computation, not <em>all</... |
14,448 | <p>Here's the most common way that I've seen letter grades assigned in undergrad math courses. At the end of the semester, the professor: 1) computes the raw score (based on homework, quizzes, and tests) for each student; 2) writes down all the raw scores in order; 3) somewhat arbitrarily clusters the scores into group... | rnrstopstraffic | 6,949 | <p>What I do is a blend of the two methods that you describe. At the beginning of the course I give students the strictest grading scale that I might possibly use (a standard 90-80-70 scale). At the end of the term, I reserve the right to lower the scale as needed. In this way, any deviations from the stated scale are ... |
332,380 | <p>The following is an excerpt from Sharpe's <em>Differential Geometry - Cartan's Generalization of Klein's Erlangen Program</em>.</p>
<blockquote>
<p>Now we come to the question of higher derivatives. As usual in modern
differential geometry, we shall be concerned only with the skew-symmetric
part of the high... | Pedro Lauridsen Ribeiro | 11,211 | <p>The meaning of higher-order derivatives in differential geometry is better understood through <em>jet bundles</em>. The covariant derivative <span class="math-container">$\nabla\phi$</span> of (say) a smooth section <span class="math-container">$\phi:M\rightarrow E$</span> of a vector bundle <span class="math-contai... |
569,300 | <p>Let $G$ be a finite group and assume it has more than one Sylow $p$-subgroup.</p>
<p>It is known that order of intersection of two Sylow p-subgroups may change depending on the pairs of Sylow p-subgroups.</p>
<blockquote>
<p>I wonder whether there is a condition which guarantees that intersection of any two Sylo... | mesel | 106,102 | <p>I have made up such a condition.</p>
<p>We know that the action of $G$ on $Syl_p(G)$ by conjugation is transitive. If this action is double transitive then the intersection of any two Sylow p-subgroups is conjugate and they must have same order.</p>
<p>Proof: Let $P,Q,R,S$ be elements of $Syl_p(G)$ such that $P\ne... |
548,470 | <p>Prove $$(x+1)e^x = \sum_{k=0}^{\infty}\frac{(k+1)x^k}{k!}$$ using Taylor Series.</p>
<p>I can see how the $$\sum_{k=0}^{\infty}\frac{x^k}{k!}$$ plops out, but I don't understand how $(x+1)$ can become $(k+1)$.</p>
| Stefan | 36,189 | <p>Rewrite:
$$(x+1) e^x = xe^x + e^x $$ then use your series representation for $e^x$.</p>
|
2,239,203 | <p>Why does $\lim\limits_{x\to\infty}(x!)^{1/x}\neq 1?$</p>
<p>As far as I know, anything to the power of $0$ is $1$.</p>
<p>We have a factorial raised to $1/\infty=0$, but the limit is not $1$? I don't even know what the limit is. But it seems like infinity? Why is this?</p>
<p><a href="https://i.stack.imgur.com/hY... | marty cohen | 13,079 | <p>A useful inequality is
$n! > (n/e)^n$.
This can be proved by induction
from
$(1+1/n)^n < e$.</p>
<p>From this
$(n!)^{1/n} > n/e$
and this last is unbounded.</p>
|
1,095,918 | <p><img src="https://i.stack.imgur.com/EST8r.jpg" alt="my problem is in prop 27, cannot prove it. Can use definition before. Notice that p-closure is the closure of G in the open point topolgy"></p>
<p>For extra notations: C(E,F) is the set of all continuous functions from E to F (topological spaces).
Can anybody help... | Henno Brandsma | 4,280 | <p>Let $V$ be any entourage in the uniformity of $F$, and $x$ be a fixed point in $E$. Then let $W$ we any symmetric entourage such that $W \circ W \circ W \subseteq V$, which can be done by the standard axioms for a uniformity. Then find $U$ open in $E$ that contains $x$, and such that for all $f \in G$, and for all $... |
2,410,243 | <p>Suppose we want to
$ min_i$ median$(a_i)$</p>
<p>$a_i$ are real numbers</p>
<p>Does someone know how to pose this as an integer programming problem or point me in the direction of a resource? </p>
| prubin | 458,896 | <p>Given <em>bounded</em> variables $a_{1},\dots,a_{N}$ subject to some
constraints, you can minimize their median with $N$ additional binary
variables $z_{1},\dots,z_{N}$ and one additional continuous variable
$y$ if $N$ is odd (or if you are a bit sloppy about the definition
of ``median''). You minimize $y$ subject t... |
1,650,881 | <p>I found this problem in a book on undergraduate maths in the Soviet Union (<a href="http://www.ftpi.umn.edu/shifman/ComradeEinstein.pdf" rel="nofollow">http://www.ftpi.umn.edu/shifman/ComradeEinstein.pdf</a>):</p>
<blockquote>
<p>A circle is inscribed in a face of a cube of side a. Another circle is circumscribed... | grand_chat | 215,011 | <p>To find the maximum distance instead of the minimum distance, you can follow the same method, but interchange "maximize" with "minimize" everywhere. In particular the analog of Lemma 21.1 is</p>
<p><strong>Lemma 21.1b.</strong> Let $\alpha$ and $\beta$ be real numbers. Then
$$\min_{t\in[0,2\pi)}\alpha\cos t +\beta... |
201,807 | <p>I heard this problem, so I might be missing pieces. Imagine there are two cities separated by a very long road. The road has only one lane, so cars cannot overtake each other. $N$ cars are released from one of the cities, the cars travel at constant speeds $V$ chosen at random and independently from a probability di... | Ross Millikan | 1,827 | <p>I would assume that the cars all have distinct speeds and are released at the same time in a random order. The slowest car will accumulate all the cars behind it. The slowest car not in that group will accumulate all the cars behind it and in front of the slowest car. These are the groups at the other end.</p>
<... |
201,807 | <p>I heard this problem, so I might be missing pieces. Imagine there are two cities separated by a very long road. The road has only one lane, so cars cannot overtake each other. $N$ cars are released from one of the cities, the cars travel at constant speeds $V$ chosen at random and independently from a probability di... | Hagen von Eitzen | 39,174 | <p>A car will eventually (remember that the road is very long) be in the same group as the one before it if that car is slower from the start or is eventually slower because it has to decelerate for some car ahead. Ultimately, a car will be the first car of a cluster iff none of the cars before it has a slower initial ... |
1,377,412 | <p>I am brand new to ODE's, and have been having difficulties with this practice problem. Find a 1-parameter solution to the homogenous ODE:$$2xy \, dx+(x^2+y^2) \, dy = 0$$assuming the coefficient of $dy \ne 0$
The textbook would like me to use the subsitution $x = yu$ and $dx=y \, du + u \, dy,\ y \ne 0$
Rewriting t... | user247327 | 247,327 | <p>Yes, you get $dy/y+ 2u/(3u^2+ 1)du= 0$. </p>
<p>The integral of $dy/y$ is $ln(|y|)$ and the integral of $2u/(3u^2+ 1)du$, using the substitution $v= 3u^2+ 1$ so that $dv= 6udu$ or $2udu= dv/3$, is the integral of $1/v dv/3$ which is $$(1/3)ln(|v|)= (1/3)ln(3u^2+ 1)= (1/3)ln(3x^2/y^2+ 1)$$</p>
<p>So integrating $dy... |
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